Molecular Dynamics Simulation of Carbon Dioxide in Aqueous Electrolyte Solution
Tao Huang
Dissertation Submitted in Fulfillment of the Requirements for the Degree of Doctor of Philosophy
Faculty of Information and Communication Technologies Swinburne University of Technology 2012 Abstract
The structural properties and diffusion coefficients of the H2O+NaCl+CO2 ternary system at various NaCl concentrations, temperatures and pressures are investigated using molecu- lar simulation. A CO2 potential model is selected for the simulation of CO2 diffusion coef- ficient in aqueous solution. As the most appropriate model, it produces simulation results which are in closest agreement with experimental data. The properties of the H2O+NaCl system are examined prior to the H2O+NaCl+CO2 system, including the radial distribu- tion functions, coordination numbers and diffusion coefficients at various temperatures and pressures. Three aspects of the ternary system are studied. First, the diffusion coefficients of the ternary system at different NaCl concentrations are observed. The NaCl concentra- tion is found to have a large impact on both the diffusion coefficients of the ternary system and also the cluster pattern of ions. Second, the diffusion coefficients of the system at dif- ferent temperatures are studied. Raising the temperature increases the diffusion coefficients and facilitates formation of ions pairs. Finally, the diffusion coefficients of the ternary sys- tem at different pressures are investigated. Pressure also has impact, but to a much lesser degree. At 278 K, the higher the pressure, the greater value of the diffusion coefficient. In contrast at a temperature of 298 K, a pressure increase leads to lower diffusion coefficient. Hydrogen bonds at low temperatures may be the reason for the unusual phenomenon. The diffusion data are compared to predictions of two models proposed by Ratcliff and Holdcroft (1963). The first approach is based on activation theory and results in both a linear and exponential relationship. They preferred a linear model over an exponential
ii model due to the limiting experimental data. However, we demonstrate that the exponential model is more suitable for predicting diffusion coefficient with the help of simulation data. The other approach is based on the relation between diffusion coefficient and viscosity, whereby the diffusion coefficient of the gas in electrolyte solution is derived from a given viscosity. The diffusion coefficients obtained from molecular simulation agree with the results from the two equations, demonstrating the accuracy of the two prediction equations.
iii Acknowledgement
I would like to acknowledge and thank my supervisors Prof. Richard Sadus and Prof. Billy Todd for their positive direction and continuing support. In particular, I am very grateful to my principal supervisor Prof. Richard Sadus. I cannot thank him enough for his constructive guidance, innovative ideas and remarkable patience. During the past few years, I have learned from them the best attributes of a researcher and I believe these attributes will continue to be of great help to my career and life. I would also like to thank Prof. Feng Wang for her advice and encouragement. I would like to thank my wife, Jing, my mother and father, whose love continues to encourage me, as it has always done. I would not have completed this thesis without their encouragement and support. I gratefully acknowledge Dr. Zhongwu Zhou and Dr. Ming Liu for their encouragement and thoughtful discussions. I also thank Dr. Junfang Li, Dr. Jianhui Li, Dr. Jarek Bosko, Dr. Alex Bosowski, Dr. Liping Li for their generous help. I also like to acknowledge the support I have received from my fellow colleagues at CMS. My special thanks go to the staff of FICT and Swinburne Research for their contin- uing support with the highest professional standard possible. I am grateful to Swinburne University of Technology for providing me financial support through a Swinburne University Postgraduate Research Award (SUPRA). I also appreciate the Australian Partnership for Advanced Computing who generously provided an allocation of computing time to perform the simulation.
iv Declaration
I hereby declare that the thesis entitled “Molecular Dynamics Simulation of Carbon Diox- ide in Aqueous Electrolyte Solution” , and submitted in fulfillment of the requirements for the Degree of Doctor of Philosophy in the Faculty of Information and Communication Technologies of Swinburne University of Technology, is my own work and that it contains no material which has been accepted for the award to the candidate of any other degree or diploma, except where due reference is made in the text of the thesis. To the best of my knowledge and belief, it contains no material previously published or written by another person except where due reference is made in the text of the thesis.
Tao Huang 2012
v Contents
Abstract ii
Acknowledgement iv
Declaration v
1 Introduction 1
2 Diffusion Theories 4 2.1 Diffusion Theories ...... 5 2.1.1 Hydrodynamic Theory ...... 5 2.1.2 Activated State Theory ...... 7 2.1.3 Free Volume Theory ...... 9 2.1.4 Kinetic Theory ...... 10 2.2 Diffusion Theories of Electrolyte Solutions ...... 11 2.2.1 Basic Equations of Diffusion in Solution ...... 14 2.2.2 Diffusion for Single Electrolyte —Nernst-Hartley Equation . . . . 15 2.2.3 Electrophoretic Effect in Diffusion—Onsager-Fuoss Equation . . . 17 2.2.4 Diffusion for Partly Ionized Electrolytes ...... 19 2.2.5 Self-Diffusion in Electrolyte Solutions—Onsager Limiting Law . . 19
vi 2.2.6 Self-Diffusion in Multicomponent Aqueous Electrolyte Systems in Wide Concentration Ranges ...... 20 2.3 Summary ...... 24
3 Molecular Simulation 26 3.1 Introduction ...... 26 3.2 Molecular Dynamics ...... 28 3.2.1 Force Field ...... 29 3.2.2 Lennard-Jones Reduced Units ...... 31 3.2.3 Periodic Boundary Conditions ...... 32 3.2.4 Equation of Motion ...... 32 3.2.5 Time Integration Algorithm ...... 34 3.2.6 Constant Temperature ...... 36 3.2.7 Electrostatic Force ...... 37 3.3 The Applications of Molecular Dynamics ...... 41 3.3.1 Trajectory Analysis ...... 41 3.3.2 Radial Distribution Function ...... 42 3.3.3 Correlation Function ...... 44 3.3.4 Mean Square Displacement ...... 44
4 Diffusion Coefficients of Carbon Dioxide in Water using Different CO2 Models 47 4.1 Introduction ...... 47 4.2 Intermolecular Potential between Water and Carbon Dioxide ...... 49 4.2.1 Water Models ...... 49 4.2.2 Carbon Dioxide Models ...... 53
4.3 Diffusivities of CO2 in Water under Different Potential Models ...... 57 4.3.1 Simulation Details ...... 57 4.3.2 Results and Discussion ...... 58
vii 5 Structural Properties and Diffusion Coefficients of NaCl Aqueous Solutions 63 5.1 Introduction ...... 63 5.2 Intermolecular Potential between Water and NaCl ...... 65 5.3 Simulation Details ...... 65 5.4 Structural Properties of Binary System ...... 68 5.4.1 Temperature Dependence ...... 68 5.4.2 Pressure Dependence ...... 75 5.5 Dynamic Properties ...... 82
6 Structural Properties and Diffusion Coefficients of Carbon Dioxide in Aqueous Solutions 87 6.1 Introduction ...... 87 6.2 Simulation Systems ...... 88 6.3 Structural Properties of Ternary System ...... 91 6.3.1 Effects of Different NaCl Concentrations ...... 91 6.3.1.1 Ion-Ion distribution functions ...... 91 6.3.1.2 Solvent atom distribution functions ...... 94
6.3.1.3 Ion-Water, CO2 –Water and Ion-CO2 distribution functions 98 6.3.2 Effects of Temperatures on the Ternary System ...... 106 6.3.3 Effects of Pressures on the Ternary System ...... 112 6.4 Diffusion in the Ternary System ...... 123
7 Prediction of Diffusion Coefficient of CO2 in Electrolyte Solutions 129 7.1 Introduction ...... 129 7.2 Prediction Equation of Diffusion Coefficient Modified From Activation Theory and Perturbation Model ...... 132 7.2.1 Theory ...... 132 7.2.2 Results and Discussion ...... 135
viii 7.3 Predicting Diffusion Coefficient via Viscosity ...... 139 7.3.1 Theory ...... 139 7.3.2 Results and Discussion ...... 141 7.3.2.1 The value of k ...... 141 7.3.2.2 Comparison of different relationships between diffusion and viscosity via experimental diffusion data ...... 145 7.3.2.3 Comparison of different relationships between diffusion and viscosity via molecular dynamics diffusion data . . . 148
8 Conclusions and Recommendations 152
Bibliography 156
ix List of Tables
4.1 Summary of the parameters of various water models (Chaplin, 2011) . . . . 53
4.2 Potential function parameters of different CO2 models ...... 55
4.3 Diffusivities of H2O under different models and ensembles (Mahoney and Jorgensen, 2001) ...... 58
4.4 Comparison of diffusivities of CO2 of experimental and different CO2 mod- els. The values for the diffusion constants are given in 10−5cm2/s...... 62
5.1 Temperature dependence in NaCl solutions ...... 67 5.2 Pressure dependence in NaCl solutions ...... 67 5.3 Potential parameter used in NaCl solutions ...... 67 5.4 Peak heights and coordination numbers of ion-ion at different temperatures in the NaCl solutions ...... 71 5.5 Peak heights and coordination numbers of ion-water in different tempera- tures in the NaCl solutions ...... 72 5.6 Peak heights and coordination numbers between ions for different pres- sures in the NaCl solutions...... 77 5.7 Peak heights and coordination numbers of ions-water for different pres- sures in the NaCl solutions ...... 79
6.1 Concentration dependence settings ...... 90 6.2 Temperature dependence settings ...... 90
x 6.3 Pressure dependence settings ...... 90 6.4 Structure features of ions-ions at different NaCl concentrations ...... 92 6.5 Parameters used to model Lennard-Jones interactions of anions and cations. 94 6.6 Structure features of Figure 6.4 and Figure 6.5...... 102 6.7 Structure features of Figure 6.7 and Figure 6.8...... 105 6.8 Structure features of Figure 6.12...... 109 6.9 Structure features of Figure 6.15 and Figure 6.16 ...... 115 6.10 Structure features of Figure 6.20 and Figure 6.21 ...... 121
◦ 7.1 Diffusion coefficient of CO2 in NaCl solutions at 25 C and 1atm (Ratcliff and Holdcroft 1963) ...... 136
7.2 MD data of diffusion coefficients of CO2 in NaCl solutions obtained in this work ...... 136 7.3 Experimental data of viscosity of NaCl solution at P=1atm and T=25◦C (Kestin et al., 1981) ...... 142
xi List of Figures
2.1 Diagram showing the asymmetric effect at the electrolyte solutions which shows the ions are tended to move in the opposite direction and pull on the given ion in the direction of their motion and slow down the motion of the ion...... 13 2.2 Diagram showing the ionic atmosphere effect at the electrolyte solutions which can slow down the motion of the ion ...... 13
3.1 Diagram showing the periodic boundary conditions, minimum image con- vention and cutoff ...... 33 3.2 Diagram demonstrating the leap-frog integration ...... 36 3.3 Diagram demonstrating the radial distribution function ...... 43
4.1 Diagram showing the different types of water models (Chaplin, 2011), see text for details...... 52 4.2 Diagram showing all the carbon dioxide models used in this work have the same line structures ...... 54
4.3 Comparison of the self diffusion constant of CO2 in water obtained from the Duan potential (•) and experimental data () at different temperatures at a pressure of 1 atm...... 59
xii 4.4 Comparison of the self diffusion constant of CO2 in water obtained from the EPM2 potential (•) and experimental data () at different temperatures at a press of 1 atm...... 60
4.5 Comparison of the self diffusion constant of CO2 in water obtained from the EPM2 + Gromos potential (•) and experimental data () at different temperatures at pressure=1 atm ...... 60
4.6 Comparison between experimental() self diffusion coefficients of CO2 with results obtained from Gromos(•) at different temperatures at a pres- sure of 1 atm...... 61
5.1 Radial distribution functions for Na+-Cl−, Na+-Na+ and Cl−-Cl− (from top to bottom) in different temperatures 278 K(), 288 K(•), 298 K(N) at a pressure of 1 atm...... 70 5.2 Coordination number of Na+ −Cl− (N), Na+ − Na+ () and Cl− −Cl− (•) at different temperatures at a pressure of 1 atm...... 71 − − 5.3 Radial distribution functions for Cl − OH, and Cl − H (from top to bot- tom) at different temperatures 278 K(), 288 K(•), 298 K(N) at a pressure of 1atm...... 73 + + 5.4 Radial distribution functions for Na −OH , and Na −H (from top to bot- tom) at different temperatures 278 K(), 288 K(•), 298 K(N) at a pressure of 1 atm...... 74 − − + 5.5 Coordination numbers of Cl −H (), Cl −OH (•), Na −OH (N) and Na+ − H (H) at different temperatures at a pressure of 1atm...... 75
5.6 Radial distribution functions between OH-OH, OH-H and H-H (from top to bottom) at different temperatures 278 K(), 288 K(•), 298 K(N) at a pressure of 1atm...... 76
xiii 5.7 Radial distribution functions for Na+ − Na+, Cl− −Cl−, and Na+ −Cl− (from top to bottom) at different pressures, 1 atm(), 400 atm(•), 700 atm(N) at a temperature of 298 K...... 78 5.8 Coordination numbers of Na+ − Na+ (), Cl− −Cl− (•) and Na+ −Cl− (N) at different pressures at a temperature of 298 K...... 79 − − 5.9 Radial distribution functions for Cl − OH, and Cl − H (from top to bot- tom) at different pressures, 1atm(), 400 atm(•), 700 atm(N) at a temper- ature of 298 K...... 80 + + 5.10 Radial distribution functions for Na − OH, and Na − H (from top to bottom) at different pressures, 1 atm(), 400 atm(•), 700 atm(N) at a tem- perature of 298 K...... 81 − − + 5.11 Coordination numbers of Cl − OH (•), Cl − H (), Na − OH(N) and Na+ − H (H) at different pressures at a temperature of 298 K...... 82
5.12 Radial distribution functions between OH − OH, OH − H and H − H (from top to bottom) at different pressures, 1 atm(), 400 atm(•), 700 atm(N) at a temperature of 298 K...... 83 + − 5.13 Diffusion coefficients of Na (), Cl (•), H2O (N) at different tempera- tures and 0.42 M NaCl ...... 86 + − 5.14 Diffusion coefficients of Na (), Cl (•), H2O (N) at different pressures and 0.42 M NaCl ...... 86
6.1 Radial distribution functions for Na+-Cl−, Na+-Na+, and Cl−-Cl− (from
top to bottom) at different NaCl concentrations, XNaCl = 0.0074(), XNaCl
= 0.0515(•), XNaCl = 0.114(N), at a temperature of 298 K and a pressure of 1atm...... 95
xiv 6.2 Coordination numbers of Na+ − Na+ (), Cl− −Cl− (•) and Na+ −Cl− (N) at different NaCl concentrations at a temperature of 298 K and a pres- sure of 1 atm...... 96
6.3 Radial distribution functions for OH − OH, H − H, and OH − H (from top
to the bottom) at different NaCl concentrations, XNaCl = 0.0074(), XNaCl
= 0.0515(•), XNaCl = 0.114(N), at a temperature of 298 K and at a pressure of 1 atm...... 97 − − 6.4 Radial distribution functions for Cl − OH and Cl − H (from top to the
bottom) at different NaCl concentrations, XNaCl = 0.0074(), XNaCl =
0.0515(•), XNaCl = 0.114(N), at a temperature of 298 K and at a pressure of 1 atm...... 100 + + 6.5 Radial distribution functions for Na − OH and Na − H (from top to
the bottom) at different NaCl concentrations, XNaCl = 0.0074(), XNaCl
= 0.0515(•), XNaCl = 0.114(N), at a temperature of 298 K and at a pressure of 1 atm...... 101 − − + 6.6 Coordination numbers of Cl − OH (•), Cl − H (), Na − OH(N) and Na+ − H (H) at different NaCl concentrations, at a temperature of 298 K and at a pressure of 1 atm...... 102
6.7 Radial distribution functions for OC − OH and OC − H (from top to the
bottom) at different NaCl concentrations, XNaCl = 0.0074(), XNaCl =
0.0515(•), XNaCl = 0.114(N), at a temperature of 298 K and at a pressure of 1 atm...... 103
6.8 Radial distribution functions for C−OH and C−H (from top to the bottom)
at different NaCl concentrations, XNaCl = 0.0074(), XNaCl = 0.0515(•),
XNaCl = 0.114(N), at a temperature of 298 K and at a pressure of 1 atm. . . 104
xv 6.9 Coordination numbers of OC − OH (), OC − H (•), C − OH (N) and C −H (H) at different NaCl concentrations, at a temperature of 298 K and at a pressure of 1 atm...... 105 6.10 Radial distribution functions between Na+ and carbon dioxide at differ-
ent NaCl concentrations XNaCl = 0.0074(), XNaCl = 0.0515(•), XNaCl = 0.114(N), at a temperature of 298 K and at a pressure of 1 atm...... 107 6.11 Radial distribution functions between Cl− and carbon dioxide at differ-
ent NaCl concentrations XNaCl = 0.0074(), XNaCl = 0.0515(•), XNaCl = 0.114(N), at a temperature of 298 K and at a pressure of 1 atm...... 108 6.12 Radial distribution functions for Na+ − Na+, Cl− −Cl−, and Na+ −Cl− (from top to the bottom) at different temperatures 278 K(), 288 K(•), 298
K(N) at a pressure of 1 atm, XNaCl = 0.0074...... 110 6.13 Coordination numbers of Na+ − Na+ (), Cl− −Cl− (•) and Na+ −Cl−
(N) at different temperatures and at a pressure of 1 atm, XNaCl = 0.0074. . . 111
6.14 Radial distribution functions for OC − OC at different temperatures 278
K(), 288 K(•), 298 K(N) at a pressure of 1 atm, XNaCl = 0.0074...... 112
6.15 Radial distribution functions for C−OH and C−H (from top to the bottom) at different temperatures 278 K(), 288 K(•), 298 K(N), at a pressure of 1
atm, XNaCl = 0.0074...... 113
6.16 Radial distribution functions for OC −OH and OC −H (from top to the bot- tom) at different temperatures 278 K(), 288 K(•), 298 K(N), at a pressure
of 1 atm, XNaCl = 0.0074...... 114
6.17 Coordination numbers of OC −OH (), OC −H (•), C−OH (N) and C−H
(H) at different temperatures and at a pressure of 1 atm, XNaCl = 0.0074. . 115 6.18 Radial distribution functions between Cl− and carbon dioxide at different
temperatures 278 K(), 288 K(•), 298 K(N) at a pressure of 1atm, XNaCl = 0.0074...... 116
xvi + 6.19 Radial distribution functions between Na and carbon dioxide (Na -OC at above and Na-C at bottom) 278 K(), 288 K(•), 298 K(N) at a pressure of
1atm, XNaCl = 0.0074...... 117
6.20 Radial distribution functions for OC − OH and OC − H (from top to the bottom) at different pressures 1atm(), 400 atm(•), 700 atm(N), at a tem-
perature of 298 K, XNaCl = 0.0074...... 119
6.21 Radial distribution functions for C−OH and C−H (from top to the bottom) at different pressures 1 atm(), 400 atm(•), 700 atm(N), at a temperature
of 298 K, XNaCl=0.0074...... 120
6.22 Coordination numbers of OC −OH (), OC −H (•), C −OH (N)and C −H
(H) at different pressures, at a temperature of 298 K, XNaCl = 0.0074. . . . 121 6.23 Radial distribution functions between Na+ and carbon dioxide at different pressures 1 atm(), 400 atm(•), 700 atm(N), at a temperature of 298 K,
XNaCl = 0.0074...... 122 6.24 Radial distribution functions between Cl− and carbon dioxide at different pressures 1 atm(), 400 atm(•), 700 atm(N), at a temperature of 298 K,
XNaCl = 0.0074...... 123 + − 6.25 Diffusion coefficients of Na (), Cl (•), CO2 (N) and H2O (H) at different concentrations at a temperature of 298 K, and a pressure of 1 atm. 125 + − 6.26 Diffusion coefficients of Na (), Cl (•), CO2 (N) and H2O (H) at different temperatures at a pressure of 1 atm...... 125 + − 6.27 Diffusion coefficients of Na (•), Cl (N), H2O (H) and CO2 () at
different pressures and a temperature of 298 K, XNaCl = 0.0074...... 127 + − 6.28 Diffusion coefficients of Na (•), Cl (N), H2O (H) and CO2 () at
different pressures and a temperature of 278 K, XNaCl = 0.0074...... 127
xvii 7.1 Diffusion coefficients of Oxygen in KOH solutions as a function of KOH molality (Anderko and Lencka, 1998) ...... 131
7.2 Comparison of the diffusion coefficients of CO2 between our MD simula- tion data () and Ratcliff’s experimental data (•) at different NaCl solutions 137
7.3 Diffusion coefficients of CO2 of our MD simulation data converted to
Ln(D/D0) showing the linear relationship to the NaCl concentrations . . . 138
7.4 Diffusion coefficients of CO2 of Ratcliff’s experimental data converted to
Ln(D/D0) showing the linear relationship to the NaCl concentrations . . . 138
7.5 Experimental data of viscosity of NaCl solutions shows that Ln(µ0/µ) is linear with respect to the NaCl concentrations ...... 142 7.6 Comparison between different prediction models via viscosity and exper- imental data, Ratcliff’s prediction model via viscosity(), Funazukuri’s prediction model via viscosity(•), Our prediction model via viscosity(N), and Ratcliff’s experimental data(H) ...... 148 7.7 Comparison between different prediction models via viscosity and Rat- cliff’s linear equation, Ratcliff’s prediction model via viscosity() , Fu- nazukuri’s prediction model via viscosity(•), our prediction model via viscosity(N), and MD simulation data(H)...... 150
xviii CHAPTER 1. INTRODUCTION
Chapter 1
Introduction
Carbon dioxide is an abundant substance that has wide application in food, oil and chemi- cal industries. Carbon dioxide is cheap, non-flammable and easy to transform from gas to liquid at room temperature and pressure of about 60 bar. Examples of its utility include: life jackets containing canisters of pressured carbon dioxide for quick inflation; high concentra- tions are used to kill pests; rapid vaporization of liquid carbon dioxide is used for blasting in coal mines; and carbon dioxide is injected to or adjacent to oil wells for enhanced oil recovery. In recent times excess carbon dioxide in the atmosphere has been attributed as a cause of global warming (Magnus et al., 2011; Zevenhoven and Beyene, 2011; Gruber, 2011; Marble et al., 2011), which is now widely regarded as the biggest global-scale issue facing human beings. The issue of anthropogenic global warming leads us to the question of what, if anything, we can do to combat it. The answer is to reduce our emissions of greenhouse gases. While the answer is simple there is a significant challenge involved in carrying out such reductions. There are three main alternatives to reducing our carbon dioxide emissions without hampering economic growth. One is to use energy more efficiently, thereby reducing en- ergy consumption. The second option is to change the consumption of renewable energy
1 CHAPTER 1. INTRODUCTION
and the final option is to burn fossil fuels while capturing and storing the CO2 instead of releasing it into the atmosphere. Storing carbon dioxide under the sea-bed is a possi- ble technology which could help to reduce global warming. The solutions would involve pumping the gas miles underground and inject it under the sea floor. There is enough space for almost unlimited carbon emissions, as reported by Harrion et al. (1995). But there are also some concerns. Previous plans to store carbon under the sea have drawn criticism because of concerns over leakage and safety.
Understanding the diffusion coefficients and other thermodynamic properties of CO2 in deep sea is essential for developing the sequestration process of CO2 into the deep ocean and assessing its feasibility. It is very difficult to perform experiment measurements due to huge cost on this project. In contrast, molecular simulation can be used to obtain the relevant data cheaply and efficiently. There has been intensive research for the ternary system of water + carbon dioxide + sodium chloride system. However, most work has focused on the phase equilibrium of the ternary system (Sabil et al., 2009; Seo and Lee, 2003; Shmulovich and Plyasunova, 1993; Baseri et al., 2009; Duan et al., 2006a; Dubessy et al., 2005; Lee et al., 2002) and the solubility of CO2 in the ternary system (Marin and Patroescu, 2006; Duan et al., 2003; Kiepe et al., 2002; Shibue, 1996; Botcharnikov et al., 2007; Kamps et al., 2006; Duan et al., 2006a,b; Bando et al., 2003; Lee et al., 2002). There are few discussions covering the dynamic properties and structural properties of the ternary system. Molecular simulation (Metropolis et al., 1953; Alder and Wainwright, 1958) has been widely used for the study of the structural and physical properties of electrolyte solutions (Calero et al., 2011; Druchok and Holovko, 2011; Marcus, 2010; Mirzoev and Lyubartsev, 2011; Molina et al., 2011). Molecular simulation, however, has not yet been used to study the diffusion coefficient and structural properties of CO2 in electrolyte solution. The main aim of this work is to investigate the diffusion coefficients and other ther- modynamic properties of CO2 in aqueous electrolyte solutions. Chapter 4 introduces the
2 CHAPTER 1. INTRODUCTION
molecular dynamics simulation of the diffusion coefficient and other properties of CO2 in water. A series of CO2 molecular models were used such as EPM2 (Harris and Yung, 1995), Duan’s model (Zhang and Duan, 2005) and the Gromos model (Gunsteren and Berendsen,
1987) to simulate the diffusion coefficient of CO2 in water system. The results were then compared with experimental data to find the most suitable CO2 potential model. Chapter 5 discusses molecular dynamics simulation for the physical properties of NaCl in water at various temperatures and pressures, especially at low temperatures and high pressures which are similar to the environment of CO2 injection. All molecular model parameters are obtained from optimized experimental data. The structures of various atoms in the sys- tem have been investigated in these ranges of temperatures and pressures. The simulation data obtained here is proved in good accord with these experimental data. The diffusion coefficients of the various molecules in the NaCl aqueous solution are also studied at differ- ent temperatures and pressures. The temperature dependence and pressure dependence of the self-diffusion coefficients of Na+ and Cl− were also determined. Chapter 6 discusses the structural properties and diffusion coefficients of the H2O+NaCl+CO2 ternary system with respect of NaCl concentration, temperature dependence and pressure dependence.
Chapter 7 examines approaches to predict the diffusion coefficients of CO2 in elec- trolyte solutions. One is the approach proposed by Ratcliff and Holdcraft (1963) based on Activation Theory and a perturbation model. At that time they developed two equa- tions based on activation theory and perturbation model, namely, linear and exponential model. They prefered the linear equation over exponential equation due to the limiting ex- perimental data. However, we demonstrate that the exponential equation is more suitable for predicting diffusion coefficient with the help of simulation data. The other approach is based on the relation between diffusion coefficient and viscosity, whereby the diffusion coefficient of the gas in electrolyte solution is derived from a given viscosity. The results from the two models agree with the molecular simulation data in Chapter 6. Chapter 8 summarizes conclusion and suggestions for future research.
3 CHAPTER 2. DIFFUSION THEORIES
Chapter 2
Diffusion Theories
Molecular diffusion is caused by a chemical potential gradient, which results in the diffu- sion of species from a region of higher chemical potential to a region of lower chemical potential. However, due to the difficulty of experimentally measuring a chemical potential gradient, the diffusion coefficient is defined in terms of the concentration gradient. When a chemical potential (concentration) gradient exists for a chemical species in so- lution, Brownian motion (Einstein, 1906) of the molecules achieves a uniform chemical potential (concentration) distribution. The molecular diffusion coefficient of a chemical species in solution is a measure of its tendency to produce entropy when a chemical po- tential gradient exits for this species. The proportionality constant between the chemical potential (concentration) gradient and the molecular motion in the direction of the gradi- ent is called the molecular diffusion coefficient. If a concentration gradient exists, this relationship is given by Fick’s law (Fick, 1855).
Ji = −Di∇ci (2.1)
4 CHAPTER 2. DIFFUSION THEORIES
Where ci is the concentration of substance i and where Di is a coefficient called the diffusion coefficient of substance i. It depends on the temperature, pressure, composition, and on the identities of all substances that are present, but not on the concentration gradient. Three categories of molecular diffusion can be defined (Mortimer, 2008): interdiffu- sion, intradiffusion, and self diffusion. Interdiffusion or mutual diffusion is defined as the diffusion of a species i in a multicomponent solution. The interdiffusion coefficient is gen- erally denoted as Dijwith i6=j. It can be shown that in a solution of two species i and j, the interdiffusion coefficient of i in j has the same numerical value as the interdiffusion coef-
ficient of j in i, or Dij=Dji. The interdiffusion coefficient is the most commonly measured type of diffusion coefficient because of its relevance for mass transport calculations. The intradiffusion coefficient (Mortimer, 2008), or tracer diffusion coefficient, characterizes the diffusion of one species in a uniform multicomponent solution when a concentration gradient is created for only one species in the mixture. For example, the intradifffusion coefficient can be studied for the diffusion of labeled and unlabeled molecules of species i in a uniform multicomponent solution. Self diffusion is a special case of intradiffusion in a system that contains only the indistinguishable forms of the chemical species i. Self diffusion and intradiffusion are denoted as Diior Di. Both the interdiffusion coefficients and intradiffusion coefficients are concentration dependent. The term "self diffusion" is defined according to Mill and Lobo (1989) to cover diffusion in a pure liquid, tracer diffusion and intradiffusion in electrolyte solutions.
2.1 Diffusion Theories
2.1.1 Hydrodynamic Theory
The hydrodynamic theory (Tory, 2000; Verwoerd and Kulasiri, 2003; Kang et al., 2008; Dufreche et al., 2008; Fu et al., 2009) is based on the Nernst-Einstein equation (Einstein,
5 CHAPTER 2. DIFFUSION THEORIES
1905). It shows the relationship between the diffusion coefficient of a single particle of A through a stationary medium B:
uA kT DAB = kT = (2.2) FA f
where uA is the steady state velocity of the particle reached under the action of a force
FA, and f is the frictional coefficient of the diffusing particle. The relationship between force and velocity can be obtained for a rigid sphere from hydrodynamics accouting for slip and is given for the case of creeping flow by :
2ηB + rAβAB FA = 6πuAηBrA (2.3) 3ηB + rAβAB
2ηB + rAβAB f = 6πηrA (2.4) 3ηB + rAβAB
where ηB is the viscosity of pure B, rA is the radius of particle A and βAB is the coeffi- cient of sliding friction. Two limiting cases of the previous equation are of interest:
• If there is no slip of fluid at the interface with particle, then βAB is the infinity and we get Stokes law:
FA = 6πuAηBrA (2.5)
which leads to the Stokes-Einstein equation:
kT DAB = (2.6) 6πηBrA This approach is of interest in the case of diffusion of large spherical particles or molecules in a liquid which can be treated as a continuum.
6 CHAPTER 2. DIFFUSION THEORIES
• If there is no tendency for the fluid to stick at the interface with the particle then βAB is zero and we obtain:
FA = 4πuAηBrA (2.7)
kT DAB = . (2.8) 4πηBrA This is the so-called Stokes-Einstein formula (Einstein, 1905; Sutherland, 1905), which shows the relationship between diffusion coefficient, temperature and viscosity. This for- mula is by far the basis and standard for testing of other formulas. Many other formulas are derived from it with variation in measurement of solute molecular radius. As shown above, the diffusion coefficient is in inverse relation with viscosity. How- ever, the above relationships are only valid when the ratio of solute molecular radius to solvent molecular radius is greater than 5 (Longsworth, 1955). The formula error becomes bigger when solute molecular radius decreases. The Stokes-Einstein formula (Einstein, 1905; Sutherland, 1905) is not suitable to forecast diffusion coefficient because it takes into account the forms of resistance only, but not the interactions between atoms. In the following studies, many researchers modified the Stokes-Einstein formula to be used for more scenarios (Arkhipov, 2011; Chathoth and Samwer, 2010; Fernandez-Alonso et al., 2007; Gisladottir and Stefansson, 2009; Kooijman, 2002; May and Mausbach, 2007; Xu et al., 2009). The modifications focused on viscosity, group viscosity and solute molecular radius.
2.1.2 Activated State Theory
In this theory (Eyring, 1935), the liquid is described as a lattice in which each molecule has a position in this lattice. Only a small part of the molecules who reach the ‘activation energy’ and intermediate ‘transition state’ can move according to the statistic distribution
7 CHAPTER 2. DIFFUSION THEORIES
of the thermal energy. This theory is in varat to explain exponential dependency of the rate constants upon temperature. The diffusion coefficient consist two parts according to this theory. One is indepen- dent of temperature and the other shows an exponential dependence on temperature. The diffusion coefficient in liquids can be calculated as follows according to Erying’s theory:
λ 2 kT 1/2 E D = exp − (2.9) AB 1/3 2πm RT Vf
In this equation, Vf is molecular volume, λ is an elementary ‘jump distance’ in the order of intermolecular distance, R is gas constant and E is the diffusion activation energy. It appears that the diffusion coefficients can be derived directly from the activated state theory. The theory, however, may not be applied to the prediction from first principles because of some difficulties with using equation: there are no reliable methods to estimate distance λ and activation energy E for the equation. First, this approach has conceptual difficulties as pointed out by Tyrrell and Harris (1984), and Alder and Hilderbrand (1973). The activated state theory states that only small amount of molecules are in the activated state. The activated state theory states that only small amount of molecules are in the activated state. In the contrast, evidence shows that the potential field encountered by molecules, either activated or inactivated, is almost uniform rather than potentially different. Furthermore, there are a large proportion of molecules in the activated state at one time as the activation energies are observed as low as on the order of 10 kJ/mol. Therefore, Tyrrell and Harris (1984) suggest that little physical significance be given to the value of the observed activation energies. Second, some experimental evidence suggests that the basis for the activated state the- ory may be incorrect. Ruby et al. (1975) have found the jump distance for an iron isotope in solution with a liquid is even less than a molecular diameter. Clifford and Dickinson (1977) have conducted dynamics simulations of diffusion and confirmed this.
8 CHAPTER 2. DIFFUSION THEORIES
Several new models have been established based on Eyring’s absolute reaction rate theory although one basic assumption is not valid for the liquid state. These new models, using other physical properties as well, are intended to describe the correlation between concentration and diffusion coefficient as a function of the diffusion coefficient at infinite dilution. Fei and Bart (1998; 2001) predicted the MS-diffusion coefficient via their group contri- bution method, which was based on Eyring’s theory. The average deviation of this method is said to be 5%. They estimated the free volume and distance parameter by the diffusional areas of the composing groups of the mixture at once. They estimated the activation energy in a similar way and also determined group parameter of technically relevant systems such as sulfolane systems. Bosse and Bart (2006) based their model for the Maxwell Stefan diffusion coefficient on the Eyring’s absolute reaction rate theory. This model investigates the concentration dependence of diffusion coefficient at infinite dilution and additional excess Gibbs energy contribution. The energy part makes it possible to consider thermodynamic non idealities explicitly when modelling the transport property.
2.1.3 Free Volume Theory
Batchiniski (1913) found that in his study of a few dozen of non-associated liquids, the relationship between the viscosity and molar volume is linear. The formula is:
1 V −V = B · B η (2.10) η Vη
where B is a constant which depend on the solvent, VB is the liquid molar volume and
Vη is the hypthetical liquid molar volume at infinite viscosity.
9 CHAPTER 2. DIFFUSION THEORIES
Hildebrand (1971; 1977) applied this reasoning to diffusion coefficient and observed that both the self-diffusion coefficient and the infinite dilution coefficient may be expressed by a similar relationship:
V −V D = B0 · B D (2.11) VD
where VD is the molar volume at its melting point at which diffusion is considered to cease. Free volume theory has been used intensively in predicting the diffusivities of gases in polymers (Heuchel et al., 2004; Kucukpinar and Doruker, 2003; Lim et al., 2003; Pavel and Shanks, 2003; Shanks and Pavel, 2002; Kwag et al., 2001; Tanaka et al., 2000; Thran et al., 1999; Barbari, 1997; Sha and Harrison, 1992; Terada et al., 1992; Vieth et al., 1991; Bennun and Levine, 1995; Sato et al., 2001)
2.1.4 Kinetic Theory
For simplifying the kinetic theory of diffusion coefficient, molecules in this theory are treated as hard spheres moving around and the molecular collisions, which occur at low density, are treated as bimolecular. The kinetic theories (Clausius, 1857) are proved suc- cessful in explaining the behavior of gases. The kinetic theory of gases treats gas as a large number of small particles in the forms of atoms or molecules in constant and random motion. The particles move rapidly and collide with each other and the container wall fre- quently. Kinetic theory explains macroscopic properties of gases (pressure, temperature or volume) in terms of their molecular composition and motion. An assumption of the theory is that pressure is due to collisions between molecules traveling at different velocities rather than due to static repulsion between molecules. The pollen grains or dust particles making up a gas can be seen vibrating rapidly under microscope although they are too tiny to be
10 CHAPTER 2. DIFFUSION THEORIES
visible for human eyes. The jittering motion, also known as Brownian motion, is caused by the collisions between gas molecules and the particle. Kinetic theory is most widely used for the perdition of gas diffusion coefficient. Orig- inated from kinetic theory of gases, the kinetic theory of liquids is proposed for increased densities and molecular interactions. The molecular interaction of non-polar gas in the dense gas system has been modeled successfully by the Enskog-Throne diffusion coeffi- cient (Chapman and Cowling, 1970).
r E 3 m1 + m2 kT 1 D12 = 2 (2.12) 8n2σ12 m1m2 2π g(σ12)
where n2is the number density, σ12 is the average hard sphere diameter of the molecules, m1, m2 are the molecular masses and g(σ12) is the radial distribution function. Although kinetic theory was first used for prediction of diffusion coefficients of gas, it has been for many other areas. Davis (1987) used the Enskog’s kinetic theory of dense hard sphere fluids and modified it allow long-ranged attractive interactions in a mean field sense to derive the tracer diffusion of the inhomogeneous fluid. Adland and Mikkelsen (2003) used kinetic theory to approach the diffusion of two-segmented macromolecules with a ball-socket joint.
2.2 Diffusion Theories of Electrolyte Solutions
Solution diffusion coefficient reflects the characteristic of mass transport process in the solution. The diffusion properties have been studied over hundred years since Fick’s law. Most existing diffusion theories (Clausius, 1857; Einstein, 1905; Tory, 2000; Verwoerd and Kulasiri, 2003; Kang et al., 2008; Bosse and Bart, 2006) are for non-electrolyte system, whereas diffusion in electrolyte solutions is less widely studied. The difference comparing to non-electrolyte solution is electrolyte solution diffusion must consider the electrophoretic effect or the relaxation effect in electrolyte solutions, and
11 CHAPTER 2. DIFFUSION THEORIES
ionic association and the like complex problems. Concentration of the other ions impacts the mobility and friction coefficient of a given ion. There are three major effects (Mortimer, 2008) : relaxation effect, electrophoretic effect and solvation effect. The relaxation effect is due to change of the ion atmosphere of an ion when it moves. “Ion atmosphere” is the excess charge of the opposite sign surrounding an ion. When an ion moves, the ion atmosphere changes and relaxes to become centred on the new position of the ion. The motion of the ion is also slowed down. From a given ion, ions of the opposite charge move in the opposite direction and pull on the given ion in the direction of their motion. This electrophoretic effect also slows down ion motion. The solvation effect occurs because ions have to compete with each other to attract solvent molecules at high concentrations and they can attract full complement of solvent molecules at low concentrations. Some solvent molecules are strongly attracted to it and can move with an ion. As a result, the mobility of an ion is affected by any change in the solvation. At low concentrations, electrophoretic effect and relaxation effect disappear and solvation effect becomes concentration free. As a result, ion mobilities and friction coefficients become constant values in the limit of infinite dilution. The earlier theory of electrolyte solution diffusion was established based on Debye- Hückel’s theory (Debye and Huckel, 1923) , i.e,. Nernst-Hartley Equation which becomes Nernst Equation in case of infinite dilution. From then on, Fuoss and Onsager studied the electrophoretic effect in diffusion, and got Fuoss-Onsager equation (Fuoss and Onsager, 1957). Robinson and Stokes (1965) got diffusion equation suitable for concentrated solu- tion. In recent years, theories (Anderko and Lencka, 1997, 1998; Anderko et al., 2002) of diffusion coefficient based on the MSA (Bernard et al., 1992b) has been developed; mean- while, investigations of molecule simulations (Miyata et al., 2002; Shi et al., 2004, 2005) are in process.
12 CHAPTER 2. DIFFUSION THEORIES
Figure 2.1: Diagram showing the asymmetric effect at the electrolyte solutions which shows the ions are tended to move in the opposite direction and pull on the given ion in the direction of their motion and slow down the motion of the ion
(a) (b)
Figure 2.2: Diagram showing the ionic atmosphere effect at the electrolyte solutions which can slow down the motion of the ion
13 CHAPTER 2. DIFFUSION THEORIES
2.2.1 Basic Equations of Diffusion in Solution
If the concentration in solution is homogeneous, then molecular in the solution only take irregular, random Brownian motion, not entire motion. But if the solute has a concentration gradient in solution, then the solute molecules will move from the high concentration region to the low concentration region. This is diffusion. Although solute molecules move from the low concentration region to the high concentration region at the same time, the number is less than the former; therefore the general result is the net move of solute molecules from high concentration region to low concentration region. Since the solvent concentra- tion is lower in high concentration region, solute molecules diffusion accompanies solvent molecule’s motion in reverse direction, i.e., motion from low solute concentration region to high solute concentration region. The two processes will proceed until the concentration is even. If traces of certain ion diffuse in a large amount of supporting electrolyte solution, and the latter’s concentration remains unchanged, it is self-diffusion. The typical example is the diffusion of trace radioisotope ion in the stable isotope ionic salt. Apparently the concentration of the latter is much greater than the former. The equation of liquid diffusion is Fick’s First Law and Second Law. Fick’s First Law indicated flux of substance is proportional with its concentration gradient, we have introduced it in the Eq.(2.1). Fick’s First Law is only suitable for diffusion in steady state, i.e., concentration gradient does not change with time. If concentration c and concentration gradient change with time, then Fick’ Second Law applies.
∂c ∂ ∂c A = D A (2.13) ∂t ∂x ∂x
∂cA ∂cA If it is steady diffusion, i.e. ∂t = 0, then D ∂x is constant, last equation reduces to Eq.(2.1).
14 CHAPTER 2. DIFFUSION THEORIES
2.2.2 Diffusion for Single Electrolyte —Nernst-Hartley Equation
From thermodynamic view, the actual reason of diffusion caused by concentration gradient is the existence of chemical potential gradient, which is the same for electrolyte and non- electrolyte. This view was first proposed by Hartley (1931). The difference of electrolyte diffusion and non-electrolyte diffusion is that electrolyte solution keeps electric neutral- ity; and the difference between electrolyte diffusion and conductance is that positive and negative ions are driven by electric field to move in reverse directions during conductance; while positive and negative ions move in the same direction during diffusion. In a single electrolyte solution, in order to keep electric neutrality, positive and negative ions diffuse at the same velocity. Mobility of positive and negative ions are different, which shows that dragging forces on the ions are different in the same electric field, since the dragging forces are different, then the speeds of positive and negative ions driven by the same concentration gradient are different. Therefore, it seems to cause the solution electric neutrality’s viola- tion. The explanation of this apparent difficulty is that a local electric field is generated due to different speeds of positive and negative ions. The local electric field slows down the fast ions while accelerating slow ions, and the positive and negative ions move eventually at the same velocity. In a single electrolyte solution, there are only one kind of positive ion and one kind of negative ion. If an electrolyte molecule decomposes as v1 of positive ions and v2 of negative ions, their charges are z1 and z2 respectively, the electrolyte chemical potential uB equals:
µB = v1µ1 + v2µ2 (2.14)
The force generated by chemical potential gradient on a single ion is − 1 ∂ µ1 and NA ∂x − 1 ∂ µ2 , in which N is Avogadro constant and the minus sign stands for the motion NA ∂x A in the direction with chemical potential decrease. The forces, driven by electric field that is
15 CHAPTER 2. DIFFUSION THEORIES
caused by differences of positive and negative charges electric mobility, acting on positive and negative charges, are z1eE, and z2eE, where e is the proton charge and E is electric field strength. This means the forces on positive and negative ions are :
1 ∂ µ1 f1 = − + z1eE (2.15) NA ∂x and
1 ∂ µ2 f2 = − + z2eE (2.16) NA ∂x The ion’s absolute mobility u0 is the velocity of ion’s motion driven by unit force, so the ion’s velocity is υ = f u0. As earlier expression, velocity of positive and negative ions 0 0 is the same, then υ = f1u1 = f2u2, so :
0 1 ∂ µ1 0 1 ∂ µ2 υ = u1 − + z1eE = u2 − + z2eE (2.17) NA ∂x NA ∂x
The whole solution is electrically neutral, which means v1z1 = −v2z2, put this in the Eq.(2.17):
υ 1 ∂ µ1 υ 1 ∂ µ2 v1 0 + = −v2 0 + (2.18) u1 NA ∂x u2 NA ∂x Put Eq.(2.14) and Eq.(2.18) together :
0 0 1 u1u2 ∂ µB υ = − 0 0 (2.19) NA v1u2 + v2u1 ∂x Since J = cυ,
0 0 u1u2 c ∂ µB ∂c J = − 0 0 (2.20) v1u2 + v2u1 NA ∂c ∂x Now we put Eq(2.20) into Eq.(2.1):
16 CHAPTER 2. DIFFUSION THEORIES
0 0 u1u2 1 ∂ µB D = 0 0 (2.21) v1u2 + v2u1 NA ∂ lnc Combing electrolyte solutin theories of activity and chemical potential with the above equation, we get so-called Nernst-Hartley Equation (Robinson and Stokes, 1965) :
(v + v )l0l0RT ∂ lny D = 1 2 1 2 1 + (2.22) 0 0 2 v1 | z1 | l1 + l2 F ∂ lnc o where l is the ionic limiting molar conductivity, | z1 |is the absolute value of the charge, and F is the Faraday constant.
∂ lny In the limit at infinite dilution, ∂ lnc = 0, the Nernst-Hartley Equation can be simplied to Nernst Equation:
(v + v )l0l0RT D = 1 2 1 2 (2.23) 0 0 2 v1 | z1 | l1 + l2 F During the derivation of the Nernst-Hartley Equation, solvent molecule motion has not been considered. It also neglects the solution viscosity, and interaction between ion and wa- ter molecules and like interactions. Onsager and Fuoss (1932) considered electrophorestic effect, and hence improved Nernst-Hartley Equation as well as better result.
2.2.3 Electrophoretic Effect in Diffusion—Onsager-Fuoss Equation
During conductance process, the interaction between ions generates two effects, i.e. relax- ation effect and electrophoretic effect. In single electrolyte diffusion, positive and negative charges move at the same velocity and symmetry of ion atmosphere has not changed, there- fore there is no relaxation effect but electrophoretic effect. It is because ions’ motion needs to pass through solvent and the motion direction of solvent molecule and that of ions are reverse. Electrophoretic effect has correlation with electrolyte concentration. Onsager and
17 CHAPTER 2. DIFFUSION THEORIES
Fuoss (1932) have done some research and introduced it into diffusion coefficient calcula- tions. Since the electrophoretic effect is correlated with concentration, it should take correc- tion on ionic mobility, i.e. modifying the ionic limiting conductivity in Eq.(2.22) to the following equation so it can be seen that expressions of symmetric electrolyte diffusion co- efficient and asymmetric electrolyte diffusion coefficient are different. Onsager and Fuoss (1932) recommended applying Eq.(2.24) to any situation. This equation is called Onsager- Fuoss Equation (Onsager and Fuoss, 1932) .
∂ lny D = D0 + 4 + 4 1 + (2.24) 1 2 ∂ lnc
0 where D is Nesrnst limiting diffusion coefficient, 41 and 42 are :
kT 2 κ 4 = − t0 −t0 (2.25) 1 6πη 2 1 1 + κa
| z |2 ε2 ∆2 = 2 φ2 (κa) (2.26) 12πηDea
where De is a dielectric constant of the medium, a is the closest distance between ions, e is the elementary charge, κ is the reciprocal of the thickness of the ionic atmosphere according to the Debye-Hückel theory, and
2 Z ∞ −2κr 2 eκa e φ2 (κa) = (κa) dr (2.27) 1 + κa a r The the background of this equation is related to the Onsager’s theory of conductance.
18 CHAPTER 2. DIFFUSION THEORIES
2.2.4 Diffusion for Partly Ionized Electrolytes
Ionic association has two effects on electrolyte diffusion. Firstly, solute activity decreases due to the association, so does chemical potential gradient which causes diffusion coeffi- cient decreasing; secondly, dragging force on a particle is less than that on two, so it will cause diffusion coefficient increasing.
Let α be the degree of dissociation, u1, u2, u12 be the mobility of the positive ion, 0 negative ion, and the ion pair, D12 be the diffusion coefficient of the one ion pair in the infinite dilute solution, Nernst-Hartley equation becomes:
∂ lny D = αD0 + 2(1 − α)D0 1 + c (2.28) 12 ∂ lnc
If we include the electrophoretic effect ∆1and 42, then it becomes :
∂ lny D = α D0 + ∆ + ∆ + 2(1 − α)D0 1 + c (2.29) 1 2 12 ∂ lnc
2.2.5 Self-Diffusion in Electrolyte Solutions—Onsager Limiting Law
In pure fluids, molecules continuously take random motion (Einstein, 1905). Molecules at a certain point have certain possibility to move to the other point after certain time, which is the original meaning of self-diffusion. In normal circumstances, the diffusion cannot be measured because the molecules are chemically indistinguishable. This self-diffusion can be measured by radioisotope tracer technology. For example, the self diffusion of Na+in NaCl solution can be measured by adding radioactive Na+and measuring it. During the process mentioned above, solvent motion will not be caused by tracer ions due to its extremely small concentration, so the electrolyte effect can be ignored. Mean- while, since supporting electrolyte is even and tracer ions move in unchanged ionic envi- ronment, then tracer ion’s activity coefficient is unchanged, i.e. lny/lnx = 0 . Since tracer ions which are not controlled by ions carrying opposite charges move in non-diffused ionic
19 CHAPTER 2. DIFFUSION THEORIES
environment and its ion atmosphere is asymmetric (this is different from sole electrolyte diffusion), then relaxation effect should be considered. Onsager (1926) calculated diffusion of trace ions j in an even electrolyte solution. He obtained the diffusion coefficient of ion j :
" z2 2 # ? jε κ q Di = kTu j 1 − 1 − d u j (2.30) 3DekT where d u j is a function of charges and transfer numbers of ions in the solution. Putting the physical constants inside the Eq.(2.30), we get the Onsager limiting equation of the self-diffusion coefficient.
" # 2.81 × 106 q √ D? = D?0 1 − 1 − d u z2 I (2.31) i i 3/2 j j (DeT)
2.2.6 Self-Diffusion in Multicomponent Aqueous Electrolyte Systems in Wide Concentration Ranges
The early electrolyte diffusion theories(Onsager, 1926; Onsager and Fuoss, 1932; Hartley, 1931; Einstein, 1905) mentioned above constitute the classic theory of electrolyte solution diffusion. In recent years, considerable progress has been achieved in development of statistical mechanics and theory of electrolyte solution diffusion. Bernard, Turq, Blum and coworkers (Bernard et al., 1992a; Chhih et al., 1994; Bernard et al., 1997; Turq et al., 1992) systematically studied the diffusion coefficient of electrolyte solution while studying the conductance properties of electrolyte solution using integral function theory. Bernard and coworkers combined Continuous Equation and mean-spherical approxi- mation (MSA) Equilibrium Correlation Function, adopting primitive model to investigate transport property of electrolyte solution. But for his model, for alkali chloride solution, the concentration range that Bernard’s equation can apply is 0-1mol/l.
20 CHAPTER 2. DIFFUSION THEORIES
Andeko and his colleagues (Anderko and Lencka, 1997, 1998; Anderko et al., 2002) de- veloped a comprehensive model to compute self-diffusion coefficients in multi-component aqueous electrolyte system, which combines contributions of short-range and long-range (Coulombic) interaction. The combined model characterizes aqueous species using effec- tive radii, which depends on the ionic environment. The short-range interactions are ex- pressed by the hand-sphere model. The long-term interaction contribution, which demon- strates itself in the relaxation effect, is derived from the dielectric continuum-based MSA theory for the unrestricted primitive model. Based on phenomenological equations of nonequilibrium thermodynamics, a mixing rule has been generated for multicomponent systems. The diffusion coefficient model and thermodynamic speciation calculation are combined to address the effects of complexation. The model precisely regenerates self- diffusivities of ions and neutral species in a wide range of aqueous solutions from infinitely dilute to concentration up to 30 mol/kg of H2O. The model can also forecast diffusivities in multi-component solutions based on the data for singe-solute system.
0 Di = Di (1 + δki/ki) (2.32)
0 HS 0 Di = Di (Di /Di )(1 + δki/ki) (2.33)
To calculate the relaxation term in Eq.(2.33), Anderko used the expressions developed by Bernard et al. (1992a) and Chhih et al. (1994) for a tracer ion in an electrolyte containing one cation and one anion at infinite concentrations. The relaxation term for a tracer ion i is given by
δk 1 z2e2(κ2 − κ2 ) 1 − exp(−2κ σ) i = i di × di (2.34) 2 2 2 ki 4πε0ε 6kBTσ(1 + Γσ) κdi + 2Γκdi + 2Γ [1 − exp(−κdiσ)]
21 CHAPTER 2. DIFFUSION THEORIES
where zi is the ionic charge, e is the charge of the electron, ε0 is the permittivity of vacuum, kB is the Boltzmann constant, and ε is the dielectric constant of pure water. σ is the average ion diameter and defined by
2 2 ∑ z jρ jσ j j=1 σ = (2.35) 2 2 ∑ z jρ j j=1
where ρ j and σ j are the number density and diameter of the jth ion, respectively. The parameters κ and kdi are given by
2 2 2 e 2 κ = ∑ ρ jz j (2.36) ε0εkBT J=1
2 2 z2D0 2 e ρ j j j kdi = ∑ 0 0 (2.37) ε0εkBT j=1 Di + D j and Γ is the MSA screening parameter, calculated in the mean diameter approximation as
κ Γ = (2.38) 2(1 + Γσ) According to Eq. (2.34)-(2.38), the relaxation term can be computed if the ion diame- ters are known. Also, the density of the solution has to be known in order to calculate the number densities of ions. The hard-sphere term is calculated from expression developed by Tham and Gubbins (1972). Details are as follows. Normally, the diffusion coefficient in a hard-sphere system can be expressed as
HS Di = Di,ENS AiCi(ρ,M,σ, χ) (2.39)
22 CHAPTER 2. DIFFUSION THEORIES
where Di,ENS is the diffusion coefficient calculated from Enskog of smooth hard spheres
(Chapman and Cowling, 1970), Ai accounts for translation-rotation coupling resulting from deviations of molecular surfaces from sphericity, and Ci(ρ,M,σ, χ) is an empirical correc- tion factor that compensates for the neglection by the Enskog theory of correlated motions in hard sphere fluids. The diffusion coefficent of species i at infinite dilution :
0 0 0 0 Di = Di,ENS AiCi (ρ ,M,σ, χ) (2.40)
Ai is independent of solution composition and density, thus
D C(ρ,M,σ, χ) DHS = D0 i,ENS (2.41) i i 0 0 0 Di,ENS C (ρ ,M,σ) According to Tham and Gubbins (1972), the diffusion coefficient of a tracer ion i in a solution containing a cation j, and anion k, and a solvent s, the expression is :
−1 gi j gik gis Di,ENS = x j + xk + x j (2.42) di j dik dis
where gi j is the radial distribution function at contact for rigid spheres of diameters σi and σ j and di j is the dilute gas diffusion coefficient for a mixture of molecules i and j. The di j here is given by :
" #1/2 3 Mi + Mj RT di j = 2 (2.43) 8σi jρ 2πMiMj where ρis the number density and the average diameter σi j = σi + σ j /2. The raidal distribution function is calculated from the equation of Boublik (1970):
2 2 1 3σiσ j ζ2 σiσ j ζ2 gi j σi j = + 2 + 2 3 (2.44) 1 − ζ3 σi + σ j (1 − ζ3) σi + σ j (1 − ζ3)
23 CHAPTER 2. DIFFUSION THEORIES
where π l ζ1 = ∑xKσK. (2.45) 6 k Since no analytical theory is available for the evaluation of the C(ρ,M,σ, χ), Anderko makes an assumption that C(ρ,M,σ,χ) = 1. C0(ρ0,M,σ) Thus, Eq.(2.41) becomes:
g0 HS is D d0 i = is (2.46) 0 gi j gik gis D x j + x + x j i di j k dik dis
2.3 Summary
As mentioned above, there have been various theoretical models of the diffusion process in liquids, but none of them has proved completely successful or accurate for the prediction of diffusion coefficient in liquids. One reason is due to the strong interactions among molecules and the other reason is that all approaches are limited to a class of solvent and solute systems. It is difficult to evaluate the formulas derived from various diffusion theories and devi- ation range because: First, a theory is normally used in conjunction with others rather than alone. For in- stance, the UNIDIF model is developed by combining the Eyring’s theory and statistical thermodynamic by Hsu and Chen (1998) to compute the Fick diffusion coefficients directly. Sek (1996) combined the Basset equation with the Blake-Kozeny-Carman formula to ob- tain an expression for the force in the Nerst-Einstein expression. Woerlee model (2001) started from the kinetic theory of gas and the Eyring theory for liquids. Second, these theories can be perceived as phenomenological formulas, where the pa- rameters will change with the environments (gas or liquid) and solutions. An equation may
24 CHAPTER 2. DIFFUSION THEORIES
have small error in one system, but big error in another system. The empirical equation de- rived from experimental data can only be applied to the condition under which it is derived. For example, in the equation derived by He and Yu (1998), the deviation is reported to be 8% after obtaining 1300 data for 11 liquids in high temperature and above-critical liquid in the range of 0.66iodine in CO2 are -20% and -38% respectively.
This thesis focuses on the physical properties of CO2 in NaCl electrolyte solution. As a diffusion equation delivers different results for different systems, varying from very good to very bad when compared to the experimental data. We aim to find a dedicated diffu- sion equation or CO2 in NaCl electrolyte solution system. Ratcliff and Holdcroft (1963) obtained the diffusion coefficients of CO2 in NaCl electrolyte solution from experiments and also constructed two prediction equations for gases in electrolyte solutions. One is developed from activated state theory and perturbation theory and the other is for diffusion coefficient prediction from viscosity. The prediction results from the two equations are in good agreement with experimental data. As the same his experimental data, the prediction data are restricted to low concentration system. These two equations, however, provide a good start for correlation and construction of new equation for a greater concentration range. The physical properties of CO2 in NaCl electrolyte solution will be studied via molecular dynamics simulation method in this work. Many data, which are difficult to ob- tain from experiments, can be obtained from molecular dynamics simulation. Thus we can get diffusion coefficient data in a wide range of electrolyte solution concentrations, which will be the basis for correlation of Ratcliff’s diffusion equations.
25 CHAPTER 3. MOLECULAR SIMULATION
Chapter 3
Molecular Simulation
To develop the sequestration process of CO2 into the deep ocean and its feasibility, we have to understand the diffusion and other thermodynamic properties of CO2 at deep sea conditions. The huge cost on this project makes it very difficult to do experiment research. Molecular simulation, however, may solve this problem because it can be used to obtain relative results at any specified physical conditions. In fact molecular simulation is a kind of “experiments” to study the phenomena occurring in nature. Molecular simulation, regarded as computational statistics mechanics, makes it possible for us to accurately evaluate the properties of molecules as described by statistical mechanics.
3.1 Introduction
Molecular simulation, a generic term, encompasses both Monte Carlo and molecular dy- namics computing methods. Molecular simulation, unlike approximate solutions, was first developed to obtain exact results for statistical mechanical problems. The biggest advan- tage of molecular simulation in preference to other computing methods and approximations is that the molecular coordinates of the system are evolved according to a rigorous calcula- tion of intermolecular energies of forces. Regarded as computational statistical mechanics,
26 CHAPTER 3. MOLECULAR SIMULATION
molecular simulation can assist us in determining macroscopic properties by evaluating exactly a theoretical model of molecular behaviour through a computer program. The ac- curacy of the model can be tested by comparing simulation results with experimental data as the nature of the theoretical model used solely determined the results of a molecular simulation. The discrepancies between accurate experimental measurements and molec- ular simulation data can be unambiguously caused by the failure of the model to present molecular behaviour. The earliest non-quantum calculation method applying to a huge system is the Monte Carlo (MC) method (Frenkel and Smit, 2001). The MC method utilizes point particles ran- dom motion combining probability distribution principle of static mechanics to obtain sys- tem statistics and thermodynamics materials. Metropolis et al. (1953) introduced the Monte Carlo method and performed the first molecular simulation of a liquid on the MANIC com- puter at Los Almaos. During a MC simulation, different trial configurations are occasion- ally generated. The intermolecular interactions in the trial configuration are evaluated and probabilities are used to accept or reject the change. Later Alder and Wainwright (1958) introduced the molecular dynamics method, which solves the equations of motion for the system of molecules. Since the 1970, because of fast development of molecular dynamics, many force fields that are suitable for biochemical molecular system, polymers, metallic and nonferrous ma- terials have been systematically established, which largely improve the accuracy and ability of calculating complex systems’ structures and some thermodynamic and spectrum’s prop- erties. Molecular dynamics simulation (MD) is the calculation method that is developed based on those force fields and Newton’s mechanics principles (Frenkel and Smit, 2001). It has big advantages because particle motion in system has the correct physics basis and accuracy is high, besides the system’s dynamics and thermodynamics statistic properties can be obtained at the same time. It also can be widely applied to various systems and
27 CHAPTER 3. MOLECULAR SIMULATION
characteristics. Now MD’s calculation technique has become mature through many im- provements. Molecular dynamics differs from MC mainly in two aspects. First, as the name implies, molecular dynamics are mostly used to obtain the dynamics properties of the system al- though Monte Carlo simulation can sometimes also be used to obtain dynamic properties. That is, the molecular coordinates and momenta change according to the intermolecular forces experienced by the individual molecules. Unlike MC simulations, MD is completely deterministic and chance plays no role. The other important distinction is that MD uses in- termolecular forces to evolve the system whereas MC simulation involves primarily the calculation of changes in intermolecular energy.
3.2 Molecular Dynamics
Molecular dynamics deals with the equations of motion of the molecules to generate new configurations. As a result, MD simulation can be used to obtain time-dependent properties of the system. As such it is ideally suited for diffusion properties and it forms the basis of calculations in this thesis. Molecular simulation is applied in the following steps: First, we choose a model system, which includes N particles. Then we solve Newton’s equation of motion for this system. After the properties of the system no longer change with the time, we perform the actual measurement. Prior to the measurement of an observable quantity in a molecular dynamics simulation, this observable has to be expressed as a function of the position and momenta of the particles in the system. To study the molecular dynamic simulation of a system, we have to know all the molec- ular models of the system, interactions between the molecules, equation of motion, periodic boundary conditions, integration algorithm and ensemble process. The following details the method, system, model and algorithm used in our simulation.
28 CHAPTER 3. MOLECULAR SIMULATION
3.2.1 Force Field
Any thermodynamic average requires us to determine the kinetic and potential energies of the system. To derive the potential energy or the forcing acting between molecules, either in a Monte Carlo or molecular dynamics simulation, we normally calculate it from a pair wise additive intermolecular potential. To calculate the potential between particles, we need a proper potential function. In general, the potential energy of a system of N particles can be expressed as: