MOLECULAR SIMULATION OF THE STRUCTURE, TRANSPORT, AND

SEPARATION OF FLUID MIXTURES IN NANOPOROUS MEMBRANES

UNDER SUBCRITICAL AND SUPERCRITICAL CONDITIONS

by

Mahnaz Firouzi

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 2005

Copyright 2005 Mahnaz Firouzi

UMI Number: 3219811

32198112006

Copyright 2005 byFirouzi, Mahnaz

UMI MicroformCopyright

All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.

ProQuest Information and Learning Company 300 North Zeeb Road

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by ProQuest Information and Learning Company.

Dedication I would like to dedicate this thesis to my husband Dr. Babak Fayyaz-Najafi for his

understanding, support, and love, my parents for their support, boundless faith, and

prayers.

ii

Acknowledgements

I am grateful to many individuals who have contributed towards shaping this thesis. I

would like to express my deepest appreciation and sincere gratitude to Professor

Muhammad Sahimi and Professor Theodore T. Tsotsis for their advice during my

doctoral research. As my advisors, they constantly assisted me to remain focused on

achieving my goal. Their intelligent observations and comments helped me to

establish the overall direction of the research and to move forward with the

investigations in depth.

I wish to extend my appreciation to Professor Chi H. Mak for being my Ph.D.

committee member, and Professor Katherine S. Shing and Professor C. Ted Lee for

being my Qualifying committee members. They all gave me a lot of useful

suggestions and support. I would like to thank Dr. Lifang Xu for her help during the

initial stage of my research. I sincerely appreciate the help of Ms. Karen Woo and

Mr. Brendan Char of the Department of Chemical Engineering, and thank them for

their assistance and help. I would also like to acknowledge National Science

Foundation, the Department of Energy, the Petroleum Research Fund, and Media &

Process Technology for providing the financial support for this work, and the

University of Southern California and San Diego Supercomputer Center for

providing computational facilities.

Finally, special thanks go to Dr. Babak Fayyaz Najafi, my husband, for his

continuous support, patience, and understanding during my graduate studies.

iii

Table of Contents

Dedication ii

Acknowledgements iii

List of Tables vi

List of Figures vii

Abstrat xxii 1 Introduction 1

1.1 Nanoporous Membranes ….…………………..…………………….. 1 1.2 Models of Transport Processes in Membranes ………..................…. 6 1.3 The Potential Energy ……………………………………………… 10 1.4 Molecular Dynamics ……...……….….…………………………… 14

1.4.1 Equilibrium Molecular Dynamics …….……...............…… 19 1.4.2 Non-equilibrium Molecular Dynamics ……………………. 20

1.5 Monte Carlo and Grand Canonical Monte Carlo ………………..… 24 1.6 Thesis Outline ……………………...……………………………… 27

2 Transport and Separation of Supercritical Fluid Mixtures in a

Single Carbon Nanopore 29 2.1 Introduction ……………………………………………………...… 29 2.2 The Pore Model ………………………………………………….... 35 2.3 The DCV-GCMD Method ………………………………………… 37 2.4 Potential Models of the Molecules and the Walls ……………....… 40 2.5 Molecular Dynamics Simulations …………………………………. 41 2.6 Results and Discussions …………………………………………… 43 2.7 Summary …………………….………………………………….…. 53

3 Transport and Separation of Carbon Dioxide-Alkane Mixtures in Carbon Nanopores 74 3.1 Introduction ….…………………………………………………….. 74 3.2 Model of Carbon Nanopore ………………….……………………. 75 3.3 Molecular Models of the Gases and the Interaction Potentials ….... 76 3.4 Configurational-Bias Monte Carlo Method ……………………….. 78 3.5 Configurational-Bias Grand-Canonical Monte Carlo Method …..... 81 3.6 Nonequilibrium Molecular Dynamics Simulations ……………..… 82 3.7 Experimental Study ……………………………………………..… 85

iv

3.8 Results and Discussions ……………………………..………......… 86 3.8.1 Methane-Propane Mixtures ………………………..……… 86 3.8.2 Methane-Butane Mixtures ………………………….....…... 92 3.8.3 Carbon Dioxide-Propane Mixtures ………………….......… 94

3.9 Modeling of Long n-Alkane Chain Mixtures and their Transport and Separation in CMSM …………………………...… 98 3.10 Summary ……………………………………..………..….……….. 99

4 Sub- and Supercritical Fluids in Nanoporous Materials: Direction- Dependent Flow Properties 126 4.1 Introduction ……………………………………………………..... 126 4.2 Asymmetric Pore Models …………….………..……………….... 128 4.3 Adsorption and Transport in Asymmetric Pore ………...……...… 131 4.4 Molecular Dynamics Simulations in Asymmetric Pore………..… 132 4.5 Results and Discussions ……………………………………..…… 136 4.6 Summary …………………………………………….…………… 147

5 Transport and Separation of Carbon Dioxide-Alkane Mixtures in a

Carbon Pore Network under Sub- and Supercritical Conditions 176 5.1 Introduction ………………………………………………………. 176 5.2 Pore Network Modelof CMSMs ………………….…………...…. 178 5.3 NEMD Simulation in CMSM Pore Network ...………................... 184 5.4 Molecular Models of the Fluid and the Interaction Potentials …... 185 5.5 Results and Discussions …………..…….………………………... 187

5.5.1 Methane-Carbon Dioxide Mixtures …………………….... 187 5.5.2 Carbon Dioxide-Propane, Methane-Propane, and Methane-Butane Mixtures …………………………..…… 193

5.6 Asymmetric Pore Network Model ……..………………………… 197 5.7 Summary ……………………………………………...………….. 199

6 Bibliography 235

v

List of Tables

2.1 The conversion between the reduced and the actual units. Subscript 1 referes to the value of the parameters for CH4 ………………….….…… 36

3.1 Values of the molecular parameters used in the simulations. kB is the B

Boltzmann’s constant ……………………………………………...….… 77

vi

List of Figures

1.1 The 12-6 Lennard-Jones potential for particles i and j (Eq.1.3). The potential energy is in units of ε and the distance between i and j is in units of δ . When is positive, the interactions for the pair of particles are repulsive. When is negative, their interactions are

ijU

ijU attractive …............................................................................................… 12

2.1 Schematics of the slit pore used in the simulations. The h and l regions represent the high- and low-pressure control volumes respectively……... 55

2.2 Dimensionless temperature distribution in the pore containing an

equimolar mixture of CO2 and CH4. The upstream pressure in both cases is 120 atm, while the downstream pressures are 90 atm (top) and 20 atm (bottom). The pore size is 1.67 1σ . Dashed lines indicate

the boundaries of the pore region …………………………..…………… 56

2.3 Time-averaged density profiles in the transport direction x in a pore of size 5 1σ , containing an equimolar mixture of the two components at T = 40°C. The upstream pressure is 120 atm, while the downstream pressures are 20 atm (open triangles), 50 atm (open circles), 70 atm (solid triangles), and 100 atm (solid circles). Dashed lines indicate the

boundaries of the pore region ……...………………...………………….. 57

2.4 The distribution of CO2 (circles) and CH4 (triangles) in a pore of size 5 1σ , containing an equimolar mixture at T = 40°C. The upstream pressure is 120 atm, while the downstream pressures are 20 atm (top),

70 atm (middle) and 100 atm (bottom) ………………..………………… 58

2.5 Time-averaged density profiles in the transport direction x in a pore of size 5 1σ , containing a mixture of CO2 and CH4 with a CO2 mole fraction of x2 = 0.9 at T = 40°C. The upstream pressure is 120 atm, while the downstream pressures are 20 atm (open triangles), 50 atm (circles), and 70 atm (solid triangles. Dashed lines indicate the

boundaries of the pore region …………………………………………… 59

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2.6 Comparison of the density profiles of CO2 (top) and CH4 (bottom), computed after 3.5 million times steps (dashed curves) and 5 million time steps (solid curves). All other parameters of the system are the same as those in Figure 2.3. The upstream and downstream pressure are, respectively, 120 atm and 90 atm. Dashed lines indicate the

boundaries of the pore region ………………………….………………... 60

2.7 Time-averaged density profiles between the upper and lower walls, in three different regions of a pore of size 5 1σ , containing an equimolar mixture of CO2 (dashed curve) and CH4 (solid curve) at T = 40°C. The upstream and downstream pressures are, respectively, 120 atm and 70 atm. Also shown is the distribution of the molecules in the pore. The

arrows indicate the boundaries of the pore region ……………….……… 61

2.8 Same as in Figure 2.7, except that the upstream and downstream pressures are, respectively, 3 atm and 1 atm ……………………………. 62

2.9 Same as in Figure 2.7, except that the upstream and downstream pressures are, respectively, 120 atm and 1 atm ….…………...…...…….. 63

2.10 Same as in Figure 2.7, except that the mole fraction of CO2 in the mixture is x2 =0.9 ……………………………………..…………………. 64

2.11 The distribution of the molecular clusters after 5 × 105 (top), 106

(middle) and 1.5 × 106 (bottom) time steps, in a pore of size 5 1σ , containing an equimolar mixture of CO2 and CH4 at T = 40 °C. The upstream and downstream pressures are both 120 atm. Numbers

indicate the size of the clusters ……………………...…………………... 65

2.12 Time dependence of the size of the largest clusters in a pore of size 5 1σ (left) and 1.67 1σ (right), containing an equimolar mixture of CO2 and CH4 at T = 40°C. The upstream and downstream pressures are,

respectively, 120 atm and 70 atm ………………………..……………… 66

2.13 Time-dependence of the size distribution of the molecular clusters. All the parameters are the same as those in Figure 2.12. Numbers next to

the curves indicate the size of the clusters ……………..…………..……. 67

2.14 Cluster size distribution ns(t) in a pore of size 5 1σ , containing an equimolar mixture of CO2 and CH4 at T = 40 °C. The results are for

times 5.1 × 106 (◊), 5.2 × 106 ( ), 5.4 × 106 (ο), and 5.5 × 106 (Δ) …….. 68

viii

2.15 The dependence of the flux and permeance of CO2 and CH4, and the corresponding separation factor, on the pressure drop PΔ applied to a pore of size 1.67 1σ at T = 40°C. The mixture is equimolar, with the

upstream pressure being 120 atm ……………………………..………… 69

2.16 Same as in Figure 2.15, but for a pore of size 5 1σ ……………….….….. 70

2.17 Same as in Figure 2.15, but for a pore of size 5 1σ and CO2 mole fraction of 2x = 0.9 …………………………...………………..………… 71

2.18 Dependence of the permeance of CO2 (solid curves) and CH4 (dashed

curves), and the corresponding separation factors, on the mole fraction of CO2 in the feed, in a pore of size 1.67 1σ at T = 40°C (circles) and 100°C (triangles). The upstream and downstream pressures are,

respectively, 120 atm and 20 atm …...………………………...……….... 72

2.19 Temperature-dependence of the permeance of CO2 (solid symbols) and CH4 (open symbols), and the corresponding separation factors, for a pore of size 1.67 1σ and downstream pressures of 90 atm (squares) and 110 atm (triangles). The upstream pressure is 120 atm, and the mixture

is equimolar ……………………………………………...…………….... 73

3.1 Dimensionless temperature distribution in a pore of size H* = 5 and the two control volumes, containing mixtures of CH4 and C3H8 with methane mole fraction in the feed being 0.7 (top) and 0.5 (bottom).

Dashed lines indicate the boundaries of the pore region …………….… 101

3.2 Snapshot of the pore containing CH4 (triangles) and C3H8 (chains), at steady state. The mole fraction of CH4 in the feed is 0.9, and the pore

size is H*= 5 ……………………………………………….…………... 102

3.3 Time-averaged density profiles of CH4 (solid curves) and C3H8 (dashed curves) in the transverse direction (perpendicular to the walls), measured near the pore’s center at T = 50°C. The pore size is H* = 5, while the mole fraction of CH4 in the feed is 0.5 (top) and 0.9

(bottom) …………………………………………………………. 103

3.4 Comparison of the time-averaged density profiles of CH4 (solid curves) and C3 H8 (dashed curves) in the transverse direction (perpendicular to the walls), measured near the pore’s center at T =

50°C, in three pores. The mole fraction of CH4 is 0.7 ………………..... 104

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3.5 Time-averaged density profiles of CH4 (solid curves) and C3H8 (dashed curves) in the transport direction x in a pore of size H* = 5, in which the methane mole fraction in the feed is 0.7 (top) and 0.5 (bottom), and T = 50°C. Dashed lines indicate the boundaries of the

pore region ……………………………………………..……………… 105

3.6 Time-averaged density profiles of CH4 (solid curves) and C3H8 (dashed curves) in the transport direction x in a pore of size H* = 5 in which the CH4 mole fraction is 0.9. The upstream and downstream pressures are, respectively, 3 atm and 1 atm (top), and 30 atm and 10 atm (bottom). Dashed lines indicate the boundaries of the pore

region …….………………………………………………....………….. 106

3.7 Time-averaged density profiles of CH4 (solid curve) and C3H8 (dashed curve) in the transport direction x in a pore of size H* = 3. The CH4

mole fraction in the feed is 0.7, and T=50°C ……………….......……… 107

3.8 Comparison of the simulation results for the permeance of CH4 (open triangles) and C3H8 (open circles) with the corresponding experimental data (solid symbols). The mole fraction of CH4 in the feed in the upper Figure is 0.7, while T = 50°C in the lower Figure. The pore size is

H* = 5 ..……………………………………………………...…………. 108

3.9 Comparison of the computed separation factors (open circles) with the experimental data (solid circles) for a CH4/C3H8 mixture. The mole fraction of CH4 in the feed in the upper Figure is 0.7,while T = 50°C in

the lower Figure ………………………………………………………... 109

3.10 The effect of pore size on the permeances of CH4 and C3H8 and the corresponding separation factors in a binary mixture in which the CH4

mole fraction in the feed is 0.7, and T = 50°C ……………….......…….. 110

3.11 A snapshot of the pore containing CH4 (triangles) and C4H10, at steady state in a pore of size H* = 5 at T = 50°C. The CH4 mole fraction in

the feed is 0.9 …………………………………………….………….…. 111

3.12 Time-averaged density profile of CH4 (solid curve) and C4H10 (dashed curve), between the upper and lower walls of a pore of size H* = 5 at T = 50°C. The profiles were calculated in the middle of the pore, and

CH4 mole fraction in the feed is 0.9 ………………...…………………. 112

3.13 Same as in Figure 3.12, but in the transport direction x. Dashed vertical lines indicate the boundaries of the pore region ………….…… 113

x

3.14 The computed permeances of CH4 (triangles) and C4H10 (circles). The mole fraction of CH4 in the feed in the upper Figure is 0.9, while T =

50°C in the lower Figure …..……………………………..….………… 114

3.15 The effect of pore size on the permeances of CH4 (triangles) and C4H10 (circles), and the corresponding separation factors at T = 50°C. The

CH4 mole fraction in the feed is 0.7 …………………………..……..… 115

3.16 Time-averaged temperature distribution in a pore of size H* = 5 and the two CVs that contain a binary mixture of CO2 and C3H8. The mole fraction of CO2 is 0.7 (top) and 0.5 (bottom). Dashed vertical lines

indicate the boundaries of the pore ………………………………….…. 116

3.17 Distribution of CO2 (triangles) and C3H8 chains in a pore of size H* = 5 at T = 50°C, obtained at steady state. The CO2 mole fraction in the

feed is 0.9 …………………………………….………………………… 117

3.18 Density profiles of CO2 (solid curves) and C3H8 (dashed curves) between the upper and lower walls of a pore of size H* = 5, computed at the pore’s center and obtained at steady state. The CO2 mole fraction in the feed is 0.5 (top) and 0.9 (bottom), and T =

50°C ……………………………………………….………………..….. 118

3.19 Same as in Figure 3.18, but in the transport direction x. The CO2 mole fraction in the feed is 0.7 (top) and 0.5 (bottom), and T = 50°C.

Dashed vertical lines indicate the boundaries of the pore ……………... 119

3.20 Comparison of the computed permeances of CO2 (open triangles) and C3H8 (open circles), for a pore of size H* = 5, with the corresponding experimental data (solid symbols). The CO2 mole fraction in the feed

in the upper Figure is 0.7, while T = 50°C in the lower Figure ……...… 120

3.21 Comparison of the computed separation factors of CO2/C3H8 binary mixtures (open circles), for a pore of size H* = 5, with the experimental data (solid circles). The CO2 mole fraction in the feed in

the upper Figure is 0.7, while T = 50°C in the lower Figure ...............… 121

3.22 The effect of the pore size on the permeances of CO2 (triangles) and C3H8 (circles), and the corresponding separation factors at T = 50°C.

The CO2 mole fraction in the feed is 0.7 ………………………………. 122

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3.23 Time-averaged density profiles of CO2 (solid curves) and C6H14 (dashed curves) between the upper and lower walls, in three different regions of a pore of size H* = 2. The upstream and downstream pressures are, respectively, 3 atm and 1 atm. The CO2 mole fraction in the feed is 0.9 and T = 50°C. Also shown is the distribution of the CO2 (triangles) and C6H14 (chains) in the pore. The arrows indicate the

boundaries of the pore region ………………………………………….. 123

3.24 Snapshot of the pore containing CO2 (triangles) and C6H14 (chains) with the same parameters as figure 3.23 ……………………….………. 124

3.25 Time-averaged density profile (top) of CO2 (solid curves) and C6H14

(dashed curves) and temperature profile (bottom), in the transport direction x in a pore of size 5* =H in which the CO2 fraction is 0.9 and T=200°C. The upstream and downstream pressures are, respectively, 20 atm and 5 atm. Dashed lines indicate the boundaries

of the pore region ……………………………………………………..... 125

4.1 Asymmetric pore model with CVs inside the pore ……...…..…..…..…. 130

4.2 Asymmetric pore model with CVs outside the pore ………..…..…….... 130

4.3 Snap-shots and density profiles of CO2 in supercritical – subcritical conditions. The upstream and downstream pressures are, respectively, 82.63 (1200 psig) and 5 atm (59 psig), and temperature is 35 °C, (a) the upstream pressure is on the macro side, obtained after 1,500,000 time steps, (b) the upstream pressure is on the nano side, obtained after 4,000,000 time steps. Vertical lines indicate the boundaries of the

pore ……..............................................................................................… 149

4.4 Snapshots of the pore containing pure CO2 in 3D. The upstream and downstream pressures are, respectively, 82.63 atm (1200 psig) and 5 atm (59 psig), and temperature is 35 °C. The upstream pressure is on

the macro side (top) and nano side (bottom) ………………………...… 150

4.5 Density profiles of pure CO2 in pore’s cross sections. The upstream and downstream pressures are, respectively, 82.63 atm (1200 psig) and 5 atm (59 psig), and temperature is 35 °C. The upstream pressure is on

the macro side ………………………………………...………………... 151

xii

4.6 Density profiles of pure CO2 in pore’s cross sections. The upstream and downstream pressures are, respectively, 82.63 atm (1200 psig) and 5 atm (59 psig), and temperature is 35 °C. The upstream pressure is on

the nano side ……………………………………….………………...… 152

4.7 Density and temperature profiles of pure CO2 along the pores for subcritical - subcritical conditions. The upstream and downstream pressures are, respectively, 21.4 atm (300 psig) and 11.2 atm (150 psig), and temperature is 35 °C, (a) the upstream pressure is on the macro side, (b) the upstream pressure is on the nano side. Dashed

vertical lines indicate the boundaries of the pore ……………………… 153

4.8 Density profiles and number of the molecules of pure CO2 along the pores for subcritical - subcritical conditions. The upstream and downstream pressures are, respectively, 41.8 atm (600 psig) and 4.4 atm (50 psig), and temperature is 35 °C, (a) the upstream pressure is on the macro side, (b) the upstream pressure is on the nano side.

Dashed vertical lines indicate the boundaries of the pore ...…...………. 154

4.9 Snapshot of the pore containing equimolar mixture of CH4 (triangles) and CO2 (circles) under supercritical - subcritical conditions obtained after 4,000,000 time steps. The upstream and downstream pressures are, respectively, 82.6 atm (1200 psig) and 35 atm (500 psig), and temperature is 35 °C. The upstream pressure is on the nano side (top)

and macro side (bottom) .……………………..................…...…….……155

4.10 Time-averaged density profiles of both CH4 (solid curves) and CO2 (dashed curves) in the transport direction x in the pore with the same parameters as Figure 4.9. The upstream pressure is on the nano side (top) and macro side (bottom). Dashed vertical lines indicate the

boundaries of the pore ……………………………….……...…………..156

4.11 Time-averaged temperature distribution along the pore with the same parameters as Figures 4.9 and 4.10. The upstream pressure is on the nano side (top) and macro side (bottom). Dashed vertical lines indicate

the boundaries of the pore ……………….………………………………157

4.12 Snapshot of the pore containing CO2 under subcritical - subcritical conditions. The upstream and downstream pressures are, respectively, 62.2 atm (900 psig) and 28.2 atm (400 psig), and temperature is 35 °C. The upstream pressure is on the nano side (top) obtained after 3,125,000 time steps, and macro side (bottom) obtained after

3,410,000 time steps ………………………………………………….... 158

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4.13 Time-averaged density profiles of pure CO2 in the transport direction x in the pore with the same parameters as Figure 4.12. The upstream pressure is on the nano side (top) and macro side (bottom). Dashed

vertical lines indicate the boundaries of the pore ………...........………. 159

4.14 Time-averaged temperature distribution of pure CO2 along the pore with the same parameters as Figures 4.12 and 4.13. The upstream pressure is on the nano side (top) and macro side (bottom). Dashed

vertical lines indicate the boundaries of the pore ……………...…..…... 160

4.15 Snapshot of the pore containing CO2 under supercritical - subcritical conditions. The upstream and downstream pressures are, respectively, 82.6 atm (1200 psig) and 48.6 atm (700 psig), and temperature is 35 °C. The upstream pressure is on the nano side (top) and macro side

(bottom) ……………………………………………………………...… 161

4.16 Time-averaged density profiles of pure CO2 in the transport direction x in the pore with the same parameters as Figure 4.15. The upstream pressure is on the nano side (top) and macro side (bottom). Dashed

vertical lines indicate the boundaries of the pore …………...…………. 162

4.17 Time-averaged temperature distribution of pure CO2 along the pore with the same parameters as Figures 4.15 and 4.16. The upstream pressure is on the nano side (top) and macro side (bottom). Dashed

vertical lines indicate the boundaries of the pore ………...……………. 163

4.18 The dependence of the permeability and molar flow of pure CO2 on the upstream pressure at T=35°C, when PΔ =500 psi is applied in the two opposite directions. Continuous and dashed curves show, respectively, the results when the upstream pressure is applied on the

macropore and nanopore ……………..………………………………… 164

4.19 Snapshot of the pore containing CO2 under supercritical - subcritical conditions. The upstream and downstream pressures are, respectively, 82.6 atm (1200 psig) and 11.2 atm (150 psig), and temperature is 35 °C. The upstream pressure is on the nano side (top) obtained after 770,000 time steps, macro side (bottom) obtained after 810,000 time

steps ………............................................................................................. 165

4.20 Time-averaged density profiles of pure CO2 in the transport direction x in the pore with the parameters as Figure 4.19. The upstream pressure is on the nano side (top), macro side (bottom). Dashed

vertical lines indicate the boundaries of the pore ……….………...….... 166

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4.21 Snapshot of the pore containing equimolar mixture of CH4 (triangles) and CO2 (circles) under supercritical - subcritical conditions. The upstream and downstream pressures are, respectively, 82.6 atm (1200 psig) and 35 atm (500 psig), and temperature is 35 °C. The upstream

pressure is on the nano side (top) and macro side (bottom) ………....... 167

4.22 Time-averaged density profiles of both CH4 (solid curves) and CO2 (dashed curves) in the transport direction x in the pore with the same parameters as Figure 4.21. The upstream pressure is on the nano side (top) and macro side (bottom). Dashed vertical lines indicate the

boundaries of the pore ………………………………………………..... 168

4.23 Time-averaged temperature distribution along the pore with the same parameters as Figures 4.21 and 4.22. The upstream pressure is on the nano side (top) and macro side (bottom). Dashed vertical lines indicate

the boundaries of the pore ……..………………………...….………….. 169

4.24 Density profiles of an equimolar mixture of CH4 (solid curves) and CO2 (dashed curves) in pore’s cross sections. The upstream and downstream pressures are, respectively, 82.6 atm (1200 psig) and 35 atm (500 psig), and temperature is 35 °C. The upstream pressure is on

the macro side ……...……………….……………….………...…......… 170

4.25 Density profiles of an equimolar mixture of CH4 (solid curves) and CO2 (dashed curves) in pore’s cross sections. The upstream and downstream pressures are, respectively, 82.6 atm (1200 psig) and 35 atm (500 psig), and temperature is 35 °C. The upstream pressure is on

the nano side ………………………………………................………… 171

4.26 Density profiles of pure CO2 in pore’s cross sections. The upstream and downstream pressures are, respectively, 96.2 atm (1400 psig) and 48.6 atm (700 psig), and temperature is 35 °C. The upstream pressure

is on the macro side ……………………………..………………...…… 172

4.27 Density profile of pure CO2 in pore’s cross sections. The upstream and downstream pressures are, respectively, 96.2 atm (1400 psig) and 48.6 atm (700 psig), and temperature is 35 °C. The upstream pressure is on

the nano side …………………………………………………………… 173

xv

4.28 The dependence of the permeabilities and molar flows of CO2 (circles) and CH4 (triangles) on the upstream pressure in an equimolar mixture at T=35°C, when PΔ =700 psi is applied in the two opposite directions. Continuous and dashed curves show, respectively, the results when the upstream pressure is applied on the macropore and

nanopore …………....................................................................…..….... 174

4.29 The dependence of the permeabilities and molar flows of CO2 (circles) and CH4 (triangles) on the pressure drop in an equimolar mixture at T=35°C, when the upstream pressure is 1400 psig. Continuous and dashed curves show, respectively, the results when the upstream

pressure is applied on the macropore and nanopore …...………..………175

5.1 Two-dimensional representation of a porous membrane ………………. 176

5.2 A two-dimensional Voronoi network ……………………….…………. 181

5.3 Computed pore size distribution (PSD) for the model CMSMs. On the top is the PSD for a system in which the pores are selected randomly. On the bottom is the PSD for a system in which the pores are

generated according to their sizes, starting from the largest size …....… 182

5.4 Computed PSD for the model CMSMs for different porosities. The pores are generated according to their sizes, starting from the largest

size ........................................................................................................... 183

5.5 PSD of the membrane used in the experiments ………………………... 183

5.6 Time-averaged density profiles of both components of an equimolar mixture of CO2 (dashed curves) and CH4 (solid curves) at T=40°C in the transport direction x . The upstream pressure is 120 atm, while the downstream pressures are 20 atm, 50 atm, and 90 atm. Dashed lines

indicate the boundaries of the pore network region …………….... 201

5.7 Time-averaged temperature profiles of mixture of CO2 and CH4 with the same parameters as Figure 5.6. The upstream pressure is 120 atm, while the downstream pressures are 20 atm (top), 50 atm (middle), and 90 atm (bottom). Dashed lines indicate the boundaries of the pore

network region ………………………...…………………………….…. 202

xvi

5.8 The dependence of the flux and permeability of CO2 (dashed curves) and CH4 (solid curves), and the corresponding separation factor, on the pressure drop ΔP applied to the pore network at T=40°C. The mixture

is equimolar, with the upstream pressure being 120 atm …………....…. 203

5.9 Time-averaged density profiles of both components of a mixture of CO2 (dashed curves) and CH4 (solid curves) with a mole fraction of 90% CO2 and 10% CH4 at T=40°C in the transport direction x . The upstream pressure is 120 atm, while the downstream pressures are 20 atm, 50 atm, and 90 atm. Dashed lines indicate the boundaries of the

pore network region ................................................................................. 204

5.10 Time-averaged temperature profiles of mixture of CO2 and CH4 with the same parameters as Figure 5.9. The upstream pressure is 120 atm, while the downstream pressures are 20 atm (top), 50 atm (middle), and 90 atm (bottom). Dashed lines indicate the boundaries of the pore

network region …………………………………….……..………….…. 205

5.11 The dependence of the flux and permeability of CO2 (dashed curves) and CH4 (solid curves), and the corresponding separation factor, on the pressure drop ΔP applied to the pore network at T=40°C, in a mixture in which the mole fraction of CO2 is 0.9 and the upstream pressure is

fixed at 120 atm ....................................................................................... 206

5.12 Snapshot of the pore network containing an equimolar mixture of CH4 (circles) and CO2 (asterisks) at T=40°C, obtained after 2,220,000 time steps. The upstream and downstream pressures are, respectively, 120

atm and 90 atm ………………………………….……………..…..….... 207

5.13 Snapshot of the pore network containing CH4 (circles) and CO2 (asterisks) at T=40°C, obtained after 1,740,000 time steps. The CO2 mole fraction in the feed is 0.9. The upstream and downstream

pressures are, respectively, 120 atm and 90 atm ………...........……….. 208

5.14 Time-averaged density profiles of both components of an equimolar mixture of CO2 (dashed curves) and CH4 (solid curves) at T=40°C in the transport direction x . The upstream pressure is 120 atm, while the downstream pressures are 20 atm (top), 50 atm (middle), and 90 atm (bottom). Dashed lines indicate the boundaries of the pore network region ………………...……………...…….………..…………….......... 209

xvii

5.15 Time-averaged temperature profiles of mixture of CO2 and CH4 with the same parameters as Figure 5.14. The upstream pressure is 120 atm, while the downstream pressures are 20 atm (top), 50 atm (middle), and 90 atm (bottom). Dashed lines indicate the boundaries of the pore network region ………………….………………………..…...….…...... 210

5.16 Snapshot of the pore network containing an equimolar mixture of CH4

(circles) and CO2 (asterisks) at T=40°C obtained after 2,650,000 time steps. The upstream and downstream pressures are, respectively, 120 atm and 20 atm ………………………….………..……………....…….. 211

5.17 The dependence of the flux and permeability of CO2 (dashed curves)

and CH4 (solid curves) in an equimolar mixture on the pressure drop ΔP applied to the pore network at T=40°C. The upstream pressure is fixed at 120 atm ……………………………………….……………...... 212

5.18 Snapshot of the pore network containing CH4 (circles) and CO2

(asterisks) at T=40°C obtained after 2,315,000 time steps. The CO2 mole fraction in the feed is 0.7. The upstream and downstream pressures are, respectively, 120 atm and 20 atm ……………………..... 213

5.19 Time-averaged density profiles of both components of CO2 (dashed

curves) and CH4 (solid curves) at T=40°C in the transport direction x . The upstream and downstream pressures are, respectively, 120 atm and 20 atm. The mole fraction of CO2 in feed is 0.5 (top), 0.7 (middle) and 0.9 (bottom). Dashed lines indicate the boundaries of the pore network region ………………………………………...……………….. 214

5.20 Time-averaged temperature profiles of mixture of CO2 and CH4 with

the same parameters as Figure 5.19. The mole fraction of CO2 in feed is 0.5 (top), 0.7 (middle) and 0.9 (bottom). Dashed lines indicate the boundaries of the pore network region …….......………………..……... 215

5.21 The dependence of the flux and permeability of CO2 (dashed curves)

and CH4 (solid curves), and the corresponding separation factor, on the mole fraction of CO2 in the feed at T=40°C. The upstream and downstream pressures are, respectively, 120 atm and 20 atm …………. 216

5.22 Time-averaged density profiles of both components of CO2 (dashed

curves) and CH4 (solid curves) in the transport direction x at different temperatures. The upstream and downstream pressures are, respectively, 120 atm and 90 atm, and the mixture is equimolar. Dashed lines indicate the boundaries of the pore network region ……... 217

xviii

5.23 Time-averaged temperature profiles of mixture of CO2 and CH4 with the same parameters as Figure 5.22. Temperature is fixed at 20°C (top), 100°C (middle), and 140°C (bottom). Dashed lines indicate the boundaries of the pore network region …………...……..……………... 218

5.24 Time-averaged density profiles of both components of CO2 (solid

curves) and C3H8 (dashed curves) in the transport direction x at different temperatures. The upstream and downstream pressures are, respectively, 3 atm and 1 atm and the mole fraction of CO2 in the feed is 0.7. Dashed lines indicate the boundaries of the pore network region …………………………………………………………………... 219

5.25 Time-averaged temperature profiles of mixture of CO2 and C3H8 with

the same parameters as Figure 5.24. Temperature is fixed at 25°C (top), 50°C (middle), and 75°C (bottom). Dashed lines indicate the boundaries of the pore network region …...………………..…….…….. 220

5.26 The dependence of the flux and permeability of CO2 (solid curves) and

C3H8 (dashed curves), and the corresponding separation factor, on the temperature. The upstream and downstream pressures are, respectively, 3 atm and 1 atm, and the mole fraction of CO2 in the feed is 0.7 ………………………………………………………………..…... 221

5.27 Time-averaged density profiles of both components of CO2 (solid

curves) and C3H8 (dashed curves) at T=50°C in the transport direction x . The upstream and downstream pressures are, respectively, 3 atm and 1 atm. The mole fraction of CO2 in feed is 0.5 (top), 0.7 (middle) and 0.85 (bottom). Dashed lines indicate the boundaries of the pore network region ……………………………………………..…. 222

5.28 Time-averaged temperature profiles of mixture of CO2 and C3H8 with

the same parameters as Figure 5.27. The mole fraction of CO2 in feed is 0.5 (top), 0.7 (middle) and 0.85 (bottom). Dashed lines indicate the boundaries of the pore network region …...………...…..……………… 223

5.29 The dependence of the flux and permeability of CO2 (solid curves) and

C3H8 (dashed curves), and the corresponding separation factor, on the mole fraction of CO2 in the feed at T=50°C. The upstream and downstream pressures are, respectively, 3 atm and 1 atm ……….…..… 224

xix

5.30 Distribution of CO2 (triangles) and C3H8 chains in a pore network at T=50°C, obtained after 4,000,000 time steps. The CO2 mole fraction in the feed is 0.85. The upstream and downstream pressures are, respectively, 3 atm and 1 atm …………...………………...……...……. 225

5.31 Time-averaged temperature and density profiles of both components of

CH4 (solid curves) and C3H8 (dashed curves) at T=50°C in the transport direction x . The upstream and downstream pressures are, respectively, 3 atm and 1 atm. The mole fraction of CH4 in feed is 0.7. Dashed lines indicate the boundaries of the pore network region ……………………………………….………………...……...… 226

5.32 Time-averaged density profiles of both components in an equimolar

mixture of CO2 (solid curves) and C3H8 (dashed curves) (top), and in an equimolar mixture of CH4 (solid curves) and C3H8 (dashed curves) (bottom), at T=50°C in the transport direction x . The upstream and downstream pressures are, respectively, 30 atm and 10 atm. Dashed lines indicate the boundaries of the pore network region …………...… 227

5.33 Time-averaged temperature and density profiles of both components of

CO2 (solid curves) and C3H8 (dashed curves) at T=50°C in the transport direction x . The upstream and downstream pressures are, respectively, 120 atm and 20 atm. The mole fraction of CO2 in feed is 0.9. Dashed lines indicate the boundaries of the pore network region ………………………………………………………..….……… 228

5.34 The dependence of the flux and permeability of CO2 (solid curves) and

C3H8 (dashed curves), and the corresponding separation factor, on the pressure drop ΔP applied to the pore network at T=50°C, in a mixture in which the mole fraction of CO2 is 0.9 and the upstream pressure is fixed at 120 atm ....................................................................................... 229

5.35 Time-averaged temperature and density profiles of both components of

CH4 (solid curves) and C4H10 (dashed curves) at T=50°C in the transport direction x . The upstream and downstream pressures are, respectively, 3 atm and 1 atm. The mole fraction of CH4 in feed is 0.7. Dashed lines indicate the boundaries of the pore network region ………………………………………………………...………… 230

5.36 A three-dimensional asymmetrical Voronoi network. Poisson points

are shown in top figure and the pore polyhedra are shown in bottom figure ........................................................................................................ 231

xx

5.37 Time-averaged density profiles of CH4 (solid curves) and CO2 (dashed curves) in an equimolar mixture at T=40°C in the transport direction x . The upstream pressure is fixed at 120 atm, in left CV (top) and right CV (bottom), and the downstream pressure is 20 atm. Dashed lines indicate the boundaries of the pore network region ….………………… 232

5.38 Dimensionless temperature distribution in a pore network and the two

control volumes, containing CH4 and CO2 in an equimolar mixture at T=40°C. The upstream pressure is fixed at 120 atm, in left CV (top) and right CV (bottom), and the downstream pressure is 20 atm. Dashed lines indicate the boundaries of the pore network region ……………………………………………………..……...….…. 233

5.39 Snapshot of the pore network containing an equimolar mixture of CH4

(circles) and CO2 (asterisks) at T=40°C. The upstream pressure is fixed at 120 atm, in left CV (top) and right CV (bottom), and the downstream pressure is 20 atm ………………………..…....………….. 234

xxi

Abstract

The goal of this dissertation is to study the transport and separation properties of

fluid mixtures confined in enclosing media, i.e., by pores. In particular, molecular

dynamics simulations are carried out to study the adsorption, transport, permeation

and separation properties of fluid mixtures in nanoporous carbon molecular-sieve

membranes at the atomic scales. Molecular modeling represents a valuable tool for

the study of nanoporous membranes, and of the transport processes that take place in

their pore space. My research is intended to provide a better understanding of the

effect of confinement on the behavior of the fluids. The result of these studies will

allow the chemical and petrochemical industries to improve the design of membrane

separation technology.

Non-equilibrium molecular dynamics simulations are implemented in order

to study the behavior of the system on which an external chemical potential or

pressure gradient has been imposed. Extensive simulations are carried out to study

the effect of the pore structure and the imposed external potential on the quantities of

interest, such as the fluids' distributions in the system, transport properties and the

separation factor for broad ranges of conditions. We examine the behavior of the

fluid mixtures under both subcritical and supercritical conditions.

It is shown that under supercritical conditions, which represent a high

pressure and relatively low temperature, the fluids form dynamic molecular clusters

that travel the pore space. Their sizes vary with the time in a seemingly oscillatory

xxii

manner. We study the dynamic behavior, and the size distribution of the clusters as it

evolves with the process time. The confined structure of the pores gives rise to

adsorption and transport phenomena, as well as dynamic evolution of the fluid

clusters, that are completely absent in the same pores under subcritical conditions.

The fluid mixtures we study are the binary mixtures of carbon dioxide and n-

alkanes, as well as mixtures of n-alkanes chains. The united atom model, in which

CH2 and CH3 groups are considered as one interaction site, are used to model n-

alkane chains. Bending and torsion potentials are included, and the bond lengths are

kept constant using the RATTLE algorithm. Configurational-bias Monte Carlo

technique is used for the efficient generation of molecular model of the n-alkanes.

Two models have been used to represent the nanoporous membrane, which

are a slit pore composed of parallel planar carbon walls, and the model represented

by a three-dimensional pore space, generated, atomistically by the Voronoi

tessellation of the space, using tens of thousands of atoms. The Voronoi model

contains interconnected pores of various sizes and shapes, which is a realistic model

for the membrane and allows us to investigate the effect of morphology of the pore

space, i.e., its pore size distribution and pore connectivity, on the transport and

separation properties of fluid mixtures in a nanoporous membrane.

In addition, the possibility of asymmetry in the permeation properties of fluid

mixtures in carbon molecular-sieve membranes is investigated under sub- and

supercritical conditions. To do so, we carry out extensive nonequilibrium molecular

dynamics simulations of flow and transport of a pure fluid, as well as a binary

xxiii

mixture, through a porous material composed of a macro-, a meso-, and a nanopore,

in the presence of an external pressure gradient. We find that under supercritical

conditions, unusual phenomena occur that give rise to direction- and pressure-

dependent permeabilities for the mixture's components. Hence, the classical models

of fluid flow and transport through porous materials that are based on single-valued

permeabilities that are independent of the direction of the applied pressure gradient

are completely in error. The simulations are in qualitative agreement with the

experimental data gathered in our laboratory and provide a rational explanation for

them in terms of a non-linear flow regime, coupled with adsorption.

xxiv

Chapter 1 Introduction 1.1 Nanoporous Membranes

low, transport and behavior of fluids and their mixtures in confined media are

currently of fundamental and practical interest (Sahimi, 1993, 1995, 2003;

Torquato, 2002). Examples of such confined media include nano- and mesoporous

materials, such as catalysts, adsorbents, skin and biological tissue, and nanoporous

thin films (e.g., silica aerogels) that are utilized as low-dielectric constant

composites, optical coatings, sensors and insulating films.

F

An important class of such materials consists of nanoporous membranes,

such as biological or synthetic membranes (e.g., carbon molecular-sieve and silicon

carbide membranes). The former play a fundamental role in biological activities of

living organisms, while the latter, are under active investigations, both

experimentally and by computer simulations, for separation of fluid mixtures into

their constituent components, and for sensors that can detect trace amounts of certain

chemical compounds. These materials, depending on their pore space morphology,

contain a range of pore sizes, from nano- to meso- to macropores. However, the main

resistance to any transport process in the pore space of these materials is offered

1

mainly by the interconnected nano- and mesopores. Thus, studies of flow and

transport processes in such materials have focused on this range of pore sizes.

Here, we briefly explain what constitute meso- and nanopores, since pore

dimensions play an important role in the overall transport and thermodynamic

properties of an adsorbed fluid and, therefore, their distinction is important.

A mesopore is any pore whose diameter is on the order of 10-6 meters, which

is large compared to the size of an adsorbed fluid layer (whose size is on the order of

angstroms). In a mesopore, the fluid molecules are uniformly distributed in the pore

volume, as in the bulk fluids, and individual fluid molecules can then be ignored, i.e.

the fluid can be treated as a continuum and the fluid properties, such as density or

velocity, can be obtained by dividing the fluid into volume elements that are large at

the molecular level, but small at the macroscopic level. A nanopore is one whose

diameter is on the order of 10-9 meters, which is comparable to the size of the

molecules it adsorbs. In this type of pores, except for a small percentage of fluid

molecules at the pore center, the fluid density varies greatly over molecular distances

and the fluid molecules experience strong interactions with the pore walls over most

of the pore volume. Hence, there is no meaningful way to ignore individual

molecules and divide the fluid into molecularly large volume elements. Therefore,

the continuum approximation fails. Consequently, the molecular-level details of a

fluid confined at the nanoscale must be taken into account, making theoretical

predictions more difficult (Hansen and McDonald, 1991). Nanopores, have

diameters which are less than ten fluid diameters. Therefore, in such pores all the

2

adsorbed molecules are within five fluid diameters of the pore walls and even some

adsorbed molecules simultaneously experience strong interactions with more than

one pore wall. In this case, confinement restrictions on adsorbed fluid molecules are

extreme, and small changes in the pore geometry or in the wall interaction strength

can profoundly affect fluid properties. Therefore, the theoretical predictions of fluid

properties in a nanopore are more difficult than in a larger sized micropore.

Some of the phenomena that occur in the pore space of nanoporous material

include adsorption of gases, flow and transport of fluids, separation of a mixture of

fluids, and several other phenomena. Among these phenomena, flow, adsorption and

transport of fluids in the pore space of a nanoporous material are of particular

importance. The significance of studying flow and transport of fluids in nanoporous

materials is due to three reasons. (1) In a confined fluid, the energy dissipated by

friction can induce chemical transformations and phase transitions that are very

different from those under bulk conditions, as well as cause drastic changes in the

fluid's static and dynamical properties – phenomena that are all of fundamental

importance. (2) From a practical viewpoint, it is clearly important to understand how

flow or transport processes occur in the pore space of nanoporous materials

(Pinnavaia and Thorpe, 1995; Sahimi, 1993, 1995, 2003; Torquato, 2002; Unger et

al., 1988) so that their morphology can be optimized for such applications as

separation, purification, and storage of gases in the pore space. (3) Even when

transport through the solid matrix of nanoporous materials is of prime importance, as

is the case, for example, during electrical conduction in low-dielectric constant

3

nanoporous materials and similar composites, understanding transport and adsorption

in the materials is still critical to characterization (Pinnavaia and Thorpe, 1995;

Sahimi, 1993, 1995, 2003; Torquato, 2002; Unger et al., 1988) of their pore space,

the morphology of which greatly influences the transport and optical properties of

the solid matrix and the materials as a whole.

Much research has been carried out, both experimentally and theoretically, to

determine the principal features which govern the adsorption and transport properties

of fluids in nanoporous membranes. Current models for macroporous membranes

are sufficiently reliable for design purposes. However, in many cases, the size of the

membrane pores is commensurate with the size of the fluid molecules and significant

hindrance effects are observed. Frequently, molecular size permselectivity is highly

desirable as, for example, in heterogenous catalysis using zeolites or passive

transport in biological membranes. With their complicated geometric structure,

many factors are involved in actual nanoporous membranes, including pore size

distribution, degree of interconnectivity, and pore cross-sectional shape. In addition

to structural effects, other factors which must be taken into consideration are the

concentration, temperature, and pressure-dependence of the properties of the fluid.

The presence of adsorption force fields also significantly influences equilibrium and

transport, particularly when the fluid is a gas or vapor and to a lesser extent when the

fluid is a liquid.

Therefore, due to the exceedingly small sizes of the pores of nanoporous

materials, the behavior of fluids and their mixtures in their pore space is typically

4

different from that observed in the bulk. It has been shown experimentally (Horn

R.G., 1981), computationally (Lane and Spurling, 1979; Liu et al., 1974; Magda et

al., 1985; Snook and Van Megen, 1980; Subramanian and Davis, 1979) and

theoretically (Bratko et al., 1989; Curtin and Ashcroft, 1985; Fischer and Methfessel,

1980; Johnson and Nordholm, 1981; Tarazona, 1985) that fluid molecules near solid

surfaces exhibit collective ordering due to fluid-substrate interactions. For example,

near a flat surface, fluid molecules line up so that they are adsorbed as layers parallel

to the surface. Far from the surface, these layers are less pronounced and diminish at

a distance of five fluid molecular diameters (Israelachvili, 1985).

Owing to the complexity of these interacting effects, the principles which

govern transport properties of fluids in membrane containing pores of molecular

dimensions have not yet been firmly established. For example, since the average

pore size of many membranes and nanoporous material is typically of the order of a

few angstroms, the traditional continuum approach cannot be used for modeling such

phenomena. This is due to the fact that the pores’ small sizes imply that the

molecular interactions between the diffusing and/or flowing fluids themselves, and

between them and the pores’ walls, are important and cannot be ignored. Hence, one

must resort to molecular modeling of such phenomena and a more detailed

understanding at a molecular level has become increasingly necessary and important

in guiding experimental work and improving engineering design. The goal in any

molecular dynamics simulation of nanoporous material is to describe the material

and the phenomena that occur in it, the practical consequences of which are at

5

macroscopic length scales, by the molecular dynamics method which considers a

system at atomic scales and provide insights into the phenomena.

1.2 Models of Transport Processes in Nanoporous Membranes Membranes have gained an important place in the chemical industry and are used in

a broad range of applications. There is a great interest in developing membranes for

separation of gas and liquid mixtures by selective adsorption and permeation.

Among them, inorganic membranes, such as alumina, silica, and carbon membranes

are of special interest compared with organic membranes, due to their favorable

characteristics in terms of thermal, mechanical and structural stability, and chemical

resistance. Carbon molecular-sieve membranes (CMSMs) have been used as model

systems by many groups (Acharya et al., 1997; Afrane and Chimowitz, 1993, 1996;

Chen and Yang, 1994; Jones and Koros, 1994a, 1994b, 1995a, 1995b; Koresh and

Sofer, 1983; Linkov, Sanderson and Jacobs, 1994; Linkov, Sanderson and Rychkov,

1994; Naheiri et al., 1997; Petersen et al., 1997; Rao and Sircar, 1993a, 1993b, 1996;

Shiflett and Foley, 1999; Shusen et al., 1996; Sircar et al., 1996; Steriotis et al.,

1997), as well as our own group (Sedigh et al., 1998, 1999, 2000), in studies

involving separation of subcritical fluid mixtures. Beyond their potential practical

significance, these membranes allow one to carry out steady-state transport

investigations, thus significantly simplifying the burden of data analysis and

interpretation. In addition, these membranes can be prepared with well-controlled

6

porosity and a narrow pore size distribution. Nanoporous membranes are used to

perform separations on a molecular level. Models that have been developed for

nanoporous membranes can be divided into two groups.

(i) In one group, the membrane is represented by a single nanosize pore. The

motivation for doing so is the fact that in some membranes the pores are more or less

parallel to each other and, therefore, studying the phenomena of interest in a single

pore may suffice for understanding them in a nanoporous membrane. Many pore

shapes have been used, ranging from cylindrical and slit pores to much more

complex shapes (Düren et al., 2003).

(ii) In the second class are those that model the nanoporous membrane by a

2D or 3D network of interconnected pores. Many different types of such molecular

pore networks have been developed. In one group, the pore network is generated

based on geometrical and topological considerations, without any regards for the

physico-chemical process that generates the nanoporous membrane and, therefore,

the chemical and energetic details of creating the nanopores are ignored. As an

example, consider pillared clays (PCs), nanoporous materials that were originally

developed as a new class of catalytic materials, although it is currently believed that

their greatest potential use may be as membrane materials for separation processes,

especially separating CO2 from a mixture of gases. Hence, in recent years a number

of investigators have studied gas adsorption in PCs. As a second example, consider

CMSMs. These membranes are prepared by pyrolysis of a polymeric precursor. If

the pyrolysis of the polymeric precursor is done at high enough temperatures, the

7

resulting solid matrix of the porous material will have a structure similar to graphite.

Thus, one begins the molecular simulations of these materials with a 3D cell of

carbon atoms with a structure corresponding to graphite. We will explain this method

in details in Chapter 5.

Here, we briefly describe some of the molecular simulation and modeling

methods for adsorption, transport and separations of fluids which have been used in

nanoporous membranes.

The thermodynamic properties of simple fluids in graphitic slit-shaped pores

have been studied extensively at the molecular level via the statistical mechanical

Density Functional Theory (DFT), Monte Carlo (MC), and molecular dynamics

(MD) computer simulations (Jiang et al., 1993; Rhykerd et al., 1991). These studies

have revealed many differences between the structural and dynamical properties of

confined fluids and the corresponding bulk fluids of the same composition. In a slit-

shaped pore, for instance, simple Lennard-Jones fluids tend to organize in layers near

the impenetrable pore walls, resulting in oscillations in the average density within the

pore, in the solvation force acting between the opposing surfaces, and in the

effective, or pore-averaged, diffusion coefficient describing mass transport parallel to

the pore walls (Magda et al., 1985; Schoen et al., 1988; Somers and Davis, 1992).

More recent studies of fluid mixtures in graphitic slit-like pores have begun to

examine complex phenomena, such as the selective adsorption of one of the

molecular components (Cracknell et al., 1993; Klochko et al., 1996; Sokolowski and

Fischer, 1990). In these studies, however, the individual fluid components are still

8

modeled as Lennard-Jones particles. In addition, these simulations have, so far, been

limited to the study of the thermodynamic properties of adsorbed mixtures, whereas

separation of many mixtures, including supercritical (SC) mixtures, that are of

interest in this thesis (Chapter 2), is expected to be transport-dominated.

Other types of pore shapes, and random pore structures, which are of more

relevance to the CMS materials, have so far received only scant attention. A few

studies on triangular pores (Bojan et al., 1992), of carbon nonotubes (Maddox and

Gubbins, 1995; Takaba et al., 1995, 1996), and of a random pore space created by

stacked disks (Segarra and Glandt, 1994) have appeared. A significant advance has

been in combining experimental adsorption isotherms and DFT (Lastoskie et al.,

1993) or grand-canonical MC (GCMC) simulations to obtain realistic pore size

distributions (PSD), thus providing a bridge between the behavior at the

molecular/single pore-scale and that at the macroscopic/pore network-scale. Samios

et al. (Samios et al., 1997), for example, employed slit-shaped graphite pores to

simulate the adsorption of CO2 and assumed the microporous structure to consist of

parallel, non-intersecting pores of various sizes having a postulated PSD. On the

basis of comparison of sorption isotherms obtained by simulation and experiments,

they derived the desired PSD function. Seaton and co-workers (Liu et al., 1992,

1993; Lopez-Ramon et al., 1997; Seaton et al., 1989, 1991, 1997) and Gusev et al.

(Gusev et al., 1997) proposed a similar technique. Their novel contribution was in

using percolation theory (Sahimi, 1994), which allowed them to estimate the

connectivity of the pore network. They used this approach to study the molecular

9

sieving mechanism of carbon molecular sieves, by combining equilibrium MD

(EMD) simulations of transport of single species in slit-shaped pores, and pore

network and percolation theory. Though simple in its inception, the study showed

how powerful the combination of MD simulations and network modeling can be in

describing important aspects of kinetic separation obtained with networks having a

range of pore sizes.

1.3 The Potential Energy Adsorption, diffusion and transport properties are modeled in molecular simulations

on a molecular scale, i.e. the molecules of the fluid mixture and the atoms of the pore

walls of the adsorbents or membranes are considered individually. In order to do this,

we need to calculate the interaction energies throughout the system. The quantum

effects are normally neglected; i.e. the system is modeled as a collection of N

classical particles interacting with each other via an intermolecular potential.

Moreover, there may be external potentials which are acting on the system. Then, the

total potential energy is:

(1.1) External ParticleU U U= +

For fluid atoms adsorbed in a pore, the first term in Eq. (1.1) is due to

interactions between the fluid molecules and the pore walls or structure. The second

term is due to interactions of fluid molecules with each other, which can occur

between pairs, triplets and larger groups of molecules. The fluid interactions beyond

10

pairs of molecules are usually ignored since the potential calculations are

computationally costly when interactions between large groups of atoms are

accounted. Using this simplifying assumption, the fluid interactions are pairwise

additive, and the total potential energy due to particle interactions for a system of N

particles can be written as:

(1.2) 1

1 1( )

N NParticle

iji j i

U−

= = +

= ∑ ∑ u r

where is the pair potential between the particles i and j that are separated by )( ijru

222222 )()()( jijijiijijijij zzyyxxzyxr −+−+−=++= , and are the

particle coordinates. The indices in the double summation of Eq.(1.2) show that each

pair is counted once, which means that only distinct particle pairs contribute to the

total potential energy.

),,( iii zyx

In many MD simulations the interaction potential between a pair of particles,

the center of which are a distance apart, is represented by the classical Lennard-

Jones (LJ) 12-6 potential (Allen and Tildesley, 1987):

ijr

12 6

( ) 4ijij ij

u rr rδ δε

⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪= −⎜ ⎟ ⎜ ⎟⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭ (1.3)

given here for a like pair of particles i and j. This potential is shown in Fig. 1.1. The

terms ε and δ in Eq. (1.3) are the Lennard-Jones parameters of the fluid, where ε

is the energy parameter of the potential (the maximum energy of attraction between a

pair of molecules), or the LJ well depth, and δ is the size parameter (or the distance

at which the LJ potential passes through zero and the potential sharply rises to

11

repulsive values), also called the collision diameter. Note that δ is not the same as

molecular diameter of the molecules, although the two quantities are usually close to

each other. In Eq. (1.3) the term represents a hard-core or repulsive potential,

while the term is the attractive part. The simulation results are usually reported

in units of

12−ijr

6−ijr

ε (energy), δ (length), and m (the mass of a fluid particle).

Figure 1.1: The 12-6 Lennard-Jones potential for particles i and j [Eq. (1.3)]. The potential energy is in units of ε and the distance between i and j is in units of δ . When is positive, the interactions for the pair of particles are repulsive. When

is negative, their interactions are attractive. ijU

ijU

12

When the particles i and j in Eq. (1.3) are of different type, we needs to

approximate the Lennard-Jones parameters, δ and ε . One commonly used

approximation is the Lorentz-Berthelot mixing rules:

and 2

i jij i j ij

δ δε ε ε δ

+= = (1.4)

given here for particles of type i and j.

In order to reduce the number of interactions calculated in a molecular

simulation, the interaction potential is truncated and shifted to zero at a cut-off

distance, rcut-off )( cr . In practice, what is done in most cases is cutting the interactions

off at a distance . Typically, is set to be a multiple of the effective molecular

diameter of the largest atom in the simulations and, therefore, it is usually smaller

than half the system’s linear size. If this approach is taken, then at every step of the

integration one must check, for any particle i, the distances of all other particles from

i to see whether they are at a distance larger or smaller than . We must point out,

however, that cutting off the interaction potentials violates energy conservation,

although if is selected carefully, the effect will be small. Moreover, by shifting the

interaction potentials one can avoid violation of energy conservation altogether by

writing

cr cr

cr

cr

0 0 ij

ij

( ) ( ), if r( )

0 if r > ij c c

ijc

u r u r ru r

r

− ≤⎧⎪= ⎨⎪⎩

(1.5)

where represents the original interaction potential to be used. However, this

shift does not affect the force resulting from the shifted potential; it remains

)(0 ijru

13

discontinuous at . In order to make the force also continuous at the cutoff point, we

write

cr

00 0

( )( ) ( ) ( ) if r( )

0 if r

cc c r r

c

du ru r u r r r ru r dr

r

=⎧ − − − ≤⎪= ⎨⎪ ⟩⎩

c (1.6)

This algorithm was first suggested by Stoddard and Ford (Stoddard and Ford, 1973).

The actual number of interacting particles (i.e., those that are within a sphere of

radius , centered at the center of a given particle) is a function of the molecular

density and this method is much more efficient than a full N−body calculation. The

cut-and-shift procedure cannot be used if electric and gravitational forces are

operative in the system, since they decay only as

cr

r1 .

1.4 Molecular Dynamics Over the past two decades, MD simulations, in which atoms and molecules are

treated as classical particles and quantum-mechanical effects are neglected, have

become an important tool for investigating and predicting various static as well as

dynamical properties of materials. Knowledge of transport properties of

multicomponent gases and oil is of great economical importance in planning of

transport and in dimensioning of industrial plants and reservoir modeling. For

mixtures that consist of molecules of dissimilar sizes, shapes and polarity, traditional

prediction methods for thermal conductivity and viscosity need experimental data to

14

fit mixing rules (Monnery et al., 1995), whereas for diffusion the few existing

prediction methods deviate up to almost an order of magnitude (Helbaek et al.,

1996). One possible route to obtained better predictions is to use MD simulations

that can predict thermophysical properties from models of molecular interactions

only. We will simply mention below a few key papers appeared in the 50s and in the

60s that can be regarded as milestones in molecular dynamics.

The first paper reporting a molecular dynamics simulation was written by

Alder and Wainwright (Alder and Wainwright, 1957) in 1957, who studied a system

with only a few hundreds hard-sphere particles and discovered a fluid-solid phase

transition. The article Dynamics of radiation damage by J.B. Gibson, A. N. Goland,

M. Milgram and G. H. Vineyard , from Brookhaven National Laboratory, appeared

in 1960 (Gibson et al., 1960), is probably the first example of a molecular dynamics

calculation with a continuous potential based on a finite difference time integration

method. The calculation for a 500-atoms system was performed on a IBM 704, and

took about a minute per time step. Aneesur Rahman at Argonne National Laboratory

was well known as a pioneer of molecular dynamics. In his famous 1964 paper

correlations in the motion of atoms in liquid argon (Rahman, 1964), he studied a

number of properties of liquid Ar, using the LJ potential on a system containing 864

atoms and a CDC 3600 computer. Loup Verlet calculated in 1967 (Verlet,

1967,1968) the phase diagram of argon using the LJ potential, and computed

correlation functions to test theories of the liquid state. The bookkeeping device

which became known as Verlet neighbor list was introduced in these papers.

15

Moreover the “Verlet time integration algorithm” was used. Phase transitions in the

same system were investigated by Hansen and Verlet a couple of years later (Hansen

and Verlet, 1969).

We call molecular dynamics a computer simulation technique where the time

evolution of a set of interacting atoms is followed by integrating their equations of

motion. One uses the MD technique to obtain the dynamic properties for a collection

of classical particles. In molecular dynamics we follow the laws of classical

mechanics, and most notably Newton’s law. For a system of size N, this involves

solving a set of 3N second order differential equations (Newton’s equations of

motion):

2 2 2

2 2 2, , i 1,2,...Ni i i

i i ix i y i z i

x y zF m F m F mt t t

∂ ∂ ∂= = = =

∂ ∂ ∂ (1.7)

where is the force exerted on the ii

Fβth particle in the β direction, mi is the particle

mass, and t represents time.

The force on the ith particle is related to the potential energy by:

( )

i

ij ijx

j i ij ij

x u rF

r r≠

∂= −

∂∑ (1.8)

where the summation is over all other particles in the system. If present, forces due

to external interactions are also added to Eq.(1.8). Similar relations hold for the y and

z directions. Therefore in contrast with the MC method, molecular dynamics is a

deterministic technique: given an initial set of positions and velocities, the

subsequent time evolution is, in principle, completely determined. In more pictorial

terms, atoms will “move” into the computer, bumping into each other, wandering

16

around (if the system is fluid), oscillating in waves in concert with their neighbors,

perhaps evaporating away from the system if there is a free surface, and so on, in a

way pretty similar to what atoms in a real substance would do. Therefore, the MD

method is a way of simulating the behavior of a system as it evolves with time since,

unlike the MC method, in the MD simulations the system moves along its physical

trajectory. The main advantage of the MD method over the MC technique is that, not

only does it provide a method for computing the static properties of a system, but

also allows one to calculate and study the dynamical properties of many nanoporous

materials that are of interest to us in this thesis.

The computer calculates a trajectory in a 6N-dimentional phase space (3N

positions and 3N momenta). However, such trajectory is usually not particularly

relevant by itself. Molecular dynamics is a statistical mechanics method. Like Monte

Carlo, it is a way to obtain a set of configurations distributed according to some

statistical distribution function, or statistical ensemble. An example is the

microcanonical ensemble, corresponding to a probability density in phase space

where the total energy is a constant E:

( )H Eδ ⎡ ⎤Γ −⎣ ⎦ (1.9)

Here, is the Hamiltonian (the total energy), and ( )ΓH Γ represents the set of

positions and momenta. δ is the Dirac function, selecting out only those states

which have a specific energy E. Another example is the canonical ensemble, where

the temperature T is constant and the probability density is the Boltzmann function:

17

( )exp BH k T⎡ ⎤− Γ⎣ ⎦ (1.10)

According to statistical physics, physical quantities are represented by

averages over configurations distributed according to a certain statistical ensemble.

A trajectory obtained by molecular dynamics provides such a set of configurations.

Therefore, a measurement of a physical quantity by simulation is simply obtained as

an arithmetic average of the various instantaneous values assumed by that quantity

during MD run.

Statistical physics is the link between the microscopic behavior and

thermodynamics. In the limit of very long simulation times, one could expect the

phase space to be fully sampled, and in that limit this averaging process would yield

the thermodynamics properties. In practice, the runs are always of finite length, and

one should exert caution to estimate when the sampling may be good (“system at

equilibrium”) or not. In this way, MD simulations can be used to measure

thermodynamics properties and, therefore, evaluate, say, the phase diagram of a

specific material. Molecular simulation allow conducting computer experiments, i.e.,

by studying the system on a molecular scale it is possible to measure macroscopic

thermodynamic properties, such as adsorption isotherms or diffusion coefficients at

various process conditions. It is possible to easily change the process conditions,

such as the temperature, and to predict both pure and multicomponent adsorption

isotherms. Furthermore, molecular simulations yield much more detailed information

about the system studied than available from conventional experiments. As the

position of every molecule in the system is known explicity, it is, for example,

18

possible to measure local density profiles and to visualize how the special

distribution of heterogeneities on the pore surface influences the distribution of the

adsorbate molecules in the pore. Molecular dynamics is especially useful for

calculating the dynamic properties, such as the self-diffusion and mutual diffusion

coefficients (Jaccucci and Mcdonald, 1975; Jolly and Bearman, 1980; Schoen and

Hoheisel, 1984) for mixtures. Beyond this “traditional” use, MD is also used for

other purposes, such as studies of non-equilibrium processes, and as an efficient tool

for optimization of structures overcoming local energy minima (simulated

annealing).

We consider here a few important phenomena that occur in the pores of a

nanoporous membranes, two of which, adsorption of a single gas or a mixture of

gases (an equilibrium phenomenon), and flow and transport of fluids in the pore

space under the influence of an external potential gradient (a non-equilibrium

phenomenon), have been studied extensively using molecular modelling. In what

follows we describe atomistic modelling of both types of phenomena.

1.4.1 Equilibrium Molecular Dynamics

Equilibrium MD (EMD) is an important way to study the structure, adsorption and

thermodynamic properties of fluids near a wall surface and inside pores. One of the

most important equilibrium phenomena that can occur in the pore space of a

nanoporous membrane is adsorption of gases. There are several computational

19

techniques for molecular modeling of adsorption phenomena in the pore space of

such materials. One can naturally use the equilibrium molecular dynamics method

for simulating adsorption (Xu et al., 2001). Such simulations are typically done in

the microcanonical ensemble, i.e., one in which the total number of particles, N, the

total volume of the system, V, and its energy, E, are held fixed, which is why it is

usually referred to as the (NVE) ensemble. For example, chemical potential of

adsorbed gas on graphite (Cheng and Steel, 1990; Cracknell et al., 1995) has been

calculated using iso-kinetic MD method.

The self diffusion coefficient Ds , which is an important dynamic property, can

also be calculated by the EMD method, either from an integral over the velocity

autocorrelation function (the Green-Kubo relation), or from the time evolution of the

mean square displacement (the Einstein relation). Pressure is calculated from the

virial theorem. Temperature is also calculated from the kinetic theory:

( )2

13 1

N

i i Bi

T m v N=

= ∑ k− (1.11)

1.4.2 Non-equilibrium Molecular Dynamics

Much attention is paid to studies of membrane separation mechanisms at a molecular

level, since high performance inorganic membranes suitable for gas separations have

been developed during the past two decades. Gas permeation is essentially a non-

equilibrium phenomenon since it occurs when different pressures are applied on two

20

sides of a membrane. Equilibrium MD (EMD) simulations are applicable to systems

that, at least in principle, are amenable to treatment by statistical mechanics.

However, if we are to compute the effective flow and transport properties of fluids in

nanoporous materials, such as the permeability, diffusivity, shear viscosity, and

thermal conductivity, then EMD is not an effective tool. For example, as is well-

known, by calculating the velocity correlation functions for every distinct pair of

species in the system, one can obtain information about the microscopic motion of

the molecules. However, since the velocity correlation function decays as the size of

the system increases, use of EMD is not feasible for estimating the transport

properties of a mixture of molecules in a system which is under the influence of an

external potential gradient - a situation which is encountered in a very large number

of practical problems. One can use the velocity autocorrelation function, but this

quantity can only be used for predicting the tracer or self-diffusivity sD of a species

(that is, when the system is very dilute) via the Green-Kubo equation:

[ ]0

1

1 ( ) . (0) 3

N

s i ii

D v t vN

∞

=

= ∑∫ dt (1.12)

where is the velocity of particle i. The self-diffusivity is, however, completely

different from the transport diffusivity because tracer diffusion ignores the effect of

the collective motion of other molecules, especially in systems with a moderate or

high density. As such, EMD is not suitable for investigation of a transport process in

a system on which an external potential (pressure, voltage, chemical potential,

concentration, etc.) gradient has been imposed. Direct MD simulation of membrane

iv

21

permeation, therefore, requires two regions at different pressures or densities, which

create a driving force for molecular transport across the membrane. This technique

involves the creation of stationary chemical potential gradients between two control

volumes. The grand canonical ensemble Monte Carlo (GCMC) method is suitable

for maintaining a region to be at constant chemical potential or a constant density.

The specification of two sub-regions at different chemical potentials (or densities)

yields a new type of non-equilibrium MD (NEMD) method: the dual control-volume

grand canonical ensemble MD (DCV-GCMD) method proposed by Heffelfinger and

Swol (Heffelfinger and Van Swol, 1994), a grand canonical MD (GCMD) method by

MacElroy (MacElroy, 1994), a GCMD method by Cracknell, Nicholson and Quirke

(Cracknell, Nicholson and Quirke, 1995), and the grand canonical ensemble NEMD

( VTμ -NEMD) method by Furukawa et al. (Furukawa et al., 1996; Furukawa and

Nitta, 1997). Such NEMD simulation methods represent practical alternatives to

EMD for those systems for which the velocity correlation function is difficult, or

meaningless, to measure. They are particularly ideal for the practical situation in

which an external driving force is applied to the system. These methods have been

widely used. In particular, the GCMD method (Cagin and Pettitt, 1991; Lupkowski

and Van Swol, 1991; Sun and Ebner, 1992) combines the MC and MD simulations,

and the DCV-GCMD has also been used for studying many non-equilibrium

phenomena.(Cracknell, Nicholson and Quirke, 1995; Düren et al., 2002; Firouzi et

al., 2003, 2004; Ford and Heffelfinger, 1998; Furukawa et al., 1997; Heffelfinger

and Van Swol, 1994; Heffelfinger and Ford, 1998; Kjelstrup and Hafskjold, 1996;

22

MacElroy, 1994; Maginn et al., 1993; Nicholson et al., 1996; Pohl and Heffelfinger,

1999; Sokhan et al., 2002; Sunderrajan et al., 1996; Supple and Quirke, 2003;

Thompson et al., 1998; Xu et al., 1998, 1999; Xu, Sahimi et al., 2000; Xu, Sedigh et

al., 2000).

The DCV-GCMD method has become an effective tool for studying systems

that are under the influence of an external potential gradient. We will describe this

method in details in Chapter 2. In this method, the simulation box is divided into

three sections: two control volumes maintained at constant chemical potential, μ1 and

μ2 correspondingly, connected by a "transport region," where there is a net flux of

molecules due to the potential gradient generated by the difference μ1 - μ2. The

transport coefficients that describe the steady-state, non-equilibrium transport under

the influence of a chemical potential gradient, are obtained by directly monitoring

the net flux of molecules through the transport region of the simulation cell. This

technique has been applied in the study of the transport of mixtures in graphitic slit-

shaped micropores (Cracknell, Nicholson and Quirke, 1995; Ford and Glandt, 1995;

Ford and Heffelfinger, 1998; Furukawa et al., 1996, 1997; Furukawa and Nitta,

1997; Heffelfinger and Van Swol, 1994; Heffelfinger and Ford, 1998; Kjelstrup and

Hafskjold, 1996; Nicholson et al., 1996; Pohl et al., 1996; Pohl and Heffelfinger,

1999; Sun and Ebner, 1992; Sunderrajan, 1996; Thompson et al., 1998; Xu et al.,

1998, 1999; Xu, Sedigh et al., 2000).

Although the DCV-GCMD has been used extensively during recent years, the

systems studied did not constitute realistic models of transport of gaseous mixtures

23

through porous materials under typical experimental conditions. In most of these

studies the physical conditions of the system did not correspond to an actual

experimental situation and, therefore, no direct comparison with experimental data

was possible. Our group (Ghassemzadeh et al., 2000; Xu et al., 1998, 1999; Xu,

Sedigh et al., 2000; Xu, Sahimi et al., 2000; Xu et al., 2001) initiated the study of the

transport and separation of fluid mixtures in CMSMs under realistic chemical

potential gradient, by both experiments and DCV-GCMC simulations.

1.5 Monte Carlo and Grand Canonical Monte Carlo

In the MC simulation technique, random system perturbations are accepted or

rejected according to the theory of statistical thermodynamics (Allen and Tildesley,

1987; Hill, 1986; McQuarrie, 1976). The resulting set of fluid configurations spans

the phase space of a chosen statistical ensemble. Since, the particle trajectories are

generated at random, unlike in MD, dynamic properties of the system are not

normally calculated in MC.

In the standard MC technique of Metropolis et al. (Metropolis et al., 1953),

the equilibrium properties of a fluid at fixed temperature (T), number of particles (N)

and system volume (V) is simulated as a Markov chain of states. Each state is

generated by perturbing the preceding state. Specifically, a randomly chosen particle

undergoes a randomly sized displacement. Once the new state is produced, one

24

compares the total potential energies in the new and old states. The new state is

accepted into the Markov chain if its energy is lower than that of the old state.

Otherwise, it is accepted with probability:

expiB

Upk T

⎛ ⎞−Δ= ⎜

⎝ ⎠⎟ (1.13)

where is the difference between the total potential energy in the

new and old states, and is the Boltzmann’s constant. The quantity in Eq. (1.13) is

the ratio of the Boltzmann factor

oldnew UUU −=Δ

Bk

( )[ ]TkU B−exp

)

in the new and old states.

Consequently, the states in the Markov chain resulting from an MC simulation

belong to the canonical ensemble, and averages of properties (e.g., energy and

density distribution) over the states correspond to the mean properties of a fluid at

constant temperature, volume and number of particles.

The main advantage of the MC technique is that it is straightforward to

simulate a fluid in different ensembles; i.e., other than for fixed temperature, volume

and number of particles. For instance by modifying the Metropolis algorithm, one

can perform simulations at constant temperature, system volume and chemical

potential (μ . In this case, the ensemble sampled is the grand canonical ensemble

and the simulation technique is aptly named grand canonical Monte Carlo (GCMC).

In a grand-canonical ensemble, both the energy and density are allowed to fluctuate,

hence making GCMC a suitable technique for computing equilibrium properties of

single- and multicomponent mixtures. In particular, it has been used extensively for

25

computing the adsorption isotherms of gases and their mixtures in models of various

types of nanoporous materials (Cracknell and Gubbins, 1993; Cracknell and

Nicholson, 1993, 1994; Cracknell, Nicholson and Gubbins, 1995; Gale et al., 1990;

Ghassemzadeh et al., 2000; Ghassemzadeh and Sahimi, 2004; June et al., 1990,

1992; Keldsen et al., 1994; Maddox and Gubbins, 1994; Mezei, 1980; Miller et al.,

1987; Percus, 1986; Peterson and Gubbins, 1987; Plee et al., 1985; Skipper et al.,

1989; Yi et al., 1995, 1996, 1998). The method was first used in studies of bulk

fluids (Adams, 1975), and was then extended to adsorbed systems (Van Megen and

Snook, 1982, 1985). The GCMC method is based on utilizing a Markov chain for

generating a series of molecular configurations by determining whether to accept a

configuration into the Markov chain using the same criteria as in canonical MC. In a

GCMC simulation the new configuration is generated by the following three steps:

(1) A particle is chosen at random and given a displacement with a probability given

by Eq. (1.13).

(2) A position is chosen at random, and a particle is added at that position with a

probability

min exp( / ),11

i ci

i

Z VpN

+BU k T

⎧ ⎫= −Δ⎨ ⎬+⎩ ⎭

(1.14)

Here, is the absolute activity, 3exp( / ) /i i BZ k Tμ= iΛ iμ is the chemical potential of

component i, TkB1=β and Tkmh Bii 2π=Λ is the de Broglie wavelength (h is

26

Plank’s constant). and are the volume of the CV and the number of atoms of

component i in CV, respectively.

cV iN

(3) A particle is chosen at random and is removed from the system with a probability

min exp( / ),1ii

i c

Np UZ V

−Bk T

⎧ ⎫= −Δ⎨ ⎬

⎩ ⎭ (1.15)

In order to guarantee irreducibility of the Markov chain (i.e., each state in the chain

can be reached from any other state), which is critical to maintaining ergodicity in

the system, the second and third GCMC perturbations are attempted with equal

probability (Allen and Tildesley, 1987). For simulation of multi-component fluids,

care must be taken to attempt insertion and deletion steps for each component with

equal probability. The dimensionless form of Z can be written as:

( )( )

(33

expexp cZ

βμ )σ βμσ

= =Λ

(1.16)

Where cμ is called the configurational potential.

1.6 Thesis Outline The aim of this thesis is to further the understanding of the transport and separation

of gas mixtures in carbon molecular sieve-membranes (CMSM), modeled either as a

single slit pore, a composite pore model, or as a three-dimensional pore network with

interconnected pores. Non-equilibrium molecular dynamics simulations are used for

27

investigating the transport and separation of binary mixtures of n-alkanes, as well as

mixtures of CO2 and n-alkanes, as well as pure CO2. The effect of important factors,

such as the temperature of the system, the applied pressure gradient, feed

compositions, pore size, pore interconnectivity, and interaction of the molecules with

each other, on their adsorption, permeation, and separation through CMSM will be

investigated. The results are compared with the experimental data generated in our

laboratory. Better understanding of these phenomena will permit the tailoring of

membranes for optimum sorption or separation performance.

The thesis is divided into the following five chapters. Chapter 2 presents the

results of extensive NEMD simulations of the transport and separation properties of

supercritical binary fluid mixtures consisting of CH4 and CO2 in a slit carbon

nanopore. In Chapter 3 contains the results of NEMD simulations of transport and

separation of binary mixtures of n-alkanes, as well as mixtures of CO2 and n-alkanes,

in a slit carbon nanopore. We present in Chapter 4 the results for the composite pore

model and address whether the permeation properties of CO2 and binary mixtures of

CH4 and CO2 in the system, under sub- and supercritical conditions exhibit any

asymmetry. The results of transport and separation of binary mixture of n-alkanes, as

well as mixtures of CO2 and n-alkanes, in a carbon pore network under sub- and

supercritical conditions are presented in Chapter 5.

28

Chapter 2 Transport and Separation of Supercritical Fluid Mixtures in a Single Carbon Nanopore 2.1 Introduction

lmost all the previous theoretical and computer simulation studies of

transport of fluid mixtures in confined media have been carried out under the

condition that the fluids that enter the nanopore space are in a thermodynamic state

below their critical points. On the other hand, supercritical fluids (SCFs) – those that

are in a thermodynamic state above their critical temperature and/or pressure – have,

in recent years, attracted much attention. They are attractive media for chemical

reactions because of their unique properties. Many of the physical and transport

properties of SCFs are intermediate between those of a liquid and a gas. Such fluids

and their transport and separation properties are the focus of this chapter.

A

Consider, for example, supercritical fluid extraction (SCFE) for adsorbent

regeneration, which appears to be a promising technology. Spent adsorbents are

currently being incinerated, landfilled, or subjected to harsh thermal regeneration.

With disposal costs rapidly increasing, attention is progressively placed on more

efficient regeneration processes, and SCFE certainly shows great potential in this

area. For example, SCFE may be used for clean-up of contaminated soils. The

29

advantage of SCFE over conventional liquid extraction is that in the SC region,

where density becomes liquid-like, viscosity and diffusivity remain between gas and

liquid values and surface tension effects are negligible, thus making it easier under

such conditions to access the soil's internal pore structure, and resulting in enhanced

mass transport.

SCFE utilizing CO2 has received considerable attention in recent years (Kiran

and Sengers, 1994; Sengers and Sengers, 1986). The potential utility of this process

has been investigated for the removal of contaminants from water (Brignole et al.,

1987; Gamse et al., 1997; Meguro et al., 1996; Roop et al., 1988; Shing et al., 1988),

sludge (McGovern et al., 1987), soils (Akgerman and Yao, 1993; Brady et al., 1987;

Dooley and Knopf, 1987; Dooley et al., 1987; Erkey et al., 1993; Firus et al., 1997;

Hall et al., 1990; Liu et al., 1991; Pang et al., 1991), spent catalysts (Silva et al.,

1993), aerogels (Novak and Knez, 1997; Wawrzyniak et al., 1998), and adsorbents,

such as activated granular carbon (De Filippi et al., 1980, 1983; Macnaughton and

Foster, 1995; Tan and Liou, 1989; Tomasko et al., 1993, 1995). They are also used

for preparing nanosize particles for drug delivery. Though other compounds, such as

propane, butane, and various fluorocarbons might be better solvents under SC

conditions (especially for low volatility solutes), CO2 is generally preferred in

environmental applications because it is non-toxic and non-flammable. In addition,

SCFE by CO2 leaves no solvent residue on the contaminated soil. Compared with

thermal incineration, SCFE is less energy intensive, leaves the soil's internal

30

structure and nutrients relatively intact, and can handle far higher concentrations of

toxic components than bioremediation.

Perhaps the most promising SCFE method is one that combines a SC fluid with

another separation medium, such as a nanoporous membrane, which will

preferentially and continuously extract the solute, leaving behind a solute-depleted,

recyclable SC solvent stream, hence promising significant reductions in the cost of

the energy intensive compression/re-compression cycle, typically associated with

SCFE. Several recent studies have investigated the use of various membranes with

SC fluids (Afrane and Chimowitz, 1993, 1996; Chimowitz and Afrane, 1996; Fujii et

al., 1996; Kelley and Chimowitz, 1990; Muller et al., 1989; Nakamura et al., 1994;

Ohya et al., 1993; Sarrade et al., 1996; Semenova et al., 1992b,1992a; Tokunaga et

al., 1997), and reported several complex phenomena. For example, hysteresis in the

permeability isotherms was observed at some temperatures but not at others. The

permeabilities also exhibited a maximum as a function of the temperature, which

also depended on the pressure. Solute rejection was found to be positive or negative,

depending on the membrane type and the solutes used. Such complex behavior is

generally attributed to the combined effects of sorption of the SC fluid, their

tendency to aggregate and form clusters, and the variation of the pore structure and

surface characteristics due to membrane type and preparation procedures, as well as

the competition between equilibrium- and transport-dominated processes. There is

currently little fundamental knowledge on the transport of SC mixtures in

membranes, due to the complexities that arise as the result of the interactions

31

between the highly nonideal, compressible SC mixtures and the complex pore space

of a nanoporous membrane. Moreover, in addition to being of practical importance,

these phenomena have also raised a number of fundamental issues related to the

solvation structure and dynamics of the solutes under SC conditions.

The fundamental understanding of the behavior of fluids under SC conditions

in microporous systems, such as adsorbents, membranes and catalysts, is in general

very limited. The early studies focused on the adsorption of single gases in carbon

molecular sieves (Bojan et al., 1992; Matranga et al., 1992), in micropores (Van

Slooten et al., 1994), and in pores with structured surfaces (Cascarini de Torre et al.,

1996; Nicholson, 1994). These studies provided valuable information about such

systems. For example, Nicholson's study of adsorption of N2 in a slit-shaped pore

with corrugated potential (to represent surface heterogeneity) indicated that, the

transition associated with corrugations are likely to be suppressed in micropores in

comparison with the corresponding phenomena observed with planar surfaces.

Afrane and Chimowitz (Afrane and Chimowitz, 1996) reported on experimental

studies of SC fluids during separation processes in inorganic membranes.

Studies of the sorption of mixtures under SC conditions have only recently

appeared (Chimowitz and Afrane, 1996). Nitta (Nitta et al., 1993; Nitta and Yoneya,

1995) and Shingetta (Shigeta et al., 1996) studied the sorption of dilute benzene/SC

CO2, and butane/SC CO2 mixtures in slit-shaped graphitic pores using MC

techniques. The kinetic properties of mixtures in pores and membranes, such as the

rates of mutual diffusion, remain unexplored and little understood. Only a few

32

modeling studies (Sedigh et al., 1998; Xu et al., 1998), of direct relevance to non-

equilibrium transport in porous carbons have been published. The first study is by

Cracknell et al. (Cracknell et al., 1993), who studied simple LJ spheres in graphite

slit-shaped nanopores. More recently, Furukawa et al. (Furukawa et al., 1996, 1997;

Furukawa and Nitta, 1997) reported a study of the transport of a mixture of two LJ

spheres through a slit-shaped nanopore, followed by studies of the effect of surface

heterogeneity. A somewhat related study is by Takaba et al. (Takaba et al., 1996),

who investigated the effect of sorption affinities on CO2 separation from a CO2/N2

mixture in a slit-shaped inorganic membrane pore, using MD simulation of the

transient uptake process.

Although work is ongoing in the area of modeling the transport and sorption of

nanoporous carbon adsorbents and membranes, we are not aware of any published

study of the same phenomena under SC conditions, through CMSM in the presence

of a strong chemical potential or pressure gradients. These are subjects of our study

in this chapter. In the context of the work reviewed above, the study of transport

properties of hydrocarbon/SC CO2 mixtures in nanoporous carbon membrane

materials represents a new class of research problems with great scientific interest

and technological importance. Although a number of studies have been reported,

there is still a lack of systematic studies of the non-equilibrium transport of SC

mixtures (even at a single pore level), and a lack of integrated experimental/

theoretical/computational studies of sorption and transport spanning the micro and

macro-scales.

33

This chapter represents the first part of our work aimed at developing a

fundamental understanding of the phenomena involved during the transport of

hydrocarbon/SC CO2 mixtures in a nanoporous membrane. The results of our studies

should be useful to applications that involve the regeneration of adsorbents by SC

CO2 and the use of membranes under SC conditions. The emphasis in this study is on

identifying the factors that control the ability of the membranes for separation of SC

mixtures.

In this work we use a dual control-volume-grand-canonical MD technique

(Heffelfinger and Van Swol, 1994; MacElroy, 1994) (DCV-GCMD), which has been

used over the past few years (Cracknell, Nicholson and Quirke, 1995; Ford, 1995;

Ford and Heffelfinger, 1998; Furukawa et al., 1996, 1997; Furukawa and Nitta,

1997; Heffelfinger and Van Swol, 1994; Heffelfinger and Ford, 1998; Kjelstrup and

Hafskjold, 1996; MacElroy, 1994; Nicholson et al., 1996; Nitta, 1995; Pohl et al.,

1996; Pohl and Heffelfinger, 1999; Smith, 1986; Sun and Ebner, 1992; Sunderrajan

et al., 1996; Thompson et al., 1998). Although molecular simulations have been used

for studying SC fluids in the bulk (Chialvo and Cummings, 1999; Chung and Shing,

1992; Cummings et al., 1991; Eckert et al., 1996; Martinez et al., 1996; Nouacer and

Shing, 1989; Shing and Chung, 1987; Shing, 1991; Yoshii and Okazaki, 1997), very

few such studies have been reported in which they have been used for investigating

the behavior of SC fluids in small pores (Nicholson, 1998).

34

2.2 The Pore Model As a prelude to understanding transport and separation of SC fluid mixtures in

CMSM, which consist of a pore space of interconnected nanosized pores, we first

consider the same phenomena in a single carbon nanopore. Recent molecular

simulations by Düren et al. (Düren et al., 2002, 2003) indicate that transport in a

nanopore is hardly influenced by its shape, so long as the correct average radius,

transport length and enclosing gradient are used. Thus, we consider here a simple a

slit-pore, a schematic representation of which is shown in Figure 2.1, in which the

origin of the coordinates is at the center. The two carbon walls are located at the top

and bottom xy planes. The external driving force is a chemical potential or,

equivalently, a pressure gradient applied in the x-direction. The system is divided

into three regions. The h- and l-regions represent, respectively, the control volumes

(CVs) exposed to the bulk fluid at high and low chemical potential or pressure, while

the middle region represents the pore. The pore's length is nL with n being an

integer. In most of our calculations we used n = 1; however, longer pores with n > 1

were also simulated, and were found to have no significant effect on the results.

Periodic boundary conditions were employed only in the y-direction.

The pore's walls are assumed to be smooth (structureless), since our group’s

previous studies of subcritical gas mixtures (Firouzi et al., 2003; Xu et al., 1998,

1999; Xu, Sedigh et al., 2000; Xu, Sahimi et al., 2000) indicated that the walls'

atomistic structure has little effect on the transport and separation properties. We

35

consider a mixture of CH4 (component 1) and CO2 (component 2), which are

represented as Lennard-Jones (LJ) spheres, characterized by effective LJ size and

energy parameters, σ and ε. We will present the results of our study for the mixtures

of SC CO2 and heavier alkane chains in Chapter 5. All the quantities of interest were

made dimensionless with the help of σ1 and ε1; the dimensionless groups are listed in

Table 2.1. As in our group’s work with subcritical gas mixtures (Xu, Sedigh et al.,

2000), we also considered a more realistic model for CO2, consisting of three LJ

interaction sites on the three atoms, plus point charges to account for the quadrupole

moment of CO2 molecules. However, we found no significant effect on the results.

Table 2.1: The conversion between the reduced and the actual units. Subscript 1

referes to the value of the parameters for CH4. Variable Reduced Form Length L 1/* σLL = Energy U 1/* εUU = Mass M 1/* MMM = Density ρ 3

1* ρσρ = Temperature T 1/* εTkT B= Pressure P 1

31 / εσP

Time t 2/12111 )/(* σε Mtt =

Flux J 2/111

31 )/(* εσ MJJ =

Permeability K 12/1

11 /)(* σεMKK =

36

The dimensions of the pore used were, W = 20 1σ and L = 40 1σ . Two pore

sizes (heights), H = HM = 5 1σ and H = Hm = 1.67 1σ , were used - narrow enough

sizes for which the carbon membranes may exhibit molecular sieving properties. Our

group’s simulations with subcritical gas mixtures (Xu et al., 1999; Xu, Sedigh et al.,

2000; Xu, Sahimi et al., 2000) indicated that the pore size Hm is optimal for

separation of binary gas mixtures into its components, yielding the highest separation

factor.

2.3 The DCV-GCMD Method In the simulations we combined MD moves in the entire system with GCMC

insertions and deletions in the two CVs that we described earlier. In the MD

simulations the Verlet velocity algorithm was used to solve the equations of motion.

Iso-kinetic conditions were maintained by rescaling the velocity independently in all

the three directions. It is essential to maintain the densities of each component in the

two CVs at some fixed values, which are in equilibrium with two bulk phases, each

at a fixed pressure and fluid concentration. The densities, or the corresponding

chemical potentials of each component in the CVs, were maintained by carrying out

a sufficient number of GCMC insertions and deletions of the particles. The

probability of inserting a particle of component i is given by

37

min exp( / ),11

i ci

i

Z VpN

+BU k T

⎧ ⎫= −Δ⎨ ⎬+⎩ ⎭

(2.1)

where is the absolute activity at temperature T, and 3exp( / ) /i i BZ k Tμ= iΛ iΛ iμ

are, respectively, the de Broglie wavelength and chemical potential of component i,

the Boltzmann's constant, U the potential energy change resulting from creating

or removing a particle, and and the volume of the CV and number of atoms of

component i in each CV, respectively. The probability of deleting a particle is given

by

Bk

cV iN

min exp( / ),1ii

i c

Np UZ V

−Bk T

⎧ ⎫= −Δ⎨ ⎬

⎩ ⎭ (2.2)

When a particle is inserted in a CV, it is assigned a thermal velocity selected from

the Maxwell-Boltzmann distribution at the given T. An important parameter of the

simulations is the ratio Ρ 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 Ρ was typically 10. During the MD

calculations particles crossing the outer boundaries of the CVs were removed. The

number of such molecules was, however, very small, typically about 1% of the total

number of molecules that were deleted during the GCMC simulations. 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, consistent with the

38

presence of bulk pressure/chemical potential gradients along the flow direction. 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 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 (Heffelfinger and Van Swol, 1994; Kjelstrup and

Hafskjold, 1996; Lupkowski and Swol, 1991; Maginn et al., 1993; Papadopoulou et

al., 1993). However, the actual streaming velocities of the molecules in the transport

pore region were still determined by the MD simulations. To study the transport of a

mixture due to a pressure gradient, the temperature of the system must be held

constant in order to eliminate any contribution of the temperature gradient to the

transport; hence special care was taken to achieve this. Figure 2.2 shows the

dimensionless time-averaged temperature profiles along a pore of size H = Hm =

1.67σ1 (the smallest pore used in the simulations; see above) containing an

equimolar mixture of CO2 and CH4. The upstream pressure in both cases is 120 atm,

while the downstream pressures are 90 atm and 20 atm. Both profiles, even in the

39

pore for which a pressure drop of 100 atm was applied, are reasonably smooth,

indicating the accuracy of the NEMD simulations.

2.4 Potential Models of the Molecules and the Walls As already mentioned, CH4 (component 1) and CO2 (component 2) were assumed to

be LJ spheres to which we assigned effective values of the LJ size and energy

parameters, σ and ε. We used σc = 3.4 Å , and εc/ = 28 K for the carbon atoms

constituting the pore's wall, and σ

Bk

1 = 3.81 Å, ε1/ = 148.1 K, σBk 2 = 3.79 Å, and

ε2/ = 225.3 K . For the cross-term LJ parameters, the Lorentz-Berthelot

combining rules were used:

Bk

2112 εεε = , and )(21

2112 σσσ += . We confirmed, by

carrying out equilibrium molecular simulations, that such a molecular representation

of CO2 produces a phase diagram which is in qualitative agreement with the

experimental data, under both subcritical and supercritical conditions.

The molecule-molecule interactions were modeled with the cut-and-shifted

LJ potential described earlier [Eq. (1.6)]. The cut-off distance was taken to be 2.5σ1

and long-range corrections were not applied. We utilized smooth pore walls for

which the 10-4-3 potential of Steele,

10 4 4

23

2( ) 25 3 (0.61 )

iw iw iwiw c iw iwU z

z z zσ σ σπρ ε σ

⎧ ⎫⎪ ⎪⎛ ⎞ ⎛ ⎞= Δ − −⎨ ⎬⎜ ⎟ ⎜ ⎟ Δ Δ +⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭ (2.3)

40

was used to calculate the interaction between a molecule and the wall, where

= 0.335 nm is the space between the adjacent carbon layers, Δ cρ = 114 nm-3 is the

number density of carbon atoms in the layer, z is the distance from the wall, and iwσ

and iwε are the LJ parameters between the walls and molecule i.

2.5 Molecular Dynamics Simulations Unless otherwise specified, all of the simulations were carried out with the upstream

CV being in equilibrium with a bulk fluid at a total pressure of 120 atm. The

temperature of the system was either T = 40°C or higher. Such pressure and

temperatures are above the critical conditions for CO2 and its mixture with CH4. For

the downstream CV the pressure of the bulk phase with which it is in equilibrium

was varied between 1 and 110 atm in order to assess the effect of the applied

pressure gradient on the phenomena. The same mixture compositions were used in

both CVs.

We computed several quantities of interest, including the density profiles of the

component i along the x- and z-directions, (x) and ziρ x

iρ (z), respectively. To

calculate (x) the simulation box was divided in the x-direction into grids of size ziρ

1σ , and for each MD step the density profiles (x) were obtained by averaging the

number of particles of component i over the distance

ziρ

1σ . A similar procedure was

used for computing xiρ (z), with the averaging done over a small distance which was

41

about 0.16 1σ . As discussed below, these quantities are important to understanding

adsorption and transport properties of the fluids between the two pores walls.

For each component i we also calculated its flux by measuring the net

number of its particles crossing a given yz plane of area Ayz:

iJ

MD

LR RLi i

iyz

N NJN tA

−=

Δ (2.4)

where and are the number of the molecules of type i moving from the left

to the right and vice versa, respectively,

LRiN RL

iN

tΔ is the MD time step (we used * = 5 ×

10

tΔ

-3, i.e., 0.00685 ps, where t* is the dimensionless time), and N≈Δt MD is the

number of the MD steps over which the average was taken (we typically used NMD =

70,000). The system was considered to have reached steady state when the fluxes

calculated at various yz planes were within 5% from the averaged values, after which

the fluxes were calculated at the center of the transport region. The equations of

motion were integrated with up to 5 × 106 time steps. The steady state was typically

reached after 3 × 106 time steps.

The permeability of species i was calculated using iK

/i

ii i

iJ nLJKP nL P

= =Δ Δ

(2.5)

where is the partial pressure drop for species i along the pore, with xPxP ii Δ=Δ i

being the mole fraction of component i, and PΔ the total pressure drop imposed on

the pore. A most important property that we wish to study is the dynamic separation

factor S21 , defined as

42

221

1

KSK

= (2.6)

2.6 Results and Discussions Figure 2.3 presents the time-averaged density profiles of CO2 and CH4, in an

equimolar mixture of the two molecules at T = 40°C, as functions of X* = x / 1σ

along the pore, defined as the region, X≤− 20 * 20≤ . The downstream pressure

ranges from 20 atm, which is a subcritical condition, to 100 atm, which is

supercritical, and the pore size is HM = 5 1σ . In this and the subsequent figures, the

dashed lines indicate the boundaries of the pore region. The density profiles are

essentially flat in the two CVs, with numerical values that match those obtained by

the GCMC method at the same conditions, indicating that the chemical potentials in

the two CVs have been properly maintained during the NEMD simulations. The

small fluctuations in the profiles in the CV regions represent numerical noise. The

small downward curvature at X* = -50 (in the CO2 profile) or upward curvature at

X* = +50 (in the CH4 profile) is due to the “leakage” of the particles out of the two

CVs. These are the particles that, as discussed in Section 2.3, cross the outer

boundaries of the CVs and leave the system. However, such deviations from a flat

profile are insignificant. In the pore region, the densities for both components

decrease along the pore, which is expected. Due to the pressure drop applied to the

43

pore, the density profiles are not necessarily linear: the total flux is the sum of the

diffusive and convective parts, which result in density profiles that decay more or

less exponentially.

These features are quite pronounced when the pressure drop along the pore is

large enough. In particular, the CO2 density profiles clearly exhibit such features.

However, as the pressure drop along the pore decreases, the differences between the

various profiles become less pronounced, particularly for CH4. The reason is that,

with an upstream pressure of 120 atm, quite larger than the critical pressures for CO2

and CH4 (which are, respectively, about 73 atm and 45 atm), the molecules will be in

a liquid-like state everywhere in the pore, if the pressure drop in the pore is not large

(that is, if the downstream pressure is close to the upstream one). In contrast, when

the pressure drop along the pore is large, with the upstream pressure still being at

120 atm, the molecules will be in a gas-like state near the downstream CV, and

therefore the density profiles take on distinct shapes for the two CVs and the pore.

These features are clearly seen in Figure 2.4, where we show snapshots of the same

pore and the distribution of the molecules in it for the same mixture as in Figure 2.3,

for three different applied pressure drops, obtained after 3.5 × 106 time steps. In all

the cases the upstream pressure is 120 atm, but the downstream pressure is different

among the three. In the case of a pore with a downstream pressure of 20 atm (the top

panel), most molecules are distributed near the walls, which is what is expected for a

gaseous state in a pore. As the downstream pressure increases, the pore is

increasingly packed with molecules, creating a liquid-like state (middle panel). For a

44

downstream pressure of 100 atm the pore is packed with a very large number of

molecules (the bottom panel).

Figure 2.5 presents the same density profiles as in Figure 2.3, but for a mixture

with a CO2 mole fraction, x2 = 0.9. All the qualitative features of the profiles are

similar to those shown in Figure 2.3 except that, due to the low mole fraction of CH4

in the mixture, the differences between its density profiles in different regions of the

system (the two CVs and the pore) are not as large as those shown in Figure 2.3.

The density profiles shown in Figures 2.3 and 2.5 exhibit some fluctuations,

although their overall shapes are what one may expect. These fluctuations will

decrease and practically vanish if the simulation time is much larger than the time

that we used typically in the simulations, namely, 3.5 × 106 time steps. To

demonstrate this, we present in Figure 2.6 the density profiles of the two components

in an equimolar mixture, obtained after 5 million time steps in a pore of size HM =

5 1σ , and upstream and downstream pressures of 120 atm and 90 atm, and compare

them with the same profiles obtained after 3.5 million time steps. Two features of

this figure are noteworthy. (1) The density profiles obtained after the longer

simulations are much smoother than the one computed after the shorter simulations.

(2) The density profiles obtained after 3.5 × 106 time steps (the dashed curves)

fluctuate around the smoother profiles obtained after over 5 million time steps. The

fluctuations are, however, small.

Figure 2.7 shows the time-averaged density profiles for CO2 and CH4 in an

equimolar mixture at T = 40°C in the direction perpendicular to the pore's walls, in

45

three different regions of the pore. The downstream pressure is 70 atm, the pore size

is 5 1σ , and Z* = z / 1σ . These profiles were computed by averaging the results over

the last 300,000 time steps of the simulations. As these results indicate, the

supercritical fluid mixtures form several liquid-like layers, packing the pore almost

completely; see the distribution of the molecules in the pore, which is also shown in

Figure 2.7. This is in contrast with subcritical gases that form only two adsorbed

layers near the a pore's walls, leaving its middle region almost empty, an example of

which is shown in Figure 2.8 for exactly the same system as in Figure 2.7, except

that the upstream and downstream pressures are 3 atm and 1 atm, respectively.

Multilayer formation persists in the entire pore so long as the downstream pressure is

high enough for the mixture to exit the pore under supercritical conditions. If,

however, the downstream pressure is low, then multilayer formation occurs only near

the upstream region of the pore. As the molecules approach the downstream region,

only two layers, one near each wall, survive, hence signaling a phase transition from

a liquid-like state to a gaseous one. An example is shown in Figure 2.9, which is for

a system completely identical with that shown in Figure 2.7, except that the

downstream pressure is only 1 atm. Such feature will not change, if the composition

of the mixture changes. For example, Figure 2.10 presents the same density profiles

for a mixture with a CO2 mole fraction of x2 = 0.9. The rest of the parameters of the

system are the same as those of Figure 2.7. All but one of the qualitative features of

the two sets of density profiles are the same. The only difference is that the size of

the maxima in the CH4 density profiles of Figure 2.10 is much smaller than those of

46

Figure 2.7, but this is expected as the mole fraction of CH4 in this mixture is only x1

= 0.1 .

An important consequence of pore packing by supercritical fluids is the

formation of fluid clusters. We consider two molecules i and j to belong to the same

cluster if the distance rij between their centers is less than a critical distance Rc, where

we take Rc = 1.1 1σ . This definition can be used to define clusters of CO2, CH4, as

well as their mixtures. We find that such clusters are formed and travel the entire

length of the pore. Figure 2.11 presents the distribution of the molecules and the

various clusters that they have formed at three different times. All the parameters of

the system are the same as in Figure 2.7, except that the downstream and upstream

pressures are both 120 atm. We find that, even though the molecules' fluxes have

reached a steady state, over the time period in which these clusters were formed,

their sizes oscillate with the time. To establish this more quantitatively, we show in

Figure 2.12 the time-dependence of the largest clusters' sizes in the pores with sizes,

H = HM = 5 1σ and H = Hm = 1.67 1σ , where we set the downstream pressure at 70

atm, which is less favorable to cluster formation than when the downstream pressure

is also 120 atm (the rest of the parameters are the same as before). Despite this, large

mixed clusters, consisting of both CO2 and CH4, have formed, with sizes that

oscillate greatly with the time.

In general, we find that there is a distribution of the cluster sizes ns(t), where

ns(t) is the number of cluster of size s at time t, and a cluster's size is defined as the

number of molecules that it contains. Figure 2.13 presents the time-dependence of

47

the various cluster sizes ns(t) for the same two pore sizes shown in Figure 2.12. The

size of all the clusters, from the smallest to largest, oscillate with the time.

In dynamic critical phenomena, the cluster size distribution ns(t) follows

certain universal scaling in both the time t and cluster size s. In the present problem,

due to the oscillatory variations of ns(t) with the time t, one cannot expect the same

type of universal scaling as in dynamic critical phenomena. We do, however, find

that for small- to moderate-size clusters,

( ) ~sn t s τ− (2.7)

which is similar to what one finds in dynamic critical phenomenon (Vicsek and

Family, 1984). This is shown in Figure 2.14, where we show the results for a pore of

size 5 1σ . The rest of the system's parameters are the same as those of Figures 2.12

and 2.13. Note that the size of the pore imposes an upper cutoff on the size of the

molecular clusters that can form, and, therefore, the largest clusters may not follow

Equation 2.7, as they are very rare. It also implies that accurate determination of the

cluster size distribution for large cluster sizes in small pores may require very large-

scale simulations. Whether the cluster size distribution ns(t) follows any universal

dynamic scaling, i.e., whether the exponent τ is universal and independent of the

system's parameters (which would then be similar to what one finds in dynamic

critical phenomena), remains to be studied with much larger-scale simulations.

From a practical view point, two of the most important properties to be studied

are the permeance (permeability per unit length) of the two components, as well as

the dynamic separation factor defined by Equation 2.6. We find that the pore packing

48

and formation of the clusters have profound consequences for separation of a

supercritical fluid mixture into its components. More specifically, we find that,

(1) in the pore with a size near the optimal size Hm = 1.67 1σ , the permeances

(and the permeabilities Ki ) of the two components vary nonmonotonically with

increasing pressure drop PΔ along the pore. This is demonstrated in Figure 2.15,

where we show the fluxes of the two components in an equimolar mixture, along

with their permeances, and together with the corresponding separation factors. In this

figure, the upstream pressure is fixed at 120 atm, while the downstream pressure

varies. These results indicate that,

(2) there is an optimal pore size and optimal pressure drop PΔ for the

separation of the two components at which the separation factor is maximum:

whereas in larger pores decreases monotonically with increasing , in the

smaller pores with a size close to H

21S

21S PΔ

m = 1.67 1σ , the separation factor reaches a

maximum value at a certain value of PΔ , beyond which it decreases monotonically

with increasing . To demonstrate this, we present in Figure 2.16 the same

quantities as in Figure 2.15, under precisely the same conditions, except that the pore

size is now H = 5

PΔ

1σ . In this case, no maximum is seen in the separation factor. This

behavior persists for other mixture compositions. For example, Figure 2.17 presents

the same results as in Figure 2.16, except that the CO2 mole fraction in the mixture is

x2 = 0.9 . The physical reason for this novel, and unexpected, result may be as

follows.

49

We have fixed the upstream pressure at 120 atm, which is significantly

above the minimum pressure for supercriticality of the mixture, and have varied

the downstream pressure over a range of values that includes both supercritical

and subcritical conditions. If , the downstream pressure, is equal to , then the

only possible mechanism of transport in the pore is molecular diffusion. However,

since under supercritical conditions the fluid mixture is in a liquid-like state, and the

pore is very small, one may have a saturation effect, in which the pore is filled with

the molecules with a very small, if any, gradient in the concentrations, as a result of

which diffusion alone is an inefficient mechanism of transport in the pore, and,

therefore, the separation factor is relatively low. Suppose that is held fixed, and

the downstream pressure is decreased from its initial value = . As soon as

, there will be a net convective flux in the pore. Since CO

1P

cP

2P

2P 1P

1P

2P 2P 1P

21 PP ≠ 2 is in a liquid-like

state in much larger amounts than CH4, transport of its clusters helps carry CO2 in

amounts that are much larger than CH4, as a result of which the separation of the two

fluids improves, and hence the separation factor increases. Thus, as the downstream

pressure decreases, hence increasing the overall pressure drop , both

convection and diffusion of fluid clusters increase, and therefore the separation

factor also increases. Thus, the separation of the two fluids must be maximum at a

downstream pressure (with a corresponding value of the overall pressure drop

) which is just above the minimum pressure for supercriticality of the

2P PΔ

2P

PΔ cP

50

mixture, since a downstream pressure ensures that the mixture will be in a

liquid-like state in the entire pore. However, if the downstream pressure

decreases below , then as the fluid molecules approach the downstream region of

the pore, they make a transition from liquid-like states to gas-like state in which most

of the CO

cPP >2

2P

cP

2 is distributed in the slow region of the pore near its walls, as a result of

which convection and diffusion of the clusters will no longer be as effective in

transporting the molecules. Therefore, for cPP <2 the separation of the fluids must

be less efficient, and thus the separation factor must decrease beyond its maximum.

This phenomenon will not be as effective in larger pores, where there is less

condensation and the saturation effect is also less severe, which explains why the

maximum in the separation factor for larger pores is not as large and distinct as the

one for the smallest pore with a size near 1.67 1σ .

We may then conclude there must indeed be an optimal pore size and an

optimal pressure drop for which the separation of the two fluids is most efficient, a

discovery with potentially very important consequences for practical applications. It

would then be very interesting to compute the optimal PΔ for a realistic model of a

nanoporous membrane, such as the one developed by our group (Ghassemzadeh et

al., 2000; Xu, Sahimi et al., 2000; Xu et al., 2001) which consisted of a molecular

network of interconnected pores with distributed sizes. Work in this direction is in

progress.

51

We also studied the temperature- and composition-dependence of the

permeances and the corresponding separation factors. Shown in Figure 2.18 are these

quantities at two temperatures and several CO2 mole fractions in the two CVs. The

downstream pressure is 20 atm, while the pore size is 1.67 1σ . The permeance of

CH4 is affected only weakly by both the temperature and composition of the mixture,

while the permeance of CO2 appears to depend rather strongly the feed composition.

The separation factor, on the other hand, depends only mildly on the feed

composition.

Figure 2.19 presents the temperature-dependence of the permeances and the

corresponding separation factors for two downstream pressures (the upstream

pressure is 120 atm). For a downstream pressure of 90 atm, the CO2 permeance

reaches a maxium at T = 100°C, beyond which it decreases with increasing

temperature, which is in agreement with the experimental observations (Afrane and

Chimowitz, 1993, 1996; Chimowitz and Afrane, 1996; Fujii et al., 1996; Kelley and

Chimowitz, 1990; Muller et al., 1989; Nakamura et al., 1994; Ohya et al., 1993;

Sarrade et al., 1996; Semenova et al., 1992b,1992a; Tokunaga et al., 1997). The

same trend is not seen for a downstream pressure of 110 atm, but this is probably

because the upstream and downstream pressures are close to each other, hence

rendering convection ineffective in transporting the molecules. The separation factor

, on the other hand, appears to be a relatively weak function of the temperature.

This is also illustrated in Figure 2.19. It appears that the maximum separation factor

is reached at T = 40°C, when the downstream pressure is 90 atm, while it decreases

21S

52

monotonically with the temperature, when the downstream pressure is 110 atm, very

close to the upstream pressure (of 120 atm), hence making convection an inefficient

mechanism of molecular transport.

2.7 Summary Extensive molecular simulations, using the dual control-volume-nonequilibrium

molecular dynamics method, were carried out to study transport and separation of

binary mixtures of supercritical CO2 and CH4 in a carbon nanopore. The driving

force was a pressure (chemical potential) gradient. The effect of the pore size, the

composition of the feed, the temperature of the system, as well as the applied

pressure gradient on the transport and separation of the mixtures were all studied.

Among these factors, the pore size and the applied pressure gradient strongly affect

the separation process. In particular, if the upstream and downstream pressures are

both above the critical pressure for the supercriticality of the mixture, then there is an

optimal pressure gradient for maximum separation of the mixture into its

components. The pore size plays a similar role, namely, there is an optimal pore size

for the separation of the fluid mixtures. These results, if confirmed by a realistic

model of a nanoporous membrane with interconnected pores, will have important

implications for the design of such membranes for use under supercritical conditions.

53

A closely related phenomenon is the formation of dynamic clusters of the

fluids' molecules that pack the pore and travel its entire length under an applied

pressure gradient. The sizes of the clusters oscillate with the time. This phenomenon

is closely tied to formation of multilayers of the fluids' molecules that pack the pore,

and is, in fact, primarily responsible for the existence of an optimal pressure gradient

for maximum separation of a supercritical fluid mixture into its components.

54

Figure 2.1: Schematics of the slit pore used in the simulations. The h and l regions represent the high- and low-pressure control volumes respectively.

55

0

1

2

3

4

Tem

pera

ture

0

1

2

3

4

-60 -40 -20 0 20 40 60X*

Tem

pera

ture

Figure 2.2: Dimensionless temperature distribution in the pore containing an equimolar mixture of CO2 and CH4. The upstream pressure in both cases is 120 atm, while the downstream pressures are 90 atm (top) and 20 atm (bottom). The pore size is 1.67 1σ . Dashed lines indicate the boundaries of the pore region.

56

0.1

0.2

0.3

0.4

-50 -30 -10 10 30 50X*

Den

sity

of C

O2

0.04

0.08

0.12

0.16

-50 -30 -10 10 30 50X*

Den

sity

of C

H4

Figure 2.3: Time-averaged density profiles in the transport direction x in a pore of size 5 1σ , containing an equimolar mixture of the two components at T = 40°C. The upstream pressure is 120 atm, while the downstream pressures are 20 atm (open triangles), 50 atm (open circles), 70 atm (solid triangles), and 100 atm (solid circles). Dashed lines indicate the boundaries of the pore region.

57

-2

0

2

Z*

-2

0

2Z*

-2

0

2

10 12 14 16 18 20X*

Z*

Figure 2.4: The distribution of CO2 (circles) and CH4 (triangles) in a pore of size 5 1σ , containing an equimolar mixture at T = 40°C. The upstream pressure is 120 atm, while the downstream pressures are 20 atm (top), 70 atm (middle) and 100 atm (bottom).

58

0.1

0.2

0.3

0.4

0.5

-50 -30 -10 10 30 50

X*

Den

sity

of C

O2

0.00

0.02

0.04

0.06

-50 -30 -10 10 30 50

X*

Den

sity

of C

H4

Figure 2.5: Time-averaged density profiles in the transport direction x in a pore of size 5 1σ , containing a mixture of CO2 and CH4 with a CO2 mole fraction of x2 = 0.9 at T = 40°C. The upstream pressure is 120 atm, while the downstream pressures are 20 atm (open triangles), 50 atm (circles), and 70 atm (solid triangles. Dashed lines indicate the boundaries of the pore region.

59

0.20

0.25

0.30

0.35

0.40

-50 -30 -10 10 30 50

X*

Den

sity

0.05

0.10

0.15

0.20

-50 -30 -10 10 30 50

X*

Den

sity

Figure 2.6: Comparison of the density profiles of CO2 (top) and CH4 (bottom), computed after 3.5 million times steps (dashed curves) and 5 million time steps (solid curves). All other parameters of the system are the same as those in Figure 2.3. The upstream and downstream pressure are, respectively, 120 atm and 90 atm. Dashed lines indicate the boundaries of the pore region.

60

0 < X* < 1

-2

-1

0

1

2

0.0 0.4 0.8 1.2 1.6 2.0 2.4

Density

-20 < X* < -19

-2

-1

0

1

2

0.0 0.4 0.8 1.2 1.6 2.0 2.4

Density

Z* Z*

19 < X* < 20

-2

-1

0

1

2

0.0 0.4 0.8 1.2 1.6 2.0 2.4

Density

Z*

-2

-1

0

1

2

-25 -20 -15 -10 -5 0 5 10 15 20 25X*

Z*

Figure 2.7: Time-averaged density profiles between the upper and lower walls, in three different regions of a pore of size 5 1σ , containing an equimolar mixture of CO2 (dashed curve) and CH4 (solid curve) at T = 40°C. The upstream and downstream pressures are, respectively, 120 atm and 70 atm. Also shown is the distribution of the molecules in the pore. The arrows indicate the boundaries of the pore region.

61

-20 < X* < -19

-2

-1

0

1

2

0 0.1 0.2 0.3 0.4

Density

Z*

Figure 2.8: Same as in Figure 2.7, except that the upstream and downstream pressures are, respectively, 3 atm and 1 atm.

62

-10 < X* < -9

-2

-1

0

1

2

0.0 0.4 0.8 1.2 1.6

Density

- 20 < X* < -19

-2

-1

0

1

2

0.0 0.4 0.8 1.2 1.6

Density

Z* Z*

0 < X*< 1

-2

-1

0

1

2

0.0 0.4 0.8 1.2 1.6

Density

Z*

-2

-1

0

1

2

25 -20 -15 -10 -5 0 5 10 15 20 25X*

Z*

-

Figure 2.9: Same as in Figure 2.7, except that the upstream and downstream pressures are, respectively, 120 atm and 1 atm.

63

-20 < X* < -19

-2

-1

0

1

2

0 0.4 0.8 1.2 1.6 2 2.4 2.8

Density

Z*

0 < X* < 1

-2

-1

0

1

2

0 0.4 0.8 1.2 1.6 2 2.4 2.8

Density

Z*

19 < X* < 20

-2

-1

0

1

2

0 0.4 0.8 1.2 1.6 2 2.4 2.8

Density

Z*

Figure 2.10: Same as in Figure 2.7, except that the mole fraction of CO2 in the mixture is x2 =0.9.

64

-2

-1

0

1

2

Z*

-2

-1

0

1

2Z*

Figure 2.11: The distribution of the molecular clusters after 5 × 105 (top), 106 (middle) and 1.5 × 106 (bottom) time steps, in a pore of size 5 1σ , containing an equimolar mixture of CO2 and CH4 at T = 40 °C. The upstream and downstream pressures are both 120 atm. Numbers indicate the size of the clusters.

147 87 85 73 72 66 61 55 53 52 47 40 39 37 36 35 31

-2

-1

0

1

2

-20 -15 -10 -5 0 5 10 15 20X*

Z*

65

0

40

80

120

160

200

5.1 5.2 5.3 5.4 5.5

Time x 10-6

Clu

ster

Siz

e

CO2

CH4

Mixture

0

20

40

60

80

5.1 5.2 5.3 5.4 5.5

Time x 10-6

Clu

ster

Siz

e

CO2

CH4

Mixture

Figure 2.12: Time dependence of the size of the largest clusters in a pore of size 5 1σ (left) and 1.67 1σ (right), containing an equimolar mixture of CO2 and CH4 at T = 40°C. The upstream and downstream pressures are, respectively, 120 atm and 70 atm.

66

0

10

20

30

40

50

5.1 5.2 5.3 5.4 5.5Time × 10-6

Num

ber o

f Clu

ster

s

6-9

26-45

4-5

10-12

13-1516-25

0

10

20

30

5.1 5.2 5.3 5.4 5.5

Time × 10-6

Num

ber

of C

lust

ers 4-5

26-45

6-9

10-12

13-1516-25

Figure 2.13: Time-dependence of the size distribution of the molecular clusters. All the parameters are the same as those in Figure 2.12. Numbers next to the curves indicate the size of the clusters.

67

0

5

10

15

20

25

30

35

40

45

50

4-5 6-9 10-12 13-15 16-25 26-45

Cluster Size

Num

ber o

f Clu

ster

s

Figure 2.14: Cluster size distribution ns(t) in a pore of size 5 1σ , containing an equimolar mixture of CO2 and CH4 at T = 40 °C. The results are for times 5.1 × 106 (◊), 5.2 × 106 ( ), 5.4 × 106 (ο), and 5.5 × 106 (Δ).

68

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

10 20 30 40 50 60 70 80 90 100

ΔP

Flux

CH4

CO2

1.5

2.0

2.5

3.0

3.5

4.0

10 20 30 40 50 60 70 80 90 100

ΔP

Sepa

ratio

n Fa

ctor

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

10 20 30 40 50 60 70 80 90 100

ΔP

Perm

eanc

e

CO2

CH4

Figure 2.15: The dependence of the flux and permeance of CO2 and CH4, and the corresponding separation factor, on the pressure drop PΔ applied to a pore of size 1.67 1σ at T = 40°C. The mixture is equimolar, with the upstream pressure being 120 atm.

69

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

10 20 30 40 50 60 70 80 90 100

ΔP

Flux

CO2

CH4

1.5

2.0

2.5

3.0

3.5

4.0

10 20 30 40 50 60 70 80 90 100

ΔP

Sepa

ratio

n Fa

ctor

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

10 20 30 40 50 60 70 80 90 100

ΔP

Perm

eanc

e

CO2

CH4

Figure 2.16: Same as in Figure 2.15, but for a pore of size 5 1σ .

70

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

50 60 70 80 90 100

ΔP

Perm

eanc

e

CO2

CH4

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

50 60 70 80 90 100

ΔP

Flux

CO2

CH4

0.9

1.2

1.5

1.8

2.1

2.4

2.7

50 60 70 80 90 100

ΔP

Sepa

ratio

n Fa

ctor

Figure 2.17: Same as in Figure 2.15, but for a pore of size 5 1σ and CO2 mole fraction of 2x =0.9.

71

0.000

0.001

0.001

0.002

0.002

0.003

0.003

0.3 0.4 0.5 0.6 0.7 0.8 0.9

Mole Fraction of CO2

Perm

eanc

e

1

2

3

4

5

6

0.3 0.4 0.5 0.6 0.7 0.8 0.9

Mole Fraction of CO2

Sepa

ratio

n Fa

ctor

Figure 2.18: Dependence of the permeance of CO2 (solid curves) and CH4 (dashed curves), and the corresponding separation factors, on the mole fraction of CO2 in the feed, in a pore of size 1.67 1σ at T = 40°C (circles) and 100°C (triangles). The upstream and downstream pressures are, respectively, 120 atm and 20 atm.

72

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 20 40 60 80 100 120 140 160

Temperature (°C)

Sepa

ratio

n Fa

ctor

0.0000

0.0005

0.0010

0.0015

0.0020

0 20 40 60 80 100 120 140 160

Temperature (°C)

Perm

eanc

e

Figure 2.19: Temperature-dependence of the permeance of CO2 (solid symbols) and CH4 (open symbols), and the corresponding separation factors, for a pore of size 1.67 1σ and downstream pressures of 90 atm (squares) and 110 atm (triangles). The upstream pressure is 120 atm, and the mixture is equimolar.

73

Chapter 3 Transport and Separation of Carbon Dioxide-Alkane Mixtures in Carbon Nanopores 3.1 Introduction

here is a considerable practical and theoretical interest in developing

predictive models and understanding the effect of chemical structure for the

separation and transport properties of alkanes. A better understanding of the

separation of alkane mixtures, as well as their mixtures with other gases, is important

for the optimization of chemical and petrochemical industries. Membrane separation

appears to be a promising candidate for alkene/alkane separation, with the process

recognized by the petrochemical industry to be a key technology.

T

Molecular dynamics simulations have been used to study the dynamic

behavior of alkane chains in bulk conditions. In this chapter, we focus on separation

of mixtures of normal alkanes (n-alkanes), and also mixtures of CO2 and an n-alkane,

by carbon molecular-sieve membranes (CMSMs). Although separation of n-alkanes

by zeolites has been studied by many groups (Funke et al., 1996; Kapteijn et al.,

1995; Kusakabe et al., 1996; Van de Graaf et al., 1999; Vroon et al., 1996), our

study, to our knowledge, is the first that investigates their separation by CMSMs. In

this chapter, we report our preliminary results of experimental studies and molecular

74

simulation of transport and separation of several binary mixtures involving n-alkanes

in CMSM. Our molecular modeling is based on nonequilibrium molecular dynamics

(NEMD) simulations. We utilize a dual control-volume grand-canonical MD (DCV-

GCMD) technique. The alkane chains are generated by the configurational-bias

Monte Carlo method. To our knowledge, this is the first time that the DCV-GCMD

method is combined with the configurational-bias Monte Carlo technique.

The plan of this chapter is as follows. In the next section, we describe the

pore model that we employ in our simulations. Next, the molecular models of the

gases are described, after which we describe the growth of the alkane chains by the

configurational-bias Monte Carlo method. We then describe the DCV-GCMD

technique. The results are then presented and analyzed.

3.2 Model of Carbon Nanopore The pore model we used here is exactly the same as in Section 2.2. The dimensions

of the pore used are, W = 20σ1 and L = 40σ1, while its height H was varied, in the

range of typical pore sizes for which CMSM exhibit molecular sieving properties, in

order to study the effect of the pore size on the results. Here, σ1 is the effective

molecular size of CH4 (see below).

75

3.3 Molecular Models of the Gases and the Interaction Potentials We consider binary mixtures of CH4/C3H8, CH4/C4H10, and CO2/C3H8. The CH4 and

CO2 molecules are represented as Lennard-Jones (LJ) spheres, characterized by

effective LJ size and energy parameters, σ and ε. All the quantities of interest are

made dimensionless with the help of the CH4 parameters, σ1 and ε1. Table 2.1 lists

the conversions between the dimensionless and dimensional quantities.

The C3H8 and C4H10 molecules are grown by a configurational-bias Monte

Carlo method, which is described in the next section. These molecules are

represented by a united-atom (UA) model (Ryckaert and Bellemans, 1978) in which

the CH2 and CH3 groups are considered as single interaction centers with their own

effective potentials. The nonbonded interactions between interaction centers of

different molecules are described with a cut-and-shifted LJ potential with rc =

11.43Å. No tail corrections were applied to this potential. The Lorentz-Berthelot

mixing rules were used in the simulations to compute the size and energy parameters

of the unlike molecules (see Section 2.4). We utilized smooth pore walls and the

Steele potential was used to calculate the interaction between a molecule and the

wall (see Section 2.4).

For simplicity the total molecular mass of alkane was equally divided

between the C atoms, and therefore CH2 and CH3 groups had equal molecular mass.

Table 3.1 lists the size and energy parameters of CH4, CO2, and those of the CH2 and

CH3 groups.

76

Table 3.1: Values of the molecular parameters used in the simulations. kB is the Boltzmann’s constant.

B

Parameter Numerical Value

2CHσ (Å) 3.905

3CHσ (Å) 3.905

4CHσ (Å) 3.810

2COσ (Å) 3.794

2

/CH Bkε (K) 59.38

3

/CH Bkε (K) 88.06

4

/CH Bkε (K) 148.1

2

/CO Bkε (K) 225.3

Bond length (Å) 1.53 kθ (K rad -2) 62,500

0θ (degrees) 112

(K) 1116 0C

(K) 1462 1C

(K) -1578 2C

(K) -368 3C

(K) 3156 4C

(K) -3788 5C

The atoms and the UA centers are connected by harmonic potentials. The

distance between the atoms is fixed at 1.53Å. The intramolecular interactions consist

77

of the contributions by bond-bending (BB) and torsional forces. For the BB term, the

van der Ploeg-Berendsen potential (Van der Ploeg and Berendsen, 1982) is used:

20

1( ) ( )2BBU kθθ θ θ= − (3.1)

where θ is the angle between the atomic bonds. For the torsional potential, the

Ryckaert-Bellemans (Ryckaert and Bellemans, 1975) potential is used:

5

0( ) cos ( )k

tor kk

U cφ φ=

= ∑ (3.2)

where φ is the dihedral angle. Numerical values of all the parameters are listed in

Table 3.1. The 10-4-3 potential of Steele was used to compute the interaction

between a molecule and the wall with the same parameter as mentioned in Section

2.4.

3.4 Configurational-Bias Monte Carlo Method Because direct generation of the n-alkanes and their insertion into the CVs that are

connected to the nanopore greatly slows down the simulations, we used the

configurational-bias Monte Carlo (CBMC) technique (De Pablo et al., 1992; Frenkel

et al., 1991; Harris and Rice, 1988; Laso et al., 1992; Macedonia and Maginn, 1999;

Mooij et al., 1992; Siepmann and Frenkel, 1992; Smit et al., 1995) to grow the

alkane molecules. We then combined the CBMC method with the grand-canonical

MC technique to insert the grown alkane chains in the two CVs and, therefore, in the

nanopore. The atom-by-atom growth of the molecules is done in such a way that

78

regions of favorable energy are identified, and overlap with other molecules is

avoided, hence speeding-up the computations greatly.

More specifically, we consider the potential energy of an atom as the sum of

two contributions: (1) the internal energy uint, which includes parts of the

intramolecular interactions, and (2) the external energy uext, which contains the

intermolecular interactions and those intramolecular interactions that are not part of

the internal energy. The division is, of course, to some extent arbitrary and depends

on the details of the model. The following procedure is then used to grow an n-

alkane atom by atom (Smit et al., 1995).

(i) We insert the first atom at a random position, and compute the energy

u1(n) along with a quantity w1, which is similar to a Boltzmann factor,

[ ]1 1( ) exp ( )w n u nβ= − (3.3)

where , with being the Boltmann’s constant, T is the temperature of the 1( )Bk Tβ −= Bk

system, and n indicates the new state in which the system is in.

(ii) We then generate k trial orientations, denoted by { } 1 2, ,... kk=b b b b , in

order to insert the next atom l. These orientations are generated with a probability

which is a function of the internal energy: int ( )l ip b

int int1( ) exp ( )l i l ip b uC

β⎡ ⎤= −⎣ ⎦b (3.4)

where C is a normalization factor. For each of these trial orientations, the external

energy is also computed, along with the quantity ex ( )tl iu b

79

(3.5) 1

( ) exp[ ( )]k

extl

jw n uβ

=

= −∑ bl j

We then select one orientation, out of the k trial positions, with the probability

1( ) exp[ ( )]( )

ext extl i l i

l

p b uw n

β= − b (3.6)

We typically generated five trial orientations for propane and butane chain

molecules, since our preliminary simulations with as many as k = 20 trial

orientations did not result in large differences.

(iii) We repeat step (ii) M - 1 times until the entire alkane molecule is grown,

and the Rosenbluth factor (Rosenbluth and Rosenbluth, 1955) W(n) of the molecule

is calculated:

(3.7) 1

( ) ( )M

ll

W n w n=

= ∏

As mentioned above, this algorithm biases the insertion of a molecule in such a way

that regions with favorable energy are found, and overlap with other atoms is

avoided. Any given molecular conformation is generated with a probability given by

{ }int int

2 2

1

1( ) ( ) ( ) exp [ ( ) ( )]( )

1 exp[ ( )]( )

M Mext ext

l l l ll l l

M

P n p n p n u n u nCw n

U nC W n

β

β

= =

−

= = − +

= −

∏ ∏ (3.8)

where the total energy of the inserted molecule is, . int

1 1( )

M Mext

l l ll l

U u u u= =

= = +∑ ∑

The above algorithm constitutes the CBMC technique. It must be

supplemented with acceptance rules that remove the bias from the insertion step.

80

These rules depend on the type of the move and of the ensemble used in the

simulations.

3.5 Configurational-Bias Grand-Canonical Monte Carlo Method After generating the n-alkanes and accepting them, the next step is to insert them into

the CVs. To do this, we combine the CBMC method described above with a grand-

canonical MC method and refer to it as the CBGCMC technique. This is a method of

computing the sorption thermodynamics of linear-chain molecules when the sorbates

are represented with a UA force field and have flexible dihedral and bond angles.

This method consists of two steps. First, it generates the chain configurations one

atom at a time by the CBMC method described above. Second, as the chain molecule

is generated, the Rosenbluth weight (Rosenbluth and Rosenbluth, 1955) W is

accumulated and utilized in the acceptance rule of the GCMC method for insertion of

the molecules into the system.

The probability of adding a single chain to a system of Ni chains is given by

(Macedonia and Maginn, 1999),

3

exp( )min ( ),1( 1)

i ci

i i

VpNβμ+ W n

⎧ ⎫= ⎨ ⎬Λ +⎩ ⎭

(3.9)

where µi is the chemical potential of chain i, V is the volume of the CV, and iΛ is

the thermal de Broglie wavelength of component i. Equation (3.9) is completely

similar to the probability of inserting a molecule into a system in a standard GCMC

81

computations, 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, [ ]1 1( ) exp ( )w o u oβ= − , '

1( ) exp ( )

kext

l l jj

w o uβ=

⎡ ⎤= −⎣ ⎦∑ b 1( ) ( )Ml lw o== Π, and W 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

'{ }kb

3 1min ,1

exp( ) ( )i i

ii c

NpV W oβμ

− ⎧ ⎫Λ= ⎨ ⎬

⎩ ⎭ (3.10)

which, aside from the Rosenbluth weight, is again similar to that of a standard

GCMC computation. To insert the CO2 and CH4 molecules in the two CVs, the

probabilities and are computed according to the standard GCMC method,

namely, the Rosenbluth factor W is replaced by

ip+ip−

exp( )Uβ− Δ , where is the

potential energy change of the system as a result of adding or removing a particle to

the CVs.

UΔ

3.6 Non-equilibrium Molecular Dynamics Simulations The NEMD simulations consist of integration of Newton’s equation of motion in the

entire system, combined with the CBGCMC insertions and deletions in the two CVs.

In the MD simulations the Verlet velocity algorithm was used to solve the equations

82

of motion. During the motion of the n-alkanes in the system, the RATTLE algorithm

(Andersen, 1983) was used to satisfy the constraints imposed on the n-alkane chains.

Iso-kinetic conditions were maintained by rescaling the velocity independently in all

the three directions. It is essential to maintain the densities of each component in the

two CVs at some fixed values, which are in equilibrium with two bulk phases, each

at a fixed pressure and fluid concentration. The densities, or the corresponding

chemical potentials of each component in the CVs, were maintained by carrying out

a sufficient number of CBGCMC insertions and deletions of the particles, as

described above. The chemical potentials were converted to equivalent pressures

using a LJ equation of state (Johnson et al., 1993).

When a molecule is inserted in a CV, it is assigned a thermal velocity

selected from the Maxwell-Boltzmann distribution at the given temperature T. An

important parameter of the NEMD simulations is the ratio Ρ of the number of

CBGCMC insertions and deletions in each CV to the number of MD steps between

successive CBGCMC 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 pore region. In our

simulations, Ρ was typically 10. During the MD computations molecules crossing

the outer boundaries of the CVs were removed. The number of such molecules was,

however, very small, typically about 1% of the total number of molecules that were

deleted during the CBGCMC simulations with a probability that is given by

Equation 3.10. In addition, for each component we allowed for a nonzero streaming

83

velocity (the ratio of the flux to the concentration of each component) in the pore

region, consistent with the presence of bulk pressure/chemical potential gradients

along the flow direction. 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 the

pore region slows down the computations. To address this problem, a very small

streaming velocity was added to the thermal velocity of all the inserted molecules

within each CV that were located within a very small distance from the boundaries

between the CVs and the pore (Arya et al., 2001; Firouzi et al., 2003; Martin et al.,

2001; Xu et al., 1998, 1999; Xu, Sedigh et al., 2000; Xu, Sahimi et al., 2000). In the

case of the n-alkanes, this was done when the lead atom or UA center was within that

small distance, in which case the small velocity was assigned to all of the atoms and

UA center of the alkanes. However, the actual streaming velocities of the molecules

in the transport pore region were still determined by the MD simulations. To study

the transport of a mixture due to a pressure gradient, the temperature of the system

must be held constant in order to eliminate any contribution of the temperature

gradient to the transport; hence special care was taken to achieve this (see also

below).

We computed several quantities of interest, including the density profiles of

the component i along the x- and z-directions, ( )zi xρ and , respectively. To

calculate

( )xi zρ

( )zi xρ and we used the same method as discussed in Chapter 2 (Sec. ( )x

i zρ

84

2.5). For each component i we also calculated its flux Ji by measuring the net

number of its particles crossing a given yz plane of area Ayz as discussed in Chapter 2

(Sec. 2.5).

We used a dimensionless time step, tΔ * = 5 × 10-3, which is equivalent to,

= 0.00685 ps, and NtΔ MD = 30,000. The system was considered to have reached a

steady state, when the fluxes calculated at various yz planes were within 5% from the

averaged values, after which the fluxes were calculated at the center of the pore

region. The equations of motion were integrated with up to 3 × 106 time steps. The

permeability Ki of species i and dynamic separation factor S21 were calculated using

the same equations as defined in Chapter2 (Sec. 2.5)

3.7 Experimental Study Transport and separation properties of three mixtures, namely, CH4/C3H8, CO2/CH4,

and CO2/C3H8 mixtures, have also been measured in a CMSM by our colleague Kh.

Molaai Nezhad. The average pore size of the membrane is about 4.5Å, which is in

the range of the pore sizes used in our simulations. The techniques for preparation of

the membrane, as well as measuring its transport and separation properties, have all

been described previously (Sedigh et al., 1998, 1999, 2000), and need not be

repeated here. All the experiments were carried out at T = 50°C. The pressure drop

85

that was applied to the membrane was the same as the one used in the molecular

simulations.

3.8 Results and Discussions We have carried out extensive NEMD simulations of three binary mixtures involving

an n-alkane. In what follows, we present and discuss the results for each mixture

separately, and compare the results with our experimental data, when possible.

Unless otherwise specified, in all the cases discussed below, the pressures in the

upstream and downstream CVs, that are in equilibrium with the two bulk states, are 3

atm and 1 atm, respectively.

3.8.1 Methane-Propane Mixtures Figure 3.1 presents the time-averaged distribution of the dimensionless temperature

in the carbon nanopore, for two different binary mixtures of CH4 and C3H8. In this

and the subsequent figures, the dashed lines indicate the boundaries of the pore

region. The pore size is 4

* / CHH H 5σ= = , the mole fractions of CH4 in the two

mixtures are 0.5 and 0.7 and, in both cases, the temperature of the system is set at T

= 50°C. As can be seen, the temperature in both cases is constant throughout the pore

and its two CVs and, moreover, the two systems’ temperatures are equal, as they

should be.

86

Figure 3.2 shows a snapshot of the mixture in a pore of size at T =

50°C, in which the mole fraction of CH

* 5H =

4 in the feed is 0.9. Both components are

mostly distributed near the pores’ walls by forming two condensed layers there.

These results are also consistent with equilibrium molecular simulations. The

densities of both components decrease from left to right. To understand better the

distribution shown in Figure 3.2, we present in Figure 3.3 the density profiles of the

two components, in two different mixtures, over the cross-section of a pore of size

, computed near its center. In the mixture in which the mole fraction of CH* 5H = 4

in the feed is 0.9 (the bottom panel), the two density profiles are almost identical,

except that the CH4 density is somewhat larger near the center, which is consistent

with the snapshot of the system shown in Figure 3.2. However, in the equimolar

mixture (top panel), the density of propane is larger than that of CH4 everywhere in

the pore. This is due to the much larger density of C3H8 under the bulk conditions (in

the two CVs), which results in much larger amounts of propane entering the pore. At

the same time, the C3H8 chains have a “shielding” effect in that; they prevent CH4

from entering the pore region. Reducing the size of the pore shifts the profiles to the

pore’s center, since the location of the fluid-wall potential minima changes: Shown

in Figure 3.4 is the comparison between such density profiles for a mixture in which

the CH4 mole fraction in the feed is 0.7, in pores of sizes *H = 5, 3 and 1.75 at T =

50°C. In all the cases, the density of C3H8 is much larger than that of CH4, since the

shielding effect mentioned above is even stronger for tighter pores. The fact that, in

all cases, there is great overlap between the two density profiles is indicative of the

87

mixture’s tendency not to segregate into two distinct regions, each essentially filled

with one of the components.

Another way of understanding the density profiles shown in Figures 3.3 and

3.4 is as follows. If we were to represent C3H8 as a simple LJ sphere, then its

effective energy parameter 3 8C Hε would be larger than

4CHε by a factor of about 1.6.

Since the energy parameters control the interaction of the molecules with the pore’s

walls, it becomes clear why the density of C3H8 near the walls is much larger than

that of CH4, since it is energetically more favorable for C3H8 to adsorb on the pore’s

walls than CH4. On the other hand, the effective LJ size parameter of C3H8 is not

much larger than that of CH4, hence explaining why the locations of the peaks in the

density profiles of the two components are not far apart.

To shed further light on the properties of this system, we also study the

density profiles in the two CVs and the pore along the transport (x-) direction. These

are shown in Figure 3.5, where we present the time-averaged density profiles for

both components in two different mixtures in a pore of size . The density

profiles are essentially flat in the two CVs (in the region -60 < X* < -20 and 20 < X*

< 60), with numerical values that match those obtained by the standard GCMC

method under the same conditions, indicating that the chemical potentials in the two

CVs have been properly maintained during the NEMD simulations. The small

fluctuations in the profiles in the CV regions represent numerical noise. The small

downward curvature at X* = -60 (in the C

* 5H =

3H8 profile) is due to the “leakage” of the

molecules out of the two CVs. These are the molecules that, as described above,

88

cross the outer boundaries of the CVs and leave the system. However, such

deviations from a flat profile are insignificant. Note that even when the mole fraction

of C3H8 in the feed is only 0.3 (top panel), its density in the two CVs is much larger

than that of CH4. In the pore region (-20 < X* < 20), the two profiles decrease from

left to right, which is expected. Due to the existence of the overall bulk pressure

gradient (or an overall nonzero streaming velocity in the pore), however, the density

profiles are not linear, and as the total flux is the sum of the diffusive and convective

parts, resulting in nonlinear profiles for C3H8 in both mixtures. For CH4, on the other

hand, the convective effect is much weaker, and, therefore, the decrease in its density

and the associated nonlinearity are also much weaker. The qualitative aspects of the

profiles shown in Figure 3.5 will not change if we impose a larger pressure gradient

on the pore system. Shown in Figure 3.6 are the density profiles of the two

components along the transport direction for two different systems. In one (the top

panel) the upstream and downstream pressures are, respectively, 3 atm and 1 atm,

while the corresponding pressures in the second pore system (the bottom panel) are

ten times larger, 30 atm and 10 atm. The pore size for both cases is . The CH* 5H = 4

mole fraction in the feed for both cases is 0.9, and the simulations were carried out at

T = 50°C. All the qualitative aspects of Figures 3.5 and 3.6 are similar, except that

the densities and their fluctuations in the pore under the larger pressure gradient are

somewhat larger, as one might expect. These fluctuations will eventually vanish if

the simulations are continued for much longer times. Our simulations also indicate

that the separation factor of this system is insensitive to the applied pressure

89

gradient. At the same time, the qualitative features of these profiles are not very

sensitive to the pore size (unless, of course, the pore is too small). We show in Figure

3.7 the density profiles in a pore of size * 3H = . The CH4 mole fraction in the feed is

0.7, with the rest of the parameters being the same as in Figures 3.5 and 3.6. The

similarities between these profiles and those shown in Figures 3.5 and 3.6 are clear.

Figures 3.8 present the dependence of the permeance (permeability per unit

length of the pore) of the two components on the temperature and feed composition,

and compare them with the experimental data. Several features of these figures are

noteworthy:

(1) The experimental permeances are about two orders of magnitude smaller

than those obtained by the molecular simulations. This is expected, as the CMSM

possesses a tortuous three-dimensional pore space, and therefore the gas permeances

of such a pore space must be smaller than those of a single straight pore with no

spatial tortuosity.

(2) Both the simulations and experiments indicate that the permeances are not

very sensitive to the composition of the mixture in the feed. For example, as the mole

fraction of C3H8 in the feed increases from 0.5 to 0.9, its permeance (both computed

and measured) changes by a factor which is less than 2, and an even smaller change

is seen in the permeance of CH4.

(3) The permeance of CH4 is practically independent of the temperature,

while that of C3H8 decreaeses essentially linearly with increasing temperature.

90

Figures 3.9 present the separation factor 3 8 421S /C H CHK K= for the mixture and

its dependence on the temperature and feed composition. Although both the

simulations and experiments suggest that the separation factor is not very sensitive to

the feed composition, the experimental data do not agree with the simulation results.

The reason is that, the most effective mechanism of separating a CH4/C3H8 mixture

into its components is by molecular sieving and kinetic effects (the rates of transport

of the two components), as it is much more difficult for the C3H8 chains to pass

through the membrane with the same rate as the CH4 molecules.

To study the effect of the pore size on the separation factors, we carried out

extensive simulations using a range of pore sizes. Figures 3.10 show the results for

the permeances and separation factors and their dependence on the pore size. The

mole fraction of CH4 in the feed is 0.7, and T = 50°C. Only when the pore size is

H* = 1.7, does the separation factor fall below one (its value for a pore of size

H* = 1.7 is about 0.8) and in the range of the experimental values (which, for this

mixture, is about 0.5). This critical pore size is, in fact, the same as what we

previously determined (Xu, Sedigh et al., 2000; Xu, Sahimi et al., 2000) to be the

optimal size for the separation of certain binary mixtures, such as CO2 and CH4.

Note also that, had we represented the C3H8 chains as LJ spheres, the critical pore

size would have been about H* = 2.15, indicating the significance of the proper

model of these molecules.

Therefore, to obtain quantitative results for the transport and separation

properties of binary mixtures of CH4 and C3H8, one must resort to 3D molecular pore

91

network models of CMSM (Ghassemzadeh et al., 2000; Xu, Sahimi et al., 2000; Xu

et al., 2001), in which interconnected pores of various shapes and sizes are

distributed in the space. In such a system, one may have a type of phase separation

such that the smaller pores carry CH4, while larger pores contain the C3H8 chains.

However, if the pores are not interconnected, then, no flux of the two components

can pass through the pore network. Therefore, the interconnectivity of the pores

plays a fundamental role in the separation of the two components.

3.8.2 Methane-Butane Mixtures Figure 3.11 presents a snapshot of the system at steady state in a pore of size H* = 5,

in which the mole fraction of methane in the feed is 0.9. Clearly, despite the mixture

being rich in CH4, there is a lot more C4H10 in the pore than CH4, which is again

attributed to the shielding effect described above. The corresponding time-averaged

density profiles of CH4 and n-C4H10 at T = 50°C and in the same system are shown

in Figure 3.12. Similar to the CH4/C3H8 mixtures, and consistent with Figure 3.11,

the density of n-butane is much larger than that of CH4. This feature of the system

can again be explained by defining an effective LJ energy parameter for n-butane,

which would be larger than that of CH4 by a factor of about 1.5, hence making it

energetically more favorable for n-C4H10 to form a thick layer near the pore’s walls.

At the same time, the effective LJ size parameter of n-butane is about twice as large

as that of CH4. This, together with the fact that the actual molecular structure of

92

n-butane is a chain with four atoms or UA centers, imply that as the components of

the mixture attempt to enter the pore from a CV, there is a shielding effect that

prevents many of the CH4 molecules from entering the pore, as a result of which the

density of n-butane inside the pore is much larger than that of CH4.

Figure 3.13 depicts the time-averaged density profiles of the two components

in the transport direction, both inside the two CVs and a pore of size H* = 5. The

densities of the two molecules are essentially constant in the two CVs, with the

density of n-C4H10 being much larger than that of CH4. In between the two CVs, the

two densities decrease from the upstream area to the downstream area in a nonlinear

fashion, with the nonlinearity being due to the convective effect imposed on the

system by the applied pressure gradient.

We show in Figure 3.14 the dependence of the two components’ permeances

on the temperature and feed composition, in a pore of size H* = 5. The qualitative

features of these results are similar to those for the CH4/C3H8 mixture, shown in

Figure 3.8. Once again, the CH4 permeance is essentially independent of its mole

fraction in the feed, as well as the temperature of the system, while the permeance of

n-C4H10 appears to vary linearly with both variables. As a result, the separation

factor of the pore system, defined as4 10 421S /C H CHK K= , also varies essentially

linearly with the feed composition and the temperature. In addition, the simulations,

for a range of pore sizes, yield large separation factors in favor of C4H10. Although

we do not yet have experimental data for this mixture, we suspect, based on the

CH4/C3H8 mixtures discussed above and the physics of the problem, that in real

93

membranes the separation factor would be in favor of CH4, the opposite of what the

simulations indicate. To show this, we carried out extensive simulations to

investigate the effect of the pore size on the permeances and the separation factors.

Figure 3.15 presents the results, obtained at T = 50°C, for a mixture in which the

CH4 mole fraction is 0.7. Even for a pore as small as H* = 1.7, the separation factor

is still about 2.4, indicating that a single pore is a gross representation of a

membrane, at least so far as this mixture is concerned.

Let us mention in passing that insertion of n-alkanes in a tight pore by the

configurational bias Monte Carlo method is difficult, since the atom or UA center of

the chain that is closest to the pore’s walls interacts with the walls much more

strongly than the rest of the chain. As a result, the standard CBMC method fails

when the pores are too tight and the alkane chains are initially inserted into the

system near the walls. To address this problem, we initially inserted the alkane

chains into the system only near its center.

3.8.3 Carbon Dioxide-Propane Mixtures In previous works (Firouzi et al., 2003; Sedigh et al., 1998, 1999, 2000; Xu et al.,

1998, 1999; Xu, Sedigh et al., 2000; Xu, Sahimi et al., 2000) we studied the

CO2/CH4 mixtures in carbon nanopores. Figure 3.16 presents the (time-averaged)

dimensionless temperature distribution in a pore of size H* = 5 and the two CVs,

obtained with two different mixtures of CO2 and C3H8. Once again, the temperature

94

is essentially constant throughout the entire system, indicating that the isothermal

condition has been maintained. A snapshot of the same system that contains a binary

mixture in which the CO2 mole fraction in the feed is 0.9 is shown in Figure 3.17,

indicating that most of the CO2 molecules are near the pore’s walls, forming a layer

near each wall. This is confirmed by the (time-averaged) density profiles shown in

Figure 3.18 for two different mixtures of CO2 and C3H8 in a pore of size H* = 5. The

temperature of the system is T = 50°C, and the data are collected at the center of the

pore. In the equimolar mixture (the top panel), the density of C3H8 near the walls is

much larger than that of CO2, since there is much more C3H8 in the pore than is CO2.

However, when the C3H8 mole fraction decreases to only 0.1 (the bottom panel), its

density inside the pore is much smaller than that of CO2, which, together with the

strong affinity of carbon dioxide for adsorption on carbon surfaces, mean that C3H8

can be present in the two layers near the walls only in relatively small amounts.

Figure 3.19 depicts the (time-averaged) density profiles of the two

components, for two different binary mixtures, in the transport direction of the same

pore as in Figure 3.18. In both cases, the densities of the two components are

constant in the two CVs, while they decrease nonlinearly from the upstream region

to the downstream region. Although, the mole fraction of CO2 in the feed is larger

(the top panel) than that of propane, the density of C3H8 is larger in both cases. This

can be explained again based on the higher density of C3H8 under the bulk condition

(in the two CVs) and the shielding effect described earlier.

95

The experimental data for the dependence of the two components’

permerances on the temperature and feed composition are compared in Figure 3.20

with the simulation results. The size of the pore is H* = 5. There are two noteworthy

features in these results:

(1) The CO2 permeance is essentially independent of the temperature and

feed composition. This is due to the affinity of this molecule to adsorb on carbon

surfaces, which means that CO2 does not really “see” the pore structure. At the same

time, the permeance of C3H8 varies essentially linearly with the temperature and

composition.

(2) Although there is quantitative agreement between the simulation results

and the experimental data, so far as the order of magnitude of their numerical values

is concerned, the trends in the two sets of results do not agree with each other: While

the simulations indicate that3 8 2C H COK K> , the experimental data indicate the opposite

trend.

This discrepancy is reflected in the separation factor, defined as,

. Shown in Figure 3.21 are the separation factors, obtained for a

pore of size H*= 5, where they are compared with the experimental data. Whereas

the simulations indicate that, S

3 8 221 /C H COS K K=

21 > 1, the data indicate the opposite. To further study

the effect of the pore size, we carried out extensive simulations at T = 50°C, using a

mixture in which the CO2 mole fraction in the feed was 0.7, and varied the pore size.

The results are presented in Figure 3.22. Only when the pore size H* is below 1.97,

does the separation factor fall below 1. For H* = 1.95, one has a separation factor of

96

about 0.94, still about one order of magnitude larger than the experimental value.

Had we represented C3H8 as a LJ sphere, the corresponding critical pore size would

have been H* = 2.28, indicating again the significance of a proper model of n-

alkanes.

Hence, similar to the mixtures of CH4/C3H8 and CH4/C4H10, the results for

CO2/C3H8 mixtures indicate that a single pore is inadequate for modeling a

membrane which has a 3D pore space consisting of interconnected pores of various

shapes and sizes. That is, for the mixtures considered in this study, the

morphological characteristics of a membrane - its topology, or pore connectedness,

and its geometry representing the pores’ shapes and sizes - control its separation

properties.

Let us mention that our previous studies (Sedigh et al., 1998, 1999; Xu,

Sedigh et al., 2000; Xu, Sahimi et al., 2000) of CO2/CH4 mixtures in a single carbon

nanopore, as well as in a molecular pore network, indicated qualitative agreement

between the simulations results and the experimental data. This is due to the fact that

for this mixture energetic effects dominate the separation process, as a result of

which even a single pore model is adequate for obtained qualitative insight into these

phenomena. However, as our study described in this chapter indicates, while

transport and separation of mixtures involving CO2 and n-alkanes (n > 1), and also

mixtures of various n-alkanes (and presumably many other mixtures) in a single

pore, with almost any realistic size, are dominated by energetic effects, the same

phenomena in a real membrane are controlled by its morphology, implying that a

97

single pore is not a reasonable model of a real membrane, although such a model has

been used extensively in the past.

3.9 Modeling of Long n-Alkane Chain Mixtures and their

Transport and Separation in CMSM We carried out non-equilibrium molecular dynamics simulation of transport and

separation properties of mixtures of carbon dioxide and long n-alkane chains, such as

n-hexane in a slit carbon nanopore. The n-hexane molecules have been represented

by a united-atom (UA) model. We used configurational-bias Monte Carlo method for

efficient generation of the n-hexane molecules. Since the length of hexane chain

molecules are longer than propane or butane chains, we generated eight trial

orientations for n-hexane in this method. The Rattle algorithm for constrained

dynamics of chain molecules has been applied. We used the Lennard-Jones potential

for intramolecular contribution of those sites which are connected by more than four

bonds inside the alkane chain molecules.

Figure 3.23 shows the time-average density profiles for CO2 and C6H14 in a

mixture of 90% CO2 and 10% C6H14 at T=50°C in the direction perpendicular to the

pore’s walls, in three different regions of a pore of size 2* =H . The upstream and

downstream pressures are, respectively, 3 atm and 1 atm. The distribution of the

molecules in the pore is also shown in Figure 3.23. The density of C6H14 is much

smaller than that of CO2 inside the pore. This is due to the higher mole fraction of

98

CO2 in the feed which is 0.9 and also the size of the pore, 2* =H which is very

small to accommodate long hexane chain molecules inside the pore. Figure 3.24

shows this distribution of the molecules at the larger scale around the pore center. If

we increase the size of the pore to be large enough to accommodate inserted hexane

chain molecules inside CVs, then since the adsorption of hexane is much higher than

CO2, the density of hexane becomes larger than CO2. In order to study the effect of

pore size on density, we increased the size of the pore to 5* =H . Figure 3.25 shows

the density profiles of the two components of a mixture of CO2 and C6H14 at

T=200°C, along the transport direction of a pore of size 5* =H . The CO2 mole

fraction in the feed is 0.9, and the upstream and downstream pressures are,

respectively, 20 atm and 5 atm. As we can see in this figure, the density of hexane is

larger than CO2 even though the mole fraction of CO2 in feed is much higher than

hexane. The temperature profile of the mixture is also shown in figure 3.25. As can

be seen, the temperature is constant throughout the pore and its two CVs.

3.10 Summary Extensive molecular simulations, combining the configurational-bias Monte Carlo

method and the dual control-volume-nonequilibrium molecular dynamics technique,

were carried out to study transport and separation of binary mixtures of n-alkanes,

and also those involving CO2 and an n-alkane, in a carbon nanopore. The driving

force was a pressure (chemical potential) gradient. The effect of the composition of

99

the feed and the size of the pore, as well as that of the temperature of the system, on

the transport and separation of the mixtures were studied, and were compared with

the experimental data. Our study indicates that, in a real membrane, transport and

separation of the mixtures considered in this chapter are dominated by the

geometrical and topological characteristics of the membrane. As a result, a single

carbon nanopore, in which only energetics of the system mostly control the transport

and separation phenomena, is a grossly inadequate model and, therefore, one must

resort to full three-dimensional molecular pore network models for modeling these

phenomena in a real membrane. We will present our simulation results using three-

dimensional molecular pore network model for the membrane in Chapter 5.

100

0

1

2

3

4

Tem

pera

ture

0

1

2

3

4

-60 -40 -20 0 20 40 60

X*

Tem

pera

ture

Figure 3.1: Dimensionless temperature distribution in a pore of size H* = 5 and the two control volumes, containing mixtures of CH4 and C3H8 with methane mole fraction in the feed being 0.7 (top) and 0.5 (bottom). Dashed lines indicate the boundaries of the pore region.

101

-2

-1

0

1

2

-20 -15 -10 -5 0 5 10 15 20

X*

Z*

Figure 3.2: Snapshot of the pore containing CH4 (triangles) and C3H8 (chains), at steady state. The mole fraction of CH4 in the feed is 0.9, and the pore size is H*= 5.

102

-2

-1

0

1

2

0 0.1 0.2 0.3 0.4 0.5 0.6

Density

Z*

-2

-1

0

1

2

0 0.04 .08 0.12 0.16

Density

Z*

0

Figure 3.3: Time-averaged density profiles of CH4 (solid curves) and C3H8 (dashed curves) in the transverse direction (perpendicular to the walls), measured near the pore’s center at T = 50°C. The pore size is H* = 5, while the mole fraction of CH4 in the feed is 0.5 (top) and 0.9 (bottom).

103

H*=5

-2.5

-1.5

-0.5

0.5

1.5

2.5

0 0.1 0.2 0.3 0.4 0.5 0.6

Density

Z*H*=3

-1.5

-0.5

0.5

1.5

0.0 0.5 1.0 1.5

Density

Z*

H*=1.75

-1.0

-0.5

0.0

0.5

1.0

0.0 0.5 1.0 1.5 2.0

Density

Z*

Figure 3.4: Comparison of the time-averaged density profiles of CH4 (solid curves) and C3H8 (dashed curves) in the transverse direction (perpendicular to the walls), measured near the pore’s center at T = 50°C, in three pores. The mole fraction of CH4 is 0.7.

104

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

-60 -40 -20 0 20 40 60

X*

Den

sity

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Den

sity

Figure 3.5: Time-averaged density profiles of CH4 (solid curves) and C3H8 (dashed curves) in the transport direction x in a pore of size H* = 5, in which the methane mole fraction in the feed is 0.7 (top) and 0.5 (bottom), and T = 50°C. Dashed lines indicate the boundaries of the pore region.

105

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Den

sity

0.00

0.04

0.08

0.12

0.16

-60 -40 -20 0 20 40 60

X*

Den

sity

Figure 3.6: Time-averaged density profiles of CH4 (solid curves) and C3H8 (dashed curves) in the transport direction x in a pore of size H* = 5 in which the CH4 mole fraction is 0.9. The upstream and downstream pressures are, respectively, 3 atm and 1 atm (top), and 30 atm and 10 atm (bottom). Dashed lines indicate the boundaries of the pore region.

106

0.00

0.05

0.10

0.15

0.20

0.25

-60 -40 -20 0 20 40 60

X*

Den

sity

Figure 3.7: Time-averaged density profiles of CH4 (solid curve) and C3H8 (dashed curve) in the transport direction x in a pore of size H* = 3. The CH4 mole fraction in the feed is 0.7, and T=50°C.

107

0.00

0.02

0.04

0.06

0.08

0.4 0.5 0.6 0.7 0.8 0.9 1

CH4 Mole Fraction in the Feed

Perm

eanc

e ( s

imul

atio

n )

0.0000

0.0004

0.0008

0.0012

0.0016

0.0020

Perm

eanc

e ( e

xper

imen

t )

0.00

0.02

0.04

0.06

0.08

20 40 60 80 100 120

Temperature(°C)

Perm

eanc

e ( s

imul

atio

n )

0.0000

0.0004

0.0008

0.0012

0.0016

0.0020

Perm

eanc

e ( e

xper

imen

t )

Figure 3.8: Comparison of the simulation results for the permeance of CH4 (open triangles) and C3H8 (open circles) with the corresponding experimental data (solid symbols). The mole fraction of CH4 in the feed in the upper Figure is 0.7, while T = 50°C in the lower Figure. The pore size is H* = 5.

108

0

3

6

9

12

15

0.4 0.5 0.6 0.7 0.8 0.9 1

CH4 Mole Fraction in the Feed

Sepa

ratio

n Fa

ctor

( si

mul

atio

n )

0.0

0.5

1.0

1.5

2.0

Sepa

ratio

n Fa

ctor

( ex

perim

ent )

0

3

6

9

12

15

20 40 60 80 100 120

Temperature(°C)

Sepa

ratio

n Fa

ctor

( si

mul

atio

n )

0.0

0.5

1.0

1.5

2.0

Sepa

ratio

n Fa

ctor

( ex

perim

ent )

Figure 3.9: Comparison of the computed separation factors (open circles) with the experimental data (solid circles) for a CH4/C3H8 mixture. The mole fraction of CH4 in the feed in the upper Figure is 0.7, while T = 50°C in the lower Figure.

109

0.00

0.02

0.04

0.06

0.08

0.10

1 2 3 4 5 6

Pore Size

Perm

eanc

e

0

5

10

15

20

25

30

1 2 3 4 5 6

Pore Size

Sepa

ratio

n Fa

ctor

Figure 3.10: The effect of pore size on the permeances of CH4 and C3H8 and the corresponding separation factors in a binary mixture in which the CH4 mole fraction in the feed is 0.7, and T = 50°C.

110

-2

-1

0

1

2

-5 -4 -3 -2 -1 0 1 2 3 4 5

X*

Z*

Figure 3.11: A snapshot of the pore containing CH4 (triangles) and C4H10, at steady state in a pore of size H* = 5 at T = 50°C. The CH4 mole fraction in the feed is 0.9.

111

-2

-1

0

1

2

0 0.1 0.2 0.3 0.4 0.5 0.6

Density

Z*

Figure 3.12: Time-averaged density profile of CH4 (solid curve) and C4H10 (dashed curve), between the upper and lower walls of a pore of size H* = 5 at T = 50°C. The profiles were calculated in the middle of the pore, and CH4 mole fraction in the feed is 0.9.

112

0.00

0.02

0.04

0.06

0.08

0.10

0.12

-60 -40 -20 0 20 40 60

X*

Den

sity

Figure 3.13: Same as in Figure 3.12, but in the transport direction x. Dashed vertical lines indicate the boundaries of the pore region.

113

0.00

0.05

0.10

0.15

0.20

20 40 60 80 100 120

Temperature(°C)

Perm

eanc

e

0.00

0.05

0.10

0.15

0.20

0.4 0.5 0.6 0.7 0.8 0.9 1

CH4 Mole Fraction in the Feed

Perm

eanc

e

Figure 3.14: The computed permeances of CH4 (triangles) and C4H10 (circles). The mole fraction of CH4 in the feed in the upper Figure is 0.9, while T = 50°C in the lower Figure.

114

0.00

0.02

0.04

0.06

0.08

0.10

1 2 3 4 5 6

Pore Size

Perm

eanc

e

0

5

10

15

20

25

30

35

40

45

1 2 3 4 5 6

Pore Size

Sepa

ratio

n Fa

ctor

Figure 3.15: The effect of pore size on the permeances of CH4 (triangles) and C4H10 (circles), and the corresponding separation factors at T = 50°C. The CH4 mole fraction in the feed is 0.7.

115

0

1

2

3

4

-60 -40 -20 0 20 40 60

X*

Tem

pera

ture

0

1

2

3

4

Tem

pera

ture

Figure 3.16: Time-averaged temperature distribution in a pore of size H* = 5 and the two CVs that contain a binary mixture of CO2 and C3H8. The mole fraction of CO2 is 0.7 (top) and 0.5 (bottom). Dashed vertical lines indicate the boundaries of the pore.

116

-2

-1

0

1

2

-20 -15 -10 -5 0 5 10 15 20

X*

Z*

Figure 3.17: Distribution of CO2 (triangles) and C3H8 chains in a pore of size H* = 5 at T = 50°C, obtained at steady state. The CO2 mole fraction in the feed is 0.9.

117

-2

-1

0

1

2

0 0.1 0.2 0.3 0.4 0.5 0.6

Density

Z*

-2

-1

0

1

2

0 0.08 .16 0.24 0.32

Density

Z*

0

Figure 3.18: Density profiles of CO2 (solid curves) and C3H8 (dashed curves) between the upper and lower walls of a pore of size H* = 5, computed at the pore’s center and obtained at steady state. The CO2 mole fraction in the feed is 0.5 (top) and 0.9 (bottom), and T = 50°C.

118

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Den

sity

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

-60 -40 -20 0 20 40 60

X*

Den

sity

Figure 3.19: Same as in Figure 3.18, but in the transport direction x. The CO2 mole fraction in the feed is 0.7 (top) and 0.5 (bottom), and T = 50°C. Dashed vertical lines indicate the boundaries of the pore.

119

0.00

0.02

0.04

0.06

0.08

0.4 0.5 0.6 0.7 0.8 0.9 1

CO2 Mole Fraction in the Feed

Perm

eanc

e ( s

imul

atio

n )

0.00

0.01

0.02

0.03

0.04

Perm

eanc

e ( e

xper

imen

t )

0.00

0.02

0.04

0.06

0.08

20 40 60 80 100 120

Temperature(°C)

Perm

eanc

e ( s

imul

atio

n )

0.00

0.01

0.02

0.03

0.04

Perm

eanc

e ( e

xper

imen

t )

Figure 3.20: Comparison of the computed permeances of CO2 (open triangles) and C3H8 (open circles), for a pore of size H* = 5, with the corresponding experimental data (solid symbols). The CO2 mole fraction in the feed in the upper Figure is 0.7, while T = 50°C in the lower Figure.

120

0

3

6

9

12

15

20 40 60 80 100 120

Temperature(°C)

Sepa

ratio

n Fa

ctor

( si

mul

atio

n )

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Sepa

ratio

n Fa

ctor

( ex

perim

ent )

0

3

6

9

12

15

0.4 0.5 0.6 0.7 0.8 0.9 1

CO2 Mole Fraction in the Feed

Sepa

ratio

n Fa

ctor

( si

mul

atio

n )

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Sepa

ratio

n Fa

ctor

( ex

perim

ent )

Figure 3.21: Comparison of the computed separation factors of CO2/C3H8 binary mixtures (open circles), for a pore of size H* = 5, with the experimental data (solid circles). The CO2 mole fraction in the feed in the upper Figure is 0.7, while T = 50°C in the lower Figure.

121

0.00

0.02

0.04

0.06

0.08

0.10

1 2 3 4 5 6

Pore Size

Perm

eanc

e

0

3

6

9

12

15

1 2 3 4 5 6

Pore Size

Sepa

ratio

n Fa

ctor

Figure 3.22: The effect of the pore size on the permeances of CO2 (triangles) and C3H8 (circles), and the corresponding separation factors at T = 50°C. The CO2 mole fraction in the feed is 0.7.

122

-20<X*<-19

-1.0

-0.6

-0.2

0.2

0.6

1.0

0 1 2 3X*

Z*

0<X*<1

-1.0

-0.6

-0.2

0.2

0.6

1.0

0 1 2 3X*

Z*

-0.3

-0.1

0.1

0.3

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Z*

19<X*<20

-1.0

-0.6

-0.2

0.2

0.6

1.0

0 1 2 3X*

Z*

Figure 3.23: Time-averaged density profiles of CO2 (solid curves) and C6H14 (dashed curves) between the upper and lower walls, in three different regions of a pore of size H* = 2. The upstream and downstream pressures are, respectively, 3 atm and 1 atm. The CO2 mole fraction in the feed is 0.9 and T = 50°C. Also shown is the distribution of the CO2 (triangles) and C6H14 (chains) in the pore. The arrows indicate the boundaries of the pore region.

123

-0.3

-0.1

0.1

0.3

-5 -4 -3 -2 -1 0 1 2 3 4 5

X*

Z*

Figure 3.24: Snapshot of the pore containing CO2 (triangles) and C6H14 (chains) with the same parameters as figure 3.23.

124

0.00

0.02

0.04

0.06

0.08

0.10

0.12

-60 -40 -20 0 20 40 60

X*

Den

sity

0

2

4

6

-60 -40 -20 0 20 40 60

X*

Tem

pera

ture

Figure 3.25: Time-averaged density profile (top) of CO2 (solid curves) and C6H14 (dashed curves) and temperature profile (bottom), in the transport direction x in a pore of size in which the CO5* =H 2 fraction is 0.9 and T=200°C. The upstream and downstream pressures are, respectively, 20 atm and 5 atm. Dashed lines indicate the boundaries of the pore region.

125

Chapter 4 Sub- and Supercritical Fluids in Nanoporous Materials: Direction-Dependent Flow Properties 4.1 Introduction

anoporous membranes have been used for separating gas mixtures. These

membranes offer high selectivities and sufficient mechanical under a variety

of conditions. In particular, nanoporous membranes have been considered for

separation of CO2 /alkane mixtures under sub- and supercritical conditions. We have

been investigating use of carbon molecular-sieve membranes (CMSMs) for this

purpose(Linkov, Sanderson and Rychkov, 1994; Sedigh et al., 1998, 1999, 2000;

Shiflett and Foley, 1999; Steriotis et al., 1997). In practice, however, such

membranes consist of a porous support made of at least two layers of macro- and

mesopores, and a nanoporous film deposited on the support. Flow and transport of

fluids in such composite porous materials have rarely been studied.

N

In this Chapter, we present the results of the first atomistic simulation of flow

and transport of fluid mixtures in a composite porous material that consists of three

distinct pores, referred to as the macro-, meso-, and nanopores. The effect of

assymetry in the permeation properties of fluid in nanoporous membrane have been

studied. In order to understand how the permeation properties of the membranes

126

depends on its structure, we investigate the impact of the multilayer structure of a

CMSM on transport of fluids under sub- and supercritical conditions. The results of

extensive nonequilibrium molecular dynamics simulations of flow and transport of a

pure fluid, as well as a binary fluid mixture, through a porous material composed of a

macro-, a meso-, and a nanopore, in the presence of an external pressure gradient, are

presented. In practice, a fluid mixture passes through a porous membrane by

applying a pressure gradient to two opposing external surfaces of the membrane. To

simulate this process we use the dual control-volume grand-canonical MD (DCV-

GCMD) simulation technique which is most suitable for simulating transport

processes in systems that operate under an extermal potential gradient (see Section

2.3). One goal of this Chapter is to understand the effects of the membrane structure,

and the pressure gradient applied to the membrane, on the flow properties of the fluid

mixture passing through the membrane.

We find that under supercritical conditions, unusual phenomena occur that

give rise to direction- and pressure-dependent permeabilities for the mixture's

components. Hence, the classical models of fluid flow and transport through porous

materials that are based on single-valued permeabilities that are independent of the

direction of the applied pressure gradient are completely in error.

127

4.2 Asymmetric Pore Models Real porous membranes consist of support and several layers, therefore the cross-

section of pores in real membranes may change as fluid transfer from the support to

the different layers of the membrane. A simple model for this change in the cross-

section is a pore with a step that consisits of two or more slits pores with different

sizes. Recent MD simulations (Düren et al., 2003) indicate that atomistic-scale

transport in a pore is hardly influenced by the pore’s shape. Thus, as a prelude to

understanding flow and transport of fluid mixtures in a real membrane, we consider

the same phenomena in the composite pore system shown in Figure 4.1, which

consists of three slit pores in series. (A somewhat similar pore model was utilized by

Düren et al. (2003) in their study of gas transport through a membrane.) Each pore

represents one layer of a three-layer supported membrane. The membrane is

connected to two control volumes (CVs) that are exposed to the bulk fluid at high

and low chemical potentials μ or pressures . The external driving force is a

chemical potential or pressure gradient applied in the

P

x − direction.

We studied this phenomena by considering two models of the composite

membrane as illustrated in Figure 4.1 and 4.2. In the first model, as shown in Figure

4.1, the control volumes are placed inside the pore at the two end parts of the pore,

therefore the size of the the control volumes is different in this model. Periodic

boundary conditions are used only in the y − direction in the pore and two CVs. In the

second model, as shown in Figure 4.2, two bulks have been considered as the control

128

volumes at the two end parts of the pore. Periodic boundary conditions are used only

in the direction in the pore and in both yy − − and z − directions in the two CVs.

Additionally, in the first model we considered three layers of graphite as the pore

wall, while in the second model, all the layers of graphite in the simulation cell have

been considered as the pore wall, which needs much more computation but

represents more realistic model of the composite membarne structure. In the wall of

the pore, carbon atoms were packed with the structure corresponding to graphite, so

that the number density of the carbon atoms was 114 nm-3, and the spacing between

the adjacent graphite layer in the z − direction was 3.35 Å. The total number of

carbon atoms in the simulation cell was 42,328 in the first model, and 56,472 in the

second model. The pores’ heights, H1, H2 and H3 as shown in Figure 4.1 and Figure

4.2, are 77, 23 and 10 Å respectively, while they all have the same length, about 43Å

in both models. The length of the two CVs are the same and equal to 63.90 Å. The

width of the pores and the two CVs are equal to 63.95 Å.

129

Figure 4.1: Asymmetric pore model with CVs inside the pore.

Figure 4.2: Asymmetric pore model with CVs outside the pore.

130

4.3 Adsorption and Transport in Asymmetric Pore Real membranes consist of pores with different sizes and shapes, which are a

significant source of heterogeneity in a membrane. As the pore size decreases, the

potential energy well becomes progressively deeper and its effect becomes stronger

and goes on until a limiting radius is reached where the repulsive interaction being to

dominate. Therefore, pores which are slightly larger than limiting radius present

very strong adsorption site, while for the larger pores, the influence of the pore walls

on particles in the center of the pore decreases and finally vanishes and only fluid

molecules located to the pore walls feel its influence. In asymmetric pore model, the

favorable adsorption sites at the step show that the most attractive potential, which is

the smallest value, is found in the corner of the steps.

In asymmetric pore model, different transport mechanisms contribute in

different regions. The diffusion mechanisms are classified according to the

interactions of the fluid molecules with the pore wall. In real adsorbents consisting

of networks of interconnected pores with varying diameters and shape, the

mechanisms cannot be distinguished so clearly. In the largest (macro-) pore, gas

permeation takes place by convective flow. In the mesopore, gas permeation is by

Knudsen diffusion (and some surface diffusion if the pressure is very large). Surface

diffusion is the main mechanism in the smallest (nano-) pore. Adsorption on the

walls becomes important when the pore diameter < 100 Å.

131

If the pore diameter is large in comparison to the mean free path of the fluid

molecules, collisions between diffusing molecules occur far more frequently than

collisions between the molecules and the pore walls. The influence of the pore wall

is small. The diffusion mechanism is the same as in the bulk and is called molecular

diffusion. In small pores and at low pressure when the mean free path is larger than

the pore diameter, the fluid molecules collide more often with the pore wall than

with each other. Knudsen diffusion takes place. Knudsen diffusion and molecular

diffusion often occur together. In the so-called transition region, the mean free path

is comparable to the pore diameter. Molecule-molecule as well as molecule-wall

collisions take place.

4.4 Molecular Dynamics Simulations in Asymmetric Pore We have carried out extensive NEMD simulations to study flow and transport

properties of pure CO2, as well as a mixture of CH4 and CO2, in the membrane. The

two components, as well as the carbon atoms that the pores’ walls consist of, are

represented by Lennard-Jones (LJ) spheres and characterized by effective LJ size and

energy parameters, σ and ε, respectively. All the quantities of interest are made

dimensionless with the help of the CH4 parameters, σ1 and ε1. Table 2.1 lists the

conversions between the dimensionless and dimensional quantities. We used the

same value for the size and energy parameters of CH4, CO2 molecules, and carbon

atoms that the pores’ walls consist of, as we used in Section 2.4. The Lorentz-

132

Berthelot mixing rules were used in the simulations to compute the size and energy

parameters of the unlike molecules (see Section 2.4)(Allen and Tildesley, 1987).

As mentioned earlier, we use the DCV-GCMD method (Cracknell,

Nicholson and Quirke, 1995; Ford and Heffelfinger, 1998; Heffelfinger and Van

Swol, 1994; Xu et al., 1998, 1999; Xu, Sedigh et al., 2000; Xu, Sahimi et al., 2000)

which combines the MD method in the entire pore system with the grand-canonical

Monte Carlo (GCMC) insertions and deletions of the molecules in the CVs.

Therefore, to mimic the experimental conditions, the densities, or the corresponding

chemical potentials, of the components in the CVs were maintained using a sufficient

number of the GCMC insertion and deletions (see Section 2.3). The two CVs are

well mixed and in equilibrium with the two bulk phases that are in direct contact

with them. Typically, 10 GCMC insertions and deletions in each CV were followed

by one MD integration step. The molecule-molecule interactions were modeled with

the cut-and-shifted LJ 6-12 potential with a cut-off distance 14cr δ= . To calculate the

interactions between the fluids’ molecules and the walls, we used the LJ potential for

the interactions between the molecules and the individual carbon atoms on the walls,

arranged as in graphite. The cut-off distance between the molecules and the carbon

atoms on the wall was 13.5cr δ= . The interaction between the gas molecules with the

entire carbon pore wall was taken to be the sum of the LJ potentials between the gas

molecules and each individual carbon atoms in the wall. In computer simulation of

confined fluids, this is the most realistic solid wall model which is an assembly of

atoms constrained to remain at the lattice sites of a crystal. However, this type of

133

wall is computationally expensive to simulate, since interactions between each fluid

and wall atom must be calculated.

In order to make the computation faster, at the beginning of each simulation

run, the simulation cell was devided into a large number of small subcells and a test

particle was assigned to each subcell. The simulation cell was discretized into

grid points along the three directions. In the first model as we shown in

Figure 4.1, we considered

zyx nnn ××

444=xn , 112=yn , and 158=zn , thus resulting in over

7,850,000 small subcells. In the second model as we shown in Figure 4.2, we

considered the same values for and , but since we do not have graphite atoms

in the CVs the value of is smaller and we considered

yn zn

xn 283=xn , thus resulting in

over 5,000,000 small subcells for this model. The test particle-solid wall interaction

energies and their three derivaties, associated with all the subcells, were then

calculated and recorded. To reduce the simulation time for calculation the interaction

between a gas molecule and all the carbon atoms in the wall, we used a 3D piecewise

cubic Hermite interpolation (Kahaner et al., 1989; Schultz, 1973) (which interpolates

a function and its first three derivatives) to compute the potential energy and forces

for the gas particle at any position using the previously recorded information at the

grid points. zyx nnn ××

The Verlet velocity algorithm was used to integrate the (dimensionless)

equations of motion with a dimensionless time step, 3* 5 10t −Δ = × (i.e., 0.00685

ps). The equations of motion were integrated with up to 1.2

tΔ ≈

× 107 time steps to

134

ensure that the steady state has been reached. Molecules that crossed the outer

boundaries of the CVs were removed. The number of such molecules was, however,

small, typically about 1% of the total number of molecules that were deleted during

the GCMC simulations. In addition, for each component we allowed for a nonzero

streaming velocity (the ratio of the component's flux and concentration) in the pore

system, consistent with the presence of a bulk pressure/chemical potential gradient

along the x − direction. In the CVs, however, the overall streaming velocity was

zero. Iso-kinetic conditions were maintained by rescaling the velocity independently

in the three directions.

Two important quantities of interest are the density profiles along the

x − direction, the direction along which the chemical potential gradient μ∇ is

imposed on the membrane, and in the yz planes that are perpendicular to the

direction of μ∇ . The density profile (x) of component i along the ziρ x − direction

was computed by dividing the simulation box in that direction into grids of size,

11.12l σ= . For each MD step, (x) was computed by averaging the number of

particles of type i over the distance l . A similar procedure was used for computing

the density profile

ziρ

xiρ (z) in the planes that are perpendicular to the direction of yz

μ∇ , with the averaging done over a small distance which was about 0.67 1σ ,

0.21 1σ , and 0.09 1σ for macro-, meso- and nanopore, respectively. As discussed

below, these quantities are important to understanding adsorption and transport

properties of the fluids in a composite membrane.

135

In addition, a most important characteristic property of a membrane is the

permeability of a fluid passing through the membrane. Thus, for each component i

we calculated its flux and permeability in the direction of the applied chemical

potential or pressure gradient. We calculated the flux and permeability in the

composite membrane with the same method as we descried earlier in Section 2.5. We

computed the permeabilities for two cases. In one, the upstream condition (higher

pressure) was maintained in the CV connected to the macropore, while in the second

case the upstream condition was maintained in the nanopore.

iJ iK

In what follows, we present and discuss the results of our simulation. The

temperature was held constant in order to eliminate any contribution of the

temperature gradient to the transport. Unless otherwise specified, in all the cases

discussed below, the temperature is fixed at T=35°C, and for the mixture of CH4 and

CO2, the mixture is equimolar.

4.5 Results and Discussions Figures 4.3 to 4.11 represent our simulation results when CVs are considered inside

the pore. The schematic of the system that we simulate has been shown earlier in

Figure 4.1. We apply a pressure drop PΔ to the system in one direction and

measure the fluid’s fluxes at steady state, and then reverse the direction of and

repeat the simulations. Figure 4.3 represents the molecules’ distribution and the

PΔ

136

density profiles (x) for COziρ 2 in the two CVs and the pore along the transport ( )x −

direction under supercritical - subcritical conditions for both directions of applied

pressure gradient. The upstream and downstream pressures are, 82.63 atm (1200

psig) and 5 atm (59 psig), respectively. These results have been obtained after

1,500,000 time steps when the upstream pressure is on the macro side, and after

4,000,000 time steps when the upstream pressure in on the nano side. The densities

in the two CVs regions are constant, as they should be and the density decreases

along the pore from upstream pressure to downstream pressure. Figure 4.4 shows

the snapshots of CO2 molecules in 3D for both direction of applied pressure gradient

along the pore. The upstream and downstream pressures are the same as Figure 4.3.

The snapshots show that most of the CO2 molecules are near the pore’s wall and the

corner of the steps. Figure 4.5 shows the time-averaged density profiles xiρ (z) for

CO2 in the direction perpendicular to the pore’s walls, at different regions of the

pore. The upstream and downstream pressures are, respectively, 82.63 atm and 5

atm and the upstream pressure is applied on the macropore. As these results

indicate, fluid packed the pore almost completely. The density is high everywhere

inside the macropore. In mesopore fluid forms several liquid-like layers and by

decreasing the pore size in nanopore, fluid forms only two adsorbed layers near the

pore’s walls. Note that the density profile has lots of fluctuations in the area which

are close to the corner of each steps. Fig. 4.6 represents the same results as Figure

4.5, except that the pressure gradient has been applied in the opposite direction along

the pore. There is a phase transition from liquid-like to gaseous one, as fluid

137

transport from upstream pressure in the nanopore to downstream pressure in the

macropore.

We have also studied these phenomena under subcritical conditions. Fig. 4.7

shows the time-averaged density and temperature profiles in the two CVs and the

pore along the transport ( )x − direction under subcritical - subcritical conditions

obtained after 5,100,000 time steps for both directions of applying pressure gradient.

The upstream and downstream pressures are, respectively, 21.4 atm (300 psig) and

11.2 atm (150 psig). Since the pressure gradient which is applied along the pore is

not very large the density is higher in nanopore for both cases due to the adsorption.

As can be seen, the temperature is constant along the pore and in the two CVs as it

should be. When the upstream pressure is on the macro side, the molar flow is about

1.4 times larger than when the upstream pressure is on the nano side. We also

increased the pressure gradient along the pore to see how it affects on the molar flow

when we apply the pressure gradient from both directions along the pore under

subcritical conditions. Fig. 4.8 represents the time-averaged density and number of

the molecules profiles in the two CVs and the pore in the transport ( )x − direction

for both directions of applied pressure gradient, in which the upstream and

downstream pressures are, respectively, 41.8 atm (600 psig) and 4.4 atm (50 psig).

Note that the time-averaged number of the molecules is the same as density except

that it has not been divided by the volume. When the upstream pressure is on the

macro side, although the average number of the molecules in nanpore is less than

138

macropore, but the density in the nanopore is higher than the macropore. This is due

to contribution of cross-sectional area of the pore in density calculation. When the

upstream pressure is on the macro side, the molar flow is about 2.3 times larger than

when the upstream pressure is on the nano side.

The binary mixture of CH4 and CO2 was also simulated. We considered the

equimolar mixture of CH4 and CO2 at T=35°C. Figure 4.9 represents the distribution

of CH4 and CO2 molecules in the pore and the two bulk regions under supercritical -

subcritical conditions, obtained after 4,000,000 time steps when the pressure drop

applied in the two opposite directions. The upstream and downstream pressures are,

respectively, 82.6 atm (1200 psig) and 35 atm (500 psig). Due to affinity of CO2 for

carbon surfaces, there are more CO2 molecules inside the pore than CH4 molecules.

Figure 4.10 depicts the time-averaged density profiles of two components in the two

CVs and the pore in the transport ( )x − direction with the same upstream and

downstream pressures as Figure 4.9. The time-averaged distribution of the

dimensionless temperature has been shown in figure 4.11. The permeabilities of

both components in the mixture are larger when the upstream pressure is applied on

the macropore than when the upstream pressure is applied on the nanopore. When

the upstream pressure is on the macro side the permeabilities of both CH4 and CO2

components in the mixture are, respectively, 1.14E-06 and 2.09E-06

(grmole.cm/min.psi.cm2), while when the upstream pressure is on the nano side, the

permeabilities of both CH4 and CO2 components are, respectively, 3.89E-08 and

9.22E-08 (grmole.cm/min.psi.cm2).

139

In the previously discussed model for the composite membrane in which the

CVs has been considered inside the pore, the size of the two CVs is not the same.

Since the size of the CV which is connected to the nanpore is much smaller, it will

saturate after reaching to some pressures. Therefore, the smaller CV can not

accommodates enough molecules to reach to high pressures which can affect on the

flux and permeability of the fluids when we apply the pressure gradient from both

opposite directions in the asymmetric pore model.

In order to study a more realistic model, the CVs were placed outside of the

asymmetric pore. Figures 4.12 to 4.29 represent our simulation results, considering

two bulks outside of the pore, as the CVs. The schematic of the system that we

simulate has been shown earlier in Figure 4.2. As in the previous model, we apply a

pressure drop to the system in one direction and measure the fluid’s fluxes at

steady state, and then reverse the direction of

PΔ

PΔ and repeat the simulations. Figure

4.12 shows the molecules’ distribution in the pore containing CO2 under subcritical -

subcritical conditions obtained after about 3,500,000 time steps for both direction of

applied pressure gradient. The upstream and downstream pressures are, respectively,

62.2 atm (900 psig) and 28.2 atm (400 psig). To understand better the molecules’

distribution in Figure 4.12, the density profile for CO2 in the two CVs and the pore

along the transport ( )x − direction have been shown in Figures 4.13 for both

directions of applied pressure gradient. The densities in the two bulk regions are

constant, as they should be. As one moves from the macro- to meso- to nanopore,

the densities, regardless of the direction of PΔ , increase since the pores’ sizes

140

decrease. Figure 4.12 and 4.13 and close inspection of the densities in the pores

indicate that, for the applied PΔ , one has a gas-like (very low density) fluid in the

macropore in both cases, followed by a transition to a liquid-like fluid (due to

condensation) in the mesopore, while the liquid-like fluid packs the nanopore

completely. The position of the transition depends on the pores’ sizes and the

direction of . The time-averaged temperature profiles of COPΔ 2 in the two CVs and

the pore along the transport ( )x − direction have been shown in Figure 4.14 for both

directions of applied pressure gradient with the same condition as Figure 4.12 and

4.13. As can be seen, the temperature in both cases is constant throughout the pore

and its two bulk regions.

In order to study the effect of supercriticality, we increased the upstream

pressure to the value above the critical pressure of CO2 having the same pressure

gradient along the pore. Figure 4.15 shows the snapshot of the pore system and the

distribution of CO2 molecules in it under supercritical - subcritical conditions. The

upstream and downstream pressures are, respectively, 82.6 atm (1200 psig) and 48.6

atm (700 psig). In both cases, the state of CO2 in the nano- and mesopores is liquid-

like, caused by condensation. The state of CO2 in the macropore, on the other hand,

depends on where the upstream condition is maintained. If the upstream condition is

maintained in the CV connected to macropore, then the CO2 density in the pore is

high almost everywhere, resembling a liquid-like state. If, on the other hand, the

upstream condition is maintained in the CV connected to the nanopore, the CO2

density in the macropore near its entrance to the mesopore is high, but decreases

141

somewhat as one gets away from this region towards the CV on the left side of the

figure. These are consistent with the time-averaged density profile ρ (x) of CO2

shown in Figure 4.16. The time-averaged temperature profiles of CO2 in the two CVs

and the pore along the transport ( )x − direction have been shown in Figures 4.17 for

both directions of applied pressure gradient.

Figure 4.18 presents the CO2 permeabilities and molar flows as a function of

the upstream pressure, when a pressure drop PΔ = 500 psi was applied to the pore

system. The direction-dependence of K is striking, with the permeabilities in the

two opposite directions differing by a factor which can be as large as nearly four.

Moreover, the trends for the two cases are opposite of each other. Whereas K

decreases when the upstream is on the macropore side, it increases when the

direction of is reversed. The reason can be understood by considering Figures

4.15 and 4.16 at a constant overall

PΔ

PΔ , with increasing the upstream pressure on the

macropore side, practically the entire pore system is packed with CO2 molecules.

This makes the passage of the molecules from the macropore to the mesopore very

difficult, resulting in reduced values of K . On the other hand, at the same overall

, increasing the upstream pressure when applied on the nanopore side moves the

transition point between a gas-like and liquid-like state to inside the macropore,

hence making the passage of the molecules from the mesopore to the macropore

easier, which increases

PΔ

K in that direction.

The effect of pressure gradient along the pore has been also investigated.

Figure 4.19 represents the snapshot of the pore containing CO2 molecules under

142

supercritical - subcritical conditions for both directions of applied pressure gradient,

in which the upstream and downstream pressures are, respectively, 82.6 atm (1200

psig) and 11.2 atm (150 psig). As we can see in this figure, the high pressure

gradient forces the molecules to move in the transport direction in macro- meso and

nanopores, even the adsorbed molecules to the horizontal carbon walls seems to be

carried away. There are not many molecules which stay inside the pore because the

stream velocity in the pore is very high. The time-averaged density profile of CO2 in

the two CVs and the pore along the transport ( )x − direction has been shown in

Figure 4.20 for both directions of applied pressure gradient along the pore. As it has

been shown in this figure the density is very low inside the pore, expect close to the

corner of the steps where the molecules can not move forward and accumulate due to

presence of pore walls.

We also increased the length of macro-, meso- and nanopore to investigate

the effect of the pore length on the transport and permeation properties in

asymmetric pore. We found no significant effect on the results.

The binary mixture of CH4 and CO2 was also simulated. We considered the

equimolar mixture at T=35°C. Figure 4.21 presents a snapshot of the pore system

and the distribution of CO2 and CH4 molecules in it, after the steady state has been

reached, with the upstream and downstream pressures being 82.6 atm (1200 psig)

and 35 atm (500 psig), respectively. Figure 4.22 presents the time-averaged density

profiles iρ (x) of the two components for the two upstream conditions. The densities

in the two bulk regions are constant, as they should be. As one moves from the

143

macro- to meso- to nanopore, the densities, regardless of the direction of (or the

upstream condition), increase since the pores' sizes decrease. A closer inspection of

the densities in the pores indicates that, for the applied

PΔ

PΔ , one has a gas-like (low

density) mixture in much of the macropore in both cases, followed by a transition to

a liquid-like mixture (due to condensation) which packs the meso- and nanopores

completely at high densities. The position of the transition line from the gas-like to

liquid-like mixture depends on the pores' sizes and the direction of . The density

profiles shown in Figure 4.22 are consistent with the snapshot of the system shown

in Figure 4.21, as they should be. Figure 4.23 shows the (dimensionless) temperature

throughout the pore system, which indicates that it remains constant. Hence, all the

possible effects due to a temperature gradient have been eliminated. To understand

the distributions of the two molecules in the pores better, we present in Figures 4.24

and 4.25 the time-averaged densities

PΔ

xiρ (z) for the two molecules at six different

planes that are perpendicular to the direction of PΔ (the coordinates' center is on the

centerline that passes through the three pores). Figure 4.24 shows the density profiles

when the upstream condition is maintained in the CV which is connected to the

macropore. In plane 1 near the pore mouth connected to the CV, two layers of each

type of molecule have been formed. One, with high densities, is near the walls, while

the second one with lower densities is closer to the center. Near the macropore

mouth that connects it to the mesopore (denoted by 2 in the Figure 4.24), the density

profiles look chaotic, with several layers of the two molecules forming. This is

caused by the entrance effect whereby, due to the size of the mesopore which is

144

much smaller than that of the macropore, a large number of molecules accumulate at

the macropore's entrance to the mesopore. But, if we inspect the density profiles just

inside the mesopore (denoted by 3 in the figure), we find again that two layers of

each type of molecules have been formed inside the mesopore. The molecules'

distributions in the region where the mesopore is connected to the nanopore (denoted

by 4 in the figure) are qualitatively similar to those in plane 2, and are again

dominated by the entrance effects. The very small size of the nanopore allows only

monolayer formation. As a result, one obtains the density profiles shown in Figure

4.24 for planes 5 and 6 shown in the figure. The same qualitative patterns are

obtained when the upstream condition is held in the CV connected to the nanpore

(see Figure 4.25), but with one difference: Only one layer of each type of molecules

has been formed in plane 1, where the macropore is connected to the CV. This is

clearly caused by the low downstream pressure which gives rise to a gas-like state in

that region, and is also consistent with the snapshot of the pore system shown in

Figure 4.21.

Figure 4.26 shows the time-averaged densities ρ (z) of pure CO2 at nine

different planes that are perpendicular to the direction of PΔ . The upstream and

downstream pressures are, respectively, 96.2 atm (1400 psig) and 48.6 (700 psig)

and the upstream condition is held in the CV connected to the macropore. Figure

4.27 represents the same results as Figure 4.26, except that the upstream condition is

held in the CV connected to the nanopore.

145

Figure 4.28 presents the permeabilities and molar flows of the two

components in an equimolar mixture with PΔ = 700 psi, computed when the

pressure drop was applied in the two opposite directions along the pore, and the

upstream pressure was varied. The same qualitative patterns are obtained when other

values of pressure drops are applied, and the upstream and downstream conditions

are such that one crosses from a subcritical region to a supercritical one (the critical

pressure for CO2 is about 73 atm). In addition to the fact that the pressure-

dependence of and is in qualitative agreement with experimental data (Ohya

et al., 1993; Sarrade et al., 1996; Tokunaga et al., 1997; Molaai Nezhad et al.),

another noteworthy feature of Figure 4.28 is that the permeabilities differ

significantly, with - that of CO

1K 2K

2K 2 - being larger. The reason is that, due to affinity

of CO2 for carbon surfaces, there is significant flow of CO2 on or near the walls

which is not the case for CH4. This is particularly important, as the nano- and

mesopores are packed with molecules and, therefore, the molecules' motion in them

is exceedingly slow. Hence, surface flow becomes important.

Figure 4.29 presents the molar flow and permeabilities of the two

components in an equimolar mixture at T=35°C versus PΔ , which were measured

when the pressure drop was applied in the two opposite directions along the pore,

and the upstream pressure was 1400 psig. As can be seen, by increasing , the

permeabilities for both components increase which is more pronounced when the

upstream pressure is on the macro side.

PΔ

146

The results for the permeabilities, for both pure CO2 and those in the mixture,

are explained as follows. Transport in a macropore is dominated by convection,

which gives rise to a permeability independent of the direction of the applied

pressure gradient, even in a gaseous state (Sahimi, 1993, 1995; Torquato, 2002). In a

mesopore, transport is by a combination of convection and Knudsen diffusion

(Sahimi, 1993, 1995; Torquato, 2002), whereas in a nanopore, due to its small size,

transport occurs mostly through surface flow. In both meso- and nanopores, the

permeabilities depend in a complex manner on the upstream and downstream

pressures (Sahimi, 1993, 1995; Torquato, 2002). This fact, and packing of the meso-

and nanopores caused by condensation, imply that the overall permeability of a

porous membrane depends on both the upstream and downstream pressures and the

direction along which the pressure gradient is applied to the membrane.

4.6 Summary The effect of orientation of the external potential gradient on the permeation

properties of the composite CMSMs has been studied by nonequilibrium molecular

dynamics simulations under sub- and supercritical conditions. The results indicate

the significance of the pore structure and the fluids' state to their transport through a

porous material. In particular, aside from being in qualitative agreement with the

preliminary experimental data obtained by our group (Molaai Nezhad et al.), the

results have two important implications. (1) That the classical modeling of transport

147

of fluids through porous membranes based on a single effective permeability,

independent of the direction of the applied pressure gradient, is completely wrong.

Therefore, optimization of a separation process using nanoporous membranes

requires consideration of the direction-dependant permeation properties of the

membrane. (2) Unlike the popular practice, a single pore is a gross and inadequate

model of an actual membrane.

In practice, supercritical fluid extraction using CO2 is utilized when the

mixture contains heavier hydrocarbons, such as pentane and hexane. In such cases,

the molecular structure of the hydrocarbons and their motion through the nanopores

give rise to additional complexities, such as freezing phenomena whereby the

mixture does not move appreciably even over long periods of times (Firouzi et al.).

148

(a) (b) Figure 4.3: Snap-shots and density profiles of CO2 in supercritical – subcritical conditions. The upstream and downstream pressures are, respectively, 82.63 (1200 psig) and 5 atm (59 psig), and temperature is 35 °C, (a) the upstream pressure is on the macro side, obtained after 1,500,000 time steps, (b) the upstream pressure is on the nano side, obtained after 4,000,000 time steps. Vertical lines indicate the boundaries of the pore.

149

Transport Direction

Transport Direction

Figure 4.4: Snapshots of the pore containing pure CO2 in 3D. The upstream and downstream pressures are, respectively, 82.63 atm (1200 psig) and 5 atm (59 psig), and temperature is 35 °C. The upstream pressure is on the macro side (top) and nano side (bottom).

150

Figure 4.5: Density profiles of pure CO2 in pore’s cross sections. The upstream and downstream pressures are, respectively, 82.63 atm (1200 psig) and 5 atm (59 psig), and temperature is 35 °C. The upstream pressure is on the macro side.

151

Figure 4.6: Density profiles of pure CO2 in pore’s cross sections. The upstream and downstream pressures are, respectively, 82.63 atm (1200 psig) and 5 atm (59 psig), and temperature is 35 °C. The upstream pressure is on the nano side.

152

0.0

0.1

0.2

0.3

0.4

0.5

0.6

-35 -25 -15 -5 5 15 25 35

X*

Den

sity

0.0

0.1

0.2

0.3

0.4

0.5

0.6

-35 -25 -15 -5 5 15 25 35

X*

Den

sity

300 psig300 psig 150 psig 150 psig

0

1

2

3

(a) (b) Figure 4.7: Density and temperature profiles of pure CO2 along the pores for subcritical - subcritical conditions. The upstream and downstream pressures are, respectively, 21.4 atm (300 psig) and 11.2 atm (150 psig), and temperature is 35 °C, (a) the upstream pressure is on the macro side, (b) the upstream pressure is on the nano side. Dashed vertical lines indicate the boundaries of the pore.

4

-35 -25 -15 -5 5 15 25 35

X*

Tem

pera

ture

01234

-35 -25 -15 -5 5 15 25 35

X*Te

mpe

ratu

re 150 psig300 psig 150 psig 300 psig

153

0

0.1

0.2

0.3

0.4

0.5

0.6

-35 -25 -15 -5 5 15 25 35

X*

Den

sity

0

0.1

0.2

0.3

0.4

0.5

0.6

-35 -25 -15 -5 5 15 25 35

X*

Den

sity

(a) (b) Figure 4.8: Density profiles and number of the molecules of pure CO2 along the pores for subcritical - subcritical conditions. The upstream and downstream pressures are, respectively, 41.8 atm (600 psig) and 4.4 atm (50 psig), and temperature is 35 °C, (a) the upstream pressure is on the macro side, (b) the upstream pressure is on the nano side. Dashed vertical lines indicate the boundaries of the pore.

0

20

40

60

80

100

-35 -25 -15 -5 5 15 25 35

X*

Num

ber o

f mol

ecul

es

0

20

40

60

80

100

-35 -25 -15 -5 5 15 25 35

X*

Num

ber o

f mol

ecul

es

600 psig600 psig 50 psig50 psig

600 psig50 psig600 psig 50 psig

154

Figure 4.9: Snapshot of the pore containing equimolar mixture of CH4 (triangles) and CO2 (circles) under supercritical - subcritical conditions obtained after 4,000,000 time steps. The upstream and downstream pressures are, respectively, 82.6 atm (1200 psig) and 35 atm (500 psig), and temperature is 35 °C. The upstream pressure is on the nano side (top) and macro side (bottom).

155

1200 psig 500 psig

0

0.1

0.2

0.3

0.4

0.5

-35 -28 -21 -14 -7 0 7 14 21 28 35

X*

Dens

ity500 psig 1200 psig

0

0.1

0.2

0.3

0.4

0.5

Den

sity

Figure 4.10: Time-averaged density profiles of both CH4 (solid curves) and CO2 (dashed curves) in the transport direction x in the pore with the same parameters as Figure 4.9. The upstream pressure is on the nano side (top) and macro side (bottom). Dashed vertical lines indicate the boundaries of the pore.

156

500 psig 1200 psig

0

2

4

6

8Te

mpe

ratu

re

1200 psig 500 psig

0

2

4

6

8

-35 -28 -21 -14 -7 0 7 14 21 28 35

X*

Tem

pera

ture

Figure 4.11: Time-averaged temperature distribution along the pore with the same parameters as Figures 4.9 and 4.10. The upstream pressure is on the nano side (top) and macro side (bottom). Dashed vertical lines indicate the boundaries of the pore.

157

Figure 4.12: Snapshot of the pore containing CO2 under subcritical - subcritical conditions. The upstream and downstream pressures are, respectively, 62.2 atm (900 psig) and 28.2 atm (400 psig), and temperature is 35 °C. The upstream pressure is on the nano side (top) obtained after 3,125,000 time steps, and macro side (bottom) obtained after 3,410,000 time steps.

158

400 psig 900 psig

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Den

sity

900 psig 400 psig

0.00

0.10

0.20

0.30

0.40

0.50

0.60

-35 -28 -21 -14 -7 0 7 14 21 28 35

X*

Den

sity

Figure 4.13: Time-averaged density profiles of pure CO2 in the transport direction x in the pore with the same parameters as Figure 4.12. The upstream pressure is on the nano side (top) and macro side (bottom). Dashed vertical lines indicate the boundaries of the pore.

159

400 psig 900 psig

0

2

4

6

Tem

pera

ture

900 psig 400 psig

0

2

4

6

-35 -28 -21 -14 -7 0 7 14 21 28 35

X*

Tem

pera

ture

Figure 4.14: Time-averaged temperature distribution of pure CO2 along the pore with the same parameters as Figures 4.12 and 4.13. The upstream pressure is on the nano side (top) and macro side (bottom). Dashed vertical lines indicate the boundaries of the pore.

160

Figure 4.15: Snapshot of the pore containing CO2 under supercritical - subcritical conditions. The upstream and downstream pressures are, respectively, 82.6 atm (1200 psig) and 48.6 atm (700 psig), and temperature is 35 °C. The upstream pressure is on the nano side (top) and macro side (bottom).

161

1200 psig 700 psig

0

0.1

0.2

0.3

0.4

0.5

0.6

-35 -28 -21 -14 -7 0 7 14 21 28 35

X*

Den

sity

700 psig 1200 psig

0

0.1

0.2

0.3

0.4

0.5

0.6

Den

sity

Figure 4.16: Time-averaged density profiles of pure CO2 in the transport direction x in the pore with the same parameters as Figure 4.15. The upstream pressure is on the nano side (top) and macro side (bottom). Dashed vertical lines indicate the boundaries of the pore.

162

700 psig 1200 psig

0

2

4

6

Tem

pera

ture

1200 psig 700 psig

0

2

4

6

-35 -28 -21 -14 -7 0 7 14 21 28 35

X*

Tem

pera

ture

Figure 4.17: Time-averaged temperature distribution of pure CO2 along the pore with the same parameters as Figures 4.15 and 4.16. The upstream pressure is on the nano side (top) and macro side (bottom). Dashed vertical lines indicate the boundaries of the pore.

163

0

2

4

6

8

10

500 600 700 800 900 1000 1100 1200 1300

Upstream pressure (psig)

Perm

eabi

lity

(gm

ole.

cm/m

in. p

si. c

m2 )

x 1

07

Macro

Nano

0

2

4

6

8

10

500 600 700 800 900 1000 1100 1200 1300

Upstream pressure (psig)

Mol

ar fl

ow (g

mol

e/m

in) x

10

11

Nano

Macro

Figure 4.18: The dependence of the permeability and molar flow of pure CO2 on the upstream pressure at T=35°C, when PΔ =500 psi is applied in the two opposite directions. Continuous and dashed curves show, respectively, the results when the upstream pressure is applied on the macropore and nanopore.

164

Figure 4.19: Snapshot of the pore containing CO2 under supercritical - subcritical conditions. The upstream and downstream pressures are, respectively, 82.6 atm (1200 psig) and 11.2 atm (150 psig), and temperature is 35 °C. The upstream pressure is on the nano side (top) obtained after 770,000 time steps, macro side (bottom) obtained after 810,000 time steps.

165

150 psig 1200 psig

0.0

0.1

0.2

0.3

0.4

0.5

Den

sity

1200 psig 150 psig

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-35 -28 -21 -14 -7 0 7 14 21 28 35

X*

Den

sity

Figure 4.20: Time-averaged density profiles of pure CO2 in the transport direction x in the pore with the parameters as Figure 4.19. The upstream pressure is on the nano side (top), macro side (bottom). Dashed vertical lines indicate the boundaries of the pore.

166

Figure 4.21: Snapshot of the pore containing equimolar mixture of CH4 (triangles) and CO2 (circles) under supercritical - subcritical conditions. The upstream and downstream pressures are, respectively, 82.6 atm (1200 psig) and 35 atm (500 psig), and temperature is 35 °C. The upstream pressure is on the nano side (top) and macro side (bottom).

167

500 psig 1200 psig

0.0

0.1

0.2

0.3

0.4

0.5

Den

sity

1200 psig 500 psig

0.0

0.1

0.2

0.3

0.4

0.5

-35 -28 -21 -14 -7 0 7 14 21 28 35

X*

Den

sity

Figure 4.22: Time-averaged density profiles of both CH4 (solid curves) and CO2 (dashed curves) in the transport direction x in the pore with the same parameters as Figure 4.21. The upstream pressure is on the nano side (top) and macro side (bottom). Dashed vertical lines indicate the boundaries of the pore.

168

500 psig 1200 psig

0

2

4

6

8

Tem

pera

ture

1200 psig 500 psig

0

2

4

6

8

-35 -28 -21 -14 -7 0 7 14 21 28 35

X*

Tem

pera

ture

Figure 4.23: Time-averaged temperature distribution along the pore with the same parameters as Figures 4.21 and 4.22. The upstream pressure is on the nano side (top) and macro side (bottom). Dashed vertical lines indicate the boundaries of the pore.

169

Figure 4.24: Density profiles of an equimolar mixture of CH4 (solid curves) and CO2 (dashed curves) in pore’s cross sections. The upstream and downstream pressures are, respectively, 82.6 atm (1200 psig) and 35 atm (500 psig), and temperature is 35 °C. The upstream pressure is on the macro side.

170

Figure 4.25: Density profiles of an equimolar mixture of CH4 (solid curves) and CO2 (dashed curves) in pore’s cross sections. The upstream and downstream pressures are, respectively, 82.6 atm (1200 psig) and 35 atm (500 psig), and temperature is 35 °C. The upstream pressure is on the nano side.

171

Figure 4.26: Density profiles of pure CO2 in pore’s cross sections. The upstream and downstream pressures are, respectively, 96.2 atm (1400 psig) and 48.6 atm (700 psig), and temperature is 35 °C. The upstream pressure is on the macro side.

172

Figure 4.27: Density profile of pure CO2 in pore’s cross sections. The upstream and downstream pressures are, respectively, 96.2 atm (1400 psig) and 48.6 atm (700 psig), and temperature is 35 °C. The upstream pressure is on the nano side.

173

0

2

4

6

8

800 900 1000 1100 1200 1300 1400 1500

Upstream pressure (psig)

Perm

eabi

lity

(gm

ole.

cm/m

in. p

si. c

m2 )

x 1

0 7

0

1

2

3

4

5

6

7

800 900 1000 1100 1200 1300 1400 1500

Upstream pressure (psig)

Mol

ar fl

ow (g

mol

e/m

in) x

10

11

Figure 4.28: The dependence of the permeabilities and molar flows of CO2 (circles) and CH4 (triangles) on the upstream pressure in an equimolar mixture at T=35°C, when =700 psi is applied in the two opposite directions. Continuous and dashed curves show, respectively, the results when the upstream pressure is applied on the macropore and nanopore.

PΔ

174

0

2

4

6

8

10

12

600 700 800 900 1000

Δ P (psi)

Perm

eabi

lity

(gm

ole.

cm/m

in. p

si. c

m2 )

x 1

0 7

0

2

4

6

8

10

600 700 800 900 1000

Δ P (psi)

Mol

ar fl

ow (g

mol

e/m

in) x

10

11

Figure 4.29: The dependence of the permeabilities and molar flows of CO2 (circles) and CH4 (triangles) on the pressure drop in an equimolar mixture at T=35°C, when the upstream pressure is 1400 psig. Continuous and dashed curves show, respectively, the results when the upstream pressure is applied on the macropore and nanopore.

175

Chapter 5 Transport and Separation of Carbon Dioxide-Alkane Mixtures in a Carbon Pore Network under Sub- and Supercritical Conditions 5.1 Introduction

he results presented in Chapters 2 and 3, using a single carbon nanopore,

provided qualitative insight into the transport and separation of mixtures in

nanoporous materials. However, such simple pore models cannot quantitatively

represent real nanoporous membranes, a two-dimensional example of which is

shown in Figure 5.1.

T

Figure 5.1 Two-dimensional representation of a porous membrane.

176

Transport and separation properties of fluid mixtures in a real membrane are strongly

influenced, and sometimes even dominated, by the morphology of a membrane,

which consists of the connectivity of its pores, and their shapes, sizes, and surface

characteristics. Therefore, a single nanopore is an inadequate model of a membrane.

Many real nanoporous membranes have a polycrystalline or amorphous structure.

Fine crystals of various sizes create grains of different forms and sizes. Cracks,

fissures, cavities, and other defects occur on the pore surface. Yet, the main source of

heterogeneity for nanoporous membranes is their complex porous structure which

contains nanopores of different sizes and shapes, including pores with straight and

snapped parts, and contracting and diverging channels. When such single pores are

connected, they forms pore networks. The structure of real disordered porous solids,

such as activated carbon, is very complicated and, therefore, cannot be represented

by a single pore.

Chapter 4 attempted to improve the single pore model by considering a three-

pore model that were in series. The results indicated unexpected phenomena, such as

the asymmetry of the permeabilities. Therefore, the results presented in Chapter 4

indicated the significance of a representative pore network model for representing a

nanoporous membrane, in addition to the effect of the various controlling

parameters, such as the temperature, the applied pressure gradient, and the

composition of the mixture in the feed.

177

To address the problem of representing the morphology of a nanoporous

membrane by a more realistic model, we use in this chapter a three-dimensional (3D)

molecular pore network model for carbon molecular-sieve membranes (CMSMs)

based on the Voronoi tessellation. As described below, in this model the pores have

completely irregular shapes and sizes. The model allows us to investigate the effect

of the morphology of the pore space, i.e., its pore size distribution and pore

connectivity, on the transport and separation properties of fluid mixtures in a

nanoporous membrane. After describing the Voronoi model, we describe the

transport and separation properties of sub- and supercritical mixtures in the model.

5.2 Pore Network Model of CMSMs In order to generate the molecular pore network model, first we create a 3D

simulation box of carbon atoms with a structure corresponding to graphite, so that

the number density of the carbon atoms is 114 nm-3 and the spacing between the

adjacent graphite layers in the z- direction is 0.335 nm. If the pyrolysis of the

polymeric precursor is done at high enough temperatures, the resulting matrix of the

pore space has a structure similar to that of graphite and, therefore, graphite structure

can be used for the membrane atomistic structure. We then tessellate the graphitic

box by inserting in it a given number of Poisson (randomly and uniformly-selected)

points, each of which is the basis for a Voronoi polyhedron. Each polyhedron is that

178

part of the box which is nearer to its Poisson point than to any other Poisson point.

The pore space is created by specifying the desired porosity and then selecting a

number of the polyhedra in such a way that their total volume fraction equals

specified porosity. The chosen polyhedra are then designated as the membrane pores

by removing the carbon atoms inside them, as well as those that are connected to

only one neighboring carbon atom (the dangling atoms), since it is impossible to

actually have such atoms connected to the surface of the pores. The remaining

carbon atoms constitute the membrane's solid matrix, while the pore space consists

of interconnected pores of various shapes and sizes. Figure 5.2 shows a 2D Voronoi

network with 50% porosity, in which the gray polygons represent the pores of the

system.

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 box is large enough, the size distribution of the polyhedra will always

be Gaussian, regardless of the porosity of the pore space. This, however, is not

realistic from a practical view point, because often the membranes that are used in

practice 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 can

mimic that of a real membrane, which is typically skewed. To obtain such PSDs, we

first sort and list the polyhedra in the box according to their sizes, from the largest to

smallest. The size of each polyhedron is taken to be the radius of a sphere that has

179

the same volume as the polyhedron. We then designate the polyhedra as the pores

according to their sizes, starting from the largest ones in the list.

Figure 5.3 compares the PSDs of the pore networks generated by the two

methods with 2500 Poisson points, and 50% porosity. Figure 5.4 shows the PSDs of

the pore networks generated by the biased method with 2500 Poisson points for

several porosities. The PSDs and the average pore sizes that are generated with the

bias towards the largest pores are, of course, dependent upon the porosity, and

resemble to some extent the experimental PSD (Sedigh et al., 1998, 1999, 2000; Xu

et al., 1999; Xu, Sedigh et al., 2000). Figure 5.5 shows the PSD of a typical CMSM

that has been measured experimentally by our group. Note that, unlike the traditional

pore networks that are used in the simulation of flow and transport in porous media

(Sahimi, 1993, 1995), the pore networks generated here are molecular networks in

which the interaction of the gas molecules with all the atoms in the network are

taken into account. While the PSD of the system in which the pore polyhedra are

selected at random is independent of the porosity, the PSD obtained with the bias

toward the largest pores is dependent upon the porosity. 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.

180

Figure 5.2: A two-dimensional Voronoi network.

181

Random

0.00

0.05

0.10

0.15

0.20

0.25

2 4 6 8 10 12

Pore Diameter (°A)

Pore

Siz

e D

istr

ibut

ion

Biased

0.00

0.05

0.10

0.15

0.20

0.25

2 4 6 8 10 12

Pore Diameter (°A)

Pore

Siz

e D

istr

ibut

ion

Figure 5.3: Computed pore size distribution (PSD) for the model CMSMs. On the top is the PSD for a system in which the pores are selected randomly. On the bottom is the PSD for a system in which the pores are generated according to their sizes, starting from the largest size.

182

Biased

0.00

0.05

0.10

0.15

0.20

0.25

2 4 6 8 10 12

Pore Diameter (°A)

Pore

Siz

e D

istr

ibut

ion

50%25%70%

Figure 5.4: Computed PSD for the model CMSMs for different porosities. The pores are generated according to their sizes, starting from the largest size.

Figure 5.5: PSD of the membrane used in the experiments.

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5.3 NEMD Simulation in CMSM Pore Network Non-equilibrium molecular dynamics simulations have been implemented to study

the transport and separation properties of binary mixtures consisting of CO2 and n-

alkanes, under both sub- and supercritical conditions using a 3D molecular pore

network model for the CMSM that we described above. The dimensions zyx LLL ××

of the simulation cell that we used are 65.6395.6390.63 ×× Å. The total initial

number of carbon atoms in the simulation cell was 29,640. We used the biased

method of generating the pore space with 50% porosity to construct the pore

network. We inserted 5200 Poisson points in the cell and constructed a 3D Voronoi

tessellation of the cell containing the same number of polyhedra. We generated three

different pore network structures with 50% porosity with the biased method. The

simulation results of three different structures were averaged to make the results

more uniform. Since the Poisson points in each structure were generated randomly,

the pore network structures are different. The average pore size for the pore networks

is about 5.1Å, if the pore polyhedra are selected with a bias toward the largest sizes.

As can be seen in Figure 5.5, the experimental PSD for CMSM also shows a peak

around 5 Å. Periodic boundary conditions were used to construct the pore network

model in all the three directions.

We considered two models in our work in order to study by NEMD

simulations the transport of the fluids in the pore network model. First, we

considered two bulk regions as the control volumes (CVs) at the two opposite

184

surfaces of the pore network. In the second approach, we extended the graphite

atoms positions of the pore network structure at the interfaces of the pore and CVs,

and inserted the molecules inside the empty spaces in the CVs. The latter model

creates isolated channels of carbon atoms in the CVs which originate from pore

polyhedra at the enterance and the exit of the pore network structure.

5.4 Molecular Models of the Fluid and the Interaction Potentials The transport and separation properties of the following binary mixtures were

studied using the pore network model: CH4/CO2, CH4/C3H8, CO2/C3H8, and

CH4/C4H10. The CH4 and CO2 molecules are represented as Lennard-Jones (LJ)

spheres, characterized by effective LJ size and energy parameters, σ and ε . All the

quantities of interest are made dimensionless with the help of the CH4 parameters,

1σ and 1ε . Table 2.1 lists the conversions between the dimensionless and

dimensional quantities. The C3H8, and C4H10 molecules are represented by united-

atom (UA) model (Ryckaert and Bellemans, 1978) as we described earlier in Section

3.3, and grown by a configurational-bias Monte Carlo method (De Pablo et al., 1992;

Frenkel et al., 1991; Harris and Rice, 1988; Laso et al., 1992; Macedonia and

Maginn, 1999; Mooij et al., 1992; Siepmann and Frenkel, 1992; Smit et al., 1995)

(see Section 3.4). The nonbonded interactions between interaction centers of

different molecules are described with cut-and-shifted LJ potential. No tail

corrections were applied to the potential. The Lorentz-Berthelot mixing rules were

185

used in the simulations to compute the size and energy parameters of the unlike

molecules (see Section 2.4).

For simplicity, the total molecular mass of alkane was equally divided

between the C atoms and, therefore, CH2 and CH3 groups had equal molecular mass.

Table 3.1 lists the size and energy parameters of CH4, CO2, and those of the CH2 and

CH3 groups. The atoms and the UA centers are connected by harmonic potentials.

The distance between the atoms is fixed at 1.53 Å. The intramolecular interactions

consist of the contributions by bond-bending (BB) and torsional forces. For the BB

term, the van der Ploeg-Berendsen potential and for the torsional potential, the

Ryckaert-Bellemans potential is used (see Section 3.3). The cut off distance for the

interaction between the gas molecules was taken to be 4

5.3 CHσ for CH4/C3H8,

CO2/C3H8, and CH4/C4H10 mixtures, and 4

0.4 CHσ for CH4/CO2 mixture. The cut off

distance for the interaction between the gas molecules and carbon wall atoms was

taken to be 4

5.3 CHσ for all of the above mentioned mixtures. The interaction

between the gas molecules with the entire carbon pore wall was taken to be the sum

of the LJ potentials between the gas molecules and each individual carbon atoms in

the wall. In order to calculate the potential between the gas molecules and carbon

atoms in the wall, the simulation cell was discretized into zyx nnn ×× grid points

along the three directions, using the same method as we described in Section 4.4.

186

5.5 Results and Discussions We have carried out extensive NEMD simulations of several binary mixtures

mentioned above in the 3D molecular pore network model of CMSM, under sub- and

supercritical conditions. In what follows we present and discuss the results for each

mixture separately.

5.5.1 Methane-Carbon Dioxide Mixtures Figure 5.6 presents the density profiles ( )z

i xρ for an equimolar mixture of CH4 and

CO2, for the case in which two bulk regions are connected to two opposite surfaces

of the pore network. In order to compute the density profiles, the system was, as

described above, divided into zyx nnn ×× small subcells with ,216=xn 140=yn ,

and , thus resulting in over 4,233,600 subcells. This allowed us to

accurately compute the profiles. To study the effect of the external pressure gradient

and supercriticality, we fixed the upstream pressure at 120 atm, which is

significantly above the minimum pressure for supercriticality of the mixture, and

varied the downstream pressure over a range of values that includes both

supercritical and subcritical conditions. Figure 5.6 shows the time-averaged density

profiles for both components at T = 40°C, in the two CVs and the pore along the

transport direction. The downstream pressures are 20 atm, 50 atm, and 90 atm.

The density profiles are essentially flat in the two CVs (in the region

140=zn

( −x )

187

38.8*15.25 −<<− X and 15.25*38.8 << X ). The downward curvatures at

and are attributed to two facts: (i) the leakage of the

molecules out of the two CVs, and (ii) no molecules have been inserted in a 0.5

15.25* −=X 15.25* =X

4CHσ

distance from the end of CVs. However, such deviations from a flat profile are

insignificant. In the pore region ( )38.8*38.8 <<− X , the two profiles decrease from

left to right, which is expected. The fluctuations in the density profile in the pore

network are caused by adsorption on the pores' walls, and the inhomogeneous

structure of the pore network. Due to the shapes of the pores in a Voronoi structure,

which are in the form of polyhedra made of insecting planes, one may have

significant accumulation of the molecules that attempt to pass from one such

polyhedron to another with a smaller pore mouth. The density profiles show that by

increasing the downstream pressure to 90 atm the pore network is almost packed,

with the fluids being in a liquid-like state, indicating by their high densties.

To show that the fluctuations in the density profiles of Figure 5.6 are not due

to non-isothermal effects, we show in Figure 5.7 the time-averaged distribution of

the dimensionless temperature for the same equimolar mixture as that in Figure 5.6,

indicating that the temperature is constant throughout the pore network and two CVs.

Hence, there is no contribution by a temperature gradient to the distribution of the

molecules in the pore network. Moreover, all the three systems' temperatures are

equal, as they should be.

The fluxes of CO2 and CH4 in their equimolar mixture, along with their

permeabilities and the corresponding separation factors are exhibited in Figure 5.8.

188

In this case, the upstream pressure is fixed at 120 atm, while the downstream

pressure varies. Figure 5.8 shows that by increasing PΔ , the fluxes' and

permeabilities' of both components will increase, while separation factor stays

essentially constant. The separation factors are comparable with those that we

obtained in Chapter 2 with a single pore. This results is somewhat surprising, since

one expects to obtain higher separation factors with a pore space of interconnected

pores as molecular sieving, which is absent in a single-pore model, is present in a

Voronoi network. However, the porosity of the Voronoi network is relatively high,

which implies that the pore space is relatively open and, therefore, the actual mean

pore size is larger than the nominal mean pore size of 5Å.

Figure 5.9 shows the time-averaged density profiles ( )zi xρ for both

components in a mixture in which the mole fraction of CO2 is 0.9, and T = 40°C.

Once again, the upstream pressure is 120 atm, while the downstream pressures are 20

atm, 50 atm, and 90 atm. As expected, the density of CO2 is much larger than CH4.

The corresponding time-averaged temperature profiles for the same system and

mixtures are shown in Figure 5.10. Figure 5.11 presents the corresponding fluxes and

permeabilites of the two components of the same mixture, along with the

corresponding separation factors. The surprising aspect of these results is that, the

separation factor attains a maximum when the downstream pressure is equal to 50

atm. Recall that the same type of maximum was also obtained when transport of such

mixtures was simulated in a single pore (see Chapter 2), except that in the present

case the maximum is very pronounced. The reason for the existence of this

189

maximum was already explained in Chapter 2, where we reasoned that it is due to

the optimal condition for convection/diffusion of the two components of the mixture.

Figure 5.12 shows a snapshot of the system for an equimolar mixture at T =

40°C, obtained after 2,220,000 time steps, where the upstream and downstream

pressures are, respectively, 120 atm and 90 atm, while Figure 5.13 presents a

snapshot of a mixture in which the mole fraction of CO2 is 0.9, obtained after

1,740,000 time steps.

Since the pore sizes in the pore network are very small, with their average

size being about 5Å, it is very difficult for the molecules to be transported inside the

pore network. This is particularly true for the alkane chains. In fact, the fluid

molcules adsorb on the carbon structure and block the pore entrances. In addition,

due to the pore entrance and exit effects, controlling the temperature and holding it

constant are difficult, especially when alkane chains are present in the system. This

effect leads to hot spots and noticeable peaks in the temperature profile at the

interfaces between the pore network and bulk regions. In order to resolve this

problem, we extended the pores that are connected to two opposing surfaces of the

network. The extended part of such pores than act as the CVs for them that are in

equilibrium with the bulk regions. In effect, for each pore that is connected to the

external surface of the system, we create its own CV, similar to the single-pore

model studied in Chapter 2. The molecules are then inserted in such CVs using the

GCMC method. Since each CV acts independent of all other CVs, the GCMC part of

the computations becomes amenable to parallel processing and computations. In

190

such a model, the fluid molecules go directly inside the actual pores in the pore

network that are connected to the external surface, hence helping to eliminate the

accumulation of the fluid molecules at the pores' mouth. In the what follows we

present our simulation results using this model. For the results that we describe

below, we used ,456=xn 140=yn , and 140=zn to create the small subcells and

compute the density profiles.

Figures 5.14 represents the resulting density profiles ( )zi xρ for a binary

mixture of CH4 and CO2 at T = 40°C. The upstream pressure is 120 atm, while the

downstream pressures are 20 atm, 50 atm, and 90 atm. Qualitatively, the density

profiles shown in Figure 5.14 are similar to those in Figure 5.6, except that they are

much smoother. Moreover, (i) the density of CO2 in two CVs is much higher due to

the adsorption of CO2 molecules in the created graphite channel, and (ii) the two

peaks at the entrance and exit of the pore network that are seen in Figure 5.6 no

longer exist in Figure 5.14. The corresponding temperature profile is shown in

Figure 5.15, while a snapshot of the molecules' distribution in the pore network,

obtained after 2,650,000 time steps, is shown in Figure 5.16.

The dependence of the components' fluxes and permeabilities, in an

equimolar mixture, on the pressure drop PΔ applied along the pore is shown in

Figure 5.17. The upstream pressure is fixed at 120 atm, and T = 40°C. Numerically,

the results shown in Figure 5.17 are similar to those in Figure 5.8, which might be

expected, but the shapes of the curves are somewhat different, which might be due to

191

less fluctuations in the density profiles of the two components, and the absence of

hot spots in the system.

Using the same simulation technique, we also studied the effect of mixture's

composition on its transport and separation properties, in order to see whether any

discernable differences exist between these systems and those described above when

we connect the pore network to two large CVs in equilibrium with two bulk regions.

Figure 5.18 shows the snapshot of the pore network and the distribution of the

molecules in it, in which the CO2 mole fraction in the feed is 0.7. The snapshot was

obtained after 2,315,000 time steps. The upstream and downstream pressures are,

respectively, 120 atm and 20 atm and T = 40°C, which should be compared to Figure

5.16.

Figure 5.19 shows the time-averaged density profiles of the two components

of three mixtures with different compositions at T = 40°C. The upstream and

downstream pressures are, respectively, 120 atm and 20 atm, and the mole fraction

of CO2 in the feed is 0.5, 0.7, and 0.9. The corresponding time-averaged distributions

of the dimensionless temperature are shown in Figure 5.20, indicating the

temperature is constant throughout the pore network and CVs.

The dependence of the fluxes and permeabilities of CO2 and CH4, and the

corresponding separation factor, on the mole fraction of CO2 in the feed are shown in

Figure 5.21. The rest of the parameters of the system are the same as before. It is

somewhat surprising that as the CO2 mole fraction increases, the separation factor

decreases, since one might expect that, due to high affinity of CO2 for adsorption on

192

carbon surfaces, a mixture with more CO2 can be better separated than one which is

leaner in CO2.

Figure 5.22 present the time-averaged density profiles ( )zi xρ of both

components in an equimolar mixture at three different temperatures. The upstream

and downstream pressures are, respectively, 120 atm and 90 atm. Increasing the

temperature reduces adsorption and, therefore, the densities decrease, since there

would be a larger flux of molecules leaving the pore network. The corresponding

time-averaged temperature profiles are shown in Figure 5.23.

5.5.2 Carbon Dioxide-Propane, Methane-Propane, and Methane-Butane Mixtures We now present and discuss the simulation results for the CO2/C3H8, CH4/C3H8, and

CH4/C4H10 mixtures. In these simulations we used the second method of creating

CVs for those pores that are directly connected to the external surface of the pore

network, namely, extending the graphite walls of the pores and creating independent

CVs for each pore.

Figure 5.24 demonstrates the time-averaged density profiles ( )zi xρ of CO2

and C3H8 at three different temperatures. The upstream and downstream pressures

are, respectively, 3 atm and 1 atm, and the mole fraction of CO2 in the feed is 0.7. As

can be seen, by increasing the temperature not only the density of both components

decreases due to reduction in the amount of adsorption, but they also become

193

increasingly similar. However, note that despite the fact that the mixture is rich in

CO2, the propane density is higher, due to adsorption and shielding effects described

in Chapter 3 for alkane chain molecules. The corresponding average temperature

distributions are shown in Figure 5.25, which show that the dimensionless

temperature of the mixture is almost constant throughout the system.

Temperature-dependence of the flux and permeability of the components in a

mixture of CO2 and C3H8 in which the CO2 mole fraction is 0.7, and the

corresponding separation factors are shown in Figure 5.26. The upstream and

downstream pressures are, respectively, 3 atm and 1 atm. Increasing the temperature

reduces adsorption. This results in the fluxes and permeabilities of both components

becoming increasingly similar. As a result, the separation factor decreases.

The effect of the composition on the transport and separation of CO2/C3H8

mixtures was also studied. Figure 5.27 shows the time-averaged density profiles

( )zi xρ of both components at T = 50°C for three composition of the mixture. The

upstream and downstream pressures are, respectively, 3 atm and 1 atm. In the

equimolar feed mixture the density of propane is higher in the pore network,

signifying the effect of stronger tendency of the graphite atoms to attract propane due

to its energy parameter which is larger than that of CO2. Even if we increase the

mole fraction of CO2 in the feed to higher values, the trends remain the same. The

corresponding time-averaged distribution of the dimensionless temperature with the

same parameters as in Figure 5.27 are shown in Figure 5.28.

194

Figure 5.29 shows the dependence of the fluxes and permeabilities of CO2

and C3H8, and the corresponding separation factors, on the mole fraction of CO2 in

the feed at T = 50°C. The upstream and downstream pressures are, respectively, 3

atm and 1 atm. While the flux of CO2 increases with increasing its mole fraction in

the feed, the opposite is true about the flux of C3H8. But, the permeabilities and,

therefore, the separation factors are almost independent of the mixture's composition.

Figure 5.30 shows a 3D snapshot of the mixture in the pore network at T =

50°C, obtained after 4,000,000 time steps. The CO2 mole fraction in the feed is 0.85,

and the upstream and downstream pressures are, respectively, 3 atm and 1 atm. The

snapshot has been also shown at a larger scale to show better the positions of C3H8

and CO2 molecules.

Figure 5.31 presents the time-averaged density ( )zi xρ and temperature

profiles for a mixture of CH4 and C3H8 at T = 50°C. The upstream and downstream

pressures are, respectively, 3 atm and 1 atm, with CH4 mole fraction in the feed

being 0.7. Even though the mixture is rich in CH4, the density of C3H8 is higher, both

under the bulk conditions and in the pore network.

Figure 5.32 shows the time-averaged density profiles ( )zi xρ in equimolar

mixtures of CO2/C3H8 and CH4/C3H8 at T = 50°C. The upstream and downstream

pressures in both cases are 30 atm and 10 atm, respectively. Once again, the density

of C3H8 is higher than those of CH4 and CO2 in both mixtures, due to adsorption and

the shielding effect as described earlier in Chapter 3. We then increased the upstream

pressure to above the supercritical pressure for a mixture of CO2 and C3H8. Figure

195

5.33 presents the resulting time-averaged temperature and density profiles ( )zi xρ if

the upstream pressure is 120 atm, above the supercritical value, at T = 50°C, while

the downstream pressure is 20 atm. The mole fraction of CO2 in feed is 0.9. More

interesting from a practical view point are the fluxes and permeabilites of the two

components in the mixture and the corresponding separation factors. These are

shown in Figure 5.34. They indicate that the fluxes of both components increase by

increasing the pressure gradients imposed on the pore network. The separation factor

achieves a maximum where the permeabilities attain their minimum values. This

happens when the downstream pressure is equal to 50 atm.

Figure 5.35 represents the time-averaged density ( )zi xρ and temperature

profiles of CH4 and C4H10 at T = 50°C. The upstream and downstream pressures are,

respectively, 3 atm and 1 atm, with the mole fraction of CH4 in the feed being 0.7.

Once again, the density of C4H10 is higher than that of CH4, due to the effect of

adsorption and shielding effect.

Since the pore sizes in the pore network are very small, with an average of

about 5Å, it is diffult for long alkane chains, such as butane and hexane, to move in

the pore network, especially when the pressure is high. Experiments performed by

our group indicate that as the pressure gradient applied to a nanoporous CMS

membrane increases, the motion of n-alkanes in the membrane becomes increasingly

more difficult. At some point, there appears to be a freezing transition whereby the

n-alkane chains “freeze” inside the nanopores and do not move, hence completely

196

plugging the membrane. We have observed this phenomenon in our simulation as

well.

5.6 Asymmetric Pore Network Model In Chapter 4, we studied the effect of asymmetry on the permeation properties of a

model membrane when a pressure gradient is applied to the membrane, its

permeation properties are merasured, and then the direction of the pressure gradient

is reversed. Recall from Chapter 4 that the membrane was modeled as three single

carbon slit pores in series, with different sizes, representing the macro-, meso- and

nanopores, such that each of the slit pores represented one layer of the membrane.

In the present chapter a pore network is developed to model the morphology

of a membrane with different layers, each having different pore structures, which

represents more realistically a supported membrane. To do so, we modeled the

support and the membrane layer by using a pore network that has different average

pore sizes for each of the layers. To our knowledge, this is the first study that

considers both the support and membrane layers together as two continuous

connected pore network models. To this end, and as a first trial, we divided the

simulation cell into two equal sections. Then, we randomly inserted 40 Poisson

points in the first section and 2100 Poisson points in the second section in order to

construct each of the pore networks. Since the number of Poisson points in the first

section is less than the second part, its corresponding average size of the constructed

197

polyhedra is larger than that of the second part. By the same token, the average pore

size of the pore network in the support layer is larger than that of the membrane

layer. The dimensions zyx LLL ×× of the simulation cell are Å.

The carbon atoms were packed with a structure corresponding to graphite, with the

total initial number of carbon atoms in the simulation cell being 29640. In order to

make the pores uniformly in our entire cell, we used the Gaussian method of

generating the pore space in which the pore polyhedra are selected at random. Note

that since the average pore size in the support layer is larger than that of the

membrane part, we cannot use the biased method of generating the pore space. The

total porosity to construct the pore network was 0.5. The average pore sizes for the

first and second layers were about 14.4 Å and 5.2 Å, respectively. The resulting

three-dimensional pore network and the Poisson points are shown in Figure 5.36.

65.6395.6390.63 ××

After constructing the asymmetric 3D molecular pore network, the non-

equilibrium molecular dynamics method was used to expose the system to an

external pressure gradient in a fixed direction. The pore network was connected to

CVs in equilibrium to two bulk regions. We studied transport and separation of an

equimolar mixture of CH4 and CO2 at T = 40°C. The upstream and downstream

pressures are 120 atm and 20 atm, respectively. The simulation results were

computed after time steps. 6103×

Figure 5.37 presents the time-averaged density profiles of both components

when the pressure gradient is applied in two opposite directions. The corresponding

time-averaged dimensionless temperature of the system for both cases are shown in

198

Figure 5.38. Figure 5.39 shows the snapshots of the mixture in the asymmetric pore

network when the pressure gradient is applied in the two opposite directions, which

shows the fluid molecules are more packed in upstream. The fluxes and, hence, the

permeabilities of the two components are not the same when the pressure gradient is

applied in the two opposite directions, although the difference in the permeances for

the two components are small. However, if we model the support layer with larger

average pore size, e.g., 70-80 Å, which would be more appropriate than 14.4 Å that

we used in the present simulations, the difference between the average pore size of

the support and the membrane layer will increase. In that case, the difference

between the permeances should increase, if our results presented in Chapter 4 are

any indications. However, in order to construct such a model, one must use much

larger simulation cells which would require massive computations that are not

possible at the moment due to the computational facility limitations.

5.7 Summary

Extensive molecular dynamics simulations, using the dual control-volume

nonequilibrium molecular dynamics method, were carried out to study the transport

and separation of binary mixtures consisting of CO2 and n-alkane chains, under both

sub- and supercritical conditions in nanoporous CMS membranes. The membrane

was represented by a three-dimensional pore space, generated atomistically by the

199

Voronoi tessellation of the space, using tens of thousands of atoms. The Voronoi

model contains interconnected pores of various sizes and shapes, which is a realistic

model of the membrane and allows us to investigate the effect of morphology of the

pore space, i.e., its pore size distribution and pore connectivity, on the transport and

separation properties of fluid mixtures. The effect of the composition of the feed, the

temperature of the system, as well as the applied pressure gradient on the transport

and separation of the mixtures were all studied.

200

P2 =20 atm

0.00

0.05

0.10

0.15

0.20

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Den

sity

P2 =50 atm

0.00

0.05

0.10

0.15

0.20

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Den

sity

P2 =90 atm

0.00

0.05

0.10

0.15

0.20

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Den

sity

Figure 5.6: Time-averaged density profiles of both components of an equimolar mixture of CO2 (dashed curves) and CH4 (solid curves) at T=40°C in the transport direction x . The upstream pressure is 120 atm, while the downstream pressures are 20 atm, 50 atm, and 90 atm. Dashed lines indicate the boundaries of the pore network region.

201

0

1

2

3

4

Tem

pera

ture

0

1

2

3

4

Tem

pera

ture

0

1

2

3

4

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Tem

pera

ture

Figure 5.7: Time-averaged temperature profiles of mixture of CO2 and CH4 with the same parameters as Figure 5.6. The upstream pressure is 120 atm, while the downstream pressures are 20 atm (top), 50 atm (middle), and 90 atm (bottom). Dashed lines indicate the boundaries of the pore network region.

202

0

20

40

60

80

10 20 30 40 50 60 70 80 90 100 110

ΔP (atm)

Flux

( gr

mol

e/m

in.c

m 2

)

CH4

CO2

1.0

1.5

2.0

2.5

3.0

10 20 30 40 50 60 70 80 90 100 110

ΔP (atm)

Sep

arat

ion

Fact

or

0

2

4

6

8

10

10 20 30 40 50 60 70 80 90 100 110

ΔP (atm)

Perm

eabi

lity

( gr

mol

.cm

/min

.atm

.cm

2 ) x

10

7

CO2

CH4

Figure 5.8: The dependence of the flux and permeability of CO2 (dashed curves) and CH4 (solid curves), and the corresponding separation factor, on the pressure drop ΔP applied to the pore network at T=40°C. The mixture is equimolar, with the upstream pressure being 120 atm.

203

P2 =20 atm

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Den

sity

P2 =50 atm

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Den

sity

P2 =90 atm

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Den

sity

Figure 5.9: Time-averaged density profiles of both components of a mixture of CO2 (dashed curves) and CH4 (solid curves) with a mole fraction of 90% CO2 and 10% CH4 at T=40°C in the transport direction x . The upstream pressure is 120 atm, while the downstream pressures are 20 atm, 50 atm, and 90 atm. Dashed lines indicate the boundaries of the pore network region.

204

0

1

2

3

4

Tem

pera

ture

0

1

2

3

4

Tem

pera

ture

0

1

2

3

4

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Tem

pera

ture

Figure 5.10: Time-averaged temperature profiles of mixture of CO2 and CH4 with the same parameters as Figure 5.9. The upstream pressure is 120 atm, while the downstream pressures are 20 atm (top), 50 atm (middle), and 90 atm (bottom). Dashed lines indicate the boundaries of the pore network region.

205

0

2

4

6

8

10 20 30 40 50 60 70 80 90 100 110

ΔP (atm)

Per

mea

bilit

y (

grm

ole.

cm/m

in.a

tm.c

m 2

) x 1

0 7

CO2

CH4

1.0

2.0

3.0

4.0

5.0

6.0

10 20 30 40 50 60 70 80 90 100 110

ΔP (atm)

Sep

arat

ion

Fact

or0

20

40

60

80

10 20 30 40 50 60 70 80 90 100 110

ΔP (atm)

Flux

( gr

mol

e/m

in.c

m 2

) CO2

CH4

Figure 5.11: The dependence of the flux and permeability of CO2 (dashed curves) and CH4 (solid curves), and the corresponding separation factor, on the pressure drop ΔP applied to the pore network at T=40°C, in a mixture in which the mole fraction of CO2 is 0.9 and the upstream pressure is fixed at 120 atm.

206

Figure 5.12: Snapshot of the pore network containing an equimolar mixture of CH4 (circles) and CO2 (asterisks) at T=40°C, obtained after 2,220,000 time steps. The upstream and downstream pressures are, respectively, 120 atm and 90 atm.

207

Figure 5.13: Snapshot of the pore network containing CH4 (circles) and CO2 (asterisks) at T=40°C, obtained after 1,740,000 time steps. The CO2 mole fraction in the feed is 0.9. The upstream and downstream pressures are, respectively, 120 atm and 90 atm.

208

0.0

0.1

0.2

0.3

0.4

Dens

ity

0.0

0.1

0.2

0.3

0.4

Dens

ity

0.0

0.1

0.2

0.3

0.4

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Dens

ity

Figure 5.14: Time-averaged density profiles of both components of an equimolar mixture of CO2 (dashed curves) and CH4 (solid curves) at T=40°C in the transport direction x . The upstream pressure is 120 atm, while the downstream pressures are 20 atm (top), 50 atm (middle), and 90 atm (bottom). Dashed lines indicate the boundaries of the pore network region.

209

0

1

2

3

4

Tem

pera

ture

0

1

2

3

4

Tem

pera

ture

0

1

2

3

4

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Tem

pera

ture

Figure 5.15: Time-averaged temperature profiles of mixture of CO2 and CH4 with the same parameters as Figure 5.14. The upstream pressure is 120 atm, while the downstream pressures are 20 atm (top), 50 atm (middle), and 90 atm (bottom). Dashed lines indicate the boundaries of the pore network region.

210

Figure 5.16: Snapshot of the pore network containing an equimolar mixture of CH4 (circles) and CO2 (asterisks) at T=40°C obtained after 2,650,000 time steps. The upstream and downstream pressures are, respectively, 120 atm and 20 atm.

211

0

20

40

60

80

10 30 50 70 90 110

ΔP (atm)

Flux

( gm

ole/

min

.cm

2 )

0

2

4

6

8

10

10 30 50 70 90 110

ΔP (atm)

Per

mea

bilit

y (

gmol

e.cm

/min

.atm

.cm

2 ) x

107

Figure 5.17: The dependence of the flux and permeability of CO2 (dashed curves) and CH4 (solid curves) in an equimolar mixture on the pressure drop ΔP applied to the pore network at T=40°C. The upstream pressure is fixed at 120 atm.

212

Figure 5.18: Snapshot of the pore network containing CH4 (circles) and CO2 (asterisks) at T=40°C obtained after 2,315,000 time steps. The CO2 mole fraction in the feed is 0.7. The upstream and downstream pressures are, respectively, 120 atm and 20 atm.

213

0.0

0.1

0.2

0.3

0.4

0.5

Den

sity

0.0

0.1

0.2

0.3

0.4

0.5

Dens

ity

0.0

0.1

0.2

0.3

0.4

0.5

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Dens

ity

Figure 5.19: Time-averaged density profiles of both components of CO2 (dashed curves) and CH4 (solid curves) at T=40°C in the transport direction x . The upstream and downstream pressures are, respectively, 120 atm and 20 atm. The mole fraction of CO2 in feed is 0.5 (top), 0.7 (middle) and 0.9 (bottom). Dashed lines indicate the boundaries of the pore network region.

214

0

1

2

3

4

Tem

pera

ture

0

1

2

3

4

Tem

pera

ture

0

1

2

3

4

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Tem

pera

ture

Figure 5.20: Time-averaged temperature profiles of mixture of CO2 and CH4 with the same parameters as Figure 5.19. The mole fraction of CO2 in feed is 0.5 (top), 0.7 (middle) and 0.9 (bottom). Dashed lines indicate the boundaries of the pore network region.

215

0

20

40

60

80

0.4 0.5 0.6 0.7 0.8 0.9 1.0

CO2 Mole Fraction in the Feed

Flux

( gm

ole/

min

.cm

2 )

0

2

4

6

8

10

0.4 0.5 0.6 0.7 0.8 0.9 1.0

CO2 Mole Fraction in the Feed

Per

mea

bilit

y (

gmol

e.cm

/min

.atm

.cm

2 ) x

107

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.4 0.5 0.6 0.7 0.8 0.9 1.0

CO2 Mole Fraction in the Feed

Sep

arat

ion

Fact

or

Figure 5.21: The dependence of the flux and permeability of CO2 (dashed curves) and CH4 (solid curves), and the corresponding separation factor, on the mole fraction of CO2 in the feed at T=40°C. The upstream and downstream pressures are, respectively, 120 atm and 20 atm.

216

T = 20 °C

0.0

0.1

0.2

0.3

0.4

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Den

sity

T = 100 °C

0.0

0.1

0.2

0.3

0.4

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Den

sity

T = 140 °C

0.0

0.1

0.2

0.3

0.4

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Den

sity

Figure 5.22: Time-averaged density profiles of both components of CO2 (dashed curves) and CH4 (solid curves) in the transport direction x at different temperatures. The upstream and downstream pressures are, respectively, 120 atm and 90 atm, and the mixture is equimolar. Dashed lines indicate the boundaries of the pore network region.

217

0

1

2

3

4

Tem

pera

ture

0

1

2

3

4

Tem

pera

ture

0

1

2

3

4

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Tem

pera

ture

Figure 5.23: Time-averaged temperature profiles of mixture of CO2 and CH4 with the same parameters as Figure 5.22. Temperature is fixed at 20°C (top), 100°C (middle), and 140°C (bottom). Dashed lines indicate the boundaries of the pore network region.

218

T=25°C

0.00

0.02

0.04

0.06

0.08

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Den

sity

T=50°C

0.00

0.02

0.04

0.06

0.08

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Dens

ity

T=75°C

0.00

0.02

0.04

0.06

0.08

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Den

sity

Figure 5.24: Time-averaged density profiles of both components of CO2 (solid curves) and C3H8 (dashed curves) in the transport direction x at different temperatures. The upstream and downstream pressures are, respectively, 3 atm and 1 atm and the mole fraction of CO2 in the feed is 0.7. Dashed lines indicate the boundaries of the pore network region.

219

0

1

2

3

4Te

mpe

ratu

re

0

1

2

3

4

Tem

pera

ture

0

1

2

3

4

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Tem

pera

ture

Figure 5.25: Time-averaged temperature profiles of mixture of CO2 and C3H8 with the same parameters as Figure 5.24. Temperature is fixed at 25°C (top), 50°C (middle), and 75°C (bottom). Dashed lines indicate the boundaries of the pore network region.

220

0

5

10

15

20

10 30 50 70 90

Temperature (°C)

Flux

( gm

ole/

min

.cm

2

0

20

40

60

80

100

120

140

10 30 50 70 90

Temperature (°C)

Perm

eabi

lity

( gm

ole.

cm/m

in.a

tm.c

m 2

) x 1

07

)

0

2

4

6

8

10 30 50 70 90

Temperature (°C)

Sep

arat

ion

Fact

or

Figure 5.26: The dependence of the flux and permeability of CO2 (solid curves) and C3H8 (dashed curves), and the corresponding separation factor, on the temperature. The upstream and downstream pressures are, respectively, 3 atm and 1 atm, and the mole fraction of CO2 in the feed is 0.7.

221

0.00

0.02

0.04

0.06

0.08

Den

sity

0.00

0.02

0.04

0.06

0.08

Dens

ity

0.00

0.01

0.02

0.03

0.04

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Dens

ity

Figure 5.27: Time-averaged density profiles of both components of CO2 (solid curves) and C3H8 (dashed curves) at T=50°C in the transport direction x . The upstream and downstream pressures are, respectively, 3 atm and 1 atm. The mole fraction of CO2 in feed is 0.5 (top), 0.7 (middle) and 0.85 (bottom). Dashed lines indicate the boundaries of the pore network region.

222

0

1

2

3

4

Tem

pera

ture

0

1

2

3

4

Tem

pera

ture

0

1

2

3

4

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Tem

pera

ture

Figure 5.28: Time-averaged temperature profiles of mixture of CO2 and C3H8 with the same parameters as Figure 5.27. The mole fraction of CO2 in feed is 0.5 (top), 0.7 (middle) and 0.85 (bottom). Dashed lines indicate the boundaries of the pore network region.

223

0

5

10

15

20

0.4 0.5 0.6 0.7 0.8 0.9 1.0

CO2 Mole Fraction in the Feed

Flux

( gm

ole/

min

.cm

2 )

0

20

40

60

80

100

120

0.4 0.5 0.6 0.7 0.8 0.9 1.0

CO2 Mole Fraction in the Feed

Perm

eabi

lity

( gm

ole.

cm/m

in.a

tm.c

m 2

) x 1

07

0

2

4

6

8

0.4 0.5 0.6 0.7 0.8 0.9 1.0

CO2 Mole Fraction in the Feed

Sepa

ratio

n Fa

ctor

Figure 5.29: The dependence of the flux and permeability of CO2 (solid curves) and C3H8 (dashed curves), and the corresponding separation factor, on the mole fraction of CO2 in the feed at T=50°C. The upstream and downstream pressures are, respectively, 3 atm and 1 atm.

224

Transport Direction

Figure 5.30: Distribution of CO2 (triangles) and C3H8 chains in a pore network at T=50°C, obtained after 4,000,000 time steps. The CO2 mole fraction in the feed is 0.85. The upstream and downstream pressures are, respectively, 3 atm and 1 atm.

225

0

0.01

0.02

0.03

0.04

0.05

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Dens

ity

0

2

4

6

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Tem

pera

ture

Figure 5.31: Time-averaged temperature and density profiles of both components of CH4 (solid curves) and C3H8 (dashed curves) at T=50°C in the transport direction x . The upstream and downstream pressures are, respectively, 3 atm and 1 atm. The mole fraction of CH4 in feed is 0.7. Dashed lines indicate the boundaries of the pore network region.

226

0.00

0.05

0.10

0.15

0.20

0.25

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Den

sity

0.00

0.05

0.10

0.15

0.20

0.25

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Den

sity

Figure 5.32: Time-averaged density profiles of both components in an equimolar mixture of CO2 (solid curves) and C3H8 (dashed curves) (top), and in an equimolar mixture of CH4 (solid curves) and C3H8 (dashed curves) (bottom), at T=50°C in the transport direction x . The upstream and downstream pressures are, respectively, 30 atm and 10 atm. Dashed lines indicate the boundaries of the pore network region.

227

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Den

sity

0

2

4

6

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Tem

pera

ture

Figure 5.33: Time-averaged temperature and density profiles of both components of CO2 (solid curves) and C3H8 (dashed curves) at T=50°C in the transport direction x . The upstream and downstream pressures are, respectively, 120 atm and 20 atm. The mole fraction of CO2 in feed is 0.9. Dashed lines indicate the boundaries of the pore network region.

228

0

1

2

3

4

5

6

7

8

10 30 50 70 90 110

ΔP (atm)

Sepa

ratio

n Fa

ctor

0

10

20

30

40

10 30 50 70 90 110

ΔP (atm)

Flux

( gm

ole/

min

.cm

2

0

2

4

6

8

10

10 30 50 70 90 110

ΔP (atm)

Per

mea

bilit

y (

gmol

e.cm

/min

.atm

.cm

2 ) x

107

)

Figure 5.34: The dependence of the flux and permeability of CO2 (solid curves) and C3H8 (dashed curves), and the corresponding separation factor, on the pressure drop ΔP applied to the pore network at T=50°C, in a mixture in which the mole fraction of CO2 is 0.9 and the upstream pressure is fixed at 120 atm.

229

0.00

0.03

0.06

0.09

0.12

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Den

sity

0

2

4

6

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Tem

pera

ture

Figure 5.35: Time-averaged temperature and density profiles of both components of CH4 (solid curves) and C4H10 (dashed curves) at T=50°C in the transport direction x . The upstream and downstream pressures are, respectively, 3 atm and 1 atm. The mole fraction of CH4 in feed is 0.7. Dashed lines indicate the boundaries of the pore network region.

230

Figure 5.36: A three-dimensional asymmetrical Voronoi network. Poisson points are shown in top figure and the pore polyhedra are shown in bottom figure.

231

0

0.04

0.08

0.12

0.16

0.2

Dens

ity

0

0.04

0.08

0.12

0.16

0.2

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Den

sity

Figure 5.37: Time-averaged density profiles of CH4 (solid curves) and CO2 (dashed curves) in an equimolar mixture at T=40°C in the transport direction x . The upstream pressure is fixed at 120 atm, in left CV (top) and right CV (bottom), and the downstream pressure is 20 atm. Dashed lines indicate the boundaries of the pore network region.

232

0

1

2

3

4

-25 -20 -15 -10 -5 0 5 10 15 20 25

X*

Tem

pera

ture

0

1

2

3

4

Tem

pera

ture

Figure 5.38: Dimensionless temperature distribution in a pore and the two control volumes, containing CH4 and CO2 in an equimolar mixture at T=40°C. The upstream pressure is fixed at 120 atm, in left CV (top) and right CV (bottom), and the downstream pressure is 20 atm. Dashed lines indicate the boundaries of the pore network region.

233

Figure 5.39: Snapshot of the pore network containing an equimolar mixture of CH4 (circles) and CO2 (asterisks) at T=40°C. The upstream pressure is fixed at 120 atm, in left CV (top) and right CV (bottom), and the downstream pressure is 20 atm.

234

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