Thermodynamic Modeling of Electrolyte Solutions

Thermodynamic Modeling of Electrolyte Solutions: Bridging Classical Macroscopic

Models and Molecular Simulations

by

Sina Hassanjani Saravi

A Dissertation

In

Chemical Engineering

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfilment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

Dr. Chau-Chyun Chen Chair of Committee

Dr. Rajesh Khare

Dr. Fazle Hussain

Dr. Sindee L. Simon

Mark Sheridan Dean of the Graduate School

August 2019

Mom, Dad, I dedicate this dissertation to you…

Thank you for giving me the ambition to believe that nothing is

impossible

Texas Tech University, Sina Hassanjani Saravi, August 2019

ii

ACKNOWLEDGEMENTS

“Be Concise!”, “Sometimes More is Less!”, and a magical yet simple one-word,

“Think!”, may not seem like life-changing mottos, but they are a few short examples of

countless impactful advice I have received from my advisor, Dr. Chau-Chyun Chen, during

my PhD. Dr. Chen, not only have you been the most hardworking person I’ve ever seen,

but you have also been the greatest mentor, teacher, and an impeccable role model. Without

your mentorship, I would not have become the person I am and the person I wish to

become. Forever, I will be grateful to you.

I’d like to express my sincerest gratitude to Dr. Rajesh Khare. You are a fantastic

teacher! I learned statistical mechanics and MD simulations from you and that opened for

me a whole new world of research which made me even more enthusiastic to pursue

science. Whether giving a lecture, reviewing manuscripts, or engaging in a scientific

discussion, you’ve shown what an amazingly professional and intellectual person you are.

I would also like to thank Dr. Fazle Hussain who gave me the motivation to look at

problems profoundly until I, as he would say, “Own them!”. I’d like to express my

appreciation to Dr. Simon who taught me ‘The Advanced Thermodynamics’ course in my

very first semester as a PhD student. Also, her invaluable comments and suggestions during

my qualifying exam helped me refine my work. I would also like to acknowledge the

financial support of the Jack Maddox Distinguished Engineering Chair Professorship in

Sustainable Energy, sponsored by the J.F Maddox Foundation, for making my PhD

possible.

I want to thank one of my best friends, colleagues, and lab mates, Dr. Ashwin

Ravichandran. He is unbelievably smart, humble, and sophisticated; or at least these are


iii

what he says he is! Kidding aside, it’s been a great journey both in our friendship and in

working together. I wish him an amazing future, in personal life and science!

Islam, Harnoor, Nazir, Hla, Toni, Tanveer, Yuan, Matt, Pradeep, Yifan, Meng, Yue,

Rajasi, Samira, Sanjoy, and last but not least, my buddy Michael! Not only have you all

been my dearest lab mates and friends, you’ve been incredibly nice and supportive. What

an environment! What a great group of people! I will never forget you all. Also, a special

thanks goes to Dr. Md Islam for being such a selflessly helpful and supportive friend.

Words cannot describe the extent of my gratitude toward my small, but warm and

loving family: My mom, Homeyra Barimani, my dad, Ahmad Hassanjani Saravi, and my

only brother, Amirhossein Hassanjani Saravi. Mom! thanks for reading to me the fifth

grade’s science book as bedtime stories when I was five! You are the reason I pursued

science as a career. Dad! Thanks for teaching me math and motivating me to become an

engineer just like yourself! Amirhossein, thanks for being the best ‘Dadashi’ in the world.

Hanging out with you and talking about funny stuff for hours are among my favorite

“activities”.

Last, my wonderful wife, Soraya (and soon to be Dr. Soraya Honarparvar) whose love

and support have shaped me into the person I am. Year after year, it becomes clearer to me

what an unbelievably lucky person I am to have you in my life. For almost a decade, you’ve

been my best classmate, research collaborator, and critique, but beyond all, you’ve been

my best friend. Thank you for believing in me and making such a big deal out of my

achievements and acting like nothing happened in my failures!


iv

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ............................................................................................. ii

ABSTRACT ...................................................................................................................... ix

LIST OF TABLES .......................................................................................................... xii

LIST OF FIGURES ....................................................................................................... xiv

1. INTRODUCTION .....................................................................................................1

1.1. Background ...........................................................................................................1

1.2. Content of Dissertation .........................................................................................6

1.3. References ............................................................................................................8

2. THERMODYNAMIC MODELING OF AQUEOUS ELECTROLYTE SYSTEMS .................................................................................11

2.1. Abstract ...............................................................................................................11

2.2. Electrolyte Systems ............................................................................................12

2.3. Basic Thermodynamics of Electrolytes ..............................................................13

2.4. Thermodynamics of Vapor-Liquid Equilibrium ................................................16

2.5. Thermodynamics of Salt Precipitation ...............................................................18

2.6. Modeling Electrolyte Systems ............................................................................20

2.7. Thermodynamic Models for Electrolytes ...........................................................22


v

2.8. Example 1. Modeling Aqueous Single Electrolytes: H2O-BaCl2 Binary Solution ...................................................................................................28

2.9. Example 2: Modeling BaSO4 (s) Precipitation in a Brine Solution .....................34

2.10. Looking Ahead .................................................................................................38

2.11. Acknowledgments ............................................................................................39

2.12. References ........................................................................................................40

3. THERMODYNAMIC MODELING OF HCl-H2O BINIARY SYSTEM WITH SYMMETRIC ELECTROLYTE NRTL MODEL ................43

3.1. Abstract ...............................................................................................................43

3.2. Introduction ........................................................................................................44

3.3. Thermodynamic Framework ..............................................................................49

Solution Chemistry and Hydration ............................................................. 49

Vapor-Liquid-Liquid Equilibrium .............................................................. 51

Symmetric Electrolyte NRTL Model.......................................................... 53

Calorimetric Properties ............................................................................... 55

3.4. The Modeling Approach .....................................................................................55

Model Parameters ....................................................................................... 55

Experimental Data ...................................................................................... 59

Data Treatment, Regression, and Optimization Method ............................ 67

3.5. Results and Discussion .......................................................................................69

3.6. Conclusions ........................................................................................................83

3.7. Acknowledgements ............................................................................................84


vi

3.8. References ..........................................................................................................85

4. BRIDGING TWO-LIQUID THEORY WITH MOLECULAR SIMULATIONS FOR ELECTROLYTES: AN INVESTIGATION OF AQUOUES NaCl SOLUTION .........................................................................93

4.1. Abstract ...............................................................................................................93

4.2. Introduction ........................................................................................................94

4.3. Theoretical Background .....................................................................................97

Two-Liquid Theory for Electrolytes ........................................................... 97

The Cation-Centered Domain ................................................................... 100

The Anion-Centered Domain .................................................................... 102

The Molecule-Centered Domain............................................................... 103

4.4. Calculation Methodology .................................................................................104

4.5. Molecular Simulations ......................................................................................106

4.6. Binary Interaction Parameters from Regression ...............................................108

4.7. Results and Discussions ...................................................................................109

Regression of Experimental and Simulation Data .................................... 109

Interpretation of the Molecular Quantities in the Framework of eNRTL ....................................................................................................... 109

Prediction of Binary Interaction Parameters from MD Simulation and Validation of the eNRTL Assumptions .............................................. 116

Phase Equilibrium Predictions .................................................................. 119

4.8. Conclusions ......................................................................................................122

4.9. Acknowledgments ............................................................................................124

4.10. Supporting Information ..................................................................................125


vii

Discussions Regarding the Nature of Interactions in the Ion- Centered Domains ..................................................................................... 131

4.11. References ......................................................................................................133

5. PRDICTING PHASE EQUILIBRIA PROPERTIES OF ELECTRLYTE SOLUTIONS BY COMBINING A CLASSICAL THERMODYNAMIC MODEL AND MOLECULAR SIMULATIONS: AQUOUES XCl2 SOLUTIONS, (X=Ba, Sr, Ca, Mg) ..........138

5.1. Abstract .............................................................................................................138

5.2. Introduction ......................................................................................................140

5.3. Theoretical Framework ....................................................................................144

Thermodynamics Background of the eNRTL Model ............................... 144

Relating the τ Parameters to Liquid Structure and Energetic Interaction Quantities ................................................................................ 148

The τ Parameter in a Cation-Centered Domain ........................................ 148

The τ Parameter in an Anion-Centered Domain ....................................... 151

The τ Parameters in a Molecule-Centered Domain .................................. 151

5.4. Quantification of the τ Parameters from Molecular Simulations .....................153

The Effective Molecular Diameters (σ) .................................................... 153

The First Neighbor Shell Radii (R) ........................................................... 153

The Energetic Interactions (ϵ) ................................................................... 154

The Nonrandomness Factor (α) ................................................................ 157

The Binary Interaction Parameters (τ) ...................................................... 158

5.5. Molecular Simulation Details ...........................................................................158

5.6. Results and Discussion .....................................................................................159

The Results of the Physical Quantities of the Aqueous Solutions from MD Simulations ................................................................................ 160

Calculating the Binary Interaction Parameters from MD Simulations ................................................................................................ 173


viii

Defining Effective Binary Interaction Parameters from MD Simulations ................................................................................................ 174

Predicting Phase Equilibria Properties from τ Parameters Obtained via MD Simulations ................................................................... 181

5.7. Conclusions ......................................................................................................190

5.8. Acknowledgements ..........................................................................................191

5.9. Supplementary Information ..............................................................................192

5.10. References ......................................................................................................213

6. CONCLUSIONS AND FUTURE WORK ...........................................................217


ix

ABSTRACT Electrolyte solutions are ubiquitous in many industrial, environmental,

pharmaceutical, and geothermal processes. The crucial key in the design, optimization, and

simulations of the processes involving electrolytes is the availability of comprehensive and

versatile thermodynamic models. Correlative-based classical macroscopic thermodynamic

models have been extensively applied for identifying the solution chemistry, as well as

predicting various phase equilibria and calorimetric properties of ionic solutions. Obtaining

these properties are essential for conducting mass and energy balance calculations while

carrying out process simulations. The widespread use of the correlative thermodynamic

models is due to their simplicity and rapid computational time. Among them, the electrolyte

Non-Random Two-Liquid (eNRTL) model, introduced in the early 1980s, has been

successfully applied in modeling varieties of electrolyte systems, from simple aqueous

binary to multicomponent solutions. The model relates the excess Gibbs free energy of a

solution to the liquid structure with a set of adjustable binary interaction parameters that

are quantified by regressing them to a wide range of experimental data. Once these

adjustable parameters are obtained, the thermophysical properties required for process

simulations can be readily calculated. Therefore, the first step toward helping the engineers

implement efficiently the eNRTL model into their simulations is to build up a

comprehensive and reliable database of the model parameters.

Part of this dissertation is attributed to the development of frameworks for several

electrolyte systems to predict accurately the thermodynamic properties of vapor-liquid,

liquid-liquid, and solid-liquid equilibria, over wide ranges of concentrations and

temperatures. It is shown that the predictions provided by the eNRTL model compare well


x

with the reported experimental data for model systems including HCl-H2O binary, aqueous

BaCl2 solution, and the multicomponent solution of Na+-Ba2+-Cl--SO42--H2O.

Though employing the original eNRTL in its correlative form is straightforward to

use, there exist a number of drawbacks associated with the current state-of-the-art that

could potentially hinder the process simulations. Some examples of which include the lack

of available experimental data or the absence of accurate reports of the data uncertainties.

Especially, if too many adjustable parameters are fitted to a limited number of data points,

the resulting regressed parameters will not be well-determined. The physical significance

of the adjustable parameters is another long-standing debate, the lack of which would raise

concerns about the credibility of the predictions beyond the range of the reported

experimental data. It is worth mentioning that the regression procedures typically fail to

result in a unique set of parameters, thereby selecting the physically meaningful parameters

would require substantial experience and ‘manual tuning’.

To overcome these issues yet exploiting the rapid and accurate predictions provided

by eNRTL, a novel theoretical framework is established to bridge the classical

thermodynamic model and molecular simulations from a statistical mechanical approach.

By revisiting the statistical mechanics of two-liquid theory, the binary interaction

parameters of eNRTL are expressed as functions of the liquid structure and energy

information quantities of the solutions. These quantities are then obtained from the

molecular dynamics simulations and potential of mean force free energy calculations.

Several electrolyte systems including the aqueous NaCl, BaCl2, SrCl2, CaCl2, and MgCl2

solutions are selected for the validation of the approach.


xi

The results for the binary interaction parameters, together with the predictions of the

phase equilibria properties from the MD simulations are in satisfactory agreement with

those obtained from the regression. It is demonstrated, for the first time, that the eNRTL

model can be rendered completely predictive, circumventing the inherent shortcomings

associated with correlation. The established method, with further refinement and

improvement, can be broadly utilized for rapid predictions of the thermodynamic

properties in industrial process design.


xii

LIST OF TABLES

2.1. The most commonly used thermodynamic models are the Pitzer, OLI-MSE, and eNRTL models. .................................................................................... 27

2.2. The temperature coefficients and binary interaction parameters for the examples. .............................................................................................................. 29

2.3. The solubility data are subsequently used to identify the temperature coefficients (A, B, C, D, E) for the solubility product constant ............................ 36

3.1. Summary of thermophysical properties and model parameters. ........................... 56

3.2. Extended Antoine equation parameters for saturation pressures (Piº) for

H2O and HCl. ........................................................................................................ 57

3.3. Redlich-Kwong equation of state parameters for H2O and HCl. .......................... 58

3.4. DIPPR liquid molar density (ρil) model parameters for H2O and HCl ................. 58

3.5. DIPPR ideal gas heat capacity (cp,iig) model parameters for H2O and

HCl ........................................................................................................................ 58

3.6. DIPPR heat of vaporization (Δvaphi) model parameters for H2O and HCl ............ 59

3.7. Dielectric constants (ε) for H2O and HCl ............................................................. 59

3.8. List of experimental data used in regression and model validation for the HCl-H2O binary system .................................................................................. 60

3.9. eNRTL model parameters (τij) for molecule-electrolyte and molecule-molecule pairs. ...................................................................................................... 69

3.10. Chemical equilibrium constant parameters for unsymmetric reference state. ...................................................................................................................... 70

3.11. Chemical equilibrium constant parameters for symmetric reference state. ...................................................................................................................... 70


xiii

4.1. Binary interaction parameters from regression ................................................... 109

5.1. Effective local mole fractions in the first shell (Å) – BaCl2 (aq) ........................ 162

5.2. Effective local mole fractions in the first shell (Å) – SrCl2 (aq) ......................... 162

5.3. Effective local mole fractions in the first shell (Å) – CaCl2 (aq) ........................ 163

5.4. Effective local mole fractions in the first shell (Å) – MgCl2 (aq) ....................... 163

5.5. Binary interaction parameters from regression ................................................... 174

5.6. Effective binary interaction parameters .............................................................. 176


xiv

LIST OF FIGURES 2.1. τ values vs. temperature for the aqueous BaCl2 system ........................................ 30

2.2. Mean ionic activity coefficient vs. molality of barium chloride .......................... 31

2.3. Osmotic coefficient vs. molality of barium chloride ............................................ 32

2.4. Vapor pressure of the solution vs. molality of barium chloride at different temperatures. . ....................................................................................................... 32

2.5. Solubility of barium chloride vs. temperature. ..................................................... 33

2.6. Excess enthalpy vs. molality of barium chloride .................................................. 34

2.7. Solubility of barium sulfate vs. temperature. ........................................................ 36

2.8. Solubility of barium sulfate vs. molality of sodium sulfate at different temperatures .......................................................................................................... 37

2.9. Solubility of barium sulfate vs. molality of sodium chloride at different temperatures. ......................................................................................................... 38

3.1. Speciation and solution chemistry of the HCl-H2O binary system. ...................... 52

3.2. Model results for the HCl dissociation versus HCl wt. % at different temperatures. ......................................................................................................... 71

3.3. Model results for species mole fractions versus HCl wt. % at 273.15 K, and at different temperatures.. .............................................................................. 72

3.4. Model results for system pressure compared with experimental data at different temperatures. .......................................................................................... 73

3.5. Model results compared with experimental data for HCl compositions in vapor phase versus in liquid phase at different temperatures. .............................. 74

3.6. Model results for LLE compared with experimental data. ................................... 75

3.7. Model results for boiling point at 50 and 100 kPa versus literature smoothed curves. .................................................................................................. 76

3.8. Model results for molality scale mean ionic activity coefficient at 298.15 K and 100 kPa, compared with experimental data. .............................................. 77

3.9. Model results for water activity at 298.15 K and 100 kPa, compared with experimental data. ................................................................................................. 78


xv

3.10. Model results for osmotic coefficient at 298.15 K and 100 kPa, compared with experimental data. ......................................................................................... 79

3.11. Model results for excess enthalpy (kJ/mol) with unsymmetric reference state at 298.15 K and 100 kPa, compared with experimental data. Also shown, the boiling-point limit and the extended trend for excess enthalpy. ................................................................................................................ 80

3.12. Model results for liquid molar heat capacity of dilute solutions (kJ/kmol.K) and experimental data at different temperatures. ............................. 81

3.13. Model results for liquid molar heat capacity (kJ/kmol.K) compared with experimental data and smoothed curves at different temperatures. ...................... 82

3.14. Merkel enthalpy-concentration chart (Btu/lb) and extended trend lines at different temperatures.. ......................................................................................... 83

4.1. Schematic describing the three possible local molecular domains considered by the eNRTL model ........................................................................ 100

4.2. The local mole fraction of different species in the three domains for the 4 mol/kg aqueous NaCl solution at 298.15 K. .................................................... 114



4.5. The binary interaction parameters for the aqueous NaCl solution at 298.15 K. ............................................................................................................. 118

4.6. Comparison of the mean ionic activity coefficients of aqueous NaCl solution phase behavior at 298.15 K between MD simulations and regression results. ................................................................................................ 121

4.7. Comparison of aqueous NaCl solution phase behavior (vapor pressure and excess Gibbs free energy) at 298.15 K between MD simulations and regression results. ................................................................................................ 122

5.1. Schematic configurations of the three local domains as hypothesized by the eNRTL model. .............................................................................................. 145

5.2. The effective mole fractions calculated from the MD simulations for the aqueous BaCl2 solution at 4 m — Cation-centered domain. .............................. 164

5.3. The effective mole fractions calculated from the MD simulations for the aqueous BaCl2 solution at 4 m — Anion-centered domain ................................ 165


xvi

5.4. The effective mole fractions calculated from the MD simulations for the aqueous BaCl2 solution at 4 m — Molecule-centered domain ........................... 166

5.5. A schematic of the potential of mean force for the 4 m aqueous MgCl2 solution at 298.15 K. PMF of Mg2+ around a center Clˉ and water around a center water. ..................................................................................................... 167

5.6. A schematic of the potential of mean force for the 4 m aqueous MgCl2 solution at 298.15 K. PMF of Mg2+ around a center water and Clˉ around a center water. ..................................................................................................... 168

5.7. Potential of mean force of the Ba2+-Cl- pair in aqueous BaCl2 solution. The potential of mean force of the anion-cation pair at the concentration of 2 m from the local van der Waals, electrostatics, and the combination of both local and electrostatics contributions...................................................... 170

5.8. Potential of mean force of the Ba2+-Cl- pair in aqueous BaCl2 solution. The potential of mean force of the anion-cation pair at the concentration of 6 m from the local van der Waals, electrostatics, and the combination of both local and electrostatics contributions. ................................................... 171

5.9. Potential of mean force of the Ba2+-Cl- pair in aqueous BaCl2 solution. The potential of mean force attributed to the electrostatic forces calculated from the Debye-Hückel theory at different concentrations. .............. 172

5.10. The effective binary interaction parameters for the aqueous BaCl2 solution at 298.15 K. Regression vs. MD simulations. . ..................................... 177

5.11. The effective binary interaction parameters for the aqueous SrCl2 solution at 298.15 K. Regression vs. MD simulations. ....................................... 178

5.12. The effective binary interaction parameters for the aqueous CaCl2 solution at 298.15 K. Regression vs. MD simulations. ....................................... 179

5.13. The effective binary interaction parameters for the aqueous MgCl2 solution at 298.15 K. Regression vs. MD simulations. ....................................... 180

5.14. Mean ionic activity coefficients for the entire concentration range in the aqueous BaCl2 solution at 298.15 K. Experimental data, Regression, and MD simulations. ................................................................................................. 183

5.15. Mean ionic activity coefficients for the entire concentration range in the aqueous SrCl2 solution at 298.15 K. Experimental data, Regression, and MD simulations. ................................................................................................ 184

5.16. Mean ionic activity coefficients for the entire concentration range in the aqueous CaCl2 solution at 298.15 K. Experimental data, Regression, and MD simulations ................................................................................................... 185


xvii

5.17. Mean ionic activity coefficients for the entire concentration range in the aqueous MgCl2 solution at 298.15 K. Experimental data, Regression, and MD simulations. .................................................................................................. 186

5.18. Vapor pressure and excess Gibbs free energy in the entire concentration range in the aqueous BaCl2 solution at 298.15 K. Regression vs. MD simulations. ......................................................................................................... 187

5.19. Vapor pressure and excess Gibbs free energy in the entire concentration range in the aqueous SrCl2 solution at 298.15 K. Regression vs. MD simulations. ......................................................................................................... 188

5.20. Vapor pressure and excess Gibbs free energy in the entire concentration range in the aqueous CaCl2 solution at 298.15 K. Regression vs. MD simulations. ......................................................................................................... 189

5.21. Vapor pressure and excess Gibbs free energy in the entire concentration range in the aqueous MgCl2 solution at 298.15 K. Regression vs. MD simulations. ......................................................................................................... 190


1

CHAPTER 1. INTRODUCTION

1.1. Background

Electrolyte solutions are present in many industrial 1-3 and natural processes 4,5. Some

examples of such processes include the hydraulic fracturing in oil and gas industry 6,

environmental processes such as desalination 7, pharmaceutical manufacturing 8, and

lithium ion batteries production 9 as renewable energy storage sources. The widespread

presence of the electrolyte solutions in such a broad range of applications requires

exercising careful attention to their underlying solution chemistries, as well as the related

phase equilibria and calorimetric properties 10. The crucial bottleneck in the design and

optimization of the processes involving electrolytes is the availability of the

thermodynamic models to provide predictions of such properties to support mass and

energy balance calculations 11. Particularly, the mean ionic activity coefficient (𝛾𝛾±) is a

unique property that quantifies the non-ideality of the electrolyte solutions. Hence, the

cornerstone of most of the thermodynamic modeling studies is to obtain 𝛾𝛾± from which all

the other properties of interest can be subsequently calculated 10,12,13.

After the world war II, thermodynamicists paid particular attention toward establishing

theoretical frameworks to predict thermophysical properties that were required for process

design in industry 14. Assessing the exact microscopic liquid structure of the electrolyte

solutions was cumbersome due to the limited computer powers which would create a

bottleneck in the use of statistical mechanical approaches. Even nowadays with accessing

to the high performance computers and the advantages that the enhanced sampling methods

offer, the use of predictive models implemented by molecular simulation techniques


2

demands expensive computational time, making it inefficient to directly utilize them in

process simulations 15. That shifted the paradigm toward developing semi-empirical and

empirical correlative models for industrial process design to compensate for the ambiguous

structural characteristics of the ionic solutions 16. Most of these models are treated as

perturbation-like theories to express the excess Gibbs free energy of the electrolyte

solutions as a combination of the contributions of long-range electrostatic and short-range

van der Waals forces. Among the most widely used such models, Pitzer 17, OLI-MSE 18,

and the electrolyte Non-Random Two-Liquid (eNRTL) 12,19-21 models have been broadly

adopted and used by the industry due to their versatility and relatively inexpensive

computational procedures. A rather succinct description of these models is discussed in the

next chapter along with their advantages and disadvantages.

The eNRTL model introduced in 1982 by Chen et al. 19 has shown to provide the most

successful predictions considering the limited number of adjustable parameters, while

covering the widest ranges of ionic species, concentrations, and temperatures. The model

uses the local composition concept described by the original Non-Random Two-Liquid

theory of Brandani and Prausnitz 16 to account for the short-range interactions. The long-

range electrostatic forces, on the other hand, are expressed by a modified semi-empirical

version of the original Debye-Hückel 22 theory, i.e., Pitzer-Debye-Hückel 12,19-21.

Combining the two contributions, the excess Gibbs free energy of the electrolyte solutions

are obtained, from which the activity coefficients and consequently other thermophysical

properties can be acquired. Only two adjustable binary interaction parameters are needed

to account for the interactions between any pair of species, i.e., ion-ion, ion-molecule, or


3

molecule-molecule, which renders the model considerably more applicable compared to

the other correlative models.

The limited number of parameters employed by the eNRTL model is a good indication

of not overestimating the physical significance of the model parameters and hence not

overfitting the experimental data. The concerns about using too many adjustable

parameters in macroscopic correlative models are twofold; first, the parameters will not be

well-determined if the number of the experimental data used for regression is insufficient

compared to the number of adjustable parameters. Second, though the interpolations could

be accurate, the extrapolations would not be reliable in a model where the adjustable

parameters are not physically well-defined. A recent molecular dynamics simulations study

of the aqueous NaCl solution by Hossain et al. 13 demonstrated that the mean ionic activity

coefficients predicted by the eNRTL model show an asymptotic behavior at supersaturated

solutions, compared to those predicted by the Pitzer model which show a rapid divergence

in high concentration regions. Such findings confirm the superiority of the extrapolations

predicted by the eNRTL model compared to those of Pitzer, while employing far fewer

number of adjustable parameters.

Despite the discussed advantages of the eNRTL model over the Pitzer model, lacking

a comprehensive library of the regressed binary interaction parameters has limited the

ability of broadly utilizing the eNRTL model in process simulations. Therefore, the first

goal of this dissertation, as part of building up a comprehensive database of the eNRTL

model parameters, is to establish thermodynamic frameworks for variety of ionic solutions

for providing accurate predictions of phase equilibria behavior, calorimetric, and speciation

properties. The electrolyte systems studied here cover from binary aqueous HCl solutions


4

to binary and multicomponent solid-liquid equilibrium systems of, respectively, aqueous

BaCl2 and Na+-Ba2+-Cl--SO42--H2O solutions, which are covered in Chapters 2 and 3.

Furthermore, while not presented in this dissertation, several other multicomponent

systems have also been rigorously studied and reported in two publications to which the

reader is referred for further information 23,24.

Although the current state-of-the-art in quantifying the binary interaction parameters

through regression is straightforward and rapidly performed, there are a number of

limitations that need to be addressed. Lack of the available or reliable experimental data is

the foremost issue that one could encounter while performing regression. Even when data

are available, the inconsistency between the reported data points from different references,

the lack of reported uncertainties, and the inevitable arbitrary treatment of the data could

impose obstacles in carrying out an objective regression procedure. Furthermore, the

minimization functions employed in the regression seldom result in a unique set of

parameters. Thereby selecting the physically relevant parameters could be ambiguous and

requires manual tuning or arbitrary choosing from multiple solutions based on experience

or the quality of the yielded predictions.

Parallel to the classical correlative-based thermodynamic models, another path has

also been pursued by researchers to calculate the thermophysical properties of electrolyte

solutions from statistical mechanical approaches. Molecular dynamics (MD) and Monte

Carlo (MC) simulations have helped the researchers to study the liquid structure and free

energy profiles in electrolyte solutions. Many molecular simulation studies reported

predictions of the fundamental properties of electrolytes such as the chemical potential,

Gibbs free energy, and activity coefficients 25-31. Though valuable insight has been gained,


5

due to the expensive computations required for implementing advanced simulation

techniques, the use of molecular simulations in process design has been limited.

In order to exploit the advantages of both correlative and predictive approaches while

circumventing the associated shortcomings discussed above, it is desired to develop a

hybrid methodology by which the adjustable parameters of the classical models can be

obtained from a completely predictive method. Such an approach has seldom been pursued

by researchers in the past. A few attempts for connecting the classical macroscopic models

to molecular simulations have been reported in the literature, however, for nonelectrolyte

components 15,32-34. A novel theoretical framework is thus established to bridge the classical

eNRTL model and molecular simulations for electrolyte solutions 11. By revisiting the

statistical mechanics of two-liquid theory, which is the basis of the eNRTL model, the

binary interaction parameters are, for the first time, formulated as functions of the local

microscopic structure of the liquid and energetic interaction quantities. All of these

physical quantities are then calculated from the MD simulations and potential of mean

force (PMF) free energy calculations. The comparison between the binary interaction

parameters obtained from the established predictive approach (which is covered in chapters

4 and 5), and those identified from the regression shows satisfactory agreement. The

established approach sheds light on the physical significance of the eNRTL model

parameters which previously assumed to be merely correlative and semi-empirical.

Overall, this dissertation illustrates that the classical macroscopic models, if developed

with solid physical foundation, could be supported from statistical mechanical theories and

molecular simulations, and hence should be confidently implemented in process

simulations, specifically, where data are scarce. Furthermore, the established hybrid


6

framework can guide the regression-based methods to select physically relevant

parameters, thereby solving the problems associated with the multiple solutions of

minimization functions in regression procedures.

1.2. Content of Dissertation

The rest of this dissertation is organized as follows. In Chapter 2, a complete

description of the electrolyte solutions, their characteristics, chemistry and etc., are

discussed. Thermodynamic basics of the electrolytes are explained thoroughly, followed

by expanding on the current status of the widely used classical correlative thermodynamic

models. Various phase equilibria properties and equations including those of vapor-liquid

and solid-liquid (salt precipitation) equilibria are discussed. Strengths and weaknesses of

different models are explained and compared to one another. The chapter is then bringing

two examples on employing the eNRTL model to predict accurately the different properties

of VLE and SLE of a number of electrolyte systems, followed by conclusion and a direction

toward the future work.

Chapter 3 presents a comprehensive thermodynamic framework for the binary system

of HCl-H2O. By regressing abundant experimental data of phase equilibria and calorimetric

properties, the binary interaction parameters of the eNRTL and their temperature

dependence are quantified. Using the regressed parameters, the model is shown to provide

accurate predictions of a wide-ranging thermophysical properties over the entire

concentration range, which includes the phase separation into two liquids (LLE). The acid

is considered to be partially dissociated in the solution rather than following a simplistic

and unrealistic assumption of the complete dissociation. Thereby, the reported model is the

first ‘complete’ model for use in process design in industry.


7

In Chapter 4 a new theoretical framework is established to render the classical

thermodynamic model—eNRTL— completely predictive. By revisiting the statistical

mechanics of two-liquid theory, the binary interaction parameters of the model are

expressed as functions of the liquid structure and interaction energy quantities. Such

quantities are then obtained from the MD simulations and potential of mean force free

energy calculations. Aqueous NaCl solution is selected as the model system to test the

validity of the developed methodology. The results demonstrate that the parameters and

property predictions from the MD simulations are aligned with those obtained from the

regression of the experimental data. Furthermore, the established framework provides the

physical interpretation of the adjustable, previously known to be semi-empirical,

parameters.

In Chapter 5, to extend the established theory to account for general electrolyte

solutions, the methodology (presented in Chapter 4) is extended and generalized for

multivalent electrolyte solutions. The formulations reported in Chapter 4 have thus been

revisited and refined. Several di-univalent ionic salt solutions are selected for the validation

of the technique. These model systems include aqueous BaCl2, SrCl2, CaCl2, and MgCl2

solutions. It is successfully illustrated that the established theoretical framework is

applicable to all classes of electrolyte solutions regardless of their valence numbers.

In Chapter 6, Conclusions and future work are discussed to help the reader capture

the essence and highlights of this dissertation, as well as demonstrating the possible future

paths that can and should be pursued toward making progress in modeling electrolyte

solutions by utilizing inexpensive molecular simulations in industry.


8

1.3. References

1. Shaffer DL, Arias Chavez LH, Ben-Sasson M, Romero-Vargas Castrillón S, Yip NY, Elimelech M. Desalination and reuse of high-salinity shale gas produced water: drivers, technologies, and future directions. Environmental Science & Technology. 2013;47:9569-9583.

2. Newman SA, Barner HE, Klein M, Sandler SI. Thermodynamics of aqueous systems with industrial applications: ACS Publications, 1980.

3. Chen C-C. Toward development of activity coefficient models for process and product design of complex chemical systems. Fluid Phase Equilibria. 2006;241:103-112.

4. Sherman DM, Collings MD. Ion association in concentrated NaCl brines from ambient to supercritical conditions: results from classical molecular dynamics simulations. Geochemical Transactions. 2002;3:102-107.

5. Brodholt JP. Molecular dynamics simulations of aqueous NaCl solutions at high pressures and temperatures. Chemical Geology. 1998;151:11-19.

6. Reible DD, Honarparvar S, Chen C-C, Illangasekare TH, MacDonell M. Environmental impacts of hydraulic fracturing. In: Environmental technology in the oil industry. Springer; 2016:199-219.

7. Al-Ahmad M, Aleem FA. Scale formation and fouling problems effect on the performance of MSF and RO desalination plants in Saudi Arabia. Desalination. 1993;93:287-310.

8. Crison JR, Weiner ND, Amidon GL. Dissolution media for in vitro testing of water‐insoluble drugs: Effect of surfactant purity and electrolyte on in vitro dissolution of carbamazepine in aqueous solutions of sodium lauryl sulfate. Journal of Pharmaceutical Sciences. 1997;86:384-388.

9. Su C-C, He M, Amine R, et al. Solvating power series of electrolyte solvents for lithium batteries. Energy & Environmental Science. 2019.

10. Saravi SH, Honarparvar S, Chen C-C. Modeling aqueous electrolyte systems. Chemical Engineering Progress. 2015;111:65-75.

11. Saravi SH, Ravichandran A, Khare R, Chen C-C. Bridging Two-Liquid Theory with Molecular Simulations for Electrolytes: An Investigation of Aqueous NaCl Solution.

12. Song Y, Chen C-C. Symmetric electrolyte nonrandom two-liquid activity coefficient model. Industrial & Engineering Chemistry Research. 2009;48:7788-7797.


9

13. Hossain N, Ravichandran A, Khare R, Chen C-C. Revisiting electrolyte thermodynamic models: Insights from molecular simulations. AIChE Journal. 2018;64:3728-3734.

14. May PM, Rowland D. Thermodynamic modeling of aqueous electrolyte systems: current status. Journal of Chemical & Engineering Data. 2017;62:2481-2495.

15. Ravichandran A, Khare R, Chen C-C. Predicting NRTL binary interaction parameters from molecular simulations. AIChE Journal. 2018;64:2758-2769.

16. Brandani V, Prausnitz J. Two-fluid theory and thermodynamic properties of liquid mixtures: General theory. Proceedings of the National Academy of Sciences. 1982;79:4506-4509.

17. Pitzer KS. Thermodynamics of electrolytes. I. Theoretical basis and general equations. The Journal of Physical Chemistry. 1973;77:268-277.

18. Wang P, Anderko A, Young RD. A speciation-based model for mixed-solvent electrolyte systems. Fluid Phase Equilibria. 2002;203:141-176.

19. Chen C-C, Britt HI, Boston J, Evans L. Local composition model for excess Gibbs energy of electrolyte systems. Part I: Single solvent, single completely dissociated electrolyte systems. AIChE Journal. 1982;28:588-596.

20. Chen C-C, Evans LB. A local composition model for the excess Gibbs energy of aqueous electrolyte systems. AIChE Journal. 1986;32:444-454.

21. Chen C-C, Song Y. Generalized electrolyte‐NRTL model for mixed‐solvent electrolyte systems. AIChE Journal. 2004;50:1928-1941.

22. Debye P, Hückel E. De la theorie des electrolytes. I. abaissement du point de congelation et phenomenes associes. Physikalische Zeitschrift. 1923;24:185-206.

23. Honarparvar S, Saravi SH, Reible D, Chen C-C. Comprehensive thermodynamic modeling of saline water with electrolyte NRTL model: A study on aqueous Ba2+-Na+-Cl−-SO4

2− quaternary system. Fluid Phase Equilibria. 2017;447:29-38.

24. Honarparvar S, Saravi SH, Reible D, Chen C-C. Comprehensive thermodynamic modeling of saline water with electrolyte NRTL model: A study of aqueous Sr2+-Na+-Cl−-SO4


25. Paluch AS, Jayaraman S, Shah JK, Maginn EJ. A method for computing the solubility limit of solids: Application to sodium chloride in water and alcohols. The Journal of Chemical Physics. 2010;133:124504.

26. Moucka F, Lísal M, Škvor Ji, Jirsák J, Nezbeda I, Smith WR. Molecular simulation of aqueous electrolyte solubility. 2. Osmotic ensemble Monte Carlo methodology


10

for free energy and solubility calculations and application to NaCl. The Journal of Physical Chemistry B. 2011;115:7849-7861.

27. Aragones J, Sanz E, Vega C. Solubility of NaCl in water by molecular simulation revisited. The Journal of Chemical Physics. 2012;136:244508.

28. Mester Z, Panagiotopoulos AZ. Mean ionic activity coefficients in aqueous NaCl solutions from molecular dynamics simulations. The Journal of Chemical Physics. 2015;142:044507.

29. Mester Z, Panagiotopoulos AZ. Temperature-dependent solubilities and mean ionic activity coefficients of alkali halides in water from molecular dynamics simulations. The Journal of Chemical Physics. 2015;143:044505.

30. Orozco GA, Moultos OA, Jiang H, Economou IG, Panagiotopoulos AZ. Molecular simulation of thermodynamic and transport properties for the H2O + NaCl system. The Journal of Chemical Physics. 2014;141:234507.

31. Jiang H, Mester Z, Moultos OA, Economou IG, Panagiotopoulos AZ. Thermodynamic and transport properties of H2O + NaCl from polarizable force fields. Journal of Chemical Theory and Computation. 2015;11:3802-3810.

32. Neiman M, Cheng H, Parekh V, Peterson B, Klier K. A critical assessment on two predictive models of binary vapor–liquid equilibrium. Physical Chemistry Chemical Physics. 2004;6:3474-3483.

33. Jónsd SÓ, Rasmussen K, Fredenslund A. UNIQUAC parameters determined by molecular mechanics. Fluid Phase Equilibria. 1994;100:121-138.

34. Sum AK, Sandler SI. Use of ab initio methods to make phase equilibria predictions using activity coefficient models. Fluid Phase Equilibria. 1999;158:375-380.


11

CHAPTER 2. THERMODYNAMIC MODELING OF AQUEOUS

ELECTROLYTE SYSTEMS1

2.1. Abstract

First-principles-based process simulation of electrolyte systems is a key enabling

technology for chemical engineers to design, debottleneck, and optimize chemical

processes with electrolytes. In the development of process simulation models for

electrolytes, a key challenge is the availability of accurate and rigorous thermodynamic

models.

This article introduces the fundamental thermodynamics of electrolyte systems, and

identifies critical thermodynamic parameters and the equations and relationships used to

determine them. It also outlines the steps to develop thermodynamic models of electrolyte

systems and provides two examples to illustrate these steps.

1 This chapter is reproduced from the paper published as: Saravi SH, Honarparvar S, Chen C-C. Modeling aqueous electrolyte systems. Chemical Engineering Progress. 2015, 111:65-75.


12

2.2. Electrolyte Systems

Electrolyte systems are involved in a wide range of industrial processes, including

hydraulic fracturing, gas sweetening, oil and gas production, fluegas desulfurization, CO2

capture and sequestration, water desalination, nuclear waste processing, energy storage,

basic chemicals manufacturing, and pharmaceuticals manufacturing.

Electrolytes — substances that dissociate into pairs of charged ions and counter-

charged ions in a solution — are categorized based on their extent of dissociation. Strong

electrolytes (e.g., NaCl, KBr, CaCl2) dissociate completely in aqueous solutions, whereas

weak electrolytes (e.g., carbonates, phosphates, and carboxylates) only partially dissociate.

Although some compounds are considered strong electrolytes at high dilution, they may be

weak electrolytes at low dilution. For example, strong acids such as hydrochloric acid,

nitric acid, and sulfuric acid dissociate nearly completely at high dilution, but only partially

dissociate at high acid concentrations; thus, these acids should be considered weak

electrolytes or mixed-solvent electrolytes.

In addition to dissociation, other reactions, such as hydration, acid-base reactions, and

complex-ion formation, can occur in electrolyte systems. Therefore, solution chemistry is

the primary factor controlling the physical and chemical properties of electrolyte solutions

1. Also, regardless of whether electrolytes dissociate completely or partially,

electroneutrality is always maintained for electrolyte solutions.

The presence of ions is responsible for another unique characteristic of electrolyte

solutions — i.e. , the long-range ion-ion Coulombic electrostatic interaction. The short-

range ion-molecule interaction and molecule-molecule interaction also contribute to

solution nonideality.


13

Data are available on the thermodynamic properties of electrolyte solutions, the most

common of which are mean ionic activity coefficient, osmotic coefficient, boiling point

elevation, freezing point depression, vapor pressure, enthalpy of solution or excess

enthalpy, partial molal heat capacity, solution pH, gas solubility, salt solubility, and true

species speciation. Solution pH and true species speciation data are particularly useful in

discerning the solution chemistry and the underlying thermodynamic constants 2,3.

Transport properties such as electrical conductivity could also help develop a proper

understanding on the nature of electrolyte solutions.

From a process simulation perspective, the most critical thermodynamic properties of

interest are the phase equilibrium properties, such as vapor pressure, gas solubility, and salt

solubility; calorimetric properties, such as liquid enthalpy and heat capacity; and speciation

such as pH and species concentrations. Accurate and consistent representation of these

thermodynamic properties is essential to support heat and mass balance calculations and

rate-based process simulation.

2.3. Basic Thermodynamics of Electrolytes

The fundamental thermodynamic property describing the behavior of component 𝑖𝑖 in

a multicomponent system is its chemical potential (𝜇𝜇𝑖𝑖). The chemical potential provides a

springboard to the properties needed for process simulation, as it can be put into molecular

thermodynamics models to calculate phase equilibrium, calorimetric, and speciation.

The chemical potential of component 𝑖𝑖 can be expressed in terms of a reference

chemical potential (𝜇𝜇𝑖𝑖0) and the component’s activity (𝑎𝑎𝑖𝑖):

𝜇𝜇𝑖𝑖 = 𝜇𝜇𝑖𝑖0 + 𝑅𝑅𝑇𝑇𝑙𝑙𝑙𝑙(𝑎𝑎𝑖𝑖) (2.1)


14

The activity describes the concentration of species 𝑖𝑖. It can be expressed on several

different concentration scales (e.g., mole fraction, mole molality) in terms of the product

of either the mole fraction (𝑥𝑥𝑖𝑖) and the mole-fraction-scale activity coefficient (𝛾𝛾𝑖𝑖), or the

molality (𝑚𝑚𝑖𝑖) and the molality scale activity coefficient (𝛾𝛾𝑖𝑖(𝑚𝑚)):

𝑎𝑎𝑖𝑖 = 𝛾𝛾𝑖𝑖𝑥𝑥𝑖𝑖 (2.2)

𝑎𝑎𝑖𝑖 = 𝛾𝛾𝑖𝑖(𝑚𝑚)𝑚𝑚𝑖𝑖 (2.3)

Two different reference-state conditions are used with electrolyte systems. The

symmetric reference state, which assumes that the activity coefficient of a pure component

is unity, is typically used to describe the activity coefficient of the solvent (i.e., water) at

system temperature and pressure. In contrast, an unsymmetric reference state, which

assumes that the activity coefficients of a component at infinite dilution is unity, is often

chosen for electrolytes and molecular solutes, because the concentrations of these solutes

are low relative to that of the solvent water. The molality scale activity coefficient is often

used in the literature for aqueous dilute electrolyte solutions. However, the mole-fraction-

scale activity coefficient is a more practical choice for aqueous concentrated electrolytes

and mixed-solvent electrolyte systems.

As electrolytes undergo dissociation (complete or partial), the species are in chemical

equilibrium:

∑ 𝜈𝜈𝑖𝑖𝜇𝜇𝑖𝑖𝑖𝑖 = 0 (2.4)

where 𝜈𝜈𝑖𝑖 is the reaction stoichiometric coefficient for species 𝑖𝑖 involved in the electrolyte

reaction.

Consider an electrolyte in the form of 𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈− that dissociates into 𝜈𝜈+ cations of 𝑀𝑀

each with a charge of 𝑧𝑧+, and 𝜈𝜈− anions of 𝑋𝑋 each with a charge 𝑧𝑧− 4:


15

𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈−(𝑎𝑎𝑎𝑎) ↔ 𝜈𝜈+𝑀𝑀𝑧𝑧+(𝑎𝑎𝑎𝑎) + 𝜈𝜈−𝑋𝑋𝑧𝑧−

(𝑎𝑎𝑎𝑎)

From Eq. 2.4, the chemical potential of the electrolyte is:

𝜇𝜇𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈− = 𝜈𝜈+𝜇𝜇𝑀𝑀𝑧𝑧+ + 𝜈𝜈−𝜇𝜇𝑋𝑋𝑧𝑧− (2.5)

By substituting Eq. 2.1 for the chemical potential of the ionic species in Eq. 2.5, the

chemical potential of the electrolytes can be calculated from:

𝜇𝜇𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈− = 𝜇𝜇𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈−0 + 𝜈𝜈+𝑅𝑅𝑇𝑇𝑙𝑙𝑙𝑙�𝛾𝛾𝑀𝑀𝑧𝑧+

(𝑚𝑚)𝑚𝑚𝑀𝑀𝑧𝑧+� +

𝜈𝜈−𝑅𝑅𝑇𝑇𝑙𝑙𝑙𝑙�𝛾𝛾𝑋𝑋𝑧𝑧−(𝑚𝑚)𝑚𝑚𝑋𝑋𝑧𝑧−�

(2.6)

where 𝛾𝛾𝑀𝑀𝑧𝑧+(𝑚𝑚) and 𝛾𝛾𝑋𝑋𝑧𝑧−

(𝑚𝑚) are the molality-scale ionic activity coefficients of 𝑀𝑀𝑧𝑧+ and 𝑋𝑋𝑧𝑧−,

respectively, which can be calculated with electrolyte activity coefficient models.

The following fundamental equations for stoichiometric number (Eq. 2.7), mean ionic

activity coefficient (Eq. 2.8), and the mean molality (Eq. 2.9) can be substituted into Eq.

2.6 to express the chemical potential of the electrolyte in terms of mean properties (Eq.

2.10).

𝜈𝜈 = 𝜈𝜈+ + 𝜈𝜈− (2.7)

𝛾𝛾±(𝑚𝑚) = (𝛾𝛾𝑀𝑀𝑧𝑧+

(𝑚𝑚)𝜈𝜈+𝛾𝛾𝑋𝑋𝑧𝑧−(𝑚𝑚)𝜈𝜈−)1 𝜈𝜈� (2.8)

𝑚𝑚± = (𝑚𝑚𝑀𝑀𝑧𝑧+𝜈𝜈+𝑚𝑚𝑋𝑋𝑧𝑧−𝜈𝜈−)1 𝜈𝜈� (2.9)

𝜇𝜇𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈− = 𝜇𝜇𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈−0 + 𝜈𝜈𝑅𝑅𝑇𝑇𝑙𝑙𝑙𝑙(𝛾𝛾±

(𝑚𝑚)𝑚𝑚±) (2.10)

Equations 2.3, 2.8, and 2.9 can be combined to express the mean activity (𝑎𝑎±) in terms

of mean properties:

𝑎𝑎± = (𝑎𝑎𝑀𝑀𝑧𝑧+𝜈𝜈+𝑎𝑎𝑋𝑋𝑧𝑧−𝜈𝜈−)1 𝜈𝜈� = 𝛾𝛾±(𝑚𝑚)𝑚𝑚± (2.11)


16

To calculate the calorimetric properties in a form that is thermodynamically consistent

with the activity coefficients, molar liquid enthalpy (ℎ𝑙𝑙) and molar heat capacity (𝑐𝑐𝑝𝑝) of the

liquid solution are calculated by the following thermodynamic relationships:

ℎ𝑙𝑙 = 𝑥𝑥𝑤𝑤ℎ𝑤𝑤𝑜𝑜 + ∑ 𝑥𝑥𝑖𝑖ℎ𝑖𝑖∞,𝑎𝑎𝑎𝑎

𝑖𝑖 + ℎ∗,𝑒𝑒𝑒𝑒 (2.12)

𝑐𝑐𝑝𝑝 = �𝜕𝜕ℎ𝑙𝑙

𝜕𝜕𝜕𝜕�𝑝𝑝

(2.13)

where 𝑥𝑥𝑤𝑤 is the mole fraction of water, ℎ𝑤𝑤𝑜𝑜 is the molar liquid enthalpy of water at the

system temperature, ℎ𝑖𝑖∞,𝑎𝑎𝑎𝑎 is the aqueous-phase infinite-dilution reference state molar

enthalpy of the liquid mixture, which accounts for the nonideal behavior of the solution.

The enthalpy of solution 𝑖𝑖 in the aqueous phase at infinite dilution (ℎ𝑖𝑖∞,𝑎𝑎𝑎𝑎) can be

calculated by:

ℎ𝑖𝑖∞,𝑎𝑎𝑎𝑎 = ∆𝑓𝑓ℎ𝑖𝑖

∞,𝑎𝑎𝑎𝑎 + ∫ 𝑐𝑐𝑝𝑝,𝑖𝑖∞,𝑎𝑎𝑎𝑎𝜕𝜕

298.15 𝑑𝑑𝑇𝑇 (2.14)

where ∆𝑓𝑓ℎ𝑖𝑖∞,𝑎𝑎𝑎𝑎 is the enthalpy of formation in the aqueous phase at infinite dilution at

298.15 K, and 𝑐𝑐𝑝𝑝,𝑖𝑖∞,𝑎𝑎𝑎𝑎 is the heat capacity of solute 𝑖𝑖 in the aqueous phase at infinite dilution.

The molar excess enthalpy of the liquid mixture (ℎ∗,𝑒𝑒𝑒𝑒) is:

ℎ∗,𝑒𝑒𝑒𝑒 = −𝑅𝑅𝑇𝑇2 ∑ 𝑥𝑥𝑖𝑖𝜕𝜕𝑙𝑙𝜕𝜕𝛾𝛾𝑖𝑖𝜕𝜕𝜕𝜕𝑖𝑖 (2.15)

Molar liquid enthalpy and molar heat capacity are essential thermodynamic properties

used in heat balance and heat-duty calculations, and in heat exchanger rating and design,

among other applications, in process simulators.

2.4. Thermodynamics of Vapor-Liquid Equilibrium

Vapor-liquid equilibrium in electrolyte systems is important in a range of chemical

processes, and needs to be considered when modeling electrolyte systems.


17

At equilibrium, the chemical potentials of component 𝑖𝑖 in the vapor and liquid phases

are equal:

𝜇𝜇𝑖𝑖𝑉𝑉 = 𝜇𝜇𝑖𝑖𝐿𝐿 (2.16)

Alternatively, the equilibrium condition can be expressed in terms of the fugacities, 𝑓𝑓𝑖𝑖.

𝑓𝑓𝑖𝑖𝑉𝑉 = 𝑓𝑓𝑖𝑖𝐿𝐿 (2.17)

The fugacity of component 𝑖𝑖 in the vapor phase is:

𝑓𝑓𝑖𝑖𝑉𝑉 = 𝜑𝜑𝑖𝑖𝑦𝑦𝑖𝑖𝑃𝑃 (2.18)

where 𝜑𝜑𝑖𝑖 is the vapor-phase fugacity coefficient of component 𝑖𝑖 (which can be calculated

by an equation of state), 𝑦𝑦𝑖𝑖 is the vapor-phase mole fraction, and 𝑃𝑃 is the system pressure.

The fugacity of component 𝑖𝑖 in the liquid phase is:

𝑓𝑓𝑖𝑖𝐿𝐿 = 𝛾𝛾𝑖𝑖𝑥𝑥𝑖𝑖𝑓𝑓𝑖𝑖0 (2.19)

where 𝛾𝛾𝑖𝑖 is the liquid-phase activity coefficient, 𝑥𝑥𝑖𝑖 is the liquid-phase mole fraction, and

𝑓𝑓𝑖𝑖0 is the liquid-phase reference fugacity.

The liquid-phase reference fugacity can be calculated from:

𝑓𝑓𝑖𝑖0 = 𝑝𝑝𝑖𝑖0𝜑𝜑𝑖𝑖0𝜃𝜃𝑖𝑖0 (2.20)

where 𝑝𝑝𝑖𝑖0 is the saturation vapor pressure of component 𝑖𝑖 at the system temperature; 𝜑𝜑𝑖𝑖0 is

the vapor-phase fugacity coefficient at the system temperature and 𝑝𝑝𝑖𝑖0; and 𝜑𝜑𝑖𝑖0 is the

Poynting pressure correction from 𝑝𝑝𝑖𝑖0 to the system pressure.

For volatile molecular solutes (e.g., nitrogen, methane, and carbon dioxide), Henry’s

law should be used to calculate the liquid-phase fugacity:

𝑓𝑓𝑖𝑖𝐿𝐿 = 𝐻𝐻𝑖𝑖𝛾𝛾𝑖𝑖∗𝑥𝑥𝑖𝑖 (2.21)

where 𝐻𝐻𝑖𝑖 is the Henry’s law constant for component 𝑖𝑖, and 𝛾𝛾𝑖𝑖∗ is the unsymmetric activity

coefficient of component 𝑖𝑖.


18

The Henry’s law constant of component 𝑖𝑖 can be calculated from the Henry’s law

constant for solute 𝑖𝑖 in solvent 𝑗𝑗 (𝐻𝐻𝑖𝑖,𝑗𝑗) and a weighting factor (𝑤𝑤𝑖𝑖,𝑗𝑗):

𝐻𝐻𝑖𝑖 = ∑ 𝑤𝑤𝑖𝑖,𝑗𝑗𝐻𝐻𝑖𝑖,𝑗𝑗𝑗𝑗 (2.22)

The unsymmetric activity coefficient (𝛾𝛾𝑖𝑖∗) is the ratio of the liquid-phase activity

coefficient (𝛾𝛾𝑖𝑖) to the liquid-phase infinite dilution activity coefficient (𝛾𝛾𝑖𝑖∞):

𝛾𝛾𝑖𝑖∗ = 𝛾𝛾𝑖𝑖𝛾𝛾𝑖𝑖∞ (2.23)

These relationships (Eqs. 2.17-2.23) can be solved for vapor-liquid equilibrium and,

therefore, the distributions of volatile solvents and molecular solutes in the vapor phase

and the liquid phase.

2.5. Thermodynamics of Salt Precipitation

Salt precipitation is another phenomenon encountered in chemical processes. It is a key

separation technology commonly used in basic chemicals and pharmaceuticals

manufacturing. It can also be a concern in industrial processes; for example, the

precipitation of low-solubility salts, such as barium sulfate (BaSO4(s)), which is discussed

in Example 2, can cause scaling in pipes.

Salt precipitation is often treated as solid-liquid phase equilibrium 5, in which the solid

electrolyte (𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈−) is in equilibrium with the aqueous ions (𝑀𝑀𝑧𝑧+ and 𝑋𝑋𝑧𝑧−):

𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈−(𝑠𝑠) ⟷ 𝜈𝜈+𝑀𝑀𝑧𝑧+(𝑎𝑎𝑎𝑎) + 𝜈𝜈−𝑋𝑋𝑧𝑧−

(𝑎𝑎𝑎𝑎)

Combining the thermodynamic relationships discussed previously (Eqs. 2.1 and 2.11),

the chemical potentials of the electrolyte as a solid crystal and in the aqueous form at salt

saturation can be written as:


19

𝜇𝜇𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈−(𝑠𝑠)= 𝜇𝜇𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈−(𝑎𝑎𝑎𝑎)

= 𝜇𝜇𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈−0 + 𝜈𝜈𝑅𝑅𝑇𝑇𝑙𝑙𝑙𝑙(𝛾𝛾±

(𝑚𝑚) ∙ 𝑚𝑚±) (2.24)

The equilibrium constant for salt precipitation is:

𝐾𝐾𝑒𝑒𝑎𝑎 =

�𝑎𝑎𝑀𝑀𝑧𝑧+�𝜈𝜈+�𝑎𝑎𝑋𝑋𝑧𝑧−�

𝜈𝜈−

𝑎𝑎𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈−(𝑠𝑠)=

�𝛾𝛾𝑀𝑀𝑧𝑧+(𝑚𝑚) 𝑚𝑚𝑀𝑀𝑧𝑧+�

𝜈𝜈+�𝛾𝛾𝑋𝑋𝑧𝑧−

(𝑚𝑚)𝑚𝑚𝑋𝑋𝑧𝑧−�𝜈𝜈−

𝑎𝑎𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈−(𝑠𝑠)=

(𝛾𝛾±(𝑚𝑚)∙𝑚𝑚±)𝜈𝜈

𝑎𝑎𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈−(𝑠𝑠)

(2.25)

By defining the solid pure crystal as its own reference state and therefore setting its

activity to unity, the solubility product constant (𝐾𝐾𝑠𝑠𝑝𝑝), which is the product of the activities

of the dissolved species that make up the solid crystal, becomes:

𝐾𝐾𝑒𝑒𝑎𝑎 (𝑎𝑎𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈−(𝑠𝑠)) = 𝐾𝐾𝑠𝑠𝑝𝑝 = �𝛾𝛾±(𝑚𝑚)𝑚𝑚±�

𝜈𝜈 (2.26)

The temperature and pressure dependency of 𝐾𝐾𝑠𝑠𝑝𝑝 is often expressed as 5:

𝑙𝑙𝑙𝑙𝐾𝐾𝑠𝑠𝑝𝑝 = 𝐴𝐴 + 𝐵𝐵𝜕𝜕

+ 𝐶𝐶𝑙𝑙𝑙𝑙𝑇𝑇 + 𝐷𝐷𝑇𝑇 + 𝐸𝐸 �𝑃𝑃−𝑃𝑃𝑟𝑟𝑟𝑟𝑟𝑟𝑃𝑃𝑟𝑟𝑟𝑟𝑟𝑟

� (2.27)

where A, B, C, D and E are determined by fitting Eq. 2.27 to experimental solubility data,

and 𝑃𝑃𝑟𝑟𝑒𝑒𝑓𝑓 is the reference pressure of 1 bar.

For cases in which salt hydrate crystals are formed, the chemical potential of hydrate

salts (e.g., 𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈− ∙ nH2O) is calculated from:

𝜇𝜇𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈−.𝜕𝜕𝐻𝐻2𝑂𝑂(𝑠𝑠) = 𝜇𝜇𝑀𝑀𝜈𝜈+𝑋𝑋𝜈𝜈−(𝑎𝑎𝑎𝑎)+ 𝑙𝑙𝐻𝐻2𝑂𝑂𝜇𝜇𝐻𝐻2𝑂𝑂 (2.28)

where the chemical potential of the water is:

𝜇𝜇𝐻𝐻2𝑂𝑂 = 𝜇𝜇𝐻𝐻2𝑂𝑂0 + 𝑅𝑅𝑇𝑇𝑙𝑙𝑙𝑙�𝑎𝑎𝐻𝐻2𝑂𝑂� (2.29)


20

2.6. Modeling Electrolyte Systems

Electrolyte solution nonideality is, in general, dominated by the solution chemistry. A

qualitatively correct model for electrolyte systems can be readily developed if the solution

chemistry has been properly represented. Therefore, the first task in modeling electrolyte

systems is to accurately account for the solution chemistry, including complete and partial

dissociation, acid-base reactions, complex-ion formation, salt precipitation, and hydration.

A note of caution: Modelers will sometimes introduce hypothesized and

unsubstantiated reactions and speciation into the model to improve the fit of the model to

experimental data, at the cost of unduly expanding the model complexity and degrading

the model fundamentals. The addition of such hypothesized reactions and speciation that

cannot be supported by experimental evidence should be carefully scrutinized and avoided.

Mathematically, modeling the solution chemistry means solving the chemical

equilibrium problem for the various reactions involved in the aqueous phase. This

transforms the solution composition in terms of electrolytes (apparent component

composition) into the solution composition in terms of ionic species, molecular species,

and salt precipitates in chemical equilibrium (true species composition).

The apparent component composition and the true species composition for a given

electrolyte system are equivalent, as both represent the same system. While process

simulation of electrolyte systems can be performed in both ways, the apparent-component

approach is generally preferred, because it involves a smaller set of molecular species. This

is not always possible, however, since the process information may be available only in

true species compositions.


21

Modeling the physical ion-ion, ion-molecule, and molecule-molecule interactions in

the aqueous solution with electrolyte thermodynamic models is the essential step to

upgrade a qualitative solution chemistry model to a robust, quantitative thermodynamic

model for electrolyte systems. Numerous electrolyte thermodynamic models have been

proposed to account for the solution nonideality resulting from such physical interactions.

Coupled with proper representation of solution chemistry, these models provide

comprehensive thermodynamic frameworks to correlate and calculate all thermodynamic

properties for electrolyte solutions.

All of the existing electrolyte thermodynamic models are correlative models designed

to provide a theoretical framework for data interpolation and extrapolation. They

incorporate adjustable binary and, in some cases, ternary interaction parameters to correlate

available experimental data and capture the solution nonideality as functions of solution

composition and temperature. The quality and usability of these models are best measured

by the ranges of concentrations for which the models are applicable, and the number and

type of adjustable parameters required to correlate experimental data within acceptable

accuracy.

To be used as a tool for process simulation, these models should provide robust

prediction capability for multicomponent electrolyte systems, involve only binary

interaction parameters, and cover the solution nonideality preferably up to high salt

concentrations (salt saturation or pure fused salts). furthermore, to support heat and mass

balance calculations, these models should account for the temperature dependence of the

solution nonideality and related calorimetric properties, and do so reliably with a

manageable number of temperature coefficients for the interaction parameters.


22

Electrolyte systems are chemically complex and thus difficult to model. As mentioned

previously, avoid incorporating unsubstantiated reactions and speciation with the solution

chemistry. Equally important, recognize the uncertainty and potential low quality of

experimental measurements and avoid over-fitting experimental data with excessive

adjustable parameters. Simpler models with fewer parameters that properly represent the

general behavior of a system are far better engineering tools than complex equations that

seemingly duplicate experimental data with expanding lists of adjustable parameters of

diminishing physical significance.

For the vapor phase, depending on the system pressure, various equations of state can

be applied to calculate thermodynamic properties of solvents and volatile solutes.

2.7. Thermodynamic Models for Electrolytes

Due to long-range ion-ion Coulombic interactions, electrolyte solutions are nonideal

even at low electrolyte concentrations 4,5. Using well-established concepts from classical

electrostatics, Debye and Hückel derived the well-known limiting law for the activity

coefficients of ions:

𝑙𝑙𝑙𝑙𝛾𝛾±(𝑚𝑚) = −𝐴𝐴𝛾𝛾|𝑧𝑧+𝑧𝑧−|√𝐼𝐼 (2.30)

𝐼𝐼 = 12∑ 𝑚𝑚𝑖𝑖𝑖𝑖 𝑧𝑧𝑖𝑖2 (2.31)

where 𝐼𝐼 is the ionic strength and 𝐴𝐴𝛾𝛾 is the Debye-Hückel parameter. The equation properly

accounts for the fact that ions with higher valence numbers have a stronger effect on the

activity coefficients than those with smaller valence numbers.

For more concentrated electrolyte solutions, i.e., with ionic strength up to 1 molality,

various extended Debye-Hückel equations have been proposed. Two common ones are:


23

𝑙𝑙𝑙𝑙𝛾𝛾±

(𝑚𝑚) = −𝐴𝐴𝛾𝛾|𝑍𝑍+𝑍𝑍−|√𝐼𝐼

1 + √𝐼𝐼 (2.32)

𝑙𝑙𝑙𝑙𝛾𝛾±

(𝑚𝑚) = −𝐴𝐴𝛾𝛾|𝑍𝑍+𝑍𝑍−|√𝐼𝐼

1 + √𝐼𝐼+ 𝑏𝑏𝐼𝐼 (2.33)

where 𝑏𝑏 is an adjustable parameter determined from experimental data.

Beyond the Debye-Hückel limiting law and extended Debye-Hückel equations, many

semi-empirical models have been proposed for electrolyte solutions. These semi-empirical

equations typically rely on the assumption that the molar excess Gibbs free energy (𝑔𝑔𝑒𝑒𝑒𝑒),

or excess Gibbs free energy (𝐺𝐺𝑒𝑒𝑒𝑒), of electrolyte solutions is the sum of two contributions,

one arising from long-range ion-ion electrostatic interactions (𝐺𝐺𝑒𝑒𝑒𝑒,𝐿𝐿𝐿𝐿) and the other from

short-range ion-ion, ion-molecular, and molecule-molecule interactions (𝐺𝐺𝑒𝑒𝑒𝑒,𝑆𝑆𝐿𝐿):

𝑔𝑔𝑒𝑒𝑒𝑒 = 𝑔𝑔𝑒𝑒𝑒𝑒,𝐿𝐿𝐿𝐿 + 𝑔𝑔𝑒𝑒𝑒𝑒,𝑆𝑆𝐿𝐿 (2.34)

𝑙𝑙𝑙𝑙𝛾𝛾± = 𝑙𝑙𝑙𝑙𝛾𝛾±𝐿𝐿𝐿𝐿 + 𝑙𝑙𝑙𝑙𝛾𝛾±

𝑆𝑆𝐿𝐿 (2.35)

where 𝐺𝐺𝑒𝑒𝑒𝑒,𝐿𝐿𝐿𝐿 is typically calculated by a Debye-Hückel type equation, and 𝐺𝐺𝑒𝑒𝑒𝑒,𝑆𝑆𝐿𝐿 can be

calculated using various proposed models.

Three thermodynamic modes (i.e., engineering expressions for 𝑔𝑔𝑒𝑒𝑒𝑒 or 𝐺𝐺𝑒𝑒𝑒𝑒) have been

extensively used in process simulators for modeling electrolyte systems: the Pitzer ion-

interaction model 6,7, the OLI mixed-solvent electrolytes (MSE) model 8, and the electrolyte

NRTL (eNRTL) model 9. Among these, the Pitzer model remains the most popular for the

thermodynamic treatment of aqueous electrolyte solutions in the academic community. In

industry, the eNRTL model is preferred if ample data are available to develop

comprehensive models, while the OLI-MSE model is used if a preliminary study is the

objective. Once, a thermodynamic model has been developed for an electrolyte system, the

excess Gibbs free energy (𝐺𝐺𝑒𝑒𝑒𝑒) can be used to calculate 𝛾𝛾𝑖𝑖:


24

ln 𝛾𝛾𝑖𝑖 =

1𝑅𝑅𝑇𝑇

�𝜕𝜕𝐺𝐺𝑒𝑒𝑒𝑒

𝜕𝜕𝑙𝑙𝑖𝑖�𝜕𝜕,𝑃𝑃,𝑗𝑗≠𝑖𝑖

(2.36)

The activity coefficient (𝛾𝛾𝑖𝑖) can then be used to determine vapor-liquid equilibria,

enthalpy, heat capacity, and salt precipitation, among other thermodynamic properties.

The Pitzer model for an electrolyte solution is a virial expansion of ionic molalities:

𝐺𝐺𝑒𝑒𝑒𝑒∗

𝑅𝑅𝑇𝑇𝑤𝑤𝑠𝑠= 𝑓𝑓(𝐼𝐼) + ��𝑚𝑚𝑖𝑖𝑚𝑚𝑗𝑗𝜆𝜆𝑖𝑖𝑗𝑗(𝐼𝐼)

𝑗𝑗𝑖𝑖

+ ��𝑚𝑚𝑖𝑖𝑚𝑚𝑗𝑗𝑚𝑚𝑘𝑘Λ𝑖𝑖𝑗𝑗𝑘𝑘(𝐼𝐼) + ⋯𝑘𝑘𝑗𝑗𝑖𝑖

(2.37)

where 𝐺𝐺𝑒𝑒𝑒𝑒∗ is the aqueous-phase infinite-dilution reference state excess Gibbs free energy;

𝑤𝑤𝑠𝑠 is the amount of solvent; 𝑓𝑓(𝐼𝐼) is the Pitzer-Debye-Hückel formula for long-range

electrostatic interactions (which depends on the ionic strength, temperature, and solvent

properties); 𝑚𝑚𝑖𝑖, 𝑚𝑚𝑗𝑗, 𝑚𝑚𝑘𝑘, … are the molalities of ionic solute species 𝑖𝑖, 𝑗𝑗, 𝑘𝑘, … respectively.

𝜆𝜆𝑖𝑖𝑗𝑗(𝐼𝐼) accounts for the contribution of the two-body ion interaction; and Λ𝑖𝑖𝑗𝑗𝑘𝑘(𝐼𝐼) represents

the three-body ion interactions.

The Pitzer model further provides empirical expressions of ionic strength dependence

with binary and ternary interaction parameters suitable for describing the thermodynamic

properties of aqueous electrolyte solutions up to ionic strengths of approximately 6 to 10

molal. However, the Pitzer model falls short in process simulation applications due to the

very large number of adjustable binary and ternary interaction parameters required to

model multicomponent electrolyte systems. Furthermore, the interaction parameters of the

Pitzer model do not account for temperature dependence, so up to eight temperature

coefficients per interaction parameter may be required to cover temperatures from 0℃ to

200℃ 7,10.


25

Additionally, the Pitzer model is only applicable to aqueous electrolyte solutions. It can

be cumbersome to extend this equation to model molecular solutes or nonaqueous solvents

in aqueous solutions or mixed-solvent electrolyte systems 10.

The OLI-MSE model 8 provides a semi-empirical expression for the molar excess Gibbs

free energy:

𝑔𝑔𝑒𝑒𝑒𝑒 = 𝑔𝑔𝑒𝑒𝑒𝑒,𝐿𝐿𝐿𝐿 + 𝑔𝑔𝑒𝑒𝑒𝑒,𝑆𝑆𝐿𝐿 + 𝑔𝑔𝑒𝑒𝑒𝑒,𝑀𝑀𝐿𝐿 (2.38)

where 𝑔𝑔𝑒𝑒𝑒𝑒,𝑀𝑀𝐿𝐿 is the middle-range contribution resulting from ion-ion and ion-molecule

interactions that are not accounted for by 𝑔𝑔𝑒𝑒𝑒𝑒,𝐿𝐿𝐿𝐿. For Eq. 2.38, 𝑔𝑔𝑒𝑒𝑒𝑒,𝐿𝐿𝐿𝐿 is calculated by a

Pitzer-Debye-Hückel formula, 𝑔𝑔𝑒𝑒𝑒𝑒,𝑆𝑆𝐿𝐿 is calculated by the UNIQUAC equation 11, and

𝑔𝑔𝑒𝑒𝑒𝑒,𝑀𝑀𝐿𝐿 is calculated with an ionic-strength-dependent, second-virial-coefficient-type

expression.

In the OLI-MSE model, a flexible yet inherently ambiguous formula is used to account

for ion-ion and ion-molecule interactions. Specifically, the short-range term and the

middle-range term provide two parallel sets of ion-ion and ion-molecule interaction

expressions and two parallel sets of binary interaction parameters that can be challenging

to uniquely identify from experimental data. A common way to address this challenge is to

use 𝑔𝑔𝑒𝑒𝑒𝑒,𝑀𝑀𝐿𝐿 primarily for ion-ion and ion-molecule interactions and 𝑔𝑔𝑒𝑒𝑒𝑒,𝑆𝑆𝐿𝐿 for molecule

interactions 12. However, it may be necessary to use both the short-range term and the

middle-range term for ion-ion interactions for a system as simple as NaCl in a solution of

water and methanol 8. Furthermore, the binary interaction parameters for the middle-range

term require up to five temperature coefficients per parameter.

The eNRTL model 9 is based on the assumption that the molar excess Gibbs free energy

is a sum of two terms — the long-range electrostatic interactions, and the local interaction


26

contribution resulting from short-range molecule-molecule, molecule-ion, and ion-ion

interactions (𝑔𝑔𝑒𝑒𝑒𝑒,𝐿𝐿𝑜𝑜𝐿𝐿𝑎𝑎𝑙𝑙):

𝑔𝑔𝑒𝑒𝑒𝑒 = 𝑔𝑔𝑒𝑒𝑒𝑒,𝐿𝐿𝐿𝐿 + 𝑔𝑔𝑒𝑒𝑒𝑒,𝐿𝐿𝑜𝑜𝐿𝐿𝑎𝑎𝑙𝑙 (2.39)

where 𝑔𝑔𝑒𝑒𝑒𝑒,𝐿𝐿𝐿𝐿 is represented with an extended Pitzer-Debye-Hückel expression, and

𝑔𝑔𝑒𝑒𝑒𝑒,𝐿𝐿𝑜𝑜𝐿𝐿𝑎𝑎𝑙𝑙 is derived from the eNRTL local composition model 13, which requires properly

accounting for two distinctive phenomena characteristic of electrolyte solutions — local

electroneutrality and like-ion repulsion:

𝑔𝑔𝑒𝑒𝑒𝑒,𝐿𝐿𝑜𝑜𝐿𝐿𝑎𝑎𝑙𝑙

𝑙𝑙𝑅𝑅𝑇𝑇= � 𝑥𝑥𝑘𝑘𝑐𝑐𝑘𝑘 �

∑ 𝑥𝑥𝑖𝑖𝑐𝑐𝑖𝑖𝐺𝐺𝑖𝑖𝑘𝑘𝜏𝜏𝑖𝑖𝑘𝑘𝑖𝑖

∑ 𝑥𝑥𝑗𝑗𝑐𝑐𝑗𝑗𝐺𝐺𝑗𝑗𝑘𝑘𝑗𝑗�

𝑘𝑘 (2.40)

𝐺𝐺𝑖𝑖𝑗𝑗 = exp�−𝛼𝛼𝑖𝑖𝑗𝑗𝜏𝜏𝑖𝑖𝑗𝑗� (2.41)

where 𝑐𝑐𝑖𝑖 is unity for molecular species and the absolute charge number for ionic species

𝑖𝑖, 𝛼𝛼𝑖𝑖𝑗𝑗 (= 𝛼𝛼𝑗𝑗𝑖𝑖) is the non-randomness factor, and 𝜏𝜏𝑖𝑖𝑗𝑗 (≠ 𝜏𝜏𝑗𝑗𝑖𝑖) is the binary interaction

parameter.

With the nonrandomness factor often fixed at a constant, the model requires two binary

interaction parameters for each molecule-electrolyte, electrolyte-electrolyte, and molecule-

molecule pair. The binary interaction parameters are further expressed in terms of a Gibbs-

Helmholtz type of equation with three temperature coefficients.

Overall, the eNRTL model offers a practical and yet highly versatile thermodynamic

framework to correlate phase equilibrium and thermodynamic properties of aqueous

electrolytes, mixed-solvent electrolytes, nonaqueous electrolytes, and ionic liquids 2-3, 9, 14-

17.

Table 2.1 summarizes the strengths and weaknesses of the three electrolyte

thermodynamic models. Ultimately, the choice of models depends on the availability of

model parameters for specific electrolyte systems and specific applications. Model


27

parameters may be found from literature, simulator databanks, or proprietary databanks.

As process simulation results depend on the reliability of the model parameters, it is

important to validate the model parameters against available experimental data and confirm

that the model results are reliable.

To assist engineers in modeling electrolyte systems of industrial interest, commercial

process simulators provide proven electrolyte thermodynamic models, model parameter

databanks, and extensive experimental data and data sources for electrolyte systems — all

of which can be useful in validating thermodynamic models and performing

thermodynamic calculations for process simulation.

For the following examples, Aspen Plus V8.4 was used to perform regression of

pertinent adjustable parameters and thermodynamic calculations.

Table 2.1. The most commonly used thermodynamic models are the Pitzer, OLI-MSE, and eNRTL models.

Model

Strengths

Weaknesses

Pitzer

Strong theoretical basis

Empirical temperature and composition dependency

Large research community following

More than 24 parameters per electrolyte

Numerous scientific publications

Applies to aqueous electrolytes (maximum of 6-10 molality) only


28

Table 2.1. Continued.

Model Strengths

Weaknesses

OLI-MSE Strong commercial focus

Ambiguity in model formulation

Comprehensive parameter database for the entire periodic table

Empirical temperature and composition dependency

About 16 parameters per

electrolyte

eNRTL Semi-empirical phenomenological model

Lack of parameter database

About 6 parameters per electrolyte

Successful industrial applications for aqueous and mixed-solvent electrolytes

2.8. Example 1. Modeling Aqueous Single Electrolytes: H2O-BaCl2 Binary Solution

This example illustrates the steps in developing a comprehensive model for an aqueous

barium chloride system with the eNRTL model.

First, the model parameters for each aqueous single-electrolyte system must be

determined. The key thermodynamic model parameters for this system are the binary

interaction parameters for the water-BaCl2 pair, and the solubility product constants (𝐾𝐾𝑠𝑠𝑝𝑝)

for precipitating salts and their temperature coefficients.

For concentrations up to saturation, BaCl2 dissolves in water and dissociates

completely to the Ba2+ and Cl- ions:


29

𝐵𝐵𝑎𝑎𝐶𝐶𝑙𝑙2(𝑎𝑎𝑎𝑎) → 𝐵𝐵𝑎𝑎2+(𝑎𝑎𝑎𝑎) + 2 𝐶𝐶𝑙𝑙−(𝑎𝑎𝑎𝑎)

For concentrations above saturation, Ba2+ and Cl- ions precipitate as BaCl2(s):

𝐵𝐵𝑎𝑎𝐶𝐶𝑙𝑙2(𝑠𝑠) ↔ 𝐵𝐵𝑎𝑎2+(𝑎𝑎𝑎𝑎) + 2 𝐶𝐶𝑙𝑙−(𝑎𝑎𝑎𝑎)

Thermodynamic data for aqueous BaCl2 solutions from various sources — mean ionic

activity coefficient 18-20, osmotic coeffcient 21-23, vapor pressure 24 , and excess Enthalpy 25

— are simultaneously fitted to the equation shown in Table 2.2 to calculate the temperature

coefficients for the eNRTL binary interaction parameters (𝜏𝜏𝑖𝑖𝑗𝑗). Figure 2.1 is a plot of 𝜏𝜏𝑖𝑖𝑗𝑗

as a function of temperature. Solubility data 26 are then used to calculate the temperature

coefficients (𝐴𝐴, 𝐵𝐵, 𝐶𝐶, 𝐷𝐷, 𝐸𝐸), which can then be substituted into Eq. 2.27 to calculate the

BaCl2(s) solubility product constant.

Table 2.2. The temperature coefficients and binary interaction parameters for the examples.

Component i Component j Cija Dij

a Eija 𝜏𝜏𝑖𝑖𝑗𝑗 at 98.15K

Example 1

H2O (Ba2+,Cl-) 5.106 834.460 0 7.904

(Ba2+,Cl-) H2O -3.249 -285.728 0 -4.207

Example 2

H2O (Na+, Cl-) 7.427b 429.0b -3.251b 8.865

(Na+, Cl-) H2O -4.350b -56.9b 3.110b -4.541

H2O (Na+, SO42-) 2.325b 1695.5b 12.020b 8.012

(Na+, SO42-) H2O -2.208b -505.5b -1.837b -3.903

(Na+, Cl-) (Na+, SO42-) -0.440b 370.7b 0b 0.804

(Na+, SO42-) (Na+, Cl-) -0.122b -152.7b 0b -0.634


30

Table 2.2. Continued

Component i Component j Cija Dij

a Eija 𝜏𝜏𝑖𝑖𝑗𝑗 at 98.15K

(Na+,SO42-) (Ba2+,SO4

2-) 0 0 0 0

(Ba2+,SO42-) (Na+,SO4

2-) -8.231 1003.866 0 -4.864 a Parameters used to calculate the temperature-dependent binary-interaction parameters (𝜏𝜏𝑖𝑖𝑗𝑗) from

the following equation:

+

−++=

ref

refij

ijijij T

TT

TTE

TD

C lnτ

Source: Ref 17.

Figure 2.1. τ values vs. temperature for the aqueous BaCl2 system: H2O: (Ba2+,Cl-) pair ( , (Ba2+,Cl-): H2O pair ( ).

Figures 2.2-2.6 compare the model results to experimental data for several

thermodynamic properties — molality-scale mean ionic activity coefficients, osmotic

coefficients, vapor pressure, solubility, and excess enthalpy. The model is in excellent

agreement with the experimental data.


31

This example demonstrates that the thermodynamic model is capable of representing

a wide range of thermodynamic properties for the aqueous BaCl2 solution. Used in process

simulators, the model provides accurate and robust calculations for boiling point rise,

solution enthalpy, solution heat capacity, and salt solubility, which are essential for heat

and mass balance calculations. The model further provides a basis for modeling aqueous

multicomponent electrolyte systems in which BaCl2 is an electrolyte of the solution. More

examples on modeling aqueous single-electrolyte systems are available in the literature 14,

17.

Figure 2.2. Mean ionic activity coefficient vs. molality of barium chloride, model results (), experimental data ( ) 18, ( ) 19, ( ) 20.


32

Figure 2.3. Osmotic coefficient vs. molality of barium chloride, model results ( ), experimental data ( ) 21, ( ) 22, ( ) 23.

Figure 2.4. Vapor pressure of the solution vs. molality of barium chloride at different temperatures, model results at 303.15 K ( ), 313.15 K ( ), 323.15 K ( ),


33

333.15 K ( ), 343.15 K ( ), experimental data at 303.15 K ( ), 313.15 K ( ), 323.15 K ( ), 333.15 K ( ), 343.15 K ( ) 24.

Figure 2.5. Solubility of barium chloride vs. temperature, model results ( ), experimental data ( ) 26.


34

Figure 2.6. Excess enthalpy vs. molality of barium chloride, model results ( ), experimental data ( ) 25.

2.9. Example 2: Modeling BaSO4 (s) Precipitation in a Brine Solution

The precipitation of barium sulfate (BaSO4(s)) in geological brine (typically NaCl

solution) is an important issue for shale-gas industry. This example identifies the steps

needed to develop a thermodynamic model for this system — an aqueous of four ions (Na+,

Ba2+, Cl-, and SO42-) and four electrolytes (NaCl, Na2SO4, BaCl2, and BaSO4), which as

strong electrolytes, dissociate completely to ions:

𝑁𝑁𝑎𝑎𝐶𝐶𝑙𝑙(𝑎𝑎𝑎𝑎) → 𝑁𝑁𝑎𝑎+(𝑎𝑎𝑎𝑎) + 𝐶𝐶𝑙𝑙−(𝑎𝑎𝑎𝑎)

𝑁𝑁𝑎𝑎2𝑆𝑆𝑆𝑆4(𝑎𝑎𝑎𝑎) → 2 𝑁𝑁𝑎𝑎+(𝑎𝑎𝑎𝑎) + 𝑆𝑆𝑆𝑆42−(𝑎𝑎𝑎𝑎)

𝐵𝐵𝑎𝑎𝐶𝐶𝑙𝑙2(𝑎𝑎𝑎𝑎) → 𝐵𝐵𝑎𝑎2+(𝑎𝑎𝑎𝑎) + 2 𝐶𝐶𝑙𝑙−(𝑎𝑎𝑎𝑎)

𝐵𝐵𝑎𝑎𝑆𝑆𝑆𝑆4(𝑎𝑎𝑎𝑎) → 𝐵𝐵𝑎𝑎2+(𝑎𝑎𝑎𝑎) + 𝑆𝑆𝑆𝑆42−(𝑎𝑎𝑎𝑎)


35

When the concentration of BaSO4(aq) exceeds the solubility limit, BaSO4(s) salt

precipitates in solution:

𝐵𝐵𝑎𝑎𝑆𝑆𝑆𝑆4(𝑠𝑠) ↔ 𝐵𝐵𝑎𝑎2+(𝑎𝑎𝑎𝑎) + 𝑆𝑆𝑆𝑆42−(𝑎𝑎𝑎𝑎)

The first step in developing thermodynamic models for electrolyte systems is to

determine the model parameters for each aqueous single-electrolyte system. Two binary

interaction parameters are required for each of the four aqueous single-electrolyte systems

and water-electrolyte pairs. Additionally, two binary interaction parameters are required

for each of the four electrolyte-electrolyte pairs in which the two electrolytes share either

a common cation or a common anion.

The eNRTL thermodynamic model (Eq. 39) is used in this example. The binary

parameters for the water-BaCl2 pair are taken from Example 1. The binary parameters for

the water-NaCl pair, the water-Na2SO4 pair, and the NaCl-Na2SO4 pair are from Ref 17.

Due to the very minute solubility of BaSO4 in water, the simulator’s default values for the

binary parameters for water-electrolyte pairs at 298.15 K, which correspond to the Debye-

Hückel limiting law, are used to describe the water-BaSO4 pair. The binary parameters for

the NaCl-BaCl2 pair and the BaCl2-BaSO4 pair are set to zero as the simulator’s default

values for electrolyte-electrolyte pairs. The binary parameters for the BaSO4-Na2SO4 pair

are determined from solubility data for BaSO4 in an aqueous Na2SO4 solution.

Next, the temperature coefficients (Eq. 2.27) for the salt precipitation of BaSO4 in

water are determined from solubility data provided by Templeton 27 for a temperature range

of 298.15-368.15 K. The results are shown in Table 2.3. Once the temperature coefficients

have been determined, 𝐾𝐾𝑠𝑠𝑝𝑝 is calculated from Eq. 2.27, followed by 𝐺𝐺𝑒𝑒𝑒𝑒 from Eq. 2.39,

and then 𝛾𝛾𝑖𝑖 from Eq. 2.36. Figure 2.7 compares the model results and the experimental


36

data from Templeton for BaSO4 (s) solubility in water — illustrating the accuracy of the

eNRTL model.

Table 2.3. The solubility data are subsequently used to identify the temperature coefficients (A, B, C, D, E) for the solubility product constant (Eq. 2.27)

Component A B C

BaCl2(s) 46.83 -4062.2 -7.83

BaSO4(s) 211.17 -12836.3 -34.94

Figure 2.7. Solubility of barium sulfate vs. temperature, model results ( ), experimental data ( ) 27.

The binary interaction parameters for the BaSO4-Na2SO4 pair for the temperature

range of 273.15-353.15 K are determined by using data for BaSO4 (s) in an aqueous Na2SO4

solution 28. Figure 2.8 plots the model results and the experimental data for the solubility

of BaSO4 (s) in the aqueous solution of Na2SO4. Because both Na2SO4 and BaSO4 contain

SO42-, the higher the Na2SO4 concentration is, the lower the BaSO4 (s) solubility.


37

Figure 2.8. Solubility of barium sulfate vs. molality of sodium sulfate at different temperatures, model results at 273.15 K ( ), 293.15 K ( ), 313.15 K ( ), 333.15 K ( ), 353.15 K ( ), experimental data at 273.15 K ( ), 293.15 K ( ), 313.15 K ( ), 333.15 K ( ), 353.15 K ( ) 34.

The binary interaction parameters for the various water-electrolyte and electrolyte-

electrolyte pairs and the solubility product constant (𝐾𝐾𝑠𝑠𝑝𝑝) for BaSO4 (s) can be inserted into

the eNRTL model (Eq. 2.39). Figure 2.9 shows that the model predictions compare well

with the reported experimental data of Templeton 27 for the solubility of BaSO4 (s) in the

aqueous NaCl solution over the temperature range of 298.15-368.15 K. The BaSO4 (s)

solubility increases with the NaCl concentration, reflecting the ionic strength of the brine

solution.


38

Figure 2.9. Solubility of barium sulfate vs. molality of sodium chloride at different temperatures, model prediction at 298.15 K ( ), 308.15 K ( ), 323.15 K (), 338.15 K ( ), 353.15 K ( ), 368.15 K ( ), experimental data at 298.15 K ( ), 308.15 K ( ), 323.15 K ( ), 338.15 K ( ), 353.15 K ( ), 368.15 K ( ) 28.

Developing models such as this for several ions is essential for chemical engineers to

use process simulation to design, debottleneck, and optimize processes.

2.10. Looking Ahead

Today's electrolyte thermodynamic models provide excellent frameworks to correlate

electrolyte thermodynamic properties over wide ranges of electrolyte concentrations and

temperatures with binary interaction parameters and a minimal number of temperature

coefficients. Going forward, the near-term focus should be to develop model parameter

databases to support process simulation of the electrolyte systems most commonly used in

industrial processes. Some examples of these electrolyte systems include high-salinity

brine solutions used in oil and gas production, CO2-brine solutions in CO2 sequestration,


39

nitric acid solutions in nuclear wastewater treatment, and hydrogen bromide solutions in

flow batteries.

In addition, research should be conducted to gain critical new insights essential for

continuing the development of predictive electrolyte thermodynamic models. furthermore,

engineering models, both correlative and predictive, are lacking for all transport properties

of electrolyte systems. Research and development of practical engineering models for

transport properties should be pursued.

Given the critical importance of electrolyte systems to modern chemical engineering,

it is hoped that real and substantive progress will be made in the not-too-distant future.

2.11. Acknowledgments

The authors gratefully acknowledge the financial support of the Jack Maddox

Distinguished Engineering Chair Professorship in Sustainable Energy sponsored by the J.

F. Maddox Foundation. The work is partially supported by a donation from the Apache

Corp. The authors thank Dr. Yuhua Song for his extensive editorial review of the article.


40

2.12. References

1. Chen C-C., Mathias PM, Orbey H. Use of Hydration and Dissociation Chemistries with the Electrolyte NRTL Model. AIChE Journal., 45 (7), pp. 1576-1586 (1999)

2. Zhang Y, Chen C-C. Thermodynamic Modeling for CO2 Absorption in Aqueous MDEA Solution with Electrolyte NRTL Model. Industrial & Engineering Chemistry Research. 50 (1), pp. 163-175 (2011)

3. Que H, Chen C-C. Thermodynamic Modeling of the NH3-CO2-H2O System with Electrolyte NRTL Model. Industrial & Engineering Chemistry Research. 50 (19), pp. 11406-11421 (2011)

4. Prausnitz JM, Lichtenthaler RN, de Azevedo EG. Molecular Thermodynamics of Fluid-Phase Equilibria. 3rd Ed., Prentice Hall PTR, Upper Saddle River, New Jersey (1998)

5. Tanveer S, Hao YF, Chen C-C. Introduction to Solid-Fluid Equilibrium Modeling. Chemical Engineering Progress, 110 (9), pp. 37-47 (2014)

6. Pitzer KS. Thermodynamics of Electrolytes. I. Theoretical Basis and General Equations. The Journal of Physical Chemistry. 77 (2), pp. 268-277 (1973)

7. Rowland D, Kӧnigsberger E, Hefter G, May PM. Aqueous Electrolyte Solution Modelling: Some Limitations of the Pitzer Equations. Applied Geometry, (2014), http://dx.doi.org/10.1016/j.apgeochem.2014.09.021

8. Wang P, Anderko A, Young RD. A Speciation-Based Model for Mixed-Solvent Electrolyte Systems. Fluid Phase Equilibria. 203, pp. 141-176 (2002)

9. Song Y, Chen C-C. Symmetric Electrolyte Nonrandom Two-Liquid Activity Coefficient Model. Industrial & Engineering Chemistry Research. 48 (16), pp. 7788-7797 (2009)

10. Chen C-C. Toward Development of Activity Coefficient Models for Process and Product Design of Complex Chemical Systems. Fluid Phase Equilibria. 241, pp. 103–112 (2006)

11. Abrams DS, and Prausnitz JM. Statistical Thermodynamics of Liquid Mixtures. A New Expression for the Excess Gibbs Energy of Partly and Completely Miscible Systems. AIChE Journal. 21 (1), pp. 116-128 (1975)

12. Wang P, Anderko A, Springer RD, Young RD. Modeling Phase Equilibria and Speciation in Mixed-Solvent Electrolyte Systems: II. Liquid-Liquid Equilibria and Properties of Associating Electrolyte Solutions. Journal of Molecular Liquids. 125 (1), pp. 37-44 (2006)


41

13. Renon H, Prausnitz JM. Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures. AIChE Journal. 14 (1), pp. 135-144 (1968)

14. Bhattacharia SK, Chen C-C. Thermodynamic Modeling of KCl+H2O and KCl+NaCl+H2O Systems Using Symmetric Electrolyte NRTL Model. Fluid Phase Equilibria. 387, pp. 169-177 (2015).

15. Que H, Song Y, Chen C-C. Thermodynamic Modeling of the Sulfuric Acid-Water-Sulfur Trioxide System with the Symmetric Electrolyte NRTL Model. Journal of Chemical & Engineering Data. 56 (4), pp. 963-977 (2011)

16. Yan Y, Chen C-C. Thermodynamic Modeling of CO2 Solubility in Aqueous Solutions of NaCl and Na2SO4. The Journal of Supercritical Fluids. 55 (2), pp. 623-634 (2010)

17. Yan Y, Chen C-C. Thermodynamic Representation of the NaCl-Na2SO4-H2O System with Electrolyte NRTL Model. Fluid Phase Equilibria. 306, pp. 149-161 (2011)

18. Hamer WJ, Wu YC. Osmotic Coefficients and Mean Activity Coefficients of Uni‐univalent Electrolytes in Water at 25 °C. Journal of Physical and Chemical Reference Data. 1 (4), pp. 047-1100, (1972)

19. Lucasse WW. Activity Coefficients and Transference Numbers of the Alkaline Earth Chlorides. Journal of the American Chemical Society. 47 (3), pp. 743-754 (1925)

20. Tippetts EA, Newton RF. The Thermodynamics of Aqueous Barium Chloride Solutions from Electromotive Force Measurements. Journal of the American Chemical Society. 56 (8), pp. 1675-1680 (1934)

21. Robinson RA, The Osmotic and Activity Coefficient Data of Some Aqueous Salt Solutions from Vapor Pressure Measurement. Journal of the American Chemical Society. 59 (1), pp. 84-90 (1937)

22. Robinson RA. The Vapor Pressures of Solutions of Potassium Chloride and Sodium Chloride. Transactions of the Royal Society of New Zealand. 75 (2), pp. 203-217 (1945)

23. Robinson RA, Bower VE. Thermodynamics of the Ternary System: Water-Sodium Chloride-Barium Chloride at 25 °C. Journal of Research of the National Bureau of Standards. 69A (1), pp. 19-27 (1965)

24. Patil KR, Tripathi AD, Pathak G, Katti SS. Thermodynamic Properties of Aqueous Electrolyte Solution. 2. Vapor Pressure of Aqueous Solutions of NaBr, NaI, KCl, KBr, KI, RbCl, CsCl, CsBr, CsI, MgCl2, CaCl2, CaBr2, CaI2, SrCl2, SrBr2, SrI2, BaCl2, and BaBr2. Journal of Chemical Engineering Data. 36 (2), pp. 225-230, (1991)


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25. Wagman DD, Evans WH, Parker VB, Schumm RH, Halow I, Bailey SM, Churney KL, Nuttall RL. The NBS Tables of Chemical Thermodynamic Properties. Selected Values for Inorganic and C1 and C2 Organic Substances in SI Units. Journal of Physical and Chemical Reference Data. (1982), 11, Supplement No. 2

26. Seidell A. Solubilities of Inorganic and Organic Compounds, NYC, New York, D.Van Nostrand Company (1923)

27. Templeton CC. Solubility of Barium Sulfate in Sodium Chloride Solutions from 25 ℃ to 95 ℃. Journal of Chemical Engineering Data. 5 (4), pp. 514-516 (1960)

28. Jiang C. Solubility and Solubility Constant of Barium Sulfate in Aqueous Sodium Sulfate Solutions between 0 and 80 ℃. Journal of Solution Chemistry. 25 (1), pp. 105-111 (1996)


43

CHAPTER 3. THERMODYNAMIC MODELING OF HCL-H2O BINIARY

SYSTEM WITH SYMMETRIC ELECTROLYTE NRTL MODEL2

3.1. Abstract

A comprehensive thermodynamic model is developed for the HCl-H2O binary system

based on the electrolyte nonrandom two-liquid activity coefficient model. Partial

dissociation of electrolyte HCl to hydronium cation and chloride anion is incorporated to

account for the true species in solution. Regressed from extensive thermodynamic data, the

model parameters include molecule-molecule and molecule-electrolyte binary interactions

parameters, as well as the chemical equilibrium constant parameters of the partial

dissociation solution chemistry. The model accurately calculates all phase equilibria and

thermodynamic properties including calorimetric properties with the concentration range

from pure water to pure HCl and temperatures from 273 up to about 400 K.

2 This chapter is reproduced from the paper published as: Saravi SH, Honarparvar S, Chen C-C. Thermodynamic Modeling of HCl-H2O Binary System with Symmetric Electrolyte NRTL Model. The Journal of Chemical Thermodynamics. 2018.


44

3.2. Introduction

As a strong acid, aqueous hydrochloric acid is one of the most widely used electrolytes

in industry. It is commonly used in metals refining and ion exchangers regeneration, as

well as many environmental and geological processes 1-4. Furthermore, the study of

aqueous HCl solution at higher temperatures and pressures is of great importance in

hydrothermal processes, steam power generators, and bitumen recovery operations 5-7. The

highly non-ideal nature of this strong acid can be highlighted with the existence of

minimum-boiling azeotropic points 8, formation of two liquids at high pressures 9,

dissociation of HCl along with hydration of ions 8,10,11, etc.

The complex thermodynamic behavior of HCl-H2O binary system has attracted

numerous studies both experimentally and theoretically over the last century. As early as

1909, Rupert 9 carried out experiments to measure liquid-liquid equilibrium (LLE)

compositions of the HCl-H2O binary. Later, Zeisberg 12 reported a relatively thorough

compilation of partial pressure data for aqueous HCl solution from 2 to 46 wt. % and over

the temperature range of 273.15 to 383.15 K. These data were later included in Perry’s

Handbook 13. During the following decades, many researchers presented various

thermodynamic data for the aqueous HCl solution over wide ranges of temperature and

pressure 5,13-45. Among them are data for isothermal and isobaric vapor-liquid equilibrium

(VLE), LLE, molality scale mean ionic activity coefficient, osmotic coefficient, enthalpy

of formation, heat capacity, etc. Most of these data cover HCl concentrations up to the

commercial acid concentration practiced in the industry (~ 37 wt. % or 16 mol.kg-1 HCl).

A comprehensive review of these data is presented later.


45

Parallel to the experimental studies, there have been many thermodynamic modeling

efforts for the HCl-H2O binary. These thermodynamic studies can be categorized either as

“equation of state-based” or “activity coefficient-based” models 8. Another dimension to

these thermodynamic studies is whether the electrolyte is treated either as “no

dissociation”, “complete dissociation”, or “partial dissociation” 46-49. While there are

several notable modeling efforts based on equations of state (EoS), most of the

thermodynamic modeling efforts fall into the category of “activity coefficient-based”

models due to their rigorous thermodynamic formulations and more reliable results.

Gibbon and Laughton 50 presented a modified Redlich-Kwong-Soave (RKS) EoS for

the HCl-H2O binary and showed qualitatively correct results for the azeotropic behavior of

the system and the heat of solution. Presenting a modified “alpha function” for cubic EoS,

Stryjek and Vera 51 proposed the Peng-Robinson-Stryjek-Vera EoS to describe VLE of the

HCl-H2O binary. With two temperature-dependent binary parameters adjusted, they

showed acceptable results of correlating the experimental VLE data specifically for

minimum-boiling azeotropic point. Later, applying a Patel and Teja cubic EoS 52 modified

by Panagiotopoulos and Reid 53, Delano 54 showed the EoS qualitatively predicts the

system pressure at bubble points.

There are many activity coefficient-based thermodynamic modeling efforts for the

HCl-H2O binary. Based on a modified Raoults’s law, the Clausius-Clapeyron equation for

pure component vapor pressures, and a simple two-suffix Margules equation for activity

coefficients, Brandes 8 obtained the “pseudo-vapor pressures” correlations for calculating

HCl and water partial pressures as functions of temperature and HCl composition. The


46

model yielded satisfactory VLE predictions for temperatures up to 473 K and HCl

concentrations up to 55 wt %.

Using an extended Debye-Hückel theory, Greeley et al. 55 calculated mean ionic

activity coefficients, osmotic coefficients, and relative partial molal heat capacities for the

HCl-H2O binary for HCl concentrations up to 1.0 mol.kg-1 at the temperature range of

298.15 to 548.15 K. Based on a modified Debye-Hückel theory, Vega and Vera 22 applied

a semi-empirical activity coefficient correlation previously proposed by Correa and Vera

56, and showed satisfactory agreement with the experimental data of mean ionic activity

coefficient and VLE up to 373.15 K and 18.28 mol.kg-1 HCl. The reported deviations were

1 to 5 % for mean ionic activity coefficients, 10 to 20 % for water activity coefficients, and

less than 10 % for total system pressures.

By considering HCl partial dissociation and subsequent hydration of ions, Engels and

Bosen 57 applied the Wilson “local composition” activity coefficient equation 58 and

achieved satisfactory fits against VLE experimental data, including minimum-boiling

azeotropic point, up to 13.88 mol.kg-1 HCl at the temperature range of 298.15 to 373.15 K.

Liu and Grén 59 proposed an activity coefficient model based on a modified Debye-Hückel

theory for long-range interactions and a Wilson-type equation accounting for short-range

interactions. They successfully applied the model to characterize the HCl-H2O binary,

covering the temperature range of 273.15 to 383.15 K and the HCl concentration up to

21.55 mol.kg-1.

Hala et al. 60 presented their work on characterizing VLE of strong electrolytes in

concentrated solutions. They started with a mathematical definition of Gibbs free energy

in dilute solution, followed by a serial expansion leading to an explicit function of mole


47

fraction-based mean ionic activity coefficient. They later applied the equations to the HCl-

H2O binary up to 12.18 mol.kg-1 HCl at 298.15 K 61,62. Applying Hala’s equation, Wozny

and Cremer 63 successfully calculated the azeotrope and activity coefficients of the aqueous

HCl system up to 14.90 mol.kg-1 HCl, with temperatures from 273.15 to 473.15 K and

pressures from 0.01 to 2 MPa. To properly capture the temperature dependence of the

system, they employed 19 coefficients and considered the azeotropic point as the reference

state.

Many models have been developed for the aqueous HCl solution 4,64-66 based on the

Pitzer activity coefficient equation 67,68. Among them, considering formation of a neutral

ion pair from the ions, Simonson et al. 65 calculated the excess thermodynamic properties

of the aqueous HCl solution in the temperature range of 298 to 648 K, pressures up to 40

MPa, and the HCl concentration up to 2 mol.kg-1. Using an extended Pitzer model

associated with Henry’s law as the HCl reference state, Brandani et al. 4 reported their

model results showing satisfactory match with the experimental VLE data and significant

improvement over those published by Engels and Bosen 57 for the temperature range of

298.15 to 383.15 K and the HCl concentration up to 16 mol.kg-1. Carslaw et al. 66 calculated

the activities and vapor pressures of a multicomponent system including the HCl-H2O

binary using a mole fraction-based Pitzer model 69,70. Mean ionic activity coefficients with

the concentration range of 0 to 20 mol.kg-1 HCl and the temperature range of 190 to 330 K

were investigated. The experimental data used for fitting the parameters included water

activity, osmotic coefficient, partial vapor pressure, and heat capacity.

Nichols and Taylor 71 presented their results for the HCl-H2O binary as part of a

sodium bearing waste treatment simulation study. Using both Pitzer and electrolyte


48

nonrandom two-liquid (eNRTL) models 72,73, they compared the modeling results with the

experimental data of mean ionic activity coefficient from 0 to 16 mol.kg-1 HCl at 298.15

K. Despite claim on the superiority of Pitzer model over eNRTL, it seems that the presented

results of the latter model were not valid as they did not properly identify the eNRTL model

parameters from available experimental data. Incorporating the Debye-Hückel theory, the

classical Born theory, and an NRTL expression, Cruz and Renon 74 reported a model for

calculating activity coefficients of electrolytes including the HCl-H2O binary at 298.15 K

for the HCl concentration up to 18 mol.kg-1. They considered both complete and partial

dissociations of the electrolyte and achieved better results with the latter assumption.

Although they did not take into consideration hydration of the proton ion, their VLE

calculations showed good agreement with the data reported by Haase et al. 18 and Vega and

Vera 22.

Using an extended UNIQUAC model 75, Thomsen 76 reported thermodynamic

modeling of various aqueous electrolyte systems including the HCl-H2O binary. He was

able to depict the phase equilibria behavior and calorimetric properties up to 6 mol.kg-1

HCl and 383.15 K with excellent results. Wang et al. presented the OLI-MSE model 47, a

hybrid of the Pitzer model and the extended UNIQUAC model, and examined phase

equilibria behavior, enthalpy, heat capacity, and speciation for a number of electrolyte

systems including the HCl-H2O binary. They showed satisfactory agreements with the

VLE data in the temperature range of 273.15 to 343.15 K.

In summary, while much progress has been made in thermodynamic modeling of the

HCl-H2O binary, prior studies have been incomplete in terms of covering both VLE and

LLE over the entire HCl concentration range and addressing calorimetric properties such


49

as liquid molar enthalpy and heat capacity. To address this deficiency and to support

process simulation of systems involving the HCl-H2O binary, this study aims to develop a

comprehensive thermodynamic model that accurately calculates all phase equilibria,

thermodynamic and calorimetric properties over the entire acid concentration range and

temperatures up to 400 K for the HCl-H2O binary. The proposed model is based on the

symmetric eNRTL activity coefficient equation 73 which has recently been successfully

applied to develop comprehensive thermodynamic models for a number of strong acid

systems including H2SO4-H2O binary and H2SO4-H2O-SO3 ternary 77, H2SO4-H2O-SO2

ternary 78, HNO3-H2O binary 79, and HNO3-H2SO4-H2O ternary 80.

The rest of this chapter starts with a section on the eNRTL model thermodynamic

framework which covers HCl dissociation solution chemistry, vapor-liquid-liquid

equilibria, eNRTL activity coefficient model, and calorimetric properties. It is then

followed by a summary of the model parameters and a comprehensive review of pertinent

experimental data for phase equilibria, thermodynamic properties, and calorimetric

properties. The data treatment, regression and optimization method are then presented,

followed by a discussion on the model results including comparisons with the experimental

data for model validation, and the conclusions.

3.3. Thermodynamic Framework

Solution Chemistry and Hydration To generate a reliable thermodynamic model for electrolyte systems, sound

understanding, and proper representation of the underlying solution chemistry are essential

as the predominant true species and their concentrations play crucial roles in all aspects of

the thermodynamic properties. Prior studies often either treated HCl as a nonelectrolyte or


50

assumed complete dissociation of HCl to H+ and Clˉ ions. In reality, HCl is known to

undergo nearly complete dissociation in the dilute acid region while undissociated

molecular HCl becomes more prominent as the acid concentration increases. Triolo and

Narten 81 reported x-ray and neutron diffraction results on the HCl-H2O binary over a wide

range of concentrations and firmly established the existence of H3O+ and Clˉ ions. Ando

and Hynes 11 described the acid ionization of aqueous HCl using ab initio molecular orbital

methods combined with Monte Carlo simulations, resulting in similar outcomes. Later

study further confirmed proton ion hydrates with one water molecule to form hydronium

ion, an Eigen structure found to be the most probable proton complex 82.

Many prior thermodynamic modeling works for the HCl-H2O binary considered

partial dissociation of HCl in the solution 10,46,57,74,83-86. Among them, the work of Chen et

al. 86 is of particular interest as their study uniquely focused on coupling the partial

dissociation reaction with different hydration numbers to represent the nonideality of

electrolyte solutions. They reported that an optimal correlation of experimental VLE and

mean ionic activity coefficient data for the HCl-H2O binary was achieved by considering

the hydration of proton ions with three water molecules.

The present work is based on the partial dissociation of HCl and hydration of proton

ion (H+) with one water molecule to form hydronium ion (H3O+) as shown by reaction R1.

The model, while comprehensively representing the thermodynamic behavior of the

system, slightly deviates from the experimental data of mean ionic activity coefficient and

osmotic coefficient. To improve the results with lower deviation margins, this work further

presents a modified case by extending the model with the hydration of hydronium ion to

Zundel structure (H5O2+), shown by reaction (R2), to allow for more precise representation


51

of the solution chemistry. Note that the existence of H5O2+ has been previously proven

experimentally 82,87 and it was also taken into account in the work of Que et al. 77 in

modeling the H2SO4-H2O binary system.

𝐻𝐻𝐶𝐶𝑙𝑙 + 𝐻𝐻2𝑆𝑆𝐾𝐾1↔ 𝐻𝐻3𝑆𝑆+ + 𝐶𝐶𝑙𝑙− (R1)

𝐻𝐻3𝑆𝑆+ + 𝐻𝐻2𝑆𝑆𝐾𝐾2↔ 𝐻𝐻5𝑆𝑆2+ (R2)

where 𝐾𝐾1 and 𝐾𝐾2 are the chemical equilibrium constants of reactions R1 and R2

respectively, and they are calculated from Eq. 3.1 80.

𝑙𝑙𝑙𝑙 𝐾𝐾𝑖𝑖 = 𝐴𝐴𝑖𝑖 +

𝐵𝐵𝑖𝑖𝑇𝑇

=−∆𝐺𝐺𝑟𝑟𝑒𝑒𝜕𝜕°

𝑅𝑅𝑇𝑇°+∆𝐻𝐻𝑟𝑟𝑒𝑒𝜕𝜕°

𝑅𝑅 �

1𝑇𝑇°−

1𝑇𝑇� , 𝑖𝑖 = 1, 2 (3.1)

where 𝐴𝐴𝑖𝑖 and 𝐵𝐵𝑖𝑖 are parameters of the chemical equilibrium constant obtained from

regression; ∆𝐺𝐺𝑟𝑟𝑒𝑒𝜕𝜕° and ∆𝐻𝐻𝑟𝑟𝑒𝑒𝜕𝜕° are the Gibbs free energy of formation and enthalpy of

formation of the reaction at standard temperature (𝑇𝑇° = 298.15 K) and standard pressure

(100 kPa).

Vapor-Liquid-Liquid Equilibrium The criterion for VLE is the equality of component fugacity in vapor and liquid phases:

𝑃𝑃𝑦𝑦𝑖𝑖𝜑𝜑𝑖𝑖 = 𝑥𝑥𝑖𝑖𝛾𝛾𝑖𝑖𝑓𝑓𝑖𝑖° (3.2)

where P is the total system pressure; 𝑦𝑦𝑖𝑖 is the mole fraction of species 𝑖𝑖 in the vapor phase;

𝜑𝜑i is the fugacity coefficient at the system pressure P; 𝑥𝑥𝑖𝑖 is the mole fraction in the liquid

phase; 𝛾𝛾𝑖𝑖 is the activity coefficient in the liquid phase; 𝑓𝑓𝑖𝑖° is the liquid phase reference

fugacity at the system temperature and the system pressure 79 shown in Eq. 3.3.

𝑓𝑓𝑖𝑖° = 𝑝𝑝𝑖𝑖°𝜑𝜑𝑖𝑖°𝜃𝜃𝑖𝑖° (3.3)


52

where 𝑝𝑝𝑖𝑖° is the saturation pressure of pure species 𝑖𝑖 at the system temperature, 𝜑𝜑𝑖𝑖° is the

fugacity coefficient at the system temperature and 𝑝𝑝𝑖𝑖°, and 𝜃𝜃𝑖𝑖° is the Poynting factor

expressed by Eq. 3.4 79.

𝜃𝜃𝑖𝑖° = 𝑒𝑒𝑥𝑥𝑝𝑝 ( 1𝐿𝐿𝜕𝜕 ∫ � 1

𝜌𝜌𝑖𝑖𝑙𝑙� .𝑑𝑑𝑝𝑝𝑃𝑃

𝑝𝑝𝑖𝑖° ) (3.4)

where 𝜌𝜌𝑖𝑖𝑙𝑙 is the liquid molar density.

At high concentrations of HCl, e.g. > 60 wt. % at 298.15 K and > 4.7 MPa, the solution

separates to two liquids. Eq. 3.5 shows the criterion for LLE as the equality of species

fugacity between liquid phases I and II.

𝛾𝛾𝑖𝑖(𝐼𝐼)𝑥𝑥𝑖𝑖(𝐼𝐼) = 𝛾𝛾𝑖𝑖(𝐼𝐼𝐼𝐼)𝑥𝑥𝑖𝑖(𝐼𝐼𝐼𝐼) (3.5)

where γi(α) and 𝑥𝑥𝑖𝑖(𝛼𝛼) are the mole fraction-based activity coefficient and the mole fraction

of species 𝑖𝑖 in the liquid phase 𝛼𝛼 (𝛼𝛼 can be phase I or II).

Figure 3.1 depicts the schematic of the vapor-liquid-liquid equilibrium of the HCl-

H2O binary with partial dissociation chemical reaction (R1) and further hydration of

hydronium ion (R2).

Figure 3.1. Speciation and solution chemistry of the HCl-H2O binary system.


53

Symmetric Electrolyte NRTL Model Extensively covered in the literature, the eNRTL model was developed based on the

local composition model of non-random two-liquid theory together with the hypotheses of

like-ion repulsion and local electroneutrality 72,73,77,84. Shown as Eq. 3.6, the model

formulates the excess Gibbs free energy as a combination of two contributions.

𝐺𝐺𝑒𝑒𝑒𝑒 = 𝐺𝐺𝑒𝑒𝑒𝑒,𝑙𝑙𝐿𝐿 + 𝐺𝐺𝑒𝑒𝑒𝑒,𝑃𝑃𝑃𝑃𝐻𝐻 (3.6)

Considering short-range interactions at the immediate neighborhood of ions and

molecules, the short-range interaction contribution term (𝐺𝐺𝑒𝑒𝑒𝑒,𝑙𝑙𝐿𝐿) is represented with the

local composition electrolyte NRTL model. There are two binary interaction energy

parameters, 𝜏𝜏𝑖𝑖𝑗𝑗 and 𝜏𝜏𝑖𝑖𝑗𝑗 , for each molecule-molecule pair, molecule-electrolyte pair, and

electrolyte-electrolyte pair with a common ion. An additional binary parameter is the

nonrandomness factor, 𝛼𝛼𝑖𝑖𝑗𝑗, which is set to 0.2 for molecule-electrolyte and electrolyte-

electrolyte pairs and set to 0.3 for molecule-molecule pairs. Note that 𝜏𝜏𝑖𝑖𝑗𝑗 is asymmetric,

i.e. 𝜏𝜏𝑖𝑖𝑗𝑗 ≠ 𝜏𝜏𝑗𝑗𝑖𝑖, as opposed to the nonrandomness factor which is considered symmetric, i.e.

𝛼𝛼𝑖𝑖𝑗𝑗 = 𝛼𝛼𝑖𝑖𝑗𝑗. The long-range interaction contribution (𝐺𝐺𝑒𝑒𝑒𝑒,𝑃𝑃𝑃𝑃𝐻𝐻) accounts for long-range ion-

ion interactions with the Pitzer-Debye-Hückel theory. There are no adjustable parameters

associated with the long-range interaction contribution 73. The terms 𝐺𝐺𝑒𝑒𝑒𝑒,𝑙𝑙𝐿𝐿 and 𝐺𝐺𝑒𝑒𝑒𝑒,𝑃𝑃𝑃𝑃𝐻𝐻 are

calculated per Eqs. 3.7-3.10 and Eqs. 3.11-3.13, respectively:

𝐺𝐺𝑒𝑒𝑒𝑒,𝑙𝑙𝐿𝐿

𝑙𝑙𝑅𝑅𝑇𝑇= �𝑙𝑙𝑚𝑚

𝑚𝑚�∑ 𝑋𝑋𝑖𝑖𝐺𝐺𝑖𝑖𝑚𝑚𝜏𝜏𝑖𝑖𝑚𝑚𝑖𝑖

∑ 𝑋𝑋𝑖𝑖𝐺𝐺𝑖𝑖𝑚𝑚𝑖𝑖�+ �𝑧𝑧𝐿𝐿𝑙𝑙𝐿𝐿

𝐿𝐿

�∑ 𝑋𝑋𝑖𝑖𝐺𝐺𝑖𝑖𝐿𝐿𝜏𝜏𝑖𝑖𝐿𝐿𝑖𝑖≠𝐿𝐿

∑ 𝑋𝑋𝑖𝑖𝐺𝐺𝑖𝑖𝐿𝐿𝑖𝑖≠𝐿𝐿�

+ �𝑧𝑧𝑎𝑎𝑙𝑙𝑎𝑎𝑎𝑎

�∑ 𝑋𝑋𝑖𝑖𝐺𝐺𝑖𝑖𝑎𝑎𝜏𝜏𝑖𝑖𝑎𝑎𝑖𝑖≠𝑎𝑎

∑ 𝑋𝑋𝑖𝑖𝐺𝐺𝑖𝑖𝑎𝑎𝑖𝑖≠𝑎𝑎�

(3.7)

𝑋𝑋𝑖𝑖 = 𝐶𝐶𝑖𝑖𝑥𝑥𝑖𝑖 = 𝐶𝐶𝑖𝑖 �𝜕𝜕𝑖𝑖𝜕𝜕�, i = m, c, a (3.8)


54

𝑙𝑙 = �𝑙𝑙𝑖𝑖𝑖𝑖

= �𝑙𝑙𝑚𝑚𝑚𝑚

+ �𝑙𝑙𝐿𝐿𝐿𝐿

+ �𝑙𝑙𝑎𝑎𝑎𝑎

(3.9)

𝐺𝐺𝑖𝑖𝑗𝑗 = 𝑒𝑒𝑥𝑥𝑝𝑝�−𝛼𝛼𝑖𝑖𝑗𝑗𝜏𝜏𝑖𝑖𝑗𝑗� (3.10)

where 𝑙𝑙𝑖𝑖, 𝑥𝑥𝑖𝑖, and 𝐶𝐶𝑖𝑖 denote the number of moles, the mole fraction, and the absolute charge

number of species i, respectively.

𝐺𝐺𝑒𝑒𝑒𝑒,𝑃𝑃𝑃𝑃𝐻𝐻

𝑙𝑙𝑅𝑅𝑇𝑇= −

4𝐴𝐴𝜙𝜙𝐼𝐼𝑒𝑒𝜌𝜌

𝑙𝑙𝑙𝑙 �1 + 𝜌𝜌𝐼𝐼𝑒𝑒

1/2

1 + 𝜌𝜌(𝐼𝐼𝑒𝑒0)1/2� (3.11)

𝐴𝐴𝜙𝜙 =

13�

2𝜋𝜋𝑁𝑁𝐴𝐴𝑣𝑣

�1/2

�𝑄𝑄𝑒𝑒2

𝜖𝜖𝑘𝑘𝐵𝐵𝑇𝑇�3/2

(3.12)

𝐼𝐼𝑒𝑒 =12�𝑧𝑧𝑖𝑖2𝑥𝑥𝑖𝑖𝑖𝑖

=12�𝑧𝑧𝐿𝐿2𝑥𝑥𝐿𝐿𝐿𝐿

+12�𝑧𝑧𝑎𝑎2𝑥𝑥𝑎𝑎𝑎𝑎

(3.13)

where 𝐴𝐴𝜙𝜙, 𝐼𝐼𝑒𝑒, 𝐼𝐼𝑒𝑒0, and 𝜌𝜌 represent the Debye-Hückel parameter, the ionic strength, the ionic

strength at fused salt, i.e., the symmetric reference state, and the closest approach parameter,

respectively; 𝑁𝑁𝐴𝐴, 𝑣𝑣, 𝑄𝑄𝑒𝑒, 𝑘𝑘𝐵𝐵, and 𝜖𝜖 represent the Avogadro’s number, molar volume of water,

electron charge, the Boltzmann constant, and water dielectric constant, respectively; 𝑥𝑥𝑖𝑖 and 𝑧𝑧𝑖𝑖

denote the mole fraction and the charge number of species (anions and cation), respectively.

Following Eq. 3.6, the activity coefficient of species 𝑖𝑖, 𝛾𝛾𝑖𝑖, can be formulated as a

combination of two contributions, shown by Eq. 3.14.

𝑙𝑙𝑙𝑙 𝛾𝛾𝑖𝑖 = 𝑙𝑙𝑙𝑙 𝛾𝛾𝑖𝑖𝑙𝑙𝐿𝐿 + 𝑙𝑙𝑙𝑙 𝛾𝛾𝑖𝑖𝑃𝑃𝑃𝑃𝐻𝐻 = 1𝑅𝑅𝑇𝑇�𝜕𝜕𝐺𝐺

𝑒𝑒𝑥𝑥,𝑙𝑙𝑐𝑐

𝜕𝜕𝑙𝑙𝑖𝑖�𝑇𝑇,𝑃𝑃,𝑙𝑙𝑗𝑗≠𝑖𝑖

+ 1𝑅𝑅𝑇𝑇�𝜕𝜕𝐺𝐺

𝑒𝑒𝑥𝑥,𝑃𝑃𝐷𝐷𝐻𝐻

𝜕𝜕𝑙𝑙𝑖𝑖�𝑇𝑇,𝑃𝑃,𝑙𝑙𝑗𝑗≠𝑖𝑖

, 𝑖𝑖 =

𝑚𝑚, 𝑐𝑐,𝑎𝑎

(3.14)

where 𝑚𝑚, 𝑐𝑐, and a stand for molecule, cation, and anion species, respectively.

In applying the symmetric eNRTL model to the HCl-H2O binary, we choose pure

liquid as the reference state for solvent water and molecular HCl, and hypothetical pure

fused salt (𝑐𝑐𝑎𝑎) for ion pair (H3O+, Clˉ) respectively.


55

𝛾𝛾𝑚𝑚(𝑥𝑥𝑚𝑚 → 1) = 1 (3.15)

𝛾𝛾𝐿𝐿𝑎𝑎(𝑥𝑥𝐿𝐿𝑎𝑎 → 1) = 𝛾𝛾±(𝑥𝑥𝐿𝐿𝑎𝑎 → 1) = 1 (3.16)

𝑥𝑥𝐿𝐿𝑎𝑎 = 𝑥𝑥𝐿𝐿 + 𝑥𝑥𝑎𝑎 (3.17)

𝛾𝛾± = (𝛾𝛾𝐿𝐿𝜐𝜐𝑐𝑐𝛾𝛾𝑎𝑎𝜐𝜐𝑎𝑎)1𝜐𝜐, 𝜐𝜐 = 𝜐𝜐𝐿𝐿 + 𝜐𝜐𝑎𝑎 (3.18)

where 𝛾𝛾± is the mean ionic activity coefficient of the salt 𝑐𝑐𝑎𝑎 in solution; 𝜐𝜐𝐿𝐿 and 𝜐𝜐𝑎𝑎 denote

the stoichiometric coefficients of the cation and anion, respectively. It should be noted that

one could calculate unsymmetric activity coefficient 𝛾𝛾∗ if the aqueous phase infinite

dilution reference state is chosen for ionic species:

𝑥𝑥𝐿𝐿 = 𝑥𝑥𝑎𝑎 → 0 (3.19)

𝛾𝛾𝐿𝐿∗(𝑥𝑥𝑚𝑚 → 1) = 𝛾𝛾𝐿𝐿∗(𝑥𝑥𝑚𝑚 → 1) = 1 (3.20)

Calorimetric Properties Accurate calculations of calorimetric properties of electrolyte solutions, including

liquid molar enthalpy, excess enthalpy, and liquid molar heat capacity are essential for

energy balance calculations in process simulation. These calorimetric properties are

calculated from the excess Gibbs free energy expression. The underlying thermodynamic

relationships for the calorimetric properties have been extensively described by Que et al.

77 and Wang et al. 79.

3.4. The Modeling Approach

Model Parameters Table 3.1 summarizes the thermophysical properties and the model parameters

required to fully establish the comprehensive thermodynamic model for the HCl-H2O

binary. They include saturation pressures, critical properties, liquid molar densities,


56

molecular and ionic component thermophysical properties, dielectric constants, eNRTL

binary interaction parameters, and chemical equilibrium constant for HCl partial

dissociation.

Table 3.1. Summary of thermophysical properties and model parameters.

Parameters Description Component Reference

𝑝𝑝𝑖𝑖° Saturation pressure HCl, H2O Aspen Plus databank 88

𝑇𝑇𝐿𝐿 ,𝑃𝑃𝐿𝐿 ,𝛺𝛺 Critical temperature, critical pressure, and acentric factor

HCl, H2O Aspen Plus databank 88

𝜌𝜌𝑖𝑖𝑙𝑙 Liquid molar density HCl, H2O Aspen Plus databank 88

𝛥𝛥𝑓𝑓ℎ𝑚𝑚,298.15𝑖𝑖𝑖𝑖 Ideal gas enthalpy of

formation HCl, H2O Aspen Plus databank 88


57


Parameters Description Component Reference

𝑐𝑐𝑝𝑝,𝑖𝑖𝑖𝑖𝑖𝑖 Ideal gas molar heat

capacity HCl, H2O Aspen Plus databank 88

𝛥𝛥𝑣𝑣𝑎𝑎𝑝𝑝ℎ𝑖𝑖 Heat of vaporization HCl, H2O Aspen Plus databank 88

𝐶𝐶𝑝𝑝,𝑖𝑖∞,𝑎𝑎𝑎𝑎 Heat capacity of

Ionic species at infinite dilution

H3O+ Aspen Plus databank 88

Clˉ Yan and Chen 89

ε Dielectric constant HCl, H2O Aspen Plus databank 88

𝜏𝜏𝑖𝑖𝑗𝑗 Binary interaction parameter

H2O:(H3O+-Clˉ) pair Regression

HCl:(H3O+-Clˉ) pair Regression

H2O:HCl pair Regression

A, B for 𝑙𝑙𝑙𝑙 𝐾𝐾 Chemical equilibrium constants

R1, R2 Regression

Tables 3.2 to 3.7 present the physical constants, empirical equations, and coefficient

values used in the calculations of saturation pressures, critical properties, liquid molar

densities, ideal gas heat capacities, heat of vaporization, and dielectric constants.

Table 3.2. Extended Antoine equation parameters for saturation pressures (Piº) for H2O and

HCl 88.

Parameters 𝐶𝐶1 𝐶𝐶2 𝐶𝐶3 𝐶𝐶4 𝐶𝐶5 𝐶𝐶6 𝐶𝐶7 𝐶𝐶8 𝐶𝐶9

H2O 73.65 -7258.2 0 0 -7.30 4.17×10-6 2 273.16 647.1 HCl 104.27 -3731.2 0 0 -15.05 0.03 1 158.97 324.6

ln𝑝𝑝𝑖𝑖° = 𝐶𝐶1𝑖𝑖 + 𝐶𝐶2𝑖𝑖

𝜕𝜕+𝐶𝐶3𝑖𝑖+ 𝐶𝐶4𝑖𝑖𝑇𝑇 + 𝐶𝐶5𝑖𝑖 ln𝑇𝑇 + 𝐶𝐶6𝑖𝑖𝑇𝑇𝐶𝐶7𝑖𝑖 for 𝐶𝐶8𝑖𝑖 ≤ 𝑇𝑇 ≤ 𝐶𝐶9𝑖𝑖; Linear extrapolation is

performed if T falls out of the temperature bounds


58

Table 3.3. Redlich-Kwong equation of state parameters for H2O and HCl 88.

Parameters 𝑇𝑇𝐶𝐶 (K) 𝑃𝑃𝐶𝐶 (MPa) Ω

H2O 647.10 22.29 0.345

HCl 324.65 8.42 0.131

Table 3.4. DIPPR liquid molar density (ρil) model parameters for H2O and HCl 88

Parameters 𝐶𝐶1 𝐶𝐶2 𝐶𝐶3 𝐶𝐶4 𝐶𝐶5 𝐶𝐶6 𝐶𝐶7

H2O 17.86 58.61 -95.37 213.89 -141.26 273.16 647.10

HCl 3.34 0.27 324.65 0.32 0.00 158.97 324.65

𝜌𝜌𝑖𝑖𝑙𝑙 = 𝐶𝐶1𝑖𝑖/𝐶𝐶2𝑖𝑖1+(1−𝜕𝜕/𝐶𝐶3𝑖𝑖)𝐶𝐶4𝑖𝑖 for 𝐶𝐶6𝑖𝑖 ≤ 𝑇𝑇 ≤ 𝐶𝐶7𝑖𝑖, for H2O

𝜌𝜌𝑖𝑖𝑙𝑙 = 𝐶𝐶1𝑖𝑖 + 𝐶𝐶2𝑖𝑖𝜏𝜏0.35 + 𝐶𝐶3𝑖𝑖𝜏𝜏2/3 + 𝐶𝐶4𝑖𝑖𝜏𝜏 + 𝐶𝐶5𝑖𝑖𝜏𝜏4/3 for 𝐶𝐶6𝑖𝑖 ≤ 𝑇𝑇 ≤ 𝐶𝐶7𝑖𝑖, for HCl

𝜏𝜏 = 1 − 𝜕𝜕𝜕𝜕𝑐𝑐

; 𝑣𝑣𝑖𝑖° is the liquid molar volume of species i; 𝑇𝑇𝐶𝐶 is the critical temperature of species

i; Linear extrapolation is performed if T falls out of the temperature bounds

Table 3.5. DIPPR ideal gas heat capacity (cp,iig) model parameters for H2O and HCl 88


H2O 56.6 0.61204 -0.6258 0.3988 0 273.15 647.1

HCl 34.872 2.1553 -2.9128 1.2442 0 158.97 324.65

𝑐𝑐𝑝𝑝,𝑖𝑖𝑖𝑖𝑖𝑖 = 𝐶𝐶1𝑖𝑖 + 𝐶𝐶2𝑖𝑖 �

𝐶𝐶3𝑖𝑖𝜕𝜕�

𝑠𝑠𝑖𝑖𝜕𝜕ℎ(𝐶𝐶3𝑖𝑖 𝜕𝜕� )�2

+ 𝐶𝐶4𝑖𝑖 �𝐶𝐶5𝑖𝑖

𝜕𝜕�

𝑠𝑠𝑖𝑖𝜕𝜕ℎ(𝐶𝐶5𝑖𝑖 𝜕𝜕� )�2

for 𝐶𝐶6𝑖𝑖 ≤ 𝑇𝑇 ≤ 𝐶𝐶7𝑖𝑖


59

Table 3.6. DIPPR heat of vaporization (Δvaphi) model parameters for H2O and HCl 88


H2O 33363 26790 2610.5 8896 1169 100 2273.15

HCl 29157 9048 2093.8 -107 120 50 1500.00

∆𝑣𝑣𝑎𝑎𝑝𝑝ℎ𝑖𝑖 = 𝐶𝐶1𝑖𝑖(1 − 𝑇𝑇𝑟𝑟𝑖𝑖)𝐶𝐶2𝑖𝑖+𝐶𝐶3𝑖𝑖𝜕𝜕𝑟𝑟𝑖𝑖+𝐶𝐶4𝜕𝜕𝑟𝑟𝑖𝑖2+𝐶𝐶5𝜕𝜕𝑟𝑟𝑖𝑖3 for 𝐶𝐶6𝑖𝑖 ≤ 𝑇𝑇 ≤ 𝐶𝐶7𝑖𝑖; 𝑇𝑇𝑟𝑟𝑖𝑖 = 𝜕𝜕

𝜕𝜕𝑐𝑐𝑖𝑖

Table 3.7. Dielectric constants (ε) for H2O and HCl 88

Parameters 𝐴𝐴 𝐵𝐵 𝐶𝐶

H2O 78.51 31989.4 298.15

HCl 4.71 3274.0 298.15

𝜀𝜀 = 𝐴𝐴 + 𝐵𝐵 �1𝑇𝑇−

1𝐶𝐶�

The model adjustable parameters fall into two categories: 1) binary interaction

parameters of molecule-electrolyte pairs, i.e., the H2O:(H3O+-Clˉ) pair and the HCl:(H3O+-

Clˉ) pair, and of molecule-molecule pair, i.e., the H2O:HCl pair, and 2) the chemical

equilibrium constant parameters for the HCl partial dissociation. Quantifying these

parameters requires regression of pertinent experimental VLL, LLE and calorimetric data.

For this purpose, a comprehensive review of available literature data is reported next.

Experimental Data Abundant experimental data of phase equilibrium, thermodynamic, and calorimetric

properties are available for the HCl-H2O binary 5,13-45. Compiled for parameter

quantification and model validation, Table 3.8 lists experimental VLE TP-xy, isothermal

P-xy, and isobaric T-xy data 13-30, LLE TP-xx data 9,18, molality scale mean ionic activity


60

coefficient (𝛾𝛾±,𝑚𝑚) data, osmotic coefficient (𝜙𝜙) data 22,31,33-36, enthalpy of formation data

(∆𝐻𝐻𝑓𝑓°) 37, and heat capacity data (𝑐𝑐𝑝𝑝) 5,13,38-43. Also included are smoothed curves of molar

heat capacity 44 and boiling point temperature 44,45.

Table 3.8. List of experimental data used in regression and model validation for the HCl-H2O binary system

Reference Data Type

T (K) P

(kPa) HCl Concentration SD

Number of Data pointsa

MRD %

Perry’s Handbook 13

𝑝𝑝H2O-x 273.15-383.15

0.04-128

1.75-19.86 mol.kg-1 - (0) 271 7.24b


𝑝𝑝HCl-x 273.15-383.15

~ 0-129

0.56-23.36 mol.kg-1 - (0) 321 29.03b

Fritz & Fuget 14

P-xy 273.15-323.15

0.3-56.1

0.05-15.88 mol.kg-1 5% for pressure

(119) 119

4.79b

Vrevskii et al. 15

P-xy 293.15-349.05

1.2-74.8

0-16.02 mol.kg-1 - (0) 38 5.97b

Storonkin & Susarev 16

P-xy 298.15 1.8-3.6

2.92-12.18 mol.kg-1 - (0) 12 2.42

Susarev & Prokofeva 17

P-xy 298.15 2-3 2.31-10.57 mol.kg-1 - (0) 7 8.18

Haase et al. 18

P-xy 298.15 1.8-21.3

4.83-15.66 mol.kg-1 5% for pressure

(12) 12 7.58

Kao 19 P-xy 283.15-343.15

106.9-1530.6

15.11-34.90 mol.kg-1

- (0) 20 52.00b

Miller 20 P-xy 273.15 0.5-3.7

2.38-15.16 mol.kg-1 -

(0) 6 12.81

Elm et al. 21

TP-xy 304.7-343

3.4-27.4

1.29-4.84 mol.kg-1 - (0) 40 2.63b


61


Reference Data Type

T (K) P

(kPa) HCl Concentration SD

Number of Data pointsa

MRD %

Vega & Vera 22

P-xy 298.15 2.7-69 3.04-18.28 mol.kg-1 - (0) 14 12.92

Berl & Staudinger 23

T-xy 289.85-383.15

99-102

0.40-31.02 mol.kg-1 - (0) 52 11.25

Hawliczek & Synowiec 24

T-xy 323.15-381.35

100 0-17.30 mol.kg-1 - (0) 72 6.97

Lutugina & Kokovkina 2

T-xy 336.45-383.15

19-101

2.31-7.78 mol.kg-1 - (0) 14 2.32b

Othmer 26

T-xy 374.65-380.95

100 0.25-7.64 mol.kg-1 - (0) 8 0.81

Lu et al. 27 T-xy 373.33-381.73

101 0.22-7.06 mol.kg-1 - (0) 10 1.37

Bonner & Wallace 28

TP-xc 321.87-393.13

6.6-162.6

6.58-8.39 mol.kg-1 - (0) 10 2.86

Sako et al. 29

T-xy 374.2-381.7

101.3 1.26-8.42 mol.kg-1 - (0) 10 0.20

Sako et al. 30

TP-xc 323.4-394.5

8.6-392.1

1.26-13.43 mol.kg-1 - (0) 70 6.33

Rupert 9 TP-xx 278.15-308.15

3000-6000

0.45-0.48 mole fraction, L(I)d,

0.991-0.997 mole

- (0) 7 2.16e

Haase et al. 18 TP-xx

278.15-308.15

3000-6000

0.45-0.48 mole fraction, L(I)d,

0.991-0.997 mole

10% for pressure (7) 7 1.26e

Randall and Young 31

γ±,m 298.15 100 0-16 mol.kg-1 - (0) 34 15.15

Harned and Owen 33

γ±,m 298.15 100 3.04-15.42 mol.kg-1 - (0) 12 11.01


62


Reference Data Type T (K)

P (kPa)

HCl Concentration SD Number of Data pointsa

MRD %

Åkerlöf 34 γ±,m 298.15 100 0.01-1.53 mol.kg-1 - (0) 10 15.14

Åkerlöf & Teare 35

γ±,m 273.15-323.15

100 3-16 mol.kg-1 - (0) 11 10.33b

Hamer & Wu 36

γ±,m 298.15 100 0-15.93 mol.kg-1 - (0) 39 14.13

Vega & Vera 22

γ±,m 298.15 100 3.04-18.28 mol.kg-1 - (0) 14 11.00

Hamer & Wu 36

𝜙𝜙 298.15 100 0-15.93 mol.kg-1 - (0) 39 6.12

Wagman et al. 37

∆𝐻𝐻𝑓𝑓° 298.15 100 0-55.51 mol.kg-1 0.1 kJ/mol

(36) 40 0.457 kJ/molf

Allred & Wooley 38

𝑐𝑐𝑝𝑝 283.15-313.15

100 0.02-0.39 mol.kg-1 5% for Cp

(36) 36 0.09b

Pogue & Atkinson 39

𝑐𝑐𝑝𝑝 288.15-328.15

100 0.05-0.23 mol.kg-1 - (0) 30 0.13b

Tremanie et al. 40

𝑐𝑐𝑝𝑝 298.15-412.61

100-500

0.09-1.02 mol.kg-1 - (0) 39 0.11b

Fortier et al. 41

𝑐𝑐𝑝𝑝 298.15 100 0.04-1.02 mol.kg-1 - (0) 18 0.16

Saluja et al. 42

𝑐𝑐𝑝𝑝 323.15-373.15

600 0.01-0.15 mol.kg-1 - (0) 12 0.04

Wicke et al. 43

𝑐𝑐𝑝𝑝 293.15-403.15

100 0.59-1.90 mol.kg-1 - (0) 25 0.36

Sharygin & Wood 5

𝑐𝑐𝑝𝑝 302-623

28000 0.10-6.02 mol.kg-1 - (0) 60 4.01b


𝑐𝑐𝑝𝑝 273.15-333.15

100 0.0-19.40 mol.kg-1 5% for Cp

(15) 18 2.88b


63


Reference Data Type T (K)

P (kPa)

HCl Concentration SD Number of Data pointsa

MRD %

Oxy’s Handbook 44

𝑐𝑐𝑝𝑝 273.15-333.15

100 3.17-17.14 mol.kg-1 5% for Cp

-g 2.33b

Oxy’s Handbook 44

𝑇𝑇𝑏𝑏 - 100 0-18.20 mol.kg-1 - -g 5.17

DeDietrich Process Systems 45

𝑇𝑇𝑏𝑏 - 50- 100

0-23.45 mol.kg-1 - -g 4.47b

a (Number of data used in regression) total number of reported data b The average MRD value of all reported temperatures, or pressures c No report of HCl compositions in vapor phase

d L(I) : liquid phase I, L(II): liquid phase II e The reported MRD is with respect to HCl liquid phase composition f The mean absolute difference in excess enthalpy g Data are extracted from smoothed curves

Perry’s handbook 13 gathered data of HCl and water partial pressures in aqueous HCl

solution in the temperature range of 273.15 to 383.15 K and the concentration range of

0.56 to 23.36 mol.kg-1 HCl. Fritz and Fuget 14 reported isothermal VLE data (P-xy) in the

temperature range of 273.15 to 323.15 K and the concentration range of 0.05 to 15.88

mol.kg-1 HCl. Vrevskii et al. 15 gathered vapor pressure data (P-xy) in the temperature range

of 293.15 to 349.05 K with the HCl concentration range of 0 to 16.02 mol.kg-1. Also

reported are vapor pressure data (P-xy) at 298.15 K by Storonkin and Susarev 16, and a

decade later by Susarev and Prokofeva 17, with HCl concentrations from 2.92 to 12.18

mol.kg-1 and from 2.31 to 10.57 mol.kg-1, respectively.


64

Haase et al. 18 presented vapor pressure data (P-xy) at 298.15 K for the concentration

range of 4.83 to 15.66 mol.kg-1 HCl. Kao 19 measured vapor pressure data (P-xy) in the

temperature range of 263.15 to 343.15 K and the concentration range of 15.11 to 34.90

mol.kg-1 HCl. Also presented are the compiled vapor pressure data (P-xy) of Miller et al.

20 from 238.15 to 273.15 K with the concentration range of 2.38 to 15.16 mol.kg-1 HCl.

Elm et al. 21 presented TP-xy data in the temperature range of 304.7 to 343 K with the

concentration range of 1.29 to 4.84 mol.kg-1 HCl. Vega and Vera 22 reported P-xy data at

298.15 K in the concentration range of 3.04 to 18.28 mol.kg-1 HCl.

Berl and Staudinger 23 reported isobaric T-xy data with the temperature range of

289.85 to 383.15 K at around 99 to 102 kPa and HCl concentrations from 0.40 to 31.02

mol.kg-1. Hawliczek and Synowiec 24 presented T-xy data in the temperature range of

323.15 to 381.35 K at 100 kPa within the concentration range of 0 to 17.30 mol.kg-1 HCl.

Lutugina and Kokovkina 25 presented T-xy data with the temperature range of 336.45 to

383.15 K, the pressure range of 19 to 101 kPa, and the concentration range of 2.31 to 7.78

mol.kg-1 HCl. Othmer 26 reported T-xy data at 100 kPa with the temperature range of 374.65

to 380.95 K and the concentration range of 0.25 to 7.64 mol.kg-1 HCl. Lu et al. 27 reported

T-xy data from 373.33 to 381.73 K at 101 kPa with the concentration range of 0.22 to 7.06

mol.kg-1 HCl.

Bonner and Wallace 28 presented VLE data in the form of TP-x with temperatures from

321.87 to 393.13 K, pressures from 6.6 to 162.6 kPa, and HCl concentrations from 6.58 to

8.39 mol.kg-1. Sako et al. 29 reported T-xy data under atmospheric pressure. A year later,

Sako et al. 30 measured TP-x data with the temperature range of 323.4 to 394.5 K and the

concentrations from 1.26 to 13.43 mol.kg-1 HCl.


65

Haase et al. 18 reported LLE data at temperatures from 278.15 to 308.15 K and

pressures from 3 to 6 MPa. However, their measurements were limited to those of the

primary liquid phase and there were no measurements for the species compositions in the

second liquid phase. However, they reported the measurements of Rupert 9 for the second

liquid phase containing mostly pure HCl. The HCl concentration ranges for the two liquid

phases are from 0.45 to 0.48 and 0.991 to 0.997 mole fractions, respectively.

Employing the electromotive force measurements (EMF), Randall and Young 31

computed molality scale mean ionic activity coefficients of HCl for the HCl-H2O binary at

298.15 K up to 16 mol.kg-1 HCl. Using the same technique, Harned and Ehlers 32 calculated

mean ionic activity coefficients, partial molar enthalpy, and partial molar heat capacity of

HCl in the aqueous binary solution at temperatures from 273.15 to 333.15 K and

concentrations up to 4 mol.kg-1 HCl. Years later, Harned and Owen 33 extended the data

range at 298.15 K up to 15.42 mol.kg-1 HCl for the mean ionic activity coefficient. Also,

by using EMF, Åkerlöf 34 reported mean ionic activity coefficients of dilute aqueous HCl

(up to 1.53 mol.kg-1) at 298.15 K, as a part of a study of different chloride solutions in

water-methanol mixtures. Åkerlöf and Teare 35 extended their experimental work on

concentrated aqueous HCl solution and reported the mean ionic activity coefficients to

cover HCl concentrations up to 16 mol.kg-1 at temperatures from 273.15 to 323.15 K.

Hamer and Wu 36 reported mean ionic activity coefficient and osmotic coefficient data at

298.15 K and pressure of 100 kPa for concentrations from 0 to 15.93 mol.kg-1 HCl. Vega

and Vera also reported mean ionic activity coefficient data up to 18.28 mol.kg-1 HCl at

298.15 K [19].


66

Wagman et al. 37 presented enthalpy of formation data for the HCl-H2O binary from

very dilute HCl solution up to 55.51 mol.kg-1 HCl at 298.15 K and 100 kPa. These enthalpy

of formation data can be easily converted to and used as excess enthalpy data. Allred and

Woolley 38 gathered heat capacity data for the HCl-H2O binary with temperatures from

283.15 to 313.15 K at 100 kPa pressure for dilute HCl solutions from 0.02 to 0.39 mol.kg-

1 HCl. Pogue and Atkinson 39 also reported the heat capacity data for the temperature range

of 288.15 to 328.15 K at 100 kPa pressure with the concentration range of 0.05 to 0.23

mol.kg-1 HCl. Moreover, the heat capacity data from Tremanie et al. 40 covered the

temperature range of 298.15 to 412.61 K at pressures of 100 to 500 kPa and the

concentration range of 0.09 to 1.02 mol.kg-1 HCl. Fortier et al. 41 reported the heat capacity

data for the concentration range of 0.04 to 1.02 mol.kg-1 HCl at 298.15 K and 100 kPa.

Saluja et al 42 published the heat capacity data covering temperatures from 323.15 to 373.15

K at pressure of 600 kPa with the concentration range of 0.01 to 0.15 mol.kg-1 HCl. Also

presented the heat capacity data, Wicke et al 43 covered the temperature range of 293.15 to

403.15 K at 100 kPa for the HCl concentrations up to 1.90 mol.kg-1. Covering the more

concentrated HCl-H2O binary system, Perry’s handbook 13 reported the heat capacity data

with the temperature range of 273.15 to 333.15 K and the concentration range of 0 to 19.40

mol.kg-1 HCl at 100 kPa pressure. Measuring the heat capacity of aqueous HCl solution at

higher temperatures and pressures, Sharygin and Wood [5] covered temperatures from 302

to 623 K at a significant pressure of 28 MPa with the concentration range of 0.10 to 6.02

mol.kg-1 HCl. In addition, the smoothed heat capacity curves reported by Occidental

Petroleum’s hydrochloric acid handbook 44 cover the temperature range of 273.15 to

333.15 K and the concentration range of 3.17 to 17.14 mol.kg-1 at 100 kPa. The handbook


67

also reports smoothed boiling temperature curves covering up to 18.21 mol.kg-1 HCl at 100

kPa 44. The De Dietrich Process Systems reported the smoothed boiling temperature curves

up to 23.45 mol.kg-1 HCl at 100 kPa, and up to 21.70 mol.kg-1 HCl at 50 kPa 45.

Data Treatment, Regression, and Optimization Method Quantification of the adjustable model parameters is carried out by simultaneous

regression of selected sets of experimental data including those of VLE 14,18, LLE 18, excess

enthalpy 37, and heat capacity 13,38,44. All the remaining data sets, although not included in

the data regression, are used for model validation. The simultaneous regression and

subsequent validation of the model with such wide varieties of experimental data help

ensure consistency of these data sets under one comprehensive thermodynamic framework.

The regression data sets are selected to cover the widest temperature and concentration

ranges possible, and with the best reliability and accuracy. The VLE data of Fritz and Fuget

14 is selected because it covers a wide concentration range from 0.05 to 15.88 mol.kg-1 HCl

and temperatures ranging from 273.15 up to 323.15 K with high data accuracy. Although

few other data sets also cover wide temperature and concentration ranges, they are

excluded from the regression. For example, the data of Zeisberg 12 reported in Perry’s

Handbook 13 have been shown to be inconsistent with the more recent measurements of

Fritz and Fuget 14. Also the uncertainty of several literature data sets 15,23,24 could not be

identified. Among the LLE data available 9,18, the more recent and accurate data set 18 is

included in the regression along with its corresponding VLE data. The mean ionic activity

coefficient data sets are not considered in the regression due to the intrinsic assumption of

complete HCl dissociation associated with mean ionic activity coefficient data. The

osmotic coefficient data are also excluded because they are inherently water vapor pressure


68

data and carry larger standard deviations. To cover calorimetric properties, we include in

the regression the data for excess enthalpy 37 and the data 13,38 and smoothed curves 44 for

heat capacity.

Based on prior experiences in correlating these thermodynamic data, standard

deviations (SDs) are assigned depending on the type of experimental measurement.

Temperatures are treated as error-free for all data sets. The SDs of pressures are set to 5 %.

For VLE data, the SDs of compositions in mole fraction are set to zero for liquid phase and

0.01 for vapor phase with the exception of very dilute regions (< 0.01 mole fraction) where

the SDs are assigned to be 5 %. For LLE, the SDs of compositions in mole fraction are

fixed at 0.01. For liquid molar excess enthalpy and molar heat capacity, the SDs are set to

0.1 kJ/mol and 5 %, respectively.

The objective function in data regression, i.e., residual root mean square error

(RRMSE), is defined in Eq. 3.21:

𝑅𝑅𝑅𝑅𝑀𝑀𝑆𝑆𝐸𝐸 =�∑ ∑ [

𝑍𝑍𝑖𝑖𝑗𝑗 − 𝑍𝑍𝑀𝑀𝑖𝑖𝑗𝑗𝜎𝜎𝑖𝑖𝑗𝑗

]2𝑚𝑚𝑗𝑗=1

𝑘𝑘𝑖𝑖=1

𝑘𝑘 − 𝑙𝑙

(3.21)

where experimental data is represented by 𝑍𝑍𝑀𝑀𝑖𝑖𝑗𝑗; 𝑍𝑍𝑖𝑖𝑗𝑗 is the calculated value by the model,

𝜎𝜎 denotes the SDs for each set of data; 𝑖𝑖 represents the 𝑖𝑖𝑡𝑡ℎ data point; j is one of the

measured variables; k is the total number of data points; m is the total number of measured

variables; and n is the total number of adjustable parameters. Furthermore, mean relative

deviation (MRD), shown in Eq. 3.22, provides a quantitative measure for the deviation

between the experimental data and the model results.


69

𝑀𝑀𝑅𝑅𝐷𝐷 (%) =

1𝑙𝑙��

𝑥𝑥𝑚𝑚𝑜𝑜𝑚𝑚𝑒𝑒𝑙𝑙 − 𝑥𝑥𝑚𝑚𝑎𝑎𝑡𝑡𝑎𝑎𝑥𝑥𝑚𝑚𝑎𝑎𝑡𝑡𝑎𝑎

�𝜕𝜕

𝑖𝑖=1

× 100 (3.22)

where 𝑥𝑥𝑚𝑚𝑜𝑜𝑚𝑚𝑒𝑒𝑙𝑙 and 𝑥𝑥𝑚𝑚𝑎𝑎𝑡𝑡𝑎𝑎 are the model result and the experimental data, respectively.

The regression is carried out with both the symmetric reference state formulation and

the unsymmetric reference state formulation. While the choice of reference states should

affect the chemical equilibrium constant of the HCl partial dissociation, the eNRTL binary

interaction parameters are independent of the choice of reference states 80.

3.5. Results and Discussion

Tables 3.9 to 3.11 summarize the identified model parameters including the binary

interaction parameters of molecule-electrolyte and molecule-molecule pairs and the

chemical equilibrium constant parameters.

Table 3.9. eNRTL model parameters (τij) for molecule-electrolyte and molecule-molecule pairsa.

Species 𝑖𝑖 Species 𝑗𝑗 𝑎𝑎𝑖𝑖𝑗𝑗 𝑏𝑏𝑖𝑖 𝑐𝑐𝑖𝑖𝑗𝑗 𝑑𝑑𝑖𝑖𝑗𝑗 𝑒𝑒𝑖𝑖𝑗𝑗 𝜏𝜏𝑖𝑖𝑗𝑗 at

298.15 K

H2O (H3O+, Clˉ) - - 7.21±0.35 1099.4±104.2 -4.14±2.72 10.897

(H3O+, Clˉ) H2O - - -3.86±0.09 -432.9±26.5 2.35±0.79 -5.312

HCl (H3O+, Clˉ) - - 1.82±3.18 1000.0b 0 5.174

(H3O+, Clˉ) HCl - - 1.01±0.38 -500.0b 0 -0.667

H2O HCl 1.000b 0 - - - 1.000

HCl H2O 0.007±0.34

7 0 - - - 0.007


70

a 𝜏𝜏𝑖𝑖𝑗𝑗 = 𝑎𝑎𝑖𝑖𝑗𝑗 + 𝑏𝑏𝑖𝑖𝑖𝑖𝜕𝜕

for molecule-molecule pairs; 𝜏𝜏𝑖𝑖𝑗𝑗 = 𝑐𝑐𝑖𝑖𝑗𝑗 + 𝑚𝑚𝑖𝑖𝑖𝑖𝜕𝜕

+ 𝑒𝑒𝑖𝑖𝑗𝑗[𝜕𝜕𝑟𝑟𝑟𝑟𝑟𝑟−𝜕𝜕𝜕𝜕

+ 𝑙𝑙𝑙𝑙 � 𝜕𝜕𝜕𝜕𝑟𝑟𝑟𝑟𝑟𝑟

�] for

molecule-electrolyte pairs

b Fixed parameters

Table 3.10. Chemical equilibrium constant parameters for unsymmetric reference statea.

Reaction 𝐴𝐴 𝐵𝐵 𝑙𝑙𝑙𝑙 𝐾𝐾 at

298.15 K ∆𝐺𝐺𝑟𝑟𝑒𝑒𝜕𝜕 ° (𝑘𝑘𝑘𝑘/𝑚𝑚𝑚𝑚𝑙𝑙) ∆𝐻𝐻𝑟𝑟𝑒𝑒𝜕𝜕

° (𝑘𝑘𝑘𝑘/𝑚𝑚𝑚𝑚𝑙𝑙)

R1 -15.52±0.23 7424.38±69.89 9.38 -23.2550 -61.7263

a 𝑙𝑙𝑙𝑙 𝐾𝐾 = 𝐴𝐴 + 𝐵𝐵𝜕𝜕

Table 3.11. Chemical equilibrium constant parameters for symmetric reference statea.

Reaction 𝐴𝐴 𝐵𝐵 𝑙𝑙𝑙𝑙 𝐾𝐾 at

298.15 K ∆𝐺𝐺𝑟𝑟𝑒𝑒𝜕𝜕 ° (𝑘𝑘𝑘𝑘/𝑚𝑚𝑚𝑚𝑙𝑙) ∆𝐻𝐻𝑟𝑟𝑒𝑒𝜕𝜕

° (𝑘𝑘𝑘𝑘/𝑚𝑚𝑚𝑚𝑙𝑙)

R1 -11.74±0.36 4206.11±107.00 2.37 -5.8682 -34.9696

a 𝑙𝑙𝑙𝑙 𝐾𝐾 = 𝐴𝐴 + 𝐵𝐵𝜕𝜕

Figure 3.2 shows the degree of HCl dissociation to H3O+ and Cl– ions versus the HCl

concentration in wt. %. The model results suggest that HCl in the aqueous solution

dissociates nearly completely at dilute concentrations. However, at higher concentrations,

i.e., > 30 wt. % HCl, there is a sharp shift toward partial dissociation as the degree of HCl

dissociation drops precipitously. Such a trend continues until phase separation occurs and

molecular HCl forms its own liquid phase. The degree of HCl dissociation drops with rising

temperatures for the same HCl concentration. Subsequently, phase separation occurs at

lower HCl concentrations with rising temperatures. The mole fractions of true species

versus HCl wt. % are presented in Figure 3.3. The mole fraction of molecular water


71

decreases dramatically with increasing HCl concentration as water is consumed in the

hydration of protons as HCl dissociates.

Figure 3.2. Model results for the HCl dissociation versus HCl wt. % ( ) at 273.15 K; () at 298.15 K; and ( ) at 323.15 K; the dotted lines show the one-liquid phase

region limit.


72

Figure 3.3. Model results for species mole fractions versus HCl wt. %: ( ) at 273.15 K, ( ) at 298.15 K, and ( ) at 323.15 K; the dotted lines show the one-liquid phase region limit.

Figure 3.4 illustrates that the model results for total pressures of the HCl-H2O binary

are in satisfactory agreement with the experimental data compiled from literature 14-20 at

273.15, 298.15, 323.15, and 349.05 K, even though the model parameters were regressed

only from the VLE data of Fritz and Fuget 14 and Haase et al. 18. The dashed lines represent

LLE tie-lines at each temperature. Also, both the experimental data and the model results

show a minimum-boiling azeotropic point around 25 wt. % HCl. The MRD values of data

sets shown in Figure 3.4 range from 2.42 to 12.81 %, except for the data reported by Kao

19 which has an MRD value of 52.00 %. The validity of the data from Kao 19 seems

questionable, since Kao claimed his results were supported by a data point of Haase et al.

18 at 303.15 K and 1.31 MPa which has not been found. The MRD values of other VLE

data sets are also presented in Table 3.8, ranging from 0.81 to 29.03 %. To further examine


73

the model results for HCl partial pressures, Figure 3.5 compares the model results with the

experimental data 14,15,18 of y-x type for the HCl component. The temperatures are in same

order as Figure 3.4, from 273.15 to 349.05 K.

Figure 3.4. Model results for system pressure (solid lines) compared with experimental data (symbols): ( ), ( ) 14, and ( ) 20a at 273.15 K; ( ), ( ) 16a, ( ) 17a, ( ) 18, and ( ) 18 at 298.15 K; ( ), ( ) 14, and ( ) 19a at 323.15 K; ( ) and ( ) 15a at 349.05 K. Also shown, the LLE tie-lines (dashed lines). a Not used in regression.


74

Figure 3.5. Model results (solid lines) compared with experimental data (symbols) for HCl compositions in vapor phase versus in liquid phase: ( ), ( ) 14 at 273.15 K; ( ) and (

) 18 at 298.15 K; ( ) and ( ) 14 at 323.15 K; ( ) and ( ) 15a at 349.05 K. a Not used in regression.

Figure 3.6 depicts LLE tie-lines at temperatures from 278.15 to 308.15 K. The model

results show excellent agreement with the experimental data of Rupert 9 and Haase et al. 18

with the MRD value of 2.16 and 1.26 %, respectively.


75

Figure 3.6. Model results for LLE ( ) compared with experimental data ( ) 9a and ( ) 18. Also shown, the tie-lines ( ). a Not used in regression.

Figure 3.7 illustrates that the model results for boiling temperatures of the HCl-H2O

binary (with HCl concentration < 40 wt. %), at 50 and 100 kPa, are in satisfactory

agreement with the smoothed literature data 44,45 with MRD values of 5.17 and 4.47 %,

respectively.


76

Figure 3.7. Model results for boiling point ( ) at 50 and 100 kPa versus literature smoothed curves ( ) 44 and ( ) 45a. a Not used in regression.

Figure 3.8 shows that the model properly captures the trend of experimental data for

molality scale mean ionic activity coefficient up to 18 molal (~ 37 wt. % HCl) 22,36. The

mean ionic activity coefficients are calculated based on an established formula [86] which

takes into account partial dissociation and hydration. The MRD values are 11.00 % for the

data of Vega & Vera [19] and 14.13 % for the data of Hamer & Wu [31]. Also presented

in Table 3.8, the MRD values for the data reported by Randall and Young 31, Harned and

Owen 33, Åkerlöf 34, and Åkerlöf and Teare 35 are 15.15, 11.01, 15.14, and 10.33 %,

respectively. Specifically, the model results are on the low side of the data in dilute region,

i.e., < 5 molal (i.e., < 15 wt. % HCl). This discrepancy is attributed to the assumption of

proton hydration with one water molecule (R1) and it can be improved if higher hydration

number is considered according to an earlier work on hydration chemistry [86]. Figure 3.8

shows the model results with consideration of hydration of hydronium ion (R2) and it


77

significantly improves the match between the model results and the experimental data.

Here the binary interaction parameters for both the H2O:(H5O2+-Clˉ) pair and the

HCl:(H5O2+-Clˉ) pair are fixed at 8 and -4.5 for molecule-electrolyte and electrolyte-

molecule interactions, respectively. Additionally, the chemical equilibrium constants at

298.15 K (in terms of 𝑙𝑙𝑙𝑙 𝐾𝐾) for the unsymmetric reference state are regressed to be 9.40

and -1.01 for the reactions R1 and R2, respectively.

Figure 3.8. Model results for molality scale mean ionic activity coefficient ( ) at 298.15 K and 100 kPa, compared with experimental data ( ) 36a and ( ) 22a. Also shown, model results considering a second hydration ( ). a Not used in regression.

Figure 3.9 shows a comparison between the model results for water activity at 298.15

K and the experimental data in the concentration range of 0 to 40 wt. % HCl. The water

activity data are calculated from the osmotic coefficient data of Hamer and Wu 36 and the

vapor pressure data of Storonkin and Susarev 16, Haase et al. 18, and Vega and Vera 22. The

dashed line represents the model results considering the hydration of hydronium (R2) and,


78

as expected, captures the trend of experimental data more favorably. Furthermore, the

comparison between the experimental data of osmotic coefficient from Hamer and Wu 36

and the model results at 298.15 K and 100 kPa is shown in Figure 3.10. The MRD value

is 6.12 %. Consideration of R2 removes water from the solution via hydration, decreases

the water activity, increases the calculated osmotic coefficients, and improves the match

with the experimental data.

Figure 3.9. Model results for water activity ( ) at 298.15 K and 100 kPa, compared with experimental data ( ) 16a, ( ) 18, ( ) 22a, and ( ) 36a. Also shown, model results considering a second hydration ( ). a Not used in regression.


79

Figure 3.10. Model results for osmotic coefficient ( ) at 298.15 K and 100 kPa, compared with experimental data ( ) 36a. Also shown, model results considering a second hydration ( ). a Not used in regression.

Figure 3.11 shows the model results for excess enthalpy using the unsymmetric

reference state versus HCl concentration at 298.15 K and 100 kPa. The model results show

an excellent match with the experimental data of excess enthalpy calculated from the

reported enthalpy of formation (∆𝐻𝐻𝑓𝑓°) data by Wagman et al. 37. The mean absolute

difference between the data and the model results is 0.457 kJ/mol. The model results

deviate more significantly from the experimental data when the HCl concentration exceeds

45 wt. % at which the boiling point limiting line (dashed line) crosses the excess enthalpy

line. Here the boiling point limiting line represents the excess enthalpy of the aqueous HCl

solution at its boiling temperature of 100 kPa. When the excess enthalpy line at 298.15 K

crosses with the boiling point limiting line, the binary solution is at its boiling point of

298.15 K and further increase in the HCl concentration reduces the boiling point below


80

298.15 K, resulting in vaporization of HCl. In other words, above the boiling point limiting

line, the binary solution does not exist as a single liquid phase at 298.15 K and 100 kPa. It

suggests the data of Wagman et al. 37 from 45 to 60 HCl wt. % might have been taken at

pressures greater than 100 kPa to maintain the solution as liquid. It is noteworthy that the

mean absolute difference reduces to merely 0.054 kJ/mol if the data beyond 45 wt. % are

not considered.

Figure 3.11. Model results for excess enthalpy (kJ/mol) with unsymmetric reference state ( ) at 298.15 K and 100 kPa, compared with experimental data ( ) 37. Also shown, the boiling-point limit ( ) and the extended trend for excess enthalpy ( ).

Figures 3.12 and 3.13 show the liquid molar heat capacity versus HCl concentration in

dilute (< 4 wt. % HCl) and concentrated (10 to 35 wt. % HCl) regions, respectively. Figure

3.12 shows an excellent match between the model results and the experimental data 38-40

for the temperature range of 298.15 to 412.61 K and HCl concentrations below 4 wt. %.


81

The MRD is 0.09 % for the Allred and Woolley data 38, 0.13 % for the Pogue and Atkinson

data 39, and 0.11 % for the Tremanie et al. data 40. The MRDs of other data sets are 0.16 %

for Fortier et al. 41 at 298.15 K, 0.04 % for Saluja et al. 42 in the temperature range of 323.15

to 373.15 K, and 0.36 % for Wicke et al. 43 in the temperature range of 293.15 to 403.15

K. Interestingly, when compared against the high temperature heat capacity data of

Sharygin and Wood 5 in the temperature range of 323 to 623 K at the pressure of 28 MPa,

the MRD is calculated to be 4.01 %. The MRD drops to 1.40 % if the data at 623 K are not

considered. Note that the Sharygin and Wood data covers HCl concentration up to 18 wt

%.

Figure 3.12. Model results for liquid molar heat capacity of dilute solutions (kJ/kmol.K) (solid lines) and experimental data (symbols): ( ), ( )38, ( ) 39a, and ( ) 40a at 298.15 K; ( ) and ( ) 40a at 349.43 K; ( ) and ( ) 40a at 375.75 K; ( ) and ( ) 40a at 398.23 K; ( ) and ( ) 40a at 412.61 K. a Not used in regression.


82

Figure 3.13. Model results for liquid molar heat capacity (kJ/kmol.K) (solid lines) compared with experimental data (symbols) 13 and smoothed curves (dashed lines) 44: (), ( ), and ( ) at 273.15 K; ( ), ( ), and ( ) at 293.15 K; ( ) and ( ) at 313.15 K; ( ), ( ), and ( ) at 333.15 K.

To further validate the model, the liquid molar heat capacity of the HCl-H2O binary is

generated for the concentrated region (10 to 35 wt. % HCl) for which the experimental data

are relatively scarce. Figure 3.13 shows a satisfactory match between the experimental

data of Perry’s Handbook 13 and the model results with an average MRD value of 2.88 %

at the four temperatures of 273.15, 293.15, 313.15 K, and 333.15 K. Furthermore, the

smoothed curves of heat capacity 44 are also compared satisfactorily with the model results

with an average MRD value of 2.33 % at the three temperatures of 273.15, 293.15, and

333.15 K.

Figure 3.14 shows the Merkel enthalpy-concentration chart for the HCl-H2O binary.

It is presented in English units, as is the convention for the Merkel chart. The reference

states for pure HCl and water are at 32 and 68 °F, respectively. The figure shows the model


83

results for solution enthalpy at temperatures ranging from 32 to 212 °F and at

concentrations up to the boiling point limit. The Merkel chart should be very useful in

calculating the heat required or released during heating or mixing with the HCl-H2O binary

system.

Figure 3.14. Merkel enthalpy-concentration chart (Btu/lb) (solid lines) and extended trend lines (dotted lines): ( ) and ( ) at 32 °F; ( ) and ( ) at 68 °F; ( ) and ( ) at 104 °F; ( ) and ( ) at 140 °F; ( ) and ( ) at 176 °F; ( ) and ( ) at 212 °F. Also shown, the boiling-point limit (dashed line).

3.6. Conclusions

A comprehensive thermodynamic model is developed for the HCl-H2O binary with the

symmetric electrolyte NRTL model. Parameterized by regression of selected vapor-liquid

equilibrium, liquid-liquid equilibrium, molar excess enthalpy, and molar heat capacity

data, the model parameters include the temperature-dependent binary interaction


84

parameters of the H2O:(H3O+-Cl–), HCl:(H3O+-Cl–), and H2O:HCl pairs, as well as the

chemical equilibrium constant parameters for partial dissociation of HCl. The model

accurately calculates all phase equilibrium, thermodynamic, and calorimetric properties for

the HCl-H2O binary over the whole concentration range from pure H2O to pure HCl and at

temperatures from 273 to about 400 K. The model should be very useful in industrial

process simulations and heat and mass balance calculations involving the HCl-H2O binary.

The model can be further extended for mixed acids and mixed solvent electrolyte systems.

3.7. Acknowledgements


Distinguished Engineering Chair Professorship in Sustainable Energy sponsored by the J.F

Maddox Foundation.


85

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89. Yan Y, Chen C-C. Thermodynamic representation of the NaCl + Na2SO4 + H2O system with electrolyte NRTL model. Fluid Phase Equilibria. 2011;306:149-161.


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CHAPTER 4. BRIDGING TWO-LIQUID THEORY WITH MOLECULAR

SIMULATIONS FOR ELECTROLYTES: AN INVESTIGATION OF AQUOUES NACL SOLUTION3

4.1. Abstract

We re-examine the theoretical framework of electrolyte Non-Random Two-Liquid

model and, based on the two-fluid theory, formulate the binary interaction parameters as

functions of species diameter (𝜎𝜎), effective interaction strength (𝜖𝜖), the domain radius

around the center species (𝑅𝑅), and the non-randomness factor (𝛼𝛼). We show that these

quantities can be directly obtained from molecular simulations using aqueous NaCl as the

model system. The binary interaction parameters determined from the simulations are

consistent with those obtained from regression of experimental data. Our work provides a

molecular interpretation of the classical thermodynamic model and shows a way to predict

from molecular simulations the binary interaction parameters for use in process industries.

3 This chapter is reproduced from the paper published as: Saravi SH, Ravichandran A, Khare R, Chen C-C. Bridging Two-Liquid Theory with Molecular Simulations for Electrolytes: An Investigation of Aqueous NaCl Solution, AIChE Journal. 2019;65.4:1315-1324


94

4.2. Introduction

Aqueous electrolyte solutions are widely used in many industrial 1-3, environmental

4,5, geological 6,7, and pharmaceutical processes 8. The availability of reliable models for

phase equilibria calculations is the key challenge in design and optimization of these

processes 3,9. Extensive modeling efforts have been reported in the literature to calculate

phase equilibria and thermodynamic properties of aqueous electrolytes 10,11. Most of these

studies aim at quantifying the mean ionic activity coefficient (𝛾𝛾±), as it is the unique

thermodynamic property of aqueous electrolyte solutions 10,12,13.

Aqueous electrolyte solution models can be broadly categorized into correlative and

predictive models. Correlative models often relate the liquid structure to the excess Gibbs

energy of a system with a set of adjustable parameters as input. The adjustable parameters

are calculated by fitting the experimental data (including those of phase equilibria,

calorimetric properties, and speciation) to the model 10. The predictive models, on the other

hand, employ molecular simulation techniques to obtain the liquid structure and free

energy, without introducing any adjustable parameter. The quantities thus obtained are

used to predict the thermodynamic properties of the electrolyte solutions 14,15.

Correlative classical thermodynamic models are extensively used in the process

industry owing to their straightforward applicability covering a wide range of electrolyte

solutions and process conditions. Some examples of these models include, Pitzer 16,

eUNIQUAC 17, ePC-SAFT 18, and the electrolyte NRTL (eNRTL) models 12,19-21. Among

these, the Pitzer model and the eNRTL model are the two most widely applied

thermodynamic models due to their mathematical simplicity and application versatility.

The underlying concept behind these models is that the excess Gibbs free energy of an


95

electrolyte solution can be expressed as a combination of two contributions: the long-range

electrostatic interactions and the short-range local interactions. These models depend on

experimental data to obtain the adjustable parameters that are further used to calculate the

excess Gibbs free energy and other derivative thermodynamic properties. Determination

of the interaction parameters of the models from regression is not always feasible due to

the limited availability of the experimental data, the need for substantial experience in data

treatment, and the necessity of properly defining the optimization problem.

On the other hand, molecular simulations circumvent the problems associated with the

classical thermodynamic models. Several of these models have previously been shown to

predict the chemical potential of aqueous electrolyte solutions 15,22-27. Though these models

have been shown to be predictive, they are often restricted by the necessity to implement

advanced simulation techniques to calculate the free energy of the system. Implementing

such techniques can be computationally expensive thereby creating a bottleneck in directly

applying molecular simulations to industrial process design. Hence it is desirable to

formulate a hybrid methodology that can utilize the predictive capability of molecular

simulations and the rapid applicability of the classical thermodynamic models. Developing

such a framework for electrolyte systems is the objective of this work.

The molecular interpretation of the classical thermodynamic models can be achieved

through the interaction parameters which quantify the strength and nature of intermolecular

forces. There have been a few attempts for finding the binary interaction parameters of

thermodynamic models for nonelectrolyte systems from molecular simulations 28-31.

Recently, a novel methodology was proposed for predicting the NRTL binary interaction

parameters from molecular dynamics (MD) simulations 29. This technique utilizes the


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framework of two-fluid theory 32 to relate the NRTL binary interaction parameters to

molecular size and interaction strength, both of which were obtained from simulations. It

was also shown that the binary interaction parameters obtained using this approach could

predict the phase equilibria behavior and the thermophysical properties of organic binary

mixtures. In this work, we extend the previously proposed technique 29 to aqueous uni-

univalent electrolyte solutions. Such an extension is non-trivial and requires the

interpretation of molecular parameters that is consistent with the theoretical framework of

the classical thermodynamic model.

We focus on the eNRTL model as it has been extensively applied to predict the phase

equilibria and thermodynamic properties of aqueous, non-aqueous, and mixed-solvent

electrolytes 12,19-21. The model’s sound theoretical background and mathematical

simplicity offers a versatile and comprehensive framework for application in process

modeling of electrolytes. Furthermore, it has been recently shown that the prediction of

mean ionic activity coefficient of aqueous NaCl solutions by the eNRTL model is

consistent with experimental data and MD simulations as opposed to the Pitzer model that

overestimates the mean ionic activity coefficient at higher salt concentrations 13. The

eNRTL model requires two adjustable binary interaction parameters per molecule-

electrolyte pair (𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 and 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎), that quantify the nature of the interaction between solvent

molecules and electrolyte species. The binary interaction parameters are commonly

determined from regression of experimental data. The model also includes an additional

parameter, the non-randomness factor, 𝛼𝛼, which is treated as an empirical constant

typically fixed at a value of 0.2.


97

In this study, we re-examine the eNRTL model to quantify the binary interaction

parameters (𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 and 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎) by combining the statistical mechanics framework of two-

liquid theory for electrolytes with MD simulations. Furthermore, the non-randomness

factor, 𝛼𝛼, is directly obtained from simulations instead of treating it as an empirical

constant. The 𝜏𝜏 parameters are formulated as a function of species diameter (𝜎𝜎), effective

interaction strength (𝜖𝜖), and the radius of the neighbor domain (𝑅𝑅), all of which are

calculated from the MD simulations. Aqueous NaCl solution is selected as the model

system due to the availability of proven force field parameters [6] and its industrial

importance. Furthermore, we examine several inherent assumptions made by the model

regarding the nature of molecular interactions. These assumptions include the

concentration independence of the interaction parameters and equal magnitude of the

effective interaction between the cation-molecule and the anion-molecule pairs 19.

The rest of this chapter is organized as follows: Theoretical Background section covers

the fundamental concepts behind the two-liquid theory for electrolytes and the

development of our approach. Calculation Methodology and Molecular Simulations

sections detail the procedure followed to obtain the binary interaction parameters. The

significance of the approach and the interpretation of the results are provided in the Results

and Discussions section which is then followed by Conclusions.

4.3. Theoretical Background

Two-Liquid Theory for Electrolytes The short-range interactions in the eNRTL model are described using the non-random

two-liquid (NRTL) equation of Renon and Prausnitz 33, which is in turn based on the

original two-liquid theory of Scott 32. The long-range electrostatic interactions are


98

described by the Pitzer–Debye–Hückel limiting law 16. While certain theoretical aspects

of the eNRTL model are described below, we refer the reader to the original literature for

an elaborate discussion 12,19-21.

Following the work of Chen et al. 19, the concept of local composition for electrolyte

solutions can be described as follows. For a binary electrolyte solution consisting of a

solvent molecule, 𝑚𝑚, and a completely dissociated solute, 𝑐𝑐𝑎𝑎, three types of molecular

domains are possible. When the solute dissociates to cation, 𝑐𝑐, and anion, 𝑎𝑎, the three

different types of domains will have 𝑐𝑐, 𝑎𝑎, and 𝑚𝑚 at the center, respectively, surrounded by

species in such a way that the two assumptions of the eNRTL model, i.e., local

electroneutrality and like-ion repulsion are satisfied. While the former ensures local

electroneutrality (i.e., the number of oppositely charged ions in a molecule-centered

domain are equal), the latter asserts that no like-ion species are present in the first neighbor

shell of an ion-centered domain. The schematic of the domains is shown in Figure 4.1.

The local compositions are then expressed as follows 19:

𝑥𝑥𝑗𝑗𝑖𝑖𝑥𝑥𝑘𝑘𝑖𝑖

= �𝑥𝑥𝑗𝑗𝑥𝑥𝑘𝑘� exp�−𝛼𝛼(𝑖𝑖)𝜏𝜏𝑗𝑗𝑖𝑖,𝑘𝑘𝑖𝑖� , 𝑖𝑖, 𝑗𝑗,𝑘𝑘 ∈ {𝑐𝑐, 𝑎𝑎,𝑚𝑚} (4.1)

𝜏𝜏𝑗𝑗𝑖𝑖,𝑘𝑘𝑖𝑖 = (𝑔𝑔𝑗𝑗𝑖𝑖 − 𝑔𝑔𝑘𝑘𝑖𝑖)/𝑅𝑅𝑇𝑇 (4.2)

where 𝑥𝑥𝑗𝑗𝑖𝑖 and 𝑥𝑥𝑘𝑘𝑖𝑖 are the local compositions of species 𝑗𝑗 and 𝑘𝑘 around the center

species 𝑖𝑖, respectively; 𝑥𝑥𝑗𝑗 and 𝑥𝑥𝑘𝑘 are the bulk mole fractions of j and k, respectively; 𝛼𝛼(𝑖𝑖)

is the non-randomness factor in an 𝑖𝑖-centered domain. 𝜏𝜏𝑗𝑗𝑖𝑖,𝑘𝑘𝑖𝑖 is the binary interaction

parameter of a domain with species 𝑗𝑗 and 𝑘𝑘 around 𝑖𝑖, representing the effective interaction

strength necessary for forming such a domain; 𝑔𝑔𝑗𝑗𝑖𝑖 and 𝑔𝑔𝑘𝑘𝑖𝑖 are the interaction energies

between species 𝑗𝑗 and 𝑖𝑖, and species 𝑘𝑘 and 𝑖𝑖, respectively. The summation of the local


99

compositions in each domain along with the constraints, 𝑥𝑥𝐿𝐿𝐿𝐿 = 𝑥𝑥𝑎𝑎𝑎𝑎 = 0 (to satisfy the like

ion repulsion assumption of the model), lead to the equations below.

𝑥𝑥𝐿𝐿𝑚𝑚 + 𝑥𝑥𝑎𝑎𝑚𝑚 + 𝑥𝑥𝑚𝑚𝑚𝑚 = 1 (4.3)

𝑥𝑥𝑚𝑚𝐿𝐿 + 𝑥𝑥𝑎𝑎𝐿𝐿 = 1 (4.4)

𝑥𝑥𝑚𝑚𝑎𝑎 + 𝑥𝑥𝐿𝐿𝑎𝑎 = 1 (4.5)

Note that the binary interaction parameters for the two types of ions are presumed to

be equal in each domain i.e.,

𝜏𝜏𝐿𝐿𝑚𝑚 = 𝜏𝜏𝑎𝑎𝑚𝑚 = 𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 (4.6)

𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 = 𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎 = 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 (4.7)

To calculate the interaction parameters from molecular simulations, it is necessary to

develop a framework that relates these parameters to molecular quantities. For that

purpose, we extend the approach of Ravichandran et al. 29 to electrolytes. Specifically,

three different fluid domains are separately constructed by solving the combinatorial

problem of forming a molecular domain around a center species. Preferential local

interactions between the species are also considered in this construction. For consistency,

terminologies of Chen et al. 19 and Ravichandran et al. 29 are closely followed.


100

Figure 4.1. Schematic describing the three possible local molecular domains considered by the eNRTL model

The Cation-Centered Domain Consider the center cation locally coordinated by 𝑧𝑧(𝐿𝐿) number of species within the

first neighbor shell of radius 𝑅𝑅(𝐿𝐿), out of which 𝑙𝑙 is the number of anions and (𝑧𝑧(𝐿𝐿) − 𝑙𝑙) is

the number of molecules (solvent). In what follows, the total coordination number in any

domain with species 𝑖𝑖 as center (𝑧𝑧(𝑖𝑖)), is denoted as 𝑧𝑧 for the sake of simplicity in the text.

Following the like-ion repulsion assumption of the eNRTL model, we reiterate that the

cation-centered domain only includes anions and solvent molecules. The probability of

arranging 𝑙𝑙 anions and (𝑧𝑧 − 𝑙𝑙) molecules around a center cation (𝑓𝑓𝜕𝜕,𝑧𝑧𝐿𝐿 ) is as follows 29:

𝑓𝑓𝜕𝜕,𝑧𝑧𝐿𝐿 = 𝐶𝐶𝜕𝜕𝑧𝑧 𝑝𝑝𝑎𝑎

𝜕𝜕𝑝𝑝𝑚𝑚𝑧𝑧−𝜕𝜕 (4.8)

Here 𝐶𝐶𝜕𝜕𝑧𝑧 is the combinatorial factor; 𝑝𝑝𝑎𝑎 𝜕𝜕 is the probability of choosing 𝑙𝑙 number of

anions from an infinite pool of anions, cations, and molecules; and 𝑝𝑝𝑚𝑚𝑧𝑧−𝜕𝜕 is the

corresponding probability of choosing (𝑧𝑧 − 𝑙𝑙) number of molecules. Note that Eq. 4.8

represents the probability of forming a cation-centered domain without considering


101

preferential interaction between the species. To account for the preferential interactions,

the above random distribution is weighted by the Boltzmann factor with a potential

function of the square-well form (as suggested by the two-fluid theory), 𝜑𝜑𝜕𝜕,𝑧𝑧𝐿𝐿 34:

𝜑𝜑𝜕𝜕,𝑧𝑧𝐿𝐿 =

−(𝑙𝑙𝜖𝜖𝑎𝑎𝐿𝐿𝜎𝜎𝑎𝑎𝐿𝐿3 + (𝑧𝑧 − 𝑙𝑙)𝜖𝜖𝑚𝑚𝐿𝐿𝜎𝜎𝑚𝑚𝐿𝐿3 )

(𝑅𝑅(𝐿𝐿))3 (4.9)

where 𝜖𝜖𝑎𝑎𝐿𝐿 and 𝜖𝜖𝑚𝑚𝐿𝐿 are the interaction strengths between anion-cation, and molecule-cation

pairs, respectively. 𝜎𝜎𝑎𝑎𝐿𝐿 and 𝜎𝜎𝑚𝑚𝐿𝐿 denote the center to center distances between anion-cation,

and molecule-cation pairs, respectively while 𝑅𝑅(𝐿𝐿) is the radius of the first neighbor shell

in a cation-centered domain. Weighting the random probability distribution given in Eq.

4.8 by the Boltzmann factor, we arrive at the probability of forming a cation-centered

domain 𝑓𝑓𝜕𝜕,𝑧𝑧𝐿𝐿� , that accounts for the interaction between the molecular species:

𝑓𝑓𝜕𝜕,𝑧𝑧𝐿𝐿� =

𝐶𝐶𝜕𝜕𝑧𝑧 𝑝𝑝𝑎𝑎 𝜕𝜕𝑝𝑝𝑚𝑚𝑧𝑧−𝜕𝜕𝑒𝑒−𝛽𝛽𝜑𝜑𝑛𝑛,𝑧𝑧

𝑐𝑐

𝛺𝛺(𝐿𝐿) (4.10)

where 𝛺𝛺(𝐿𝐿) is the normalization factor (equivalent to the local partition function) which is

the summation over all possible configurations of the domain that varies in the number of

anions and molecules surrounding the center cation.

𝛺𝛺(𝐿𝐿) = �𝐶𝐶𝜕𝜕𝑧𝑧 𝑝𝑝𝑎𝑎

𝜕𝜕𝑝𝑝𝑚𝑚𝑧𝑧−𝜕𝜕𝑒𝑒−𝛽𝛽𝜑𝜑𝑛𝑛,𝑧𝑧𝑐𝑐

𝑧𝑧

𝜕𝜕=0

(4.11)

Here 𝛽𝛽 is 1/𝑘𝑘𝑇𝑇 where 𝑘𝑘 is the Boltzmann constant.

Substituting Eq. 4.9 into Eq. 4.10, the probability function of the domain can be

rearranged as:

𝑓𝑓𝜕𝜕,𝑧𝑧𝐿𝐿� = 𝐶𝐶𝑛𝑛𝑧𝑧

𝛺𝛺(𝑐𝑐) 𝑝𝑝𝑎𝑎 𝜕𝜕𝑒𝑒

𝛽𝛽𝑛𝑛𝜖𝜖𝑎𝑎𝑐𝑐𝜎𝜎𝑎𝑎𝑐𝑐3

(𝑅𝑅(𝑐𝑐))3 .𝑝𝑝𝑚𝑚𝑧𝑧−𝜕𝜕 𝑒𝑒𝛽𝛽(𝑧𝑧−𝑛𝑛) 𝜖𝜖𝑚𝑚𝑐𝑐𝜎𝜎𝑚𝑚𝑐𝑐

3

(𝑅𝑅(𝑐𝑐))3 (4.12)

Defining two terms, 𝑝𝑝𝑎𝑎 ′ and 𝑝𝑝𝑚𝑚

′ by Eqs. 4.13 and 4.14, Eq. 4.12 can be rewritten as Eq.

4.15.


102

𝑝𝑝𝑎𝑎 ′ = 𝑝𝑝𝑎𝑎𝑒𝑒

𝛽𝛽𝜖𝜖𝑎𝑎𝑐𝑐𝜎𝜎𝑎𝑎𝑐𝑐3(𝐿𝐿(𝑐𝑐))3 (4.13)

𝑝𝑝𝑚𝑚 ′ = 𝑝𝑝𝑚𝑚𝑒𝑒

𝛽𝛽𝜖𝜖𝑚𝑚𝑐𝑐𝜎𝜎𝑚𝑚𝑐𝑐3

(𝐿𝐿(𝑐𝑐))3 (4.14)


𝐶𝐶𝜕𝜕𝑧𝑧

𝛺𝛺(𝐿𝐿) (𝑝𝑝𝑎𝑎′ )𝜕𝜕. (𝑝𝑝𝑚𝑚′ )𝑧𝑧−𝜕𝜕 (4.15)

Taking the ratio of 𝑝𝑝𝑚𝑚 ′ and 𝑝𝑝𝑎𝑎

′ results in following relation:

𝑝𝑝𝑚𝑚′

𝑝𝑝𝑎𝑎′=𝑝𝑝𝑚𝑚𝑝𝑝𝑎𝑎

𝑒𝑒−𝛽𝛽�𝜖𝜖𝑎𝑎𝑐𝑐𝜎𝜎𝑎𝑎𝑐𝑐

3 −𝜖𝜖𝑚𝑚𝑐𝑐𝜎𝜎𝑚𝑚𝑐𝑐3

(𝐿𝐿(𝑐𝑐))3� (4.16)

This equation is equivalent to the eNRTL equation for a cation-centered domain (a variant

of Eq. 4.1).

𝑒𝑒𝑚𝑚𝑐𝑐𝑒𝑒𝑎𝑎𝑐𝑐

= 𝑒𝑒𝑚𝑚𝑒𝑒𝑎𝑎𝑒𝑒−𝛼𝛼(𝑐𝑐)𝜏𝜏𝑚𝑚𝑐𝑐,𝑎𝑎𝑐𝑐 (4.17)

Here 𝛼𝛼(𝐿𝐿) is the non-randomness factor of a cation-centered domain. The terms 𝑝𝑝𝑚𝑚′ and

𝑝𝑝𝑎𝑎′ in Eq. 4.16 are probabilities of finding a molecule and an anion around a center cation

which are equivalent to the definition of the local mole fractions, 𝑥𝑥𝑚𝑚𝐿𝐿 and 𝑥𝑥𝑎𝑎𝐿𝐿, respectively.

Note that the quantities 𝑝𝑝𝑚𝑚 and 𝑝𝑝𝑎𝑎, represent the bulk mole fractions, 𝑥𝑥𝑚𝑚 and 𝑥𝑥𝑎𝑎,

respectively.

Comparing Eq. 4.16 and Eq. 4.17, the effective binary interaction parameter of a center

cation domain, 𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿, can be expressed as:

𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 = 𝛽𝛽𝛼𝛼(𝑐𝑐)

𝜖𝜖𝑎𝑎𝑐𝑐𝜎𝜎𝑎𝑎𝑐𝑐3 −𝜖𝜖𝑚𝑚𝑐𝑐𝜎𝜎𝑚𝑚𝑐𝑐3

(𝑅𝑅(𝑐𝑐))3 (4.18)

The Anion-Centered Domain Following a similar argument as described in the previous section, an expression for

the binary interaction parameter, 𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎, in an anion-centered domain (surrounded by

molecules and cations), can be derived as:


103

𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎 = 𝛽𝛽𝛼𝛼(𝑎𝑎)

𝜖𝜖𝑐𝑐𝑎𝑎𝜎𝜎𝑐𝑐𝑎𝑎3 −𝜖𝜖𝑚𝑚𝑎𝑎𝜎𝜎𝑚𝑚𝑎𝑎3

(𝑅𝑅(𝑎𝑎))3 (4.19)

where 𝛼𝛼(𝑎𝑎) is the non-randomness factor of an anion-centered domain.

The Molecule-Centered Domain Similar to the construction of cation and anion-centered domains, we construct a

molecule-centered domain that is locally coordinated by 𝑧𝑧 species. Out of the 𝑧𝑧 species, 𝑘𝑘

is the number of surrounding molecules and (𝑧𝑧 − 𝑘𝑘) is the number of neighbor ions,

equally distributed as cations and anions. Hence, the number of surrounding anions and

cations is (𝑧𝑧 − 𝑘𝑘)/2, respectively. Note that such a construction satisfies the local

electroneutrality constraint of the eNRTL model. As in the previous cases, the non-random

probability of arranging molecules and equal number of cations and anions around a center

molecule can be formulated as:

𝑓𝑓𝑚𝑚𝑘𝑘�= 𝐶𝐶𝑘𝑘

𝑧𝑧

𝛺𝛺(𝑚𝑚) �𝑝𝑝𝑎𝑎𝑧𝑧−𝑘𝑘2 𝑒𝑒

𝛽𝛽�𝑧𝑧−𝑘𝑘2 𝜖𝜖𝑎𝑎𝑚𝑚𝜎𝜎𝑎𝑎𝑚𝑚

3

(𝑅𝑅(𝑚𝑚))3�� 𝑝𝑝𝐿𝐿

𝑧𝑧−𝑘𝑘2 𝑒𝑒

𝛽𝛽�𝑧𝑧−𝑘𝑘2 𝜖𝜖𝑐𝑐𝑚𝑚𝜎𝜎𝑐𝑐𝑚𝑚

3

(𝑅𝑅(𝑚𝑚))3�� 𝑝𝑝𝑚𝑚𝑘𝑘 𝑒𝑒

𝛽𝛽�𝑘𝑘 𝜖𝜖𝑚𝑚𝑚𝑚𝜎𝜎𝑚𝑚𝑚𝑚3

(𝑅𝑅(𝑚𝑚))3�� (4.20)

where 𝛺𝛺(𝑚𝑚) is the normalization factor with a similar definition to 𝛺𝛺(𝐿𝐿) defined in Eq. 4.11.

Also, similar to the previous cases, 𝐶𝐶𝑘𝑘𝑧𝑧 represents the combinatorial factor of arranging

molecules, cations and anions in a domain with molecule as the center. By rearranging Eq.

4.20 and defining terms 𝑝𝑝𝑎𝑎′ , 𝑝𝑝𝐿𝐿,′ and 𝑝𝑝𝑚𝑚′ as before, we arrive at Eq. 4.21.

𝑓𝑓𝑚𝑚𝑘𝑘� = 𝐶𝐶𝑘𝑘𝑧𝑧

𝛺𝛺(𝑚𝑚) (𝑝𝑝𝑎𝑎′ )𝑧𝑧−𝑘𝑘2 (𝑝𝑝𝐿𝐿′)

𝑧𝑧−𝑘𝑘2 (𝑝𝑝𝑚𝑚′ )𝑘𝑘 (4.21)

where:

𝑝𝑝𝑎𝑎′ = 𝑝𝑝𝑎𝑎 𝑒𝑒

𝛽𝛽𝜖𝜖𝑎𝑎𝑚𝑚𝜎𝜎𝑎𝑎𝑚𝑚3(𝐿𝐿(𝑚𝑚))3 (4.22)

𝑝𝑝𝐿𝐿′ = 𝑝𝑝𝐿𝐿 𝑒𝑒

𝛽𝛽𝜖𝜖𝑐𝑐𝑚𝑚𝜎𝜎𝑐𝑐𝑚𝑚3(𝐿𝐿(𝑚𝑚))3

(4.23)


104

𝑝𝑝𝑚𝑚′ = 𝑝𝑝𝑚𝑚 𝑒𝑒

𝛽𝛽𝜖𝜖𝑚𝑚𝑚𝑚𝜎𝜎𝑚𝑚𝑚𝑚3

(𝐿𝐿(𝑚𝑚))3 (4.24)

By dividing Eq. 4.22 and Eq. 4.23 by Eq. 4.24, the two following relations are obtained:

𝑝𝑝𝑎𝑎′

𝑝𝑝𝑚𝑚′=𝑝𝑝𝑎𝑎𝑝𝑝𝑚𝑚

𝑒𝑒−𝛽𝛽�𝜖𝜖𝑚𝑚𝑚𝑚𝜎𝜎𝑚𝑚𝑚𝑚

3 −𝜖𝜖𝑎𝑎𝑚𝑚𝜎𝜎𝑎𝑎𝑚𝑚3(𝐿𝐿(𝑚𝑚))3

� (4.25)

𝑝𝑝𝐿𝐿′

𝑝𝑝𝑚𝑚′=𝑝𝑝𝐿𝐿𝑝𝑝𝑚𝑚


3 −𝜖𝜖𝑐𝑐𝑚𝑚𝜎𝜎𝑐𝑐𝑚𝑚3(𝐿𝐿(𝑚𝑚))3

�

(4.26)

Similar to our previous arguments, these equations are equivalent to Eq. 4.27 and Eq.

4.28, two other variants of Eq. 4.1.

𝑒𝑒𝑎𝑎𝑚𝑚𝑒𝑒𝑚𝑚𝑚𝑚

= 𝑒𝑒𝑎𝑎𝑒𝑒𝑚𝑚𝑒𝑒−𝛼𝛼(𝑚𝑚)𝜏𝜏𝑎𝑎𝑚𝑚 (4.27)

𝑒𝑒𝑐𝑐𝑚𝑚𝑒𝑒𝑚𝑚𝑚𝑚

= 𝑒𝑒𝑐𝑐𝑒𝑒𝑚𝑚𝑒𝑒−𝛼𝛼(𝑚𝑚)𝜏𝜏𝑐𝑐𝑚𝑚 (4.28)

where 𝛼𝛼(𝑚𝑚) is the non-randomness factor of a molecule-centered domain. Thus, the binary

interaction parameters for anion and cation around a center molecule are as follows:

𝜏𝜏𝑎𝑎𝑚𝑚 = 𝛽𝛽𝛼𝛼

𝜖𝜖𝑚𝑚𝑚𝑚𝜎𝜎𝑚𝑚𝑚𝑚3 −𝜖𝜖𝑎𝑎𝑚𝑚𝜎𝜎𝑎𝑎𝑚𝑚3

(𝐿𝐿(𝑚𝑚))3 (4.29)

𝜏𝜏𝐿𝐿𝑚𝑚 = 𝛽𝛽𝛼𝛼

𝜖𝜖𝑚𝑚𝑚𝑚𝜎𝜎𝑚𝑚𝑚𝑚3 −𝜖𝜖𝑐𝑐𝑚𝑚𝜎𝜎𝑐𝑐𝑚𝑚3

(𝐿𝐿(𝑚𝑚))3 (4.30)

As mentioned earlier, the eNRTL model assumes 𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 = 𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎 and 𝜏𝜏𝐿𝐿𝑚𝑚 = 𝜏𝜏𝑎𝑎𝑚𝑚 19.

However, in this work we make no such assumptions and evaluate the interaction

parameters independently so that the assumptions of the eNRTL model can be validated.

4.4. Calculation Methodology

Quantifying the interaction parameters requires the calculation of effective species

diameters, i.e., 𝜎𝜎𝑎𝑎𝐿𝐿(= 𝜎𝜎𝐿𝐿𝑎𝑎), 𝜎𝜎𝑚𝑚𝐿𝐿(= 𝜎𝜎𝐿𝐿𝑚𝑚), 𝜎𝜎𝑚𝑚𝑎𝑎(= 𝜎𝜎𝑎𝑎𝑚𝑚), and 𝜎𝜎𝑚𝑚𝑚𝑚, the interaction strengths,

i.e., 𝜖𝜖𝑎𝑎𝐿𝐿(= 𝜖𝜖𝐿𝐿𝑎𝑎), 𝜖𝜖𝑚𝑚𝐿𝐿(= 𝜖𝜖𝐿𝐿𝑚𝑚), 𝜖𝜖𝑚𝑚𝑎𝑎(= 𝜖𝜖𝑎𝑎𝑚𝑚), and 𝜖𝜖𝑚𝑚𝑚𝑚, the first neighbor shell radii of


105

different domains, i.e., 𝑅𝑅(𝐿𝐿), 𝑅𝑅(𝑎𝑎), and 𝑅𝑅(𝑚𝑚), and the non-randomness factors, 𝛼𝛼(𝐿𝐿), 𝛼𝛼(𝑎𝑎),

and 𝛼𝛼(𝑚𝑚). We obtain each of these quantities directly from MD simulations.

The 𝜎𝜎 parameters are obtained as the position of first peaks from the radial distribution

functions (RDFs) between the pairs of interest. The molecular and ionic interaction

strength, 𝜖𝜖, are obtained from the potential of mean force (PMF) between the pairs of

interest, i.e., depths of the first minima of the PMFs between cation-anion, molecule-cation,

molecule-anion, and molecule-molecule are taken as the interaction strengths, respectively.

Such calculations provide a better representation for the net interactions in the liquid phase

rather than considering the bare pair-wise interactions 29.

The size of the molecular domain 𝑅𝑅, is usually obtained as the size of first neighbor

shell from MD simulations. While this approach has been shown to be appropriate for

organic binary mixtures 29, such a definition is not consistent with the underlying

assumptions of the eNRTL model: local electroneutrality and like-ion repulsion. Hence,

to predict the interaction parameters of the eNRTL model, we propose a new approach to

define the size of the molecular domains such that the assumptions of the model are

satisfied. For this purpose, the local compositions in domains of different radii are obtained

by integrating the RDF of different pairs involving the center species. In a molecule-

centered domain, 𝑅𝑅(𝑚𝑚) is defined as the shortest distance from the solvent molecule at

which the local compositions of anions and cations are equal. The construction as the one

described above satisfies the local electroneutrality condition. On the other hand, in cation-

centered and anion-centered domains, the size of the domains, 𝑅𝑅(𝐿𝐿) and 𝑅𝑅(𝑎𝑎) are defined as

the longest distance from the center ionic species until which no other ion with the same


106

charge as the center ion (i.e. like-ion) is present. Such a definition ensures consistency

with the like-ion repulsion assumption.

Finally, the non-randomness factor, 𝛼𝛼, is calculated from the knowledge of the local

coordination numbers. Renon and Prausnitz 33 pointed out that the non-randomness factor

is equivalent to the inverse of the coordination number (1/𝑧𝑧). This equivalency was

established by comparing the NRTL model relationship between local compositions and

bulk mole fractions with that of the Guggenheim’s quasichemical theory 35. We use this

interpretation to obtain 𝛼𝛼 directly from MD simulations. As the coordination number

around a center species can be different in cation-centered, anion-centered, and molecule-

centered domains due to the inherent differences in the local molecular structure, three

separate non-randomness factors, 𝛼𝛼(𝐿𝐿), 𝛼𝛼(𝑎𝑎), and 𝛼𝛼(𝑚𝑚) are obtained by integrating the

pertinent RDFs. Furthermore, we investigate the composition dependence of the 𝜏𝜏

parameters by performing simulations on aqueous solutions over a wide range of

concentrations. The molecular quantities of interest (described above) are obtained at each

concentration; these values are then used to examine the concentration dependence of the

binary interaction parameter.

4.5. Molecular Simulations

Simulations were carried out over wide range of concentrations from 0.5 to 16 molality

(mol/kg) of aqueous NaCl solution and at 298.15 K and 1 bar. Each simulation system

consisted of 1000 water molecules and varying number of ions as required to achieve the

desired concentration of the solution. Choosing simulations of small size facilitates rapid

prediction of the binary interaction parameters thereby enabling the use of our


107

methodology for industrial process design. However, to investigate the effect of finite

simulation system size on our calculations, a larger system (with ten times the number of

water molecules than the smaller system) was also simulated at the concentration of 4

mol/kg.

The LAMMPS package was used to perform all of the simulations 36. Initial

equilibration of the system was carried out at constant number of particles, volume, and

temperature conditions (canonical ensemble) which was followed by 10 ns long

equilibration and 40 ns long production runs in the isothermal-isobaric ensemble using the

Nośe-Hoover thermostat and barostat 37 for the smaller systems. On the other hand, the

larger system was simulated for a duration of 25 ns. The Lennard-Jones and Coulombic

interactions were truncated at distance of 14 Å. The long-range electrostatic interactions

were calculated using the particle-particle particle-mesh (PPPM) 38 method and tail

corrections were applied to correct for the long-range part of the Lennard-Jones

interactions. The SPC/E model 39 and the Kirkwood-Buff force field (KBFF) parameters

40,41 were employed to represent the water molecules and ions, respectively. This choice

of the force field was made based on its ability to predict the mean ionic activity

coefficients, as described in previous studies 13,26. The 40 ns long production runs were

divided into 10 blocks of 4 ns length, each of which was used to obtain the desired

quantities and to estimate the error in those quantities. Similarly, for the larger system, the

first 10 ns of the trajectory were discarded, and the remaining 15 ns were divided into 5

blocks of 3 ns duration each. The PMF for different interactions were calculated from the

RDFs using Eq. 4.31 42. Given that the correlation times for water molecules (determined

from the orientation correlation of the water OH bond vector), even at high salt


108

concentrations, are much shorter than 25 ps 13, our simulation trajectory includes many

such correlation times. Hence, our simulations sufficiently sample several configurations

of the aqueous system, enabling us to calculate the PMFs as per Eq. 4.31.

𝜙𝜙(𝑟𝑟) = −𝑅𝑅𝑇𝑇 ln𝑔𝑔(𝑟𝑟) (4.31)

4.6. Binary Interaction Parameters from Regression

The validity of the binary interaction parameters calculated from MD simulations is

tested by performing regression of experimental 43 and simulation 𝛾𝛾± data 13 (separately)

to independently determine the values of 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 and 𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚, and the associated uncertainties.

It is essential to quantify such uncertainties as the magnitude of the calculated mean ionic

activity coefficients is extremely sensitive to minor fluctuations in the 𝜏𝜏 values 12. First,

the experimental mean ionic activity coefficient data of Robinson and Stokes 44 at 298.15

K and 1 bar, in the concentration range of 0.1 to 6 mol/kg NaCl is correlated with the

eNRTL model by minimizing the residual root-mean-square error (RRMSE) as the

objective function. APSEN Properties package 45, version 8.8 was used for this purpose.

Furthermore, another set of values of the binary interaction parameters was also obtained

from the simulation determined 𝛾𝛾± data (calculated using the Kirkwood-Buff theory)

reported by Hossain et al. 13. Note that the simulation results of Hossain et al. employed

the same force field parameters as the present work. The 𝜏𝜏 parameters are identified using

the simulation data via regression using the same optimization technique as stated above.


109

4.7. Results and Discussions

Regression of Experimental and Simulation Data The binary interaction parameters obtained from regression in this work for the

H2O:(Na+-Clˉ) pair at 298.15 K and 1 bar are listed in Table 4.1 along with those reported

by Yan and Chen 43. It can be seen that 𝜏𝜏 values calculated in this work are in agreement

with the binary interaction parameters reported by Yan and Chen 43. Also, Table 4.1 lists

the binary interaction parameters obtained from the KB simulation data for 𝛾𝛾± by Hossain

et al. 13, along with the uncertainties. These values will be used to validate the parameters

obtained from simulations in this work.

Table 4.1. Binary interaction parameters from regression

Parameters Yan and Chen 43 This work1 KB simulation2

𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 8.86 8.89±0.06 7.23±0.18

𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 -4.54 -4.55±0.02 -3.95±0.06

1 regressed from the data reported by Robinson and Stokes 44 2 regressed from the simulation data of Hossain et al. 13

Interpretation of the Molecular Quantities in the Framework of eNRTL

The binary interaction parameters are obtained from molecular simulations as

described in the previous sections. The effective species diameters are obtained from the

location of first peaks of the corresponding RDF. For instance, 𝜎𝜎𝑎𝑎𝐿𝐿 which represents the

effective diameter between Na+ and Clˉ interactions is considered as the location of the

first peak of the cation-anion RDF. All the 𝜎𝜎 parameters thus obtained are listed in Table

4.10.1 of the Supporting Information, at each concentration. We note that the

uncertainties in 𝜎𝜎 are negligible (less than 0.25%) and hence are not reported. Furthermore,


110

we note that the 𝜎𝜎 values do not change over the concentration range studied. As expected,

the effective diameters of cation-molecule interactions are the smallest while the 𝜎𝜎 of

anion-molecule pairs are the largest, due the difference in size between Na+ and Clˉ ions.

Furthermore, the RDF of Na+ and Clˉ pair at each concentration exhibits two peaks. For

illustration purpose, one such RDF is shown in Figure 4.10.1 of the Supporting

Information for the concentration of 4 mol/kg NaCl. The first peak corresponds to the

contact ion pairs (CIP) while the second peak is due to the solvent-separated ion pairs

(SSIP). Such behavior has been previously observed and widely reported in the literature

25,40,41. The 𝜎𝜎 values of Na+-water and Clˉ-water calculated in this study as 2.3 and 3.2 Å,

respectively agree with those reported by Weerasinghe and Smith 40 and experimental

measurements 46. Similarly, the 𝜎𝜎 value of Na+- Clˉ pair calculated here as 2.7 Å is in good

agreement with that reported by Gee at al. 41. Note that the results of Weerasinghe and

Smith 40 and Gee et al. 41 are both based on the same force field parameters (SPC/E+KBFF)

as those employed in this work.

As described earlier, at each concentration, the depth of the first minimum in the PMF

of the corresponding pair is taken as the interaction strength. Figures 4.10.2 and 4.10.3 of

the Supporting Information show one such PMF that was used to obtain 𝜖𝜖 for Na+-Clˉ

and water-water pairs, and Na+-water and Clˉ-water pairs, respectively, at concentration of

4 mol/kg of NaCl. Table 4.10.2 of the Supporting Information shows the interaction

strengths (𝜖𝜖) between different pairs for all concentrations. The uncertainties in 𝜖𝜖 are the

highest at low salt concentration due to the statistical uncertainties associated with the

simulation systems at low number of ions. It can be seen from the Table 4.10.2 that the

interaction strengths between ion-water and water-water pairs decrease with an increase in


111

concentration while the interaction strength between cation-anion pairs tends to get

stronger with increasing concentration, before all the interactions reach a plateau value.

Such an observation is attributed to the ion pair screening effect with an increase in

concentration. We further note that the strength of the interaction energies obtained from

classical MD simulations are sensitive to the details of the force field used 13,26. To validate

our calculations, we compared the results of Na+-Clˉ interaction energies with those

obtained by Timko et al. 47 using ab initio Car-Parrinello MD (CPMD) simulations and

other MD calculations with different force fields. The energy barrier height of the CIP

obtained from CPMD simulations were reported as 1.40±0.28 kcal/mol. That value is

approximately 1.6-2 𝑘𝑘𝑇𝑇 smaller than the interaction energies predicted by our simulations

using classical MD force fields (SPC/E+KBFF). Other classical MD force fields studied

by Timko et al. 47 resulted in interactions energies ranging from 2.70 to 3.56 kcal/mol (~2-

2.5 𝑘𝑘𝑇𝑇 higher than CPMD predictions). The overestimation of interaction energies by

classical MD simulations is noted in the literature 48, the reason for which is often attributed

to the polarization effect in the medium which is not explicitly accounted for in MD force

fields. Nevertheless, among the different force fields studied by Timko et al. 47, the results

of our calculations are comparable to those obtained using the Smith and Dang’s

parameters 49. Furthermore, as we are interested in the differences between the interactions

energies and not the absolute values (Eqs. 4.18, 4.19, 4.29, and 4.30) of various species

(cation-water, anion-water, water-water, and cation-anion), we believe that any error due

to the force field parameters will approximately cancel out and hence the procedure should

result in consistent binary interaction parameter values for different force fields.


112

Systematic study of the effect of force field on the values of the 𝜏𝜏 parameters is beyond the

scope of this study.

As mentioned earlier, the domain radii are defined based on the assumptions of the

eNRTL model: local electroneutrality and like-ion repulsion. For this purpose, the local

number of species 𝑗𝑗 around center 𝑖𝑖 (𝑁𝑁𝑗𝑗𝑖𝑖𝑙𝑙𝑜𝑜𝐿𝐿𝑎𝑎𝑙𝑙) is calculated as a function of distance from

the center species. This is achieved first by integrating the corresponding RDF of the 𝑖𝑖-𝑗𝑗

pair, 𝑔𝑔𝑗𝑗𝑖𝑖(𝑟𝑟), using Eq. 4.32.

𝑁𝑁(𝑟𝑟)𝑗𝑗𝑖𝑖𝑙𝑙𝑜𝑜𝐿𝐿𝑎𝑎𝑙𝑙 = 4𝜋𝜋𝜌𝜌𝑗𝑗𝑏𝑏𝑏𝑏𝑙𝑙𝑘𝑘 �𝑟𝑟′2

𝑟𝑟

0

𝑔𝑔𝑗𝑗𝑖𝑖(𝑟𝑟′)𝑑𝑑𝑟𝑟′ (4.32)

Here 𝜌𝜌𝑗𝑗𝑏𝑏𝑏𝑏𝑙𝑙𝑘𝑘 is the bulk number density of species 𝑗𝑗 in the solution; 𝑟𝑟 is the distance

over which the integration is performed. The local number of species are then converted

to the corresponding local mole fractions by considering appropriate species in the domain.

To satisfy the like-ion repulsion assumption, 𝑅𝑅(𝐿𝐿) and 𝑅𝑅(𝑎𝑎) are defined as the distances until

which the like ion species remain absent (i.e., distances up to which the 𝑥𝑥𝐿𝐿𝐿𝐿𝑙𝑙𝑜𝑜𝐿𝐿𝑎𝑎𝑙𝑙 and 𝑥𝑥𝑎𝑎𝑎𝑎𝑙𝑙𝑜𝑜𝐿𝐿𝑎𝑎𝑙𝑙

remain zero, respectively). In a similar fashion, 𝑅𝑅(𝑚𝑚) is defined as the first distance at

which 𝑥𝑥𝐿𝐿𝑚𝑚𝑙𝑙𝑜𝑜𝐿𝐿𝑎𝑎𝑙𝑙 and 𝑥𝑥𝑎𝑎𝑚𝑚𝑙𝑙𝑜𝑜𝐿𝐿𝑎𝑎𝑙𝑙 are equal, to satisfy the local electroneutrality constraint. To

explain our definitions, a sample calculation to define the local domain size is shown in

Figures 4.2-4.4 at concentration of 4 mol/kg NaCl. For this concentration the size of the

domains 𝑅𝑅(𝐿𝐿), 𝑅𝑅(𝑎𝑎), and 𝑅𝑅(𝑚𝑚) are located at 2.98, 3.80, and 3.34 Å respectively. The size

of the domains obtained at all concentrations are listed in Table 4.10.3 of the Supporting

Information. As expected, the size of the cation-centered domain is the smallest while the

size of the anion-centered domain is the largest due to the size difference between Na+ and

Clˉ ions. Also, as a consequence of the size difference between the cation and anion, the


113

hydration shell of the anion is shifted further outward when compared to the cation as

shown in Figures 4.2-4.4.

The non-randomness factor, 𝛼𝛼 is calculated from the total coordination number in the

domain of interest, as 𝛼𝛼 = 1/𝑧𝑧(𝑖𝑖). Here 𝑧𝑧(𝑖𝑖) is the coordination number in a 𝑖𝑖-centered

domain. The non-randomness factors thus obtained from simulations are listed in Table

4.10.4 of the Supporting Information. The 𝛼𝛼 values obtained from simulations are

approximately between 0.1-0.2 which is in good agreement with the value of 0.2 usually

set by the eNRTL model. Using the values of molecular parameters (𝜎𝜎, 𝜀𝜀, and 𝑅𝑅) so

obtained, the four binary interaction parameters, 𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿, 𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎, 𝜏𝜏𝐿𝐿𝑚𝑚, and 𝜏𝜏𝑎𝑎𝑚𝑚, are then

calculated using Eqs. 4.18, 4.19, 4.29, and 4.30, respectively, as a function of salt

concentration as opposed to the eNRTL model that assumes the interaction parameters to

be independent of concentration.


114

Figure 4.2. The local mole fraction of different species in the three domains for the 4 mol/kg aqueous NaCl solution at 298.15 K. ( ), ( ), and ( ) denote the local mole fractions of Na+, Clˉ, and water around a center Na+, respectively.


115

Figure 4.3. The local mole fraction of different species in the three domains for the 4 mol/kg aqueous NaCl solution at 298.15 K. ( ), ( ), and ( ) denote the local mole fractions of Na+, Clˉ, and water around a center Clˉ, respectively.


116

Figure 4.4. The local mole fraction of different species in the three domains for the 4 mol/kg aqueous NaCl solution at 298.15 K. ( ), ( ), and ( ) denote the local mole fractions of Na+, Clˉ, and water around a center water, respectively.

Also, unlike the eNRTL model, we make no assumptions about the nature of

interactions between different species. In what follows, in addition to calculating 𝜏𝜏, we use

our approach as an independent framework to validate the assumptions of the eNRTL

model.

Prediction of Binary Interaction Parameters from MD Simulation and Validation of the eNRTL Assumptions

Figure 4.5 shows the binary interaction parameters in the concentration range of 0.5-

16 mol/kg. The numerical values of the parameters are given in Table 4.10.5 of the

Supporting Information for clarity. Also shown in the figure are the binary interaction

parameters obtained from regressing the simulation mean ionic activity coefficient data of


117

Hossain et al. 13. The 𝜏𝜏 values obtained from simulations are in good agreement with those

obtained from regression of the literature simulation data based on the KB-theory 13. We

note that the present work and the calculations of Hossain et al. employ the same force

field. Also, to determine the effect of finite simulation size on the 𝜏𝜏 parameter values, as

mentioned previously, the simulations were carried out for a larger model system at a

specific concentration of 4 mol/kg NaCl. The 𝜏𝜏 parameter values thus obtained are as

follows: 𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿= 7.94±0.28, 𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎= 5.41±0.18, 𝜏𝜏𝐿𝐿𝑚𝑚= -4.65±0.08, and 𝜏𝜏𝑎𝑎𝑚𝑚= -4.96±0.06.

Note that these parameter values are in close agreement (within the uncertainty range) with

the 𝜏𝜏 values obtained from the corresponding smaller model system (see Table 4.10.5).

Note that finite system size effects do not affect the calculation of the binary interaction

parameters as they depend only on local molecular properties, unlike the cases where long

range correlations are dominant 50.

While the parameters obtained from the regression of MD simulation data of Hossain

et al. 13 are independent of the salt concentration, the 𝜏𝜏 parameters determined in this work

show a weak concentration dependence, this is especially the case for the ion center

parameters (𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 and 𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎) at low concentrations. Also, note that the significantly

larger error bar associated with 𝜏𝜏 at low concentrations is due to the larger uncertainties in

the interaction energies. It must also be noted that the 𝜏𝜏 parameter as defined by eNRTL

model is strictly a local interaction parameter that includes only the van der Waals type of

interactions, while the 𝜏𝜏 parameter values obtained from simulations are based on the

potential of mean force which includes both the local electrostatic and van der Waals

interactions. Considering this fact, the concentration dependence of 𝜏𝜏 at low salt

concentrations can be rationalized as follows. Since at high concentrations, the van der


118

Waals interactions are the major contributing factor to the PMF (due to ion-pair screening

effect), the 𝜏𝜏 parameter values calculated from simulations are consistent with the 𝜏𝜏 values

obtained from regression (which only considers the bare van der Waals interaction).

However, at low concentrations, the electrostatic interactions between ions also play a

significant role thus leading to the observed deviation between the interaction parameter

values calculated from simulation and regression. We hypothesize that, subtracting the

electrostatic contribution from PMF obtained from the MD simulations and using the

adjusted PMF to calculate the 𝜏𝜏 value will lead to a better agreement between the regressed

values and the MD predicted values of 𝜏𝜏 at low concentrations. We defer such

investigations to future work.

Figure 4.5. The binary interaction parameters for the aqueous NaCl solution at 298.15 K. () denotes the equivalent binary interaction parameters (τm,ca and τca,m) regressed from

the simulation data 13 from Kirkwood-Buff theory; The binary interaction parameters from MD simulations are denoted by symbols as follows: τmc,ac ( ),τma,ca ( ),τcm ( ), and τam (

); The hatched area denotes the error margins on regression parameters.


119

The eNRTL model assumes that the strength of interaction between the cation-

molecule pair and the anion-molecule pair is equal. In terms of the two-liquid theory, this

assumption translates to the following expressions: 𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 = 𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎 and 𝜏𝜏𝐿𝐿𝑚𝑚 = 𝜏𝜏𝑎𝑎𝑚𝑚. This

assumption can be seen to hold at high concentrations for 𝜏𝜏𝐿𝐿𝑚𝑚 and 𝜏𝜏𝑎𝑎𝑚𝑚 while at low

concentrations, significant deviations are observed (see Figure 4.5). Also, the assumption

of the eNRTL model regarding the nature of interactions in the cation and anion-centered

domains (𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 and 𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎) is approximately true with 𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 systematically higher than

𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎 throughout the entire concentration range. This observation is discussed further in

the Supporting Information (Figure 4.10.4). Furthermore, note that the abovementioned

assumption of the eNRTL theory reduces the number of adjustable parameters and renders

correlation of experimental data more feasible and hence has been widely adopted. The

results described above support that the 𝜏𝜏 parameters of the eNRTL model can be obtained

directly from the information of molecular structure and interactions in the condensed

phase system. Moreover, interpreting the model from statistical mechanics framework

allows for more rigorous validation of the underlying assumptions involved.

Phase Equilibrium Predictions As a further test of the parameter values, we compare the phase equilibrium properties

of the aqueous electrolyte solution obtained using the eNRTL model and the binary

interaction parameters calculated from MD simulations. The phase equilibrium predictions

from the eNRTL model considers both the local NRTL part and the long-range Pitzer-

Debye-Hückel formulation 12. For this purpose, the mean ionic activity coefficients

obtained from the eNRTL model using the 𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 (= 8.34) and 𝜏𝜏𝐿𝐿𝑚𝑚 (= −4.54) parameters

and the corresponding 𝛼𝛼(𝐿𝐿) (= 0.18) and 𝛼𝛼(𝑚𝑚) (= 0.18) values determined in this work


120

at the concentration of 4 mol/kg NaCl are shown in Figure 4.6. Further details on the

calculation of mean ionic activity coefficient using the eNRTL model and the

corresponding binary interaction parameters are presented in the literature where the

pertinent relations are derived 19. Figure 4.6 also shows the mean ionic activity coefficient

calculated from the regressed 𝜏𝜏 parameters 43, the predicted 𝛾𝛾± from MD simulations using

Kirkwood-Buff method 13, and the experimental data reported by Robinson and Stokes 44

along with the experimental data of Tang et al. 51. In addition, the uncertainties in 𝛾𝛾±

predicted by the eNRTL model (hatched area), due to the uncertainties associated with the

regressed 𝜏𝜏 parameters, are also shown. The 𝛾𝛾± values predicted from the binary

interaction parameters calculated using MD simulations in this work are in a reasonable

agreement with the KB theory-simulation results of Hossain et al. 13, despite the fact that

the techniques employed to obtain 𝛾𝛾± in both these calculations are different. As

mentioned before, we selected the 𝜏𝜏 values from the concentration 4 mol/kg NaCl as the

prediction of the 𝜏𝜏 parameter from MD simulations at high concentrations attains a plateau

value consistent with the eNRTL framework. Also, this choice of the 𝜏𝜏 parameter provided

the best prediction for 𝛾𝛾± compared with the results of Hossain et al. Accurate prediction

of 𝛾𝛾± using the MD determined 𝜏𝜏 parameters is challenging due to sensitivity of the

predicted 𝛾𝛾± to the values of the binary interaction parameters. Methodologies to further

improve the accuracy of the calculated 𝜏𝜏 parameters should be considered.

The vapor pressure (𝑃𝑃𝑠𝑠𝑎𝑎𝑡𝑡) and excess Gibbs free energy (𝐺𝐺𝑒𝑒𝑒𝑒) of the aqueous

electrolyte system are also calculated using the binary interaction parameters. Here the

excess Gibbs free energy was directly calculated from the activity coefficients, while the

vapor pressure was obtained from the flash calculation. Figure 4.7 shows that the


121

predictions of vapor pressure and excess Gibbs free energy from the binary interaction

parameters of MD simulations are in excellent agreement with those obtained from

regression of MD simulation data of Hossain et al. 13, with the mean relative deviation

(MRD) of 1.04 % and 1.24 % for 𝑃𝑃𝑠𝑠𝑎𝑎𝑡𝑡 and 𝐺𝐺𝑒𝑒𝑒𝑒, respectively. We note that the predictions

of vapor pressure and excess Gibbs free energy are not as sensitive to the changes in 𝜏𝜏

parameters unlike 𝛾𝛾± which are very sensitive to the 𝜏𝜏 values.

Figure 4.6. Comparison of aqueous NaCl solution phase behavior at 298.15 K between MD simulations and regression results. ( ) denotes the mean ionic activity coefficients predicted from the binary interaction parameters of Yan and Chen 43, with the associated uncertainties shown as hatched area; ( ) presents the experimental data of Robinson and Stokes 44; ( ) denotes the experimental data of Tang et al. 51; the predicted mean ionic activity coefficient 13 by Kirkwood-Buff theory are presented by ( ); ( ) denotes the prediction of mean ionic activity coefficient using MD simulations results for binary interaction parameters of 4 mol/kg NaCl.


122

Figure 4.7. Comparison of aqueous NaCl solution phase behavior at 298.15 K between MD simulations and regression results. ( ) and ( ) present the predictions of aqueous NaCl solution vapor pressure and excess Gibbs free energy at 298.15 K from binary interaction parameters of regression of the simulation data 13. ( ) and ( ) represent the vapor pressure and excess Gibbs free energy obtained from the binary interaction parameters calculated from the MD simulations (this work).

4.8. Conclusions

The eNRTL model is re-examined in terms of the two-fluid theory and a recent

framework for determining the binary interaction parameters from MD simulations.

Aqueous NaCl solution is chosen as the example system to investigate the validity of the

proposed approach. The binary parameters in the eNRTL model are expressed as functions

of species diameter (𝜎𝜎), effective interaction strength (𝜖𝜖), the domain radius around the

center species (𝑅𝑅), and the non-randomness factor (𝛼𝛼). Each of these quantities is directly

calculated from MD simulations while satisfying the two assumptions of eNRTL model,

i.e., local electroneutrality and like-ion repulsion.


123

The binary interaction parameters calculated from MD simulations are in good

agreement with those obtained from data regression, especially, at moderate to high

concentrations. At low concentrations, these parameters exhibit concentration dependence

as they capture both the long-range electrostatic interactions and the short-range van der

Waals interaction. The binary interaction parameters calculated from MD at medium to

high concentrations are concentration independent, consistent with the eNRTL model and

hence the predictions from these concentrations should be used to calculate the phase

equilibrium properties.

This study provides a theoretical linkage between eNRTL, a classical thermodynamic

model for electrolytes, and molecular scale structure and interactions in the system. The

technique is useful for predicting the binary interaction parameters of the eNRTL model,

specifically in cases where the availability of experimental data is sparse to facilitate the

data regression procedure. The methodology can also guide data regression to predict

physically relevant parameters, thereby reducing the ambiguity in the parameter selection

process which otherwise requires significant manual tuning. On the other hand, challenges

remain in obtaining an accurate estimate for the phase equilibrium properties, especially

the mean ionic activity coefficient, using MD determined 𝜏𝜏 parameters. The predictions

of the binary interaction parameters obtained from our work can be refined by (1) fine-

tuning of the force fields/techniques (like CPMD) used to obtain the species interaction

energies and (2) including only the local van der Waals interaction energies between

components to calculate the 𝜏𝜏 parameters, (as discussed in the Results and Discussions

section). Our future work is aimed towards performing such calculations in addition to

extending and testing the current framework to multivalent electrolytes.


124

4.9. Acknowledgments



Maddox Foundation, United States. The authors also acknowledge the computational

resources provided by High Performance Computing Center (HPCC) at Texas Tech

University.


125

4.10. Supporting Information

Table 4.10.1. Effective species diameters (Å)

NaCl concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0

𝜎𝜎𝑎𝑎𝐿𝐿 2.70 2.71 2.70 2.70 2.70

𝜎𝜎𝐿𝐿𝑚𝑚 2.31 2.31 2.31 2.31 2.31

𝜎𝜎𝑎𝑎𝑚𝑚 3.20 3.20 3.20 3.21 3.21

𝜎𝜎𝑚𝑚𝑚𝑚 2.75 2.76 2.77 2.77 2.79

Parameters 10.0 12.0 14.0 16.0

𝜎𝜎𝑎𝑎𝐿𝐿 2.70 2.70 2.70 2.70

𝜎𝜎𝐿𝐿𝑚𝑚 2.31 2.31 2.31 2.31

𝜎𝜎𝑎𝑎𝑚𝑚 3.22 3.22 3.23 3.23

𝜎𝜎𝑚𝑚𝑚𝑚 2.80 2.81 2.82 2.83

𝑎𝑎, 𝑐𝑐, and 𝑚𝑚 denote the anion, the cation, and the molecule


126

Figure 4.10.1. Radial distribution function (RDF) of the Na+-Clˉ pair for the 4 mol/kg aqueous NaCl solution at 298.15 K.


127

Figure 4.10.2. Potential of mean force (PMF) for the 4 mol/kg aqueous NaCl solution at 298.15 K. PMF of Na+ around a center Clˉ ( ) and water around a center water (

). The symbols represent the calculated PMF and lines denote the smoothed PMF results using cubic splines; 𝜖𝜖ji is the depth of the energy barriers or effective interaction strength for species 𝑗𝑗 around a center 𝑖𝑖.


128

Figure 4.10.3. Potential of mean force (PMF) for the 4 mol/kg aqueous NaCl solution at 298.15 K. PMF of Na+ around a center water ( ) and Clˉ around a center water (

). The symbols represent the calculated PMF and lines denote the smoothed PMF results using cubic splines; 𝜖𝜖ji is the depth of the energy barriers or effective interaction strength for species 𝑗𝑗 around a center 𝑖𝑖.


129

Table 4.10.2. Interaction strengths (kcal/mol)


Parameters 0.5 2.0 4.0 6.0 8.0

𝜖𝜖𝑎𝑎𝐿𝐿 2.26±0.18 2.66±0.09 2.75±0.08 2.78±0.05 2.81±0.03

𝜖𝜖𝐿𝐿𝑚𝑚 2.59±0.06 2.51±0.03 2.45±0.02 2.38±0.02 2.31±0.01

𝜖𝜖𝑎𝑎𝑚𝑚 1.22±0.02 1.11±0.01 0.97±0.01 0.87±0.01 0.81±0.00

𝜖𝜖𝑚𝑚𝑚𝑚 0.69±0.00 0.60±0.01 0.57±0.00 0.54±0.00 0.53±0.00

Parameters 10.0 12.0 14.0 16.0

𝜖𝜖𝑎𝑎𝐿𝐿 2.81±0.02 2.81±0.02 2.78±0.02 2.76±0.01

𝜖𝜖𝐿𝐿𝑚𝑚 2.25±0.01 2.19±0.01 2.15±0.01 2.11±0.01

𝜖𝜖𝑎𝑎𝑚𝑚 0.78±0.00 0.77±0.00 0.77±0.00 0.77±0.01

𝜖𝜖𝑚𝑚𝑚𝑚 0.52±0.00 0.51±0.00 0.51±0.00 0.50±0.00

𝑐𝑐, 𝑎𝑎, and 𝑚𝑚 denote the cation, the anion, and the molecule

Table 4.10.3. First neighbor shell radii (Å)


Parameters 0.5 2.0 4.0 6.0 8.0

𝑅𝑅(𝐿𝐿) 3.14±0.13 2.98±0.06 2.98±0.06 2.98±0.06 3.00±0.00

𝑅𝑅(𝑎𝑎) 4.54±0.09 4.02±0.06 3.80±0.00 3.60±0.00 3.60±0.00

𝑅𝑅(𝑚𝑚) 3.37±0.00 3.35±0.00 3.34±0.00 3.34±0.00 3.33±0.00

Parameters 10.0 12.0 14.0 16.0

𝑅𝑅(𝐿𝐿) 3.00±0.00 3.00±0.00 3.00±0.00 3.00±0.00

𝑅𝑅(𝑎𝑎) 3.60±0.00 3.60±0.00 3.60±0.00 3.60±0.00

𝑅𝑅(𝑚𝑚) 3.33±0.00 3.33±0.00 3.32±0.00 3.32±0.00



130

Table 4.10.4. Non-randomness factors


Parameters 0.5 2.0 4.0 6.0 8.0

𝛼𝛼(𝐿𝐿) 0.18 0.18 0.18 0.19 0.19

𝛼𝛼(𝑎𝑎) 0.08 0.12 0.12 0.14 0.14

𝛼𝛼(𝑚𝑚) 0.20 0.19 0.18 0.18 0.18

Parameters 10.0 12.0 14.0 16.0

𝛼𝛼(𝐿𝐿) 0.19 0.19 0.20 0.20

𝛼𝛼(𝑎𝑎) 0.14 0.14 0.14 0.14

𝛼𝛼(𝑚𝑚) 0.18 0.18 0.18 0.18


Table 4.10.5. Binary interaction parameters


Parameters 0.5 2.0 4.0 6.0 8.0

𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 3.97±1.36 7.65±0.56 8.34±0.84 8.65±0.42 8.90±0.22

𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎 0.98±0.82 3.60±0.40 5.46±0.39 6.81±0.23 7.51±0.18

𝜏𝜏𝐿𝐿𝑚𝑚 -3.98±0.17 -4.36±0.08 -4.54±0.06 -4.55±0.06 -4.39±0.06

𝜏𝜏𝑎𝑎𝑚𝑚 -5.80±0.13 -5.68±0.09 -4.97±0.11 -4.37±0.07 -3.99±0.04

Parameters 10.0 12.0 14.0 16.0

𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 8.96±0.17 9.10±0.14 8.97±0.16 8.99±0.13

𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎 7.72±0.14 7.79±0.11 7.64±0.16 7.54±0.12

𝜏𝜏𝐿𝐿𝑚𝑚 -4.25±0.04 -4.04±0.04 -3.92±0.05 -3.73±0.05

𝜏𝜏𝑎𝑎𝑚𝑚 -3.83±0.06 -3.75±0.04 -3.73±0.05 -3.70±0.07



131

Discussions Regarding the Nature of Interactions in the Ion-Centered Domains

The assumption of the eNRTL model that 𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 = 𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎 and 𝜏𝜏𝐿𝐿𝑚𝑚 = 𝜏𝜏𝑎𝑎𝑚𝑚 (as

mentioned in the main text) is further elaborated in this section. Consider the difference in

the energies between the different species with respect to the species at the center of the

domain, i.e., 𝛿𝛿𝐸𝐸𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎 = 𝜖𝜖𝐿𝐿𝑎𝑎𝜎𝜎𝐿𝐿𝑎𝑎3 − 𝜖𝜖𝑚𝑚𝑎𝑎𝜎𝜎𝑚𝑚𝑎𝑎3 . These quantities are plotted in Figure 4.10.4.

Note that the contribution to 𝛿𝛿𝐸𝐸 in Figure 4.10.4 includes both van der Waals and

electrostatic interactions, unlike the consideration in the eNRTL model as explained in the

main text. The nature of interaction between the species at the center of the domain and

the surrounding is governed by both their size and the potential of mean force. At high

concentrations, these interactions are of similar magnitude while at low concentrations they

show deviation primarily due to the size difference between the cation and the anion.

Following these observations, we conclude that the assumption made by the eNRTL model

about the nature of these interactions are consistent with MD determined interaction

parameters at moderate to high salt concentrations.


132

Figure 4.10.4. The energy difference weighted by corresponding effective sizes between different species in the three local domains for aqueous NaCl solution at 298.15 K. () and ( ) denote the energy difference between molecule and anion around a center cation (δEmc,ac), and molecule and cation around a center anion (δEma,ca), respectively; (

) and ( ) denote the energy difference between cation around a center molecule (δEcm), and anion around a center molecule (δEam), respectively.


133

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47. Timko J, Bucher D, Kuyucak S. Dissociation of NaCl in water from ab initio molecular dynamics simulations. The Journal of Chemical Physics. 2010;132:114510.

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CHAPTER 5. PRDICTING PHASE EQUILIBRIA PROPERTIES OF

ELECTRLYTE SOLUTIONS BY COMBINING A CLASSICAL THERMODYNAMIC MODEL AND MOLECULAR SIMULATIONS: AQUOUES XCL2

SOLUTIONS, (X=BA, SR, CA, MG)4

5.1. Abstract

Process design and simulations in industry rely on the availability of thermodynamic

models to provide accurate predictions of phase equilibria properties. As a widely used

classical thermodynamic model, the eNRTL model has shown to provide reliable

predictions for electrolyte solutions by employing the binary interaction parameters (𝜏𝜏). In

this study, we develop a predictive theoretical framework to calculate the 𝜏𝜏 parameters

directly from molecular dynamic (MD) simulations by revisiting the statistical mechanics

of two-liquid theory. The 𝜏𝜏 parameters are expressed as functions of the liquid structure

and local preferential interaction quantities, which in turn are obtained from the MD

simulations and free energy calculations. The aqueous BaCl2, SrCl2, CaCl2, and MgCl2

solutions are selected as model systems for the validation of our approach. The 𝜏𝜏

parameters obtained from the MD simulations are then used to predict the mean ionic

activity coefficient, vapor pressure, and excess Gibbs free energy of the model systems in

a wide concentration range from 0.5 to 12 mol/kg. The results from the MD simulations

are in satisfactory agreement with the predictions from regression of the experimental data.

4 This chapter is based on a manuscript under preparation as: Saravi SH, Khare R, and Chen C-C., Predicting Phase Equilibria Properties of Electrolyte Solutions by Combining a Classical Thermodynamic Model and Molecular Simulations: Aqueous XCl2 Solutions, (X = Ba, Sr, Ca, and Mg)


139

The methodology can be broadly utilized for predicting the phase equilibria properties of

electrolytes, especially when data are scarce. Furthermore, the technique can provide

guidelines to select physically meaningful parameters in data regression procedures,

thereby enhancing the feasibility of using the classical thermodynamic models in industry.


140

5.2. Introduction

The study of aqueous electrolyte solutions has attracted significant attention due to

their ubiquitous presence in industrial 1-3, pharmaceutical 4, and environmental processes

5. The Design and optimization of these processes rely heavily on the accurate predictions

of phase equilibria properties 6,7. Over the decades, numerous modeling studies have

reported predictions of the essential thermodynamic properties of electrolyte solutions,

such as the mean ionic activity coefficients (𝛾𝛾±), excess Gibbs free energy, and chemical

potential. Classical molecular thermodynamic models have been the primary tool to

provide such predictions for industrial use, owing to their versatility and inexpensive

computational procedures 7.

As a widely used classical thermodynamic model, the electrolyte non-random two-

liquid (eNRTL) model 8-11 has shown to provide accurate predictions of phase equilibria

properties for electrolyte solutions 7,12. The eNRTL’s consistent thermodynamics

framework and mathematical simplicity have led the model to be extensively applied for

predicting the phase equilibria properties of variety of electrolyte solutions; from simple

aqueous uni-univalent to mixed-solvent multivalent electrolyte systems 13-19. The model

relates the excess Gibbs free energy of an electrolyte solution to the microscopic local

structure of the liquid, with a set of concentration-independent binary interaction

parameters (𝜏𝜏). The parameters quantify the effective interaction strengths between

different species of the solution, namely, anions, cations, and molecules. The two inherent

assumptions made by the model, that is the electroneutrality and like-ion repulsion,

determine the configuration of the local structures, also known as the local domains. The

former assumption states that the net electric charge in the first solvation shell around a


141

center molecule is zero, while the latter asserts that there are no like-charged ions within

the first shell around the ionic species.

The 𝜏𝜏 parameters are typically identified by regressing them to various experimental

data of phase equilibria, calorimetric, and speciation properties. Once the 𝜏𝜏 parameters are

quantified, the thermodynamic properties required for use in process simulations including

the activity coefficients can be readily calculated 8-11. Identifying the model parameters

from regression, however, could be hindered by the lack of available/reliable experimental

data. Furthermore, selecting physically relevant parameters could be ambiguous as the

minimization function employed in the regression procedure, often exhibits several local

minima and hence multiple solutions for the binary interaction parameters. Quantifying the

binary interaction parameters from first-principles-based theories, without the need for

introducing any adjustable parameters can help circumvent these drawbacks. There have

been a number of attempts for obtaining the binary interaction parameters of the classical

thermodynamic models, however only for nonelectrolytes, from predictive approaches by

utilizing molecular simulations 20-23. In particular, Ravichandran et al. 23 presented a novel

methodology to predict the 𝜏𝜏 parameters of the NRTL model from MD simulations for a

number of organic mixtures.

Extending the work of Ravichandran et al. 23, we have recently developed a theoretical

framework to identify the binary interaction parameters of the eNRTL model by bridging

the statistical mechanical framework of two-liquid theory for uni-univalent electrolytes

with molecular simulations 24. The 𝜏𝜏 parameters were expressed as functions of the liquid

structure and energetic interaction quantities of the solution, which themselves were

calculated directly from the MD simulations 24. The parameters calculated for aqueous


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NaCl solution, as a model system, were in satisfactory agreement with those obtained from

regression of the experimental data, specifically at moderate (~ 4 mol/kg) to supersaturated

(~ 16 mol/kg) concentrations. On the other hand, the parameters showed rather

considerable concentration dependence at lower salt concentrations (< ~ 2 mol/kg).

Furthermore, another assumption made by the model, that is the equality of the molecule-

cation and molecule-anion interactions was also shown to be somewhat reasonable by the

order of magnitude, however, with slight difference in quantities due to the species size

effects.

Overall, the results 24 were encouraging as a first-step toward directly utilizing the

molecular simulations in industrial process design to support phase equilibria predictions,

with relatively inexpensive computational time. Furthermore, our study provided insight

into the physical interpretation of the eNRTL binary interaction parameters, which were

previously assumed to be semi-empirical and merely correlative. The versatility of the

established methodology, on the other hand, requires further examination to assess its

generality to cover more complex systems including multivalent electrolyte solutions. Note

that the presence of the uni-univalent electrolytes was an intrinsic part of the theoretical

framework in our previous work 24.

The extension of the approach to the multivalent electrolyte solutions requires the

incorporation of the absolute charge numbers into the developed framework. In the original

eNRTL 9,11, such an extension was achieved by introducing a new concentration variable,

namely the effective mole fraction, for the treatment of the ionic species of higher valency

9. The effective mole fraction is defined as the product of the mole fraction and the absolute

charge number for the ionic species, and the mole fraction for molecular species, all of


143

which are normalized such that their summation is unity. Employing such a variable has

shown to be useful for modeling the electrolyte systems with multivalent ions 9,25.

Similarly, the absolute charge number of the ionic species must be integrated into our

statistical mechanical framework so that the developed methodology 24 can be applied for

multivalent electrolyte solutions.

In this study, we revisit our proposed methodology 24 to establish a generalized

framework to identify the binary interaction parameters of multivalent electrolyte solutions

from the MD simulations. The binary interaction parameters are expressed as functions of

the effective molecular diameters (𝜎𝜎), the interaction strengths (𝜖𝜖), the first neighbor shell

radii (𝑅𝑅), and the nonrandomness factors (𝛼𝛼), all of which are obtained from the MD

simulations. Several di-univalent electrolyte systems, namely, aqueous BaCl2, SrCl2,

CaCl2, and MgCl2 solutions are selected for the validation of the framework. By gaining

insight into the structure of the local domains from the MD simulations, we assess the

validity of the electroneutrality and like-ion repulsion assumptions for each model systems,

throughout the concentration range studied. The preferential interactions in the local

domains are quantified from the potential of mean force calculations as described in our

previous work 24. However, a second case is also presented here where the contribution of

the short-range van der Waals forces is obtained by subtracting the contribution of the

Coulombic electrostatic forces from the total potential of mean force.

The rest of this chapter is organized as follows: Theoretical Framework section which

details the underlying concepts of the eNRTL model and our developed approach to

formulate the 𝜏𝜏 parameters. It is then followed by the Quantification of the 𝜏𝜏 Parameters

from Molecular Simulations. The Molecular Simulation Details are then reported in the


144

next session. The binary interaction parameters obtained from the MD simulations, their

significance, and a comprehensive interpretation of the results are reported in the Results

and Discussion, followed by Conclusions.

5.3. Theoretical Framework

Thermodynamics Background of the eNRTL Model The eNRTL model 8-11 expresses the excess Gibbs free energy of the electrolyte

solutions as a combination of two contributions: the long-range electrostatic interactions

expressed by Pitzer-Debye-Hückel theory 26-28 and the short-range interactions, typically

of van der Waals-type, from the NRTL model developed by Renon and Prausnitz 29.

Consider the complete dissociation reaction of an electrolyte component (𝑐𝑐𝑎𝑎) to its

constituent cations (𝑐𝑐) and anions (𝑎𝑎), in the solvent molecules (𝑚𝑚). Eqs. 5.1 & 5.2 show

such a reaction and the corresponding stoichiometric relation between the species:

(𝑐𝑐𝑎𝑎) → 𝜈𝜈𝐿𝐿𝑐𝑐+𝑍𝑍𝑐𝑐 + 𝜈𝜈𝑎𝑎𝑎𝑎−𝑍𝑍𝑎𝑎 (5.1)

where

𝜐𝜐𝐿𝐿𝑍𝑍𝐿𝐿 = 𝜐𝜐𝑎𝑎𝑍𝑍𝑎𝑎 (5.2)

Here, 𝜈𝜈𝑖𝑖 and 𝑍𝑍𝑖𝑖 denote the stoichiometric coefficients and the absolute charge numbers of

ionic species, respectively. Upon the dissociation reaction of the electrolyte, it is assumed

that three local domains are formed with cation, anion, and molecule in turn at center. Each

of the center species is surrounded by solution components that are locally coordinated in

the first neighbor shell, such that the two assumptions of the model, i.e., electroneutrality

and like-ion repulsion are satisfied (see Figure 5.1).


145

Figure 5.1. Schematic configurations of the three local domains as hypothesized by the eNRTL model.

To take into account the absolute charge numbers, the effective mole fraction (𝑋𝑋) of

the species is defined following Eqs. 5.3 & 5.4 9:

𝑋𝑋′𝑗𝑗 = 𝑥𝑥𝑗𝑗𝑍𝑍𝑗𝑗 = �𝜕𝜕𝑖𝑖𝜕𝜕� 𝑍𝑍𝑗𝑗 , 𝑙𝑙 = ∑ 𝑙𝑙𝑗𝑗𝑗𝑗 = 𝑙𝑙𝐿𝐿 + 𝑙𝑙𝑎𝑎 + 𝑙𝑙𝑚𝑚 (5.3)

𝑋𝑋𝑗𝑗 =

𝑋𝑋′𝑗𝑗∑ 𝑋𝑋′𝑗𝑗𝑗𝑗

(5.4)

where 𝑥𝑥𝑗𝑗 and 𝑙𝑙𝑗𝑗 denote, respectively, the mole fraction and the mole number of the species

𝑗𝑗; 𝑍𝑍𝑗𝑗 is the absolute charge number for the ionic species and unity for the molecules. The

relation between the effective mole fractions in the three local domains can thereby be

written as follows 9:

𝑋𝑋𝑚𝑚𝐿𝐿 + 𝑋𝑋𝑎𝑎𝐿𝐿 = 1, (𝑋𝑋𝐿𝐿𝐿𝐿 = 0) Cation-centered domain (5.5)

𝑋𝑋𝑚𝑚𝑎𝑎 + 𝑋𝑋𝐿𝐿𝑎𝑎 = 1, (𝑋𝑋𝑎𝑎𝑎𝑎 = 0) Anion-centered domain (5.6)


146

𝑋𝑋𝐿𝐿𝑚𝑚 + 𝑋𝑋𝑎𝑎𝑚𝑚 + 𝑋𝑋𝑚𝑚𝑚𝑚 = 1, (𝑋𝑋𝐿𝐿𝑚𝑚 = 𝑋𝑋𝑎𝑎𝑚𝑚) Molecule-centered domain (5.7)

The effective local mole fractions in each domain are related to the effective bulk mole

fractions with the binary interaction parameters following Eqs 8-11. For molecular species

at center:

𝑋𝑋𝑗𝑗𝑖𝑖𝑋𝑋𝑖𝑖𝑖𝑖

= �𝑋𝑋𝑗𝑗𝑋𝑋𝑖𝑖� 𝑒𝑒�−𝛼𝛼(𝑖𝑖)𝜏𝜏𝑖𝑖𝑖𝑖� (5.8)

where

and for ionic species at center:

𝑋𝑋𝑗𝑗𝑖𝑖𝑋𝑋𝑘𝑘𝑖𝑖

= �𝑋𝑋𝑗𝑗𝑋𝑋𝑘𝑘� 𝑒𝑒�−𝛼𝛼(𝑖𝑖)𝜏𝜏𝑖𝑖𝑖𝑖,𝑘𝑘𝑖𝑖� (5.10)

where

𝜏𝜏𝑗𝑗𝑖𝑖,𝑘𝑘𝑖𝑖 =

(𝑔𝑔𝑗𝑗𝑖𝑖 − 𝑔𝑔𝑘𝑘𝑖𝑖)𝑅𝑅𝑇𝑇

(5.11)

Here the single-indexed 𝑋𝑋 denotes the effective bulk mole fraction and the double-indexed

𝑋𝑋 designates the effective local mole fraction, with the second subscript representing the

center species; 𝑔𝑔𝑗𝑗𝑖𝑖, the interaction energy of the 𝑗𝑗-𝑖𝑖 pair, can be interpreted as the free

energy between the species 𝑗𝑗 and 𝑖𝑖 along the reaction coordinate (here center to center

distance); 𝛼𝛼(𝑖𝑖) is the nonrandomness factor in an 𝑖𝑖-centered domain, typically considered

as a semi-empirical constant fixed at a value of 0.2.

As shown by Eqs. 5.8 & 5.10, there are four binary interaction parameters, that is

𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿, 𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎, 𝜏𝜏𝐿𝐿𝑚𝑚, and 𝜏𝜏𝑎𝑎𝑚𝑚. The sign and magnitude of these parameters indicate the

favorable or unfavorable nature of the interactions in the local domains. For example, a

positive value of the 𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 indicates that there is an energy penalty for inserting a solvent

𝜏𝜏𝑗𝑗𝑖𝑖 =

(𝑔𝑔𝑗𝑗𝑖𝑖 − 𝑔𝑔𝑖𝑖𝑖𝑖)𝑅𝑅𝑇𝑇

(5.9)


147

molecule in the first neighbor shell of a center cation, inside the lattice structure of a

hypothetical fused salt (𝑐𝑐𝑎𝑎). The model inherently assumes the effective interaction

strength of the molecule-anion pair is equal to that of the molecule-cation pair as noted

below:

𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 = 𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎 = 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 (5.12)

𝜏𝜏𝐿𝐿𝑚𝑚 = 𝜏𝜏𝑎𝑎𝑚𝑚 = 𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 (5.13)

Following such an assumption, only the two parameters 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 and 𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 are sufficient

to represent, respectively, the interaction strength of the molecule-electrolyte and

electrolyte-molecule pairs. By obtaining 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 and 𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 from regression, the activity

coefficients and therefore other thermophysical properties can be calculated. The

derivations of the thermodynamic properties from the 𝜏𝜏 parameters are presented in the

original series of articles on the eNRTL model development 8-11.

The eNRTL model further assumes 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 and 𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 are concentration independent.

Such an assumption makes the applicability of the model significantly easier as only one

set of 𝜏𝜏 parameters can predict the activity coefficients for the entire concentration range.

This was rationalized as 𝜏𝜏 parameters are expressed as functions of the local interactions,

which in turn are considered to be strictly of short-range van der Waals nature. Short-range

interactions are known not to change with concentration change. Note that the,

hypothetically, concentration-independent 𝜏𝜏 parameters are obtained by fitting them to the

experimental data from the entire range of concentration, and hence are able to predict

reasonably the thermodynamics properties at any given concentration. This assumption

gives the theory a crucial advantage over the conventional MD simulation methods, where

obtaining the activity coefficients requires implementing advanced free energy calculation


148

techniques with significantly more expensive computational time. The free energy

calculations are carried out at discrete quantities of the salt concentration, which makes the

interpolation and extrapolation cumbersome. On the other hand, quantifying one set of 𝜏𝜏

parameters representing the entire concentration range would overcome such an issue and

helps achieve rapid predictions of the thermophysical properties.

Relating the τ Parameters to Liquid Structure and Energetic Interaction Quantities

The ratio of the local and bulk effective mole fractions appeared in Eqs. 5.8 & 5.10

can be expressed in terms of probability functions of forming the local domains. Consider

that the three local domains are constructed by randomly selecting the particles from the

infinite pool of entities in the solution. Probability functions of constructing each domain

can be derived by solving the combinatorial problem, together with taking into account the

preferential interactions. Below, we explain in detail the mathematical framework for each

of the local domains, i.e., cation-centered, anion-centered, and molecule-centered. The

reader is also referred to the literature for more discussion 23,24,30.

The τ Parameter in a Cation-Centered Domain In a hypothetical lattice structure around a center cation, consider 𝑇𝑇 number of species

that are locally coordinated within the first neighbor shell radius of 𝑅𝑅(𝐿𝐿), out of which 𝑙𝑙 is

the number of anions and (𝑇𝑇 − 𝑙𝑙) is the number of molecules, following the like-ion

repulsion assumption. The probability of randomly arranging the species (𝑓𝑓𝜕𝜕,𝜕𝜕𝐿𝐿 ) in such a

domain without taking into account the effective interactions between species is as follows:

𝑓𝑓𝜕𝜕,𝜕𝜕𝐿𝐿 = 𝐶𝐶𝜕𝜕𝜕𝜕 𝑝𝑝𝑎𝑎

𝜕𝜕𝑝𝑝𝑚𝑚𝜕𝜕−𝜕𝜕 (5.14)


149

where 𝐶𝐶𝜕𝜕𝜕𝜕 is the combinatorial factor; 𝑝𝑝𝑎𝑎 𝜕𝜕 is the probability of choosing 𝑙𝑙 number of anions

and 𝑝𝑝𝑚𝑚𝜕𝜕−𝜕𝜕 is the probability of selecting (𝑇𝑇 − 𝑙𝑙) number of molecules, from the infinite

pool of solution particles.

In order to translate the random arrangement probability function to that of the

nonrandom arrangement, the preferential interactions in the local domain are taken into

account. The preferential interactions can be expressed as the net potential of the pairwise

additive forces acting on the domain’s center species, in the field of all the surrounding

particles. In lieu of the bare pairwise two-body potential for each pair, the potential of mean

force is used as it provides the free energy surface along the reaction coordinate in the

presence of all the species in the solution. This can also be inferred as the work required to

bring two particles from the infinite separation to form the first shell. In two-fluid theory

of Brandani and Prausnitz 30, the preferential interactions in a local domain were mapped

into a square-well potential as proposed by Kerley 31. Consistent with the two-fluid theory,

here the potential of mean force is projected into a square-well type expressed by the

variables of the effective molecular diameter (𝜎𝜎); the characteristic energy (𝜖𝜖), inferred as

the energy required to form or disrupt the first molecular cage; 𝑅𝑅(𝑖𝑖) the radius of the first

neighbor shell; and 𝑇𝑇𝑖𝑖, the coordination number of species in the local domain.

Weighting the probability function with the Boltzmann factor of the square-well

type potential function 30—𝜑𝜑𝜕𝜕,𝜕𝜕𝐿𝐿 (See Eq. 5.15)—we arrive at the expression for the

probability function of nonrandomly arranging the species around a center cation (𝑓𝑓𝜕𝜕,𝜕𝜕𝐿𝐿� )

following Eq. 5.16.

𝜑𝜑𝜕𝜕,𝜕𝜕𝐿𝐿 =

−∑ (𝑇𝑇𝑖𝑖𝐿𝐿𝜖𝜖𝑖𝑖𝐿𝐿𝜎𝜎𝑖𝑖𝐿𝐿3 )𝑖𝑖

(𝑅𝑅(𝐿𝐿))3, 𝑖𝑖 = 𝑎𝑎 ,𝑚𝑚 (5.15)


150

𝑓𝑓𝜕𝜕,𝜕𝜕𝐿𝐿� =

𝐶𝐶𝜕𝜕𝜕𝜕 ∏ (𝑝𝑝𝑖𝑖𝜕𝜕𝑖𝑖𝑐𝑐)𝑖𝑖 . 𝑒𝑒−𝛽𝛽𝜑𝜑𝑛𝑛,𝑇𝑇

𝑐𝑐

𝛺𝛺𝐿𝐿 , 𝑖𝑖 = 𝑎𝑎 ,𝑚𝑚 (5.16)

where 𝜑𝜑𝜕𝜕,𝜕𝜕𝐿𝐿 denotes the potential function in a cation-centered domain with 𝑙𝑙 number of

anions out of 𝑇𝑇 total coordination number; 𝑇𝑇𝑖𝑖𝐿𝐿, 𝜖𝜖𝑖𝑖𝐿𝐿, and 𝜎𝜎𝑖𝑖𝐿𝐿 denote, respectively, the

coordination number, interaction strength, and effective molecular diameter of species 𝑖𝑖

around the center cation; 𝑝𝑝𝑖𝑖 is the probability of selecting an 𝑖𝑖-type species from the infinite

pool; 𝛺𝛺𝐿𝐿 is the normalization factor as the summation over all possible configurations of

species in a cation-centered domain 24 (Equivalent to the local partition function); and 𝛽𝛽 is

the inverse of the thermal energy.

By expanding Eq. 5.16 and introducing the new terms (𝑝𝑝𝑖𝑖′), we arrive at Eq. 5.20,

following Eqs 17-19.

𝑓𝑓𝜕𝜕,𝜕𝜕𝐿𝐿� =

𝐶𝐶𝜕𝜕𝜕𝜕

𝛺𝛺𝐿𝐿�(𝑝𝑝𝑖𝑖𝜕𝜕𝑖𝑖𝑐𝑐

𝑖𝑖

. 𝑒𝑒𝛽𝛽𝜕𝜕𝑖𝑖𝑐𝑐𝜖𝜖𝑖𝑖𝑐𝑐𝜎𝜎𝑖𝑖𝑐𝑐

3

(𝐿𝐿(𝑐𝑐))3 ), 𝑖𝑖 = 𝑎𝑎 ,𝑚𝑚 (5.17)

𝑝𝑝𝑎𝑎 ′ = 𝑝𝑝𝑎𝑎𝑒𝑒

𝛽𝛽𝜖𝜖𝑎𝑎𝑐𝑐𝜎𝜎𝑎𝑎𝑐𝑐3

(𝐿𝐿(𝑐𝑐))3 (5.18)

𝑝𝑝𝑚𝑚 ′ = 𝑝𝑝𝑚𝑚𝑒𝑒

𝛽𝛽𝜖𝜖𝑚𝑚𝑐𝑐𝜎𝜎𝑚𝑚𝑐𝑐3

(𝐿𝐿(𝑐𝑐))3 (5.19)


𝐶𝐶𝜕𝜕𝜕𝜕

𝛺𝛺𝐿𝐿 (𝑝𝑝𝑎𝑎′ )𝜕𝜕. (𝑝𝑝𝑚𝑚′ )𝜕𝜕−𝜕𝜕 (5.20)

Dividing 𝑝𝑝𝑚𝑚 ′ by 𝑝𝑝𝑎𝑎

′ ,we arrive at Eq. 5.21:

𝑝𝑝𝑚𝑚′

𝑝𝑝𝑎𝑎′=𝑝𝑝𝑚𝑚𝑝𝑝𝑎𝑎

𝑒𝑒−𝛽𝛽�𝜖𝜖𝑎𝑎𝑐𝑐𝜎𝜎𝑎𝑎𝑐𝑐

3−𝜖𝜖𝑚𝑚𝑐𝑐𝜎𝜎𝑚𝑚𝑐𝑐3

(𝐿𝐿(𝑐𝑐))3� (5.21)

The terms 𝑝𝑝𝑚𝑚′

𝑝𝑝𝑎𝑎′ and 𝑝𝑝𝑚𝑚

𝑝𝑝𝑎𝑎 represent, respectively, the ratio of the probability of finding

molecules to anions in the local domain and in the bulk. Eq. 5.21 can then be expressed in

terms of the number of species in the local domain and in bulk as per Eq. 5.22.


151

(𝑁𝑁𝑚𝑚𝐿𝐿/𝑁𝑁𝑡𝑡𝑙𝑙𝐿𝐿)(𝑁𝑁𝑎𝑎𝐿𝐿𝑍𝑍𝑎𝑎/𝑁𝑁𝑡𝑡𝑙𝑙𝐿𝐿)

= �(𝑁𝑁𝑚𝑚/𝑁𝑁𝑡𝑡)

(𝑁𝑁𝑎𝑎𝑍𝑍𝑎𝑎/𝑁𝑁𝑡𝑡)� 𝑒𝑒

−𝛽𝛽�𝜖𝜖𝑎𝑎𝑐𝑐𝜎𝜎𝑎𝑎𝑐𝑐3−𝜖𝜖𝑚𝑚𝑐𝑐𝜎𝜎𝑚𝑚𝑐𝑐

3

(𝐿𝐿(𝑐𝑐))3� (5.22)

Here, 𝑁𝑁𝑡𝑡𝑙𝑙𝐿𝐿 and 𝑁𝑁𝑡𝑡 denote, respectively, the total number of species in the local domain

and in the bulk. On the other hand, Eq. 5.10 for a cation-centered domain can be expressed

explicitly as per Eq. 5.23.

𝑋𝑋𝑚𝑚𝐿𝐿𝑋𝑋𝑎𝑎𝐿𝐿

= �𝑋𝑋𝑚𝑚𝑋𝑋𝑎𝑎� 𝑒𝑒�−𝛼𝛼(𝑐𝑐)𝜏𝜏𝑚𝑚𝑐𝑐,𝑎𝑎𝑐𝑐� (5.23)

From the equivalency of Eq. 5.22 and Eq. 5.23, we arrive at the expression

representing the 𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 as a function of physical quantities.

𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 = 𝛽𝛽𝛼𝛼(𝑐𝑐)

𝜖𝜖𝑎𝑎𝑐𝑐𝜎𝜎𝑎𝑎𝑐𝑐3−𝜖𝜖𝑚𝑚𝑐𝑐𝜎𝜎𝑚𝑚𝑐𝑐3

(𝐿𝐿(𝑐𝑐))3 (5.24)

where 𝛼𝛼(𝐿𝐿) represents the nonrandomness factor in a cation-centered domain.

The τ Parameter in an Anion-Centered Domain In a similar fashion, the binary interaction parameter of an anion-centered domain,

𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎, can be expressed as follows:

𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎 = 𝛽𝛽𝛼𝛼(𝑎𝑎)

𝜖𝜖𝑐𝑐𝑎𝑎𝜎𝜎𝑐𝑐𝑎𝑎3 −𝜖𝜖𝑚𝑚𝑎𝑎𝜎𝜎𝑚𝑚𝑎𝑎3

(𝐿𝐿(𝑎𝑎))3 (5.25)

Here, 𝛼𝛼(𝑎𝑎) denotes the nonrandomness factor in an anion-centered domain.

The τ Parameters in a Molecule-Centered Domain Consider there are a total 𝑇𝑇 number of species locally coordinated around a center

molecule, out of which there are 𝑘𝑘 number of surrounding molecules. The remaining (𝑇𝑇 −

𝑘𝑘) coordination spots are distributed between the cations and anions following the local

charge neutrality as follows:


152

𝑍𝑍𝐿𝐿𝑁𝑁𝐿𝐿𝑚𝑚 = 𝑍𝑍𝑎𝑎𝑁𝑁𝑎𝑎𝑚𝑚 (5.26)

The probability function of the non-random arrangement of the molecule-centered

domain is thus expressed as per Eq. 5.27.

𝑓𝑓𝑘𝑘,𝜕𝜕𝑚𝑚� =

𝐶𝐶𝑘𝑘𝜕𝜕

𝛺𝛺𝑚𝑚�(𝑝𝑝𝑖𝑖𝑁𝑁𝑖𝑖𝑚𝑚

𝑖𝑖

. 𝑒𝑒𝛽𝛽𝑁𝑁𝑖𝑖𝑚𝑚𝜖𝜖𝑖𝑖𝑚𝑚𝜎𝜎𝑖𝑖𝑚𝑚

3

(𝐿𝐿(𝑚𝑚))3 ), 𝑖𝑖 = 𝑐𝑐,𝑎𝑎 ,𝑚𝑚 (5.27)

where 𝑁𝑁𝑚𝑚𝑚𝑚 denotes the local number of molecules around the center molecule; 𝐶𝐶𝑘𝑘𝜕𝜕 is the

combinatorial factor; and 𝛺𝛺𝑚𝑚 is the normalization factor.

By introducing the 𝑝𝑝𝑖𝑖′ terms defined by Eq. 5.28 and rearranging the relations, we

arrive at Eq. 5.29 and Eq. 5.30

𝑝𝑝𝑖𝑖′ = 𝑝𝑝𝑖𝑖 𝑒𝑒

𝛽𝛽𝜖𝜖𝑖𝑖𝑚𝑚𝜎𝜎𝑖𝑖𝑚𝑚3

(𝐿𝐿(𝑚𝑚))3 , 𝑖𝑖 = 𝑐𝑐,𝑎𝑎,𝑚𝑚 (5.28)

𝑝𝑝𝑎𝑎′

𝑝𝑝𝑚𝑚′=𝑝𝑝𝑎𝑎𝑝𝑝𝑚𝑚


3 −𝜖𝜖𝑎𝑎𝑚𝑚𝜎𝜎𝑎𝑎𝑚𝑚3

(𝐿𝐿(𝑚𝑚))3�

(5.29)

𝑝𝑝𝐿𝐿′

𝑝𝑝𝑚𝑚′=𝑝𝑝𝐿𝐿𝑝𝑝𝑚𝑚


3 −𝜖𝜖𝑐𝑐𝑚𝑚𝜎𝜎𝑐𝑐𝑚𝑚3

(𝐿𝐿(𝑚𝑚))3�

(5.30)

Similar to the previous arguments for ion-centered domains, Eq. 5.29 and Eq. 5.30 are

equivalent to the following equations as variants of the Eq. 5.8 for, respectively, cations

and anions around the center domain molecule.

𝑋𝑋𝐿𝐿𝑚𝑚𝑋𝑋𝑚𝑚𝑚𝑚

= �𝑋𝑋𝐿𝐿𝑋𝑋𝑚𝑚

� exp�−𝛼𝛼(𝑚𝑚)𝜏𝜏𝐿𝐿𝑚𝑚� (5.31)

𝑋𝑋𝑎𝑎𝑚𝑚𝑋𝑋𝑚𝑚𝑚𝑚

= �𝑋𝑋𝑎𝑎𝑋𝑋𝑚𝑚

� exp�−𝛼𝛼(𝑚𝑚)𝜏𝜏𝑎𝑎𝑚𝑚� (5.32)

Hence, we arrive at the binary interaction parameters of cations and anions around a

center molecule as follows:


153

𝜏𝜏𝐿𝐿𝑚𝑚 = 𝛽𝛽𝛼𝛼(𝑚𝑚)

𝜖𝜖𝑚𝑚𝑚𝑚𝜎𝜎𝑚𝑚𝑚𝑚3 −𝜖𝜖𝑐𝑐𝑚𝑚𝜎𝜎𝑐𝑐𝑚𝑚3

(𝐿𝐿(𝑚𝑚))3 (5.33)

𝜏𝜏𝑎𝑎𝑚𝑚 = 𝛽𝛽𝛼𝛼(𝑚𝑚)

𝜖𝜖𝑚𝑚𝑚𝑚𝜎𝜎𝑚𝑚𝑚𝑚3 −𝜖𝜖𝑎𝑎𝑚𝑚𝜎𝜎𝑎𝑎𝑚𝑚3

(𝐿𝐿(𝑚𝑚))3 (5.34)

5.4. Quantification of the τ Parameters from Molecular Simulations

In order to quantify the binary interaction parameters expressed by Eqs. 5.24 & 5.25

and Eqs. 5.33 & 5.34, the effective molecular diameters (𝜎𝜎𝑖𝑖𝑗𝑗), first neighbor shell radii of

each domain (𝑅𝑅(𝑖𝑖)), interaction strengths (𝜖𝜖𝑖𝑖𝑗𝑗), and the nonrandomness factors (𝛼𝛼(𝑖𝑖)) are

obtained from the MD simulations.

The Effective Molecular Diameters (σ) The effective molecular diameter of each pair of interest was taken as the position of

the first peak in the pertinent radial distribution function (RDF) 32, including that of the

cation-molecule (𝜎𝜎𝐿𝐿𝑚𝑚), anion-molecule (𝜎𝜎𝑎𝑎𝑚𝑚), anion-cation (𝜎𝜎𝑎𝑎𝐿𝐿), and molecule-molecule

(𝜎𝜎𝑚𝑚𝑚𝑚).

The First Neighbor Shell Radii (R) The location of the first minimum in the RDF of each pair of species, say the 𝑖𝑖 − 𝑗𝑗

pair, indicates the extent of the first neighbor shell of 𝑗𝑗-type around 𝑖𝑖 or 𝑖𝑖-type around 𝑗𝑗 23.

We designate such a distance as the 𝑅𝑅𝑓𝑓𝑚𝑚𝑖𝑖−𝑗𝑗 for the sake of simplicity. Thereby it may be

inferred that the radius of the first solvation shell around a cation-centered domain can be

taken as the larger of the two quantities, 𝑅𝑅𝑓𝑓𝑚𝑚𝑎𝑎−𝐿𝐿 and 𝑅𝑅𝑓𝑓𝑚𝑚𝑚𝑚−𝐿𝐿 (see Figure 5.1). Such a treatment,

though rationalized in molecular simulation studies 33, could potentially be in conflict with

the like-ion repulsion assumption of the eNRTL model. To overcome this issue in our

previous study 24, the like-ion repulsion assumption was strictly imposed to obtain the


154

radius of the first solvation shell of the cation- and anion-centered domains as the longest

distance beyond which, the local mole numbers of the same-charged ions were nonzero.

On the other hand, enforcing the like-ion repulsion assumption can raise inconsistencies

for some model systems such as aqueous BaCl2 solutions where the acquired radius could

be smaller than the effective molecular diameters. In order to avoid conducting subjective

studies with different treatments for various aqueous solutions, the location of the first

minimum in the RDF of the pertinent pair is taken as the first shell radius. The like-ion

repulsion assumption for each model system is then evaluated to assess its validity

throughout the entire concentration range. A similar argument can be made for the

electroneutrality assumption where instead of imposing strictly, the first solvation shell

radius for calculating the 𝜏𝜏𝐿𝐿𝑚𝑚 and 𝜏𝜏𝑎𝑎𝑚𝑚 is taken, respectively, as the average between the

largest of the two quantities, (𝑅𝑅𝑓𝑓𝑚𝑚𝐿𝐿−𝑚𝑚 and 𝑅𝑅𝑓𝑓𝑚𝑚𝑚𝑚−𝑚𝑚) and (𝑅𝑅𝑓𝑓𝑚𝑚𝑎𝑎−𝑚𝑚 and 𝑅𝑅𝑓𝑓𝑚𝑚𝑚𝑚−𝑚𝑚). The

electroneutrality assumption is then evaluated in the entire range of concentration for the

four model systems.

The Energetic Interactions (ϵ) The energetic interaction strengths of an 𝑖𝑖 − 𝑗𝑗 pair, also known as the characteristic

energies (𝜖𝜖𝑖𝑖𝑗𝑗), were obtained from the PMF of the pertinent pair. The value of 𝜖𝜖𝑖𝑖𝑗𝑗 was taken

as the first barrier height in the free energy surface of the 𝑖𝑖 − 𝑗𝑗 pair, which is the difference

between the PMF at the shell radius, 𝑅𝑅(𝑖𝑖), and that at the contact distance, 𝜎𝜎𝑖𝑖𝑗𝑗. As mentioned

previously, this difference is attributed to the free energy required to form or disrupt the

first solvation shell, consistent with the definition of the characteristic energy discussed in

two-liquid theory 30.


155

Using the total potential of mean force to determine the quantities of the characteristic

energies could raise concern about the inconsistency between the nature of the 𝜖𝜖𝑖𝑖𝑗𝑗 in the

theory, versus that in the MD simulations. The energetic interactions considered by the

eNRTL, shown in Figure 5.1, are considered to be strictly of van der Waals type, whereas

the potential of mean force from MD simulations are a combination of both electrostatics

and van der Waals contributions. In spite of this apparent discrepancy, using the total

potential of mean force is rationalized; the reason for which is two-fold. The configuration

of the local molecular domains hypothesized by the eNRTL theory depends on the structure

of the liquid, which is governed by both contributions. Second, due to the strong screening

effects predicted by the Debye-Hückel theory, the Lennard-Jones contribution in the local

structure of domains are anticipated to be predominant, even at low concentrations studied

here.

To verify such a rationale, a second case in presented here where the contribution of

the Coulombic electrostatic forces in the total PMF is approximated by solving the

Poisson’s equation. It is then subtracted from the total potential of mean force to obtain the

PMF governed only by the short-range van der Waals forces. While a brief description of

the approach is discussed here, we refer the reader to the literature for a detailed discussion

34. According to the Debye-Hückel’s first approximation, the potential of mean force of the

𝑗𝑗 − 𝑖𝑖 pair, 𝜑𝜑𝑗𝑗→𝑖𝑖(𝑟𝑟), where 𝑖𝑖 is the center species in any arbitrary domain, can be written by

the following equation:

𝜑𝜑𝑗𝑗→𝑖𝑖(𝑟𝑟) = 𝑞𝑞𝑗𝑗∅𝑖𝑖(𝑟𝑟) (5.35)


156

Here, ∅𝑖𝑖(𝑟𝑟) is the average electrostatic potential acting upon the particle 𝑖𝑖 in the presence

of all the species in the solution, in a canonical ensemble; and 𝑞𝑞𝑗𝑗 is the charge of the species

𝑗𝑗. The term ∅𝑖𝑖(𝑟𝑟) itself satisfies the Poisson’s equation 34 as follows:

∇2∅𝑖𝑖(𝑟𝑟) =−4𝜋𝜋𝜀𝜀 �𝑐𝑐𝑗𝑗𝑞𝑞𝑗𝑗𝑒𝑒

−𝛽𝛽𝜑𝜑𝑗𝑗→𝑖𝑖(𝑟𝑟)

𝑗𝑗

(5.36)

where 𝑐𝑐𝑗𝑗 is the bulk number density of the 𝑗𝑗-type species; and 𝜀𝜀 is the dielectric constant.

Following the linearization approximation of the Debye-Hückel theory, the exponential

term can be expanded as follows:

�𝑐𝑐𝑗𝑗𝑞𝑞𝑗𝑗𝑒𝑒−𝛽𝛽𝜑𝜑𝑖𝑖→𝑖𝑖(𝑟𝑟� ≈�𝑐𝑐𝑗𝑗𝑞𝑞𝑗𝑗 − 𝛽𝛽𝑗𝑗𝑗𝑗

�𝑐𝑐𝑗𝑗𝑞𝑞𝑗𝑗2𝜑𝜑𝑗𝑗→𝑖𝑖(𝑟𝑟�𝑗𝑗

(5.37)

Note that the second term in the right-hand side is zero as per the electroneutrality in the

solution. Hence the Poisson’s equation can be summarized as below:

∇2∅𝑖𝑖(𝑟𝑟) = 𝜅𝜅2𝜑𝜑𝑗𝑗→𝑖𝑖(𝑟𝑟), 𝑟𝑟 > 𝑎𝑎

(5.38)

∇2∅𝑖𝑖(𝑟𝑟) = 0, 0 < 𝑟𝑟 ≤ 𝑎𝑎 (5.39)

where

𝜅𝜅 = �

4𝜋𝜋𝛽𝛽𝜀𝜀

�𝑞𝑞𝑗𝑗2𝑐𝑐𝑗𝑗𝑗𝑗

(5.40)

and 𝑎𝑎 is referred to as the distance of closest approach between ions. In the primitive model

34, where the ions are considered to be hard spheres, the quantity 𝑎𝑎 for a pair of species can

be regarded as the arithmetic mean of their molecular diameters. However, in order to

maintain the actual structure of the system governed by both electrostatics and van der

Waals forces, the distance of closest approach between ions are obtained from the structure

of the system given by the MD simulations. For this purpose, the exponential term in Eq.


157

5.36 is replaced with the radial distribution function 35 so the Poisson’s equation can be

written as follows.

∇2∅𝑖𝑖(𝑟𝑟) =−4𝜋𝜋𝜀𝜀 �𝑐𝑐𝑗𝑗𝑞𝑞𝑗𝑗𝑔𝑔𝑗𝑗𝑖𝑖

𝑗𝑗

(𝑟𝑟) (5.41)

From Eq. 5.39 and Eq. 5.41, we can see that the quantity 𝑎𝑎 for each pair can be taken

as the distance after which, the 𝑔𝑔(𝑟𝑟) is nonzero. Finally, by solving Eqs. 5.38 & 5.39, the

potential of mean force of the 𝑗𝑗 − 𝑖𝑖 in the presence of all the charged entities in the solution

are expressed as below:

𝜑𝜑𝑗𝑗→𝑖𝑖(𝑟𝑟) = 𝑞𝑞𝑗𝑗𝑞𝑞𝑖𝑖 �1𝜀𝜀𝑟𝑟𝑖𝑖𝑖𝑖

− 𝜅𝜅𝜀𝜀(1+𝜅𝜅𝑎𝑎)

�, 0 < 𝑟𝑟 ≤ 𝑎𝑎 (5.42)

𝜑𝜑𝑗𝑗→𝑖𝑖(𝑟𝑟) = 𝑞𝑞𝑗𝑗𝑞𝑞𝑖𝑖𝑒𝑒−𝜅𝜅(𝑟𝑟𝑖𝑖𝑖𝑖−𝑎𝑎)

𝜀𝜀𝑟𝑟𝑖𝑖𝑖𝑖(1+𝜅𝜅𝑎𝑎), 𝑟𝑟 > 𝑎𝑎 (5.43)

Note that the obtained PMF is only due to the average electrostatic potential in the system,

while maintaining the structure of the solution.

The Nonrandomness Factor (α) The nonrandomness factor (𝛼𝛼) for electrolytes is typically considered as an empirical

constant set to be 0.2. As discussed in our previous study 24, Renon and Prausnitz 29

suggested that from the analogy between the non-random two-fluid theory and the quasi-

chemical theory of Guggenheim 36 the nonrandomness factor can be inferred as 1/𝑧𝑧(𝑖𝑖)

where 𝑧𝑧(𝑖𝑖) denotes the total coordination number of species in an 𝑖𝑖-center domain. A

definition as such provides the possibility of calculating the 𝛼𝛼(𝑖𝑖) directly from the MD

simulations by reciprocating the corresponding coordination numbers.


158

The Binary Interaction Parameters (τ) Finally, the binary interaction parameters were calculated using the quantities

discussed above using Eqs. 5.24 & 5.25 and Eqs. 5.33 & 5.34.

5.5. Molecular Simulation Details

The molecular simulations were performed separately for aqueous BaCl2, SrCl2,

CaCl2, and MgCl2 solutions using the LAMMPS package 37. For each system, the

simulations were carried out over a wide concentration range from 0.5 to 12 mol/kg at

298.15 K and 1 bar. Hereafter, we designate by 𝑚𝑚 the concentration of the salt solutions in

molality. To be consistent with our previous study for uni-univalent electrolytes 24 and

enabling rapid computations for practical industrial use, a relatively small system size

consisted of 1000 water molecules were considered for the simulations, with the number

of electrolytes varying as per the salt concentration. It was already shown 24 that the finite

size effect for calculating the binary interaction parameters are negligible.

Initial equilibration was carried out at a constant number of particles, volume, and

temperature (canonical ensemble) followed by an isothermal-isobaric (NPT) equilibration

of 10 ns long using Nośe-Hoover thermostat and barostat 38. The NPT production runs were

then carried out for the duration of 40 ns. The particle-particle particle-mesh (PPPM) 39

was used to calculate the long-range electrostatic interactions and tail corrections were used

for the long-range part of Lennard-Jones interactions. Lennard-Jones and Coulombic

interactions were truncated at the distance of 14 Å. The SPC/E 40 model was used to

represent the water molecules. For ionic species, the Kirkwood-Buff force field (KBFF)

parameters were employed from a recent study by Naleem et al. 41 reporting the KBFF of

several alkaline halide salts, including those selected as our model systems. KBFF


159

parameters have been previously shown to predict successfully the mean ionic activity

coefficients of electrolyte solutions 12,24,42.

The trajectories of the 40 ns production run were divided into 10 blocks of 4 ns length

and the physical quantities required for identifying the 𝜏𝜏 parameters were then calculated

at each block. The quantity of each variable was calculated by averaging over the blocks

and the associated standard deviations were taken as the statistical uncertainties.

The potential of mean force of each pair of interest was obtained from the pertinent

RDF as per Eq. 5.44 43.

𝜙𝜙(𝑟𝑟) = −𝑅𝑅𝑇𝑇 ln𝑔𝑔(𝑟𝑟) (5.44)

Previously, Hossain et al. 12 reported that the correlation times for water molecules are

shorter than 25 ps. Since our simulations include sufficient number of correlations as

compared to those required for equilibration of water molecules, obtaining the PMF from

the RDF is rationalized and enables rapid free energy calculations, in lieu of employing

any types of enhanced sampling techniques.

5.6. Results and Discussion

Each of the liquid structure and energetic interaction quantities required for calculating

the 𝜏𝜏 parameters are obtained from the molecular simulations as explained previously.

These quantities include the effective molecular dimeter of the species (𝜎𝜎), the radius of

the first solvation shell of each local domain (𝑅𝑅), characteristic energies (𝜖𝜖), and the

nonrandomness factors (𝛼𝛼). The results and a comprehensive discussion on the physical

significance of the variables are reported herein, while more details are presented in the

Supplementary Information.


160

The Results of the Physical Quantities of the Aqueous Solutions from MD Simulations

The effective molecular diameters—𝜎𝜎𝑎𝑎𝐿𝐿, 𝜎𝜎𝐿𝐿𝑚𝑚, 𝜎𝜎𝑎𝑎𝑚𝑚, and 𝜎𝜎𝑚𝑚𝑚𝑚—for aqueous BaCl2,

SrCl2, CaCl2, and MgCl2 solutions are reported in Tables 5.9.1-5.9.4, in the

Supplementary Information, obtained from the location of the first peak in the RDF of

each pair of interest. Furthermore, Figures 5.9.1-5.9.4 in the Supplementary

Information, as examples, illustrate one such RDF for each of the Ba2+-Clˉ, Sr2+-Clˉ, Ca2+-

Clˉ, and Mg2+-Clˉ pairs at the salt concentration of 4 m. As it can be observed from the

RDFs, the dominancy and sharpness of the first peaks, attributed to the contact ion pairs

(CIP), are reduced as the charge density of the cation increases. In the aqueous solution

with the smallest cationic size (Mg2+), there is only one prominent peak at the distance of

4.50 Å attributed to solvent-separated ion pair (SSIP). On the other hand, there exists no

peak corresponding to the contact ion pair (CIP). Such an observation has been confirmed

by both simulation 41,44,45 and experimental 46 studies. This can be rationalized due to the

high charge density of the Mg2+, which causes the cations to bound strongly to the water

molecules, forming fully hydrated complex ions. The CIP peak in the aqueous MgCl2

remains absent even at the highest concentration studied, i.e., 12 m salt.

With increasing the size of the cation, and hence a decrease in the charge density, the

anion is substituted with water molecules forming contact ion pairs. The CIP peak is thus

observed in the RDF of the cation-anion pair in the aqueous CaCl2 solution, however, not

significantly at lower concentrations. This explains the shift in the reported 𝜎𝜎 values from

4.94 Å (SSIP) to 2.90 Å (CIP) beyond the concentration of 4 m. In this study, the CIP peaks

with the 𝑔𝑔 (𝑟𝑟) values below unity are neglected in calculating the binary interaction

parameters and subsequently the SSIP peaks are used. In the aqueous SrCl2 solution, the


161

RDF of the anion-cation pair illustrates the CIP peak above 2 m concentrations. Finally, in

the aqueous BaCl2 solution, the predominant peak is attributed to the CIP, throughout the

entire concentration range.

The first neighbor shell radii of the cation-, anion-, and molecule-centered domains—

𝑅𝑅(𝐿𝐿), 𝑅𝑅(𝑎𝑎), and 𝑅𝑅(𝑚𝑚)—are reported in Tables 5.9.5-5.9.8 in the Supplementary

Information for the four model systems at each concentration. In order to examine the

like-ion repulsion and electroneutrality assumptions, the related effective mole fractions

(X) are calculated as follows. First, the mole numbers of the species in the three local

domains are calculated by integrating over the RDF of pertinent pairs involving the center

species 𝑖𝑖, from the following equation:

𝑁𝑁(𝑟𝑟)𝑗𝑗𝑖𝑖𝑙𝑙𝑜𝑜𝐿𝐿𝑎𝑎𝑙𝑙 = 4𝜋𝜋𝜌𝜌𝑗𝑗𝑏𝑏𝑏𝑏𝑙𝑙𝑘𝑘 �𝑟𝑟′2

𝑟𝑟

0

𝑔𝑔𝑗𝑗𝑖𝑖(𝑟𝑟′)𝑑𝑑𝑟𝑟′ (5.45)

where 𝜌𝜌𝑗𝑗𝑏𝑏𝑏𝑏𝑙𝑙𝑘𝑘 is the bulk number density of 𝑗𝑗-type species; and 𝑁𝑁𝑗𝑗𝑖𝑖𝑙𝑙𝑜𝑜𝐿𝐿𝑎𝑎𝑙𝑙 is the integrated

number of 𝑗𝑗-type species around the center species 𝑖𝑖, over the distance of 𝑟𝑟. Using Eqs. 5.3

& 5.4, the effective mole fractions (𝑋𝑋𝑖𝑖𝑗𝑗) are then obtained for each local domain. Following

Eqs. 5.5 & 5.6 for the like-ion repulsion assumption, 𝑋𝑋𝐿𝐿𝐿𝐿 and 𝑋𝑋𝑎𝑎𝑎𝑎 are calculated at each

concentration within the first solvation shell radii and reported in Tables 5.1-5.4 in

percentage. Such quantities demonstrate the extent to which the structure of the local

domains hypothesized by the eNRTL model deviates from that specified from the MD

simulations. We reiterate the quantities 𝑋𝑋𝐿𝐿𝐿𝐿 and 𝑋𝑋𝑎𝑎𝑎𝑎 are assumed to be zero in the eNRTL

model.

According to the tables, the like ion-repulsion assumption for the cation-centered

domain is shown to be reasonable to a great extent for all of the model systems. Within the


162

first solvation shell identified from the MD simulations, the 𝑋𝑋𝐿𝐿𝐿𝐿 % is shown to be zero for

aqueous BaCl2 and SrCl2 solutions in the entire concentration range studied. For aqueous

CaCl2 and MgCl2 solutions, the largest 𝑋𝑋𝐿𝐿𝐿𝐿 % is reported to be 2.4 and 4.4 %, respectively.

For the anion-centered domain, on the other hand, the like-ion repulsion assumption is only

qualitatively satisfied.

Table 5.1. Effective local mole fractions in the first shell (Å) – BaCl2 (aq)

Table 5.2. Effective local mole fractions in the first shell (Å) – SrCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

% 𝑋𝑋𝐿𝐿𝐿𝐿 0.0 0.0 0.0 0.0 0.0 0.0 0.0

% 𝑋𝑋𝑎𝑎𝑎𝑎 0.0 31.8 31.9 32.0 31.3 31.8 32.5

% |𝑋𝑋𝑎𝑎𝑚𝑚 − 𝑋𝑋𝐿𝐿𝑚𝑚| 0.4 0.7 1.4 2.0 3.6 5.2 2.5


Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

% 𝑋𝑋𝐿𝐿𝐿𝐿 0.0 0.0 0.0 0.0 0.0 0.0 0.0

% 𝑋𝑋𝑎𝑎𝑎𝑎 0.8 0.0 0.1 2.8 13.3 15.5 20.5

% |𝑋𝑋𝑎𝑎𝑚𝑚 − 𝑋𝑋𝐿𝐿𝑚𝑚| 0.3 0.4 1.6 4.5 7.0 8.0 8.6



163

Table 5.3. Effective local mole fractions in the first shell (Å) – CaCl2 (aq)

Table 5.4. Effective local mole fractions in the first shell (Å) – MgCl2 (aq)

To examine the electroneutrality assumption, the absolute difference between the

effective mole fractions of anions and cation around a center molecule, |𝑋𝑋𝑎𝑎𝑚𝑚 − 𝑋𝑋𝐿𝐿𝑚𝑚|, is

reported in percentage. Note that in the eNRTL model, it is assumed that the net electric

charge in a molecule-centered domain is zero. Following the reported quantities in the

table, the electroneutrality assumptions is shown to be reasonable for all of the aqueous salt

solutions. Figures 5.2-5.4 illustrate the effective mole fraction of the species in the three

local domains together with the location of the first solvation shell.

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

% 𝑋𝑋𝐿𝐿𝐿𝐿 0.0 0.1 0.5 2.4 0.0 0.0 0.0

% 𝑋𝑋𝑎𝑎𝑎𝑎 0.8 4.1 8.7 13.2 5.9 12.9 17.5

% |𝑋𝑋𝑎𝑎𝑚𝑚 − 𝑋𝑋𝐿𝐿𝑚𝑚| 0.3 0.6 0.1 1.3 5.0 8.9 11.0


Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

% 𝑋𝑋𝐿𝐿𝐿𝐿 0.0 0.0 0.0 0.1 1.3 3.5 4.4

% 𝑋𝑋𝑎𝑎𝑎𝑎 0.7 3.5 7.8 12.1 16.1 18.9 21.5

% |𝑋𝑋𝑎𝑎𝑚𝑚 − 𝑋𝑋𝐿𝐿𝑚𝑚| 0.3 0.8 3.7 3.2 3.8 3.6 7.0



164

Figure 5.2. The effective mole fractions calculated from the MD simulations for the aqueous BaCl2 solution at 4 m — Cation-centered domain, where Xji denotes the effective mole fraction of j-type around a center i-type.


165

Figure 5.3. The effective mole fractions calculated from the MD simulations for the aqueous BaCl2 solution at 4 m — Anion-centered domain, where Xji denotes the effective mole fraction of j-type around a center i-type.


166

Figure 5.4. The effective mole fractions calculated from the MD simulations for the aqueous BaCl2 solution at 4 m — Molecule-centered domain, where Xji denotes the effective mole fraction of j-type around a center i-type.

The interaction strength (𝜖𝜖 value) of each pair is obtained from the PMF plots as

described previously. Figures 5.5 and 5.6 demonstrate, as examples, the PMF plots of the

Mg2+-Clˉ and water-water pairs, and Mg2+-water and Clˉ-water pairs, respectively, at the

concentration of 4 m. Tables 5.9.9-5.9.12 in Supplementary Information list the 𝜖𝜖 values

and their associated uncertainties for the four aqueous solutions at 298.15 K.


167

Figure 5.5. A schematic of the potential of mean force for the 4 m aqueous MgCl2 solution at 298.15 K. PMF of Mg2+ around a center Clˉ ( ) and water around a center water (

). The symbols represent the calculated PMF and lines denote the smoothed PMF results using cubic splines; ϵji is the depth of the energy barriers or effective interaction strength for species j-type around a center i-type.


168

Figure 5.6. A schematic of the potential of mean force for the 4 m aqueous MgCl2 solution at 298.15 K. PMF of Mg2+ around a center water ( ) and Clˉ around a center water (

). The symbols represent the calculated PMF and lines denote the smoothed PMF results using cubic splines; ϵji is the depth of the energy barriers or effective interaction strength for species j-type around a center i-type.

At low concentrations, the interaction strength between the anion-cation pair increases

with an increase in the salt concentrations, expectedly, as the dipole moments are enhanced.

However, at higher concentrations, the 𝜖𝜖𝑎𝑎𝐿𝐿 exhibits a plateau behavior due to the ion

screening effect which has been addressed in our previous study 24. This changing in the

trend of the interaction strengths can also be observed for 𝜖𝜖𝐿𝐿𝑚𝑚 and 𝜖𝜖𝑎𝑎𝑚𝑚. Adding more salts

to the solution will lead the interaction strengths to reach a plateau in a fully screened

structure as the local domains are saturated. Unexpectedly, 𝜖𝜖𝑚𝑚𝑚𝑚 in the aqueous MgCl2

solution shows a monotonic increase throughout the concentration range. Though this may

be due to the fact that polarized water molecules present in the hydrated cation shells tend

to attract each other more strongly, the reliability of the 𝜖𝜖𝑚𝑚𝑚𝑚 values at high concentrations


169

is not ensured. We note that the force field parameters failed to capture the experimental

data for concentrations beyond 4 m 41.

Furthermore, the contribution of the local van der Waals forces in the total potential of

mean force is also calculated as explained previously. Figures 5.7 and 5.8 show the PMF

of the anion-cation pair as a function of center to center distance, in the aqueous BaCl2

solution at the concentration of 2 m and 6 m, respectively. Also shown in the figure are the

contributions of both electrostatic and local van der Waals forces in the total PMF. As

demonstrated, the depth of the free energy barrier of the short-range PMF is smaller than

that of the total PMF due the subtraction of the electrostatic forces. Also, by comparing

Figures 5.7 and 5.8, it is observed that by increasing the concentration from 2 m to 6 m,

the electrostatic forces are screened out significantly; thus, the local structure is governed

predominantly by the local short-range forces. That confirms our hypothesis that the

structure of the local domains is attributed to the short-range van der Waals forces. Figure

5.9 further demonstrates the effect of adding salts to the decrease of the electrostatic forces.

It can be observed that the so-called long-range interactions at high concentrations are

essentially diminished within the first solvation shell.


170

Figure 5.7. Potential of mean force of the Ba2+-Cl- pair in aqueous BaCl2 solution. The potential of mean force of the anion-cation pair at the concentration of 2 m from the local van der Waals (—), electrostatics (—), and the combination of both local and electrostatics (—) contributions.


171

Figure 5.8. Potential of mean force of the Ba2+-Cl- pair in aqueous BaCl2 solution. The potential of mean force of the anion-cation pair at the concentration of 6 m from the local van der Waals (—), electrostatics (—), and the combination of both local and electrostatics (—) contributions.


172

Figure 5.9. Potential of mean force of the Ba2+-Cl- pair in aqueous BaCl2 solution. The potential of mean force attributed to the electrostatic forces calculated from the Debye-Hückel theory at different concentrations: (—) 0.5 m, (—) 2 m, (—) 4 m, (—) 6 m, (—) 8 m, (—) 10 m, (—) 12 m.

Tables 5.9.13-5.9.16 in the Supplementary Information list the 𝛼𝛼 value of each

domain throughout the concentration range. As reported, the nonrandomness values lie

between 0.1 and 0.2 for the aqueous BaCl2 and SrCl2 solutions, showing qualitative

agreement between the empirical constant value of the regression-based method, typically

fixed at 0.2, and those from the MD simulations. Similar values are observed for the

aqueous CaCl2 solution at moderate to high concentrations. At lower concentrations, the

nonrandomness factors for the cation- and anion-centered domains are significantly

smaller. That is due to the hydration of the ions and consequently the larger quantities of

the first solvation shells, enhancing the total coordination number of species and thus small

numbers for the nonrandomness factors.


173

Calculating the Binary Interaction Parameters from MD Simulations The four binary interaction parameters, 𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿, 𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎, 𝜏𝜏𝐿𝐿𝑚𝑚, and 𝜏𝜏𝑎𝑎𝑚𝑚, are calculated

from the quantities thus obtained from the MD simulation results. By employing the Eqs.

5.24 & 5.25 and Eqs. 5.33 & 5.34, the binary parameters are calculated at each

concentration. In order to make a more meaningful comparison between the binary

interaction parameters from the MD simulations and those from regression, the value of

the nonrandomness factors were fixed at 0.2. Note that the product of 𝛼𝛼𝜏𝜏 determines the

structure of the local domains according to Eqs 8 & 10.

Tables 5.9.17-5.9.20 list the binary interaction parameters for the four aqueous salt

solutions at 298.15 K, over the entire concentration range. As it can be seen from the tables,

the sign of the 𝜏𝜏 parameters are as anticipated, i.e., positive for ion-centered domains and

negative for molecule-centered domains. The only exception is that while the 𝜏𝜏𝐿𝐿𝑚𝑚 and 𝜏𝜏𝑎𝑎𝑚𝑚

in aqueous MgCl2 exhibit negative values at lower concentrations, they eventually become

positive above 4 m concentration. That is due to the monotonic increase of the 𝜖𝜖𝑚𝑚𝑚𝑚

discussed previously. Figures 5.9.5-5.9.8 in Supplementary Information also show the

binary interaction parameters, together with those from regression of the experimental data.

The binary interaction parameters from regression (𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 and 𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚) for the model systems

of our study are readily available in literature 13-16. Honarparvar et al., presented the 𝜏𝜏

values for the aqueous BaCl2 13 and SrCl2 14 solutions. Tanveer and coworkers have

reported the 𝜏𝜏 values for the aqueous CaCl2 and 15 MgCl2 16 solutions. Table 5.5 lists the

𝜏𝜏 values from regression for each aqueous solution at 298.15 K in the entire range of

concentration. These values are used for the validation of our results, as they have shown


174

to accurately predict the mean ionic activity coefficients and other phase equilibria

properties 13-16.

Table 5.5. Binary interaction parameters from regression

Parameters BaCl2(aq) SrCl2(aq) CaCl2(aq) MgCl2(aq)

𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 7.42a 9.13b 10.48c 10.85d

𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 -4.03a -4.75b -5.29c -5.41d


Refs: a 13, b 14, c 15, d 16

Defining Effective Binary Interaction Parameters from MD Simulations

The calculated 𝜏𝜏 parameters from the MD simulations demonstrate that the assumption

of the equal magnitude between the cation-molecule and anion-molecule interactions made

by the eNRTL model are qualitatively reasonable. However, the quantities of 𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 and

𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎 do show a slight difference in quantities, as do the parameters 𝜏𝜏𝐿𝐿𝑚𝑚 and 𝜏𝜏𝑎𝑎𝑚𝑚. In order

to provide a clear guideline to use the parameters thus obtained, we introduce average-type

𝜏𝜏 parameters, hereafter called the effective binary interaction parameters designated as

𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 and 𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚, for the molecule-electrolyte and electrolyte-molecule pairs, respectively.

In the three local domains, the anions and cations are replaced by the hypothetical

electrolyte components (ca). Following Eqs. 5.24-5.25 and Eqs. 5.33-5.34, the binary

interaction parameters for the electrolyte-molecule (𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎) and molecule-electrolyte

(𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚) are then defined as follows.

𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 = 𝛽𝛽𝛼𝛼(𝑐𝑐𝑎𝑎)

𝜖𝜖𝑎𝑎𝑐𝑐𝜎𝜎𝑎𝑎𝑐𝑐3−𝜖𝜖𝑚𝑚,𝑐𝑐𝑎𝑎𝜎𝜎𝑚𝑚,𝑐𝑐𝑎𝑎3

(𝐿𝐿(𝑐𝑐𝑎𝑎))3 (5.45)


175

𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 = 𝛽𝛽𝛼𝛼(𝑚𝑚)

𝜖𝜖𝑚𝑚𝑚𝑚𝜎𝜎𝑚𝑚𝑚𝑚3−𝜖𝜖𝑎𝑎𝑐𝑐,𝑚𝑚𝜎𝜎𝑐𝑐𝑎𝑎,𝑚𝑚

3

(𝐿𝐿(𝑚𝑚))3 (5.46)

Here, 𝜖𝜖𝑚𝑚,𝐿𝐿𝑎𝑎 = 𝜖𝜖𝐿𝐿𝑎𝑎,𝑚𝑚 = �𝜖𝜖𝑚𝑚𝐿𝐿𝜖𝜖𝑚𝑚𝑎𝑎; 𝜎𝜎𝑚𝑚,𝐿𝐿𝑎𝑎 = 𝜎𝜎𝐿𝐿𝑎𝑎,𝑚𝑚 = (𝜎𝜎𝑐𝑐𝑚𝑚+𝜎𝜎𝑎𝑎𝑚𝑚)2

; 𝑅𝑅(𝐿𝐿𝑎𝑎) = (𝐿𝐿(𝑐𝑐)+𝐿𝐿(𝑎𝑎))2

; and

𝛼𝛼(𝐿𝐿𝑎𝑎)=2𝛼𝛼(𝑐𝑐)𝛼𝛼(𝑎𝑎)

𝛼𝛼(𝑐𝑐)+𝛼𝛼(𝑎𝑎). The nonrandomness factor of the hypothetical electrolyte-centered domain

is obtained from the inverse of the average coordination numbers in cation- and anion-

centered domains.

Using the above equations, the two binary interaction parameters, 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 and 𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚, are

obtained over the concentration range studied for the four aqueous solutions at 298.15 K.

Table 5.6 lists these parameters with their associated uncertainties. Also Figures 5.10-5.13

depict the parameters together with those obtained from the regression of the experimental

data. In the aqueous BaCl2 solution, the binary interaction parameters seldom show

concentration dependency, consistent with the eNRTL theory. Also, the quantities 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎

and 𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 are in very good agreement with their counterparts from regression. In the

aqueous SrCl2 solution, the binary parameters demonstrate concentration dependency at

lower concentrations before reaching a plateau at medium to high concentration. Such a

behavior has been previously observed and explained for the aqueous NaCl solution 24. At

higher concentrations, due to the strong screening effects discussed previously the structure

of the local domains becomes saturated and is only governed by the local van der Waals

forces (See Figure 5.9).


176

Table 5.6. Effective binary interaction parameters

Concentration (m)

Systems Parameters 0.5 2.0 4.0 6.0

BaCl2(aq) 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 5.01±0.81 7.25±1.21 6.37±0.48 6.02±0.32

𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 -6.79±0.16 -3.09±0.15 -3.33±0.12 -3.49±0.12

SrCl2(aq) 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 2.02±0.55 5.70±1.41 7.61±0.78 8.44±0.57

𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 -6.30±0.34 -6.11±0.08 -4.27±0.42 -3.55±0.04

Systems Parameters 8.0 10.0 12.0

BaCl2(aq) 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 5.52±0.42 5.04±0.27 7.19±0.81

𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 -3.25±0.15 -2.86±0.58 -1.27±0.45

SrCl2(aq) 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 6.46±0.75 6.46±0.75 6.38±0.27

𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 -3.18±0.12 -3.18±0.12 -3.10±0.13

Systems Parameters 0.5 2.0 4.0 6.0

CaCl2(aq) 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 2.47±0.63 2.73±0.08 4.46±1.44 7.03±1.12

𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 -6.51±0.23 -6.05±0.25 -4.86±0.35 -4.85±0.26

MgCl2(aq) 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 3.42±0.36 3.74±0.19 4.64±0.06 5.51±0.27

𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 -3.56±0.32 -2.76±0.06 -2.26±0.05 -0.90±0.05

Systems Parameters 8.0 10.0 12.0

CaCl2(aq) 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 5.52±0.42 5.04±0.27 7.19±0.81

𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 -3.25±0.15 -2.86±0.58 -1.27±0.45

MgCl2(aq) 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 6.46±0.75 6.46±0.75 6.38±0.27

𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 -3.18±0.12 -3.18±0.12 -3.10±0.13



177

Figure 5.10. The effective binary interaction parameters for the aqueous BaCl2 solution at 298.15 K. ( ) denotes the regressed binary interaction parameters (τm,ca and τca,m) from literature: 13 in 3.a, 14 in 3.b, 15 in 3.c, and 16 in 3.d; ( ) and ( ) denote the effective binary interaction parameters (τm,ca and τca,m) from the MD simulations.


178

Figure 5.11. The effective binary interaction parameters for the aqueous SrCl2 solution at 298.15 K. ( ) denotes the regressed binary interaction parameters (τm,ca and τca,m) from literature: 13 in 3.a, 14 in 3.b, 15 in 3.c, and 16 in 3.d; ( ) and ( ) denote the effective binary interaction parameters (τm,ca and τca,m) from the MD simulations.


179

Figure 5.12. The effective binary interaction parameters for the aqueous CaCl2 solution at 298.15 K. ( ) denotes the regressed binary interaction parameters (τm,ca and τca,m) from literature: 13 in 3.a, 14 in 3.b, 15 in 3.c, and 16 in 3.d; ( ) and ( ) denote the effective binary interaction parameters (τm,ca and τca,m) from the MD simulations.


180

Figure 5.13. The effective binary interaction parameters for the aqueous MgCl2 solution at 298.15 K. ( ) denotes the regressed binary interaction parameters (τm,ca and τca,m) from literature: 13 in 3.a, 14 in 3.b, 15 in 3.c, and 16 in 3.d; ( ) and ( ) denote the effective binary interaction parameters (τm,ca and τca,m) from the MD simulations.

In the aqueous CaCl2 solution, 𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚 shows almost no concentration dependency and

demonstrates an excellent agreement with that reported from regression. 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 on the other

hand, displays a slight increase by adding salts at lower concentrations before reaching a

plateau. A systematic underestimation of the 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 is observed with respect to that reported

from regression. A similar behavior is shown for the aqueous MgCl2 solution in which

𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎, though demonstrating almost no concentration dependency, only agrees

qualitatively with that reported from regression. Such underestimations are expected to be

observed in electrolyte systems where the ionic species show the tendency toward forming

hydrated complex ions. In other word, the ionic species that form strong CIP peaks, such

as in aqueous BaCl2 solution and to some extent in aqueous SrCl2 solution, the results of


181

the 𝜏𝜏 parameters from the MD simulations are in more favorable agreement with those

from regression. In order to improve the calculation of the 𝜏𝜏 parameters from MD

simulations and consequently reaching better thermophysical predictions, it is required to

take into account explicitly the complex ions into our framework. We defer such studies to

future work.

To examine the effect of separating out the contribution of the electrostatic forces from

the total PMF on the magnitude of the 𝜏𝜏 parameters, we calculate 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 for each model

system using the short-range 𝜖𝜖𝑎𝑎𝐿𝐿 thus obtained. Here, a water molecule is considered as

one united neutral particle. Thereby, the interaction energies involving water molecules,

i.e., 𝜖𝜖𝐿𝐿𝑚𝑚, 𝜖𝜖𝑎𝑎𝑚𝑚, and 𝜖𝜖𝑚𝑚𝑚𝑚 are assumed to be unaffected by the electrostatic forces.

Consequently, the calculations for finding the short-range interactions do not alter 𝜏𝜏𝐿𝐿𝑎𝑎,𝑚𝑚.

Table 5.9.21 in Supplementary Information lists the calculated 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 using the short-

range van der Waals interactions. The binary interaction parameters at moderate to high

concentrations are not significantly different (within the statistical uncertainties) compared

to those reported in Table 5.6, where the parameters are obtained using the total PMF. At

low concentrations, on the other hand,(0.5 and 2 m) the differences are more significant

where the electrostatic forces are relatively more effective in determining the configuration

of the first solvation shell.

Predicting Phase Equilibria Properties from τ Parameters Obtained via MD Simulations

To further assess the validity of our framework, a number of essential thermodynamic

properties including the mean ionic activity coefficients, vapor pressure, and excess Gibbs

free energy in the four aqueous solutions are predicted using the 𝜏𝜏 parameters obtained

from the MD simulations. The molality-based mean ionic activity coefficients (𝛾𝛾±) are


182

calculated by combining the activity coefficients of the long-range electrostatic forces—as

a function of concentration—and those of the short-range van der Waals forces—as

functions of the binary interaction parameters. We refer the reader to the literature where

the derivation of 𝛾𝛾± is described 8-11.

To facilitate the use of our methodology for enabling rapid predictions in industry, we

are interested in finding a pair of 𝜏𝜏 parameters that can be used for the entire range of

concentrations. Due to the tendency of the binary interaction parameters to reach a plateau,

it was observed in our previous work 24 that the 𝜏𝜏 parameters from such a region yield the

best predictions. Specifically, the parameters at the concentrations studied between 4 to 6

m. The 𝜏𝜏 parameters at concentrations beyond 6 m, though qualitatively predict the

thermophysical properties, are not as accurate. That is due to the inherent limitations

associated with the force filed parameters to reliably reproduce the data at higher

concentrations. Figures 5.14-5.17 show the predictions of the mean ionic activity

coefficients at 298 K for the aqueous BaCl2, SrCl2, CaCl2, and MgCl2 solutions, using the

𝜏𝜏 parameters at 6 m. The exception is for the aqueous MgCl2 solution where due to the

monotonic increase observed for the water-water interactions throughout the salt

concentrations, we concluded that the results at or beyond the 2 m concentration should

not be trusted. That is aligned with the results reported by Naleem et al. 41, in which the

force field parameters did not reproduce well the experimental Kirkwood-Buff integral in

such a region. Hence, here the prediction for the aqueous MgCl2 solution was achieved

from the MD results at the lowest concentration studied.


183

Figure 5.14. Mean ionic activity coefficients for the entire concentration range in the aqueous BaCl2 solution at 298.15 K. ( ) shows the prediction from the binary interaction parameters from regression of the experimental data; ( ) denotes the experimental data of Robinson and Stokes 47; ( ) demonstrates the predictions using the binary interaction parameters obtained from the MD simulations.


184

Figure 5.15. Mean ionic activity coefficients for the entire concentration range in the aqueous SrCl2 solution at 298.15 K. ( ) shows the prediction from the binary interaction parameters from regression of the experimental data; ( ) denotes the experimental data of Robinson and Stokes 47; ( ) demonstrates the predictions using the binary interaction parameters obtained from the MD simulations.


185

Figure 5.16. Mean ionic activity coefficients for the entire concentration range in the aqueous CaCl2 solution at 298.15 K. ( ) shows the prediction from the binary interaction parameters from regression of the experimental data; ( ) denotes the experimental data of Robinson and Stokes 47; ( ) demonstrates the predictions using the binary interaction parameters obtained from the MD simulations.


186

Figure 5.17. Mean ionic activity coefficients for the entire concentration range in the aqueous MgCl2 solution at 298.15 K. ( ) shows the prediction from the binary interaction parameters from regression of the experimental data; ( ) denotes the experimental data of Robinson and Stokes 47; ( ) demonstrates the predictions using the binary interaction parameters obtained from the MD simulations.

Using the activity coefficients obtained from the MD simulations, the excess Gibbs

free energy of the solutions are predicted and compared to those calculated from the

regression-based method. Furthermore, the vapor pressures are computed from the flash

calculations and compared to those calculated using the 𝜏𝜏 parameters from regression.

Figures 5.18-5.21 demonstrate that the results from the MD simulations can reasonably

predict the vapor pressure and excess Gibbs free energy of the model systems.


187

Figure 5.18. Vapor pressure and excess Gibbs free energy in the entire concentration range in the aqueous BaCl2 solution at 298.15 K. ( ) and ( ) denote respectively the predictions of the vapor pressure and excess Gibbs free energy using the binary interaction parameters obtained from regression of the experimental data; ( ) and ( ) demonstrate respectively the prediction of the vapor pressure and excess Gibbs free energy from the binary interaction parameters obtained from the MD simulations.


188

Figure 5.19. Vapor pressure and excess Gibbs free energy in the entire concentration range in the aqueous SrCl2 solution at 298.15 K. ( ) and ( ) denote respectively the predictions of the vapor pressure and excess Gibbs free energy using the binary interaction parameters obtained from regression of the experimental data; ( ) and ( ) demonstrate respectively the prediction of the vapor pressure and excess Gibbs free energy from the binary interaction parameters obtained from the MD simulations.


189

Figure 5.20. Vapor pressure and excess Gibbs free energy in the entire concentration range in the aqueous CaCl2 solution at 298.15 K. ( ) and ( ) denote respectively the predictions of the vapor pressure and excess Gibbs free energy using the binary interaction parameters obtained from regression of the experimental data; ( ) and ( ) demonstrate respectively the prediction of the vapor pressure and excess Gibbs free energy from the binary interaction parameters obtained from the MD simulations.


190

Figure 5.21. Vapor pressure and excess Gibbs free energy in the entire concentration range in the aqueous MgCl2 solution at 298.15 K. ( ) and ( ) denote respectively the predictions of the vapor pressure and excess Gibbs free energy using the binary interaction parameters obtained from regression of the experimental data; ( ) and ( ) demonstrate respectively the prediction of the vapor pressure and excess Gibbs free energy from the binary interaction parameters obtained from the MD simulations.

5.7. Conclusions

In this study, we established a theoretical framework to quantify the binary interaction

parameters (𝜏𝜏) of the eNRTL model for electrolytes from molecular dynamics (MD)

simulations. By revisiting the statistical mechanics of two-liquid theory, the 𝜏𝜏 parameters

were formulated as functions of the microscopic scale liquid structure and energetic

interaction quantities, which in turn were obtained from the MD simulations and free

energy calculations. The validity of the developed framework was assessed by examining

several di-univalent electrolyte solutions including BaCl2 (aq), SrCl2 (aq), CaCl2 (aq), and


191

MgCl2 (aq) in a wide range of concentrations from 0.5 to 12 mol/kg. The 𝜏𝜏 parameters

quantified from the MD simulations were in satisfactory agreement with those identified

from the regression of the experimental data. The 𝜏𝜏 parameters did not demonstrate a strong

concentration dependency, specifically at moderate to high concentrations where the 𝜏𝜏

parameters reached a plateau. The 𝜏𝜏 parameters obtained in the moderate concentration

region (4-6 mol/kg in this study) were shown to provide the best predictions of the phase

equilibria properties including the mean ionic activity coefficients, vapor pressure, and

excess Gibbs free energy. The exception was for the aqueous MgCl2 solution where the

binary interaction parameters calculated at concentrations higher that 2m were considered

to be unreliable due to the limitations of the employed force field parameters. Thereby the

predictions were achieved by using the 𝜏𝜏 parameters at the lowest concentration studied.

Overall, the results demonstrate the possibility of rendering the eNRTL, as a classical

thermodynamic model, completely predictive to enhance its feasibility for use in industrial

applications. Furthermore, this technique can circumvent the drawbacks associated with

the regression procedures, such as data scarcity and multiple solutions for the binary

interaction parameters.

5.8. Acknowledgements



Maddox Foundation, United States. The computational resources were provided by the

High-Performance Computing Center (HPCC) at Texas Tech University.


192

5.9. Supplementary Information

Figure 5.9.1. Radial distribution function (RDF) of the Ba2+-Clˉ pair in the aqueous BaCl2 at 4 m concentration and 298 K.


193

Figure 5.9.2. Radial distribution function (RDF) of the Sr2+-Clˉ pair in the aqueous SrCl2 at 4 m concentration and 298 K.


194

Figure 5.9.3. Radial distribution function (RDF) of the Ca2+-Clˉ pair in the aqueous CaCl2 at 4 m concentration and 298 K.


195

Figure 5.9.4. Radial distribution function (RDF) of the Mg2+-Clˉ pair in the aqueous MgCl2 at 4 m concentration and 298 K.


196

Figure 5.9.5. The binary interaction parameters for the aqueous BaCl2 solution at 298.15 K. ( ) denotes the regressed binary interaction parameters (τm,ca and τca,m) from literature 13; ( ), ( ), ( ), and ( ) represent the binary interaction parameters from MD simulations, τmc,ac, τma,ca, τcm, and τam, respectively.


197

Figure 5.9.6. The binary interaction parameters for the aqueous SrCl2 solution at 298.15 K. ( ) denotes the regressed binary interaction parameters (τm,ca and τca,m) from literature: 14; ( ), ( ), ( ), and ( ) represent the binary interaction parameters from MD simulations, τmc,ac, τma,ca, τcm, and τam, respectively.


198

Figure 5.9.7. The binary interaction parameters for the aqueous CaCl2 solution at 298.15 K. ( ) denotes the regressed binary interaction parameters (τm,ca and τca,m) from literature: 15; ( ), ( ), ( ), and ( ) represent the binary interaction parameters from MD simulations, τmc,ac, τma,ca, τcm, and τam, respectively.


199

Figure 5.9.8. The binary interaction parameters for the aqueous MgCl2 solution at 298.15 K. ( ) denotes the regressed binary interaction parameters (τm,ca and τca,m) from literature: 16; ( ), ( ), ( ), and ( ) represent the binary interaction parameters from MD simulations, τmc,ac, τma,ca, τcm, and τam, respectively.


200

Table 5.9.1. Effective species diameters (Å) – BaCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝜎𝜎𝑎𝑎𝐿𝐿 3.10 2.74 2.70 2.70 2.70 2.70 2.86

𝜎𝜎𝐿𝐿𝑚𝑚 2.9 2.9 2.9 2.90 2.9 2.9 2.9

𝜎𝜎𝑎𝑎𝑚𝑚 3.30 3.70 3.70 3.70 3.70 3.70 3.50

𝜎𝜎𝑚𝑚𝑚𝑚 2.70 2.70 2.70 2.70 2.70 2.70 2.70


Table 5.9.2. Effective species diameters (Å) – SrCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝜎𝜎𝑎𝑎𝐿𝐿 3.98 3.12 2.90 2.90 2.90 2.90 2.90

𝜎𝜎𝐿𝐿𝑚𝑚 2.50 2.50 2.50 2.50 2.5 2.50 2.50

𝜎𝜎𝑎𝑎𝑚𝑚 3.28 3.30 3.30 3.30 3.30 3.30 3.30

𝜎𝜎𝑚𝑚𝑚𝑚 2.70 2.70 2.90 2.90 2.92 2.92 2.90



201

Table 5.9.3. Effective species diameters (Å) – CaCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝜎𝜎𝑎𝑎𝐿𝐿 4.94 4.90 4.50 2.90 2.90 2.90 2.90

𝜎𝜎𝐿𝐿𝑚𝑚 2.50 2.50 2.50 2.50 2.34 2.30 2.30

𝜎𝜎𝑎𝑎𝑚𝑚 3.30 3.28 3.16 3.26 3.30 3.30 3.30

𝜎𝜎𝑚𝑚𝑚𝑚 2.70 2.90 2.90 2.90 2.90 2.90 2.90


Table 5.9.4. Effective species diameters (Å) – MgCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝜎𝜎𝑎𝑎𝐿𝐿 4.50 4.50 4.50 4.50 4.48 4.30 4.30

𝜎𝜎𝐿𝐿𝑚𝑚 1.90 1.90 1.90 1.90 1.84 1.70 1.70

𝜎𝜎𝑎𝑎𝑚𝑚 3.14 3.10 3.10 3.10 3.10 3.10 3.10

𝜎𝜎𝑚𝑚𝑚𝑚 2.70 2.70 2.70 2.70 2.70 2.70 2.70



202

Table 5.9.5. First neighbor shell radii (Å) – BaCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝑅𝑅(𝐿𝐿) 3.90 3.78 3.76 3.86 3.76 3.70 3.72

±0.13 ±0.10 ±0.09 ±0.08 ±0.09 ±0.00 ±0.06

𝑅𝑅(𝑎𝑎) 3.94 4.62 4.62 4.62 4.68 4.74 4.70

±0.08 ±0.10 ±0.10 ±0.10 ±0.06 ±0.08 ±0.09

𝑅𝑅(𝑚𝑚) 3.78 4.06 4.06 4.06 4.15 4.42 5.38

±0.04 ±0.05 ±0.05 ±0.05 ±0.08 ±0.44 ±0.38


Table 5.9.6. First neighbor shell radii (Å) – SrCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝑅𝑅(𝐿𝐿) 5.08 4.00 3.70 3.70 3.86 3.86 3.80

±1.06 ±0.71 ±0.00 ±0.00 ±0.08 ±0.08 ±0.10

𝑅𝑅(𝑎𝑎) 5.08 4.12 3.90 4.16 4.62 4.62 4.64

±1.06 ±0.66 ±0.00 ±0.09 ±0.10 ±0.10 ±0.09

𝑅𝑅(𝑚𝑚) 3.90 3.90 4.16 4.30 4.62 4.62 4.64

±0.00 ±0.00 ±0.13 ±0.00 ±0.10 ±0.10 ±0.09



203

Table 5.9.7. First neighbor shell radii (Å) – CaCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝑅𝑅(𝐿𝐿) 6.00 6.08 5.56 3.50 3.50 3.52 3.60

±0.10 ±0.06 ±1.08 ±0.09 ±0.00 ±0.06 ±0.10

𝑅𝑅(𝑎𝑎) 6.00 6.08 5.66 3.90 4.10 4.30 4.36

±0.10 ±0.06 ±0.88 ±0.00 ±0.00 ±0.00 ±0.09

𝑅𝑅(𝑚𝑚) 3.67 3.70 3.79 3.80 3.93 4.11 4.18

±0.05 ±0.00 ±0.03 ±0.00 ±0.05 ±0.05 ±0.10


Table 5.9.8. First neighbor shell radii (Å) – MgCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝑅𝑅(𝐿𝐿) 5.46 5.44 5.50 5.54 5.70 5.66 5.50

±0.12 ±0.09 ±0.00 ±0.08 ±0.00 ±0.08 ±0.00

𝑅𝑅(𝑎𝑎) 5.46 5.44 5.50 5.54 5.70 5.66 5.50

±0.12 ±0.09 ±0.00 ±0.08 ±0.00 ±0.08 ±0.00

𝑅𝑅(𝑚𝑚) 3.60 3.60 3.40 3.43 3.41 3.71 3.98

±0.00 ±0.00 ±0.00 ±0.05 ±0.03 ±0.20 ±0.04



204

Table 5.9.9. Interaction Strengths (kcal/mol) – BaCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝜖𝜖𝑎𝑎𝐿𝐿 3.05 4.90 4.79 4.77 4.46 4.28 4.22

±0.21 ±0.31 ±0.28 ±0.15 ±0.12 ±0.10 ±0.11

𝜖𝜖𝐿𝐿𝑚𝑚 3.21 2.67 2.71 2.76 2.59 2.51 2.41

±0.08 ±0.11 ±0.05 ±0.04 ±0.03 ±0.03 ±0.02

𝜖𝜖𝑎𝑎𝑚𝑚 1.07 0.41 0.43 0.44 0.45 0.48 0.48

±0.01 ±0.02 ±0.00 ±0.00 ±0.01 ±0.01 ±0.01

𝜖𝜖𝑚𝑚𝑚𝑚 0.59 0.65 0.63 0.60 0.57 0.57 0.67

±0.00 ±0.01 ±0.00 ±0.00 ±0.00 ±0.03 ±0.04



205

Table 5.9.10. Interaction Strengths (kcal/mol) – SrCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝜖𝜖𝑎𝑎𝐿𝐿 1.73 3.39 3.99 4.36 4.28 4.28 4.20

±0.86 ±085 ±0.21 ±0.15 ±0.28 ±0.28 ±0.07

𝜖𝜖𝐿𝐿𝑚𝑚 3.80 3.76 3.90 4.01 4.05 4.05 4.05

±0.08 ±0.07 ±0.04 ±0.06 ±0.07 ±0.07 ±0.11

𝜖𝜖𝑎𝑎𝑚𝑚 1.13 1.08 0.98 0.87 0.88 0.88 0.86

±0.02 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01 ±0.01

𝜖𝜖𝑚𝑚𝑚𝑚 0.62 0.48 0.47 0.50 0.51 0.51 0.51

±0.00 ±0.00 ±0.00 ±0.00 ±0.01 ±0.01 ±0.00



206

Table 5.9.11. Interaction Strengths (kcal/mol) – CaCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝜖𝜖𝑎𝑎𝐿𝐿 0.94 1.04 1.74 3.67 3.83 3.97 3.86

±0.11 ±0.02 ±0.97 ±0.28 ±0.14 ±0.11 ±0.07

𝜖𝜖𝐿𝐿𝑚𝑚 3.78 3.84 3.92 4.06 4.29 4.51 4.69

±0.09 ±0.06 ±0.07 ±0.05 ±0.08 ±0.09 ±0.10

𝜖𝜖𝑎𝑎𝑚𝑚 1.14 1.10 1.06 0.97 0.93 0.93 0.94

±0.02 ±0.01 ±0.01 ±0.02 ±0.01 ±0.01 ±0.01

𝜖𝜖𝑚𝑚𝑚𝑚 0.64 0.55 0.61 0.65 0.63 0.57 0.53

±0.00 ±0.00 ±0.00 ±0.00 ±0.00 ±0.01 ±0.01



207

Table 5.9.12. Interaction Strengths (kcal/mol) – MgCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝜖𝜖𝑎𝑎𝐿𝐿 1.09 1.13 1.34 1.54 1.60 1.54 1.47

±0.04 ±0.02 ±0.01 ±0.01 ±0.03 ±0.03 ±0.03

𝜖𝜖𝐿𝐿𝑚𝑚 4.04 3.87 3.76 3.58 3.04 3.33 3.86

±0.15 ±0.09 ±0.07 ±0.08 ±0.09 ±0.05 ±0.06

𝜖𝜖𝑎𝑎𝑚𝑚 1.10 1.04 1.01 1.01 0.97 0.90 1.03

±0.01 ±0.01 ±0.01 ±0.00 ±0.01 ±0.02 ±0.01

𝜖𝜖𝑚𝑚𝑚𝑚 0.71 0.82 1.02 1.29 1.47 1.51 1.61

±0.00 ±0.00 ±0.00 ±0.01 ±0.03 ±0.01 ±0.03


Table 5.9.13. Nonrandomness factors – BaCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝛼𝛼(𝐿𝐿) 0.11 0.16 0.16 0.16 0.16 0.16 0.16

±0.00 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00

𝛼𝛼(𝑎𝑎) 0.13 0.13 0.13 0.13 0.13 0.13 0.13

±0.00 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00

𝛼𝛼(𝑚𝑚) 0.14 0.15 0.15 0.15 0.15 0.15 0.14

±0.00 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00



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Table 5.9.14. Nonrandomness factors – SrCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝛼𝛼(𝐿𝐿) 0.08 0.11 0.13 0.13 0.15 0.14 0.15

±0.00 ±0.00 ±0.00 ±0.00 ±0.01 ±0.00 ±0.01

𝛼𝛼(𝑎𝑎) 0.08 0.12 0.12 0.11 0.12 0.11 0.12

±0.01 ±0.01 ±0.01 ±0.01 ±0.00 ±0.00 ±0.00

𝛼𝛼(𝑚𝑚) 0.14 0.13 0.13 0.13 0.14 0.13 0.13

±0.00 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00


Table 5.9.15. Nonrandomness factors – CaCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝛼𝛼(𝐿𝐿) 0.03 0.03 0.05 0.14 0.14 0.15 0.16

±0.00 ±0.00 ±0.04 ±0.00 ±0.00 ±0.00 ±0.00

𝛼𝛼(𝑎𝑎) 0.03 0.03 0.05 0.12 0.12 0.11 0.11

±0.00 ±0.00 ±0.04 ±0.00 ±0.00 ±0.00 ±0.00

𝛼𝛼(𝑚𝑚) 0.15 0.14 0.13 0.13 0.13 0.14 0.14

±0.00 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00



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Table 5.9.16. Nonrandomness factors – MgCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝛼𝛼(𝐿𝐿) 0.04 0.04 0.04 0.05 0.05 0.05 0.05

±0.00 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00

𝛼𝛼(𝑎𝑎) 0.04 0.04 0.04 0.04 0.04 0.04 0.04

±0.00 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00

𝛼𝛼(𝑚𝑚) 0.15 0.14 0.16 0.15 0.15 0.15 0.15

±0.00 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00 ±0.00


Table 5.9.17. Binary interaction parameters – BaCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 1.76 5.71 4.45 3.91 3.93 3.83 6.59

±0.78 ±2.17 ±0.74 ±0.37 ±0.65 ±0.33 ±1.23

𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎 7.27 6.88 6.22 6.15 5.36 4.76 6.37

±0.91 ±0.73 ±0.49 ±0.44 ±0.29 ±0.29 ±0.69

𝜏𝜏𝐿𝐿𝑚𝑚 -10.41 -6.58 -6.79 -6.99 -6.14 -5.09 -2.59

±0.20 ±0.44 ±0.23 ±0.28 ±0.32 ±1.01 ±0.87

𝜏𝜏𝑎𝑎𝑚𝑚 -4.16 -0.97 -1.18 -1.31 -1.35 -1.32 -0.44

±0.18 ±0.09 ±0.05 ±0.04 ±0.07 ±0.08 ±0.03



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Table 5.9.18. Binary interaction parameters – SrCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 1.21 4.63 6.05 7.26 6.03 6.03 6.03

±0.33 ±1.29 ±0.80 ±0.58 ±1.01 ±1.01 ±0.50

𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎 2.88 6.77 8.81 8.81 6.24 6.24 6.04

±0.32 ±1.08 ±0.75 ±0.71 ±0.72 ±0.72 ±0.39

𝜏𝜏𝐿𝐿𝑚𝑚 -7.86 -7.58 -5.83 -5.36 -4.82 -4.82 -4.75

±0.22 ±0.16 ±0.57 ±0.10 ±0.20 ±0.20 ±0.20

𝜏𝜏𝑎𝑎𝑚𝑚 -4.61 -4.53 -2.81 -2.03 -1.81 -1.81 -1.74

±0.44 ±0.05 ±0.19 ±0.03 ±0.08 ±0.08 ±0.08



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Table 5.9.19. Binary interaction parameters – CaCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 2.14 2.34 3.81 5.17 7.55 8.12 6.73

±0.63 ±0.09 ±1.46 ±1.40 ±0.97 ±0.63 ±0.50

𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎 2.84 3.13 4.93 7.95 7.35 6.73 6.19

±0.63 ±0.10 ±1.32 ±0.92 ±0.40 ±0.28 ±0.42

𝜏𝜏𝐿𝐿𝑚𝑚 -7.96 -7.78 -7.21 -7.30 -5.51 -4.98 -5.11

±0.37 ±0.14 ±0.16 ±0.12 ±0.59 ±0.23 ±0.15

𝜏𝜏𝑎𝑎𝑚𝑚 -4.86 -4.26 -2.88 -2.74 -2.52 -2.36 -2.39

±0.14 ±0.37 ±0.38 ±0.34 ±0.08 ±0.12 ±0.08



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Table 5.9.20. Binary interaction parameters – MgCl2 (aq)

Concentration (m)

Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

𝜏𝜏𝑚𝑚𝐿𝐿,𝑎𝑎𝐿𝐿 3.73 3.99 4.88 5.77 5.71 4.94 4.96

±0.36 ±0.21 ±0.06 ±0.28 ±0.27 ±0.27 ±0.11

𝜏𝜏𝑚𝑚𝑎𝑎,𝐿𝐿𝑎𝑎 3.41 3.76 4.65 5.48 5.26 4.44 4.36

±0.39 ±0.19 ±0.05 ±0.26 ±0.34 ±0.26 ±0.12

𝜏𝜏𝐿𝐿𝑚𝑚 -2.48 -1.90 -1.24 0.20 2.11 2.25 1.69 ±0.19 ±0.11 ±0.10 ±0.10 ±0.51 ±0.35 ±0.06

𝜏𝜏𝑎𝑎𝑚𝑚 -3.62 -2.69 -2.20 -1.00 0.02 0.49 0.12 ±0.44 ±0.05 ±0.05 ±0.04 ±0.10 ±0.13 ±0.07


Table 5.9.21. Effective binary interaction parameters using the local van der Waals PMF approximation

Concentration (m)

Systems Parameters 0.5 2.0 4.0 6.0 8.0 10.0 12.0

BaCl2(aq) 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 0.68± 2.03

6.68± 0.96

7.54± 0.65

7.60± 0.38

7.23± 0.55

6.86± 0.36

7.34± 0.31

SrCl2(aq) 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 1.61± 1.46

2.72± 0.29

5.04± 1.17

6.68± 0.54

5.23± 0.74

5.57± 0.74

5.60± 0.26

CaCl2(aq) 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 1.82± 0.79

2.71± 0.08

4.03± 1.10

5.61± 1.19

6.90± 0.45

6.91± 0.40

6.14± 0.28

MgCl2(aq) 𝜏𝜏𝑚𝑚,𝐿𝐿𝑎𝑎 0.90± 1.10

3.59± 0.19

4.62± 0.06

5.51± 0.27

5.40± 0.30

4.58± 0.26

4.52± 0.11



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CHAPTER 6. CONCLUSIONS AND FUTURE WORK

This dissertation presents an in-depth study of the thermodynamic modeling of

electrolyte solutions from both macroscopic classical models and statistical mechanical

approaches. Due to the ubiquitous presence of the electrolytes in many industrial and

environmental processes, it is essential for the engineers to have access to the reliable and

rigorous thermodynamic models. Such models enable rapid predictions of phase equilibria,

calorimetric, and speciation properties to support mass and energy balance calculations.

Predictive models based on various statistical mechanical theories, by employing

molecular simulations, provide insight into the liquid structure and free energy information

of the solution, from which the macroscopic thermophysical properties can be calculated.

Although molecular simulations make it possible to understand the underlying phenomena

taking place at the microscopic molecular level, the expensive computational time prevents

the rapid predictions of properties, thus creating a bottleneck in directly applying such

methods in process simulations for industrial applications. Correlative models, on the other

hand, have been widely applied in modeling the electrolytes in industry, owing to their

versatility and rapid computational procedures. In particular, the eNRTL model has shown

to be successful in predicting thermophysical properties of wide variety of electrolyte

solutions with only two adjustable ‘binary interaction parameters’ per pair of species. The

binary interaction parameters (𝜏𝜏) demonstrate the favorable or unfavorable nature of the

interactions between any pair of entities in the solution.

In the first part of this dissertation, the eNRTL model has been applied in several

electrolyte solutions and the adjustable 𝜏𝜏 parameters have been quantified from regression


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of various experimental data. A comprehensive thermodynamic framework is developed

for the HCl-H2O binary system which predicts accurately the VLE and LLE properties, as

well as the calorimetric properties such as excess enthalpy and heat capacity, within the

entire concentration range of acid and temperatures up to 400 K. The model has also been

applied to describe the aqueous BaCl2 solution and the aqueous quaternary solution of Na+-

Ba2+-Cl--SO42-, as part of the investigation of the scale precipitation phenomena in oil and

gas industry.

The binary interaction parameters of eNRTL have been conventionally identified from

regression of the experimental data. However, relaying merely on regression could

potentially cause a number of drawbacks, including the lack of available experimental data.

Also, selecting physically relevant parameters could be ambiguous as in most cases the

regression procedures fail to result in a unique set of parameters. To overcome these issues

and to expand the applicability of the model, a new statistical mechanical framework has

been established wherein the interaction parameters are described in terms of the size of

the species, the size of the local solvation shells, and the local interaction energies between

the particles. All of these physical quantities are obtained via MD simulations and potential

of mean force (PMF) free energy calculations. In order to test the validity of the proposed

approach, several electrolyte solutions including aqueous NaCl, BaCl2, SrCl2, CaCl2, and

MgCl2 solutions have been selected as model systems. The results for the binary interaction

parameters from the MD simulations are in good agreement with their counterpart from

regression. The model parameters from MD were then used in predicting a number of

essential thermodynamic properties such as mean ionic activity coefficient, excess Gibbs


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free energy, and vapor pressure, all of which demonstrated satisfactory agreement with

predictions from regression.

Overall, the results demonstrated that the eNRTL model can be rendered completely

predictive which enhances its feasibility for use in industry. Also, the established

methodology can help guide the regression procedure to select physically meaningful

parameters. This study lays the groundwork for connecting the classical macroscopic

models to their underlying statistical mechanics, and hence helps distinguish the correlative

models with strong physical background from the unsubstantiated empirical models.

Systematic studies have to be carried out to expand the applicability of the approach

toward building a comprehensive yet inexpensive theoretical framework and simulation

tool for chemical engineers. An example of the possible future research directions in this

area is studying the temperature dependence of the binary interaction parameters from

molecular simulations. Note that most of the classical force field parameters that are used

in molecular simulations are known to be valid up to only moderately higher temperatures

(~ 60 ºC). Furthermore, the challenges still exist to improve the predictions of mean ionic

activity coefficients which are highly sensitive to the slight changes in the quantity of the

𝜏𝜏 parameters. More accurate and exact PMF calculations could potentially improve the

results, especially if the free energy profile is taken explicitly into the formulations, in lieu

of the simplistic assumption of projecting the PMF onto a square-well type. Finally, the

hydration of the ionic species with high charge densities should be taken into account by

considering the hydrated complex ions formed in the aqueous solutions as united particles.

All of these possible improvements could significantly enhance the versatility of the


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established methodology, thus providing a new horizon to capture, explain, and model the

complex behavior of the electrolyte solutions.

Thermodynamic Modeling of Electrolyte Solutions

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