molecular dynamics simulation of carbon dioxide in aqueous ... · i hereby declare that the thesis...

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

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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


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7.3 Predicting Diffusion Coefficient via Viscosity . . . . . . . . . . . . . . . . 139

7.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139


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


the EPM2 potential (•) and experimental data (�) at different temperatures

at a press of 1 atm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60


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

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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


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

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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


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


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


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


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 i 6=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


1905). It shows the relationship between the diffusion coefficient of a single particle of A

through a stationary medium B:

DAB = kTuA

FA=

kTf

(2.2)

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 :

FA = 6πuAηBrA

[2ηB + rAβAB

3ηB + rAβAB

](2.3)

f = 6πηrA

[2ηB + rAβAB

3ηB + rAβAB

](2.4)

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:

DAB =kT

6πηBrA(2.6)

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


• 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)

DAB =kT

4πηBrA. (2.8)

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


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:

DAB =λ 2

V 1/3f

(kT

2πm

)1/2

exp[− E

RT

](2.9)

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


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:

1η

= B ·VB−Vη

Vη

(2.10)

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


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:

D = B′ · VB−VD

VD(2.11)

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


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).

DE12 =

38n2σ2

12

√m1 +m2

m1m2

kT2π�

1g(σ12)

(2.12)

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


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


Figure 2.1: Diagram showing the asymmetric effect at the electrolyte solutions whichshows the ions are tended to move in the opposite direction and pull on the given ionin 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 whichcan slow down the motion of the ion

13


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.

∂cA

∂ t=

∂

∂x

(D

∂cA

∂x

)(2.13)

If it is steady diffusion, i.e. ∂cA∂ t = 0, then D∂cA

∂x is constant, last equation reduces to

Eq.(2.1).

14


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 − 1NA

∂ µ1∂x and

− 1NA

∂ µ2∂x , in which NA is Avogadro constant and the minus sign stands for the motion

in the direction with chemical potential decrease. The forces, driven by electric field that is

15


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 :

f1 =−1

NA

∂ µ1

∂x+ z1eE (2.15)

and

f2 =−1

NA

∂ µ2

∂x+ z2eE (2.16)

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

is the same, then υ = f1u01 = f2u0

2, so :

υ = u01

(− 1

NA

∂ µ1

∂x+ z1eE

)= u0

2

(− 1

NA

∂ µ2

∂x+ z2eE

)(2.17)

The whole solution is electrically neutral, which means v1z1 = −v2z2, put this in the

Eq.(2.17):

v1

(υ

u01+

1NA

∂ µ1

∂x

)=−v2

(υ

u02+

1NA

∂ µ2

∂x

)(2.18)

Put Eq.(2.14) and Eq.(2.18) together :

υ =− 1NA

u01u0

2

v1u02 + v2u0

1

∂ µB

∂x(2.19)

Since J = cυ ,

J =−u0

1u02

v1u02 + v2u0

1

cNA

∂ µB

∂c∂c∂x

(2.20)

Now we put Eq(2.20) into Eq.(2.1):

16


D =u0

1u02

v1u02 + v2u0

1

1NA

∂ µB

∂ lnc(2.21)

Combing electrolyte solutin theories of activity and chemical potential with the above

equation, we get so-called Nernst-Hartley Equation (Robinson and Stokes, 1965) :

D =(v1 + v2) l0

1 l02RT

v1 | z1 |(l01 + l0

2)

F2

(1+

∂ lny±∂ lnc

)(2.22)

where lo is the ionic limiting molar conductivity, | z1 |is the absolute value of the charge,

and F is the Faraday constant.

In the limit at infinite dilution, ∂ lny±∂ lnc = 0, the Nernst-Hartley Equation can be simplied

to Nernst Equation:

D =(v1 + v2) l0

1 l02RT

v1 | z1 |(l01 + l0

2)

F2(2.23)

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


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) .

D =(D0 +41 +42

)(1+

∂ lny±∂ lnc

)(2.24)

where D0 is Nesrnst limiting diffusion coefficient,41 and42 are :

41 =−kT

6πη

(t02 − t0

1)2 κ

1+κa(2.25)

∆2 =| z |2 ε2

12πηDea2 φ2 (κa) (2.26)

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

φ2 (κa) = (κa)2(

eκa

1+κa

)2 ∫ ∞

a

e−2κr

rdr (2.27)

The the background of this equation is related to the Onsager’s theory of conductance.

18


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,

negative ion, and the ion pair, D012 be the diffusion coefficient of the one ion pair in the

infinite dilute solution, Nernst-Hartley equation becomes:

D =[αD0 +2(1−α)D0

12](

1+ c∂ lny±∂ lnc

)(2.28)

If we include the electrophoretic effect ∆1and42, then it becomes :

D =[α(D0 +∆1 +∆2

)+2(1−α)D0

12](

1+ c∂ lny±∂ lnc

)(2.29)

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


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 :

D?i = kTu j

[1−

z2jε

2κ

3DekT

(1−√

d(u j))]

(2.30)

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.

D?i = D?0

i

[1− 2.81×106

(DeT )3/2

(1−√

d(u j))

z2j

√I

](2.31)

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


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.

Di = D0i (1+δki/ki) (2.32)

Di = D0i (D

HSi /D0

i )(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

δki

ki=

14πε0ε

z2i e2(κ2−κ2

di)

6kBT σ(1+Γσ)2 ×1− exp(−2κdiσ)

κ2di +2Γκdi +2Γ2[1− exp(−κdiσ)]

(2.34)

21


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

σ =

2

∑j=1

z2jρ jσ j

2

∑j=1

z2jρ j

(2.35)

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

κ2 =

e2

ε0εkBT

2

∑J=1

ρ jz2j (2.36)

k2di =

e2

ε0εkBT

2

∑j=1

ρ jz2jD

0j

D0i +D0

j(2.37)

and Γ is the MSA screening parameter, calculated in the mean diameter approximation

as

Γ =κ

2(1+Γσ)(2.38)

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

DHSi = Di,ENS AiCi(ρ,M,σ ,χ) (2.39)

22


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 :

D0i = Di,

0ENS AiC0

i (ρ0,M,σ ,χ) (2.40)

Ai is independent of solution composition and density, thus

DHSi = D0

iDi,ENS

D0i,ENS

C(ρ,M,σ ,χ)

C0(ρ0,M,σ)(2.41)

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 :

Di,ENS =

[x j

gi j

di j+ xk

gik

dik+ x j

gis

dis

]−1

(2.42)

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 :

di j =3

8σ2i jρ

[(Mi +M j

)RT

2πMiM j

]1/2

(2.43)

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):

gi j(σi j)=

11−ζ3

+3σiσ j(

σi +σ j) ζ2

(1−ζ3)2 +2

(σiσ j

σi +σ j

)2ζ 2

2

(1−ζ3)3 (2.44)

23


where

ζ1 =π

6 ∑k

xKσlK. (2.45)

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)

C0(ρ0,M,σ)= 1.

Thus, Eq.(2.41) becomes:

DHSi

D0i

=

g0is

d0is

x jgi jdi j

+ xkgikdik

+ x jgisdis

(2.46)

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


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 0.66<Tt<1.78 and 0.22<ρr<2.62. Due to not considering the influence of liq-

uid viscosity and concentration on diffusion, the equations may generate bigger deviations

when they are used for the liquid with different viscosity and concentration from the above

liquid tested. For instance, the deviations for chloroform and iodine 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


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


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


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:

EP = ∑i

u1 (ri)+∑i

∑j>i

u2(ri,r j

)+∑

i∑j>i

∑k> j>i

u3(ri,r j,rk

)+ . . . . . . (3.1)

Where the first term represents the effect of an external fields and other terms repre-

sent interaction between particles. For example, u2 shows the potential between pairs of

particles, and u3 is the potential between particle triples.

It is the most time consuming and crucial in any molecular simulations to calculate the

potential energy or force of intermolecular interaction. In addition, we always assume two-

body interaction or pair wise potential part as the biggest contributor to particle interaction

and system energy. As a result, we will ignore higher order interactions and truncate after

the second term. Three-body interaction, however, in some cases may be important .

Although a pairwise additive potential, the potential reflects spherically averaged effects

of all orders of many-body interactions.

In molecular systems, the potential functions can be subdivided into three parts:

• Non-bonded: Lennard-Jones or Buckingham, and Coulomb or modified Coulomb

The non-bonded interactions are computed on the basis of a neighbour list, a list of

non-bonded atoms within a certain radius, in which exclusions are already removed.

• Bonded: covalent bond-stretching, angel-bending, improper dihedrals and property

dihedrals. These are computed on the basis of fixed lists.

29


• Restraints: position restraints, angle restraints, distance restraints, orientation re-

straints and dihedral restraints.

These are all based on fixed lists. In this work, we only consider non-bond interactions.

The following non-bonded interactions are frequently used:

Lennard –Jones interaction :

ULJ(ri j)= 4εi j

{(σi j

ri j

)12

−(

σi j

ri j

)6}

(3.2)

The Lorentz- Bertelot combination rules are commonly used. An arithmetric average

is used for sigma’s, while a geometric average is used for the epsilon’s:

σi j =12(σii +σ j j

)(3.3)

εi j =(εiiε j j

)1/2 (3.4)

Buckingham potential :

The Buckingham potential has a more flexible and realistic repulsion term than the

Lennard-Jones interaction, but also more expensive to compute. The potential form is:

Ubh(ri j)= Ai jexp

(−Bi jri j

)−

Ci j

r6i j

(3.5)

The force derived from this is:

Fi(ri j)=

[Ai jBi jexp

(−Bi jri j

)−6

Ci j

r7i j

]ri j

ri j(3.6)

If the atoms in a molecule or an ion carry part of charges, then there is electrostatic

attraction or repulsion between atoms. Potential energy term describing electrostatic effect

is called columbic interaction term.

30


Uel = ∑i, j

qiq j

Dri j(3.7)

where qi and q j are the ith ion and charge carried by the jth ion in a molecule; r is

distance; D is effective dielectric constant. As molecules without ions, columbic effective

term is primarily dipole effective. Due to its complexity and importance for the electrolyte

solutions which we will research in the coming chapters , we will give more details in 3.2.7.

3.2.2 Lennard-Jones Reduced Units

All quantities in this thesis use reduced Lennard-Jones units.

r∗i j = ri j/σ (3.8)

ρ∗= ρσ3 (3.9)

T ∗ = kBT/ε (3.10)

P∗ = Pσ3/ε (3.11)

η∗ =

(σ

4/mε)1/2

η (3.12)

We take the mass of the molecules as a fundamental unit, i.e. set mi = 1 with both ε and

σ being assigned a value of one. Here ρ is the system density, T is the kinetic temperature,

and P is the pressure tensor.

31


3.2.3 Periodic Boundary Conditions

In this thesis, molecular simulations are all conducted in a cubic box.

It is noted that atoms near the boundary would have less neighbors than atoms inside.

No matter how big the simulation system and cubic box are, the number of atoms contained

in the system is much less than the number of atoms contained in a macroscopic piece of

matter. The ratio between the number of surface atoms and the total number of atoms

would be much larger than in reality, causing surface effects to be much more important

than what they should.

A solution to this problem is to use periodic boundary conditions. Please see Fig 3.1.

In our case here, the atoms of the system, for example, in the CO2 + NaCl aqueous sys-

tem, CO2 molecule, water molecule, Na+ and Cl− are all put in a box, which is sur-

rounded by translated copies of itself. There are no boundaries in the system. The artifact

caused by boundaries is now replaced by the artifact of periodic conditions. Therefore

those molecules moving out of the simulation box will be replaced by their periodic im-

ages entering the box from the opposite side. In this way, the boundary issue will be solved

in some degree.

3.2.4 Equation of Motion

Take a system of N particles for example where the intermolecular potential determines

the interaction between these particles. In a molecular dynamics simulation, Newton’s

equations of motion for classical systems need to be solved:

Fi = miri (3.13)

The microscopic coordinates and momenta of all the molecules should be determined,

as they are used to obtain thermodynamic and transport properties via statistical mechanics.

If the simulations are carried out for a sufficient length of time, the resulting time averages

32


Figure 3.1: Diagram showing the periodic boundary conditions, minimum image conven-tion and cutoff

33


are equivalent to ensemble averages due to periodicity. It is assumed that the motion of

atoms is governed by Newton’s classical equations of motion given by:

ri =dri

dt=

Pi

m(3.14)

Pi =dPi

dt= Fi (3.15)

Where Fi and Pi are the force and momentum of atom i respectively. Fi is the total force

acting on the particle .

3.2.5 Time Integration Algorithm

A time integration algorithm is used to integrate the equations of motion of the interacting

particles and follow their trajectory. Time evolution of interacting particles is followed by

integrating their equations of motion and any changes in particle position, the strength of

inter-particle forces, velocities and accelerations. Time integration algorithms are based on

finite difference methods, where time is discredited on a finite grid, the time step being the

distance between consecutive points on the grid. According to the positions and some time

derivatives at time t (the exact details rely on the type of algorithm), the integration scheme

gives these quantities at a later time t + δ t. By repeating the procedure, the time evolution

of the system can be followed for long times. Due to truncation, these schemes are certainly

error associated. Truncation is associated with the accuracy of the finite difference method

with respect to the true solution. Finite difference methods are usually based on a Taylor

Expansion truncated at some term. These errors are intrinsic to the algorithm and not

related to the implementation.

Round-off error is another source of error, which is related to a particular implemen-

tation of the algorithm. The finite number of digits used in the computer simulation is an

example. Both errors can be reduced by reducing the time step δ t.

34


Numerous integration algorithms suitable for molecular dynamics have been proposed.

The most popular are the Verlet method (Verlet, 1967) and various predictor-corrector al-

gorithms.

In our work, the so-called leap-frog algorithm (Hockney, 1970) is used for the inte-

gration of the equation of motion. The algorithm is called leap-frog because r and v are

leaping like frogs over each others back.

The early Velert method (Verlet, 1967) expanded particle positions in Taylor’s way:

r (t +δ t) = r (t)+ddt

r (t)δ t +12!

d2

dt2 r (t)(δ t)2 (3.16)

By replacing the δ t with −δ t , we will obtain

r (t−δ t) = r (t)− ddt

r (t)δ t +12!

d2

dt2 r (t)(δ t)2 (3.17)

By adding Eq.(3.16) and Eq.(3.17) , we get:

r (t +δ t) =−r (t−δ t)+2r (t)+d2

dt2 r (t)(δ t)2 (3.18)

since d2

dt2 r (t) = a(t), so the position of time t + δ t can be predicted by the position of

time t and time t−δ t.

If we subtract Eq.(3.17) to Eq.(3.16). The velocity can be gained:

υ (t) =drdt

=1

2δ t(r (t +δ t)− r (t−δ t)) (3.19)

The Velert equation is limited by the 1/δ t term in Eq.(3.19) because error is easily

cauased in case of small value of δ t. To address this issue, Velert developed leap frog

method. The equation calculates velocity and position as:

~υi

(t +

12

δ t)= ~υi

(t− 1

2δ t)+~ai (t)δ t (3.20)

35


Figure 3.2: Diagram demonstrating the leap-frog integration

~ri (t +δ t) =~ri (t)+~υi

(t +

12

δ t)

δ t (3.21)

Assume ~υi(t− 1

2δ t)

and~ri (t) are known, we can calculate the force on particles and ac-

celerated velocity from the position~ri (t) at time t. Then we can use Eq.(3.20) and Eq.(3.21)

to predict the velocity ~υi(t + 1

2δ t)

at time t + 12δ t. Thus the velocity at time t can be ob-

tained from the following equation:

~υi (t) =12

(~υi

(t +

12

δ t)+~υi

(t− 1

2δ t))

(3.22)

By using this Verlert leap-frog method, the storage space can be saved since it only

needs information about ~υi(t− 1

2δ t)

and~ri (t). Another advantage of this method is due to

its high accuracy and simplicity.

Fig 3.2 is the graphic presentation of the leap-frog intergration. It shows that leapfrog

integration is equivalent to calculating positions and velocities alternately, at alternate time

points, so that they ‘leap-frog’ over each other.

3.2.6 Constant Temperature

There are various alternatives to maintaining constant temperatures in MD simulations.

• Berendsen temperature coupling

36


The Berendesen temperature coupling algorithm (Berendsen et al., 1984) imitates weak

coupling with first-order kinetics to an external heat bath with given temperature T0. The

effect of this algorithm is that a deviation of the system temperature from T0 is slowly

corrected according to:

dTdt

=T0−T

τ(3.23)

This means that a temperature deviation decays exponentially with a time constant τ .

This method of coupling has the advantage that the strength of the coupling can be varied

and adapted to the user requirement: for equlibrium purposes the coupling time can be

taken quite short to 0.01 ps, but for reliable equilibrium runs it can be taken much long in

which case it hardly influences the conservative dynamics.

• Nose-Hoover temperature coupling

d2ri

dt2 =Fi

mi−ξ

dri

dt(3.24)

where the equation of motion (Nose, 1984; Hoover, 1985) for the heat bath parameter

ξ is :

dξ

dt=

1Q(T −T0) (3.25)

3.2.7 Electrostatic Force

In a space, when the decrease rate of the interaction between atoms is greater than r−d

(the space dimension), the interaction is called long range force. Thus Coulomb force is a

typical long range force.

Uq =q1q2

4πε0r2i j

(3.26)

37


In simulation the Coulomb force will be a problem as its impact goes beyond the box

by half of the box width. There are several solutions to this problem.

• Ewald Sum

Ewald Sum was proposed by Ewald (Ewald, 1921) in 1921 to calculate energy of ionic

crystal. An initial box is first selected and the particles in the box interact with others in

the box and also with all the particles in the mirror image. In the order of the surrounding

boxes, when the mirror image boxes increase to a maximum number, the whole area looks

like a ball, which is Ewald ball. Ewald sum is based on Ewald ball (convergence) and a

certain dielectric constant.

Suppose an initial box (centre box) is a cube with L side length and N particles in it.

The position of the mirror image box can be represented as (±iL , ±jL , ±kL), where (I, j, k

= 0, 1, 2, 3 . . . ). The coulomb interaction between the particles in the initial box is:

U =12

N

∑i=1

N

∑j=1

qiq j

4πε0ri j(3.27)

There are totally 6 boxes closest to the initial box in the positions of (L, 0, 0), (-L, 0,

0), (0, L, 0), (0, -L, 0), (0, 0, L), (0, 0, -L). The inter-particles coulomb interaction between

the initial box and the 6 boxes is:

U =12

6

∑box=1

N

∑i=1

N

∑j=1

qiq j

4πε0|~ri j +~rbox|(3.28)

The position vector of the surrounding mirror image box can be represented ~n =

(nxL,nyL,nzL), of which nx, ny and nz are all integers. In the same way the coulomb

interactions between the particles in the initial box and all the surrounding boxes can be

calculated.

U =12

6

∑~n

N

∑i=1

N

∑j=1

qiq j

4πε0|~ri j +~rn|(3.29)

38


Considering the interaction of the particles themselves in the initial box, the above

equation can be converted to :

U =12

6

∑~|n|=o

′N

∑i=1

N

∑j=1

qiq j

4πε0|~ri j +~rn|(3.30)

In the equation, the symbol ”′” on the right top hand of ∑ represents the scenario when

~|n|= 0 , that is, when the self- interaction of particles in the initial box is excluded.

Via Laplace transformation, the Coulomb term is divided to two and one of them can

be converged fast in real space. The expression is :

Ureal =12

N

∑i=1

N

∑j=1

∞

∑|~n|=0

′qiq j

4πε0

er f c(a|~ri j +~n|

)|~ri j +~n|

(3.31)

er f c is complementary error function, which is:

er f c(x) =2√π

∫∞

xexp(−t2)dt (3.32)

The convergence rate of complementary error function is so fast that convergence is

achievable when only a few |n| is calculated in coulomb summation. The summing rate is

determined by a in the expression. The bigger a is, the quicker the convergence is. In the

real calculation, we can select big a value. When |n| = 0, only the electric charge in the unit

cell needs to be calculated.

The second term can be converged in reciprocal space:

Urep = ∑k 6=0

N

∑i=1

N

∑j=1

1πL3

qiq j

4πε0

4π2

k2 exp(− k2

4a2

)cos(~k · ~ri j

)(3.33)

where~k is the reciprocal vector of the particles in the initial box, which can be expressed

as~k = 2π~n/L2.

39


It is noted that the self-interaction of electrical charge needs to be deducted from the

time of reciprocal space summation. This term is independent from the atom position; so

for a given atom it is a constant.

Usel f =−a√π

N

∑k=1

q2K

4πε0(3.34)

The other term is related to the medium nature around the Ewald ball. For conductive

medium, this term is not needed; for vacuum medium, it is needed:

Ucorrection =2π

3L3 |N

∑i=1

qi

4πε0~ri|2 (3.35)

So, based on the Ewald method, the long-range Coulomb potential energy can be cal-

culated via:

U =Ureal +Urep +Usel f +Ucorrection (3.36)

Ewald summation is one of the most feasible way to calculate static coulomb interac-

tion, especially for higly electrified system. Ewald sum calculation takes more time than

normal molecular simulation because it involves many addition terms.

Some faster static summation methods such as particle-mesh Ewald (PME) (Essmann

et al., 1995; Darden et al., 1993) and particle-particle particle-mash Ewald (PPPM) (Hock-

ney and Eastwood, 1981; Essmann et al., 1995) were created to speed up Ewald calculation.

In this thesis, we used PME method for calculation of static coulomb interaction due to

its high accuracy and efficiency. Alternative methods, described briefly below, have some

limitations.

• Reaction Field

Reaction field is a simplified way to calculate Coulomb potential energy. It calculates the

coulomb static interaction between all the charges within the transversal radius only while

40


treating the outside of the transversal radius as homogeneous medium with medium con-

stant of εs . The medium within the transversal radius forms an electric field. The potential

energy of a molecule can be derived from calculation of the dipole vector interaction of

the molecules in the electric field. A limitation of this method is that: when the number

of molecules within the transversal radius changes, the energy calculated will represent

discontinuation. The solution is to use switch function frequently to avoid it.

• Solvent Dielectric Model

In this model, dielectric constant ε is a valid dielectric constant and its value is related

to the distance between particles. The closer the distance is, the less effect the solvent

molecules show; the farther the distance is, the greater effect the solvent molecules present.

The dielectric constant is almost the same as that of the solvent system.

3.3 The Applications of Molecular Dynamics

The application of molecular simulation has been discussed in the introduction part. This

part will introduce the principles of a few applications to be covered in this thesis.

3.3.1 Trajectory Analysis

Molecular dynamics calculation can provide the coordinate and velocity of each atom (or

group) in a system at each step. The change of atom coordinate with time presents the

moving routes of atoms in the system. The moving route is called trajectory. In molecular

dynamics calculation, the velocity of atoms reflects moving speed and direction. Normally,

the coordinates and velocity of all atoms in the access system are calculated and saved once

every 10 or 20 steps for the purpose of analysis. From the calculation of some physical

properties based on this saved trajectory, a great amount of information can be obtained

such as heat, statistical and dynamic information.

41


3.3.2 Radial Distribution Function

Radial distribution function (RDF or alternatively g(r)) is illustrated in Fig.3.3. In this

figure, a molecule in a fluid system, is named as the target molecule. The distance to the

centre is determined by the number of molecules dN between r and r+ dr. The definition of

g(r) is :

ρg(r)4πr2 = dN (3.37)

Where, ρ is the system density. Suppose the number of molcules in the system is N,

then from the above Equation:

∫∞

0ρg(r)4πr2dr =

∫ N

0dN = N (3.38)

and

g(r) =dN

ρ4πr2dr(3.39)

RDF function can be expressed as the ratio of system local density and bulk density.

The local density close to target molecule (small r value) is different from system bulk

density, but they are equal when it is far from target molecule. That is, when r value is big,

the value of RDF function is close to 1. The RDF function equation is:

g(r) =1

ρ4πr2dr

T

∑t=1

N

∑j=1

∆N (r→ r+δ r)

N×T(3.40)

Where, N is the number of molecules, T is time (steps), δ r is the set distance difference,

DN is the number of molecules in r→r+δ r.

The applications of RDF function are widely used. Besides the well-known structure

application, it is also used to calculate the mean potential energy and pressure:

42


Figure 3.3: Diagram demonstrating the radial distribution function

UN

=12

ρ

∫∞

0u(r) ·g(r)4πr2dr (3.41)

P = ρT − 13

12

ρ2∫

∞

0

du(r)dr· r ·g(r)4πr2dr (3.42)

where U = ∑i< j

u(ri j)

43


3.3.3 Correlation Function

Suppose the MD system includes N molecules, and n steps was saved, then :

For the auto correlation function of velocity in Leanard-Jones liquid system, when time

is long enough (t value is big enough), particle velocity is not correlated to its initial ve-

locity anymore and thus the value of the correlation function is closing to zero. On the

contrary, when they are correlated strongly, the value becomes big. Since particle velocity

is a vector, the negative sign represents the direction is opposite to initial velocity. As the

most important function in statistic mechanics, correlation function in various forms can

be used to calculate the average value of time and many physical properties.

For example, the velcocity auto correlation function (Allen and Tildesley, 1987) can

also be used to calculate the diffusion coefficient :

D =13

∫∞

0〈υ (t) · υ (0)〉dt (3.43)

3.3.4 Mean Square Displacement

In the molecular dynamics calculation system, atoms keep moving from their initial po-

sition to a different position at each point of time. Take ri (t) as the position of particle

i at time t. The average of the square of the particle displacement is called mean square

displacement MSD (Allen and Tildesley, 1987).

MSD = R(t) =⟨|r (t)− r (0)|2

⟩(3.44)

In this equation, angle brackets < > denotes ensemble average. According to statistical

principles, if the number of molecules is infinitely big and time is infinitely long, any

moment of the system can be viewed as time zero and the average values calculated are

always the same. Thus for the MSD calculation from access trajectory, each trajectory is

viewed as zero point and the average values should be the same. Suppose there are n-step

44


trajectories in a molecular dynamics calculation, and the positon vectors of the steps are

r (1) , r (2) , · · · , r (n). The trajectory is normally split to two parts and the average of n/2

group data is selected as MSD. The two parts of trajectory are:

r (1) , r (2) , · · · , r (n/2) and r (n/2+1) , r (n/2+2) , · · · , r (n)

Set the time interval as δ t . As any moment can be viewed as zero point, so

R(δ t) =|r (2)− r (1)|2 + |r (3)− r (2)|2 + · · ·+ |r (n/2+1)− r (n/2)|2

n/2(3.45)

R(2δ t) =|r (3)− r (1)|2 + |r (4)− r (2)|2 + · · ·+ |r (n/2+2)− r (n/2)|2

n/2(3.46)

· · · · · · · · ·

R(mδ t) =|r (m+1)− r (1)|2 + |r (m+2)− r (2)|2 + · · ·+ |r (m+n/2+1)− r (n/2)|2

n/2(3.47)

· · · · · · · · ·

R(nδ t/2) =|r (n/2)− r (1)|2 + |r (n/2+2)− r (2)|2 + · · ·+ |r (n)− r (n/2)|2

n/2(3.48)

The above is the calculation of the MSD of a certain particle, and the MSD of all

particles in a system can be derived by averaging the number of particles.

Normally, when t is very small, R(t) increases exponentially, but when t is getting

larger, R(t) increases almost linearly. According to the Einstein diffusion theory:

45


limt→∞

⟨|r (t)− r (0)|2

⟩= 6Dt (3.49)

where D is the diffusion constant. Eq. (3.49) was used to calculate diffusion constants

in this work.

46

CHAPTER 4. DIFFUSION COEFFICIENTS OF CARBON DIOXIDE IN WATER USING DIFFERENT CO2 MODELS

Chapter 4

Diffusion Coefficients of Carbon Dioxide

in Water using Different CO2 Models

This chapter investigates the diffusion coefficients in water using a series of molecular

dynamic simulations. The SPC/E model (Berendsen et al., 1987) for water is used and

different carbon dioxide potential models have been used to compare with experimental

data. The purpose of this chapter is to find the most suitable potential model of carbon

dioxide for further research in the coming chapters.

4.1 Introduction

To calculate potential energy, we have to assume the nature of attraction and repulsion

between molecules. Intermolecular interaction is the result of both short- and long-range

effects. Theoretically, we may calculate the intermolecular interactions from first princi-

ples or ab-initio approach. Such an approach in practice is only applicable to relatively

simple system. More commonly, some type of intermolecular potential is used to express

the influence of intermolecular. Historically many effective potentials have been devel-

oped and applied to atoms, and empirical approach was used with the parameters of the

47


potential, which are obtained from experimental data such as second viral coefficients, vis-

cosities, molecular beam cross sections. We can evaluate the accuracy of pair potential by

comparing the properties predicted by the potential with experiment.

Although computer simulation makes evaluation of the accuracy of intermolecular po-

tentials possible, molecular simulation is rarely applied to testing potentials extensively on

different conditions. Effective potentials for atoms are often incorporated into the molecu-

lar simulation of polyatomic molecules and increasingly macromolecules. The atomic pair

potential thus becomes an important staring basis for predicting high quality molecular

properties.

The topic of this chapter is to investigate the physical properties of carbon dioxide in

water. Water and carbon dioxide, two important molecules, which have attracted extensive

attention of researchers. Many appropriate intermolecular potential models have been pro-

posed. The parameters of these models however are obtained in specific conditions and by

comparing experimental data. Thus the evaluation the accuracy of these models in variable

temperatures and pressures becomes necessary. It is especially important when we conduct

simulation in deep sea in special conditions like high pressure or low temperature. We can

not simply judge whether a potential model is better without specified conditions. Some

are better than the others in one specific condition, while some perform undesirably under

other conditions. Therefore, in this chapter we will test some well-known effective pair

potential for the prediction of the diffusion of carbon dioxide in water using molecular dy-

namic simulation. One of the aims of the chapter is to find the best suitable potential for

exploring the physical properties of carbon dioxide in water so that it can help us further

explore these properties of carbon dioxide in the sea water.

48


4.2 Intermolecular Potential between Water and Carbon

Dioxide

4.2.1 Water Models

Water, especially, has been intensively researched and many different models have been

proposed. The first molecular simulation was reported in 1953 by Metrepolis et al. (1953)

who presented the Monte Carlo sampling scheme and the first simulation of liquid water

was reported in 1969 by Baker and Watts (1969) and Rahman and Stillinger (1971) then

conducted molecular dynamics simulation at 1971. In 1976 Matsuokay et.al (1976) derived

a pair potential for water from ab inito potential. Prior to 1983, the potentials like BF

(Bernal and Fowler, 1933), BNS and ST2 (Stillinger and Rahman, 1974) potential functions

were generally used in liquid water simulation, and in 1983 the TIP3P and TIP 4P models

(Jorgensen et al., 1983)were introduced becoming the most commonly used models today

along with SPC (Berendsen et al., 1981)and SPC/E (Berendsen et al., 1987)models.

Water model can be categorized by the following broad types:

• Rigid: SPC, SPC/E, TIP3P, TIP4P, WK, NSPCE, TIP5P, SPC/HW, DEC

• Flexible: CF, DCF, RWK, BJH, SPC/F, NCF

• Polarizable: SPCP, MCHO, NCC, NEMO, PTIP4P, PSRWK

A model may have more than one characteristic. For instance, CKL is both flexible and

polarisable.

A water model can also be described by the number of points used, i.e. 3 points, (SPC,

TIP3P, SPC/E), 4 points (BF, TIP4P), 5 points (BNS, ST2, TIP5P)

A rigid model is the simplest model, and flexible model allows intra-molecular vibra-

tion. A flexible model is normally obtained by adding bond stretching and angle bending

terms to rigid model like SPC/E model which derives from SPC model.

49


Polarisable and flexible models are more detailed, they take more computing resources

and time especially for complicated molecular system. As a result, rigid model is popular

for complicated system. Polarisable and flexible models, however, will be more and more

used in the future.

The SPC model, proposed by Berendsen in 1981, is regarded as the first accurate and

simple pair potential for liquid water. TIP3P was proposed by Jorgensen and TIP4P by

Jorgensen and Mahoney in 1983. The SPC model is a 3 point model with each atom having

a point charge. The negative charge of the oxygen atom is twice as the positive charge of

a single H atom, but only the oxygen atoms interact via a Lennard-Jones type potential.

That is, we only need to compute the interaction between oxygen atoms to decide van der

Waals force. In contrast to other 3-point model which uses geometry matching the known

geometry of the water molecular (H-O-H angle is 104.5), SPC models assume water is of

ideal tetrahedral shape (H-O-H angle is 109.47).

SPC treats water as a rigid molecule with fixed bond length and bond angle. The force

between two water molecules includes van der Waals forces and Coulomb forces. Van der

Waals forces only exist between oxygen atoms. Every atom is with charge and the coulomb

forces exist not only betwen the atoms from diferent molecules but also between the atoms

from the same water molecule.

The force field of SPC is :

UOO (roo) =qoqoe2

roo+4ε

[(σoo

roo

)12

−(

σoo

roo

)6]

(4.1)

UOH (rOH) =qoqHe2

rOH(4.2)

UHH (rHH) =qHqHe2

rHH(4.3)

50


SPC/E model stems from SPC model. The SPC model is unable to accurately simulate

water properties in condensed state as it ignores polarizability, which impacts intermolec-

ular bonding in a great sense and thus is crucial to liquid water. Average polarization

correction to the potential energy function is added to SPC model and thus SPC/E model is

created.

The average polarization correction added to the potential function of SPC/E is :

Upol =12 ∑

i

(µ−µ0)2

αi(4.4)

where µ is the dipole of the effectively polarized water molecule (2.35 D for the SPC/E

model), µ0 is the dipole moment of an isolated water molecule (1.85 D from experiment),

and αi is an isotropic polarizability constant, with a value of 1.608 × 10−40 F m. Since

the charges in the model are constant, this correction just results in adding 1.25 kcal/mol

(5.22 kJ/mol) to the total energy. The SPC/E model results in a better density and diffusion

constant than the SPC model (Mahoney and Jorgensen, 2001).

TIP3P is also a 3-point model for electrostatic interactions. Like the SPC model, point

charge exists on each atom of TIP3P model and H atom is ignored in the computation of

van der waals interaction. TIP3P, however, has different charge parameters and van der

Waals parameters. The biggest difference is geometry that TIP3P is 104.52.

For TIP4P model (4-point model), a dummy atom is placed near oxygen along the bi-

sector of the H-O-H angle 0.15A, and the site of negative charge is placed on the dummy

atom so that oxygen atom carries no charge. The purpose of the dummy atom is to increase

electrostatic distribution around the water molecular. TIP4P model contains different pa-

rameters.

Chaplin (Chaplin, 2011) has summarized the most useful water models. In Fig.4.1

model types a, b and c are all planar whereas type d is almost tetrahedral. The mid-point

site (M) in c and the lone pair sites (L) in d are labeled q2. One negative charge, q2, is placed

51


Figure 4.1: Diagram showing the different types of water models (Chaplin, 2011), see textfor details.

52


Table 4.1: Summary of the parameters of various water models (Chaplin, 2011)

Model Type σ ε I1 I2 q1 q2 θ 0 φ 0

A kJ/mol A A e eSPC a 3.166 0.650 1.000 - +0.410 -0.82 109.47 -SPC/E a 3.166 0.650 1.000 - +0.4238 -0.8476 109.47 -SPC/HW a 3.166 0.650 1.000 - +0.435 -0.8700 109.47 -TIP3P a 3.15061 0.6364 0.957 - +0.4170 -0.8340 104.52 -TIP3P/Fw a 3.1506 0.6368 0.960 - +0.4170 -0.8340 104.5 -PPC b 3.2340 0.6000 0.934 0.06 +0.5170 -1.0340 106.00 127.00TIP4P c 3.15365 0.6480 0.957 0.15 +0.52 -1.0400 104.52 52.26TIP4P-Ew c 3.16435 0.68095 0.957 0.125 +0.5242 -1.0484 104.52 52.26TIP-FQ c 3.15365 0.6480 0.957 0.15 +0.63 -1.26 104.52 52.26TIP4P/Ice c 3.1668 0.8822 0.957 0.158 +0.5897 -1.1794 104.52 52.26ST2 d 3.1000 0.31694 1.000 0.80 +0.2436 -0.2436 109.47 109.47TIP5P d 3.1200 0.6694 0.957 0.70 +0.2410 -0.2410 104.52 109.47TIP5P-Ew d 3.097 0.7448 0.957 0.70 +0.2410 -0.2410 104.52 109.47

on the mass centre of oxygen (O) atom in model a, while two equivalent point charges, q1,

are placed on two hydrogen (H) centres respectively. The length of O-H bonding is l1.

SPC/E and other three-site models commonly follow this model. As shown in model b and

c, negative charge (q2) and the mass centre of O are separated, and the negative charge is

placed on the bisector of the H-O-H angle. Model d employs two negative partial charges,

which is typical for the latest rigid water potentials such as TIP5P, and etc. A summary of

the parameters of various water models are given in Table 4.1.

4.2.2 Carbon Dioxide Models

Carbon dioxide has attracted a lot of attention due to its peculiar attributes. Carbon dioxide

models are developed to investigate carbon dioxide in various conditions. Some models

are suitable for investigating its crystal structure and spectrum, whereas other models have

been developed for the liquid and super criutial conditions.

53


Figure 4.2: Diagram showing all the carbon dioxide models used in this work have thesame line structures

Some popular carbon dioxide models are listed herein. As shown in the Fig. 4.2, all of

the models discussed here are rigid and the angle of O=C=C is also fixed as 180◦.

• EPM and EPM2 Model

The elementary physical model (EPM) is a Lennard-Jones (12-6) force fields model. Harris

and Yung (Harris and Yung, 1995) introduced this model in 1995. EPM is a rigid, 3-

molecular model which has three Lennard-Jones sites with charges centered at each atom.

The carbon-oxygen bonds are rigid and 1.163 A long. EPM model is unique in that it uses

point charges centered at atom sites and has a quadrupole moment of 4.3×10−26 esu which

is insignificantly different from the experimental of 4.1× 10−26 esu. A coexistence curve

and critical properties which are quite close to the experimental values have been predicted

by this model. The critical temperature predicted by EPM model is 313.4± 0.7K which

is in reasonable agreement with the experimental value at 304K. EPM2 (Harris and Yung,

1995) was also introduced to reproduce the liquid-vapor coexistence curve more accurately

by rescaling the potential parameters of the EPM. A flexible EPM model with rigid bonds,

but a flexible angle exhibits idential coexistence properties to within the sensitivity of their

calculations. So normally, for simplicity, the EPM is used in rigid bonds and rigid angle.

54


Table 4.2: Potential function parameters of different CO2 models

εc−c σc−c εo−o σo−o qc α

kJ/mol A kJ/mol A eEPM 0.24127 2.785 0.69 3.064 +0.6645EPM2 0.234 2.757 0.6698 3.033 +0.6512

Errington 0.242 2.753 0.692 3.029 +0.6466 14.0Zhang and Duan 0.23999 2.7918 0.688 3.0 +0.5888

Gromos 0.51369 3.2618 0.81057 3.0145 +0.6172

The potential functions includes short-range interaction between three atoms and

coulombic interaction between charges:

U (1,2) =Ushort (1,2)+Ucoul (1,2) = ∑i∈{1}

∑j∈{2}

Ushort(ri j)+ ∑

i∈{1}∑

j∈{2}

qiq j

ri j(4.5)

where ri j is the separation between two atoms and qi and q j are the partial charges

designated on the center of each atom.

The EPM model and EPM2 model both use Lennard-Jones function to calculate the

short-range interactions (see Eq.(3.2)). The potential function parameters for CO2 used in

EPM and EPM2 are given in Table 4.2.

• Errington model

This model was developed by Errington (Errington, 1999). This model delivers good result

in high temperature and high pressure. This model, however, hasn’t been widely used due

to its Buckingham exponent-6 form. The most popular water models today are lennard-

Jones potential.

The potential function of Errington model is also with the same format like Eq.(4.5) ,

but with a different function for short-range interactions:

55


Uexp−6short

(ri j)=

εi j

1−6/α

[6α

exp(

α

[1−

ri j

rm

])−(

rm

ri j

)6]

(4.6)

where εi j is the well depth for short-range interactions, rm and α are parameters for

exponential-6 potential. The parameters of Errington model are summarized in Table 4.2.

• Zhang and Duan Model

This model is a 3-point rigid model with 180 degree angle of O=C=O. It was developed

by Zhang and Duan (Zhang and Duan, 2005) based on MSM and EPM2 model. Zhang

and Duan found out that the average deviation of this new model from experimental data

for the saturated liquid densities, vapor densities, vapor pressures and heats of vaporiza-

tion are around 0.1%, 2.3%, 0.7% and 1.9% respectively. The calculated critical point is

almost pinpointed by the new model. The experimental radial distribution functions rang-

ing from 240.0 to 473.0 K are well reproduced as compared to neutron-diffusion measures.

The predicted self-diffusion coefficients are in good agreements with the nuclear-magnetic-

resonance measures.

The potential function used in Zhang and Duan model has the same format like EPM

and EPM2 ,but with different parameters.

The parameters used in Zhang and Duan’s model is summarized in Table 4.2 .

• Gromos Force Field

Gromos (Gunsteren and Berendsen, 1987) is a force field for molecular dynamics simula-

tion developed at the University of Groningen and at Computer-Aided Chemistry Group at

the Laboratory for Physical Chemistry at the ETH Zurich. Strictly speaking, this is not a

specific CO2 model like the models we discussed above. We just used the Lennard-Jones

parameters from the Gromos manual, the partial charges q on CO2 were chosen such that

the CO2 quadroupole moment was−4.0 DA. The reason we use this force field is that Pan-

huis et al. (1998) reported good results for the translational diffusion coefficient and free

56


energy of solvation determined from molecular dynamics. For simplicity, we will call this

Gromos model in the subsequent sections. As there is only one CO2 in the box in Panhuis’s

simulations, there is no CO2−CO2 interactions. The other parameters are in the Table 4.2

.

4.3 Diffusivities of CO2 in Water under Different Poten-

tial Models

4.3.1 Simulation Details

A series of molecular simulation models were used to study carbon dioxide in water using

molecular dynamics package Gromacs. A big system was established, which included 2180

water molecules and 1 CO2 molecule. A leap-frog algorithm (Chapter 3.2.5) was used

for integrating Newton’s equations of motion and fast Particle-Mesh Ewald electrostatics

(Chapter 3.2.7) for computing coulomb forces. The LJ potential was decreased over the

whole range and the forces decay smoothly to zero between 0.8 and 1.0. Temperature

coupling was used with a Berendsen-thermostat (Chapter 3.2.6) to a bath with designated

temperature. Exponential relaxation pressure coupling was used with time constant 0.1ps.

Temperature for the Maxwell distribution was designated to 298 K.

SPC/E potential was the water model in simulation due to its simplicity and adaptability

to large range of temperature and pressure. Water self-diffusion coefficients under SPC/E

model are in good agreement with experimental data with very small error. For example,

under 1atm and 298K, the actual self-diffusion of water is 2.30. The data under TIP4P

and TIP3P is 3.29+/- 0.05 cm2/s× 10−5 and 5.06+/- 0.09 cm2/s× 10−5(Jorgensen et al.,

1983) and for SPC it is 3.89 +/- 0.09 cm2/s× 10−5 (Mahoney and Jorgensen, 2001). For

more data , please refer to the Table 4.3. The function form for the interaction parameters

57


Table 4.3: Diffusivities of H2O under different models and ensembles (Mahoney and Jor-gensen, 2001)

Ensemble T (K) P(Atm) D(10−5cm2/s)SPC NPT 298 1 3.85+/−0.09

SPC/E NPT 298 1 2.49+/−0.05TIP3P NPT 298 1 5.19+/−0.08TIP4P NPT 298 1 3.31+/−0.08TIP3P NVT 298 1 5.06+/−0.09TIP4P NVT 298 1 3.29+/−0.05TIP5P NVT 298 1 2.62+/−0.04

are given as a Lennard-Jones potential and the normal mixing rules apply for all the inter-

molecular molecules(please refer to Eq 3.2, Eq.3.3 and Eq. 3.4). Since there is just one

CO2 molecules, so we do not consider CO2−CO2 interactions . We have studied the the

diffusion of CO2 in water at five different temperatures(293K, 298K, 303K, 313K, 333K).

The parameters are given in the Table 4.1, and Table 4.2.

4.3.2 Results and Discussion

Fig 4.3 provides a comparison of CO2 diffusivities in water obtained from the simulations

using different Duan potential model and experimental data. Duan model is first selected

due to its excellence. As an improvement of MSM and EPM2 model, the data obtained

agree with experimental data well, including CO2 diffusion, saturated liquid densities, va-

por densities, vapor pressures and heats of vaporization (Zhang and Duan, 2005). As shown

in this figure, the Duan model delivers higher CO2 diffusion coefficient data in water than

experimental data at low temperature, but lower diffusion coefficient data in high temper-

ature. Although more close to experimental data in medium temperature, the simulation

data are still not ideal. It is then concluded that Zhang and Duan model, although almost

perfect for pure CO2 simulation, is not suitable for simulation of CO2 diffusion in water.

During the optimization process, Zhang and Duan selected the best parameters to fit in the

58

CHAPTER 4. DIFFUSION COEFFICIENTS OF CARBON DIOXIDE IN WATER USING DIFFERENT CO2 MODELS�290 300 310 320 330 3401.52.02.53.03.54.04.5D(10-5 cm2 /s)

T(K) Experimental Duan

Figure 4.3: Comparison of the self diffusion constant of CO2 in water obtained from theDuan potential (•) and experimental data (�) at different temperatures at a pressure of 1atm.

experimental data for densities and vapor pressures. In these systems, the interactions be-

tween molecules are between CO2 and CO2 . So the Lennard-Jones parameters and partial

charges parameters were all fitted to get the best results for CO2 and CO2 interactions rather

than CO2 and H2O or other kinds of molecules.

Fig 4.4 compares the CO2 diffusion coefficient in water obtained from the EPM2 model

simulation and experimental data. It is observed that when the temperature equals 313K

, diffusion data from simulation is almost as the same as experimental data, but higher at

other temperatures. The simulation error increases with temperature increase. Considering

the existence of error even under different experiments, EPM2 model result, however, is

acceptable.

Fig 4.5 compares calculations using Lennard-Jones parameters from EPM2 model and

point charge obtained from the parameter which was optimised by Panhuis et al. (1998)

at 298 K and 1 atm. It has been reported that both Lennard-Jones parameters and partial

charges have the big influence on the diffusion results Mahoney and Jorgensen (2001).

59

CHAPTER 4. DIFFUSION COEFFICIENTS OF CARBON DIOXIDE IN WATER USING DIFFERENT CO2 MODELS�290 300 310 320 330 3401.52.02.53.03.54.04.55.05.5

D(10-5 cm2 /s)

T(K) Experimental EPM2

Figure 4.4: Comparison of the self diffusion constant of CO2 in water obtained from theEPM2 potential (•) and experimental data (�) at different temperatures at a press of 1 atm.�

290 300 310 320 330 3401.52.02.53.03.54.04.5D(10-5 cm2 /s)

T(K)

Experimental EPM2+GromosFigure 4.5: Comparison of the self diffusion constant of CO2 in water obtained from theEPM2 + Gromos potential (•) and experimental data (�) at different temperatures at pres-sure=1 atm

60

CHAPTER 4. DIFFUSION COEFFICIENTS OF CARBON DIOXIDE IN WATER USING DIFFERENT CO2 MODELS�290 300 310 320 330 3401.52.02.53.03.54.04.5

D(10-5 cm2 /s)T(K)

Experimental Gromos

Figure 4.6: Comparison between experimental(�) self diffusion coefficients of CO2 withresults obtained from Gromos(•) at different temperatures at a pressure of 1 atm.

Here, we just finished the simulation from EPM2 model and obtained the good results.

Panhuis et al. (1998) also reported good results for diffusion of CO2 in water by using the

Gromos model. It is helpful to try combing two good models (by using the Lennard-Jones

parameters from Panhuis model and using partial charges parameters from EPM2 model)

to see whether it will generate even better results. Unfortunately the data obtained are the

worst.

Finally, the Gromos model is used at 298 K and 1 atm in Fig 4.6. That is, Gromos

model parameters are used for Lennard-Jones and point charge parameters from Panhuis

et al. (1998) are used. It is found that Gromos model data agrees with experimental data

very closely, especially at low temperatures. Gromos simulation data error increases at

high temperatures.

It doesn’t necessarily mean that Gromos model is not applicable to high temperature

because generally high temperature brings bigger error and furthermore two experimental

data may have such error.

The simulation data has been summarized in the Table 4.4.

61


Table 4.4: Comparison of diffusivities of CO2 of experimental and different CO2 models.The values for the diffusion constants are given in 10−5cm2/s.

Experimental Duan EPM2 EPM2+Gromos Gromos293K 1.76 2.2+/-0.4 2.01+/-0.25 2.1+/-0.32 1.76+/-0.18298K 1.94 2.26+/-0.28 2.09+/-0.15 2.3+/-0.36 1.95+/-0.15303K 2.2 2.4+/-0.25 2.45+/-0.25 3.0+/-0.4 2.21+/-0.16313K 2.93 3.15+/-0.27 2.932+/-0.36 3.2+/-0.35 2.92+/-0.15333K 4.48 4.1+/-0.4 4.5+/-0.5 3.75+/-0.35 4.12+/-0.2

All the above simulations prove that Gromos model is most applicable to simulation of

CO2 diffusion in water based on the SPC/E model for water. So we will use this model in

the coming chapters to find out the diffusion coefficient of CO2 in electrolyte solutions.

62

CHAPTER 5. STRUCTURAL PROPERTIES AND DIFFUSION COEFFICIENTS OF NACL AQUEOUS SOLUTIONS

Chapter 5

Structural Properties and Diffusion

Coefficients of NaCl Aqueous Solutions

5.1 Introduction

The previous chapter discussed molecular dynamics simulation of CO2 in water. This

chapter will discuss NaCl and water structures and the next chapter covers structure change

of NaCl and water after CO2 is added.

The sodium chloride + water binary mixture is probably a Type I system with a contin-

uous vapour-liquid critical curve and no liquid-liquid immiscibility under any conditions.

The high estimated critical temperature of sodium chloride (∼3400 K) and the corrosive

nature of this system mean that its phase behaviour has not been fully determined exper-

imentally. At low temperatures, it is generally accepted that the high dielectric constant

of water means that sodium chloride behaves at two distinct ions in solutions whereas ion

pairing occurs at supercritical conditions where the dielectric constant is low. In this work,

as noted above, sodium chloride is modelled as distinct ions to reflect its solution behaviour.

The nature of the sodium chloride + water mixture also poses severe challenges for ei-

ther X-ray scattering or neutron diffraction studies. Therefore, there is a limited amount of

63


experimental data for the structural properties of this mixture. Instead, our understanding

of the structural depends largely on pair-distribution data obtained from molecular simu-

lation studies. The results obtained from molecular simulation obviously depend on the

nature of the intermolecular potential used as the underlying model of the system. A com-

parison of simulation results for this system display different quantitative outcomes for the

structural properties. Some times, there are even qualitative disagreements between the

different models.

For instance, there have been some work on NaCl system under different concentra-

tions that show totally different results (Chen and Pappu, 2007; Chowdhuri and Chandra,

2001). One of the discoveries is that the fraction of contact ion pairs increase and that of

solvent separated ion pairs decrease with increasing ion concentration. Also, with an in-

crease of ion concentration, significant structural changes are found in the oxygen-oxygen

radial distribution function of water molecules. In concentrated solutions, some of the wa-

ter molecules occupy interstitial positions and, as a result, no well defined second hydration

shell is found around a central molecule. In this situation, the Na+−Cl− radial distribution

function shows that the first peak of Na+ and Cl− increases in height with NaCl concen-

tration increasing (Chowdhuri and Chandra, 2001).

Another discovery is that the RDFs depend on electrolyte concentration in counterin-

tuitive ways. Specifically, the first peak of cation-anion correlations is found to decrease

with increasing concentration, while both the cation-cation and anion-anion RDFs exhibit

the onset of short range correlations (Chen and Pappu, 2007). We will discuss about why

there are two different results more in the next chapter.

The above discussion tells that although there have been many studies about NaCl aque-

ous solution (Patra and Karttunen, 2004; Chen and Pappu, 2007; Chowdhuri and Chandra,

2001), they are not perfect and even contradict to each other. To study the ternary physical

properties of CO2 in NaCl aqueous solution, we have better to study NaCl aqueous solution

first at similar situations start.

64


5.2 Intermolecular Potential between Water and NaCl

One important factor for molecular simulation is the choice of intermolecular potential.

The reliability of molecular simulation results is ultimately dependent upon the efficacy of

the intermolecular interactions that underpin the simulation. Consequently, development of

accurate and reliable intermolecular potentials is critical for improved simulation capabil-

ities. Extensive validation of developed intermolecular potentials is needed to ensure that

simulation results are accurate and reproducible over a range of systems and conditions. As

we discussed at the above, when using the same molecular dynamics simulation to simulate

the same system, people can get very contrary results.

Potential water models have been discussed in the previous chapter. There are many

focre fields available for NaCl ,like Gromas, Charmm-22/x-plor, Charmm-2, Amber-1999

and OPLS-AA force fields, and the paramerization by Smith and Dang. But all of them are

only suitable for some particular systems. Patra and Karttunen (2004) has done a series of

molecular dynamics simulations to test the widely used force fields.

5.3 Simulation Details

A series of molecular simulation based on Gromacs were used to study NaCl properties in

water. A leap-frog algorithm (Chapter 3.2.5) was used for integrating Newton’s equations

of motion. Fast particle-Mesh Ewald electrostatics (Chapter 3.2.7) was used for calculating

Coulomb force. The LJ potential was decreased over the whole range and the forces decay

smoothly to zero between 0.8 and 1.0. Temperature coupling with a Berendsen-thermostat

(Chapter 3.2.6) to a bath was set at a temperature, exponential relaxation pressure coupling

with time constant 0.1 ps. Temperature for Maxwell distribution was set to 298 K. All

bonds were converted to constraints by shake algorithm. A cutoff of 10 A was set for the

evaluation of Lennard-Jones potential. A total time 200 ps was spent on calculation and

65


1ns is taken on each step. SPC/E potential was used as water model in the series simulation

due to its adaptability to wide range of temperature and pressure. It fits the study of CO2

properties in electrolyte solutions at low temperature and high pressure, which is the topic

of the following two chapters. In the previous chapter, we have found the most suitable

model for studying the diffusion of CO2 in water, namely, SPC/E. SPC/E is known to yield

a reasonable description of the properties of water and it is computationally convenient.

Most importantly, Patra and Karttunen (2004) has shown that SPC/E is an ideal model for

study of NaCl solution. Thus in this chapter, SPC/E will be used as the water model and

Gromacs ion model as the NaCl ion model.

There are three reasons we chosed the Gromacs ion model. First, during the simulation

study of NaCl aqueous solution (Patra and Karttunen, 2004), the results appear to be most

sensitive to the nature of intermolecular potential for water. The water model is more criti-

cal. Since we have selected the most suitable water model, ion model seems less important.

Second, the study of the Na+−Cl− radial distribution function shows two different results.

One is from the results by using SPC/E + Dang model used by Chowdhuri and Chandra

(2001) and the other is from TIP3P+OPSL/AA+Aqvist model used by Chen and Pappu

(2007). The SPC/E model used in our discussion is the same as Chowdhuri, but a different

ion model was used to discover the ion force field influence on the results. The third reason

is that part of the rationale of this work was to compare and contrast the difference between

the ternary and binary systems. The choice of intermolecular potential is not critical to

achieving this goal.

The functional form for the interaction parameters were given as a Lennard-Jones po-

tential and the normal mixing rules applied for all the intermolecular interactions as detailed

in (Chapters 3 and 4).

The first system we used was to study the diffusion and radial distribution function

change of the water+ NaCl binary system with changing of temperature. The percentage

66


Table 5.1: Temperature dependence in NaCl solutions

T (K) Number of Water Number of Na+ Number of Cl− P(Atm)278 18406 135 135 1288 18406 135 135 1298 18406 135 135 1

Table 5.2: Pressure dependence in NaCl solutions

P(Atm) Number of Water Number of Na+ Number of Cl− T (K)A 1 18406 135 135 298B 400 18406 135 135 298C 700 18406 135 135 298

of water and NaCl of the system and pressure were kept unchanged, as detailed in Table

5.1.

The second system we used was to study the diffusion and radial distribution function

change of the water+NaCl binary system with changing of pressure. The percentage of

water and NaCl of the system and temperature all kept unchanged as detailed in Table 5.2.

For the potential parameters used in this chapter, please refer to Table 5.3.

Table 5.3: Potential parameter used in NaCl solutions

Site σ

(A)

ε (KJ/mol) charge(e)

Na+ 2.572 0.0148 +1.0Cl− 4.448 0.1064 -1.0H - - +0.4238

OH 3.166 0.65016 -0.8476

67


5.4 Structural Properties of Binary System

5.4.1 Temperature Dependence

Figure 5.1 shows that at 1atm pressure, with temperature increasing, the first peak height

of Na+−Cl− increases obviously, the position of the first peak is always at 0.26 nm,

unchanged with temperature.

The RDF data for different force fields at 298 K and 1atm are available in the literature

(Patra and Karttunen, 2004). Our results are consistent with the data obtained using SPC

water model and the Gromacs ion model, in terms of the position and height of the first

peak. The data for other force field can also be found in the literature (Patra and Karttunen,

2004) , like SPC/E+ AMBR, TIP4P+SMIT, TIP3P+XPLR. These data, however, are not

consistent with the first peak height changing from 8 to 56 and position from 0.25 nm to

0.3 nm. It further demonstrates that the simulation results are sensitive to the force field.

The increase of the first peak height with temperature increasing demonstrates that op-

positely charged ions are more easily associated with each other with temperature increas-

ing, which is consistent with experiment. Zimmerman (1995) used a flow conductance ap-

paratus to detect NaCl conductance at various temperatures. With temperature increasing,

conductance reduces rapidly. From the viewpoint of experiment, with temperature increas-

ing the oppositely charged ions are indeed more easily associated with each other. The

reason, we think, is the change of water dielectric properties. Water dielectric properties

have a big impact on electrolyte dissociation and solvation of ions. When temperature goes

up, the dielectric properties of water reduces rapidly. This is also found in other computer

simulation (Cui and Harris, 1995; Wallqvist and Berne, 1985).

Radial distribution function can, in principle, be measured by X-ray diffraction because

they are related to the fourier transformation of the structure factor. To get a strong enough

68


signal, the substances have to be of sufficiently high concentration. This makes the experi-

mental determination of Na+−Na+, or Cl−−Cl− rdfs impossible. Thus, few experimental

data are available and molecular simulation is a great help in this regard.

When developing empirical force fields, however, one matches certain experimental

quantities with their counterparts, as determined by numerical simulations. The choice is

determined by the availability of high-quality experimental data and physical significance

of that quality. The parameters of a force field thus depend critically on the choices of

quantities that are being compared. The force fields discussed here have been parameter-

ized using the available experimental information on aqueous NaCl, and therefore repro-

duce those experimental values reasonable well. In contrast, structural properties without

known experimental values vary significantly between force fields. This uncertainty has

become a significantly problem recently.

With more powerful computing capability and more advanced molecular models avail-

able, we studied the radial distribution function of Na+−Na+, Cl−−Cl− under different

temperatures and pressures in a qualitative way.

Fig 5.1 and Table 5.4 show that with temperature increasing and pressure being un-

changed, the radial distribution function of Na+−Na+, Cl−−Cl− represents a little bit

increase with temperature, but the scatter with radial distribution function proves there is

very little interaction between Na+−Na+ and Cl−−Cl−. Table 5.4 has shown that the

coordination number between Na+−Na+ and Cl−−Cl− is so small that they can barely

mean anything.

As shown in Figure 5.3 and Figure 5.5, under a certain pressure, the first peak height of

Cl− and water reduces with temperature increasing. It shows that when temperature rises,

the correlation between Cl− and water is weakened. Figure 5.1 and Figure 5.2 show that

Na+−Cl− get together more easily with temperature increasing and pressure unchanged.

When temperature rises, the ions of opposite and same charge occupy the space around Cl

ions, and thus the water molecules around Cl ions are reduced.

69


�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.505101520253035g(r)

r(nm) 1atm 278K 1atm 288K 1atm 298KNa-Cl

�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.00.20.40.60.81.01.21.4g(r)

r(nm)

Na-Na

�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.00.20.40.60.81.01.21.4g(r)

r(nm)Cl-Cl

Figure 5.1: Radial distribution functions for Na+-Cl−, Na+-Na+ and Cl−-Cl− (from top tobottom) in different temperatures 278 K(�), 288 K(•), 298 K(N) at a pressure of 1 atm.

70


Table 5.4: Peak heights and coordination numbers of ion-ion at different temperatures inthe NaCl solutions

1st 1st CoordinationMaximum(nm) Minimum(nm) numbers

Na+−Cl− (278 K) 0.26(0.01) 0.36(0.01) 0.1Na+−Cl− (288 K) 0.26(0.01) 0.36(0.01) 0.119Na+−Cl− (298 K) 0.26(0.01) 0.37(0.01) 0.168Na+−Na+ (278 K) 0.35(0.01) 0.48(0.01) 0.023Na+−Na+ (288 K) 0.35(0.01) 0.48(0.01) 0.028Na+−Na+ (298 K) 0.35(0.01) 0.49(0.01) 0.035Cl−−Cl− (278 K) 0.44(0.01) 0.46(0.01) 0.016Cl−−Cl− (288 K) 0.44(0.01) 0.47(0.01) 0.018Cl−−Cl− (298 K) 0.44(0.01) 0.47(0.01) 0.02

Coordination numbers were determined at 1st minimum in the according g(r). See textfor discussion. Numbers in brackets are uncertainties in nm. —Features of Figure 5.2.

�275 280 285 290 295 3000.000.020.040.060.080.100.120.140.160.180.200.220.24

CNT(K)

Na-Na Cl-Cl Na-Cl

Figure 5.2: Coordination number of Na+−Cl− (N), Na+−Na+ (�) and Cl−−Cl− (•) atdifferent temperatures at a pressure of 1 atm.

71


Table 5.5: Peak heights and coordination numbers of ion-water in different temperatures inthe NaCl solutions


Cl−−H (278 K) 0.22(0.01) 0.31(0.01) 6.92Cl−−H (288 K) 0.22(0.01) 0.30(0.01) 6.68Cl−−H (298 K) 0.22(0.01) 0.30(0.01) 6.66

Cl−−OH (278 K) 0.32(0.01) 0.39(0.01) 7.38Cl−−OH (288 K) 0.32(0.01) 0.39(0.01) 7.36Cl−−OH (298 K) 0.32(0.01) 0.39(0.01) 7.3Na+−H (278 K) 0.29(0.01) 0.37(0.01) 13.7Na+−H (288 K) 0.29(0.01) 0.36(0.01) 12.95Na+−H (298 K) 0.29(0.1) 0.36(0.1) 12.9

Na+−OH (278 K) 0.23(0.1) 0.31(0.1) 5.364Na+−OH (288 K) 0.23(0.1) 0.31(0.1) 5.3165Na+−OH (298 K) 0.23(0.1) 0.31(0.1) 5.23

Numbers in brackets are uncertainties in nm. Features of Figure 5.3 and Figure 5.4.

As shown in Figure 5.4 and Figure 5.3, under a certain pressure, the first peak height

of Cl− and water reduces with temperature increasing. It shows that when temperature

rises, the correlation between Cl− and water is weakened. The gCl−O (r) and gCl−H (r) pair

correlation functions show two well pronounced peaks at 0.32 and 0.22 nm respectively.

The examination of Cl−−water radial pair correlation functions show that the hydrogen

atoms point toward the Cl−, whereas the oxygen atoms face bulk water. That is to say, the

correlation function between Cl− and water hydrogen is mostly dependant on hydrogen

atoms and will be greatly weakened with the increasing of temperature. As a result, the

number of hydration in the first shell around Cl− is changed.

In Figure 5.4 we find that the first peaks in the Na+−H distribution functions lie farther

away than the first maximums in the Na+−O radial distribution functions. This indicates

that the oxygen atoms point toward the ion and the hydrogen atoms face bulk water.

Figure 5.6 depicts the radial distribution functions of O−H and H −H in water in

different temperatures. The two peaks in O−H correspond to the two hydrogen atoms of

72


�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.00.51.01.52.02.53.03.54.04.5g(r)

r(nm)

1atm 278K 1atm 288K 1atm 298KCl-OH

�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.51.01.52.02.53.03.54.0

r(nm)

g(r)

Cl-H

Figure 5.3: Radial distribution functions for Cl−−OH , and Cl−−H (from top to bottom)at different temperatures 278 K(�), 288 K(•), 298 K(N) at a pressure of 1atm.

73


�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00123456789

r(nm)

g(r)

1atm 278K 1atm 288K 1atm 298KNa-OH

�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.51.01.52.02.53.03.54.0g(r)

r(nm)

Na-H

Figure 5.4: Radial distribution functions for Na+−OH , and Na+−H (from top to bottom)at different temperatures 278 K(�), 288 K(•), 298 K(N) at a pressure of 1 atm.

74

CHAPTER 5. STRUCTURAL PROPERTIES AND DIFFUSION COEFFICIENTS OF NACL AQUEOUS SOLUTIONS�

275 280 285 290 295 30024681012141618

CNT(K)

Cl-H Cl-OH Na-OH Na-H

Figure 5.5: Coordination numbers of Cl−−H (�) , Cl−−OH (•) , Na+−OH (N) andNa+−H (H) at different temperatures at a pressure of 1atm.

water molecule in the first shell .The first peaks corresponds to the hydrogen bonding peak.

O−H radial distribution function shows that under a certain pressure, the first peak height

reduces slightly with temperature increasing, indicating that hydrogen boning network is

breaking down with increasing temperature. It is echoed by the evident reduction of the

first H−H peak height. The phenomena are consistent with the literatures (Smith et al.,

2005).

5.4.2 Pressure Dependence

In Figure 5.7, at 298 K the first peak height of Na+−Cl− reduces greatly when pressure

increases. The first peak always appears at 0.26 nm. The equation suggested by Bradley and

Pitzer (1979) for the region above saturation pressure and temperatures below 350 K shows

that under a certain temperature, the dielectric properties of water increases with pressure.

He draws a curve indicating the water dielectric properties increase with pressure at 298

K. The curve is even and facing inward. That is to say, the water dielectric increases with

75


�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.51.01.52.02.53.03.5g(r)

r(nm)

1atm 278K 1atm 288K 1atm 298KOH-OH

�0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.20.40.60.81.01.21.41.61.82.0

g(r) of OH-Hr(nm)

OH-H

�0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.20.40.60.81.01.21.41.6

g(r)r(nm)

H-H

Figure 5.6: Radial distribution functions between OH-OH , OH-H and H-H (from top tobottom) at different temperatures 278 K(�), 288 K(•), 298 K(N) at a pressure of 1atm.

76


Table 5.6: Peak heights and coordination numbers between ions for different pressures inthe NaCl solutions.


Na+−Cl− (1 atm) 0.26(0.01) 0.37(0.01) 0.168Na+−Cl− (400 atm) 0.26(0.01) 0.37(0.01) 0.14Na+−Cl− (700 atm) 0.26(0.01) 0.37(0.01) 0.12Na+−Na+ (1 atm) 0.35(0.01) 0.49(0.01) 0.035

Na+−Na+ (400 atm) 0.34(0.01) 0.47(0.01) 0.03Na+−Na+ (700 atm) 0.34(0.01) 0.47(0.01) 0.021

Cl−−Cl− (1 atm) 0.44(0.01) 0.47(0.01) 0.02Cl−−Cl− (400 atm) 0.44(0.01) 0.47(0.01) 0.012Cl−−Cl− (700 atm) 0.44(0.01) 0.47(0.01) 0.01

Coordination numbers were determined at 1st minimum in the according g(r). See textdiscussion. Numbers in brackets are uncertainties in nm. —Features of Figure 5.7.

temperature increasing, but slowly. Especially it increases slower and slower with pressure

increasing. Our findings are similar. When pressures changes from 1atm to 100 atm, the

reduction of the first peak height of Na+−Cl− radial distribution function is more evident

or bigger than pressure changing from 100 atm to 300 atm. It further demonstrates the

dielectric properties of water have big impact on ion structure, dissociation and salvation.

Figure 5.7 also shows that with pressure increasing and temperature remaining unchanged,

the first peak height of the Na+−Na+, Cl−−Cl− radial distribution function decreases

slightly when pressure changes from 1 atm to 700 atm.

As shown in Figure 5.9, Figure 5.10 and Figure 5.11, at a certain temperature, the co-

ordination number between Cl− and water, Na+ and water increases with pressure increas-

ing. Figure 5.7 and Figure 5.8 show that Na+−Cl− is more difficult to get together with

pressure increasing while temperature unchanged, so the coordination number decreases

significantly with increase pressure. When pressures rises, the ions of opposite and same

charge get other, thus they do not have as big number as low pressure to occupy the space

around water, and thus the water molecules around ions are increased.

77

CHAPTER 5. STRUCTURAL PROPERTIES AND DIFFUSION COEFFICIENTS OF NACL AQUEOUS SOLUTIONS�0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0051015

20253035g(r)r(nm)

298K 1atm 298K 400atm 298K 700atmNa-Cl

�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.00.20.40.60.81.01.21.4g(r)

r(nm)

Na-Na

�0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.00.20.40.60.81.01.21.4

g(r)r(nm)

Cl-Cl

Figure 5.7: Radial distribution functions for Na+−Na+, Cl−−Cl−, and Na+−Cl− (fromtop to bottom) at different pressures, 1 atm(�), 400 atm(•), 700 atm(N) at a temperature of298 K.

78

CHAPTER 5. STRUCTURAL PROPERTIES AND DIFFUSION COEFFICIENTS OF NACL AQUEOUS SOLUTIONS�-100 0 100 200 300 400 500 600 700 8000.000.020.040.060.080.100.120.140.160.18

CNP(Atm)

Na-Na Cl-Cl Na-Cl

Figure 5.8: Coordination numbers of Na+−Na+ (�), Cl−−Cl− (•) and Na+−Cl− (N)at different pressures at a temperature of 298 K.

Table 5.7: Peak heights and coordination numbers of ions-water for different pressures inthe NaCl solutions


Cl−−H (1 atm) 0.22(0.01) 0.30(0.01) 6.66Cl−−H (400 atm) 0.22(0.01) 0.30(0.01) 6.726Cl−−H (700 atm) 0.22(0.01) 0.30(0.01) 6.78Cl−−OH (1 atm) 0.32(0.01) 0.39(0.01) 7.3

Cl−−OH (400 atm) 0.32(0.01) 0.39(0.01) 7.42Cl−−OH (700 atm) 0.32(0.01) 0.39(0.01) 7.48

Na+−H (1 atm) 0.29(0.01) 0.36(0.01) 12.9Na+−H (400 atm) 0.29(0.01) 0.37(0.01) 13.7Na+−H (700 atm) 0.29(0.01) 0.37(0.01) 13.76Na+−OH (1 atm) 0.23(0.01) 0.31(0.01) 5.23

Na+−OH (400 atm) 0.23(0.01) 0.31(0.01) 5.29Na+−OH (700 atm) 0.23(0.01) 0.31(0.01) 5.31

Coordination numbers were determined at 1st minimum in the according g(r). See textdiscussion. Numbers in brackets are uncertainties in nm.

Features of Figure 5.9 and 5.10.

79


�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.00.51.01.52.02.53.03.54.0g(r)

r(nm)

298K 1atm 298K 400atm 298K 700atmCl-OH

�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.51.01.52.02.53.03.54.0g(r)

r(nm)

Cl-H

Figure 5.9: Radial distribution functions for Cl−−OH , and Cl−−H (from top to bottom)at different pressures, 1atm(�), 400 atm(•), 700 atm(N) at a temperature of 298 K.

80


�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00123456789

r(nm)

g(r)

298K 1atm 298K 400atm 298K 700atmNa-OH

�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.51.01.52.02.53.03.5

r(nm)

g(r)

Na-H

Figure 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 temperature of 298 K.

81

CHAPTER 5. STRUCTURAL PROPERTIES AND DIFFUSION COEFFICIENTS OF NACL AQUEOUS SOLUTIONS�

-100 0 100 200 300 400 500 600 700 8002345678910111213

1415161718CN

P(Atm)


Figure 5.11: Coordination numbers of Cl−−OH (•) , Cl−−H (�) , Na+−OH(N) andNa+−H (H) at different pressures at a temperature of 298 K.

Figure 5.12 shows the hydrogen atom has strong tendency to form linear bonds. The

gOH−H(r) shows the first peak around 0.175 nm, which corresponds to the average length

of the hydrogen bonds between water molecules. The experimental H bond between two

water molecules is 0.185 nm, which is considerably less than the intermolecular distance

expected from the van der Waals radii for O and H, while greater than the O−H bond

length of 0.1 nm. Also in Figure 5.12, the radial distribution function of O−O in water

shows that at a certain temperature, with pressure increasing, the first peak height of the

O−O radial distribution function represents little change or very slightly.

5.5 Dynamic Properties

Usually, when pressure is applied to a liquid, the molecules are forced together, holes and

defects vanish, and the viscosity of the liquid rises. In liquid water below about 33°C, the

82


�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.51.01.52.02.53.03.5g(r)

r(nm)

298K 1atm 298K 400atm 298K 700atmOH-OH

�0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.20.40.60.81.01.21.41.61.8

r(nm)

g(r) OH-H

�0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.20.40.60.81.01.21.41.6

g(r)r(nm)

H-H

Figure 5.12: Radial distribution functions between OH −OH , OH −H and H −H (fromtop to bottom) at different pressures, 1 atm(�), 400 atm(•), 700 atm(N) at a temperature of298 K.

83


situation is reversed. This decreasing viscosity with increasing pressure can therefore be

classed as one of this substance’s most interesting anomalies.

But what is about the NaCl aqueous solution? Will the diffusion of water increase with

pressure or decrease with pressure. Do the hydrogen bonds still play an important role

to decide the diffusion properties? A lot of study is available about NaCl aqueous solu-

tion, but mostly around the physical property change of NaCl aqueous solution at different

concentrations. Changes in properties such as diffusion at different pressures is seldom

covered.

Figure 5.13 depicts the self diffusion coefficient of water, Na+ , and Cl− at 0.42 M

NaCl solution under a certain pressure and different temperatures. Chowdhuri and Chandra

(2001) studied NaCl diffusion coefficient and water diffusion coefficient in various NaCl

concentration at 298 K based on SPC/E water model and Dang ion model. Our study here

uses SPC/E water model but the Gromacs ion model. They studied NaCl concentration of

0.0 M, 0.88 M (human NaCl), 2.2 M and 3.35 M, but not the same concentration as we

have done here. As one of our purpose is to mimic the sea enviroment as best as we can.

So, we chosed the 0.5 M as the concentration here.

Their data demonstrates that with the increasing of NaCl concentration, the diffusion

of Na+, Cl− and water molecule all decreases. These data are very helpful to our study

as 0.42 M is between 0 M and 0.88 M. The water diffusion at 298 K in our system is

close to the data of 0.88 M, which is in a good agreement with their data. The diffusion

coefficient of Na+ in this figure also increases with increasing temperature, showing a

large dependence on temperature. The self diffusion of Na+ at 0.42 M, however, is a bit

smaller than Chowdhuri’s data (Chowdhuri and Chandra, 2001). The difference is caused

by different ion models. Due to lack of experimental data, it is hard to say which ion model

is better. Water diffusion, however, is quite similar as their data.

Figure 5.14 shows the water diffusion at 298 K at different pressures. It is found that

with pressure increasing, there is very small change to the water diffusion coefficient within

84


allowable statistic error range. To be specific, water diffusion coefficient decreases very

slowly. Increased pressure and unchanged temperature certainly affect the free movement

of water molecules and in fact decrease diffusion. Pressure change actually affects water

diffusion to far less a degree than temperature change. The reason is that temperature

increase stimulates activity of water molecule, while pressure change is more likely to

change structure or density.

Generally the diffusion of a liquid decreases with pressure increasing. Water, however,

is a kind of very special liquid. In many studies, the diffusion of liquid water is even

found slightly increasing with temperature increasing, and reduces gradually under very

big pressure. The special feature of water is because of hydrogen bonding.

An interesting phenomenon is found in our study. With pressure increasing, water dif-

fusion reduces slowly like other liquid instead of increasing. This may caused by NaCl,

which breaks down the hydrogen in water and invalidate the cause of liquid water being

special. Liquid water then has the feature of an ordinary liquid. The change of Cl− diffu-

sion coefficient represents small decrease within allowable error range.

Patra and Karttunen (2004) have undertaken systematic research on ion solubility

in NaCl solution using a series of combinations of water and ion models including

TIP3P+XPLR, SPC/E+AMBR, SPC+GROM, TIP4P+SMIT etc. Salt concentration of 0.87

mol was specified in this study. Infinite dilution experimental data was used in their work

as no adequate NaCl diffusion experimental data of different concentrations is available.

The following experimental data was used: diffusion of Na+= 11.334 •10−5 cm2/s and

Cl−=2.032•10−5 cm2/s. As indicated by Patra and Karttunen’s study, ion solubility varies

significantly with concentration, and thus Patra and Karttunen were unable to determine the

best model for ion diffusion simulation. There is a very important conclusion from Patra

and Karttunen’s research that dynamics is determined by the water model while the contri-

bution of the ionic force field is negligible, which is maybe the reason why our results are

familar with the reults from Chowdhuri’s results since we have used the same water model.

85


�

275 280 285 290 295 3000.00.20.40.60.81.01.21.41.61.82.02.22.42.62.83.0

D(10-5 cm2 /s)

T(K)

Na Cl H2O

Figure 5.13: Diffusion coefficients of Na+ (�), Cl− (•) , H2O (N) at different temperaturesand 0.42 M NaCl

�

-100 0 100 200 300 400 500 600 700 8000.00.51.01.52.02.53.03.5

Na Cl H2OD(10-5 cm2 /s)

P(atm)Figure 5.14: Diffusion coefficients of Na+ (�), Cl− (•) , H2O (N) at different pressuresand 0.42 M NaCl

86

CHAPTER 6. STRUCTURAL PROPERTIES AND DIFFUSION COEFFICIENTS OF CARBON DIOXIDE IN AQUEOUS SOLUTIONS

Chapter 6

Structural Properties and Diffusion

Coefficients of Carbon Dioxide in

Aqueous Solutions

6.1 Introduction

Seawater and water in saline aquifers is a multi component mixture containing many dif-

ferent salts (Bottcher et al., 2007; Horita et al., 1991). However, sodium chloride is by far

the dominant salt. This means that, at least to a first approximation, the carbon dioxide +

seawater system can be usefully modelled by the ternary carbon dioxide + water +sodium

chloride mixture. This ternary mixture is more realistic than a simple binary carbon dioxide

+ water model and it incorporates the influence of three dissimilar interactions.

The behaviour of ternary water + carbon dioxide + sodium chloride system is influenced

by the interplay of interaction both between like and dissimilar molecules. An understand-

ing of the like-interactions can be obtained by studying pure component properties. In

contrast, there are two contributions to dissimilar interactions, namely the contribution of

87


the three binary sub-systems (water + carbon dioxide, water + sodium chloride, sodium

chloride + carbon dioxide) plus the various ternary combinations of the mixture compo-

nents. The pure component properties of water, carbon dioxide, and to a lesser extent

sodium chloride, have been investigated extensively in the literature. Therefore, the focus

of this work will be on the ternary mixture and binary sub-systems.

In the past two chapters, we have studied the binary sub-systems of carbon + water,

water + sodium chloride. In this chapter, we will focus on the ternary system of water +

carbon dioxide + sodium chloride.

There has been intensive research about the ternary system of water + carbon dioxide

+ sodium chloride. Most of them discuss the phase equilibrium of the ternary system

(Sabil et al., 2009; Seo and Lee, 2003; Shmulovich and Plyasunova, 1993; Duan et al.,

2006a; Dubessy et al., 2005; Lee et al., 2002; Baseri et al., 2009) 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., 2006b,a; Bando

et al., 2003; Lee et al., 2002). In contrast, there are few discussions covering the dynamic

properties and structural properties of the ternary system.

This chapter will cover two parts. Part one discusses the effect of NaCl concentration

on the NaCl + water + CO2 ternary system at various temperature and pressure, i.e. radial

distribution functions and coordination number. Part two discusses the effect of NaCl

concentration on the diffusion of the NaCl + water + CO2 ternary system under varying

temperature and pressure.

6.2 Simulation Systems

A series of molecular simulation based on Gromacs were used to study properties in wa-

ter. A leap-frog algorithm (Chapter 3.2.5) was used for integrating Newton’s equations of

motion. Fast particle-Mesh Ewald electrostatics (Chapter 3.2.7) was used for calculating

88


Coulomb force. The LJ potential was decreased over the whole range and the forces decay

smoothly to zero between 0.8 and 1.0. Temperature coupling with a Berendsen-thermostat

(Chapter 3.2.6) to a bath was set at a temperature, exponential relaxation pressure coupling

with time constant 0.1ps. Temperature for Maxwell distribution was set to 298 K. All bonds

are converted to constraints by shake algorithm. A cut-off of 10 A was used for evaluation

of Lennard-Jones interactions. The total time 200 ps was spent on calculation and 1ns was

taken on each step.

SPC/E potential was used as the water model in the series of simulations due to its

simplicity and adaptability to wide range of temperature and pressure. The Gromos CO2

model was selected which is proved to be in good agreement with experimental data of

solubility of CO2 in water. The functional form for the interaction parameters were given

as a Lennard-Jones potential as the previous chapters and so were the mixing rules for all

the intermolecular molecules.

In this chapter, we will discuss the effect of temperature, pressure and NaCl concentra-

tion on the ternary system via three systems.

The first system we adopt is to study the change of diffusion and radial distribution

function caused by NaCl concentration. In this system, we are about to look at the infinite

diluted CO2. If there is only one CO2 molecule, diffusion will show violent fluctuation. In

our research, 15 CO2 molecules and nearly 20,000 water molecules were used to constitute

an infinite dilution system. In order to study the effect of NaCl on the ternary system, we

kept CO2, T and P unchanged during the simulation. Please refer to Table 6.1 for simulation

settings.

The second system we used is to study the diffusion and radial distribution function

change of the water + NaCl+ CO2 ternary system with the change of temperature. The

percentage of water, NaCl and CO2 of the system and pressure were all kept unchanged.

For this system, refer to Table 6.2 for the simulation settings.

89


Table 6.1: Concentration dependence settings

NaCl Number Number Number Number T PMole fraction of Water of Na+ of Cl− of CO2 (K) (Atm)

0.0074 18608 139 139 15 298 10.0515 17036 925 925 15 298 10.114 15036 1925 1925 15 298 1

Table 6.2: Temperature dependence settings

Temperature Number Number Number Number P(K) of Water of Na+ of Cl− of CO2 (Atm)278 18608 139 139 15 1288 18608 139 139 15 1298 18608 139 139 15 1

The third system we used is to study the diffusion and radial distribution function

change of the water + NaCl + CO2 ternary system with the change of pressure. The per-

centage of water, NaCl and CO2 of the system and temperature were all kept unchanged.

For this system, refer to Table 6.3 for the simulation settings and parameters.

Table 6.3: Pressure dependence settings

Pressure Number Number Number Number T(Atm) of Water of Na+ of Cl− of CO2 (K)

1 18608 139 139 15 298400 18608 139 139 15 298700 18608 139 139 15 298

90


6.3 Structural Properties of Ternary System

6.3.1 Effects of Different NaCl Concentrations

6.3.1.1 Ion-Ion distribution functions

Figure 6.1 depicts the radial distribution function of Na+−Na+, Cl−−Cl− , and Na+−

Cl− in different NaCl concentrations. It is found that with NaCl concentration increasing,

the first peak of Na+−Na+, Cl−−Cl− radial distribution function rises while the first

peak of Na+−Cl− decreases.

Table 6.4 shows the structure features of Figure 6.1 and the hydration numbers which

were determined at 1st minimum according to g(r). From Table 6.4, it is easy to find out that

at low concentration XNaCl = 0.0074, there is almost no Na+−Na+ , Cl−−Cl− connection.

The coordination numbers are 0.04 and 0.02, respectively. When concentration becomes

higher, the coordination numbers between Na+−Na+, Cl−−Cl− increase sharply. When

mole fraction is XNaCl = 0.114, the coordination number between them is 2.03 and 1.776.

This shows that clustering is beginning. At a concentration of XNaCl = 0.0074, the Na+−

Cl− coordination number is 0.16, which shows that even at low concentration, there is still

connection between Na+ and Cl−. With NaCl concentration increasing, the Na+−Cl−

coordination number increases slower than Cl−−Cl− coordination numbers. (Refer to

Figure 6.2 for more information ).

The reason may lie in that the origin of increasing correlations between like-charged

ions must results from the increased likelihood of populating ion triplets and higher or-

der clusters at higher concentrations. This is consistent with Chen and Pappu’s finding

(2007). In their study, the Na+−Cl− radial distribution function also shows that the first

peak of Na+ and Cl− decreases in height with NaCl concentration increasing. Chen and

Pappu concluded that triples or higher order clusters are formed between ions. In addition,

the triplets consist of two like-charged ions bridged by an intervening ion of the opposite

91


Table 6.4: Structure features of ions-ions at different NaCl concentrations

Mole fraction of NaCl 1st Maximum 1st Minimum Coordination Number(nm) (nm)

Na+−Na+(0.0074) 0.37(0.01) 0.49(0.01) 0.04Na+−Na+(0.0515) 0.43(0.01) 0.50(0.01) 0.496Na+−Na+(0.114) 0.43(0.01) 0.54(0.01) 2.03Cl−−Cl−(0.0074) 0.44(0.01) 0.48(0.01) 0.022Cl−−Cl−(0.0515) 0.44(0.01) 0.49(0.01) 0.48Cl−−Cl− (0.114) 0.44(0.01) 0.50(0.01) 1.776Na+−Cl−(0.0074) 0.26(0.01) 0.36(0.01) 0.16Na+−Cl−(0.0515) 0.26(0.01) 0.36(0.01) 0.78Na+−Cl−(0.114) 0.26(0.01) 0.36(0.01) 1.53

Coordinate numbers were determined at 1st minimum in the according g(r). See textdiscussion. Numbers in brackets are uncertainties in nm.

charge, which leads to an apparent correlation between like-charged ions. Our result, how-

ever, is different from Chowdhuri and Chandra’s (2001). In their study, the fraction of

contact ion pairs increase and that of solvent separated ion pairs decreases with increasing

ion concentration. In this situation, the Na+−Cl− radial distribution function shows that

the first peak of Na+ and Cl− increases in height with NaCl concentration increasing.

There are two explanations for the different findings of Chen and Pappu (2007) and

Chowdhuri and Chandra (2001). The first reason is that different concentrations were used.

In Chowdhuri’s study, NaCl concentrations were 0.88 M, 2.2 M, 3.35 M and 4.5 M. In

Chen’s study, concentrations are 100 mM, 250 mM, 500 mM, 750 mM and 1 M. Different

NaCl concentrations may change the interaction and structure of Na+ and Cl−, Na+ and

Na+ , and Cl− and Cl− ion pairs. For example, in a certain situation, ions have cation-

anion correlation; in another situation, ion triplets or higher order clusters are easily formed.

Although our research ranges from infinite dilution to XNaCl = 0.114, the radial distribution

of Na+−Na+, Cl−−Cl−, Na+−Cl− is quite consistent with Chen’s study. Thus, the

reason is not the different concentrations, but different ion models.

92


The existence of the two different phenomena is perhaps because of use of different

intermolecular potential. It is found that different intermolecular potential can change the

Na+−Cl− radial distribution function significantly (Patra and Karttunen, 2004). For ex-

ample, the Na+−Na+ radial distribution function shows that short correlation is evident

between Na+ and Na+ under Na+−Cl− ion Gromacs molecular model and SPC water

molecular model; however, the Na+−Na+ radial distribution function shows opposite

change under AMBER ion model and SPC/E water molecular model that no short corre-

lation between Na+ and Na+ is found. Chowdhuri and Chandra (2001) used the SPC/E

water molecular model and Dang ion model (Dang, 1992; Dang and Garrett, 1993). Chen

and Pappu (2007) used TIP3P water model (Jorgensen et al., 1983) and ions were modelled

using parameters that are part of the all atom OPSL/AA force field (Jorgensen et al., 1996).

Some cation parameters in the force field are adaptations of values originally obtained by

Aqvist (1990).

A further investigation finds that the biggest difference of our ion model from Chen and

Pappu (2007), Chowdhuri and Chandra (2001) lies in Na ion parameters, especially the

electrostatic interaction potential parameters of Na+. 0.4184 KJ/mol was used by Chowd-

huri and Chandra while 0.01159 KJ/mol was used by Chen for ε . After closely comparing

Chen’s model parameters and Chowdhuri and Chandra’s, we found that they used almost

the same Cl− Lennard-Jones parameters and electrostatic interaction potential parameters

only with very slight difference. This is also the same for Na+ Lennard-Jones parameters,

but difference exists between Na+ electrostatic interaction potential. This demonstrates

that in aqueous solution, ion electrostatic interaction potential parameter has evident im-

pact on ion clustering pattern. It can decide whether ions are more likely grouped together

only by opposite ions or more likely be grouped together by ion groups, like, populating

ion triplets and higher order clusters. By comparing our study and Chen and Pappu (2007),

Chowdhuri and Chandra (2001), we are more certain about the correctness of our conclu-

sion. The water molecule model in our study is SPC/E, which is as the same as Chowdhuri

93


Table 6.5: Parameters used to model Lennard-Jones interactions of anions and cations.

Ion σ

(A)

ε (KJ/mol) charge(e) Literature reference

Na 2.583 0.4184 +1.0 (Chowdhuri and Chandra, 2001)Na 3.33045 0.01159 +1.0 (Chen and Pappu, 2007)Na 2.572 0.0148 +1.0 This researchCl 4.4 0.4184 -1.0 (Chowdhuri and Chandra, 2001)Cl 4.41724 0.492 -1.0 (Chen and Pappu, 2007)Cl 4.448 0.1064 -1.0 This research

and Chandra, while different from Chen and Pappu. But our radial distribution functions

between ions shows consistent change to Chen and Pappu, but different from Chowdhuri

and Chandra. This demonstrates that the nature of ions themselves, in some sense, decides

the assembly status, simple cation-anion assembly between two ions or cluster of more

ions. Please refer to Table 6.5 for the parameters of different models.

6.3.1.2 Solvent atom distribution functions

Figure 6.3 shows the radial distribution functions for OH−OH , H−H, and OH−H (from

top to bottom) at different concentrations.

It is obvious that with NaCl concentration increasing, the first H−H peak height de-

creases. The diminishing of this peak shows that decrease in the orientation ordering of

a pair of hydrogen-bonded water molecules. We also find that with NaCl concentration

increasing, the first H −H minimum increases. This shows a decrease in the orientation

ordering of the pair of hydrogen-bonded water molecules.

With NaCl concentration increasing, the first OH −OH peak height decreases dramat-

ically. Like the decrease of the first H −H peak, it is because of water disruption. The

second OH −OH peak (r = 0.44 nm) is the signature of a high degree of correlation be-

tween second neighbor molecules at positions corresponding to vertexes of a tetrahedron

with a first neighbor molecule at the geometrical center of the tetrahedron. With NaCl

concentration increasing, the second peak diminishes at r is 0.44 nm till almost disappears.

94


�

0.0 0 .1 0 .2 0.3 0.4 0.5 0.6 0 .7 0 .8 0 .9 1 .0 1.1 1.2 1.3 1 .4 1 .505101520253035

r(nm ) X NaC l=0 .0074 X NaC l=0 .0515 X NaC l=0 .114

g(r) Na-C l

�0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .0 1 .1 1 .2 1 .3 1 .4 1 .5-0 .20 .00 .20 .40 .60 .81 .01 .21 .41 .61 .82 .02 .2

r ( n m )

g(r) N a -N a

�

0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .0 1 .1 1 .2 1 .3 1 .4 1 .5-0 .20 .00 .20 .40 .60 .81 .01 .21 .41 .61 .82 .02 .22 .42 .62 .83 .03 .23 .43 .6g(r)

r (n m )C l-C l

Figure 6.1: Radial distribution functions for Na+-Cl−, Na+-Na+, and Cl−-Cl− (from topto 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

CHAPTER 6. STRUCTURAL PROPERTIES AND DIFFUSION COEFFICIENTS OF CARBON DIOXIDE IN AQUEOUS SOLUTIONS�

0.00 0.02 0.04 0.06 0.08 0.10 0.120.00.51.01.52.0 CN

Mole fraction of NaCl

Na-Na Cl-Cl Na-Cl

Figure 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 pressure of 1 atm.

That is to say, short range order associated with tetrahedral coordination has been disrupted.

This was also found by other researchers. Mountain and Thirumalai (2004) used molecular

dynamics simulations to probe the ion-induced changes in the water structure as the con-

centration of ion is varied. As the concentration of sodium ions, Na+, is increased the water

structure was greatly perturbed. When the mole fraction of water is less than about 0.93,

the tetrahedral network of water is affected and there is a total disruption of the tetrahde-

dral structure of water, just as found in water at high pressures. The number of water-water

hydrogen bonds per water molecule is greatly diminished as Na+ concentration increases.

In Mountain’s system, he only used Na+ without Cl−. Sherman and Collings (2002) also

studied the physical properties of NaCl system at different temperatures and pressures and

found at 298 K, 0.1 MPa, near saturation, the O−O radial distribution function shows

similar trend that the second peak almost disappears.

96


�

0 . 0 0 .1 0 .2 0 . 3 0 .4 0 .5 0 . 6 0 .7 0 .8 0 . 9 1 . 00 . 00 . 51 . 01 . 52 . 02 . 53 . 0r ( n m )

X N a C l= 0 .0 0 7 4 X N a C l= 0 .0 5 1 5 X N a C l= 0 .1 1 4

g(r) O H -O H�

0 . 0 0 .1 0 .2 0 . 3 0 .4 0 .5 0 . 6 0 .7 0 .8 0 . 9 1 . 00 . 00 . 20 . 40 . 60 . 81 . 01 . 21 . 41 . 61 . 8

r ( n m )g(r) H -H

�

0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 .00 .00 .20 .40 .60 .81 .01 .21 .41 .61 .82 .0

r(nm )g(r)

O H -H

Figure 6.3: Radial distribution functions for OH−OH , H−H, and OH−H (from top to thebottom) 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


The first gOH−H(r) peak represents the hydrogen bonds between water molecules. With

NaCl concentration increasing, the first peak is clearly found decreasing. It also demon-

strates that with NaCl concentration increasing, the hydrogen bonds between water are

disrupted.

Leberman and Soper (1995) used neutron diffraction to compare the effects of applied

pressure and high salt concentrations on the hydrogen-bonded network of water. They

found that the ions induce a change in structure equivalent to the application of high pres-

sures, and that the size of the effect is ion-specific. They proposed that these changes may

be understood in terms of the partial molar volume of the ions, relative to those of water

molecules. We have found the similar phenomena in this research that the ions induce a

change in structure equivalent to the application of high pressures. For more details, please

refer to the chapter 6.3.2.

6.3.1.3 Ion-Water, CO2 –Water and Ion-CO2 distribution functions

As shown in Figure 6.4, the first peak height of Cl−−OH and Cl−−H reduce drastically

with concentration increasing. It shows that when concentration rises, the correlation be-

tween Cl− and water is weakened. Figure 6.1 has shown that Na+−Cl−, Na+−Na+ form

cluster more easily with concentration increasing. When concentration rises, the ions of

opposite and same charge occupy the space around Cl ions, and thus the water molecules

around Cl ions are reduced. It can be further demonstrated by the coordination numbers of

Cl−−H (Table 6.6), which reduce from 6.66 to 3.94. However, the coordination number

of Cl−OH does not reduce like the coordinate number of Cl−−H. On the contrary, it rises

from 7.16 to 8.61 from low concentration to high concentration (Table 6.6 and Figure 6.6).

This happens because Cl− shows a slight expansion of the first hydration shell from 0.39

nm to 0.44 nm (Table 6.6). The geometric definition of the first hydration shell becomes

ambiguous due to this shift of the position of the minimum of the Cl−−H radial distribu-

tion function. This is similar with the phenomena found by Driesner and Trionni (1998). In

98


their studies, Cl− shows a slight expansion of the first hydration shell by about 0.02 from

ambient to near critical temperatures. This causes confusion for the definition of the first

hydration of the shell. It also makes the proper definition of the hydration number more

ambiguous. These results show that current empirical approaches for modelling aqueous

fluids using simple electrostriction concepts do not adequately mimic the properties of the

actual microscopic structure at high temperatures. The solution which may solve this is to

take the position of the first minimum at ambient as the criterion. In this way, Driesner and

Trionni (1998) can get the right result.

If we take a further look, clearly, from Table 6.6 and Figure 6.6, it shows that all the

coordination numbers between ions and water are reducing except Cl−−H. So, can we

choose the Cl−−OH rather than Cl−−H to decide the hydration number? It is easy to find

that the first peaks in the Cl−−H radial distribution functions lie farther away than the first

maximums in the Cl−−OH radial distribution functions. This indicates that the oxygen

atoms point toward the ion and the hydrogen atoms face bulk water. So, when we try to

calculate the hydration number or try to define the first shell of the hydration, we may still

have to use Cl−−OH rather than Cl−−H.

Figure 6.5 shows the first peak heights of Na+−OH and Na+−H reduce drastically

with concentration increasing. Figure 6.6 and Table 6.6 further prove that when concentra-

tion rises, the correlation between Na+ and water is weakened.

Figure 6.7 shows the Radial distribution functions for OC−OH (top) and OC−H (bot-

tom) at different concentrations. It is found that with NaCl concentration increasing, both

the height of first peak of OC−OH and the height of first peak of OC−H height increases.

Figure 6.8 shows the radial distribution functions for C−OH (top) and C−H (bottom) in

different concentrations. The same trends can be found that both the height of first peak

of C−OH and the height of first peak of C−H height increase. These data clearly tell

us that with NaCl concentration increasing, the bond between CO2 and water molecules is

reinforced. Table 6.7 and Figure 6.9 can further prove this.

99


�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.51.01.52.02.53.03.54.0

r(nm)

XNaCl=0.0074 XNaCl=0.0515 XNaCl=0.114g(r) Cl-OH

�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.51.01.52.02.53.03.5g(r)

r(nm)

Cl-H

Figure 6.4: Radial distribution functions for Cl−−OH and Cl−−H (from top to the bot-tom) 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


�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0012345678910 XNaCl=0.0074 XNaCl=0.0515 XNaCl=0.114

r(nm)g(r)

Na-OH

�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.51.01.52.02.53.03.5g(r)

r(nm)

Na-H

Figure 6.5: Radial distribution functions for Na+−OH and Na+−H (from top to thebottom) 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


Table 6.6: Structure features of Figure 6.4 and Figure 6.5.

Mole fraction of NaCl 1st Maximum 1st Minimum Coordination Number(nm) (nm)

Cl−−H(0.0074) 0.24(0.01) 0.31(0.01) 6.66Cl−−H(0.0515) 0.23(0.01) 0.30(0.01) 5.12Cl−−H(0.114) 0.23(0.01) 0.30(0.01) 3.94

Cl−−OH(0.0074) 0.33(0.01) 0.39(0.01) 7.16Cl−−OH(0.0515) 0.33(0.01) 0.42(0.01) 8.14Cl−−OH(0.114) 0.33(0.01) 0.44(0.01) 8.61Na+−H(0.0074) 0.29(0.01) 0.36(0.01) 13.1Na+−H(0.0515) 0.29(0.01) 0.36(0.01) 11.65Na+−H(0.114) 0.30(0.01) 0.37(0.01) 9.79

Na+−OH(0.0074) 0.23(0.01) 0.32(0.01) 5.22Na+−OH(0.0515) 0.23(0.01) 0.31(0.01) 3.92Na+−OH(0.114) 0.23(0.01) 0.31(0.01) 2.93

Coordination numbers were determined at 1st minimum in the according g(r). See textdiscussion. Numbers in brackets are uncertainties in nm.�

0.00 0.02 0.04 0.06 0.08 0.10 0.1224681012

1416CN



Figure 6.6: Coordination numbers of Cl−−OH (•) , Cl−−H (�) , Na+−OH(N) andNa+−H (H) at different NaCl concentrations, at a temperature of 298 K and at a pressureof 1 atm.

102


�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.00.20.40.60.81.01.21.41.61.82.0

r(nm)

XNaCl=0.0074 XNaCl=0.0515 XNaCl=0.114g(r) OC-OH

�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.00.20.40.60.81.01.21.41.6g(r)

r(nm)

OC-H

Figure 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


�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.00.20.40.60.81.01.21.41.61.82.0 XNaCl=0.0074 XNaCl=0.0515 XNaCl=0.114

g(r)r(nm)

C-OH

�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.00.20.40.60.81.01.21.41.61.82.0

r(nm)g(r)

C-H

Figure 6.8: Radial distribution functions for C−OH and C−H (from top to the bottom) atdifferent 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


Table 6.7: Structure features of Figure 6.7 and Figure 6.8.

1st Maximum 1st Minimum Coordination Number(nm) (nm)

OC−OH(0.0074) 0.34(0.01) 0.47(0.01) 13.84OC−OH(0.0515) 0.34(0.01) 0.48(0.01) 13.87OC−OH(0.114) 0.34(0.01) 0.48(0.01) 14.43OC−H(0.0074) 0.36(0.01) 0.54(0.01) 40.2OC−H(0.0515) 0.36(0.01) 0.54(0.01) 40.3OC−H(0.114) 0.36(0.01) 0.55(0.01) 43.9

C−OH(0.0074) 0.40(0.01) 0.54(0.01) 20.98C−OH(0.0515) 0.40(0.01) 0.55(0.01) 21.4C−OH(0.114) 0.40(0.01) 0.55(0.01) 22C−H(0.0074) 0.37(0.01) 0.54(0.01) 43.34C−H(0.0515) 0.38(0.01) 0.55(0.01) 43.54C−H(0.114) 0.38(0.01) 0.55(0.01) 44.82


0.00 0.02 0.04 0.06 0.08 0.10 0.1215202530354045505560

CN OC-OH OC-H C-OH C-H

Mole fraction of NaClFigure 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


Figure 6.10 and Figure 6.11 are the radial distribution functions of Na+−CO2 and

Cl−−CO2. It is found that at low NaCl concentration, the radial distribution functions

show frequent small random change. This is because few NaCl and CO2 molecules exist

in the system. With NaCl concentration increasing, it is more and more stable. At NaCl

concentration being XNaCl = 0.114, even still in infinite dilution status, we can find that

the radial distribution function of NaCl−CO2 shows more and more stable and regular

change.

From Figure 6.10 and Figure 6.11, we also find that the change of NaCl concentration

has little impact on the radial distribution function of NaCl−CO2. The possible reason is

that the system mainly consists of many water molecules and also that the NaCl and CO2

has loose bond. It is also found that Na+ and CO2 cannot form strong bond, however, Cl−

and CO2 form a certain level bond.

6.3.2 Effects of Temperatures on the Ternary System

The NaCl + water + CO2 ternary system is very complicated and changes with temperature

and pressure. The solvent water is peculiar. For instance, while the density of a simple

liquid, such as argon, decreases monotonically with increasing temperature, H2O exhibits

a density maximum at T = 277 K and D2O around T = 283 K. Similarly, other properties,

such as translational and rotational diffusion present unexpected extrema when the pressure

is varied at constant temperature. There are few studies concerning the physical properties

of ternary system under different temperatures and pressures.

We will study the radial distribution function and diffusion of ternary system under

different temperatures and pressures in this part and the following part. For simulation

details, please refer to 6.2 Simulated Systems and for components of the system, please

refer to Table 6.2 temperature dependence.

106


�

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.01.21.4 XNaCl=0.0074 XNaCl=0.0515 XNaCl=0.114

r(nm)g(r)

Na-OC

�

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.01.21.4

g(r)

r(nm)

Na-C

Figure 6.10: Radial distribution functions between Na+ and carbon dioxide at differentNaCl concentrations XNaCl = 0.0074(�), XNaCl = 0.0515(•), XNaCl = 0.114(N), at a tem-perature of 298 K and at a pressure of 1 atm.

107


�

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.01.21.41.6 XNaCl=0.0074 XNaCl=0.0515 XNaCl=0.114

r(nm)g(r)

Cl-OC

�

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.01.21.41.6

r(nm)g(r)

Cl-C

Figure 6.11: Radial distribution functions between Cl− and carbon dioxide at differentNaCl concentrations XNaCl = 0.0074(�), XNaCl = 0.0515(•), XNaCl = 0.114(N), at a tem-perature of 298 K and at a pressure of 1 atm.

108


Table 6.8: Structure features of Figure 6.12.


Na+−Na+(278 K) 0.36(0.01) 0.49(0.01) 0.033Na+−Na+(288 K) 0.36(0.01) 0.49(0.01) 0.035Na+−Na+(298 K) 0.37(0.01) 0.49(0.01) 0.04Cl−−Cl−(278 K) 0.44(0.01) 0.47(0.01) 0.017Cl−−Cl−(288 K) 0.44(0.01) 0.48(0.01) 0.02Cl−−Cl−(298 K) 0.44(0.01) 0.48(0.01) 0.022Na+−Cl−(278 K) 0.26(0.01) 0.36(0.01) 0.12Na+−Cl−(288 K) 0.26(0.01) 0.36(0.01) 0.156Na+−Cl−(298 K) 0.26(0.01) 0.36(0.01) 0.165

Coordination numbers were determined at 1st minimum in the according g(r). See textdiscussion. Numbers in brackets are uncertainties in nm.

From Figure 6.12, Figure 6.13 and Table 6.8, we find that with temperature increasing,

the radial distribution function of Na+−Na+, Cl−−Cl− increases too, but to a far less

extent with NaCl concentration increasing. This is also found in the Na+−Cl− radial dis-

tribution function. In this case, ions are connected because of opposite electric properties

of Na+ and Cl− and do not form clusters, thus taking up much room. The big difference

is that when NaCl concentration changes, they form clusters, like Na+−Na+, Cl−−Cl−

and when temperature increases, the ions formed not in clusters, but more in a simple way,

Na+−Cl−. The increase of the first peak height with increasing temperature demonstrates

that oppositely charged ions are more easily associated with each other. This is consis-

tent with experiment. With temperature increasing, conductance reduces rapidly. From

the viewpoint of experiment, with temperature increasing the oppositely charged ions are

indeed associated with each other more easily. The reason, we think, is the change of water

dielectric properties. Water dielectric properties have big impact on electrolyte dissociation

and solvation of ions. When temperature goes up, the dielectric properties of water reduces

rapidly. It is also found in other computer simulation (Cui and Harris, 1995; Wallqvist and

Berne, 1985).

109


0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.00.20.40.60.81.01.21.4

r(nm)g(r)

278K 288K 298KNa-Na

�0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.00.20.40.60.81.01.21.4

r(nm)g(r)

Cl-Cl

�0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0051015

20253035r(nm)

g(r) Na-Cl

Figure 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 apressure of 1 atm, XNaCl = 0.0074.

110


275 280 285 290 295 3000.000.020.040.060.080.100.120.140.160.180.200.220.240.26

T(K)CN

Na-Na Cl-Cl Na-Cl Linear Fit of Na-Na Linear Fit of Cl-Cl Linear Fit of Na-Cl

Figure 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.

From the irregular fluctuation of Figure 6.14, we can tell that CO2 and CO2 have no

interaction in the solution.

As shown by Figure 6.15, the first peak radial distribution function of C−OH , C−H

reduces with temperature increasing; it is the same for for OC −OH , OC −H as shown

in Figure 6.16. It all concludes that with temperature increasing, the cohesion between

CO2 and water is weakened. The reason is perhaps because temperature increasing weak-

ens the forces of hydrogen bonds; temperature decreasing helps to form a more complete

network because of stronger hydrogen bonds, creating a more favorable environment for

CO2 existing. That is to say, CO2 and water are bonded more closely and also because

of the existence of CO2 molecules in the network, CO2 molecules form a closer and more

orderly cohesion by means of hydrogen bonds at lower temperatures. The connections be-

tween CO2 and water also become stronger due to the help of this hydrogen network. With

temperature increasing, the connections between hydrogen bonds in water breaks up and

111


0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.01.21.41.61.82.02.22.42.62.83.03.23.43.63.84.0

T=278K T=288K T=298KOC-OC

Figure 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.

this causes the whole hydrogen network collapse a bit. As a result, the connections be-

tween CO2 and water also become weaker due to the breaks down of the hydrogen network

where they stay in. This also can explain why the first peak radial distribution function of

OC−OH , OC−H reduces with temperature increasing.

Figure 6.18 and Figure 6.19 show the radial distributions between CO2 and NaCl. With

temperature increasing, the CO2−NaCl radial distribution functions show similar change,

but still around 1. That is to say, there is no strong force between CO2 and NaCl.

The NaCl+water+CO2 ternary system contains very little CO2 or infinitely diluted, the

existence of CO2 has very limited effect on the system. Here we just briefly discuss the

change between ions and water. For more details please refer to Chapter 5.

6.3.3 Effects of Pressures on the Ternary System

Water is very special. For example, large dielectric constant, the high melting temperature,

and the density (Eisenberg and Kauzmann., 1969). It is because the electronic nature of

112


�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.00.20.40.60.81.01.21.41.61.82.0g(r)

r(nm)

T=278K T=288K T=298KC-OH

�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.00.20.40.60.81.01.21.41.61.8g(r)

r(nm)

C-H

Figure 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


�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.20.40.60.81.01.21.41.61.82.0g(r)

r(nm)

T=278K T=288K T=298KOC-OH

�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.20.40.60.81.01.21.41.6g(r)

r(nm)

OC-H

Figure 6.16: Radial distribution functions for OC−OH and OC−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.

114


Table 6.9: Structure features of Figure 6.15 and Figure 6.16


OC−OH(278 K) 0.34(0.01) 0.47(0.01) 13.96OC−OH(288 K) 0.34(0.01) 0.47(0.01) 13.86OC−OH(298 K) 0.34(0.01) 0.47(0.01) 13.8OC−H(278 K) 0.36(0.01) 0.58(0.01) 51.98OC−H(288 K) 0.36(0.01) 0.57(0.01) 49.2OC−H(298 K) 0.36(0.01) 0.54(0.01) 40.22C−OH(278 K) 0.39(0.01) 0.56(0.01) 22.1C−OH(288 K) 0.39(0.01) 0.54(0.01) 21.36C−OH(298 K) 0.40(0.01) 0.54(0.01) 20.98C−H(278 K) 0.37(0.01) 0.54(0.01) 46.34C−H(288 K) 0.37(0.01) 0.54(0.01) 45.54C−H(298 K) 0.37(0.01) 0.54(0.01) 43


275 280 285 290 295 3001520253035404550556065

CNT(K)

OC-OH OC-H C-OH C-H

Figure 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


�

0.0 0.5 1.0 1.5 2.0 2.50.00.20.40.60.81.01.21.41.6

r(nm)

g(r)

T=278K T=288K T=298KCl-OC

�

0.0 0.5 1.0 1.5 2.0 2.50.00.20.40.60.81.01.21.41.6

r(nm)

g(r)

Cl-C

Figure 6.18: Radial distribution functions between Cl− and carbon dioxide at differenttemperatures 278 K(�), 288 K(•), 298 K(N) at a pressure of 1atm, XNaCl = 0.0074.

116


�

0.0 0.5 1.0 1.5 2.0 2.50.00.20.40.60.81.01.21.41.6

r(nm)

g(r)

T=278K T=288K T=298KNa-OC

�

0.0 0.5 1.0 1.5 2.0 2.50.00.20.40.60.81.01.21.41.6

r(nm)

g(r)

Na-C

Figure 6.19: Radial distribution functions between Na and carbon dioxide (Na+-OC atabove and Na-C at bottom) 278 K(�), 288 K(•), 298 K(N) at a pressure of 1atm, XNaCl =0.0074.

117


water molecules mutually making the hydrogen bonds in the tetrahedral structure. This

hydrogen-bond forming nature also reflects in the properties of water as a solvent. Water

can react with strongly polar molecules and ion species, which can easily dissolve in water

(Tissandier et al., 1998; Pearson, 1986). On the contrary, the non-polar molecules are hard

to dissolve in water because water molecules form strong hydrophobic repulsion (Ben-

Naim and Marcus, 1984). An ice like structure of water is a source of phenomena called

the “structure making” and the “structure breaking” appearing in ion hydrations (Lynden-

Bell and Rasaiah, 1997). The ternary system contains easily dissolved ions and also more

difficult to dissolve CO2. It is an interesting to study the structural change of the compo-

nents of the complicated system with varying pressure. Please refer to 6.2 and Table 6.3

for more details.

The NaCl + water + CO2 ternary system contains very little CO2 or infinitely diluted,

the existence of CO2 has very limited effect on the system. So most of this part is similar

with the pressure part of Chapter 5, for more discussions, refer to Chapter 5. Here we just

focus on the discussion about the pressures effects on the CO2 in ternary system.

Figure 6.23 and Figure 6.24 are the radial distributions between CO2 and NaCl. With

pressure increasing, the CO2−NaCl radial distribution functions show similar change, but

still around 1. That is to say, there is no strong force between CO2 and NaCl.

.

118


�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.00.20.40.60.81.01.21.41.61.82.0

r(nm)

g(r)

1 atm 400 atm 700 atmOC-OH

�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.00.20.40.60.81.01.21.41.6g(r)

r(nm)

OC-H

Figure 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 temperature of 298 K, XNaCl= 0.0074.

119


�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.00.20.40.60.81.01.21.41.61.82.0

r(nm)g(r)

1 atm 400 atm 700 atmC-OH

�

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.50.00.20.40.60.81.01.21.41.61.8

r(nm)

g(r)

C-H

Figure 6.21: Radial distribution functions for C−OH and C−H (from top to the bot-tom) at different pressures 1 atm(�), 400 atm(•), 700 atm(N), at a temperature of 298 K,XNaCl=0.0074.

120


Table 6.10: Structure features of Figure 6.20 and Figure 6.21


OC−OH(1 atm) 0.34(0.01) 0.47(0.01) 13.8OC−OH(400 atm) 0.34(0.01) 0.48(0.01) 13.91OC−OH(700 atm) 0.34(0.01) 0.48(0.01) 14.86

OC−H(1 atm) 0.36(0.01) 0.54(0.01) 40.2OC−H(400 atm) 0.36(0.01) 0.55(0.01) 42.39OC−H(700 atm) 0.36(0.01) 0.55(0.01) 42.88C−OH(1 atm) 0.40(0.01) 0.54(0.01) 20.98

C−OH(400 atm) 0.40(0.01) 0.54(0.01) 21.23C−OH(700 atm) 0.40(0.01) 0.54(0.01) 21.6

C−H(1 atm) 0.37(0.1) 0.54(0.1) 43C−H(400 atm) 0.37(0.1) 0.57(0.1) 47.74C−H(700 atm) 0.37(0.1) 0.57(0.1) 48.4


-100 0 100 200 300 400 500 600 700 8000510152025303540455055606570

OC-OH OC-H C-OH C-HCNP(atm)

Figure 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


0.0 0.5 1.0 1.5 2.0 2.50.00.20.40.60.81.01.21.41.6

r(nm)

g(r)

1 atm 400 atm 700 atmNa-OC

�

0.0 0.5 1.0 1.5 2.0 2.50.00.20.40.60.81.01.21.41.6

r(nm)

g(r)

Na-C

Figure 6.23: Radial distribution functions between Na+ and carbon dioxide at differentpressures 1 atm(�), 400 atm(•), 700 atm(N), at a temperature of 298 K, XNaCl = 0.0074.

122


0.0 0.5 1.0 1.5 2.0 2.50.00.20.40.60.81.01.21.41.6

r(nm)

g(r)

1 atm 400 atm 700 atmCl-OC

�

0.0 0.5 1.0 1.5 2.0 2.50.00.20.40.60.81.01.21.41.6

r(nm)

g(r)

Cl-C

Figure 6.24: Radial distribution functions between Cl− and carbon dioxide at differentpressures 1 atm(�), 400 atm(•), 700 atm(N), at a temperature of 298 K, XNaCl = 0.0074.

6.4 Diffusion in the Ternary System

Figure 6.25 shows the diffusion coefficients of CO2+NaCl+water at different concentra-

tions. As concentration rises, CO2 diffusion coefficient decreases. The reason may be that

123


the increase of concentration increases the viscosity of this ternary system (Chandra and

Bagchi, 2000). Both Chowdhuri and Chandra (2001) and Bouazizi et al. (2006) studied the

physical properties of the system at different concentration, but neither of which involves

CO2. They used different force field from this thesis. Chowdhuri and Chandra (2001) used

SPC/E as water force field and Dang ion model; Bouazizi et al. (2006) used SPC as water

model and Dang ion model. They found that as concentration increases, the diffusion of

Na+, Cl− and water molecule become smaller. In spite of different models, we have the

similar findings that Na+, Cl− and water diffusion decrease with concentration increasing

in infinite diluent CO2 system. We also obtained quite similar data to them in similar con-

centration. The reason is obvious that low concentration and charge-free CO2 is unable

to produce strong electric field and thus unable to bring evident change to Na+, Cl− and

water diffusions.

Another interesting aspect is that the diffusion of Na+ and Cl− were getting closer

and closer with the concentration of NaCl getting higher. The reason is with the high

concentration of NaCl, Na+ and Cl− get more chance to get in touch with each other.

Therefore, they will bind together and move with the same speed by the electric force once

they are getting closer.

124


0.00 0.05 0.10 0.15 0.20 0.250.00.20.40.60.81.01.21.41.61.82.02.22.42.62.83.0 Na Cl CO2 H2O

D(10-5 cm2 /s)Mole fraction of NaCl

Figure 6.25: Diffusion coefficients of Na+ (�) , Cl− (•), CO2 (N) and H2O (H) at differentconcentrations at a temperature of 298 K, and a pressure of 1 atm.�

275 280 285 290 295 3000.20.40.60.81.01.21.41.61.82.02.22.42.62.8 Na Cl CO2 H2O

D(10-5 cm2 /s)T(K)

Figure 6.26: Diffusion coefficients of Na+ (�) , Cl− (•), CO2 (N) and H2O (H) at differenttemperatures at a pressure of 1 atm.

125


In general, with pressure increasing, the interstitials between molecules are reduced

and thus the diffusion coefficient is reduced accordingly. The water system, however, is

different because of hydrogen bonds. There has been many discussions in this regard. For

example, self- diffusion coefficients of water have been measured by the steady –gradient

spin-echo method at pressure up to 2380 bar and temperatures between 2 and -20◦C in the

stable liquid range using both strengthened glass and beryllium-copper pressure vessels

(Angell et al., 1976). Angell et al. (1976) found that at low temperature water diffusion

represents different change from other liquid which is reduced with pressure increasing.

On the contrary, water diffusion increases gradually when pressure increases, and then

reduces gradually when pressure increases to a certain level.

The unusual property of water diffusion largely disappears at temperatures more than a

few degrees above the density maximum temperature (Prielmeier et al., 1987). The possible

reasons are as explained as below. These water properties change with the thermodynamic

condition. At high temperature and high pressure conditions, water loses the properties

originated from the hydrogen bonds and becomes like a simple liquid (Noriyuki et al.,

1998). For example, the hydration enthalpy of a noble gas molecule changes from negative

at 300 K to positive at the high temperature (Guillot and Guissani, 1993). This is due to

a disappearance of an “iceberg” structure, which exists around the noble gas molecule at

ambient condition.

126


-100 0 100 200 300 400 500 600 700 800 9000.81.01.21.41.61.82.02.22.42.62.83.0

CO2 Na Cl H2OD(10-5 cm2 /s)

P(atm)

T=298K

Figure 6.27: Diffusion coefficients of Na+ (•) , Cl− (N) , H2O (H) and CO2 (�) at differentpressures and a temperature of 298 K, XNaCl = 0.0074.�

-100 0 100 200 300 400 500 600 700 8000.60.81.01.21.41.61.82.0D(10-5 cm2 /s)

P(atm)

CO2 Na Cl WaterT=278K

Figure 6.28: Diffusion coefficients of Na+ (•) , Cl− (N) , H2O (H) and CO2 (�) at differentpressures and a temperature of 278 K, XNaCl = 0.0074.

127


Figure 6.27 and Figure 6.28 show the change of water diffusion with pressure chang-

ing at 278 K and 298 K. At 278 K, water diffusion first increases slowly with pressure

increasing; at 298 K, it starts decreasing slowly with pressure increasing. As per previous

discussion, we know that at low temperature, water diffusion increases with pressure in-

creasing; and at high temperature, it decreases with pressure increasing. Our findings here

are consistent with pervious studies. In general, water is more like normal liquid between

300 K and 330 K in that water diffusion decreases with pressure increasing. Our finding,

however, is that water diffusion starts decreasing with pressure increasing at 298 K. Rea-

sons may be: first, model parameters can reflect the special change of water qualitatively,

but not quantitatively or accurately. Second, the density of water is dependent on the dis-

solved salt content as well as the temperature of the water. Ice still floats in the oceans;

otherwise they would freeze from the bottom up. However, the salt content of oceans low-

ers the freezing point by about 2◦C and lowers the temperature of the density maximum

of water to the freezing point. That is why, in ocean water, the downward convection of

colder water is not blocked by an expansion of water as it becomes colder near the freezing

point. Existing study tells that the unusual property of water diffusion largely disappears at

temperatures more than a few degrees above the density maximum temperature (Prielmeier

et al., 1987).

In the existence of NaCl, water at 300 K or less will represent the feature of normal liq-

uid. It is because the existence of NaCl lowers the required temperature. Figure 6.27 and

Figure 6.28 also show changes of diffusion coefficients of Na+, Cl−, and CO2 at different

pressures at different temperatures. At 298 K, all these diffusion coefficients start decreas-

ing slowly with pressure increasing. At 278 K, diffusion coefficients of all the components

increase slowly with pressure increasing. Na+, Cl− and CO2 all represent the same change

as water. This demonstrates that water plays a major and critical role in the system and

decides the diffusion change of other components.

128

CHAPTER 7. PREDICTION OF DIFFUSION COEFFICIENT OF CO2 IN ELECTROLYTE SOLUTIONS

Chapter 7

Prediction of Diffusion Coefficient of

CO2 in Electrolyte Solutions

7.1 Introduction

The physical qualities such as diffusion coefficient are always critical to engineers. It is

always difficult to gather experimental data and the insufficient experimental data cannot

reflect the complication of projects. Thus, it becomes very important to provide engineers

with the methods of forecasting the physical qualities. The previous chapter discusses

the use of molecular dynamics simulation to forecast the diffusion coefficient of CO2 in

NaCl. This method, however, can only generate very limited data and demands lots of time

and rigid conditions. In practice the use of formula seems a quick and accurate way for

engineers in forecasting.

Ratcliff and Holdcroft (1963) first measured the absorption rates of the gas in liquids

flowing over spheres and then determined the diffusion coefficients of carbon dioxide in

the NaCl. They also experimentally measured the diffusion coefficient of CO2 in Na2SO4,

MgCl2, Mg(NO3)2 and MgSO4. They then developed two models for prediction of diffu-

sion coefficient. One model was developed by combining Eyring theory and perturbation

129


model proposed by Podolsky (1958). The other model was based on the power relationship

between diffusion and fluidity. The first model deriving from Eyring theory an Podolsky’

perturbation models is:

D = D0 (1−ac) (7.1)

In this equation, D represents the diffusion coefficient of gas in electrolyte solution. D0

denotes the diffusion coefficient of gas in pure water, c denotes concentration and “a” is a

constant related to system. Thus, according to Ratcliff ’s model, the diffusion coefficient

of gas in any electrolyte solution can be predicted if the diffusion coefficient of gas in pure

water and “a” are known. It is an ideal situation if prediction can be done in this way.

Ratcliff compared his experimental data and model data and found good agreement.

The second model derived from Ratcliff is :

ddc

(DD0

)=−0.03+0.55

(∆

φ0

)(7.2)

where ∆ = φ(c=1)− φ0, i.e. ∆ is the difference in fluidity between a 1 molar solution

and pure water.

Although being derived from the relationship with friction, the second model is also

partially based on the first model. That is, the accuracy of the second model depends on

the accuracy of the first model. It is noted that even Ratcliff himself was aware his model

is probably not applicable to high concentrated electrolyte solution. In his experiment, the

highest concentrated electrolyte solution is 3.776 g mol/l NaCl.

The previous chapter studied the diffusion coefficient of CO2 in NaCl solution of vari-

ous concentrations via molecular dynamics. Our data shows that CO2 diffusion coefficient

does represent linear decrease at low NaCl concentration, but this is not the case at rela-

tively high concentration. In the latter case, the CO2 diffusion coefficient decreases steadily,

which is different from Ratcliff ’s model.

130


Figure 7.1: Diffusion coefficients of Oxygen in KOH solutions as a function of KOH mo-lality (Anderko and Lencka, 1998)

Our data obtained from molecular dynamic simulation are consistent with experimen-

tal data at low NaCl concentration, but there is no experimental data of CO2 at higher

NaCl concentration. However Tham et al. (1970) obtained the diffusion coefficient of O2

in KOH at concentrations ranging from very low up to as high as 15M, which could be

used for guidance. Anderko (Anderko and Lencka, 1997, 1998; Anderko et al., 2002)

developed a model which combines contributions of long-range (Coulombic) and short-

range interactions. The long-range interaction contribution, which manifests itself in the

relaxation effect, is obtained from the dielectric continuum-based mean-spherical approxi-

mation (MSA) theory for the unrestricted primitive model. The short-range interactions are

represented using the hard-sphere model. He used his model to predict the self-diffusion

coefficient of O2 diffusion in KOH and compared his results with the experimental data

from Tham and Gubbins.

As shown in Figure 7.1, their data support our findings in the oxygen diffusion in KOH.

It shows linear decrease in low concentration, but steady decrease in high concentration,

131


which are consistent with our findings for CO2 in NaCl solution. Thus it becomes very

meaningful to explore the basis of Ratcliff’s model and also enhance the accuracy of the

model via molecular dynamic simulation. In fact it has been recognized to improve the

empirical model by using the data from molecular dynamic simulation.

7.2 Prediction Equation of Diffusion Coefficient Modified

From Activation Theory and Perturbation Model

7.2.1 Theory

A lattice model of liquid state was used by Eyring and his co-worker (Glasstone et al.,

1941) in the application of their theory of absolute reaction rates to transport processes

in liquids. Molecules move from one site to the next site of lattice and an activation free

energy is required.

Formulae have been established relating the self-diffusion coefficient, D, the fluidity, φ

and the ionic mobility, µ , to this activation energy, i.e.,

π = kπλ2exp

(−∆G∗◦

RT

)(7.3)

Where π is D, φ or µ , where kπ is a constant for the particular rate process considered,

and λ is the distance between lattice sites.

The lattice model for aqueous electrolytes was used by Podolsky (1958) to establish

the relation of the fluidity of the solution to the self diffusion coefficient of water and

other properties. Good agreement was found between theory and properties. Podolsky also

developed a perturbation theory. Consider water molecules and a single ith ion surrounds

a solute molecule. Assume DG∗ to be the corresponding activation energy for diffusing

solute in pure water, and DG∗o to be perturbed from DG∗ to DG∗+ δi by the presence of

132


ion. It is also assumed that the perturbation of DG∗ is proportional to the number of ith

ions surrounding it.

Based on the two assumptions, Ratcliff used the perturbation theory to predict the dif-

fusion coefficient of CO2 in electrolyte solutions.

• The lattice spacing remains that of pure water regardless of the existence of the elec-

trolyte. This assumption may hold for relatively small ions, but not for large ions

such as tetraethylammounium. But the assumption may be less restrictive as it seems

because evidence was found by Podolsky that the change in the latticing spacing

caused by an electrolyte does not impose as big effect on transport processes as the

change in the activation energy does.

• If D is considered to be the average activation energy of the species under study,

Eq.(7.3) will apply to any transport process. The D over all species available must

be averaged for fluidity, but only over the species diffusing for diffusion coefficient.

From Podolsky’s perturbation theory, Ratcliff derived the following process of gas in the

electrolyte solution:

Let ∆G∗ be the corresponding activation energy for diffusing solute in pure water. Con-

sider a solute molecule surrounded by water molecules and by a single ith ion. ∆G∗◦ will be

assumed to be perturbed from ∆G∗ to ∆G∗+δi by the presence of the ion. We shall assume

that the perturbation of ∆G∗ will be proportional to the number of ith ions surrounding it.

Except in very strong electrolyte solutions, each solute molecule will have no more than

one ith ion as immediate neighbor.

We now have the problem of averaging ∆G. In a non-ideal system such as aqueous

electrolyte, the relative proportion of water molecules and ions in the solution immediately

surrounding the solute molecules may differ from that in the solution as a whole. However,

since no detailed information is available, we are forced to assume that they are the same.

133


Let a molecule of the solute interact with n surrounding water molecules or ions. Let

the concentration of the electrolyte solution be c, and let one mole of electrolyte give v1

mole of ion 1 and v2 moles of ions 2. We have a total of [18c/(1000ρ− cM)] ions per

molecules of water, where M is the molecular weight of the electrolyte and ρ the density of

the solution. Thus, on the average, each solute molecule interacts with 18cv1n1000ρ−cM+18c(v1+v2)

ith ions. Hence:

∆G∗◦ = ∆G∗+18cn(v1δ1 + v2δ2)

1000ρ− cM+18c(v1 + v2)(7.4)

Hence from Eq.(7.3):

D = kDλ2exp

{−∆G∗

RT− 1

RT·[

18cn(v1δ1 + v2δ2)

1000ρ− cM+18c(v1 + v2)

]}(7.5)

But for diffusion in pure water:

D0 = kDλ2exp

(−∆G∗

RT

)(7.6)

Hence:

D = D0exp{− 1

RT·[

18cn(v1δ1 + v2δ2)

1000ρ− cM+18c(v1 + v2)

]}(7.7)

Now activation energies for diffusion are small, and the perturbation δi will be even

smaller. Furthermore, in dilute solutions,1000ρ − cM + 18c(v1 + v2) ' 1000. Hence, in

dilute solutions:

D' D0

[1− cn(v1δ1 + v2δ2)

55.5RT

](7.8)

For a given solute and electrolyte at a given temperature:

D = D0 (1−ac) (7.9)

134


Where a = n(v1δ1+v2δ2)55.5RT , and is constant.


The derivation Ratcliff given here was based on one mathematic approximation which is :

limac→0

exp(−ac)1−ac

= 1 (7.10)

The prerequisite of this approximation to be true is that the solute concentration which

is c is small enough. This is the reason why this equation can not apply to the condition

when the concentration of the electrolyte solution is high. However, the Eq.7.7 can be

written as:

DD0

= exp(−ac) (7.11)

where a = 18n(v1δ1+v2δ2)RT [1000ρ−cM+18c(v1+v2)]

. Even in relative high concentrations, it is still

safe to assume that 1000ρ − cM + 18c(v1 + v2) ' 1000. a = 18n(v1δ1+v2δ2)RT [1000ρ−cM+18c(v1+v2)]

≈−n(v1δ1+v2δ2)

55.5RT , and is constant.

Ratcliff and Holdcroft (1963) preferred a linear model over a exponential model due to

the limiting experimental data. However, since we have got the diffusion coefficient data of

high concentrations of electrolyte solutions via MD simulation, It is good to compare two

models to determine which is the more accurate model to provide better prediction results

at wider range of concentrations.

Table 7.1 shows the diffusion coefficient of CO2 in the electrolyte solutions of various

concentrations. Table 7.2 shows the diffusion coefficient of CO2 in the electrolyte solutions

of various concentrations via molecular simulation MD.

Figure7.2 compares the two sets of data. By comparing the MD simulation data and

experimental data in Figure 7.2, we can find out two things:

135


Table 7.1: Diffusion coefficient of CO2 in NaCl solutions at 25◦C and 1atm(Ratcliff and Holdcroft 1963)

NaCl concentration Diffusion coefficient of CO2 Viscosity of solutionC (gmol/l) D

(cm2/s×105) µ = 1/φ (cp)

0 1.92 0.8931.041 1.73 0.991.939 1.61 1.082.754 1.5 1.193.776 1.3 1.37

Table 7.2: MD data of diffusion coefficients of CO2 in NaCl solutions obtained in this work

NaCl concentration Diffusion coefficient of CO2C (mole f raction) D

(cm2/s×105)

1×10−4 1.960.00733 1.9080.01477 1.710.0515 1.4520.0596 1.2460.0807 1.06

0.08695 0.9120.11339 0.87560.13295 0.723980.153 0.654330.1742 0.5325

0.21855 0.34415

136

CHAPTER 7. PREDICTION OF DIFFUSION COEFFICIENT OF CO2 IN ELECTROLYTE SOLUTIONS�

0.00 0.05 0.10 0.15 0.20 0.25 0.300.00.20.40.60.81.01.21.41.61.82.0

MD simulation data Ratcliff and Holdcraft(1963)D(10-5 cm2 /s)

Mole fraction of NaClFigure 7.2: Comparison of the diffusion coefficients of CO2 between our MD simulationdata (�) and Ratcliff’s experimental data (•) at different NaCl solutions

• MD simulations data agree well with the available experimental data.

• The diffusion of CO2 does not decrease linearly when the NaCl concentraion in-

creases. If we still use the prediction method of Ratcliff which is D = D0 (1−ac),

the error will get quite large when the concentration gets higher. Other researchers

have found the similar phenomena of O2 in KOH, see Figure 7.1.

Figure 7.3 shows the molecular dynamics simulation diffusion of CO2 in aqueous solution.

It shows clearly an almost perfectly straight line when the data is modified to ln(D/D0)

even at very high concentrations of the aqueous solutions. Figure 7.4 modified the experi-

mental data from the Ratcliff to ln(D/D0), a clearly straight line also can be found in this

Figure.

Based on the theory basis and with the help of simulation data, we regard the expo-

nential model as the more suitable model for predicting the diffusion coefficient of CO2 in

NaCl solution.

137


0.00 0.05 0.10 0.15 0.20 0.25

-2.2

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

Ln

(D/D

0)


MD simulation data

CO2

Figure 7.3: Diffusion coefficients of CO2 of our MD simulation data converted toLn(D/D0) showing the linear relationship to the NaCl concentrations

�

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-0.5-0.4-0.3-0.2-0.10.0

Ln(D/D0)


Ratcliff and Holdcroft (1963)

Figure 7.4: Diffusion coefficients of CO2 of Ratcliff’s experimental data converted toLn(D/D0) showing the linear relationship to the NaCl concentrations

138


7.3 Predicting Diffusion Coefficient via Viscosity

7.3.1 Theory

To use Eq.(7.9) to predict diffusivities, we need some way of relating the constant a to

known physical properties of the system. That is the reason why Ratcliff had the second

model to predict diffusion coefficient via viscosity and the reason for us to modify it so it

can be used for high aqueous electrolyte solutions.

As we discussed before, although being derived from the relationship with friction, the

second model is partially based on the first model. That is, the accuracy of the second

model depends on the accuracy of the first model. Here we will modify this relationship

equation based on exponential model.

By following the same appoximations as the first model Ratcliff has made to use the

lattice model to relate the fluidity of an electrolyte solution to its concentration. For a given

solute and electrolyte at a given temperature, He obtained:

φ = φ0 (1−bc) (7.12)

where:

b =(1+ r)(v1δ ′1n1 + v2δ ′2n2)

55.5RT(7.13)

Where φ is the fluidity, φ0 is the fluidity of pure solvent, δ ′1 is the perturbation of

activation energy for transport of solvent by presence of ith ion, r equals to εi/δ ′1 , εi is

the perturbation of activation energy for transport of ith ion by an adjacent water molecule

and ni is the effective number of surrounding solvent molecules with which one ith ion

interacts.

139


The original relationship between fluidtity of an electrolyte solution and its concentra-

tion is logarithmic (Podolsky, 1958). The linear relationship Ratcliff got here was based on

one mathematic approximation which is limbc→0exp(−bc)

1−bc = 1.

The prerequisite of this approximation to be true is that the solute concentration which

is c is small enough. This is the reason why this equation can not apply to the condition

when the concentration of the electrolyte solution is high. Thus, we make a change to this

equation so it can apply to the scenario when the solution concentrations are high.

So we simply make the same modification to this equation like we did to Ratcliff’s first

model.

For a given temperature, we can write:

φ

φ0= exp(bc) (7.14)

ln(

φ

φ0

)= bc (7.15)

φ = 1/µ, so Eq. (7.15) equals to:

ln(

µ0

µ

)= bc (7.16)

The deviation of Eq. (7.16) based on concentation is:

d{

ln(

µ0µ

)}dc

= b (7.17)

and from the Eq.(7.11), we can get:

ln(

DD0

)= ac (7.18)

Hence:

140


d{

ln(

DD0

)}dc

= a (7.19)

Eq. (7.18) and Eq (7.16) can be tranfromed to:

ln(D/D0)

ln(µ0/µ)=

ab= k (7.20)

Hence:

ln(

DD0

)= kln

(µ0

µ

)= ln

(µ0

µ

)k

= ln(

µ

µ0

)−k

(7.21)

which meansDD0

=

(µ0

µ

)k

=

(µ

µ0

)−k

(7.22)

This is the relationship between diffusion and viscosity and the way to predict diffusion

coefficient via viscosity.

The way to get the value of k will be based on dividing Eq.(7.19) to Eq. (7.17) :

d {ln(D/D0)}/dcd {ln(µ0/µ)}/dc

=ab= k (7.23)


7.3.2.1 The value of k

Table 7.3 shows the correlated experimental data of the viscosity of the electrolyte solution

at a wide-range NaCl concentrations up to 6 mol/kg (Kestin et al., 1981, 1978a). These

experimental data are obtained by using osillating-disk method (Kestin et al., 1978b,a) and

presented in Fig.7.5. Horizontal axis represents the mole fraction of NaCl and vertical axis

is ln(µ0/µ). It is found that dln(µ0/µ)/dc displays a straight line.

141


Table 7.3: Experimental data of viscosity of NaCl solution at P=1atm and T=25◦C (Kestinet al., 1981)

NaCl Concentration Viscosity of SolutionC (mole f raction) µ = 1/φ (cp)

0 0.890.00884 0.92860.01768 0.9720.02651 1.020.03535 1.07450.04419 1.1340.05303 1.20.06186 1.2720.0707 1.35

0.07954 1.4360.08838 1.52890.09721 1.62930.10605 1.7375

Figure 7.5: Experimental data of viscosity of NaCl solutions shows that Ln(µ0/µ) is linearwith respect to the NaCl concentrations

142


From figure 7.5, we can get the slope of dln(µ0/µ)/dc is −6.03 , which is the value

of b, and from figure 7.3, we can get the slope of d {ln(D/D0)}/dc is −7.6, which is is

the value of a. So k equals a/b = 1.26, which means the relationship between diffusion

coefficient and viscosity is:

DD0

=

(µ

µ0

)−1.26

=

(µ0

µ

)1.26

(7.24)

This kind of relationship has been found later by other researchers as a well-known

modification equation to the Stokes-Einstein equation (Hayduk and Cheng, 1971; Davis

et al., 1980).

Now recall Ratcliff’s paper. He used the linear equations of D = D0(1− ac) and

φ = φ0 (1−bc) and get the relationship equation as ddc

(DD0

)= 0.642

(∆

φ0

)which is not

satisfiable in the high concentrations. Then he deduced a empirical equation purely based

on the data rather on the solid theory which is:

log10

(D0

D

)= 0.637log10

(µ

µ0

)(7.25)

By deriving this relationship equation Eq. (7.25), we can get:

DD0

=

(µ0

µ

)0.637

(7.26)

Now Let us compare Eq.(7.22) and Eq. (7.26). It is clearly shown that Ratcliff has

already found the similar relationship between diffusion and viscosity of the aqueous elec-

trolyte solution with us, only the parameters are different. His parameter is 0.637 and ours

is 1.26.

There is another researcher from Japan (Funazukuri and Nishio, 1995) who published a

paper to predict the diffusion of CO2 in water and in aqueous electrolytic solutions purely

base on experimental method. In his paper, a empirical correlation given by:

143


D12/T = 1.013×10−14µ−0.9222 (7.27)

where D12 is diffusion coefficient, T is temperature , and µ is viscosity.

The average absolute deviation AAD for this correlation is 5.6% for the literature data

on D12 of CO2 in water with number of data points N of 79 and 3.9%(N=103) for those in

aqueous electrolytic solutions. He claimed that the above correlation obtained in his study

is more accurate than the various empirical correlations including the relationship equation

from Ratcliff. Funazukuri used binary diffusion coefficient and Ratcliff used self diffusion

coefficient, however, this did not stop Funazukuri to compare two results. The reason is

that when the solubility of gas is small, the binary diffusion coefficient of gas in solution is

almost equal to the self-diffusion coefficient of gas. The proof can be found in (Zhou et al.,

2005).

Eq.(7.27) can be wirten as follows:

D0/T = 1.013×10−14µ−0.92220 (7.28)

and

D/T = 1.013×10−14µ−0.9222 (7.29)

where D represents diffusion coefficient of gas in electrolyte solution. D0 denotes the

diffusion coefficient of gas in pure water, µ is the viscosity, µ0 is the fluidity of pure solvent.

Divide Eq.(7.29) by Eq.(7.28):

DD0

=

(µ

µ0

)−0.922

=

(µ0

µ

)0.922

(7.30)

144


Compare Eq. (7.24), and Eq. (7.30). It shows the same form of the equation, only the

parameters are different. However, Eq. (7.30) was obtained purely by empirical modifica-

tion of the experimental data which further prove the correctness of the foundation of our

theory.

7.3.2.2 Comparison of different relationships between diffusion and viscosity via ex-

perimental diffusion data

Our aim is to predict diffusion coefficients via viscosity. The experimental viscosity data

were used for various prediction models to obtain diffusion coefficients. The diffusion

coefficients were then compared with the real experimental data to determine the most

accurate prediction model.

As shown in Fig.7.6, at the concentration between 0 and 0.02 mole fraction, the dif-

fusion coefficients predicted via viscosity based on our model are almost as the same as

experimental data, Funazukuri’s results are close to experimental data too, and Ratcliff’s

results show the biggest difference among the three models. As concentration increases be-

tween 0.02 and 0.04 mole fraction, Funazukuri’s results become the most consistent with

the experimental data, followed by our data. Likewise, Ratcliff’s results show the biggest

difference from the experimental data. As concentration increases ever further from 0.04

to 0.06 mole fraction, Funazukuri’s data slightly deviate from experimental data while our

data go slight close to experimental data. Ratcliff’s data disagree most with experimental

data.

In summary, the diffusion coefficients derived from Funazukuri’s model and our model

via viscosity are both in good agreement with experimental data. Funazukuri’s data are

probably even closer to the current experimental data than our prediction. It is hard to say

which one is better in the allowable error range. Among the three models, Ratcliff’s data

are in least agreement with experimental data. The possible reasons are as follows:

145


• First, Funazukuri and Ratcliff models are all developed from experimental data, but

Funazukuri used broader range of experimental data (Funazukuri and Nishio, 1995).

Apart from the experimental data used by Ratcliff (Ratcliff and Holdcroft, 1963),

Funazukuri also used many other researchers’data such as (Onda and Yamaji, 1960).

Therefore, Funzukuri’s equation is more accurate than Ratcliff’s.

• Second, Ratcliff intended to establish a common relationship equation for diffusion

coefficient and viscosity and therefore he used the data of many electrolyte solutions

of NaCl, NaNO3, Na2SO4, MgCl2, Mg(NO3)2 and MgSO4. These data are very

disperse and the mean parameters obtained from these data is just about 1/2 of the

parameter if only the NaCl experimental data was consided alone. As our research

focuses on establishing an equation for prediction of the diffusion of CO2 in the NaCl

solutions, the data in a wider range are used without compromising data accuracy to

adapt to other types of electrolyte solutions like MgCl2 etc. Plus, a similar formula

was found by other researchers (Hayduk and Cheng, 1971) when they undertook

empirical modification to Stokes-Einstein formula.

Dµc1 = c2 (7.31)

Where c1 and c2 are found to be the functions of solutions properties which means

different solutions have different c1 and c2. So it is better to have a unique parameter to

different solutions.

• Third, juding from the second point, it is better to have a unique parameter to different

solutions, why are the data from Funzzukri’s model in better agreement with exper-

imental data since this model is also based on the experimental data of even more

various electrolyte solutions? C1 in Hayduk’s formula is only related to solution fea-

tures if there are no solute-solvent interactions. It was found that the solution-related

parameter only relates to molecular diameter (Davis et al., 1980). Ratcliff only used

146


a few kinds of electrolyte solution to obtain his empirical equation for diffusion co-

efficient prediction via viscosity. His experimental data of Mg(NO3)2 diffusion co-

efficients is very different from NaCl diffusion coefficients. Therefore, his equation

based on these experimental data will produce very different results from the real

diffusion coefficients data of CO2 in NaCl. Funazukuri, however, obtained his exper-

imental data from many kinds of electrolyte solution and used over 100 data points.

As many of the electrolyte solutions have similar diameters to NaCl, Funazukuri’s

empirical equation can make accurate prediction on the diffusion coefficient of CO2

in NaCl, but not on the diffusion coefficient of CO2 in Mg(NO3)2. As concluded by

Funazukuri, the diffusion coefficient of CO2 in NaCl has an average absolute devia-

tion (AAD) less than 2%, whereas that of CO2 in Mg(NO3)2 has about 10% AAD.

It is worth noting that AAD is 1.2% when NaCl concentration is 2.8mol/l and it

increases to 1.7% when concentration is 3.78mol/l. It is shown in Fig.7.6.

• Finally, in our calculation via viscosity, diffusion data from wider range of electrolyte

solutions are used during the derivation of the empirical parameter. The experimental

data of viscosity (Kestin et al., 1981) we used is up to 6 mol/kg whereas the NaCl

concentration in Ratcliff’s study is only up to 3.776 mol/l (Ratcliff and Holdcroft,

1963). Our diffusion coefficients data are obtained from MD simulation at over 10

mol/l and Ratcliff’s data are obtained at low concentrations. So our empirical param-

eter in the prediction equation via viscosity is more accurate. In addition, different

kinds of electrolyte solution have different parameters. Our equation parameters are

all from the experimental data of pure NaCl solution, and thus our equation is more

accurate.

147


0.00 0.02 0.04 0.06 0.08 0.10 0.120.50.60.70.80.91.01.11.21.31.41.51.61.71.81.92.02.12.2

Ratcliff's prediction model via viscosity Funazukuri's prediction model via viscosity Our prediction model via viscosity Ratcliff's experimental dataD(10-5 cm2 /s)

Mole fraction of NaClFigure 7.6: Comparison between different prediction models via viscosity and experimen-tal data, Ratcliff’s prediction model via viscosity(�), Funazukuri’s prediction model viaviscosity(•), Our prediction model via viscosity(N), and Ratcliff’s experimental data(H)

7.3.2.3 Comparison of different relationships between diffusion and viscosity via

molecular dynamics diffusion data

Two issues were identified in the process of comparing the different relationships between

diffusion and viscosity:

• First, the real experimental data of diffusion coefficient are limited. The experimental

viscosity data were obtained from the electrolyte solution at concentrations from pure

water to 6 mol/kg NaCl solutions (Kestin et al., 1981). So ideally the experimental

data of diffusion coefficients are in the same range. However, the experimental data

of diffusion coefficient is only up to 3.776 gmol/l.

• Second, the real experimental data of diffusion coefficient is at the low range of con-

centrations, therefore, failed to present the special diffusion coefficient features of the

electrolyte solution at high concentrations. Earlier in this chapter, our simulation data

and many other researchers’ study (Tham et al., 1970; Anderko and Lencka, 1997,

148


1998; Anderko et al., 2002) find that the diffusion coefficients of gas in electrolyte

solution shows exponential decrease as concentration increases. It may be caused by

strong relaxation effect. It means that the diffusion coefficient of electrolyte solu-

tion shows very special change, especially when concentration increases. Therefore,

comparison of diffusion coefficients of low-concentrated electrolyte solution is in-

sufficient. For example, Ratcliff’s linear prediction model produces good result at

low concentration, but big errors at high concentration.

There are two ways to obtain the data: One is Ratcliff’s experimental data (Ratcliff and

Holdcroft, 1963) of concentrations range (from pure water to 3.776 gmol/l NaCl solu-

tions), smaller range than the predicted electrolyte solutions; the other is the data derived

from the molecular dynamic simulation at a broad range of concentrations of electrolyte

solutions. So, to address the above two issues, we compare the diffusion coefficient models

via viscosity in the following two steps:

• Step 1, compare the diffusion coefficients data obtained from different prediction

models by using viscosity with experimental data. The experimental data are ob-

tained at low concentrations while some prediction models involve high concentra-

tions. Fig.7.6 gives more details. We have done that at the (Chapter 7.3.2.2).

• Step 2, compare the diffusion coefficients data with the MD simulation. The aim is

to determine the accuracy of the models to predict diffusion coefficients via viscosity

at high concentrations. The assumption is that the MD simulation data represents

the real diffusion coefficients as real experimental data is not available. For more

information, please see Fig. 7.7.

Fig.7.7 illustrates step 2, in which the diffusion coefficient data predicted by different mod-

els via viscosity are compared with MD simulation data. The diffusion coefficient otained

via viscosity and the MD simulation data shown in good agreement. The reason is that

149


0.00 0.02 0.04 0.06 0.08 0.10 0.120.50.60.70.80.91.01.11.21.31.41.51.61.71.81.92.02.12.2

Ratcliff's prediction model via viscosity Funazukuri's prediction model via viscosity Our prediction model via viscosity MD simulation dataD(10-5 cm2 /s)

Mole fraction of NaClFigure 7.7: Comparison between different prediction models via viscosity and Ratcliff’slinear equation, Ratcliff’s prediction model via viscosity(�) , Funazukuri’s predictionmodel via viscosity(•), our prediction model via viscosity(N), and MD simulation data(H).

molecular simulation data are used to obtain the empirical parameter in the prediction equa-

tion via viscosity. Both Funazukuri’s data and Ratcliff’s data, which are predicted by using

viscosity, deviate from the MD simulation data farther with concentration increasing. Rat-

cliff’s data via viscosity show the biggest deviation.

Therefore it is concluded that our prediction model via viscosity provides better predic-

tion results at wider range of concentration. There are two reasons.

• The experimental data Funazukuri used to obtain the empirical equation were in low

concentrations about up to 1 mol/l while our data has far higher ranger up to more 10

mol/l.

150


• Our model is developed by using the diffusion coefficient data of CO2 in NaCl while

Funazukuri and Ratcliff’s diffusion coefficient data are from different types of elec-

trolyte solutions. The parameter is specific to one kind of electrolyte solution (Hay-

duk and Cheng, 1971; Davis et al., 1980), and thus the selection of one common

parameter for all electrolyte solutions compromises the accuracy of prediction.

151

CHAPTER 8. CONCLUSIONS AND RECOMMENDATIONS

Chapter 8

Conclusions and Recommendations

This thesis discusses a series of molecular dynamics simulations on the CO2 in aqueous

NaCl aqueous solution systems at different pressures and temperatures, especially high

pressures and low temperatures, trying to make it as real as sea water enviroment. The

purpose is to obtain various physical properties of the complicated system consisting of

water, NaCl and CO2.

Molecular dynamics simulation was first undertaken for CO2 in water using the appro-

priate water and CO2 models. The SPC/E model has been proved for its effectiveness at

low temperatures and high pressures by other researchers. It is thus used for water model.

A few CO2 models are tested for simulation of physical properties of CO2 in water system,

including EPM2, Duan’s model and Gromos model. Gromos CO2 model is found the most

appropriate one in this case as the simulation results agree with experimental data well.

Molecular dynamics simulations are then conducted to obtain the physical properties

of NaCl in water. All molecular model parameters were obtained from optimized experi-

mental data. Due to the specialty of ion solution and the lack of experimental data, there

hasn’t been an ideal ion molecular model arising by far. The Gromos ion model is used to

simulate the structural properties of various atoms in the system at different temperatures

and pressures. Some reliable experimental data of the NaCl aqueous solution system are

152


available, including the RDF between Na+ and water, Cl− and water , hydrogen and hydro-

gen of water (H –H), hydrogen and oxygen of water (H–OH), oxygen and oxygen of water

(OH–OH). The simulation data is proved in good accord with these experimental data. The

diffusion coefficients of various atoms in the NaCl aqueous solution are also studied at dif-

ferent temperatures and different temperatures. A very interesting phenomenon arises. At

298 K, the diffusion coefficients of water, Na+ and Cl− decrease with pressure increases.

The ternary system is examined as follows. First, this thesis discusses the effect of

NaCl concentrations on the diffusivities and structural properties CO2, NaCl and water

molecules in the ternary system. It is found that as NaCl concentration rises, not only

Na+ and Cl−, but also the same charge ions, like Na+-Na+, Cl−-Cl− are more closely

bonded. With NaCl concentration increasing to a certain level, the Na+-Cl− coordination

number increases slower than the Na+-Na+, Cl−-Cl− coordination numbers. This suggests

that the same charge ions in NaCl are clustered faster than Na+ and Cl− are paired. With

NaCl concentration increasing, it is also found that the water hydrogen bond is destroyed.

Diffusion coefficients of CO2 , Na+, Cl− and water all decrease significantly. Moreover,

when NaCl concentration increases to a certain level, the diffusion coefficient values of

Na+ and Cl− becomes very close, further indicating they are clustered.

Second, the effect of changing temperatures and pressures on the diffusities and stru-

cural properties of the components in NaCl aqueous solutions were analyzed. The radial

distribution functions and coordination numbers indicate that with temperature increasing,

Na+ and Cl− become closer because the dielectric decreases significantly. Na+-Na+, Cl−-

Cl− become closer too, but in a very minor and unobvious way. It is because in the very

low NaCl concentration, the ions of the same charge are vey hard to cluster. With pres-

sure increasing, it is clearly found that the interaction and cluster between Na+ and Cl−

diminish. The reason is that increased pressure leads to the increased dielectric, which is

different from temperature. Like the NaCl + H2O binary system, in the NaCl + H2O + CO2

ternary system water, Na+ and Cl− diffusivities reduce with pressure increasing at 298 K;

153


CO2 diffusion coefficient reduces with pressure increasing too. At 278 K, water, Na+ and

Cl− diffusivities increase with pressure increasing and it is the same with CO2 diffusion

coefficient. We know that at low temperatures, water diffusion coefficient increases with

pressure increasing; and at high temperatures, it decreases with pressure increasing. Our

findings here are consistent with other researchers in this field. In general, at lower tem-

peratures, water shows abnormal properties, and it starts to exhibit normal liquid properties

between 300 K and 330 K which means water diffusion coefficient decreases with pressure

increasing. Our finding, however, is that water diffusion coefficient starts decreasing with

pressure increasing at 298 K, which is outside the normal range. Reasons may be: first,

model can only reflect the change of water properties in a qualitative way rather than in

a accurately quantitative way. Second, previous study has found that the salt content of

oceans lowers the freezing point by about 2◦C and lowers the temperature of the density

maximum of water to the freezing point. The unusual change of water diffusion coefficient

largely disappears at temperatures a few degrees above the density maximum temperature.

So, in the existence of NaCl, water at 300 K or less temperature will exhibit the feature of

normal liquid.

Two equations are examined for prediction of diffusion coefficient of gas in electrolyte

solution, which allows prediction in a greater range of electrolyte solution concentrations.

One model is developed by combining Eyring theory and perturbation model. The diffu-

sion coefficient of CO2 does not decrease linearly according with Ratcliff’s theory when

concentration of aqueous solution gets higher. If we still use the prediction method of Rat-

cliff which is D = D0 (1−ac), the error will get quite large when the concentration gets

higher. Other researchers has found the similar phenomena of O2 in KOH. Our approach

enables a more accurate prediction in higher concentrated aqueous electrolytic solutions.

The other model is based on relationship between diffusion coefficient and viscosity. The

diffusion coefficients data obtained from molecular simulation agree with the result from

the two equations, demonstrating the accuracy of the two prediction equations.

154


The work is intended to aid the study of the physical properties of CO2 in sea water.

To simulate the sea water, we have discussed the CO2 molecular model and the effect of

NaCl concentration on CO2 diffusion coefficients. However, sea water is a complex system

including not only NaCl but also MgCl2, MgSO4, CaSO4 and so on. These electrolytes

definitely affect the physical properties of CO2 in sea water due to the LJ and electrostatic

interactions. We can further study the physical properties of these electrolytes using molec-

ular simulation in the future.

To store CO2 in sea water, we need to consider the physical properties like the phase

equilibrium of CO2 in the sea water and diffusion coefficient etc, viscosity data of aqueous

solutions of CO2 at high pressures are also required for simulation of proposed deep-sea

storage of waster CO2 (Haugan et al., 1995; Liro et al., 1992). There have been some

experimental data, but very limited. If we can use molecular simulation to obtain viscosity

data in a wider range, it will help with the study of the storage model of CO2 in sea water.

This work has studied the impact of different NaCl concentrations on the NaCl + H2O +

CO2 ternary system. It is found that the diffusion coefficient of CO2 decreases. However,

this work does not study the effect of CO2 concentration on the physical properties like

diffusion coefficients in this NaCl + H2O + CO2 ternary system. Some research shows

that the change of CO2 concentration does change the physical properties of the system

like viscosity (Kumagai and Yokoymam, 1998, 1999). It is also worth further study in this

respect.

155

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