energy consumption and salt adsorption in capacitive

ENERGY CONSUMPTION AND SALT ADSORPTION IN CAPACITIVE DEIONIZATION

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MECHANICAL ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Ali Hemmatifar September 2018

© 2018 by Ali Hemmatifar. All Rights Reserved. Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons AttributionNoncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/sd486jn9838

ii

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Juan Santiago, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Christopher Chidsey

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Ali Mani

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii

© Copyright by Ali Hemmatifar 2018 All Rights Reserved

ii

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

(Juan G. Santiago)

Principal Adviser


(Ali Mani)


(Christopher E. D. Chidsey)

Approved for the Stanford University Committee on Graduate Studies

iii

Abstract Challenges for clean water are global and diverse, and flexible water treatment approaches are needed. To date, over 4 billion people, more than half of the world’s population, live in water scarce areas where withdrawn water surpasses the amount the region can sustainably support for at least one month per year. Slightly under 4 billion live in areas of severe scarcity conditions. Aggravating the situation further, naturally occurring toxins, chemical contamination introduced by human activities, and high salinity make many already limited water sources unsafe for consumption. Water desalination and disinfection provides viable solution to deficit of clean water. Seawater desalination is currently the main source of clean water production, however, availability of seawater is geographically uneven and thus not an available option targeted towards inland regions. Brackish water (water with a low to moderate salt content), however, is relatively common and can provide an appealing source of potable water with appropriate treatment technologies. Capacitive desalination or capacitive deionization (CDI) is an electrosorptive desalination method that leverages porous and conductive electrodes for electrostatic ion adsorption. Upon application of a small voltage (order 1 V) across each electrode pair, salt ions are removed from feed water and electrostatically held within the electrode pores. The CDI cell is then regenerated by removing or reversing the voltage, which spontaneously releases the ions and forms brine solution. CDI has a number of advantages over common desalination techniques. Most importantly, it does not require high pressure or temperature to operate, is widely scalable, and thus relevant for distributed applications (as investment and infrastructure cost is low and is directly proportional to plant capacity). CDI is potentially energy efficient and cost effective for brackish water desalination, since the energy cost per volume of treated water roughly scales with the amount of removed

iv

salt (rather than volume of treated water). CDI is thus the most advantageous in brackish water desalination as well as water recycling and reuse where salt content is far below that of seawater. In addition, CDI has the great potential for selective removal of ionic species based on ion valence, hydrated ion size and pore size, surface chemistry, and pH environments. We first focus on electrosorptive desalination energy, in both theory and practice. We present a general top-down approach to show minimum energy of ion separation is indeed Gibbs free energy of separation for most known EDLs irrespective of EDL geometry and thickness. We fabricate a low series resistance CDI cell, operate the cell at various current and flow rates, and demonstrate low-energy desalination with unprecedented 9% thermodynamic efficiency and only 4.6 kT energy requirement per removed ion. We further experimentally quantify individual loss mechanisms and show resistive and Faradaic losses as two main loss mechanisms. We show the two loss mechanisms favor different charging rates: resistive losses are dominant at high charging currents, but Faradaic losses are dominant at low charging rates, as the cell spends longer time at high voltage. Our results provide a powerful tool for optimizing CDI operation. In addition to study of desalination energy, we study charge and species transport of electrosorption process. We formulate and solved the first two-dimensional model of a CDI cell coupling external electrical network, charge conservation, and mass conservation in bimodal pore structure electrodes. We fabricate a lab-scale CDI cell, experimentally calibrate the model, and show a good agreement between model results and experimental data. Our results show CDI process has two distinct phases: a fast adsorption step at the beginning of charging followed by a slow salt removal step. Finally, we study the effect of surface functional groups on pH dependent salt adsorption and ion selectivity by developing theory and performing controlled experiments. To this end, we expanded the current surface charge models by coupling a double layer model with acid-base equilibria theory and further validate the model by well-controlled titration experiments. The fitted model with one acidic and one basic surface group showed a very good agreement with the experiments. Our results show (1) specific adsorption of cations and expulsion of anions at electrolyte pH values higher than pK of acidic groups, and (2) specific adsorption of anion and expulsion of cations at pH values lower than pK of basic groups.

v

Acknowledgements I would like to first thank my amazing PhD adviser, Prof. Juan G. Santiago. Juan’s passion, energy, and enthusiasm for teaching on any occasion is beyond words. During my time in Stanford Microfluidics Lab, I learned a great deal from him, professionally and personally, from his critical thinking and writing and presentation skills to his honesty and character. His continuous guidance and technical advice shaped my research work in the best way. Juan, I am forever grateful for all of it. I would like also to thank our collaborator Dr. Michael Stadermann and his research group, Dr. Steven Hawks, Dr. Patrick Campbell, and many others. Thank you Michael for your continuous support and advice on my research and for your help with my career development. I am also lucky to know and work with Dr. James Palko and Dr. Kang Liu in Stanford Microfluidics Lab. I am grateful for everything I learned from their broad spectrum of knowledge and problem-solving skills. I would also like to thank my reading and oral examination committee members, Prof. Ali Mani, Prof. Christopher Chidsey, and Prof. Ilenia Battiato for fruitful discussions and their genuine feedback on my dissertation. I also sincerely thank CC and Linda for always being there and making the tricky administrative tasks easy. I sincerely acknowledge the financial support of Stanford School of Engineering fellowship as well as Stanford Graduate Fellowship for giving me the opportunity and freedom to pursue my research. I am thankful of Santiago’s group alumni and current members that I had the chance to work with. Thank you Drs. Yatian Qu, Charbel Eid, Viktor Shkolnikov, Crystal Han, and Denitsa Milanova for being such great senior labmates. Thank you Diego Oyarzun, Ashwin, Diego Huyke, Gabrielle, Erica, Byunghang, Heungdong, Hyekyung, and Joseph for being great friends inside and outside the

vi

lab and making my PhD life enjoyable. You are awesome! Special thanks to all my Stanford friends and the PSA family for the unforgettable memories and countless indoor and outdoor activities. Thank you Hojat, Milad, Ali, Koosha, Mona, and 2017-2018 PSA board of directors. I am afraid I cannot mention all the names here, but let me say this: I cherish my friendship with every single one of you. Lastly, I am immensely grateful of the continuous support and unconditional love I received from my parents and my sister. Thank you for always being there for me, and thank you for all the sacrifice you made, because I could not be where I am now without it.

vii

Contents Abstract

iv

Acknowledgements

vi

1 Introduction

1

1.1

The challenge of fresh water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Review of brackish water desalination methods . . . . . . . . . . . . . . . . . . . . .

2

1.2.1

Thermal desalination processes . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.2

Membrane-based desalination processes . . . . . . . . . . . . . . . . . . . . .

4

1.2.3

Electrical desalination processes

. . . . . . . . . . . . . . . . . . . . . . . . .

5

Capacitive deionization for brackish water treatment . . . . . . . . . . . . . . . . . .

5

1.3.1

Introduction to capacitive deionization . . . . . . . . . . . . . . . . . . . . . .

6

1.3.2

Advantages of capacitive deionization . . . . . . . . . . . . . . . . . . . . . .

7

Scope of dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.3

1.4

2 Thermodynamics of Ion Separation by Electrosorption

13

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.2

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.2.1

Problem definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.2.2

Free energy functional and electrical work of electrosorption . . . . . . . . . .

16

2.2.3

Alternative approach: direct integration of electrical work . . . . . . . . . . .

20

2.2.4

Effect of chemical or electrochemical reactions

21

viii

. . . . . . . . . . . . . . . . .

2.2.5

Thermodynamic efficiency of electrosorption in practice . . . . . . . . . . . .

22

Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.3.1

CDI cell design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.3.2

Cell characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.3.3

Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.4.1

Importance of series resistance energy loss . . . . . . . . . . . . . . . . . . . .

27

2.4.2

Operation for high thermodynamic efficiency . . . . . . . . . . . . . . . . . .

28

2.4.3

Trade-off between thermodynamic efficiency and throughput . . . . . . . . .

29

2.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.6

Addendum to: Thermodynamics of Ion Separation by Electrosorption . . . . . . . .

32

2.6.1

Equivalency of minimum electrical work and free energy of separation . . . .

32

2.6.2

Direct integration of minimum electrical work . . . . . . . . . . . . . . . . . .

33

2.6.3

Free energy of separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

2.6.4

Voltage and concentration profiles . . . . . . . . . . . . . . . . . . . . . . . .

36

2.6.5

CDI cell performance

37

2.3

2.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Energy breakdown in CDI

41

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

3.2

Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3.2.1

CDI cell design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3.2.2

Energy pathway in CDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.2.3


46

Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

3.3.1

Voltage profile and energy breakdown . . . . . . . . . . . . . . . . . . . . . .

48

3.3.2

In-situ series resistance measurement . . . . . . . . . . . . . . . . . . . . . . .

50

3.3.3

Energy losses in CDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

3.3.4

Energy and salt adsorption performance in CDI . . . . . . . . . . . . . . . . .

55

3.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

3.5

Addendum to: Energy breakdown in CDI . . . . . . . . . . . . . . . . . . . . . . . .

60

3.3

ix

3.5.1

Voltage profile under constant current conditions . . . . . . . . . . . . . . . .

60

3.5.2

Effluent concentration measurement . . . . . . . . . . . . . . . . . . . . . . .

61

3.5.3

In-situ series resistance measurement . . . . . . . . . . . . . . . . . . . . . . .

61

3.5.4

Energy and cycle time measurements . . . . . . . . . . . . . . . . . . . . . . .

63

3.5.5

Energetic operational metric (EOM) . . . . . . . . . . . . . . . . . . . . . . .

65

3.5.6

Time scales in fbCDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

4 A Two-Dimensional Porous Electrode Model for CDI

69

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

4.2

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

4.2.1

Double layer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

4.2.2

Validity of mD model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

4.2.3

Transport equations in flow-between CDI cell . . . . . . . . . . . . . . . . . .

76

4.2.4

Equilibrium solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

Experimental setup and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

4.3.1

CDI cell design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

4.3.2


85

Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

4.4.1

Parameter extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

4.4.2

Charge efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

4.4.3

Temporal results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

Spatiotemporal results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

4.5.1

Macropore concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

4.5.2

Macropore concentration for high inlet concentration . . . . . . . . . . . . . .

94

4.5.3

Micropore concentration and evidence of a concentration shock . . . . . . . .

95

4.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

4.7

Addendum to: A Two-Dimensional Porous Electrode Model for CDI . . . . . . . . . 100

4.3

4.4

4.5

4.7.1

Effect of electrode compression on contact resistance . . . . . . . . . . . . . . 100

4.7.2

Macropore concentration plots for low and high inlet concentration . . . . . . 100

4.7.3

Micropore concentration plots and evidence of shockwave . . . . . . . . . . . 103

x

4.7.4

Diffusion flux and salt adsorption rate . . . . . . . . . . . . . . . . . . . . . . 103

4.7.5

Macropore potential and micropore charge state . . . . . . . . . . . . . . . . 104

4.7.6

Note on potential boundary condition . . . . . . . . . . . . . . . . . . . . . . 105

5 Equilibria model for pH variations and ion adsorption in CDI electrodes

108

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.2

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.3

5.4

5.2.1

Multi-equilibria surface charge model . . . . . . . . . . . . . . . . . . . . . . . 110

5.2.2

Electric double-layer model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.2.3

Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.3.1

Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.3.2

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.4.1

Model results and physical insights . . . . . . . . . . . . . . . . . . . . . . . . 116

5.4.2

Experimental results and model verification . . . . . . . . . . . . . . . . . . . 125

5.4.3

Prediction of N-ACC surface charge . . . . . . . . . . . . . . . . . . . . . . . 127

5.4.4

Proposed generalization for charge efficiency of CDI systems

. . . . . . . . . 127

5.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.6

Addendum to: Equilibria model for pH variations and ion adsorption in CDI electrodes131 5.6.1

Ideal solution limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.6.2

Titration model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.6.3

Salt adsorption and micro-to-macropore equilibrium for asymmetric carbon . 133

5.6.4

Determination of point of zero charge . . . . . . . . . . . . . . . . . . . . . . 134

5.6.5


6 Conclusions 6.1

. . . . . . . . . 137 141

Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.1.1

Identification and quantification of energy loss mechanisms . . . . . . . . . . 141

6.1.2

Charge and mass transport in two-dimensions . . . . . . . . . . . . . . . . . . 142

xi

6.1.3 6.2

Effect of surface functional groups on salt adsorption . . . . . . . . . . . . . . 142

Recommendations for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.2.1

Long term performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.2.2

Temperature effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.2.3

Surface conduction effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.2.4

Species-specific adsorption and selectivity . . . . . . . . . . . . . . . . . . . . 144

A Notes on fabrication and test of CDI cells A.1 Cell design and fabrication

145

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

A.1.1 Sources for materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 A.1.2 fbCDI cell designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 A.2 Test of CDI cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 A.2.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 A.2.2 Pre-testing and troubleshooting . . . . . . . . . . . . . . . . . . . . . . . . . . 148 A.2.3 Serial communication with devices . . . . . . . . . . . . . . . . . . . . . . . . 149 B Electrochemical measurements of CDI cells

155

B.1 Cyclic voltammetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 B.2 Electrochemical impedance spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 157 C Use of CNC machine for fabrication CDI cells

161

C.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 C.2 Hardware setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 C.3 Software settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 C.4 Notes on operation of CNC machine . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

xii

List of Tables 1.1

Review of desalination methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.1

Comparison between the low resistance cell and the conventional cell. . . . . . . . .

25

4.1

Parameters used in Figure 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

4.2

Parameter settings for the two-dimensional model for an fbCDI cell incorporating mD EDL model and operated in constant external voltage and flow rate. . . . . . . . . .

89

5.1

Model parameters used in titration model and Figure 5.2 . . . . . . . . . . . . . . . 118

5.2

Parameters used in model for N-ACC samples . . . . . . . . . . . . . . . . . . . . . . 126

5.3

Parameters used in Figure 5.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

A.1 Recommended materials and providers for fabrication of CDI cells. . . . . . . . . . . 146

xiii

List of Figures 1.1

Schematic of a typical fbCDI cell operation with two conductive and porous electrodes held apart by a non-conductive porous spacer. (a) Upon application of voltage across the electrodes (charging step), ions are sequestered in their respective electrodes and fresh water is collected at the outlet. (b) Once the applied voltage is lifted, ions immediately leave the electrode pores and form brine stream, which is later collected as waste water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

7

Volumetric energy cost of desalination for CDI (red curves) and RO (solid grey curves) 3

in units of kWh/m versus feed concentration for salt removal down to 5 mM solution with water recovery of (a) WR = 50% and (b) WR = 75%. Thermodynamic limit (dashed curve) shows minimum theoretical limit. Shown here are estimates for productivity values of 20 and 50 L/m2 /h. For seawater concentrations, the energy required by RO is significantly lower than CDI. However, CDI outperforms RO for brackish water desalination for initial concentrations of around 20 mM and below. CDI has thus great potential for increased energy efficiency for the globally important sector of brackish water desalination. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1

9

(a) Schematic of an electrochemical desalination system with a pair of porous, conductive electrodes (only one shown) in contact with a reservoir. (b) Solid line is an ion separation cycle (on cions vs. c∞ plane) wherein charging and discharging are performed with respect to two reservoirs. Dashed lines depict some arbitrary charging/discharging relative to a time-varying value of reservoir concentration. . . .

xiv

16

2.2

Schematic of embedded titanium mesh sheet into the activated carbon electrode using 100 C hot press. Images of top view and cross-section side view after hot press step show details of embedded titanium mesh. . . . . . . . . . . . . . . . . . . . . . . . .

2.3

25

(a) Cyclic voltammetry (CV) test of the assembled cell at 0.2 mV/s scan rate and 20 mM NaCl electrolyte solution with −0.6 to 0.6 V voltage window. CV test shows about 17 F cell capacitance or about 60 F/g specific capacitance. (b) Electrochemical impedance spectroscopy (EIS) test of the assembled cell again with 20 mM NaCl electrolyte solution. The setup resistance (ionic resistance in separators, electric resistance of current collectors and wires) is only 0.33 Ω and the contact resistance is negligible.

2.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

Measured volumetric energy consumption (Ev ) versus estimated resistive energy consumption (Ev,resist ) for a wide variety of CDI cell designs operated under constantcurrent (CC) operation. Ev,resist provides a very good estimate of minimum Ev observed in practice, as all data points lay near and above the line set by the resistive limit Ev = Ev,resist . This shows decreasing series resistance is essential for improvement of thermodynamic efficiency.

2.5

. . . . . . . . . . . . . . . . . . . . . . .

28

(a) Measurements of volumetric energy consumption Ev , (b) energy consumption per ion and salt removed in units of kT/ion and kJ/mol, and (c) the thermodynamic efficiency for CC operation and 0.4-1 V voltage window at various currents and flow rates. Mid-range values of current density (4 A/m2 ) and flow rate (0.44 ml/min) result in thermodynamic efficiency of up to 9%, unprecedentedly high for traditional CDI designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6

29

Thermodynamic efficiency versus (a) average effluent concentration reduction ∆c and (b) productivity (in units of L/h/m2 ) for our cell and selected published CDI cells under constant current charge/discharge. The thermodynamic efficiency generally correlates with high ∆c and low throughout productivity. . . . . . . . . . . . . . . .

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30

2.7

(a) Variation of the Gibbs free energy of separation normalized by feed concentration, ∆gsep /c0 vs. concentration reduction normalized by feed concentration, ∆c/c0 for varying recovery ratio values γ = 0.25, 0.5, and 0.75. Solid black lines represent the exact variation from thermodynamic principles (Equation 2.26), and the dashed grey lines represent a curve fit from the approximation of the form, ∆gsep /c0 = a(∆c/c0 )n , where a and n are fitting parameters. The fit parameters a and n depend solely on the recovery ratio, and their variation is shown in (b). . . . . . . . . . . . . . . . . .

2.8

36

Selected voltage and effluent concentration profiles for constant current experiments with 20 mM NaCl inlet salt solution. Each column corresponds to experiments at a fixed current density (2, 4, and 8 A/m2 , respectively), and labels in each panel shows values of flow rate used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.9

37

Measured values of CDI performance indicators: (a) Average salt adsorption rate (ASAR), (b) Energy normalized adsorbed salt (ENAS), (c) Average concentration reduction (∆c), and (d) Gibbs free energy of separation (∆gsep ), versus operating current (CC operation) for flow rates of 0.22, 0.44, 0.89, 1.73 and 3.47 ml/min. In all cases, the cell voltage window was 0.4-1 V. Dashed line in (a) represents an asymptotic variation of ASAR with current for all flow rates. . . . . . . . . . . . . . . . . . . . .

3.1

40

(a) Schematic of circular fbCDI cell with five pairs of activated carbon electrodes (only two pairs shown here). The stack was housed inside a clamshell structure (not shown here) and sealed with O-rings and fasteners. Arrows indicate flow paths. (b) Schematic of energy pathway in a typical CDI system. A fraction of input energy Ein R P during charging is dissipated via resistive (Ein ) and parasitic (Ein ) processes and the

rest is stored in the cell (Ecap ). A portion of stored energy is then dissipated during R P discharging (Eout and Eout ) and remaining energy is recovered (Eout ). . . . . . . . .

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46

3.2

(a) Measured voltage profile of the cell vs. time under 2 mL/min flow rate at 200 mA current and limit voltage of Vmax = 1.2 V. Inset shows RC circuit analogy of the cell, where Rs and Rp respectively model series and parallel resistances in CDI. (b) Power input/generation of the cell for the conditions identical to those of (a). Shaded areas labeled as Ein and Eout show energy input and recovered during charging and discharging in a single cycle. Diagonal hatched areas show series resistive energy loss R,S R,S P (Ein and Eout ), and vertical hatched areas show parasitic energy losses (Ein and P Eout ). Inset shows measured parasitic current vs. Vcap as obtained from independent,

constant voltage experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3

49

Measured series resistance vs. ∆Vcap during charging and discharging for 25, 50, 100, 150, 200, and 300 mA currents and Vmax = 1.2 V (each loop corresponds to a fixed current). At low currents, Rs does not vary considerably throughout the cycle, while it varies more strongly at high currents due to significant salt removal. The inset presents series resistance data vs. time (normalized by cycle time tcycle ) for one cycle. 51

3.4

(a) Measured energy loss per cycle vs. ∆Vcap for 25, 50, 100, 150, 200, and 300 mA currents. Energy loss increases with both ∆Vcap and I0 . At low currents, energy loss varies approximately exponential with ∆Vcap , while it is almost linear at high currents. (b) Measured parasitic loss per cycle vs. ∆Vcap . Parasitic losses (likely associated with Faradaic reactions) vary exponentially with ∆Vcap (see inset). (c) Resistive loss (series and non-series) in one cycle for experimental conditions identical to those of (a). Resistive loss increases almost linearly with both ∆Vcap and I0 . (d) Calculated stored energy is well described as the square of ∆Vcap . . . . . . . . . . . .

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54

3.5

(a) Ratio of resistive to total energy loss in one cycle vs. ∆Vcap for 25-300 mA currents. Resistive loss dominates total loss at high charging current and small ∆Vcap cases. Parasitic loss, however, is dominant at low current and high ∆Vcap (see shaded area in which parasitic > 50% of total loss). (b) Ratio of stored charge to total energy loss in one cycle vs. ∆Vcap for the same data as in (a). This ratio quantifies the effectiveness of energy storage in the cell and is generally greater at lower currents. Results show this ratio has an optimum at small currents (25 and 50 mA), and this optimum coincides with ∆Vcap at which (series plus non-series) resistive loss and parasitic loss are comparable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6

55

(a) Normalized salt adsorption (Γads ) in units of µmole/cm2 and mg/g, (b) average salt adsorption rate (ASAR) in units of µmole/cm2 /min and mg/g/min, and (c) energy normalized adsorbed salt (ENAS) in units of µmole/J and mg/J, each as a function of ∆Vcap . Results are for the experimental conditions identical to those of Figure 3.4. The interplay between resistive effects and parasitic effects results in maxima in ASAR and ENAS for low-to-midrange applied currents. . . . . . . . . . .

3.7

57

Contour plots of interpolated (a) average salt adsorption rate (ASAR) in units of µmole/cm2 /min and (b) energy normalized adsorbed salt (ENAS) in units of µmole/J vs. current I0 and ∆Vcap . (c) ASAR versus ENAS for the same data as in (a) and (b). The arrow shows direction of increase of ∆Vcap . ASAR versus ENAS respectively quantify desalination speed and energetic performance of the cell. Results show a trade-off between the two: ASAR is greatest at high currents and high ∆Vcap , while ENAS is generally greater in low currents and low ∆Vcap . In very low currents (i.e. 25 and 50 mA), however, ENAS shows an abrupt drop as ∆Vcap passes a certain limit. The optimum value of ENAS at lowest currents corresponds to where resistive and parasitic losses are comparable.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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58

3.8

Measured external voltage versus time for 25-300 mA charging/discharging current (I0 ) and limit voltages 0.2-1.2 V (Vmax ). Galvanostatic voltage increases until the pre-set limit voltage is reached and current is reversed. The jumps just after current reversals are associated with the fast response associated with purely serial resistive response. Profiles shown here are all under DSS condition and each profile is an overlay of 2 to 4 successive cycles.

3.9

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

Raw measurements of effluent concentration profiles measured via an in-line conductivity meter. Each curve is an overlay of two or more cycles under DSS condition. Dashed lines show influent concentration level. . . . . . . . . . . . . . . . . . . . . .

61

3.10 Measured (a) voltage and (b) effluent concentration for I0 = 200 mA and Vmax = 0.8 V during the first six cycles. DSS condition is established after first few cycles. .

62

3.11 (a) An example of 2 mA amplitude AC current probe signal (δI) with a fixed 200 mA DC component (I0 ) used for in-situ resistance measurement of our fbCDI cell. (b) Voltage response of the cell for current signal shown in (a). The response consists of saw-tooth with the underlying linear component associated with the charging of electrodes. To calculate resistance, we subtract the underlying linear signal variation from voltage response and divide the amplitude of resulting signal (δV ) by δI. . . .

63

3.12 Time resolved series resistance for various currents (25-300 mA) and limit voltages (0.2-1.2 V) during cell operation. Each curve is an overlay of resistance measurements for 2 to 4 cycles under DSS condition. . . . . . . . . . . . . . . . . . . . . . . . . . .

64

3.13 Series resistance during cell operation as a function of ∆Vcap for various currents (25-300 mA) and limit voltages (0.2-1.2 V). Resistance values form a closed loop for each experimental condition. The loop becomes wider and more asymmetric under higher currents and higher ∆Vcap , which we attribute to the effect of depletion and enrichment of ions from the spacer region (substantial salt removal and enrichment during charging and discharging, respectively). . . . . . . . . . . . . . . . . . . . . .

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64

3.14 (a) Input and (b) recovered electrical energy of our fbCDI cell as a function of ∆Vcap for currents in the range of 25-300 mA. Both energies increase monotonically with ∆Vcap , but current magnitude has opposite effects on input and recovered energy (see arrows in (a) and (b)). (c) Cycle time versus ∆Vcap under experimental conditions similar to those of (a) and (b). Cycle time and ∆Vcap are approximately linearly dependent.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

3.15 Contour plot of EOM (defined as ASAR·ENAS) as a function of current I0 and ∆Vcap . Dashed curve and the associated parasitic dominant shaded region are consistent with Figure 3.7 of Chapter 3. As with our earlier observations, EOM is maximized roughly in the regions with comparable resistive and parasitic losses. . . . . . . . . . . . . . .

66

3.16 (a) Schematic of effluent concentration profile and voltage profile for an fbCDI cell under constant current and constant flow rate conditions. (b) Regime map corresponding to plateau mode (upper-left region) with low current and high flow rate, and triangular mode (lower-right region) with relatively high current and low flow rate. 68 4.1

(a) Schematic of a slit micropore with width of 2L and Stern layer thickness of λSt . The electrode potential φe is distributed between Stern layer potential drop ∆φSt , diffuse layer potential drop ∆φdif f , and Donnan potential drop ∆φD . (b) Diffuse layer potential drop (normalized by thermal voltage) on σ ∗ vs. λ∗ plot. The shaded region below the curve ∆φdif f = 0.1 is the validity region of the mD model, with the potential drop across diffuse layer is negligibly small compared to the thermal voltage VT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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77

4.2

(a) Schematic of elements of our two-dimensional adsorption-desorption and flow model for an fbCDI cell. Two porous carbon electrodes of thickness Le are sandwiched between two current collectors. Electrodes are separated by a distance Ls to provide space for flow of water (left to right). We lump all contact resistances, e.g. wire-to-current collector and current-collector-to-electrode connections, into a purely resistive element Rc . (b) Not-to-scale schematic of computational domain and boundary and interface conditions in non-dimensional form. The dash-dot-dash line denotes the geometric symmetry line. The computation domain includes inlet and outlet regions for the flow of length L/8. . . . . . . . . . . . . . . . . . . . . . . . . .

4.3

78

Three-dimensional drawing of our CDI cell clamshell structure. We sandwiched a pair of activated carbon electrodes inside the upper and lower clamshells and sealed the desalination cell using gasket and fasteners (not shown). Electrodes have dimension of 100 × 20 mm with 0.68 mm thickness. Our design had inlet and outlet plenum chambers in order to uniformly introduce salt solution to and collect desalted water from the CDI cell. See Figure 4.10 for more details. . . . . . . . . . . . . . . . . . . .

4.4

85

(a) Near-equilibrium adsorbed salt and transferred charge (per mass of electrodes) and (b) charge efficiency, Λ = F Γeq /Mw Σeq , for 2 h of constant voltage operation in the range of 0 to 1 V, and flow rate of 0.42 mL/min. Feedwater inlet concentration is 20 mM KCl. Open circles and diamonds are measured data and solid lines are model prediction. Charge efficiency approaches unity as voltage increases, indicating stronger asymmetry in the micropore counter-ion adsorption and co-ion expulsion. .

4.5

89

(a) Measured and (b) model predicted normalized effluent salt concentration vs. time (normalized by diffusion time L2e /De ) for constant voltage operation at 0.4, 0.6, 0.8, and 1 V and constant flow rate of 0.42 mL/min. Feedwater concentration is 20 mM KCl, and time is normalized by diffusion time scale across the electrode thickness. .

4.6

91

(a) Cumulative stored salt Γ (t) (mg/g) and (b) electric charge Σ (t) (C/g) vs. normalized time for 2 h of charging for constant external voltage operation at 1 V and flow rate of 0.42 mL/min. Each is reported per electrode dry mass. . . . . . . . . . .

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92

4.7

Normalized ion concentration fields in the gap and macropores for the symmetric top half of the cell (not-to-scale) for constant voltage (a) charging (1 V) and (b) discharging (0 V) (as per Table 4.2). The dashed lines indicate the electrode-solution interface. Time is normalized by the diffusion time scale associated with the electrode thickness. In charging, a depletion region forms near and propagates upward from the electrodesolution interface. This rapid charging is followed by a slow, diffusion-limited uptake of new ions from the gap. At the beginning of discharge, formerly adsorbed ions (now have same sign of charge as electrode) are quickly electrostatically repelled from micropores in the near-gap region. This forms a near-gap depletion region whose boundary then propagates upward. Discharge then slows down significantly as the cell proceeds into a diffusion-limited transport of ions out of the electrode. . . . . . .

4.8

94

Ion concentration fields in gap and macropores normalized by a 200 mM inlet concentration. Shown is the not-to-scale top half of the cell for constant voltage (a) charging (1 V) and (b) discharging (0 V). All other parameters are similar to those of Figure 4.7. In a similar manner to Figure 4.7, a depletion region rapidly forms inside the electrodes and charging rate is then limited by diffusion. However, the diffusion-limited charging process occur faster than 20 mM case. We hypothesize this difference is associated with electrode starvation, which is only observed in the low concentration (20 mM) case. Discharge process shows an analogous trend, i.e. rapid but brief discharge followed by slow diffusion. . . . . . . . . . . . . . . . . . . . . . .

4.9

96

Normalized micropore concentration profile in the upper electrode. Note that plots are not-to-scale. Micropores in the regions near solution-electrode interface (lower edge of squares) are almost completely charged quickly, while the rest of micropores remain partially-charged. As time goes on, the interface between low and high concentration regions becomes thinner and propagates toward the collector boundary (upper edge of squares) until all the micropores are completely charged. . . . . . . . . . . . . . .

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97

4.10 (a) Three-dimensional drawing of CDI cell clamshell structure. We sandwiched a pair of activated carbon electrodes inside the upper and lower clamshells and sealed the desalination cell using gasket and fasteners. Electrodes have dimension of 100×20 mm with 0.68 mm thickness. Our design had compression slots we used to compress outer regions of the electrode upon assembly. This technique led to 3- to 4-fold decrease in contact resistance. (b) Cross-section schematic of stack of activated carbon electrodes between stainless steel current collectors. Current collectors were bonded to backside of the electrodes by silver epoxy. We placed thin frames of plastic between the electrodes to avoid the electrodes touching each other. (c) Cross sectional view of the assembled cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.11 Normalized ion concentration in gap and macropores along the line x = 1/2 in constant voltage (a) charging and (b) discharging under conditions similar to those of Figure 4.7 of Chapter 4. Vertical dashed line denoted the electrode interface and results are shown for upper-half of the system. Gray lines represent concentration profiles shortly after starting charging or discharging, and black lines show concentration evolution at longer times. Each line is labeled with corresponding non-dimensional time (scaled by L2e /De ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.12 Ion concentration in gap and macropores normalized by 200 mM inlet concentration along x = 1/2 for constant voltage (a) charging and (b) discharging under conditions similar to those of Figure 4.8 of Chapter 4. The label for each curve corresponds to time non-dimensionalized by L2e /De . In contrast to Figure S3a, electrode starvation is not observed. However, early charging as well as discharging have time scales analogous to 20 mM case. Moreover, the concentration lines for the discharge process are quite similar to 20 mM case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

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4.13 Normalized micropore counter-ion concentration along x = 1/2 for constant voltage (a) charging and (b) discharging (under the charging conditions of Figure 4.3). The degree of charge of micropores near the interface quickly increases to 85% (corresponding to c+ m ≈ 30). Counter-ion diffusion (from bulk) and adsorption to partiallycharged micropores sharpens gradient between the low- and high-charge state zones, and apparently forming a moving shock which slowly propagates toward the collector. The process continues until micropores are filled. Discharging shows an initially rapid expulsion of ions, followed by a slow, diffusive expulsion. . . . . . . . . . . . . . . . . 104 4.14 Normalized (a) diffusion flux in y-direction and (b) local salt adsorption rate along the line x = 1/2 during charging process. (a) Shows the formation of a step-like distribution of diffusion flux (after time t = 1), and this feature propagates to the right. Coincident with this step feature is a corresponding local peak in adsorption rate. This peak in adsorption rate occurs at the left boundary of the shrinking starvation region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.15 Gap and Macropore potential (normalized by thermal voltage) along x = 1/2 during (a) charging and (b) discharging processes (under the charging conditions of Figure 4.11). In charging, initial macropore potential at the near-collector region quickly drops to ∼ 7VT and remains approximately pinned near this value from t = 1 to 4 (consistent with partially charged micropores). Potential drops to zero throughout the electrode as micropores fully charge (at about t = 8). During discharge, macropore potential quickly increases and is positive as net charge is released. Macropore then relaxes back to zero as the electrode is discharged. . . . . . . . . . . . . . . . . 106 4.16 Normalized potential in the inlet, outlet, and spacer regions along y = Ls /4 at time t = 20 s for four possible choice of BCs at inlet and outlet. Results show overlap of all solutions. Parameter settings are similar to those in Figure 4.7 of Chapter 4. . . . 107

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5.1

Schematic of our acid-base equilibria model for activated carbon with bimodal (micropore and macropore) pores. Here, the carbon is electrically floating (not connected to an electrical source) and in equilibrium with the solution in a beaker. Electrolyte (outside carbon) and macropores (pathways for ion transport) are electroneutral. Micropores, however, are ion adsorbing regions wherein electronic, ionic, and surface charges (due to acid-base equilibrium) are in balance with zero net charge. We here depict a special case of one acidic and one basic surface functional group in the micropores. Note the global net neutrality of surface charges, charge in solution, and net surface electrons within the micropore. . . . . . . . . . . . . . . . . . . . . . . . . 114

5.2

Summary of characteristic shapes of predicted titration curves for samples with weak acids and weak bases for porous carbon samples. (a) and (b) each consider a single acidic functional group, and respective pKX are 4 and 10, while (c) and (d) each consider a single basic functional group with respective pKY of 4 and 10. Titrants are 1 M HCl and 1 M NaOH. We show results for cX,0 , cY,0 = 0.1, 0.5, and 1 M (solid lines). Dashed lines indicate blank titrations (cX,0 = cY,0 = 0). Insets show change in concentration of anion/cation (∆c± = c± − c±,0 ) as a function of pHf . Initial salt concentration and solution volume are c0 = 20 M and vs ol = 25 mL. Other parameters are listed in Table 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.3

Ionic strength affects the equilibrium between carbon micropores and the electrolyte (outside the micropore). (a) Final micropore pH (pHm,f ) versus final electrolyte pH (pHf ) for titration model with c0 = 0, 1, 20, 100, and 500 mM. Surface parameters are cX,0 = 1 M, pKX = 4, cY,0 = 1 M, and pKY = 10, and other parameters are similar to those of Figure 5.2. pHm,f is lower (higher) than pHf above (below) PZC (pHPZC = 7 here), and pHm,f approaches pHf only at high ionic strengths. (b) Concentration of charged acidic and basic groups (cX− and cYH+ ) versus pHf and (c) versus pHm,f . Curves for cX− and cYH+ collapse onto a single curve when plotted against pHm,f , as per our formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 121

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5.4

Effect of ionic strength and micropore pH environment on salt adsorption dynamics. (a) Net adsorbed salt Γ = Γ+ +Γ− as a function of pHm,f for a carbon with one acidic (cX,0 = 1 M, pKX = 4) and one basic (cY,0 = 1 M, pKY = 10) surface groups for c0 = 0, 20, 100, and 500 mM. Other parameters are identical to those of Figure 5.2. Salt adsorption is considerable at pHm,f < pHX and pHm,f > pHY . Further, the magnitude of salt adsorption decreases with increasing ionic strength (or equivalently with increasing c0 ). The insets show individual cationic and anionic salt adsorption (Γ+ and Γ− ) for c0 =20 and 500 mM. (b) Chemical charge efficiency versus pHm,f for the same c0 values as in (a). At low ionic strengths, counter-ion adsorption (cations at high pHm,f and anions at low pHm,f ) dominates co-ion expulsion and so |Λchem |→ 1. At high ionic strengths, however, counter-ion adsorption and co-ion expulsion are of the same order (ion swapping) and hence |Λchem |→ 0. . . . . . . . . . . . . . . . . . 124

5.5

Model and experimental results of pH shift and ion adsorption.

(a) Final elec-

trolyte pH (pHf ) versus initial pH (pH0 ) for N-ACC samples (circles), control samples (squares), and fitted model (solid line). Dashed line shows pHf = pH0 . pHf for NACC samples is higher than pH0 for pH0 < 5, and is lower than pH0 for pH0 > 5. N-ACC therefore is an ion adsorbent with pHPZC ≈ 5. (b) Change in concentration of sodium and chloride ions versus pHf for experiments and fitted model. Results show specific adsorption of chloride ions at pHf < pHPZC and strong adsorption of sodium ions otherwise. We modeled N-ACC with cX,0 = 1.17 M, cY,0 = 1.1 M, pKX = 4, and pKY = 8. Other parameters used in the model are listed in Table 5.2. . . . . . . . . 126 5.6

Prediction of net, acidic, and basic surface charge densities (σchem , σX− , and σYH+ ) of N-ACC samples in units of C cm−3 (left axis) and M (right axis) versus (a) micropore pH (pHm,f ) and (b) final electrolyte pH (pHf ). Net surface charge density is sum of acidic and basic surface densities. pHPZC (where σchem = 0) is ∼ 5 for our N-ACC. Parameters used in calculation of charge densities are listed in Table 5.2. . . . . . . . 128

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5.7

Effect of ionic strength on micropore-to-macropore equilibrium for asymmetric carbon. (a) pHm,f versus pHf for titration of an asymmetric carbon with cX,0 = cY,0 = 1 M, pKX = 4, and pKY = 8 at initial concentrations c0 = 0, 1, 20, 100, and 500 mM. pHm,f is lower (higher) than pHf above (below) pHPZC = 6. (b), (c) Concentration of charged acidic and basic groups (cX− and cYH+ ) versus pHf and versus pHm,f . Similar to symmetric carbon, each cX− and cYH+ collapses on a single curve when plotted against pHm,f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.8

Effect of ionic strength and micropore pH environment on salt adsorption dynamics for carbon with asymmetric surface chemistry parameters of cX,0 = cY,0 = 1 M, pKX = 4, and pKY = 8. (a) Net adsorbed salt Γ = Γ+ + Γ− as a function of pHm,f for c0 = 0, 20, 100, and 500 mM. Salt adsorption is considerable at pHm,f < pKX and pHm,f > pKY and is much lower otherwise. Moreover, decreases with ionic strength (or with c0 ). (b), (c) Cationic and anionic salt adsorption (Γ+ and Γ− ) for c0 = 20 and 500 mM cases. (d) Chemical charge efficiency Λchem versus pHm,f . General trend of Λchem is similar to the case discussed in Chapter 5, however, Λchem is not symmetric about PZC (pHPZC = 6) here. . . . . . . . . . . . . . . . . . . . . . . . . 135

5.9

pHPZC as a function of α1 and α2 for pKX = 4 and pKY = 10 (solid lines). pHPZC decreases with α1 and increases with α2 . At small values of α1 and α2 (i.e., cX,0 = cY,0 ), pHPZC approaches (pKX + pKY )/2. At high values of α1 , on the other hand, pHPZC varies almost linearly with log (α1 ) as pHPZC = pKX − log (α1 ) and so pHPZC is independent of pKY . At high values of α2 , pHPZC is independent of pKX and varies linearly with log(α2 ) as pHPZC = log (α2 ) − pKY . Dashed lines are extrapolation of pHPZC for cases where α1 α0 and α2 α0 . . . . . . . . . . . . . . . . . . . . . . . 137

5.10 Volumetric adsorbed salt and electronic charge (in units of M) vs external voltage Vext at various (a) analytical concentration of acidic and basic surface charges, (c) dissociation constant of acidic functional groups, (e) salt concentrations, and (g) electrolyte pH values. (b), (d), (f), (h) Charge efficiency vs Vext for parameters as in (a), (c), (e), and (g). Other parameters used are listed in Table 5.3. . . . . . . . . . . . . 139 A.1 Schematic and a photo of a CDI cell for quick fabrication. . . . . . . . . . . . . . . . 147

xxvii

A.2 3D design and a photo of a CDI cell made by CNC machine. . . . . . . . . . . . . . 147 A.3 Schematic of our experimental setup consisting a solution reservoir, a peristaltic pump, eDAQ conductivity and pH sensors, Keithley sourcemeter, and a CDI cell. . . 148 B.1 Schematic of three-point CV and EIS measurements using a potentiostat and reference electrode. Use of a reference electrode is optional. In this work, we used a two-point measurement instead, without reference electrode. For more information, refer to Figure B.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 B.2 Schematic of two-point CV and EIS measurements using a potentiostat. We used two-point (and not three-point) measurement method for CV and EIS measurements throughout this work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 B.3 An example of Gamry Instruments Framework software settings for cycling voltammetry measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 B.4 An example of CV measurement for high to low scan rates at s = 5, 1, and 0.2 mV/s. Scan rate of 5 mV/s is too high (CV curve is leaf-shaped) and should not be used to approximate the capacitance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 B.5 An example of Gamry Instruments Framework software settings to perform electrochemcial impedance spectroscopy test. . . . . . . . . . . . . . . . . . . . . . . . . 160 C.1 Photo of spindle of the CNC machine with endmill mounted. . . . . . . . . . . . . . 162 C.2 Photo of hand-held controller (left), used for setting the Z-coordinate origin. The Z-sensor is placed on top of work place and endmill is lowered to about 1 cm above the sensor (top right). Make sure that the sensor is properly connected to the machine front panel (bottom right) before pressing Z0 SENSE button. . . . . . . . . . . . . . 163

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Chapter 1

Introduction 1.1

The challenge of fresh water

Water is the essential element of life. Our planet Earth is the only life-sustaining place (so far), because it holds vast amounts of water and oxygen. However, although 75% of the Earth surface is covered with water, only a small fraction of it is readily available for our consumption. To put this into perspective, only about 2.5% of the water on the Earth is fresh water, and the rest is salt water in the oceans and seas. Of the total available fresh water, the majority is frozen ice or snow, almost a third is in the form of groundwater, and only 0.6% (of 2.5% total fresh water) is surface fresh water in lakes and rivers. [1] The fresh water is thus a very limited resource with uneven spread across the globe. In fact, dry regions with scarce fresh water are becoming even drier. [2] To exacerbate the issue, according to the United Nations 2018 world water development report, [2] the available surface water resources at the continent level remains relatively constant as opposed to the global water use, which has increased 6-fold over the past 100 years and is estimated to grow by %1 annually. [3]. This increase in water demand as well as increase in per capita fresh water demand due to urbanization and industrialization suggests more and more fresh water shortage in years to come. A well-cited index of local water shortage is perhaps “water scarcity” (WS) defined as the ratio of water footprint over local water availability. [4] The former refers to the fresh surface water and groundwater that is withdrawn and not returned, and the latter is the total fresh water available

1

CHAPTER 1. INTRODUCTION

2

locally. Water is said to be scarce if WS > 1 and is severely scarce if WS > 2. To date, around two billion people (almost one-fifth of the world’s population) live in water-scarce areas. [5] In the U.S., one-third of all counties in the 48 contiguous states will face high risks of water scarcity by mid-century. [6] Besides the quantity, the quality of available water has also seen a decline. [7] Currently, almost one-sixth of the world’s population relies on unsafe or unstable water supplies. [8] Examples include chemical contamination, increased salinity, and presence of bacteria and other pathogen. Bacteria and pathogen polluted water sources may bring communities various waterborne diseases, including amebiasis, diarrhea, chlorea, etc. [9] Further, naturally occurring ionic contaminants including fluoride, as in China, [10] or arsenic, a major problem in India, [11] Southeast Asia, and South America, [12–14] create severe health issues from chronic exposure. Increased salinity of groundwater (making otherwise viable water sources unusable) can be caused by natural mineral deposits or by an overdraft of groundwater sources, which can lead to saltwater intrusion near the coast. [15] The salinity of these waters (also known as brackish water) is far below sea water, but they are still undrinkable. [16] Brackish water is relatively common and provides an appealing source of potable water with appropriate treatment technologies.

1.2

Review of brackish water desalination methods

Despite significant improvements in the access to water sources and desalination facilities over the last two decades, much still remains to be accomplished to effectively meet the demand for fresh water. A number of desalination techniques are available for brackish water treatment, each with their own advantages and shortcomings, as we will discuss later in this chapter. Table 1.1 lists most known desalination methods and their operation principles. These processes can be categorized roughly into three main groups: thermal, membrane-based, and electrical desalination methods. We here provide a review of working principles of each process.


3

Table 1.1: Review of desalination methods. Method† FD MSF MED HDH MD UF NF RO FO ED Shock ED CDI MCDI iCDI BDI

Thermal Cooling Heating X X X X X

Semipermeable membranes

X X X X X X

X X

Electric field

Surface groups

X X X X X X

X X X X X

Electrochem. reactions

X

†

FD: Freezing desalination, [17] MSF: Multi-stage flash distillation, MED: Multiple-effect distillation, HDH: Humidification dehumidification, [18] MD: Membrane distillation, NF: Nanofiltration, UF: Ultrafiltration, RO: Reverse osmosis, FO: Forward osmosis, [19] ED: Electrodialysis, Shock ED: Shock electrodialysis, [20] CDI: Capacitive deionization, MCDI: Membrane CDI, iCDI: Inverted CDI, and BDI: Battery CDI. [21]

1.2.1

Thermal desalination processes

Thermal desalination can be achieved through phase-change processes using either heating or cooling the saline water, with heating processes being used much more frequently. Multi-stage flash distillation (MSF), multiple-effect distillation (MED), humidification dehumidification (HDH), and membrane distillation (MD) are examples of heating thermal desalination techniques. In MSF, saline water is heated and flashed into steam (by passing through throttling devices) in multiple stages. The steam in each stage is condensed and collected as fresh water. MED consists of multiple stages or “effects”. Saline water is heated (usually by spraying onto steam tubes) and the vapor is transferred to the next stage. The vapor is then used for evaporation of even more saline water in subsequent stages. HDH uses a carrier gas (usually air) for desalination. A preheating element (usually solar heater) is used to elevate the gas temperature and thus extract vapor from saline water and increase carrier gas humidity level. The humid gas then carries the vapor to a condenser (or dehumidifier) and the fresh water is extracted. In MD, saline and fresh water solutions are


4

separated by hydrophobic membranes of vapor-filled pores. Water evaporates at liquid-gas interface in the pores, permeates through the membrane, and subsequently condenses on a colder surface. Freezing desalination (FD) process, on the other hand, separates fresh water from saline solution as ice crystals. FD can be beneficial compared to other thermal processes as the latent heat of fusion is about seven times lower than the latent heat of vaporization. Currently, MSF and MED processes produce over one-third of global desalination capacity. [22] Although these techniques are commonly used, they are mainly designed to target highly concentrated waters and not energetically favorable for removing low-concentration ionic contaminants, also known as brackish water (a salinity far below sea water yet undrinkable). Most importantly, these are optimized for large centralized operations and require significant investment and infrastructure, as they require high temperatures and large energy expenditures to desalt. [23] These processes are effective for seawater desalination but not energy efficient for brackish water. For example, MSF and MED used for seawater desalination consume respectively 10-16 and 5.5-9 kWh/m3 (energy per volume of produced fresh water). [24] This is significantly higher than membrane-based and electrical desalination processes for treating brackish water.

1.2.2

Membrane-based desalination processes

Membrane-based methods use semi-permeable or ion exchange membranes for water desalination. Ultrafiltration (UF), nanofiltration (NF), reverse osmosis (RO), and forward osmosis (FO) are examples of membrane-based processes. In UF, NF, and RO, saline water passes through semi-permeable membranes by an applied pressure larger than the osmotic pressure. Water molecules permeate through the membrane but microparticles, bacteria, microorganisms, and ions (in case of RO) are blocked by the membrane. UF, NF, and RO have similar working principle, but they differ in membrane pore size. They use membranes with pore size of 10-100 nm, 1-10 nm and ∼0.1 nm, respectively. [25] RO desalination consumes less energy compared to distillation processes, 3-4 kWh/m3 for seawater and 0.5-2.5 kWh/m3 for brackish water. [26] Membrane-based processes can be combined by either thermal or electrical desalination methods. Membrane distillation (MD) and electrodialysis (ED) are two examples. MD is discussed in Section 1.2.1, and ED will be introduced in the following section.


1.2.3

5

Electrical desalination processes

Electric field can also be driving force for desalination. Electrodialysis (ED), shock electrodialysis (shock ED), and family of capacitive deionization (CDI) methods are among these processes. In ED, an external electric potential is applied to pairs of electrodes to transport ions from one solution to another through ion-exchange membranes. ED cells consist of at least one brine and one fresh water flow channels, and alternate anion and cation exchange membranes (placed between a pair of electrodes) separate the channels. Shock ED, on the other hand, uses ion-exchange membranes and leverages ion concentration polarization zones formed in the pores close to the membranes. [20] Family of CDI techniques, as we will discuss in detail in Section 1.3, uses porous and conductive electrodes and electric potential for electrostatic sequestration of ions into electric double layers of the electrodes. CDI can be combined with ion-exchange membranes (membrane CDI or MCDI). In MCDI, anion and cation exchange membranes are placed between the electrodes. Membranes prevent re-adsorption of co-ions into the electrodes and so increase cell overall performance compared to conventional CDI cells. The main disadvantage of MCDI is relatively high cost of membranes. Additionally, CDI electrodes can be physically [27] or chemically [28] treated to obtain functional groups (inverted CDI or iCDI). In iCDI, ion removal can be achieved without application of electric potential (i.e. passive adsorption due to surface adsorption). Recently, intercalation-type electrodes (battery electrodes) have been used in devices named battery CDI (BDI) and hybrid CDI. [21, 29–31] BDI has been shown to have considerably higher adsorption capacity compared to CDI. [21]

1.3

Capacitive deionization for brackish water treatment

Brackish water is currently over 70% of the U.S. desalination capacity, and interestingly, the U.S. contributes to over 36% of global brackish water desalination capacity. [32] More than 90% of this desalination capacity is provided by energy intensive RO and distillation processes. So, alternative brackish water treatment strategies with energy efficient, low equipment and operation cost, and negligible infrastructure requirements have a great potential in matching supply and demand distributions of brackish water in the targeted areas. Additionally, reuse and recycle of brackish


6

wastewater can be more energy and cost efficient compared to direct desalination of seawater. Municipal, industrial, or agricultural wastewater, which usually has salt content far below seawater, can thus be treated at lower energy costs. One alternative is the CDI process.

1.3.1

Introduction to capacitive deionization

Capacitive desalination, also known as capacitive deionization (CDI), is an electrosorptive desalination technique that leverages porous electrodes for electrostatic ion adsorption, and has potential as an energy efficient and cost effective method of the desalination of water with a low to moderate salt content. [16, 33] The active component of CDI is a pair of engineered porous carbon electrodes, as shown in Figure 1.1. Upon application of voltage (∼ 1 V) or current across these electrodes (the charging process as shown in Figure 1.1a), salt ions are removed from feed water and held electrostatically within the electric double layers (EDLs) inside the pores. A CDI cell is then regenerated by removing or reversing the voltage (the discharging process as shown in Figure 1.1b); wherein, ideally in absence of Faradaic reactions, all of the capacitive charge can be recovered. [34, 35] Salt ions are spontaneously released from the pores and form the brine solution. CDI cells can be categorized into two groups based on their flow structure: flow through electrode and flow between electrode. [36] In flow-through CDI (ftCDI) cells, the water stream flows directly through the electrode pores, enabling faster cycling times. [36] In the more common flow-between CDI (fbCDI) architecture, the primary flow is through a gap (or porous spacer composed of a porous dielectric bulk material with pores filled with the aqueous electrolyte) between electrode pairs. In addition to the operation showed in Figure 1.1, a new operation scheme called iCDI has recently been reported, as mentioned earlier. [28, 37] In iCDI, the surface chemistry of the electrodes allows ion adsorption and desorption to occur on the electrodes in an inverted manner with respect to conventional CDI operation (adsorption in the absence of external voltage and regeneration by application of voltage). iCDI possesses the advantages of traditional CDI as mentioned above and has demonstrated improved salt removal performance.


7

Figure 1.1: Schematic of a typical fbCDI cell operation with two conductive and porous electrodes held apart by a non-conductive porous spacer. (a) Upon application of voltage across the electrodes (charging step), ions are sequestered in their respective electrodes and fresh water is collected at the outlet. (b) Once the applied voltage is lifted, ions immediately leave the electrode pores and form brine stream, which is later collected as waste water.

1.3.2

Advantages of capacitive deionization

Scalability and simplicity of operation CDI has a number of advantages over common desalination technologies mentioned above. First, the investment and infrastructure cost is low and is directly proportional to plant capacity. This makes CDI widely scalable and highly relevant for distributed applications. Second, in contrast to membrane or thermal desalination methods, CDI does not require high pressures or temperatures to operate. A small voltage of around 1 V required for CDI makes it very attractive for distributed and solar-powered applications.

Low energy cost of brackish water desalination In addition, CDI has a great potential to decrease the energy cost (per treated volume) of brackish water desalination. In CDI, in contrast to RO and distillation where energy scales mostly with volume of treated water, the energy cost per volume of treated water roughly scales with the amount of removed salt due to the electrosorptive nature of CDI. We here present a rough energy cost analysis of CDI, taking into account the initial salt concentration of feed water and the amount of salt removed, for various practical throughput and water recovery (WR) values. Water recovery is defined as volume ratio of fresh water produced from feed water. The estimates are based on the experimentally validated model for operation of CDI and RO described in references [38–40]. 3

Figure 1.2 shows volumetric energy cost (Ev ) in units of kWh/m versus feed concentration for a final


8

diluate concentration of 5 mM for WR values of 50% and 75%. Dashed curve is the reversible energy lower limit based on thermodynamic limitations of the Gibbs free energy of solvation. [41, 42]. The energy cost Ev for CDI can potentially be 10× lower than that of RO in the low salt content limit of 20 mM and below. In calculation of Ev for CDI, we assume constant current CDI operation with energy recovery during discharging step. This figure also shows that higher productivity (defined as volume of produced water per cycle time per total electrode area [39]) results in higher energy costs for both RO and CDI, but CDI remains more energy efficient for brackish water. Therefore, CDI is the most advantageous in brackish water desalination. As discussed above, brackish water desalination is an immense world issue and critical to providing clean drinking water, especially for distributed purification systems where scalability, energy cost, and availability are an essential limiting factor. The models used for estimations in Figure 1.2 are as follows. For CDI, we assume that the energy of operation is predominantly resistive loss, estimated as Ev,CDI ≈ I 2 Rs /γQ, where γ is water recovery, Q is flow rate, I is applied current and Rs is the effective series resistance of the CDI cell. Energy 2 F (c0 −cf ) cost can also be re-cast in terms of productivity (Prod) as Ev,CDI = Prod Rs A where γΛ F is Faraday’s constant, c0 is initial concentration, cf = 5 mM is final concentration, and Λ is cycle efficiency [39] (see details in Chapter 2). For RO energy estimates, we assume the main contribution to operation costs is the pressure work performed to enable fluid flow across the membrane. The energy cost can be modeled using Jw = Lp (P − ∆π), where P is the applied pressure, ∆π is the osmotic pressure difference which depends on the concentration difference across the membrane, Jw is the water flux across the membrane in units of m/s, and Lp is the membrane hydraulic permeability in units of m/s/Pa. [40] The applied pressure is related to the volumetric energy by Ev,RO ≈ P/γ. This energy can be written in terms of productivity as Ev,RO ≈ γ1 Prod with R Lp + n(c0 − cf )RT and T being universal gas constant and temperature. We used the following values of parameters in Figure 1.2. For RO estimate, we used Lp = 4 × 10−12 m/s/Pa. For CDI estimate, we used electrode surface area of 50 cm2 , constant charge efficiency of Λ = 0.8, and a concentration-dependent series resistance of Rs [Ω] = 0.4 + 50/c0 (c0 being inlet salt concentration in units of mM).


9

Figure 1.2: Volumetric energy cost of desalination for CDI (red curves) and RO (solid grey curves) 3 in units of kWh/m versus feed concentration for salt removal down to 5 mM solution with water recovery of (a) WR = 50% and (b) WR = 75%. Thermodynamic limit (dashed curve) shows minimum theoretical limit. Shown here are estimates for productivity values of 20 and 50 L/m2 /h. For seawater concentrations, the energy required by RO is significantly lower than CDI. However, CDI outperforms RO for brackish water desalination for initial concentrations of around 20 mM and below. CDI has thus great potential for increased energy efficiency for the globally important sector of brackish water desalination. Water recycle and reuse CDI, as a cost-effective technology for brackish water desalination, is a viable solution for water recycling/reuse and reduction of water demand. Examples include a wide range of applications. The industries that use water cooling for heat exchange processes can greatly benefit from CDI. CDI can be used in post-treatment of municipal wastewater reclamation to reduce salt content and remove ionic contaminants (and to reduce water hardness). Another application is in food processing industry, where CDI can be utilized for treatment of water used in washing step for subsequent reuse. CDI applications are significantly wide and span from power plants to golf course irrigation.

Selective ion removal In addition to substantially reducing energy requirements for brackish water desalination, CDI has the great potential for selective removal of ionic species and impurities from water. Selective ion removal is critical for the extraction of ionic contaminants and/or rare metal ions (e.g. calcium, magnesium, nitrate, [27, 43] arsenate, fluoride, [44, 45] lithium, [46, 47] etc.) from aqueous solutions


10

and reduce the cost of treating water. Current technologies for contaminant removal (e.g. reverse osmosis, [48, 49] forward osmosis (FO), [50] electrodialysis (ED), and ionic exchange resins [51]) have limited applicability or drawbacks for selective ion removal. Although membrane technologies such as RO or FO often have high salt rejection (98% or higher [52, 53]), they are not readily optimized for removal of specific ions in the solution. [54] Ion exchange resins, as alternative method, use charged surface groups on a solid substrate to exchange low-affinity ions (e.g. chloride) with high-affinity ions (e.g. nitrate). The main disadvantage of ion exchange resins, however, is the requirement the regeneration with high concentration chemicals during and subsequent disposal of these chemicals. Family of CDI methods (conventional, inverted, and membrane CDI) have potential in selective ion removal. [27, 55–57] Ion selectivity has been attributed ion valence, [58] steric effects, [59–61] hydrated ion size and nanoporous size, [62, 63] and surface chemistry and functional groups, [27, 64, 65] as well as pH environment. [66] Recently, Oyarzun et al. [27] demonstrated an inverted CDI cell with surfactant-treated electrodes with significant selectivity towards nitrate (compared to chloride).

1.4

Scope of dissertation

In Chapter 2, we present our study on thermodynamic limit of desalination energy with CDI (as an example of electrosorptive salt separation). To this end, We use variational form of free energy formulation. We analytically show the minimum energy required for most known electric double layers, regardless of geometry or thickness, is equivalent to free energy of separation. We then discuss importance of inevitable irreversibilities present in practice (including Faradaic and resistive losses) and their effect on desalination thermodynamic efficiency. We fabricate a CDI cell with reduced cell resistance and operate the cell and operate the cell between 0.4 to 1 V to limit Faradaic losses. We demonstrate an unprecedented 9% thermodynamic efficiency and only 4.6 kT energy requirement per removed ion (NaCl solution) at the cost of medium to low productivity. Contributions of each collaborator (A. Hemmatifar (AH), A. Ramachandran (AR), K. Liu (KL), D. I. Oyarzun (DIO), M. Z. Bazant (MZB), and J. G. Santiago (JGS)) are as follows. AH, AR, and JGS conceived the idea. AH and KL fabricated the CDI cell. AH and AR performed the experiments, and with DIO, analyzed the results. AH, AR, MZB, and JGS developed the theoretical framework. In Chapter 3, we delve deeper into energy losses associated with the electrosorption. We quantify


11

individual loss mechanisms in CDI charging and discharging under different charging current and cell voltage window. We measure series resistance for the cell and quantify parasitic losses as a function of current and voltage window. We show these two loss mechanisms favor different charging rates: resistive losses are minimized at low charging currents, but Faradaic losses are minimized at higher current rates (mainly because the cell spends less time at high voltage). We introduce two figures of merit, average salt adsorption rate and energy-normalized adsorbed salt, which characterize the performance of a CDI cell in terms of throughput and energy efficiency, respectively. We show that these figures of merit provide a powerful tool for optimizing CDI operation. Contributions of each collaborator (AH, J. W. Palko (JWP), Michael Stadermann (MS), and JGS) are as follows. AH, JWP, MS, and JGS conceived the idea. AH fabricated the cell and performed the desalination experiments. AH and JWP analyzed the data. In Chapter 4, we present our efforts in formulating and solving the first 2D model of a CDI cell. Our model includes coupled species and mass conservation in bimodal pore structure electrodes. We fabricate a flow-between cell (fbCDI) and perform series of experiments at constant voltage and constant flow rate. We show a reasonable agreement between model results and experimental data and use the experiments to calibrate the model. The measured effluent concentration suggest that desalination process in fbCDI cells has two distinct phases: a fast adsorption step at the beginning of charging followed by a slow, diffusion-limited salt removal step. We show, numerically, that low inlet concentration can lead to electrode starvation. The coupling of diffusion-limited transport and adsorption can result in a sharpening of the gradient between high and low charge concentration. This sharpening apparently results in an ion concentration shock wave in the charge concentration propagating through the electrode. Contributions of each collaborator (AH, MS, and JGS) are as follows. AH, MS, and JGS conceived the idea and designed the experiments. AH fabricated the cell, performed the experiments, and analyzed the data. In Chapter 5, we focus on multi-species transport of ions in surface-treated activated carbon electrodes under steady state conditions. We expand the current CDI-based surface charge models by coupling the EDL model with acid-base equilibria theory. Surface functional groups on the surface can protonate or deprotonate based on their individual pK values and local pH environment. We validate our model by comparing predictions to experimental measurements from controlled titration


12

experiments of activated carbon samples. Our fitted model show a very good agreement with the experiments. Our results show (1) specific adsorption of cations and expulsion of anions at electrolyte pH values higher than pK of acidic groups, and (2) specific adsorption of anion and expulsion of cations at pH values lower than pK of basic groups. Contributions of each collaborator (AH, DIO, JWP, S. A. Hawks (SAH), MS, and JGS) are as follows. AH, MS, and JGS conceived the idea and designed the experiments. AH and DIO fabricated the cell and performed the experiments. AH, DIO, and JWP analyzed the data. AH and SAH developed the equilibria model. In Chapter 6, we summarize the main findings of our work and provide suggestions for future work related in capacitive water desalination.

Chapter 2

Thermodynamics of Ion Separation by Electrosorption Sections of this Chapter are based on article published in Environmental Science & Technology [42] and are reproduced here with some modifications.

2.1

Introduction

Ion separation (desalination) processes of any type have long been known to require at least a minimum energy equivalent to the (Gibbs) free energy of separation. The minimum energy consumption corresponds to an ideal, extremely slow reversible process free of resistive losses. In systems with electrostatic interactions, such as supercapacitors, capacitive deionization (CDI), and capacitive mixing (CAPMIX), [67–70] a common practice to analyze this limit is to start from mechanistic details of electrostatic charge attraction, including specific treatments of the electric double layer (EDL) structure, and then proceed to formulate work integrals to calculate energy. [69–72]. For example, Wang et al. [72] showed equivalency of minimum electrical work and the Gibbs free energy of separation in a four-stage electrosorption cycle, although they did not analytically derive this and instead used a numerical approach. A significant limitation of this bottom-up approach is that it cannot be readily generalized to wide range of EDL models, and more importantly, it provides

13

CHAPTER 2. THERMODYNAMICS OF ION SEPARATION BY ELECTROSORPTION

14

limited insight into the various assumptions and approximations associated with each EDL model. In particular, numerical integration of electrosorptive work generally obscures the effects of compact versus diffuse portions of the EDL structure, effects of finite ion size, and the geometry and relative thickness of EDLs (e.g. whether the EDLs are overlapped or non-overlapped). In this work, we present an alternative “top-down” theoretical framework that naturally reveals the thermodynamic limits of electrosorption (including supercapacitors, CDI, CAPMIX, etc.), starting from first principles by variation of the electrochemical free energy functional. Similar variational approaches have been used to calculate electric double-layer surface forces [73–75] and to formulate modified Poisson-Boltzmann models of diffuse charge profiles, [76–80] implicit solvent models [81, 82] and phase-field models of ionic liquids and solids. [83–86] We shall see that the variational approach to electrochemical modeling is also convenient to describe the thermodynamic properties of CDI and yield conclusions applicable across the full range of EDL models and chemistries. The goal of our analysis is to provide a general framework which can be used (and also be expanded) to better understand how various physics such as surface functional groups, ion crowding effects, Faradaic reactions, etc. can add to the electrical work (on top of the free energy of separation) and how thermodynamic efficiency of the process (η, defined as the ratio of free energy of separation to the actual work input) will decrease accordingly. Such analysis with the additional aforementioned physics can help to understand energy requirements in a given phase of ion separation. One can then couple this analysis with detailed models for irreversibilities to track the extent and sources of energy losses associated with the ion separation process. One important measure of ion separation in practice is the thermodynamic efficiency of the process (η). Without a careful system design, energy consumption can be more than an order of magnitude larger than free energy of separation (i.e. η 10%). Various sources of irreversibility include ionic resistive and electronical resistive losses which scale with the temporal rate of adsorption/desorption, parasitic losses of Faradaic charge-transfer reactions, dispersion effects and associated flow mixing, and pressure work toward pumping electrolytes. A low resistance and optimized operation design is thus critical for a successful ion separation system. Although numerous studies focus on increasing thermodynamic efficiency of mature technologies such as reverse osmosis (RO) [87–89] and thermal desalination, [90, 91] there have been only a few


15

studies on the thermodynamics of emerging electrosorptive technologies such as CDI. In fact, only a handful of published studies on CDI even report thermodynamic efficiency values. [92, 93]. Further, there has been no overview of the current typical values of CDI thermodynamic efficiency or of methods by which it can be improved.

2.2 2.2.1

Theory Problem definition

We introduce our top-down thermodynamic formulation as follows. Consider a pair of identical porous and electrically conductive electrodes which will be used to trap, transfer, and release ions from two reservoirs of fixed volume: a diluted solution and a brine reservoir. Each of the two reservoirs is filled with a binary symmetric (z : z) electrolyte of initial concentration c0 (Figure 2.1a). The volume of the electrode pore space, diluted reservoir, and brine reservoir are respectively vp , vD , and vB . For these, we define recovery ratio as γ = vD /(vD + vB ). Figure 2.1b (solid lines) shows an example ion separation cycle on c˙ions vs. c∞ plane, where we define cions = c+ + c− as total ion concentration trapped by the electrode (c± being pore concentration of cations and anions) and c∞ as the reservoir bulk concentration. The ion separation cycle can be rationalized as four stages: (1) charging stage, which stores charge in the pores and reduces diluted reservoir concentration to cD , (2) exchange of one reservoir for the other, maintaining constant cions value, (3) discharging stage, which releases salt and increases brine reservoir concentration to cB and decreases cions to cB , and (4) a return to the first reservoir at constant cions value. The latter four stages are also discussed in refs. [68, 69, 71, 72]. In this construct, mass conservation requires dcions /dc∞ = −vbulk /vp , where vbulk = vD in charging and vbulk = vB in the discharging phase. We stress that ion separation from a single diluted reservoir to a single brine reservoir (solid lines) is an idealized case useful in rationalizing CDI systems. For example, one can envision a process involving multiple intermediate reservoirs (not just two end points). In an actual CDI system with flow from a common source, charging and discharging is performed relative to some time-varying concentration associated with the effective time-varying ion density in the cell. The dashed curves in Figure 2.1b are shown here as a qualitative representation of a more general case.


16

Figure 2.1: (a) Schematic of an electrochemical desalination system with a pair of porous, conductive electrodes (only one shown) in contact with a reservoir. (b) Solid line is an ion separation cycle (on cions vs. c∞ plane) wherein charging and discharging are performed with respect to two reservoirs. Dashed lines depict some arbitrary charging/discharging relative to a time-varying value of reservoir concentration.

2.2.2

Free energy functional and electrical work of electrosorption

Our interest is the minimum electrical work required for the ion separation cycle discussed in Section 2.2.1, which corresponds to the limit of infinitely slow traversal of the charge-voltage cycle with all chemical species in quasi-equilibrium. We seek to formulate this from an energetic perspective without assumptions regarding the details of the charge distribution within the pore electrical double layers (EDLs) or the electrode matrix material. We stress that the analysis we present here is applicable in electrosorptive processes such as supercapacitors, CDI, and CAPMIX. The most general starting point is the total Gibbs free energy functional G[{ci }, φ; ρe , φe ; {crj }] of a single composite porous electrode, which is an integral over the spatial distributions of ionic concentrations {ci (x, t)}N i=1 and the electrostatic potential of mean force φ(x, t) in the electrolyte-filled pore space; the electronic charge density ρe (x, t) and electrostatic potential (Fermi level per charge) r φe (x, t) of electrons in the solid electrode matrix; and the concentration profiles {crj (x, t)}N j=1 of any

redox-active chemical species (reactants and/or products) that undergo electrochemical reactions with the ions in the pore space. In general, the redox-active species may be charged or uncharged, resulting from Faradaic, auto-ionization or complexation reactions, and may exist anywhere in the porous electrode volume, including dissolved species in the electrolyte filled pores, adsorbed species at interfaces, or inserted species within the electrode solid matrix. Although our theoretical development below does not depend on the specific form of G, it is


17

instructive to point out how various existing mathematical models of electrostatic CDI for flat [94] and porous [95, 96] blocking electrodes and Faradaic CDI with redox-active porous electrodes [29, 97, 98] are included as special cases our general thermodynamic formalism. In particular, all PoissonBoltzmann (PB) models (including the classical dilute-solution approximation as well as various modified PB models [78] for thin or thick double layers) correspond to following form of the free energy functional, which expresses the mean-field and local density approximations for a linear dielectric material (neglecting all chemical or electrostatic correlations between ions): [79] Z 2 G= gc + ρp φ − |∇φ| dVp + qe φe dAe , 2 Vp Se Z

(2.1)

where the integrals are over pore volume and the bounding surface of the solid electrode matrix, P respectively; ρp (x, t) = i zi F ci is the ionic charge density in the electrode pores; F is Faraday’s constant (mole of charge); zi is the valence of ionic species i; gc ({ci }, T ) = h − T s is the chemical portion of the Gibbs free energy density (due to short-range non-electrostatic interactions) expressed in terms of the enthalpy density h({ci }, T ) and the entropy density s({ci }, T ); qe (x, t) is the electronic surface charge density per area of the electrode solid matrix, assumed here to be a metal at constant potential φe ; and (x, t) is the permittivity of the pore solution. The free energy functional is constructed so that the physically relevant electrostatic potential, which conserves Maxwell’s dielectric displacement field, satisfies the stationarity condition of vanishing first variation. This construction leads to Poisson’s equation for bulk variations

δG δφ

= ρp + ∇2 φ = 0,

(2.2)

bulk

and the associated electrostatic boundary condition for surface variations, (δG/δφ)surf ace = qe + n ˆ· ∇φ = 0. In the mean-field approximation, the force experienced by each individual ion derives from the self-consistent electric field generated by the mean charge density via Poisson’s equation. This approximation is reflected in the form of the free energy functional by defining the electrochemical potential of any species i as its variational derivative with respect to a localized unit concentration perturbation, [85] µi =

δgc δG = + zi F φ = µΘ i + RT ln ai + zi F φ, δci δci

(2.3)


18

where µΘ i is the standard reference chemical potential and ai is the chemical activity of species i. By setting each electrochemical potential to a constant value, µi = µ∞ i (for exchange with a reservoir “at ∞ infinity”), we define the quasi-equilibrium concentration profiles, {ceq i (φ, µi )}. Inserting the quasiP eq ∞ ∞ equilibrium charge density, ρeq p (φ, {µi }) = i zi F ci (φ, µi ), into Equation 2.2 yields a modified

Poisson-Boltzmann (PB) equation for concentrated solutions (ai 6= ci ), [78] which reduces to the familiar PB equation in the limit of a dilute solution (ai = ci ): − ∇2 φ =

X i

zi F c∞ i exp

zi F (φ∞ − φ) . RT

(2.4)

Existing models of CDI describe electrosorption of ions in quasi-equilibrium diffuse EDLs via the Gouy-Chapman-Stern (GCS) and modified Donnan (mD) models, which correspond to solutions of the PB equation (Equation 2.4) with the Stern “surface capacitor” boundary condition in the limits of thin and thick double layers, respectively, [29, 95, 97, 98] and are thus included in our thermodynamic framework. More generally, we could relax the quasi-equilibrium assumption in our thermodynamic formulation by defining mass fluxes in terms gradients of the electrochemical potentials defined by Equation 2.3 in order to obtain modified Poisson-Nernst-Planck equations for diffuse-charge dynamics out of equilibrium, [78, 82, 85] which capture additional effects of non-equilibrium charging and tangential surface conduction. [99] Since such non-equilibrium phenomena introduce additional resistance into the system, however, we shall focus on quasi-equilibrium electrosorption, in order to derive a thermodynamic bound on the consumed energy. Let us define the specific free energy per volume of the diluted reservoir, g = G/vD , and assume δg macroscopic quasi-equilibrium across the electrode pores. The electrochemical potential µi = vD δc , i

is then uniform and constant for each ionic species i, where µΘ i + RT ln ai =

δgc δci

is the familiar

chemical potential and ai the activity of species i. In the case of dilute electrolyte, ai = ci . At this point, we can calculate the electrostatic work of the system for the complete ion separation cycle of R Figure 2.1b. Assume a differential electronic charge dqe = Ve δρe dVe is injected to the electrodes at fixed electrode potential φe , under quasi-equilibrium (µi uniform and constant) exchange with bulk reservoir solution, and results in a change in pore ion concentration of δci (x). This process requires a minimum electrostatic work of −2VT φ˜e dqe , where φ˜e = φe /VT and VT is the thermal voltage. Due R R to electroneutrality, ionic charge compensates for electronic charge as in Vp δρp dVp + Ve δρe dVe = 0.


19

Without loss of generality, we take φ = 0. The change in free energy of the electrode for this process R P R is then vD δg = Vp i µi δci dVp + Ve VT δρe φ˜e dVe and using electroneutrality condition we further simplify to vD δg =

X

(µi − zi RT φ˜e )dNi ,

(2.5)

i

with dNi =

R Vp

δci dVp . For a closed loop, we have

H

δg = 0, and thus the minimum volumetric

energy consumption for the cycle Ev,min is

Ev,min = −

VT vD

I

˜ e= 2φdq

2 vD

I X

µi dNi = ∆gsep ,

(2.6)

i

where ∆gsep is specific Gibbs free energy of separation. Refer to Section 2.6 for details. Equation 2.6 states that in the absence of chemical (non-electrostatic) portion of the free energy, Ev,min is equivalent to the Gibbs free energy of separation ∆gsep for any thermodynamically-consistent electrical double layer (EDL) of arbitrary geometry and thickness (overlapped or non-overlapped). This includes Poisson-Boltzmann (PB) type EDL treatments such as Gouy-Chapman (GC) model as well as modified Poisson-Boltzmann with finite ion size effects. [77, 100] EDL descriptions with constant or ci -dependent non-electrostatic potentials such as modified-Donnan (mD) model [97, 101] each also respect Equation 2.6 (since the integral of the non-electrostatic term µatt on a closed path vanishes). Importantly, as noted above, Equation 2.6 is also valid for EDLs with compact layers such as Gouy-Chapman-Stern (GCS) and Gouy-Chapman-Stern-Carnahan-Starling (GCS-CS) models. These conventional electrosorptive desalination EDL models satisfy the free energy formulation results in general, since these models can be derived variationally from the same free energy model. Equation 2.6 can be further simplified for 1 : 1 dilute electrolytes (ai = ci ) and assuming an idealized instantaneous, quasi-equilibrium ion exchange between reservoir and the pores. Under these conditions and for φ = 0 in the reservoir, µi ≈ µi,∞ = RT ln c∞ . Moreover, 2dNi = −vbulk dc∞ by mass conservation. The minimum volumetric electrical work then takes the form

Ev,min = −

2RT vD

I vbulk ln c∞ dc∞ ,

(2.7)

For the ion separation cycle of Figure 2.1b (solid lines), Ev,min can be written in the familiar form


2RT cD ln cD + γ1 − 1 cB ln cB −

2.2.3

c0 γ

20

ln c0 .

Alternative approach: direct integration of electrical work

Alternative to the approach above, for special cases of non-overlapping (e.g. GCS) and highlyoverlapped (e.g. mD) double layer models, the equivalency of Ev,min and ∆gsep can be shown by explicit calculations (direct integration of electrical work). For example, consider a model including a Stern layer assumption. For the equilibrium condition, the (non-dimensional) potential of a single electrode can be written as φ˜e = ∆φ˜St + ∆φ˜D where ∆φ˜St is the Stern layer potential drop and ∆φ˜D is either the diffuse layer (in GCS model) or Donnan potential drop (in mD model), both normalized by the thermal voltage. A close inspection of the two models reveals that they each obey the following relation between pore ionic concentration cions and pore charge density ρp for 1 : 1 electrolytes 1 ρp d = n F c∞ d∆φ˜D

cions ncn∞

(2.8)

where n = 1/2 and 1 for GCS and mD models, respectively. Using this ad-hoc relation, the electroneutrality condition (qe + vp ρp = 0), and mass conservation (dcions /dc∞ = −vbulk /vp ), the electrical work can be written as (see Section 2.6 for details of derivation)

Ev,min = 2

I I h I i VT − ∆φ˜St dqe − d qe ∆φ˜D + F vp (1/n − ln c∞ )cions + F vbulk ln c∞ dc∞ vD (2.9)

or again simply Ev,min = ∆gsep . Importantly, note the electrical work of the Stern layer (the first term on the right-hand side) vanishes on a closed loop for the classical surface-capacitor approximation introduced by Grahame, [102] where ∆φ˜St is solely a function of surface charge density qe (equal to the total nearby pore charge density by macroscopic electroneutrality). We can thus write the integrand as a total differential dESt , where ∆φ˜St = dESt /dqe , whose integral around a closed loop must vanish. The second integral must also vanish for any closed loop. In the classical picture, the Stern layer charge and discharge dissipates no energy in a cycle, since its capacitance depends only on charge, and not voltage. In contrast, diffuse layer charging cycles consume energy, since EDL capacitance is inherently voltage dependent, except in the limit of small EDL voltage drop (Debye-Huckel theory), where linear response again yields a constant capacitance, independent of


21

voltage.

2.2.4

Effect of chemical or electrochemical reactions

Although the preceding analysis is based on the electrostatic free energy functional (Equation 2.1) for electrosorption of ions in the diffuse double layers of a porous electrode, it also applies to CDI processes involving chemical or electrochemical reactions. A simple extension would include regulation of surface chemical charge of the porous electrode, e.g., by pH-dependent deprotonation or hydroxyl dissociation reactions. [66, 103] A broader class of reactive systems include Faradaic CDI, where significant electrosorption is mediated by reversible electron transfer reactions that reduce or oxidize the adsorbed ions, either to form new molecules or ions in the pores or on the solid surface [29, 83, 98] or to intercalate ions into an active solid matrix of the porous electrode. [21, 30, 31, 104–106] In R order to describe such situations, an additional term, GF = VF gF ({crj }, φ, φe )dVF , must be added to the electrostatic free energy functional (Equation 2.1), where the integral is over the “reactive volume” containing all the active species which undergo chemical or electrochemical reactions involving ions in the pores, which have concentration profiles {crj (x, t)}, homogeneous free energy density gF , and chemical potentials µj (x, t) = δGF /δcrj . In the case of Faradaic reactions with different redox species in each electrode, the open circuit voltage is generally non-zero, and the transient voltage under general non-equilibrium conditions is determined by a stoichiometric sum of the chemical potentials of the reactive ions and reduced/oxidized species, which are defined variationally from the total free energy functional according to the general thermodynamic framework of electrochemical kinetics. [85] As with electrostatic adsorption in the diffuse EDLs described above, the free energy change of any charge/discharge cycle is minimized when Faraday reactions are also in quasi-equilibrium. In the case of chemical surface charge, a suitable adsorption isotherm governs the thermodynamics. For Faradaic reactions, the quasi-equilibrium charge-voltage relation is derived from a generalized form of the Nernst equation, [85] such as the Frumkin isotherm (or regular solution model) for ion intercalation. In this limit of “fast” reactions, there is no net change in total chemical potential from reactants to products, and the total free energy is the same as in the original electrostatic adsorption model, at the same state of charge. Moreover, since the additional contribution to the


22

energy only depends on the state of charge under the assumption of quasi-equilibrium reactions, H it must also vanish under any closed charging cycle, ∆GF = vD δgF = 0. Therefore, our main result, Equation 2.6, is unchanged by the presence of any chemical or electrochemical reactions in the electrodes.

2.2.5

Thermodynamic efficiency of electrosorption in practice

Having established a general theory of minimum energy consumption, we now focus on practical relevance of this analysis in CDI cycles. First, we highlight the importance of series resistance (an important cell design parameter and fabrication challenge), and then we focus on effects of flow rate and ion separation rate (operational parameters) on thermodynamic efficiency, η. In CDI, the (ionic and electronic) resistances and Faradaic reactions are two main sources of energy loss. [107, 108] The pumping energy is usually insignificant, [107] except perhaps for flow-through CDI designs at very high flow rates. We make the effort to reduce Faradaic losses as much as possible by limiting the voltage window to 0.4 to 1 V range. The existing models that quantify Faradaic energy loss mechanisms are overly complex. [43, 97, 109, 110] We strive for simplicity, and thus in the current work, we neglect Faradaic losses and focus instead on resistive losses in quantifying the thermodynamic efficiency of CDI operation. Such an estimate provides a simple yet useful lower bound for η. Motivated by this, we later show that simple reduction of series resistance can lead to significant increase in η. In its simplest form, the volumetric resistive consumption Ev,resist for constant current operation can be estimated as Rs I 2 /γQ where Rs , I, and Q are respectively the effective series resistance, current magnitude, and flow rate. We show in Section 2.6.3 that the minimum energy of ion separation (∆g) can be well approximated as ∆gsep /c0 ≈ a(∆c/c0 )n , where a and n depend only on the recovery ratio γ, and ∆c (= c0 − cD ) is the desalination depth. The values of a and n versus the recovery ratio are given in Section 2.6.1. Importantly, the exponent n is close to 2 (between 1.5 to 2.5) for all recovery ratios. The thermodynamic efficiency can thus be estimated as ∆gsep γΛcycle a η≈ ≈ Ev,resist F IRs

∆c c0

n−1 (2.10)


23

In Equation 2.10, we have used the relation ∆c = IΛcycle /F Q, where Λcycle is the cycle charge efficiency. Also, note that the exponent n − 1 is positive (close to 1). Thus, for a given recovery ratio (i.e., fixed γ, a and n), thermodynamic efficiency of a desalination cycle can be increased by (i) decreasing the series voltage drop IRs , (ii) increasing cycle charge efficiency and, (iii) larger desalination depths, ∆c/c0 . Motivated by this, as we will see in Section 2.3, we fabricate a low resistance CDI cell and optimize the thermodynamic efficiency of CDI by careful choice of operational parameters (such as flow rate, current, and voltage window).

2.3 2.3.1

Experimental methods CDI cell design

The cell assembly was similar to that in reference [108] with a radial flow structure. Two pairs of square shaped activated carbon electrodes (Materials & Methods, PACMM 203, Irvine, CA) with 5 × 5 cm size and initial (uncompressed) thickness of 270 µm and total dry mass of 1.14 g was used. We ultrasonically-cleaned titanium meshes of 250 µm thickness (uncompressed) and used them as current collectors. Each current collector was square shaped with 5 × 5 cm size and had a tab section (1 × 3 cm) for connection to external wires. To enhance electrical contact between current collectors and electrodes, we embedded the titanium meshes intro the electrodes using a hot press at 100◦ C temperature. A schematic of a single current collector with compressed thickness of 170 µm embedded into carbon electrode is shown in Figure 2.2a. The compressed thickness of the stack was 300 µm. Electrode side, titanium mesh side, and cross-section views are shown in Figures 2.2b to 2.2d, respectively. Polypropylene meshes of 30 µm thickness (McMaster-Carr, Los Angeles, CA) acted as spacers between the electrode pairs. The assembly was then sealed with gaskets and fastened by C-clamps.

2.3.2

Cell characterization

We characterized the cell assembly by cyclic voltammetry (CV) and electrochemical impedance spectroscopy (EIS) tests and compare our cell’s characteristics with a conventional cell with 500 µm spacer thickness and compressed titanium sheets current collectors. As mentioned above, our cell


24

had 2 pairs of electrodes with total surface area of 50 cm2 . The comparison cell, on the other hand, had 5 pairs of electrodes with total electrode area of about 123 cm2 . The CV test in Figure 2.3a was performed at 0.2 mV/s scan rate with 20 mM NaCl solution and showed cell capacitance of about 17 F (or about 60 F/g specific capacitance). Results of the comparison cell (cell with titanium sheet current collectors) are also shown in Figure 2.3a. Note, for an accurate comparison, the reported capacitance in the case of comparison cell is normalized (by the total electrode surface area) to represent the capacitance of a two-pair cell (as opposed to five pairs). Results show that titanium embedding does not change the capacitance of the electrodes. However, our design decreases the series resistance, because the shape of CV plot is now closer to a square. EIS test in Figure 2.3b shows 0.33 Ω (or 16.5 Ωcm2 area-normalized) setup resistance (ionic resistance of the solution in the separators, electrical resistance of current collectors, and resistance of connecting wires). Note that, for an accurate comparison, the results for conventional cell (with five electrode pairs and 123 cm2 surface area) is normalized by electrodes surface area to represent equivalent EIS test of a two electrode pairs cell (with 50 cm2 surface area). The results show setup resistance of around 2.29 Ω. This decrease in setup resistance can be attributed mainly to the use of thinner spacers, as the other components of setup resistance are almost identical between the two cells. Most importantly, the contact resistance, usually associated with a semi-circle feature in the EIS Nyquist plot, is almost completely eliminated. Table 2.1 lists the differences between the low resistance cell (with thin spacers and embedded Ti mesh current collectors) and the comparison cell (with thicker spacers and conventional Ti sheet current collectors). Additionally, we show an equivalent circuit of the CDI cells in the inset of Figure 2.3b. Here, Rs is the setup resistance, Rct and Cct are the contact resistance and capacitance, respectively, and Ztl is the transmission line impedance. See reference [111] for details of various components. A fidelity model for finite-length transmission line impedance Ztl in porous electrodes is discussed in [112]. The model for Ztl has four parameters and can be written as

Ztl = nRtr Zw,1

1/2

h i coth (nRtr /Zw,1 )1/2 ,

(2.11)

with n being the number of parallel transport pores in the electrode and subscripts “t” and “w” respectively correspond to the transport pores (macropores) and pore walls. Here, Zw,1 =

Zs,m j j−ωCtr Zs,m /n ,


25

with ω being angular frequency and

Zs,m = (1 − j)

n2 Rst 2ωCst

!1/2

h i coth (1 + j)(ωRst Cst /2)1/2 .

(2.12)

Here, Rst and Cst are the electrode resistance and capacitance associated with storage pores (micropores). In short, transmission line impedance Ztl is a function of four parameters Rtr , Rst , Cst , and Ctr . The equivalent circuit has then seven parameters in total (including Rs , Rct , and Cct ). We used the freely available MATLAB code “Zfit” for n = 10 and calculated the values of free parameters as Rs = 2.29 Ω, Rct = 1.13 Ω, Cct = 235 µF, Rst = 0.082 Ω, Rtr = 0.17 Ω, Cst = 73.5 F, and Ctr = 52.5 F.

Figure 2.2: Schematic of embedded titanium mesh sheet into the activated carbon electrode using 100 C hot press. Images of top view and cross-section side view after hot press step show details of embedded titanium mesh.

Table 2.1: Comparison between the low resistance cell and the conventional cell. Property

Low resistance cell

Spacer thickness Setup resistance Contact resistance

30 µm 0.33 Ω Negligible

Conventional cell (two electrode pair equivalent cell) 500 µm 2.29 Ω 1.13 Ω


26

Figure 2.3: (a) Cyclic voltammetry (CV) test of the assembled cell at 0.2 mV/s scan rate and 20 mM NaCl electrolyte solution with −0.6 to 0.6 V voltage window. CV test shows about 17 F cell capacitance or about 60 F/g specific capacitance. (b) Electrochemical impedance spectroscopy (EIS) test of the assembled cell again with 20 mM NaCl electrolyte solution. The setup resistance (ionic resistance in separators, electric resistance of current collectors and wires) is only 0.33 Ω and the contact resistance is negligible.

2.3.3

Experimental procedure

The experimental setup consisted of our fbCDI cell, a 3 L reservoir filled with 20 mM NaCl solution, a peristaltic pump (Watson Marlow 120U/DV, Falmouth, Cornwall, UK), a sourcemeter (Keithley 2400, Cleveland, OH), and a flow-through conductivity sensor (eDAQ, Denistone East, Australia). We operated our cell at constant current (CC) charging and discharging and studied the effect of flow rate and current on the thermodynamic efficiency of the CDI desalination process. We used flow rates varying between 0.2 to 3.5 mL/min, and current values between I = 5 to 65 mA with closed-loop circulation in all of our experiments (flow from reservoir to cell and back to reservoir). For all experiments, we charged (at current I) and discharged (at current −I) the cell between 0.4 to 1.0 V voltage window. Further, we performed at least three complete charge/discharge cycles for each experiment to ensured dynamic steady state (DSS) operation, in which salt adsorption during charging is equal to desorption during discharging. We recorded external voltage and effluent conductivity using a Keithley sourcemeter and an eDAQ conductivity sensor (with 93 µm internal channel volume). Conductivity was converted to salt concentration using a calibration curve for NaCl. In Section 2.6.4, we present voltage and concentration measurements versus time for our experiments.


2.4 2.4.1

27

Results and discussion Importance of series resistance energy loss

To put the energetic performance of current CDI technology into perspective, we show in Figure 2.4 experimental performance data for variety of reported CDI cells. The figure shows experimental volumetric energy consumption Ev versus Ev,resist for various CDI and membrane-CDI systems under constant-current (CC) operation. [38, 39, 107, 108, 113–115] We here focus on CC operation, since the energy recovery during discharge step results in a lower energy per removed ion compared to short-circuit discharge at 0 V. [92] Particularly, Ev here is calculated based on the measured R voltage and current as V Idt with the integral over a full cycle (charge and discharge). In addition to published data, we include measurements from a cell which we constructed specifically for the current study. As discussed in Section 2.3.1, our CDI cell here features a low-resistance (for each electrode pair) design which we realized by pressing into and embedding titanium mesh current collectors into 270 µm thick commercially available porous carbon electrodes. We also lowered series resistance by using 30 µm spacers. We observe that despite the wide variation in CDI cell materials, geometry, and operational parameters (including applied current, maximum and minimum voltage settings, and flow rate), the data organize fairly well near and just above the line set by the resistive limit Ev = Ev,resist line. This shows our simple estimate of Ev,resist correctly predicts an approximate minimum for Ev in practice. Further, it confirms that work towards decreasing series resistance is essential to the improvement of thermodynamic efficiency of CDI cells. Also, we note that for cells with low series resistance, the Faradaic losses can contribute to most of the energy loss. The spread of our experimental data and their distance from the dashed line (red, filled circles in Figure 2.4) indicates that Faradaic losses are dominant for our low-resistance cell. As a result, remedies for reduction of Faradaic losses can also greatly improve the energy performance of CDI.


28

Figure 2.4: Measured volumetric energy consumption (Ev ) versus estimated resistive energy consumption (Ev,resist ) for a wide variety of CDI cell designs operated under constant-current (CC) operation. Ev,resist provides a very good estimate of minimum Ev observed in practice, as all data points lay near and above the line set by the resistive limit Ev = Ev,resist . This shows decreasing series resistance is essential for improvement of thermodynamic efficiency.

2.4.2

Operation for high thermodynamic efficiency

Consistent with the importance of series resistance, slow ion separation processes are often more thermodynamically efficient than faster processes. However, slower processes are not always superior because of the effects of Faradaic losses. We explore and demonstrate this in Figure 2.5. Shown is data given our low-resistance CDI cell for CC operation with current densities in 1-26 A/m2 (equivalent to 2.5-65 mA) and flow rates in 0.22-3.47 ml/min range. Figures 2.5a, 2.5b, and 2.5c are respectively the volumetric energy consumption Ev , energy consumption per ion and per salt removed (note two ordinates), and thermodynamic efficiency, all plotted versus applied current density. The voltage windows were 0.4-1 V to ensure high charge efficiency. [39, 116] For a given flow rate, operation at high currents is limited by resistive losses as expected. [108] However, at overly low applied current, performance is limited by Faradaic losses since the cell spends overly long times near the high voltage limit. [108] This trade-off between two very different energy loss mechanisms results in pronounced optima in η. Similarly, exceedingly low flow rate operation (e.g. 0.22 ml/min) suffers from dispersion and mixing inefficiencies associated with incomplete removal of desalted water or brine during charging and discharging, respectively. [117] Instead of a monotonic trend, we


29

Figure 2.5: (a) Measurements of volumetric energy consumption Ev , (b) energy consumption per ion and salt removed in units of kT/ion and kJ/mol, and (c) the thermodynamic efficiency for CC operation and 0.4-1 V voltage window at various currents and flow rates. Mid-range values of current density (4 A/m2 ) and flow rate (0.44 ml/min) result in thermodynamic efficiency of up to 9%, unprecedentedly high for traditional CDI designs. see that mid-range values of current and flow rate (4 A/m2 and 0.44 ml/min in this case) results in thermodynamic efficiency of about 9% and energy consumption of about 4.6 kT/ion. This thermodynamic efficiency is unprecedentedly high for traditional CDI designs (i.e. with no membranes, no ion intercalation, no inverted operation [27], etc.). We note that energy consumption per removed ion, unlike thermodynamic efficiency, is not a fundamental property. Such high thermodynamic efficiency is a result of low resistances and careful choice of operation parameters, such as voltage window, current, and flow rate. We predict that even higher η can be achieved using optimized operation and high-performance electrodes with low electrical resistance.

2.4.3

Trade-off between thermodynamic efficiency and throughput

Lastly, for completeness, we stress that the energy-efficient ion separation of our cell comes at a price. Any practical operational design is a trade-off between throughput and energy efficiency. [108, 118] Figure 2.6a shows thermodynamic efficiency of ion separation versus desalination depth (∆c) for our cell and published CC cell data. [38, 39, 107, 108, 113–115] Figure 2.6b shows the main trade-off for this thermodynamic performance is a mediocre value for productivity (defined as volume of treated solution per unit time per cell area) at around 3 L/h/m2 . [39] Clearly, at a given desalination depth ∆c, one can achieve significantly higher η by sacrificing productivity (low flow rates). Finally, we


30

Figure 2.6: Thermodynamic efficiency versus (a) average effluent concentration reduction ∆c and (b) productivity (in units of L/h/m2 ) for our cell and selected published CDI cells under constant current charge/discharge. The thermodynamic efficiency generally correlates with high ∆c and low throughout productivity. present practical considerations for efficient CC operation based on observations in Figure 2.6. The following considerations can ensure good overall performance: 1. Choice of current and flow rate: the flow rate and current can be determined based on throughput and salt removal (∆c = IΛcycle /F Q) requirements. 2. Choice of voltage limits: the maximum voltage (after accounting for series resistance voltage drop) determines the extent of Faradaic losses. The minimum voltage can be chosen so that least 4-5 cell volumes are flowed during charging. This ensure efficient recovery of fresh water at the effluent (high flow efficiency) and high cycle charge efficiency as a result. [39] 3. Avoiding extremities: Extreme operations such as (i) very low operating currents wherein the operating current is comparable to the leakage current, (ii) very low flow rates wherein dispersion and mixing inefficiencies are important, (iii) very high operating voltages wherein Faradaic losses dominate can all adversely affect overall performance.


2.5

31

Conclusions

We presented a top-down approach to calculate minimum energy requirement of electrosorptive ion separation using variational form of free energy formulation. The advantage of our formulation is its generality; it is free of any assumptions about the charge distribution in the EDLs or the profile of Faradaic reaction products, since all such processes occur in quasi-equilibrium (without internal resistance to transport or reactions). We showed, analytically, the minimum energy required for most known EDLs (irrespective of EDL geometry or thickness) is indeed equivalent to free energy of separation. These include (but not limited to) Poisson-Boltzmann type models such as GCS, GCS-CS, and mD. We focused on thermodynamic efficiency η of CDI systems as an example of electrosorptive ion separation method. We then discussed practical considerations in CDI cell design and operation and showed cell series resistance and optimization of operational parameters such as flow rate and current are two contributing factors to practical values of η. As a comparison to published studies, we fabricated and tested a cell with reduced cell resistance. We used variations in operation to identify Faraday losses, resistive losses, and flow efficiency losses as primary irreversible loss mechanisms. Our cell demonstrated an unprecedented 9% thermodynamic efficiency and only 4.6 kT energy requirement per removed ion (NaCl solution), but we show that this achievement comes at the cost of productivity. Overall, the study identifies fundamental limits for CDI as an energy efficient desalination technology as well as the significant challenges associated with approaching this limit.


2.6

32

Addendum to: Thermodynamics of Ion Separation by Electrosorption

2.6.1

Equivalency of minimum electrical work and free energy of separation

Details of derivation of minimum electrical work for ion separation of Figure 2.1b in Chapter 2 R are discussed here. Assume electronic charge dqe = Ve δρe dVe is injected to the electrode at fixed electrode potential φ˜e (normalized by thermal voltage VT ). Also, assume this process changes the pore ion concentration by δci according to a quasi-equilibrium ion exchange between reservoir and the pore electric double layers (EDLs). The chemical potential of species i, µi , thus remains constant. By definition, the work associated with injection of charge δqe is −VT φ˜e dqe . Since φ˜e remains constant and φ = 0, the change in free energy of the electrode is

δg =

where δgc =

1 vD

Z δgc dVp + Vp

VT vD

Z

δρe φ˜e dVe ,

(2.13)

Ve

P

µi δci by definition. Using electroneutrality condition, δρp dVp + δρe dVe = 0, and the P definition ρp = i zi F ci , Equation 2.13 can simplified as (note VT = RT /F ) i

1 δg = vD

Z Vp

  X  (µi − zi RT φ˜e )δci  dVp .

(2.14)

i

Since µi and φ˜e are constant and uniform throughout the electrode pore volume, we switch the order of summation and volume integral to arrive at Equation 2.6 of Chapter 2. " # Z 1 X (µi − zi RT φ˜e ) δci dVp , δg = vD i Vp with dNi =

R Vp

δci dVp . So, on a closed loop with

Ev,min =

1 vD

H

I X i

(2.15)

δg = 0 and

µi dNi = ∆gsep .

(2.16)


2.6.2

33

Direct integration of minimum electrical work

As an alternative to general approach of Chapter 2, we here prove Ev,min = ∆gsep in special case of Gouy-Chapman-Stern (GCS) and modified Donnan (mD) models (as examples of non-overlapping and highly-overlapped electrical double layers) by explicitly calculation of electrical work. Under equilibrium conditions in absence of electric currents, the electrode potential (normalized by thermal voltage VT ) is φ˜e = ∆φ˜St + ∆φ˜D . Pore ionic concentration cions and pore charge density φp in GCS and mD models can be written as     ρp    cions     ρ p    cions

= 4λD F c∞ sinh ∆φ˜D /2 = 4λD c∞ cosh ∆φ˜D /2

= 2F c∞ exp(µatt ) sinh ∆φ˜D = 2c∞ exp(µatt ) cosh ∆φ˜D

GCS model

mD model

(2.17)

(2.18)

where λD is Debye length and µatt is attraction parameter which accounts for non-electrostatic ion adsorption. One can relate cions and ρp through the single equation

1 ρp d = n F c∞ d∆φ˜D

cions ncn∞

,

n=

    1/2    1

GCS model (2.19) mD model

such that ρp d(∆φ˜D ) = F

1 dcions − cions d ln(c∞ ) , n

(2.20)

or equivalently,  ρp d(∆φ˜D ) = F d

 ! 1 − ln (c∞ ) cions + ln(c∞ )dcions  , n

(2.21)


34

and using mass conservation dcions /dc∞ = −vbulk /vp ,  ρp d(∆φ˜D ) = d

 ! 1 vbulk − ln (c∞ ) cions − ln(c∞ )dc∞  . n vp

(2.22)

Now, the differential electrical work per unit desalted volume is −2 VvDT φe dqe can be expanded as dEv,min = −2

VT VT ˜ (∆φ˜St + ∆φ˜D )dqe = −2 ∆φSt dqe + d(qe ∆φ˜D ) − qe d(∆φ˜D ) . vD vD

(2.23)

Electrical work on a closed loop can be derived by substituting Equation 2.22 into 2.23 and use of electroneutrality condition qe + vp ρp = 0,

Ev,min

  # I I " I 1 VT  ∆φ˜St dqe + d qe ∆φ˜D + vp F − ln (c∞ ) cions − F vbulk ln(c∞ )dc∞  . = −2 vD n (2.24)

In the absence of native surface charges, the Stern potential relates to electronic charge density qe as qe = CSt ∆φ˜St VT , where CSt is Stern layer capacitance. So, assuming a constant or qe -dependent Stern capacitance, Stern layer does not contribute to electrical work (first integral on the right-hand side vanishes) and thus does not consumes energy. The second term also vanishes by definition and we recover Ev,min = ∆gsep for the case of dilute 1 : 1 electrolyte as

Ev,min =

2.6.3

2RT vD

I vbulk ln(c∞ )dc∞ .

(2.25)

Free energy of separation

We here study the dependence of the thermodynamic minimum energy of ion separation (as given by the Gibbs free energy of separation, ∆gsep ) as a function of the desalination depth given by ∆c and recovery ratio γ. We show below that, to a good approximation, ∆g/c0 ∼ (∆c/c0 )n , where a and n depend only on the recovery ratio. Note, the Gibbs free energy per unit of fresh volume produced is given by " ∆gsep = 2RT cD ln cD +

# 1 c0 − 1 cB ln cB − ln c0 . γ γ

(2.26)


35

Further, using mass conservation, we have

c0 = cB (1 − γ) + cD γ

(2.27)

where, cD = c0 − ∆c. Substituting Equation 2.27 in 2.26, and using a Taylor series expansion of Equation 2.26 about ∆c/c0 = 0, we obtain ∆gsep = 2RT c0

"

1 2(1 − γ)

∆c c0

2

1 − 2γ + 6(1 − γ)2

∆c c0

3

# 4

+ O(∆c )

(2.28)

Note that the leading order term in Equation 2.28 is quadratic in ∆c/c0 . Hence, we expect that ∆gsep /c0 has a power law dependence with ∆c/c0 , where the exponent is close to 2, i.e., we hypothesize, ∆gsep ≈a c0

∆c c0

n .

(2.29)

Figure 2.7a shows the variation of ∆gsep /c0 vs. ∆c/c0 for recovery ratios of 0.25, 0.5, and 0.75, as given by Equation 2.26 and from a power law fit, ∆gsep /c0 = a(∆c/c0 )n . Note that the power law fits exceedingly well to the thermodynamic value of ∆gsep /c0 given by Equation 2.26. We show the values of the fitting parameters a and n versus the recovery ratio in Figure 2.7b. Note that these are solely dependent on the recovery ratio, and that the exponent n is close to 2 (between 1.5-2.5) for all recovery ratios. We note that for significant desalination depths ∆c/c0 → 1, higher order terms are required to approximate the Taylor series expansion about ∆c/c0 = 0 as given in Equation 2.28. Alternately, for such conditions, a Taylor series expansion around ∆c/c0 = 1 can be performed (not shown here), which results in a similar power law dependence. Finally, combining the power law (Equation 2.29) with the resistive electrical work Ev,resist , we get, ∆gsep γΛcycle a η≈ ≈ Ev,resist F IRs

∆c c0

n−1 (2.30)

In Equation 2.30, we have used the relation ∆c = IΛcycle /F Q, where Λcycle is the cycle charge efficiency. Also, note that the exponent n − 1 is positive (close to 1). Thus, for a given recovery ratio (i.e., fixed γ, a and n), thermodynamic efficiency of a desalination cycle can be increased by (i) increasing cycle charge efficiency, (ii) decreasing the series voltage drop IRs and, (iii) larger


36

Figure 2.7: (a) Variation of the Gibbs free energy of separation normalized by feed concentration, ∆gsep /c0 vs. concentration reduction normalized by feed concentration, ∆c/c0 for varying recovery ratio values γ = 0.25, 0.5, and 0.75. Solid black lines represent the exact variation from thermodynamic principles (Equation 2.26), and the dashed grey lines represent a curve fit from the approximation of the form, ∆gsep /c0 = a(∆c/c0 )n , where a and n are fitting parameters. The fit parameters a and n depend solely on the recovery ratio, and their variation is shown in (b). desalination depths, ∆c/c0 .

2.6.4

Voltage and concentration profiles

Figure 2.8 shows selected voltage and effluent concentration profiles for constant current charging and discharging of our CDI cell with 20 mM NaCl inlet salt concentration. First, second, and third column respectively correspond to operation at 2, 4, and 8 A/m2 current densities. Labels in each panel show the flow rate used in ml/min. All data correspond to dynamic-steady-state (DSS) condition wherein concentration and voltage profiles do not vary between the cycles. DSS reached after 3 to 5 cycles depending on the current and flow rate values. Voltage window for all the experiments was 0.4-1 V to achieve higher charge efficiency compared to 0-1 V voltage window. [116] For experiments at fixed charge/discharge current (each column), the value of flow rate dramatically affects both voltage and concentration profiles. Operation at low flow rates leads to diffusion-limited regime where mass transport of ions is the major limiting factor. [117] So, the cell undergoes salt depletion (starvation) and voltage sharply increases, as seen in experiments at 0.22 ml/min and 4 A/m2 as well as 0.44 ml/min and 8 A/m2 . At high enough flow rates, however, voltage profiles collapse to a single profile determined by the cell capacitance value and Faradaic losses.


37

Figure 2.8: Selected voltage and effluent concentration profiles for constant current experiments with 20 mM NaCl inlet salt solution. Each column corresponds to experiments at a fixed current density (2, 4, and 8 A/m2 , respectively), and labels in each panel shows values of flow rate used.

2.6.5

CDI cell performance

We here discuss the variation of throughput, energy and desalination performance metrics with operational variables (flowrate and current). First is the average salt adsorption rate (ASAR) in units of moles of salt per total electrode area per time (or grams of salt per gram of electrode per time) defined as ASAR =

Q N Atcycle

Z

tcharge

(c0 − c)dt

(2.31)

0

where N = 2 is number for electrode pairs, A = 25 cm2 is single electrode area, tcycle is cycle time, tcharge is charging time, Q is flow rate, and c and c0 are effluent and influent salt concentrations, respectively. Second is the energy normalized adsorbed salt (ENAS) in units of moles of salt removed (or grams of salt removed) per Joules of energy lost defined as [108]

ENAS =

Q ∆E

Z

tcharge

(c0 − c)dt, 0

(2.32)


where ∆E (=

R tcycle 0

38

IV dt) represents the net energy input during one cycle. Third is the average

concentration reduction in the freshwater produced, ∆c in units of mmol defined as R tcharge ∆c =

0

Q(c0 − c)dt . R tcharge Qdt 0

(2.33)

Here, ∆c is a measure of desalination depth in the freshwater produced. The desalted water concentration is thus cD = c0 − ∆c, corresponding to the desalinated volume VD = Qtcharge . The remaining feed water processed in a cycle results in concentrated brine solution with volume VB and concentration cB . The recovery ratio is γ = VD /(VB + VD ). Salt conservation requires that c0 = cB (1 − γ) + cD γ. Finally, the specific free energy of separation resulting from the desalination process ∆gsep in units of Wh/m3 is given by " ∆gsep = 2RT

c0 ln γ

c0 − γcD c0 (1 − γ)

− cD ln

c0 − γcD cD (1 − γ)

# ,

(2.34)

where ∆gsep represents the minimum energy required for the given desalination process. Figure 2.9 shows the variation of CDI performance indicators versus operating conditions (current and flowrate). Figures 2.9a, 2.9c, and 2.9d show that at each flow rate, the three metrics: ASAR, ∆c and ∆gsep increase, reach a maximum simultaneously and then decreases with increasing current. Further, note that the current corresponding to the maximum value in these metrics shifts to higher values for higher flow rates. Figure 2.9b shows that ENAS also increases, reaches a maximum and then decreases with increasing current. However, the current corresponding to peak ENAS is slightly lower than the current at which ASAR, ∆c and ∆gsep are maximum. Also, from Figure 2.9a note that ASAR increases monotonically with increasing flowrate, whereas in Figures 2.9c and 2.9d, the peak values of ∆c and ∆gsep (over varying current) decrease with increasing flow rate. More importantly, note in Figure 2.9b that the peak value of ENAS (over varying current) increases, reaches maximum and then decreases with increasing flowrate. This trend in ENAS is similar to the variation of thermodynamic efficiency with flowrate and current values in Figure 2.5 in Chapter 2. Further, from Figure 2.9a, note that at low currents, ASAR for all flow rates asymptote to the same line (indicated by a dashed line). This is best explained using the following relation for a CC


39

charging and discharging operation.

ASAR =

Itcharge Λcycle , F N Atcycle

(2.35)

where Λcycle = λEDL λc λf l . [39] Here, λc = tdischarge /tcharge is a cycle average Coulombic efficiency, λEDL represents a cycle average dynamic EDL efficiency which is mainly a function of the voltage window, and λf l is the flow efficiency which is a measure of salt recovery at the effluent. We note that in the limit of vanishing current I → 0, we have λf l → 1 for any finite flow rate. [39] Hence, in the limit I → 0, Equation 2.35 simplifies to

ASAR =

IγλEDL λc . FNA

(2.36)

Taking the derivative of ASAR with respect to current in this limit, we get γλEDL λc d(ASAR) = . dI FNA

(2.37)

where the right-hand side of Equation 2.37 is a constant value which determines the slope of the asymptotic line shown in Figure 2.9a. The slope (given by the right-hand side of Equation 2.37) depends on recovery ratio, and average EDL and Coulombic efficiencies for the given voltage window.


40

Figure 2.9: Measured values of CDI performance indicators: (a) Average salt adsorption rate (ASAR), (b) Energy normalized adsorbed salt (ENAS), (c) Average concentration reduction (∆c), and (d) Gibbs free energy of separation (∆gsep ), versus operating current (CC operation) for flow rates of 0.22, 0.44, 0.89, 1.73 and 3.47 ml/min. In all cases, the cell voltage window was 0.4-1 V. Dashed line in (a) represents an asymptotic variation of ASAR with current for all flow rates.

Chapter 3

Energy breakdown in CDI Sections of this chapter are based on article published in the journal of Water Research, [108] and are reproduced here with minor modifications.

3.1

Introduction

Energy has traditionally been the dominant cost component for many desalination systems such as those applying distillation, which is highly energy intensive. [119] RO has dramatically reduced the energy requirements for desalination, with modern systems achieving roughly 50% energy efficiency for treating seawater based on the thermodynamic ideal free energy of mixing. [120] However, RO fares significantly worse for water with lower concentrations of dissolved solids, such as brackish water, where it only reaches 10% or less efficiency. [121]. RO forces all treated water through the active membrane, with energy losses (and plant size) roughly corresponding to the total throughput of the plant. On the other hand, since the ions themselves are directly targeted in CDI, the energy consumption of this technique largely scales with the amount of salt removed (i.e. throughput times input concentration). This scaling promises higher energy efficiency for CDI compared to competing technologies when treating waters with lower dissolved solid concentrations than seawater (e.g. brackish water) [107]. There are a variety of operational parameters that can be tuned for CDI, including

41

CHAPTER 3. ENERGY BREAKDOWN IN CDI

42

time dependence of charging voltage or current, level of cell charging (i.e. final cell voltage), and flow rate. The choice of these can dramatically influence the energy efficiency achieved in operation, and a consistent framework for determining optimal conditions for operation of CDI cells is still lacking. We note that electric double layer capacitors, or supercapacitors, rely on very similar physics to CDI and have been optimized to maximize charge/discharge cycle efficiency and energy storage density. A number of studies have studied the loss mechanisms present in supercapacitors, [122] including series resistance [123, 124], charge redistribution loss, and parasitic reaction loss. [125] However, the design and operational regimes of supercapacitors are very different than CDI. Importantly, there is generally no electrolyte flow, and organic solvent based, high concentration electrolytes are commonly used to achieve high operating voltage windows and minimize resistance. The goal of supercapacitor operation is solely the storage and recovery of energy. Further, supercapacitors are often applied in high current applications, and this requires a focus on series resistive losses. This focus has led to substantial supercapacitor optimization and sub-milliohm equivalent series resistances are commonly achieved. [126] The promise of CDI for energy efficient processing of lower concentration inlet feeds has led to a number of studies concerning energy loss. [113, 115, 127–132] These have generally focused on the total energy loss of the process, which is useful for comparison with different technologies or among different CDI designs, but provides little insight for optimizing CDI operation or refining current CDI designs. One element that has been studied in some detail is the choice of operation of CDI cells with constant current charging versus constant voltage charging. [115, 128, 132] Constant current operation generally leads to superior energy performance with energy usage reduced by up to 30%, [132] and some studies have dealt with the specific mechanisms of loss operative in CDI. Alvarez-Gonzalez et al. [127] developed a simple model accounting for resistive and parasitic losses consisting of series and parallel resistances and parameterized this model using experimental data. They then optimized cell geometry and charging current in terms of cell energy loss using this model and showed good agreement with experiments. Detailed studies have also been conducted on the series resistance of CDI cells, e.g. by Qu et al. [111]. Improved understanding of the constituent energy loss mechanisms in CDI offers the opportunity for more efficient operation of existing cells and improved future designs, and hence, motivates this work.


43

Here, we experimentally quantify the specific energy loss mechanisms operative during CDI with constant current charging. These mechanisms separate roughly into those dominant at high or low charging currents. The mechanisms dominant at high currents motivate slow charging of the cell. We attribute these losses mostly to resistive dissipation during charge and discharge and, to a lesser degree, redistribution of accumulated charge within electrodes. We perform in situ, realtime measurements of cell series resistance as a function of charging current and time within the charging phase. The dominant losses at low charging currents, corresponding to parasitic currents in the cell, prompt acceleration of the charge phase and a reduction of charge time. We perform an independent set of constant voltage experiments to measure parasitic currents vs. cell voltage. We characterize both loss categories over a broad operational parameter space and show that balancing these losses leads to optimal energy efficiency. Total salt removed per cycle is another key parameter for CDI operation. We define two figures of merit (FOMs) relevant for practical CDI operation and plant design, salt removed per unit time and salt removed per unit energy. These provide quantitative metrics for evaluating tradeoffs between operational requirements (e.g. throughput vs. energy efficiency). We also provide relations for the investigated CDI cell identifying regimes of charging current and maximum cell voltage which allow a balance between cell throughput and energy efficiency as quantified by the product of salt removal rate and salt removed per unit energy.

3.2 3.2.1

Materials and methods CDI cell design

Figure 3.1a shows a schematic of our radial flow-between CDI (fbCDI) cell. We fabricated the cell using five pairs of activated carbon electrodes (two of which are shown here) with 6 cm diameter and 270 µm thickness and total dry mass of 4.3 g. The electrode material (Materials and Methods, PACMM 203, Irvine, CA) has been used and characterized for CDI applications extensively and is well described. [58, 103, 113, 115] We stacked the electrodes between 130 µm thick circular shaped titanium sheets, which acted as current collectors (total of six sheets). Each current collector had a tab section (1 × 5 cm) for connection to external wires (see Figure 3.1a). All the electrodes and current collectors (except the ”book end” electrode and current collector on top of the stack) had a


44

5 mm diameter opening at their center for the flow passage. We used 420 µm thick non-conductive polypropylene mesh (McMaster-Carr, Los Angeles, CA) between each electrode pair as spacers. We cut the spacers in circles slightly larger (∼ 4 mm) than electrodes and current collectors to prevent electrical short circuit. This assembly was then housed inside a CNC-machined acrylic clamshell structure and sealed with O-ring gaskets and fasteners (not shown here). Flow paths are indicated with arrows in Figure 3.1a. Feed water enters the cell via a 5 mm diameter inlet port in the upper clamshell and is radially distributed to the outer surfaces of the stack within the header. Feed solution then flows radially inward (toward the center of the stack) through the spacers and between the electrodes. This radial flow empties down into the vertical flow channel and exits via an outlet port in the lower clamshell.

3.2.2

Energy pathway in CDI

The schematic of Figure 3.1b shows the energy pathway in a typical CDI cell. To understand this, we first note that the goal in any CDI system is to increase the potential energy of electrode stack from its base level, and consequently, attract ionic species to the electrodes with electrostatic forces. This is done by transferring electrons to the cell through an external voltage and/or current source. This input energy is denoted as Ein in Figure 3.1b. However, not all transferred potential energy is used for ionic charge storage (capacitive energy, or Ecap ), as there are various loss mechanisms during R the charging process. Namely, resistive and parasitic energy losses, denoted respectively as Ein and P Ein in Figure 3.1b. The charging process continues until one or more charging criteria are met, such

as a specified maximum cell voltage or a preset amount of transferred electronic charge. Then the regeneration or discharge process starts and gradually lowers the stack’s potential energy level to its base level. The extractable or recoverable energy (Eout ), however, is smaller than Ecap as there are R P resistive and parasitic energy losses in discharge process as well (Eout and Eout respectively).

We emphasize that Ein is the total electrical energy input during charging. We measure Ein as the (unsigned) magnitude area under the voltage versus time curve during charging multiplied by the current during charging. As we shall describe, we measure Eout as the (unsigned) magnitude area under the voltage versus time curve during discharge multiplied by the current during discharge. A portion of Ein is dissipated (by internal resistance and parasitic reaction losses) and the rest of Ein


45

is stored as capacitive energy. The energy loss in the entire charge and discharge phase is thus equal to Ein − Eout . The following two equations can then describe the energy pathway in CDI systems. R R P P Ein − Eout = (Ein + Eout ) + (Ein + Eout )

(3.1)

R P Ecap = Ein − (Ein + Ein )

(3.2)

We further define resistive loss during charging and discharging as

R Ein

=

R,N S Ein

+

R,S Ein

=

R,N S Ein

tcharge

Z +

I02 Rs (t) dt

(3.3)

I02 Rs (t) dt

(3.4)

0

R,N S R,S R,N S R Eout = Eout + Eout = Eout +

Z

tcycle

tcharge R,S R,S where Ein and Eout are series resistive loss during charging and discharging, respectively.

Series resistance here corresponds to contact resistance, ionic resistance of solution in separators, R,N S R,N S are energy loss due to network of distributed and resistance of wires. Similarly, Ein and Eout

ionic resistance of solution inside the electrode pores during charging and discharging. Superscript NS stands for non-series resistance. We here will neglect the resistances of the electrode matrix as this tends to be negligible in CDI (e.g., compared to ionic resistance in electrodes). [111, 113] We separated resistive loss contributions into series and non-series resistances because of their distinct behavior, as described in the following. The equivalent circuit of a CDI cell can be described as a network of resistors and non-linear capacitors. [111, 112] Some of these resistors are electrically in series and others are parallel to capacitors. The series resistors include the external lead resistances, the current collector, and the non-series resistance associated with the electrolyte inside the pores of the (porous dielectric) spacers. The voltage (current) response of these series resistors to rapid changes in current (voltage) can be assumed to be instantaneous. In contrast, the distributed resistor/capacitor network of the porous CDI cell electrodes have significant characteristic RC (resistance-capacitance) time delays associated with charging (order 10’s of seconds or greater for significant penetration of charge into the electrode). As a result, due to its fast time response, series resistances can be measured at each time during charging and discharging (c.f. Section 3.3.2),


46

Figure 3.1: (a) Schematic of circular fbCDI cell with five pairs of activated carbon electrodes (only two pairs shown here). The stack was housed inside a clamshell structure (not shown here) and sealed with O-rings and fasteners. Arrows indicate flow paths. (b) Schematic of energy pathway in R a typical CDI system. A fraction of input energy Ein during charging is dissipated via resistive (Ein ) P and parasitic (Ein ) processes and the rest is stored in the cell (Ecap ). A portion of stored energy is P R ) and remaining energy is recovered (Eout ). and Eout then dissipated during discharging (Eout R,N S R,N S while it is not feasible to directly measure values of Ein and Eout in situ and independently.

We therefore directly measure series resistive loss (c.f. Section 3.3.2) and also quantify parasitic loss with a separate experiment. We then use Equation 3.1 to calculate the sum of non-series reR,N S R,N S sistive loss for the charging and discharge phases (Ein + Eout ). In this chapter, we perform a

series of experiments to distinguish contribution of different loss mechanisms (resistive and parasitic mechanisms) and study energetic performance in CDI.

3.2.3


The experimental setup consisted of our fbCDI cell (c.f. Section 3.2.1), a 3 L reservoir filled with 50 mM potassium chloride (KCl) solution, a peristaltic pump (Watson Marlow 120U/DV, Falmouth, Cornwall, UK), a sourcemeter (Keithley 2400, Cleveland, OH), and a flow-through conductivity sensor (eDAQ, Denistone East, Australia). We used KCl to approximate a univalent, binary, and symmetric solution. We operated our cell at constant current (CC) charging and discharging to study the energy budget introduced in Section 3.2.2. We used a fixed flow rate of 2 mL/min with closed-loop circulation in all of our experiments (flow from reservoir to cell and back to reservoir). This is equivalent to normalized flow rate of 0.014 mL/min/cm2 (flow rate divided by stack electrode


47

area, N A, where N = 5 is number for electrode pairs and A ≈ 28 cm2 is single electrode area). We continuously purged the reservoir with high purity argon gas during the experiments. We estimate < 1% change in reservoir concentration based on adsorption capacity of our cell, and approximate influent concentration as constant in time. We estimate a flush time (defined here as the time to replace one cell volume) of about 3 min. We applied external currents of 25, 50, 100, 150, 200, and 300 mA in charging and a reversed current of the same magnitude in the discharging process. These values are equivalent to current densities of 1.8, 3.6, 7.1, 10.7, 14.2, and 21.4 A/m2 (current divided by stack electrode area, N A). For each current, we charged the cell to fixed external voltage values between 0.2 to 1.2 V (with 0.2 V increments) and discharged the cell to 0 V. Higher currents had necessarily narrower working voltage because of considerable resistive voltage drop. For example, we charged the cell to 0.6, 0.8, 1, 1.2 V at highest 300 mA current. For each external current and voltage combination (total of 32 experiments), we performed at least three complete charge/discharge phases. This ensured the dynamic steady state (DSS) condition, in which salt adsorption during charging is equal to desorption during discharging. DSS was reached after a few cycles, and voltage and effluent concentration profiles did not vary between cycles thereafter. As shown by Cohen et al., [133] salt removal performance of CDI cells can be prone to degradation under prolonged experiments. This is believed to be partly due to oxidation and corrosion of positive electrode. We did not observe noticeable degradation during the course of our experiments (which were performed over a period of about 2 months). We recorded external voltage and effluent conductivity using a Keithley sourcemeter and an eDAQ conductivity sensor (with ∼ 93 µL internal channel volume). Conductivity was converted to salt concentration using a calibration curve for KCl. Refer to Sections 3.5.1 and 3.5.2 for plots of voltage and concentration measurements for the experimental conditions mentioned above. We also show the establishment of DSS condition for the case of 100 mA current and 0.8 V maximum voltage in Section 3.5.2. Additionally, in order to estimate resistive losses, we used a sourcemeter for in-situ series resistance measurement during charging and discharging. See Section 3.3.2 and Section 3.5.3 for more details.


3.3 3.3.1

48

Results and discussion Voltage profile and energy breakdown

Figure 3.2a shows voltage profiles of our cell vs. time with 200 mA charge/discharge current and limit voltage of 1.2 V and 2 mL/min flow rate (under DSS condition). Solid curve shows voltage measured by the source-meter and denoted as Vext . Dashed curve corresponds to underlying “equivalent capacitance” voltage (Vcap ), the total voltage difference across the electrodes excluding voltage drop across the series resistance. We term this Vcap as an analogy to the equivalent RC circuit shown as an inset in Figure 3.2a and we define it as Vext −I0 Rs or Vext +I0 Rs respectively during charging and discharging (Rs and I0 being series resistance and external current magnitude). The instantaneous rise/drop in Vext shown in Figure 3.2a is because of series resistance and is equal to 2I0 Rs . The prefactor 2 is consistent with the reversal of current at the start of charging or discharging. Vcap also exhibits a small, abrupt drop after current reversal as well. We hypothesize the latter effect is due to charge redistribution in the porous carbon electrodes, which has been observed in transmission line [134, 135] and high-fidelity models [136, 137] of CDI as well as in experiments. [93, 138] Refer to Figure 3.8 for plots of voltage measurements at other experimental conditions. In Figure 3.2b, we show power input/generation of our fbCDI cell under the same conditions as those of Figure3.2a. This plot is generated by multiplying Vext and Vcap by external current I0 . Positive I0 Vext values correspond to power transferred to the cell and negative −I0 Vext values are power generated by the cell, power which can ideally (in the limit of perfect transfer efficiency) be stored or used. Shaded regions show total input (Ein ) and output (recovered) energy (Eout ) of R,S the cell. Diagonal and vertical hatched areas are respectively measured series resistive loss (Ein R,S P P and Eout ) and parasitic loss (Ein and Eout ) during charging and discharging. We calculated series

resistive loss using in-situ, in-line measurement of series resistance (see Section 3.3.2 for more information). Further, we measured parasitic energy loss through an independent set of constant voltage experiments. We hypothesize that the parasitic loss is primarily due to leakage currents associated with Faradaic reactions at electrodes. To this end, we charged the cell to fixed external voltage values between 0.1 to 1.2 V (with 0.1 V increments) for 25 min each and monitored the current via sourcemeter. We attribute the remaining current at 25 min (> 10τRC , with τRC being the RC time


49

Figure 3.2: (a) Measured voltage profile of the cell vs. time under 2 mL/min flow rate at 200 mA current and limit voltage of Vmax = 1.2 V. Inset shows RC circuit analogy of the cell, where Rs and Rp respectively model series and parallel resistances in CDI. (b) Power input/generation of the cell for the conditions identical to those of (a). Shaded areas labeled as Ein and Eout show energy input and recovered during charging and discharging in a single cycle. Diagonal hatched areas show series R,S R,S P resistive energy loss (Ein and Eout ), and vertical hatched areas show parasitic energy losses (Ein P and Eout ). Inset shows measured parasitic current vs. Vcap as obtained from independent, constant voltage experiments. scale of our cell at the beginning of charging phase) mainly to the parasitic current (Ip ) at that voltage. We show the parasitic current vs. capacitance voltage in the inset of Figure 2b. We further made the assumption that parasitic current is only a function of voltage (not applied current) and used the relation below to calculate parasitic energy loss.

P Ein

Z =

tcharge

Ip Vcap dt and 0

P Eout

Z

tcycle

=

Ip Vcap dt

(3.5)

tcharge

P P Note, as a visual aid, we have exaggerated the magnitudes of Ein and Eout in Figure 2b (although

Figure 3.2a is actual experimental data to scale).


3.3.2

50

In-situ series resistance measurement

We performed in-situ, on-the-fly measurement of series resistance of the cell (Rs ) by sampling voltage response to a mid-frequency (∼ 10 Hz), small-amplitude (2 mA) AC current signal on top of the charging or discharging DC current (I0 ). We measured Rs by dividing the measured voltage amplitude by current amplitude (see Figure 3.11. Rs , as mentioned before, includes interfacial contact resistance and resistance of solution in spacers as well as external wires. For more information about resistance characterization refer to Section 3.5.3. In Figure 3.3, we show the results of series resistance measurements vs. capacitance voltage difference (Rs ) for currents of 25-300 mA and fixed limit voltage of Vmax = 1.2 V. We define capacitance voltage difference as maximum variation of capacitance voltage during a full cycle. ∆Vcap is defined as max(Vcap ) − min(Vcap ). Each data point in each loop is an average of at least two measurements in two consecutive cycles under DSS conditions. The upper (lower) half of the loops corresponds to series resistance in the charging (discharging) process (see arrows in Figure 3.3). As can be seen here, Rs in the charging step is greater than that in the discharging step. This is because salt is removed from the spacers during charging. Cell operation under high currents therefore leads to greater asymmetry in resistance plots. This is expected, as charging with high currents removes a considerable portion of influent salt, which in turn, increases solution resistance in the spacer. As an example, Figure 3.9 shows more than 80% salt removed from inlet stream with I0 = 300 mA. Note that series resistance in this case varies by only about 30% (from 0.4 to 0.52 Ω). This suggests that interfacial (between electrodes and current collectors) and wire resistances contribute the majority of series resistance. A simple analysis (not shown here) suggests that spacer resistance is about 25% of total series resistance. We estimate resistance of titanium current collectors to be < 2% of Rs . In the inset of Figure 3.3, we show the same resistance data vs. time (normalized by cycle time, tcycle ). The inset again shows a fast increase in resistance at higher currents. Refer to Section 3.5.3 for a complete set of resistance plots. In Section 3.3.3, we will use these measurements to calculate resistive loss and present a comprehensive study of energy pathways in our fbCDI cell. To summarize, we list all the loss mechanisms we study in this work below. P P 1. Parasitic loss, Ein + Eout , calculated by Equation 3.5. We hypothesize that this is primarily


51

Figure 3.3: Measured series resistance vs. ∆Vcap during charging and discharging for 25, 50, 100, 150, 200, and 300 mA currents and Vmax = 1.2 V (each loop corresponds to a fixed current). At low currents, Rs does not vary considerably throughout the cycle, while it varies more strongly at high currents due to significant salt removal. The inset presents series resistance data vs. time (normalized by cycle time tcycle ) for one cycle. due to Faradaic currents. R R 2. Resistive loss, Ein + Eout R,S R,S (a) Series resistive loss, Ein + Eout , measured using ∼ 10 Hz probe signal and includes the

following: i. Wires resistance ii. Interfacial contact resistance between electrodes and current collectors iii. Ionic resistance in spacers R,N S R,N S P P (b) Non-series resistive loss, Ein + Eout , estimated as (Ein − Eout ) − (Ein + Eout )− R,S R,S (Ein + Eout ). The primary component of this is the ionic resistance of solution within

electrode pores. Note we neglect the distributed resistance within the electrode matrix (carbon) material.


3.3.3

52

Energy losses in CDI

In Figure 3.4a, we show total energy loss per cycle (Ein − Eout ) vs. ∆Vcap for currents between 25-300 mA. At a fixed current, energy loss monotonically increases with ∆Vcap (or equivalently, with cycle time). In Figure 3.14, we show that cycle time increases almost linearly with ∆Vcap . We also include Ein and Eout vs. ∆Vcap in Figure 3.14. Figure 3.4a further shows that energy loss is generally greater at higher charging currents. We attribute this to the importance of the resistive loss which is approximately linearly proportional to current (for fixed charge transferred), and dominates the total loss at higher charging currents. We will discuss this in more detail below. P P Figure 3.4b shows calculated parasitic loss (Ein + Eout ) vs. ∆Vcap . Reduction of dissolved

oxygen at 0.69 V (vs. SHE) and oxidation of carbon electrode at 0.7-0.9 V are considered as two main sources of parasitic reactions at voltages below electrolysis potential in CDI. [139, 140] We observe an exponential relation between parasitic loss and ∆Vcap (in the inset of this figure we plot the data on a logarithmic scale). This exponential growth is consistent with power loss due to parasitic reactions on the carbon surface. As given by the Butler-Volmer equation, the currents for these reactions (e.g. oxidation of surface groups and dissolved gasses in solution such as oxygen) are usually exponential with respect to surface potential. For example, Biesheuvel et al. [98] used generalized Frumkin-Butler-Volmer model to derive an exponential relation between rate of redox reactions and Stern potential. Our results further show that parasitic loss is smaller at higher charging currents. For example, at ∆Vcap ≈ 1.2 V, parasitic loss at 100 mA is about 5 times smaller than the 25 mA case. We attribute this to the effect of series resistance voltage drop and cycle time. At high currents, cycle time is shorter and voltage drop across series resistances can be significant (see Figure 3.8). So, the electrodes experience lower voltages (compared to low current cases) for a shorter period of time. We show calculated resistive loss per cycle vs. ∆Vcap for 25-300 mA currents in Figure 3.4c. As discussed in Section 3.2.2, to arrive at resistive loss, we first independently measured energy loss R,S R,S P P (Ein − Eout ), series resistive loss (Ein + Eout ), and parasitic loss (Ein + Eout ). We next used R,N S R,N S Equation (1) to estimate energy loss due to non-series resistances (Ein + Eout ). We finally R R used Equations (3) and (4) to approximate resistive loss (Ein + Eout ). Results indicate that resistive

loss increases proportionally with ∆Vcap , or equivalently, with charging time (see Figure 3.14).


53

Figure 3.4c also shows that resistive loss increases almost linearly with charging current. Figure 3.4d shows calculated stored energy in the cell vs. ∆Vcap . We have here made an assumption that non-series resistive losses during charging and discharging are approximately equal R,N S R,N S R (Ein ≈ Eout ) and so here approximate capacitance energy as Ecap ≈ Ein − [Ein,series + 1 R 2 (Ein,ionic

R NR + Eout,ionic )] − Ein . We note that this approximation can be justified at low ap-

plied currents (where non-series resistance during charging and discharging should be almost equal), whereas at higher currents, it likely overpredicts Ecap . Results show the collapse of data to a sin2 gle quadratic-form relation between Ecap and ∆Vcap in form of Ecap = 12 C∆Vcap with C ≈ 110 F

(or 26F/g after normalizing by total electrode mass). The calculated value of Ecap here, although approximate, provides insight into energy efficiency of our CDI cell as we will discuss in the next section. To summarize, we here showed that energy losses in CDI have at least two components. First, there is an approximately linear (with voltage and/or current) resistive component (since resistive power scales as the square of current while cycle time is inverse to applied current). Second, there is an exponential (with voltage) parasitic component, and this is likely associated with parasitic reactions. We next turn our attention to the relative magnitude of these loss mechanisms. Figure 3.5a demonstrates the relative importance of energy loss mechanisms by plotting the ratio of (series plus non-series) resistive loss to total energy loss in one cycle vs. ∆Vcap for currents ranging from 25 to 300 mA. The shaded and white areas respectively correspond to parasitic dominant (> 50% parasitic) and resistive dominant (> 50% resistive) conditions. The results show that the resistive energy loss dominates the total loss at high charging current and small ∆Vcap cases. Both resistive and parasitic losses decrease with decreasing ∆Vcap , but the exponential dependence of parasitic loss on ∆Vcap makes it negligible at low ∆Vcap . Parasitic loss, however, is dominant at low current and high ∆Vcap (see shaded area where parasitic > 50% of total loss). This is because, as Figures 3.4b and 3.4c suggest, resistive loss linearly increases with current, while parasitic loss generally decreases with charging current. In Figure 3.5b, we show the ratio of capacitor energy, Ecap , over total energy loss in a cycle vs. ∆Vcap for the applied current values of Figure3.5a. This ratio is essentially an energy transfer coefficient and reflects the efficiency of energy storage in the cell. As a visual aid, the shaded region is plotted to be consistent with Figure 3.5a. Results show that this ratio is generally greater at


54

Figure 3.4: (a) Measured energy loss per cycle vs. ∆Vcap for 25, 50, 100, 150, 200, and 300 mA currents. Energy loss increases with both ∆Vcap and I0 . At low currents, energy loss varies approximately exponential with ∆Vcap , while it is almost linear at high currents. (b) Measured parasitic loss per cycle vs. ∆Vcap . Parasitic losses (likely associated with Faradaic reactions) vary exponentially with ∆Vcap (see inset). (c) Resistive loss (series and non-series) in one cycle for experimental conditions identical to those of (a). Resistive loss increases almost linearly with both ∆Vcap and I0 . (d) Calculated stored energy is well described as the square of ∆Vcap . lower charging currents. However, in the lowest charging currents (i.e. 25 and 50 mA), we observe a maximum at a voltage in which parasitic and resistive losses are comparable. We hypothesize that this optimum operating point balancing resistive and parasitic losses may hold for other CDI systems, at least for low to moderate applied current densities, although more evidence is needed before we can confirm this.


55

Figure 3.5: (a) Ratio of resistive to total energy loss in one cycle vs. ∆Vcap for 25-300 mA currents. Resistive loss dominates total loss at high charging current and small ∆Vcap cases. Parasitic loss, however, is dominant at low current and high ∆Vcap (see shaded area in which parasitic > 50% of total loss). (b) Ratio of stored charge to total energy loss in one cycle vs. ∆Vcap for the same data as in (a). This ratio quantifies the effectiveness of energy storage in the cell and is generally greater at lower currents. Results show this ratio has an optimum at small currents (25 and 50 mA), and this optimum coincides with ∆Vcap at which (series plus non-series) resistive loss and parasitic loss are comparable.

3.3.4

Energy and salt adsorption performance in CDI

We here present two performance FOMs for our cell. The first metric is average salt adsorption rate (ASAR) in units of moles of salt per total electrode area per time and can be defined as [33]

ASAR =

Γads Q = N Atcycle N Atcycle

Z

tcharge

(c − c0 ) dt

(3.6)

0

where Γads is amount of salt adsorbed during charging (in units of moles), N = 5 is number for electrode pairs, A ≈ 28 cm2 is single electrode area, tcycle is cycle time, tcharge is charging time, Q is flow rate, and c and c0 are effluent and influent salt concentrations, respectively. This metric quantifies the throughput of the desalination process. Second is energy normalized adsorbed salt (ENAS) in units of moles of salt per Joules of energy lost and is defined as

ENAS =

Q

R tcharge 0

(c − c0 ) dt , Ein − Eout

(3.7)


56

which quantifies the energetic efficiency of the desalination process. In Figure 3.6, we show values of Γads (in units of µmole/cm2 and mg/g), ASAR (in units of µmole/cm2 /min and mg/g/min), and ENAS (in units of µmole/J and mg/J) vs. ∆Vcap for various currents mentioned before. The conversion between our two forms of normalization can be performed by using a total mass of the ten individual electrodes of 4.3 g, an area of A ≈ 28 cm2 per electrode, and the KCl atomic mass of 74.55 g/mole). We have here normalized Γads and ASAR by stack electrode area (N A) and total electrode mass. Figure 3.6a shows that charging the cell with higher ∆Vcap leads to greater salt adsorption. Operating at lower charging current generally has the same effect. Salt adsorption, however, can decrease for very low currents of 25 mA and high ∆Vcap (where parasitic loss dominates). For example, at fixed ∆Vcap = 1.2 V, salt adsorption at 25 mA current results in significant charge consumed by parasitic losses, and this results in less salt adsorbed than the 50 mA case. ASAR and ENAS are shown in Figures 3.6b and 3.6c. Regions with dominant parasitic loss are indicated by grey shading. In resistive dominant regimes, both ASAR and ENAS increase with ∆Vcap , however, as parasitic loss becomes dominant, they both can decrease with ∆Vcap . An important observation in Figure 3.6a is the inefficiency of salt adsorption at high currents. That is, ASAR does not noticeably improve from 200 to 300 mA current. We attribute this retardation of salt removal rate to the relative magnitude of cell time constant (defined as ratio of cell volume to flow rate) and charging time (tcharge ). In Section 3.5, we develop a simple transport model for effluent salt concentration under CC charging condition and show that the time scale for concentration to reach a plateau can be well described by a simple cell time constant of the form tcell = vcell /Q (vcell and Q being cell volume and flow rate, respectively). At high currents (beyond 200 mA), the charging time tcharge is so short (on the order of a few minutes) that it becomes comparable to tcell . As a result, the charging phase finishes “prematurely” (discharge phase starts before effluent concentration reaches its plateau level). This is evident for the case of 300 mA charging current and limit voltage of 1.2 V, where the shape of the effluent concentration profile has no clear plateau region (see Figure 3.16). Another possible reason is the effect of re-adsorption of desorbed salt from previous discharging phase. This is specifically problematic at high currents, as the flush time and charging time are on the same order.


57

Figure 3.6: (a) Normalized salt adsorption (Γads ) in units of µmole/cm2 and mg/g, (b) average salt adsorption rate (ASAR) in units of µmole/cm2 /min and mg/g/min, and (c) energy normalized adsorbed salt (ENAS) in units of µmole/J and mg/J, each as a function of ∆Vcap . Results are for the experimental conditions identical to those of Figure 3.4. The interplay between resistive effects and parasitic effects results in maxima in ASAR and ENAS for low-to-midrange applied currents. For a better representation of ASAR and ENAS results, in Figures 3.7a and 3.7b, we show interpolated contour plots of these two metrics as functions of ∆Vcap and external current for the same data as in Figures 3.6b and 3.6c. The markers overlaid on the contour plots are the corresponding measurement points (at each current and voltage). The dashed curves are consistent with those of Figure 3.5 and 3.6 and indicate the locus of equal resistive and parasitic losses. As discussed earlier, ASAR increases as either current or ∆Vcap increases. However, as current exceeds ∼ 200 mA, ASAR remains constant or even decreases at low ∆Vcap . This, therefore, shows the best removal rate performance at mid-level currents (∼ 200 mA) and highest possible ∆Vcap (∼ 1.1 V). In contrast, Figure 3.7b shows that ENAS (indicator of energetic performance) is maximized at lower currents and mid-level voltage (∼ 0.6 V). Note that, similar to our observations of the data of Figure 3.6c, ENAS rapidly drops (with increasing ∆Vcap ) as parasitic losses begin to dominate the energy loss. This suggests that there is no operational point that simultaneously favors the two performance requirements considered, namely, removal rate and low energy cost. To elaborate this, we plot ENAS versus ASAR for different external currents in Figure 3.7c. Data points in each curve correspond to a variation in the value of ∆Vcap (as shown in Figures 3.6b and 3.6c). The results clearly show a trade-off between removal rate and energy efficiency of desalination process (ASAR and ENAS respectively quantify desalination speed and energetic performance of the cell). For example, small charging currents are generally more favorable in terms of energy performance (higher ENAS), while


58

Figure 3.7: Contour plots of interpolated (a) average salt adsorption rate (ASAR) in units of µmole/cm2 /min and (b) energy normalized adsorbed salt (ENAS) in units of µmole/J vs. current I0 and ∆Vcap . (c) ASAR versus ENAS for the same data as in (a) and (b). The arrow shows direction of increase of ∆Vcap . ASAR versus ENAS respectively quantify desalination speed and energetic performance of the cell. Results show a trade-off between the two: ASAR is greatest at high currents and high ∆Vcap , while ENAS is generally greater in low currents and low ∆Vcap . In very low currents (i.e. 25 and 50 mA), however, ENAS shows an abrupt drop as ∆Vcap passes a certain limit. The optimum value of ENAS at lowest currents corresponds to where resistive and parasitic losses are comparable. large currents have higher adsorption rate (higher ASAR). It is possible, however, to combine ASAR and ENAS into a new metric (or user-defined cost function) and optimize the resulting metric. In Section 3.5.5, we introduce an energetic operational metric (EOM) as the product of ASAR and ENAS and seek a combination of current and ∆Vcap which maximizes our EOM. Interestingly, we show that the location of maximum EOM approximately coincides with the locus of operational points where (series plus non-series) resistive loss and parasitic loss are comparable.

3.4

Conclusions

We have quantified individual loss mechanisms operative during CDI charging and discharging, and characterized their dependence on the parameters of charging current and maximum cell voltage. We identified losses dependent on cell voltage attributable to parasitic currents and losses depending on charging rate, which are dominated by cell resistances. We measured series resistance for the cell throughout charge/discharge phases for a range of input solute concentrations and a variety of charging currents and cell voltages. We also used independent experiments to quantify parasitic losses as a function of voltage in double layers. The two categories of loss favor different charging


59

rates, with resistive losses minimized at low charging currents, but parasitic losses (and associated leakage current losses) lessened for higher rates which reduce the time the cell spends at high voltage. We introduced two figures of merit, ASAR and ENAS, which characterize the performance of a CDI cell in terms of throughput and energy efficiency, respectively. We showed that these figures of merit provide a powerful tool for optimizing CDI operation.


3.5

Addendum to: Energy breakdown in CDI

3.5.1

Voltage profile under constant current conditions

60

As discussed in Section 3.2.3, we performed a series of constant current charging/discharging experiments for a range of applied currents I0 and limit voltages Vmax . That is, we charged the cell with a constant current until the pre-determined limit voltage was reached. We then discharged the cell with the same current (with reversed direction) until the voltage reached zero. We present voltage profiles for 25-300 mA charging/discharging current (I0 ) and 0.2-1.2 V limit voltage (Vmax ) in Figure 3.8. Each subplot shows the external voltage measurements for a fixed current I0 and different Vmax values. Each profile is an overlay of 2 to 4 successive cycles under dynamic steady state (DSS) condition (to show cycle-to-cycle reproducibility). As mentioned in Chapter 3, the DSS condition is the operation mode wherein voltage and effluent concentration profiles vary negligibly between cycles; DSS is reached after a few cycles. We observe that this is indeed the case for the results in Figure S1, as the voltage profiles shown here are very repeatable across cycles.

Figure 3.8: Measured external voltage versus time for 25-300 mA charging/discharging current (I0 ) and limit voltages 0.2-1.2 V (Vmax ). Galvanostatic voltage increases until the pre-set limit voltage is reached and current is reversed. The jumps just after current reversals are associated with the fast response associated with purely serial resistive response. Profiles shown here are all under DSS condition and each profile is an overlay of 2 to 4 successive cycles.


61

Figure 3.9: Raw measurements of effluent concentration profiles measured via an in-line conductivity meter. Each curve is an overlay of two or more cycles under DSS condition. Dashed lines show influent concentration level.

3.5.2

Effluent concentration measurement

We measured effluent salt concentration via a calibrated in-line conductivity meter. To this end, we created a calibration curve relating the known KCl solution concentrations to measured conductivity. In Figure 3.9, we present effluent concentration versus time for experimental conditions identical to those of Figure 3.8. Flow rate was held constant at 2 mL/min. Dashed lines in each subplot show the influent concentration level. Each curve is an overlay of 2 to 4 cycles under DSS condition. Results show very good cycle-to-cycle repeatability for effluent concentration. In Figure 3.10, we show example measured cell voltage and effluent concentration for the case of 200 mA current and 0.8 V maximum external potential from startup until six cycles. Results show establishment of DSS condition after first few cycles.

3.5.3

In-situ series resistance measurement

In Section 3.3.2, we briefly introduced the procedure for in-situ series resistance measurement. We here discuss this process in more detail. As mentioned in Section 3.3.2, we provide galvanostatic control of the cell at specified currents (and pre-set voltage limits) using a sourcemeter (Keithley 2400, Cleveland, OH). We then use the same sourcemeter to probe the CDI cell in real-time to


62

Figure 3.10: Measured (a) voltage and (b) effluent concentration for I0 = 200 mA and Vmax = 0.8 V during the first six cycles. DSS condition is established after first few cycles. measure its serial resistance response. As mentioned in Chapter 3, we use the strong disparity in time response between series and non-series resistances to real-time sample series resistance component of CDI cells. Namely, we apply a small-amplitude (2 mA) AC current signal with ∼ 10 Hz frequency during small portion of the charging or discharging current steps (within each charging or discharging half-cycle we apply a DC current I0 ). We apply this 10 Hz probing signal once the external voltage is a multiple of 50 mV (e.g. at 0.4, 0.45, 0.5 V, etc.) and sustain this for order 1 s durations at a time. We initiate such pulse modulated AC currents roughly every 15 s to 4 min during charging and discharging (which last order 4 to 90 min, depending on applied current and set voltage limit). In Figures 3.11a and 3.11b, we show example of applied current and corresponding voltage response signals for the case of I0 = 200 mA, Vmax = 1 V, and Vext = 1 V during a typical charging step. The applied current profile is a saw-tooth signal with a mean value of I0 and peak-to-peak amplitude δI of 2 mA. Voltage is a saw-tooth signal on top of an underlying approximately linear voltage response. The DC current signal (I0 ) is responsible for charging/discharging the cell and the linear feature in voltage signal. As a result, this underlying voltage feature should be subtracted from the measured signal for correct resistance measurement. We label the voltage signal amplitude about this linear trend as δV in Figure 3.11b. Series resistance of the cell can then be calculated simply by the ratio of δV to δI (Rs = δV /δI). Figure 3.12 shows the results of resistance measurement versus time with the method discussed above for experimental conditions similar to those of Figure 3.8. Each curve is an overlay of 2 to 4 cycles under DSS condition. In Figure 3.13, we present the same resistance data but here plot


63

Figure 3.11: (a) An example of 2 mA amplitude AC current probe signal (δI) with a fixed 200 mA DC component (I0 ) used for in-situ resistance measurement of our fbCDI cell. (b) Voltage response of the cell for current signal shown in (a). The response consists of saw-tooth with the underlying linear component associated with the charging of electrodes. To calculate resistance, we subtract the underlying linear signal variation from voltage response and divide the amplitude of resulting signal (δV ) by δI. them as a function of ∆Vcap . Both figures show higher (lower) series resistance during charging (discharging), consistent with the respective depletion (enrichment) of ions from (into) the spacer layer. Also, variation of resistance is greater at higher currents as expected.

3.5.4

Energy and cycle time measurements

As discussed in Section 3.3.1, we measured the input and output energy (Ein and Eout ) of the cell operation under a variety of current and voltage conditions by integrating measured instantaneous power over the charging and discharging phases. We present the results as a function of ∆Vcap in Figures 3.14a and 3.14b. Both Ein and Eout monotonically increase with ∆Vcap . In contrast, current magnitude has a competing effect on input and output energies. Figure 3.14a shows that energy input Ein is generally greater when the cell is operated at higher currents (see arrow in Figure 3.14a). Energy recovered Eout , however, decreases as discharge current magnitude increases. This leads to greater energy loss (Ein − Eout ) at high currents, as shown in Figure 3.4a of Chapter 3. We additionally plot the cycle time (in units of min) versus ∆Vcap in Figure 3.14c. This figure shows an approximately linear relation between cycle time tcycle and ∆Vcap .


64

Figure 3.12: Time resolved series resistance for various currents (25-300 mA) and limit voltages (0.2-1.2 V) during cell operation. Each curve is an overlay of resistance measurements for 2 to 4 cycles under DSS condition.

Figure 3.13: Series resistance during cell operation as a function of ∆Vcap for various currents (25-300 mA) and limit voltages (0.2-1.2 V). Resistance values form a closed loop for each experimental condition. The loop becomes wider and more asymmetric under higher currents and higher ∆Vcap , which we attribute to the effect of depletion and enrichment of ions from the spacer region (substantial salt removal and enrichment during charging and discharging, respectively).


65

Figure 3.14: (a) Input and (b) recovered electrical energy of our fbCDI cell as a function of ∆Vcap for currents in the range of 25-300 mA. Both energies increase monotonically with ∆Vcap , but current magnitude has opposite effects on input and recovered energy (see arrows in (a) and (b)). (c) Cycle time versus ∆Vcap under experimental conditions similar to those of (a) and (b). Cycle time and ∆Vcap are approximately linearly dependent.

3.5.5

Energetic operational metric (EOM)

We study the combined effect of ENAS and ASAR on desalination performance and introduce an energetic operational metric (EOM) as the product of average salt adsorption rate (ASAR) and energy-normalized adsorbed salt (ENAS). This EOM is just one arbitrary cost function, which we use here as an example global optimization used to balance the trade-off between desalination throughput and energy efficiency. Other possibilities include ASARα ENASβ form or linear form αASAR + βENAS where α and β are arbitrary positive constants determined by the user and which reflect the relative value the user places on adsorption rate versus energy loss. We presented here as one example figure of merit describing global performance. We then seek a combination of current and ∆Vcap (or equivalently, current and cycling time) to maximize EOM. Figure 3.15 shows an interpolated contour plot of the EOM as a function of ∆Vcap and external current. The markers overlaid on the contour plot are the corresponding measurement points (similar to those in Figure 3.7 in Chapter 3). The dashed curve is the locus of operational points where resistive loss equals parasitic loss. Figure 3.15 shows that the EOM dramatically decreases in the limit of low current and high ∆Vcap (dominant parasitic loss, negligible ASAR) as well as high current, low ∆Vcap (dominant resistive loss, negligible ENAS). EOM, on the other hand, is maximized at moderate values of current and ∆Vcap . The location of maximum EOM coincides with comparable


66

Figure 3.15: Contour plot of EOM (defined as ASAR · ENAS) as a function of current I0 and ∆Vcap . Dashed curve and the associated parasitic dominant shaded region are consistent with Figure 3.7 of Chapter 3. As with our earlier observations, EOM is maximized roughly in the regions with comparable resistive and parasitic losses. resistive and parasitic losses.

3.5.6

Time scales in fbCDI

We here present a simple transport model for fbCDI systems under constant current and constant flow rate operation and identify relevant time scales. Starting with a one-dimensional transport equation for salt along the flow direction and neglecting diffusion, we have

psp

∂c ∂c λ + usup = i ∂t ∂x F

(3.8)

where psp is porosity of the spacer, usup is superficial velocity in the spacer (by definition the product of usup , spacer area perpendicular to the flow, and spacer porosity psp is the volume flow rate), i is the current density, λ is differential charge efficiency, and F is Faraday’s constant. We assume a constant inlet concentration c0 , fixed differential charge efficiency, and uniform (transverse) current density along the flow. We then integrate Equation 3.8 in the direction of flow and arrive at

psp

Q λ ∂ cout = (c0 − cout ) − I0 , ∂t vcell F

(3.9)


67

where vcell is cell volume (volume of spacers), cout is effluent concentration, Q is flow rate, and I0 is applied current. Equation 3.9 can be written as ∂ c¯out ¯ − c¯out ) − I¯0 , = Q(1 ∂t

(3.10)

¯ = Q/(psp vcell ), and I¯0 = λI0 /(F psp vcell c0 ). Effluent concentration can then where c¯out = cout /c0 , Q be solved as i I¯0 h ¯ . c¯out = 1 − ¯ 1 − exp −Qt Q

(3.11)

¯ This simplified analysis shows effluent concentration profile should exhibit a time scale of τcell = 1/Q. We show the result of normalized effluent concentration in Figure 3.8a. Time scale for concertation to ¯ and (normalized) concentration change under constant current charging reach a plateau level is 1/Q, ¯ The second time scale shown in Figure 3.16a is charging time τcharge , I0 condition is ∆¯ c = I¯0 /Q. which can be approximated by the ideal capacitor equation as C∆Vcap /I0 . Equating the two time scales gives a linear relation between current and flow rate as psp vcell /(C∆Vcap ). Figure 3.16b is then a regime map constructed by plotting flow rate versus current (in logarithmic scale). This figure summarizes the two possible regimes based on relative values of τcell and τcharge . The upper-left region corresponds to a “plateau mode” regime, where the effluent concentration reaches a steady level before the charging phase ends. The lower-right regime corresponds to a “triangular-peaked” regime, where the charging phase ends prematurely, and the effluent does not reach to a plateau.


68

Figure 3.16: (a) Schematic of effluent concentration profile and voltage profile for an fbCDI cell under constant current and constant flow rate conditions. (b) Regime map corresponding to plateau mode (upper-left region) with low current and high flow rate, and triangular mode (lower-right region) with relatively high current and low flow rate.

Chapter 4

A Two-Dimensional Porous Electrode Model for CDI Sections of this Chapter are based on article published in the Journal of Physical Chemistry C, [136] and are reproduced here with minor modifications.

4.1

Introduction

Characterizing ionic transport in porous electrodes has been subject of many research efforts in diverse applications including microbial fuel cells, [141, 142] energy storage in supercapacitors and batteries, [143–147] energy recovery via salinity gradients, [148–150] and CDI. [33, 95, 99, 103, 129, 137, 151–161] The understanding of ion transport in CDI applications (specifically with the more common fbCDI architecture) is critical, as the primary flow is through a gap (or porous spacer composed of a porous dielectric bulk material with pores filled with the aqueous electrolyte) between electrode pairs. Hence, in fbCDI cells, the primary directions of ion electromigration and diffusion are commonly orthogonal to ion advection (transport of ions due to bulk flow motion). One-dimensional (1D) solutions of such systems are convenient numerically, but woefully incomplete in capturing the coupling of such multidimensional effects. There have been several recent studies using CDI with various levels of complexity. Biesheuvel

69

CHAPTER 4. A TWO-DIMENSIONAL POROUS ELECTRODE MODEL ...

70

et al. [162] developed an adsorption/desorption model tailored for CDI and based on the seminal work of Newman and co-workers on porous electrodes [155–157] and using a classic Gouy-ChapmanStern (GCS) EDL model. This formulation neglected the diffusive transport (assuming well-stirred solution in the gap) and developed a transport model analogous to Ohm’s law, where the electric field is the sole driving force. The model resulted in an ordinary differential equation (ODE) in time and provided no spatial information (spatially zero-dimensional or 0D). In subsequent work, Biesheuvel and Bazant [95] developed a volume-averaged model and explored capacitive charging in porous electrodes. They employed a GCS treatment of EDLs in the absence of Faradaic reaction and without non-electrostatic ion adsorption. The latter porous electrode model was subsequently extended to include non-electrostatic ion adsorption and a modified Donnan (mD) treatment of charge layers (e.g., suitable for bimodal porous structures and highly overlapped double layers). [97] In both of the latter studies, the models were one-dimensional, assuming a quiescent solution between the electrodes, and a fixed salt concentration far from the electrodes (e.g., modeling either a separated membrane compartment or a stagnant diffusion layer). Suss et al. [36] used a 1D porous electrode model with a simple Helmholtz treatment of electric double layers (EDLs), and predicted and measured the time evolution of concentration and potential across the gap and electrodes under constant external voltage (i.e., the electric potential applied to the system by an external source) and no-flow conditions. Subsequently, Suss et al. [34] demonstrated in-situ spatiotemporal measurements of salt concentration between porous carbon CDI electrodes using an electrically neutral fluorescent probe species and under no-flow (stagnant) conditions. They modeled their CDI cell using a 1D model based on porous electrode theory and an mD treatment of charge capacitance. Rios Perez et al. [159] developed and solved a 1D mass transport model which considered variations along the flow of fbCDI cell. They modeled the transverse flux and subsequent ion adsorption from bulk flow into the electrodes as a local sink term. Their model well predicts the outflow concentration for both low and high flow rates but provides no spatial information throughout the electrodes. 1D models and models using fixed concentration values at some boundary separated by a stagnant layer are overly simple for CDI cells operated at flowing conditions. To date, we know of no fully two-dimensional (2D) models of CDI cells. The closest approximation of multidimensional effects is


71

perhaps the recent work of Porada et al. [163] The latter group approximated the fbCDI cell under flowing conditions using six 1D electrodes coupled to individual subcells arranged in series along the flow direction. Such treatment is a step toward the capture the spatial distribution of charge states, but each subcell was assumed to be perfectly stirred-therefore neglecting the coupling of advection and orthogonal diffusive transport. To the best of our knowledge, there are no theoretical studies which solve fully coupled axial and transverse transports in CDI. There are no 2D ion and transport models in CDI. The spatiotemporal dynamics of such systems are critical to the design and optimized operation of both fbCDI and ftCDI systems. In this Chapter, we solve, for the first time, a full 2D transport model in an fbCDI cell. We first focus on the formulation of the model, its numerical solution, and experiments associated with model calibration and validation. Our model considers coupled mass transport and ionic and electronic charge transport, including an external, purely resistive parasitic resistance. We strive to simplify the model as much as possible, and our model includes three free parameters. The first is a non-electrostatic adsorption parameter which captures experimentally observed [33, 97] adsorption of salt in absence of external voltage. The second is a lumped parameter which captures volumetric ionic charge capacitance of the electrodes. The third is the micropore porosity (i.e. the volume fraction of the pores contributing to ion adsorption). To calibrate the model and validate predicted temporal trends, we fabricated a laboratory scale fbCDI cell with one pair of activated carbon electrodes. Calibration experiments consisted of charging and discharging steps as a function of external voltage under constant flow rate, and an open flow loop configuration. We obtained measurements of cumulative adsorbed salt and total electric charge, and used these integral measures to fit the associated model predictions using our free parameters. We then fixed these parameters and used the model to predict temporal dynamics of effluent charge concentrations and electric charging current, and compared these to temporal measurements. We then present model predictions of unsteady 2D ion concentrations and electric potential within the pressure-driven flow in the gap and within electrodes. Our simulations include both low (20 mM) and relatively high (200 mM) salt concentrations. We investigate the effect of advection, diffusion, and adsorption on desalination process, and identify performance limitations of fbCDI cells including ion starvation and diffusionlimited uptake of salt.


4.2

72

Theory

In this section, we first introduce a double layer submodel designed for desalination applications and then integrate it into a 2D model based on macroscopic porous electrode (MPE) theory. [155–157] We use this model to capture adsorption and desorption dynamics of charged species in multiscale porous structures.

4.2.1

Double layer model

We here take the approach of mD model [33, 98, 160] to simulate the EDL structure and charge accumulation at the surface of the porous electrode. The mD model approach is more comprehensive than the straightforward EDL model such as Gouy-Chapman-Stern (GCS). [164, 165] The GCS model describes the EDL as a combination of an inner, immobile, compact layer (also known as Stern layer) with constant area-specific capacitance, and an outer diffuse layer in which ion distribution is governed by Gouy-Chapman model. The classic GCS is applicable in the case of non-overlapping EDLs, wherein Debye length is small compared to individual pores of the electrode. mD model, on the other hand, is applicable to highly-overlapped EDLs and thus is not suitable in the case of thin EDLs. We implement a form of the mD model (which describes the electrode double layer formation) in a formulation applicable to bimodal pore structure electrodes. [34] The model allows for a numerically robust treatment [166] of porous electrodes systems which include a significant volume wherein the characteristic pore diameter is on the order of or smaller than Debye length, and hence overlapped EDLs. Using the typical terminology in CDI, the bimodal pore structure model distinguishes between so-called macropores (pore channels with order 50 nm diameter or larger) and micropores (pore channels with order < 2 nm diameter). The volume occupied by the interconnected macropores is assumed to dominate ion transport throughout the porous electrode, while the micropore volume dominates the regions occupied by EDLs and therefore the capacity for charge accumulation. [34, 97, 98] The model assumes a quasi-neutral condition (electroneutrality) for ionic concentration in the macropores (i.e., EDLs are negligible in macropore pathways). Further, micropore EDLs include a compact Stern layer immediately adjacent to electrode surfaces. Outside of the Stern layer, the micropore volume is treated as strongly overlapped EDLs (note Debye length is typically 1 − 10 nm


73

in CDI systems). As a result, this mD model assumes uniform ion concentration within micropores. This mD model includes two important deviations from the classical Donnan model: the aforementioned Stern layer, and the introduction of a treatment for non-electrostatic adsorption of ions into micropores. The latter assumption has been used in CDI electrode systems which use activated carbon. [98, 166] The motivation for this non-electrostatic treatment is that experimental observations confirm that these materials are known to be significant absorbents of ions even in the absence of external voltage. [167–169] The physics of non-electrostatic adsorption are not well understood and not well characterized, and the current and previous formulations can be considered empirical corrections of models to account for this effect. The mD model can be formulated as follows. At chemical equilibrium, there is no electrochemical potential gradient between micro- and macropores. Hence, an extended Boltzmann distribution [170] can be used to relate micro- and macropore concentration of each species i as cm,i = cM,i exp −zi ∆φD + µatt,i

(4.1)

where cm,i and cM,i are concentration of species i (subscripts m and M denote respectively micro- and macropores parameters), zi is species valence, ∆φD is the Donnan potential (normalized by thermal voltage VT = kT /e, where k and T are Boltzmann constant and temperature), and parameter µatt,i = (µM,att,i − µm,att,i )/kT accounts for a non-electrostatic adsorption potential of ions into micropores. Positive µatt,i may be described as a macro-to-micro chemical (attraction) potential difference normalized by kT energy. µatt,i may vary for different ions [166, 171] or may be a function of charging state of the system. [101] For simplicity (and to avoid increasing the number of free parameters of our model), we use the same value for all species, i.e. µatt = µatt,i , and will use it as one of three free parameters in quantitative comparison to data. Next, volumetric (net) charge density stored in the micropores 2qm (in units of moles per electrode volume) and micropore potential drop ∆φm (normalized potential difference between electrode surface and center of the micropore) can be related to micropore capacitance as

2qm F = −Cm ∆φm VT

(4.2)


74

where F is the Faraday constant. We here define Cm as the effective volume-specific capacitance of micropores. Cm can be interpreted as the total charge capacitance per unit volume of micropores and includes all EDL structures including Stern layers and/or overlapped diffuse charge layers. This is our second free parameter. Equation 4.2 shows that micropore specific surface area (i.e. total surface area per electrode volume or mass) does not directly appear in mD model formulation. Instead, we quantify accumulated charge via a volumetric micropore capacitance which characterizes the ion adsorption capacity. [97] There have been several studies presenting models for the accumulated charge in the porous electrodes of CDI cells. [33, 34, 162] Inspired by empirical observations, [172, 173] several recent studies and models have suggested that the effective micropore capacitance in CDI cell electrodes may be a function of micropore charge density and/or micropore potential drop. 2 or For example, this dependence has been empirically modelled as [98, 174] Cm = CS0 + κ1 qm

Cm = CS0 +κ2 ∆φ2m , where the first term in each relation was a constant base-value Stern capacitance (at zero net charge), and the second term was an attempt to capture effects of increasing Stern capacitance at higher charge density and/or higher voltages ( κ1 and κ2 have been assumed constant and positive for the individual cell). We here chose to not adopt the latter variable capacitance relations for two reasons. First, we know of no explicit and closed-form relation between micropore capacitance and micropore charge state in this model. Second, removing this treatment results in three (instead of four) fitting parameters for our model. We propose a self-consistent treatment for micropore capacitance as a future work. For example, we hypothesize that effects such as finite ion size may just as plausibly affect the total micropore capacitance, and should be included in capacitance models. For completeness, we note that Biesheuvel et al. [101] studied the ion size effect in mD model, but in the context of relating non-specific attraction energy to interaction between individual ions and pore surfaces. Lastly, we note our model neglects the effect of surface conduction in all pores and Faradaic reactions within the porous structure. The third free parameter is macropore porosity pm , or volume fraction of pores contributing to ion adsorption. We refer the reader to Section 4.2.3 for further discussion. We stress that the aforementioned, three-parameter mD model is just one of several EDL type charge capacity models which can be applied to CDI systems. Our formulation (see Section 4.2.3) is sufficiently modular


75

so that the current transport model can be readily modified using a modified and/or improved EDL model (e.g., using additional parameters). Overall, the formulation for the specific charge capacitance of the CDI system as formulated here has three free parameters, µatt , Cm , and pm . We extract these parameters by performing a series of experiments on a flow-between CDI cell under constant voltage conditions (see experimental methods and results below).

4.2.2

Validity of mD model

We here discuss the validity of mD model using a one-dimensional EDL model for a micropore shown in Figure 4.1a. We study a slit micropore of thickness 2L as a rough approximation of the electrode micropores. Assume the electrode potential on the micropore surface is φe and the potential difference across the Stern layer (with thickness λSt ) is ∆φSt . Further, the potential between the edge of Stern layer and the pore centerline (called diffuse layer potential) and between the pore centerline and macropore potential (called Donnan potential) are respectively ∆φdif f and ∆φD . Without loss of generality, we assume macropore potential φ (far outside the micropore) is zero. mD model assumes uniform concentration profile in the micropores and thus constant potential in the diffuse part ∆φdif f . We here model the overlapped EDL shown in Figure 4.1a for a variety of surface charge densities and pore thicknesses and seek conditions where ∆φdif f = ∆φdif f /VT 1. To this end, we follow derivations in Baldessari and Santiago’s work [175] for a symmetric and binary electrolyte (1:1). The electrode potential φ (the overbar indicates normalization by the thermal voltage VT ) in the diffuse part of EDL satisfies the following equation [175] " # 1/2 dφ √ φ − φ c = p 2 sinh2 ( ) − Ω sinh φ − φc , dx 2

(4.3)

where the derivative dφ/dx is along the pore width with x = x/λD and λ2D = kB T /(2e2 NA c0 ) being the Debye length. Here, φc is the non-dimensionalized potential at the pore center, p = 2 cosh φc , and Ω = tanh φc . The coefficient Ω measures the degree of EDL overlap: Ω → 0 indicates noninteracting EDLs and Ω → 1 indicates strongly overlapping EDLs. The coefficient p is the ratio of ionic strength in the micropore centerline to that of the macropore. Integrating Equation 4.3 from the edge of the Stern layer (x = λSt ) to the micropore centerline (x = L), we arrive at the following


76

integral equation for ∆φdif f √

p

L − λSt λD

Z

0

= ∆φdif f

"

φ − φc 2 sinh ( ) − Ω sinh φ − φc 2 2

#−1/2 dφ.

(4.4)

We then solve the above equation for ∆φdif f at a given L and φc and plug the result into Equation 4.3. The surface charge density can then be calculated as

σ = −

√ i1/2 VT p h VT dφ =− 2 sinh2 (φdif f /2) − Ω sinh φdif f . λD dx x=λSt λD

(4.5)

We solve Equations 4.3 to 4.5 at different micropore widths L and surface charges σ and calculate the resulting ∆φdif f for the parameters listed in Table 4.1. We summarize the results in Figure 4.1b, where we show ∆φdif f = 0.1 and ∆φdif f = 1 curves. The x- and y-axis are respectively the normalized Debye length λ∗ = 2λD /L and the normalized surface charge density σ ∗ = σL/VT . The shaded area below the curve ∆φdif f = 0.1 is the validity region of the mD model, where the potential drop across diffuse part of the EDL is negligibly small (compared to thermal voltage VT ). This translates to up to around 10% concentration variations in the diffuse part of the EDL. The dashed line shows typical surface charge density (σ ∗ ≈ 2) for the charged activated carbon electrodes in CDI applications. Table 4.1: Parameters used in Figure 4.1. Parameter c0 λSt λD T

4.2.3

Description Ion concentration in the macropores Stern layer thickness Debye length Temperature

Value 20 mM 0.6 nm 2.18 nm 300 K

Transport equations in flow-between CDI cell

Our main focus here is investigation of the ion and bulk flow transport dynamics in fbCDI cells. A not-to-scale schematic of our two-dimensional fbCDI cell is shown in Figure 4.2a. The cell consists of two porous carbon electrodes sandwiched between two highly conductive current collector plates (on


77

Figure 4.1: (a) Schematic of a slit micropore with width of 2L and Stern layer thickness of λSt . The electrode potential φe is distributed between Stern layer potential drop ∆φSt , diffuse layer potential drop ∆φdif f , and Donnan potential drop ∆φD . (b) Diffuse layer potential drop (normalized by thermal voltage) on σ ∗ vs. λ∗ plot. The shaded region below the curve ∆φdif f = 0.1 is the validity region of the mD model, with the potential drop across diffuse layer is negligibly small compared to the thermal voltage VT . top and bottom), and feedwater flows in the space between the electrodes from left to the right. We assume the space (or gap) region between electrodes is completely filled with feedwater; although we note it is simple to extend our model to include simple Darcy flow through a porous spacer region. As in a typical fbCDI cell, we assume the gap thickness, Ls , is much smaller than the electrode spanwise (into the page) width W . This allows us to avoid the complexity of three-dimensional velocity field, and hence, to use a fully-developed parabolic velocity profile (Poiseuille flow) in the space region. The contact resistance associated with the interface between the current collectors and electrodes can be a dominant portion of total resistance of a typical CDI cells. [111] So, we here include a purely resistive element, Rc , to model this effect (the capacitance of the contact is negligible compared to that of the electrode). We further neglect the resistance of the solid portions of the electrode matrix, as we estimate this is much smaller than typical electrolyte resistances. We lump any other external resistances (e.g., connecting wires) as in series with the contact resistance and thus part of the parameter Rc . To the best of our knowledge, this is the first solution of a full two-dimensional model for an fbCDI system.


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Figure 4.2: (a) Schematic of elements of our two-dimensional adsorption-desorption and flow model for an fbCDI cell. Two porous carbon electrodes of thickness Le are sandwiched between two current collectors. Electrodes are separated by a distance Ls to provide space for flow of water (left to right). We lump all contact resistances, e.g. wire-to-current collector and current-collector-to-electrode connections, into a purely resistive element Rc . (b) Not-to-scale schematic of computational domain and boundary and interface conditions in non-dimensional form. The dash-dot-dash line denotes the geometric symmetry line. The computation domain includes inlet and outlet regions for the flow of length L/8. In a manner similar to classical MPE theory, we formulate transport equations on a volume average basis. [155–157] Volume averaging is performed over a length scale considerably larger than macropore features, but small enough to capture the spatiotemporal variation of electric potential and species concentrations. [95, 155, 157] This avoids detailed treatment of the complex morphology of porous structures and models the ionic adsorption/desorption effects into sink/source terms within the governing equations. We start with the general form of mass transport equation without reaction for species i as ∂ ci = ∇ · ji ∂t

(4.6)

where ci is the concentration of species i and ji is its associated molar flux vector with electromigration, diffusion, and advection contributions as below

ji = ci u − zi Di ci ∇φ − Di ∇ci

(4.7)

where u is the bulk velocity vector, Di is diffusion coefficient of species i, and φ is local electric potential (normalized by thermal voltage VT ). We assume a dilute, binary, symmetric, and univalent electrolyte (zi = ±1) with equal anion and cation mobilities and thus equal diffusion coefficients. We also assume fully developed parabolic velocity profile u = u(y)ˆ x within the gap region. By


79

adding and subtracting Equation 4.6 for i = ±1 and using electroneutrality approximation in the gap, the conservation of salt and charge in this region is simply governed by the advection-diffusion equation

∂c ∂t

∂c + u(y) ∂x = D∇2 c and Ohm’s law ∇ · (c∇φ) = 0 , where c is the concentration of co-

and counter-ions in bulk solution and in macropores (i.e. c = ci ). We assume symmetric electrolyte so D = Di for i = ±1. In our bimodal pore structure electrode, ion transport in macropores can be described as [98] h i ∂ (pM cM,i + pm cm,i ) + ∇ · cM,i usup − pM De,i ∇cM,i + zi cM,i ∇φ = 0 ∂t

(4.8)

where pm and pM are respectively the porosity of micro- and macropores, and are related to total electrode porosity p as pm +pM = p. We measured the total porosity of our electrode as approximately 0.7 using dry/wet-saturated mass measurement. Equation 4.8 is consistent with porosities are defined as pore volume per the total macroscopic volume of electrode and not per volume of electrode solid (carbon). In the macropores, electroneutrality holds and thus c = cM,i for i = ±1. In micropores, however, the difference between co- and counter-ion concentrations balances the wall surface charge density. usup is a superficial flow velocity throughout the pores, defined as the flow rate per total cross-section area. De,i is the effective diffusion coefficient of species in macropores and in general differs from that of the free solution due the effect of porosity and tortuosity. [176–179] For simplicity, we here assume a value of De,i /Di = 0.5 for both species. This is an approximate but reasonable estimate according to the reported empirical correlations for porous materials, including porous electrodes, with porosities in the range of 0.6 to 0.8. [176, 180] In the macropores, we set usup = 0. This means we neglect all bulk flow (e.g., pressure-driven Darcy-type flow or any electrohydrodynamic flow) within the electrode. The latter assumption is equivalent to assuming the permeability of the electrodes is negligible compared to the effective (open channel) permeability of the gap region. For simplicity, we further neglect electroosmotic flow (and any pressure-driven flow associated with non-uniform electroosmotic flow) within the porous electrodes. We hypothesize that the latter may be important in some systems and suggest it as a topic for future work. With these assumptions, the governing equations for salt and charge balance


80

(derived from Equation 4.8 by adding and subtracting equations for i = ±1) can be expressed as ∂c pm ∂ wm = D e ∇2 c − ∂t p − pm ∂t

(4.9)

pm ∂ qm p − pm ∂t

(4.10)

De ∇ · (c∇φ) =

− + − where 2wm = c+ m + cm is the volumetric ions concentration, and 2qm = cm + cm is the (net) charge

density in micropores (each with units of moles per electrode volume). Assuming electrochemical equilibrium for micropore double layers throughout the electrode, anion (cation) expulsion in the micropores of negative (positive) electrode leads to an electric potential mismatch between macroand micropores. This potential difference is the well-known Donnan potential ∆φD . The potential difference between electrode matrix and pore solution is related to micropore potential drop ∆φm and Donnan potential as φe − φ = ∆φm + ∆φD

(4.11)

where φe is time varying but spatially uniform potential of the upper electrode. Due to symmetry, the other electrode has potential −φe . Thus, the cell voltage (voltage difference between the upper and lower electrode) can be written as Vcell = 2φe . Next, we non-dimensionalize the governing equations as x = Lx, y = Le y, t = tL2e /De , c = c0 c, qm = c0 q m , and wm = c0 wm , where L, Le , and c0 are length and thickness of electrodes, and the inlet and initial concentration of solution, respectively. The final set of equations describing adsorption/desorption process inside the electrodes are ∂c pm ∂ w m 2 = De ∇ c − p − pm ∂t ∂t

(4.12)

pm ∂ q m p − pm ∂t

(4.13)

De ∇·(c∇φ) =

q m = −c exp(µatt ) sinh(∆φD )

(4.14)

wm = c exp(µatt ) cosh(∆φD )

(4.15)

φe − φ = −

qm + ∆φD Cm

(4.16)


81

and the following apply to the gap region ∂c ∂c 2 + αe P ee u = d∇ c ∂x ∂t

(4.17)

∇·(c∇φ) = 0

(4.18)

Non-dimensional effective micropore capacitance, C m , is defined as C m = Cm VT /2F c0 , u = u/U is bulk flow velocity normalized by the mean flow velocity, αe = Le /L is the geometric aspect ratio of the electrode-to-flow region length, P ee = U Le /De is Peclet number based on the electrode thickness, and d = D/De is ratio of diffusion coefficients in the gap to the pore region. Based on Equations 4.12, 4.13, and 4.17, there are three fundamental timescales associated with the fbCDI system. First, is the diffusion time of ions across the electrode, L2e /De . Second, a resistive-capacitive (RC) timescale associated with micropore charging. The latter can be written as pm /pM and normalized by the diffusion time across the electrode thickness. This RC timescale characterizes the time evolution of charge and salt adsorption within the electrode (see Equations 4.12 and 4.13). Third, is an advective timescale associated with the feedwater flow (and ion advection) within the gap. The parameter αe P ee in Equation 4.17 is simply the ratio of (transverse) diffusion time across the electrode thickness to the advection time of the stream. This compares the rates of axial advection and transverse diffusion. A schematic of computational domain is shown in Figure 4.2b. Our model includes inlet and outlet sections of lengths L/8 in the computational domain. These enable us to impose inlet and outlet boundary conditions sufficiently far from electrode edges. We set concentration as uniform at the inlet and assume zero concentration gradient at the outlet. The inlet and outlet boundaries assume zero electric potential. We performed a limited number of numerical experiments using inlet and outlet regions of longer length to confirm that the L/8 additions to the computational domain were sufficient to avoid end effects. Boundary and interface conditions (also shown in Figure 4.2b) are as follows. (1) Zero mass and current flux across sidewalls of the inlet and outlet sections and across the outer boundaries of the electrodes (left, right, and top sides of the electrode in Figure 4.2b), (2) uniform concentration at the inlet, (3) advective boundary condition for concentration at the outlet, (4) zero electric potential at


82

the inlet and outlet (refer to Section 4.7.6 for further discussion of these boundary conditions), and ∂c ∂c (5) continuity of concentration and potential as well as (6) salt flux (i.e. pM ( ∂y )e = d( ∂y )s ) and

electric current (i.e. pM ( ∂∂yφ )e = d( ∂∂yφ )s ) at the electrode-solution interface. The interface conditions for salt flux and electric current at the solution-side are multiplied by pM in order to correct for the effect of macropore porosity. Our assumption of a binary, symmetric electrolyte results in a geometric symmetry about the midplane of the gap region. This and the boundary conditions result in symmetry of the spatiotemporal concentration fields and an anti-symmetry of potential. We take advantage of this to only model the geometrically symmetric (top) half of the CDI cell. We couple the desalination system to an external circuit by incorporating a contact resistance. Kirchhoff’s law relates the external voltage and current as

V cell = V ext − rc iext

(4.19)

where V cell = Vcell /VT , V ext = Vext /VT , iext = Iext /I0 , and rc = Rc /R0 are non-dimensional cell voltage, external voltage, external current, and contact (and other external series) resistance, respectively, and R0 = VT /I0 = VT Ls /2F c0 DLW . To close the system of equations, we seek a relation between external current iext as a cumulative quantity to other parameters of the model. We note that macropore ionic current is not divergence-free throughout the electrode, as there is a net charge transfer to and from the micropores. Macropore ionic current i = F z(j+ − j− ), after qm correcting for porosities, can be related to micropore charge flux as pM ∇ · i = −2pm F ∂∂t . External

current Iext can thus be found, equivalently, by integrating micropore charge flux over the entire RR ∂ qm electrode volume as 2pm F ∂t W dx dy or ionic current passing through the electrode-solution R ∂φ interface as 2pM F De ∂y W dx. For the latter, we have simply used divergence theorem to recast the surface integral to a line integral along the interface. So, the non-dimensional form of external current can be written as

iext =

pm d

ZZ

∂ qm ∂t

dx dy = −

pM d

Z 0

1

c ∂∂yφ dx

(4.20)

The surface integral above simply states that total electron current to the system is equal to the


83

volume integral of time derivative of spatiotemporal adsorbed charge within the micropores. Equations 4.19 and 4.20 couple the external circuit to the desalination system, and along with Equations 4.12 to 4.18 fully describe adsorption/desorption dynamics. The governing equations in this section are also applicable to alternate treatments of the EDL and/or micropore charge relations by replacing Equations 4.14 to 4.16 with a set of equations describing the alternate model. Further, Equations 4.19 and 4.20 should also allow for modeling adsorption/desorption dynamics under any constant or time dependent voltage- and current-controlled operation.

4.2.4

Equilibrium solution

In this section, we focus on a solution of the model discussed above at equilibrium (i.e. at infinitely long time). The equilibrium solution is governed solely by the EDL submodel, external voltage, and the inlet concentration. Other parameters such as the system geometry, flow rate, or shape of the velocity profile do not affect the equilibrium behavior of the system. At a time much longer than the largest time scale of the system, salt concentration becomes uniform throughout the gap and macropores. Moreover, the electric potential of solution and total electric current both approach zero. By combining Equations 4.2, 4.14, 4.16, and 4.19, and using symmetry assumption, we arrive at the following transcendental equation for equilibrium Donnan potential Vext 2F c0 = ∆φD + exp(µatt ) sinh(∆φD ) 2VT Cm VT

(4.21)

where c0 is inlet salt concentration. Equation 4.21 has a guaranteed unique solution (for which we define as ∆φ∞ D ) as the right hand side is strictly monotonic with respect to Donnan potential. We ∞ ∞ term the resulting (characteristic) micropore salt and charge wm and qm , respectively. The near-

equilibrium total adsorbed salt and transferred electronic charge (in units of moles and Coulombs, respectively) can then be written as

∞ 0 2pm (wm − wm )Ve = 2c0 pm exp(µatt ) cosh(∆φ∞ D ) − 1 Ve

(4.22)

∞ 2pm qm Ve = −2F c0 pm exp(µatt ) sinh(∆φ∞ D )Ve

(4.23)


84

0 Here, Ve is the electrode volume and wm = c0 exp µatt is initial adsorbed charge at zero external

voltage due to non-electrostatic adsorption of ions. A close inspection reveals that, for the equilibrium solution, our three fitting parameters (µatt , Cm , and pm ) reduce to just two independent groups: pm Cm and pm exp µatt = exp µ∗att . As will be described in Section 4.4.1, we use experimental equilibrium conditions to extract these two groups from our measurements. As described in Section 4.4.1, we subsequently use dynamic measurements (including the temporal profile of effluent salt concentration) to fix the third independent parameter pm .

4.3

Experimental setup and methods

We designed and fabricated a custom-built, meso-scale fbCDI cell using a pair of activated carbon electrodes and performed a series of constant voltage, continuous-flow experiments. We used these experiments to validate selected trends predicted by our two-dimensional model and to extract free parameters µatt , Cm , and pm .

4.3.1

CDI cell design

We fabricated an fbCDI cell using a pair of activated carbon electrodes (Material Methods, PACMM, Irvine, CA) with dimensions of 100 × 20 × 0.68 mm and total dry mass of 1.42 g. We stacked the electrodes between two 100×20×0.2 mm stainless steel plates, which acted as current collectors (see Figure 4.3). To improve electric contact between the porous electrodes and current collectors, [111] we adhered the electrodes to these plates using silver conductive epoxy (MG Chemicals, cat No. 8331-14G, Surrey, BC, Canada). In addition to silver epoxy, we observed that electrode compression significantly reduces the electrode-current collector resistance. Our design had four compression slot features (see Figure 4.10) to compress the peripheral regions of electrode. We used plastic shim features (not shown in Figure 4.3) between the compressed parts of electrodes to avoid electrical shorting. We refer reader to Figure 4.10 for more details. We also adhered the connecting wires to the current collectors with silver epoxy. This assembly was then housed inside a 3D-printed (ProJet HD 3000 Plus, 3D Systems, Rock Hill, SC) clamshell structure and sealed with gaskets and fasteners. The housing and seals held the electrodes a distance of 0.8 mm apart in the absence of


85

Figure 4.3: Three-dimensional drawing of our CDI cell clamshell structure. We sandwiched a pair of activated carbon electrodes inside the upper and lower clamshells and sealed the desalination cell using gasket and fasteners (not shown). Electrodes have dimension of 100 × 20 mm with 0.68 mm thickness. Our design had inlet and outlet plenum chambers in order to uniformly introduce salt solution to and collect desalted water from the CDI cell. See Figure 4.10 for more details. any porous spacer. The upper clamshell was connected to two 1/8 in outer diameter stainless steel tubes (Scanivalve, Liberty Lake, WA) which served as water inlet and outlet. Our cell had inlet and outlet plenum chambers which were used, respectively, to distribute salt solution from inlet port to electrode gap and to guide desalted water from the gap toward the outlet port (see Figure 4.3). Minimizing volume of the chambers is essential in any CDI cell design, as this reduces the dispersion effect (a phenomenon in which salt and fresh water solution actively mix to each other and produce a solution with intermediate salt concentration). We estimated the dead volume of our cell (volume of inlet and outlet plenum chambers) to be < 0.1 mL while total gap volume was 1.6 mL. This suggests that the effects of dispersion within the inlet and outlet chambers was negligible.

4.3.2


We used potassium chloride (KCl) to approximate a binary, symmetric, and univalent solution. Potassium and chloride ions have roughly equal electrophoretic mobility (76.2 × 10−9 and 67 × 10−9 m2 /V/s in free solution and room temperature [181, 182]). KCl has been used in other work for CDI experiments for this reason. [34] We used 20 mM influent salt concentration for two reasons. First,


86

this low concentration enabled investigation of regimes associated with very significant ion depletion, and hence strong, easily measurable removal of ions. It also enables more detailed study of diffusionlimited desalination regimes (typical of fbCDI). Second, desalination of moderate salt concentrations (10 to 100 mM) is common in experimental CDI studies, [93, 116, 129, 132, 133, 162, 166, 183–186] and these serve as comparison cases. In Section 4.5.2, we present simulations in higher concentrations (200 mM) and compare the results to low concentration case. We performed a series of preliminary wash and charge/discharge cycles prior to the experiments. The washing procedure was as follows. We filled the cell with 20 mM KCl and let it stand for 24 h prior to starting experiments. To reduce the effect of non-specific interaction of ions with pristine electrodes, we performed 18 consecutive charge-discharge cycles by applying 1.2 V (for 30 min) for charging and 0 V (for 1 h) for discharging (Keithley 2440 sourcemeter, Cleveland, OH) at 0.42 mL/min flow rate (Watson Marlow peristaltic pump, Wilmington, MA). After this aging process, we pumped 20 mM KCl solution again for 6 h with no external potential in an effort to flush out loosely adsorbed ions. The feedwater used was not purged with nitrogen gas. At this point, we started desalination experiments. We used 0.42 mL/min constant flow, and we applied external voltage of 0.4, 0.6, 0.8, and 1 V (in ascending order) across the cell. For each voltage, we charged the cell for 2 h, and discharged the cell at 0 V for 3 h. The charging and discharging times were chosen such that the effluent concentration asymptotes to around 90% of the inlet value. We performed two successive charge/discharge cycles for each external voltage. For each experiment, we recorded a time series of electric current and effluent conductivity using respectively the Keithley sourcemeter and a calibrated flow-through conductivity sensor (eDAQ, Denistone East, Australia). The conductivity sensor had ∼93 µL internal channel volume. Conductivity was converted to salt concentration using a calibration curve for KCl. As described below, we used these current and concentration measurements to evaluate the free parameters of our fbCDI cell (see Section 4.4.1 for more information).

4.4

Experimental results

Here, we first define two desalination metrics and show their experimental values. We compare these measured quantities to model predictions. We then find the values of free parameters by minimizing


87

the error between experiments and model results. We performed all numerical simulations with COMSOL Multiphysics (version 4.5, COMSOL Inc., Burlington, MA) using transport of diluted species and equation-based modeling interfaces.

4.4.1

Parameter extraction

We used electric current and effluent concentration measurements to estimate near-equilibrium adsorbed salt and transferred electric charge during CDI charging. (i.e., total electronic charge supplied from the external power source). To this end, we performed the aforementioned set of experiments to quantify these as a function of external voltage. We then compared these measured quantities to corresponding values predicted using our model. We first describe our estimate of cumulative salt adsorbed during charging. For a finite charging time t, salt absorbed can be expressed as follows: 1 Γ (t) = mtot

Z

t

Q c0 − hcout (τ )i dτ

(4.24)

0

where Q is the water flow rate, hcout i is the outlet-area-averaged effluent concentration (defined as R R ucout dy/ u dy), and mtot is total dry mass of the electrodes. For our predictions, we integrate the outlet concentration for each simulated time to calculate hcout i under constant flow rate conditions. In the experiments, our downstream conductivity sensor directly quantifies hcout i, also under constant flow rate conditions. The adsorbed salt (per electrodes mass) can also be expressed in volume integral as follows 1 Γ (t) = mtot

ZZ

0 2pm (wm (t) − wm )W dx dy

(4.25)

At steady state, these two forms are indeed numerically equivalent (and we have confirmed this with our numerical model). At each external voltage, we integrated the measured cumulative salt over time during 2 h charging. We call this near-equilibrium salt (Γeq ) and show the corresponding measured values in Figure 4.4a. This figure indicates that salt adsorption capacity increases (almost linearly) with external voltage. Next, we define cumulative transferred electronic charge (or equivalently, cumulative adsorbed


88

ionic charge) relative to zero external voltage on a per electrodes mass basis as

Σ (t) =

1 mtot

Z

t

(Iext (τ ) − Ileak ) dτ

(4.26)

0

where leakage current, Ileak , is a non-zero current measured even long after starting the charging step, and likely associated with unwanted Faradaic reactions at the electrode surface. [187] Cumulative charge can also be presented as follows

Σ (t) =

1 mtot

ZZ 2F pm qm (t)W dx dy

(4.27)

We again integrated the measured cumulative charge over 2 h charging time. We call this value near-equilibrium charge (Σeq ) and also show its experimental measurements in Figure 4.4a. Similar to Γeq , variation of Σeq is also almost linear with external voltage. Further, we note that Γeq and Σeq are only a function of external voltage and not the operational parameters such as flow rate. As discussed in Section 4.2.4, the equilibrium results do not provide sufficient information for parameter extraction. As a result, we used effluent salt concentration data as well. We first used measured equilibrium salt and electrical current to evaluate two unknown parameter groups (see Section 4.2.4 for more details). To this end, we varied pm Cm and pm exp µatt (as two independent groups) and minimized the sum-of-squares of model prediction and measured equilibrium values of salt and charge. Subsequent to this, we used the dynamic data to obtain pm . To this end, we varied pm (while maintaining pm Cm and pm exp µatt constant) such that the effluent salt concentration from model matched that of the recorded data. Our final estimated parameters are as follows: µatt = 1.5, Cm = 150 F cm−3 , and pm = 0.3. All predictions presented below use the same three values of this parameter. These values are close to those reported for similar activated carbon electrode materials [115] and for hierarchical carbon aerogel monolith (HCAM) electrodes. [34] A complete list of parameter settings used in the model is presented in Table 4.2. As shown in Figure 4.4a, there is a reasonable agreement between data and model predictions for these model parameters. We see a deviation between model and experimental data for external voltages higher than about ∼ 1.2 V. We attribute this discrepancy primarily to the effects of Faradaic reactions at these higher potentials (which is not captured by the present model).


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Figure 4.4: (a) Near-equilibrium adsorbed salt and transferred charge (per mass of electrodes) and (b) charge efficiency, Λ = F Γeq /Mw Σeq , for 2 h of constant voltage operation in the range of 0 to 1 V, and flow rate of 0.42 mL/min. Feedwater inlet concentration is 20 mM KCl. Open circles and diamonds are measured data and solid lines are model prediction. Charge efficiency approaches unity as voltage increases, indicating stronger asymmetry in the micropore counter-ion adsorption and co-ion expulsion. Table 4.2: Parameter settings for the two-dimensional model for an fbCDI cell incorporating mD EDL model and operated in constant external voltage and flow rate. Parameter µatt Cm pm pM D De c0 Rc U Ls L Le W

Description Non-electrostatic adsorption parameter Micropore capacitance Micropore porosity Macropore porosity Diffusion coefficient in the gap Effective diffusion coefficient in macropores Influent salt concentration Contact resistance Mean flow velocity Gap thickness Electrode length Electrode thickness Electrode width

Value 1.5 150 0.3 0.4 1.9 × 10−9 0.95 × 10−9 20 4.7 4.38 × 10−4 0.8 100 0.68 20

Units Fcm−3

m2 /s m2 /s mM Ω m/s mm mm mm mm


4.4.2

90

Charge efficiency

As mentioned above, Figure 4.4a shows near-equilibrium adsorbed salt Γeq and transferred charge Σeq (the associated cumulative quantities evaluated at the end of 2 h charging process) as a function of external voltage for both model and experiments. In Figure 4.4b, we show the ratio of these metrics, Λ = F Γeq /Mw Σeq , known as charge efficiency (Mw being the salt molecular weight). The electric current applied to the cell results in both accumulation of counter-ions (within micropores) and repulsion of co-ions. Hence, charge efficiency characterizes the ratio of accumulated charge (as bound ions) to the integral of electric current. [71, 188] For small external voltage (e.g., in DebyeH¨ uckel limit), the amount of counter-ion adsorption into the diffuse double layer is approximately the same as the amount of co-ion expulsion. However, higher potentials break this symmetry and so increase charge efficiency. The latter trend is reflected in the data of Figure 4.4b.

4.4.3

Temporal results

Figure 4.5 shows evolution of effluent salt concentration (normalized by 20 mM KCl inlet concentration) in charging and discharging processes for experiments and model. We here use the parameter setting identical to those in Table 4.2 without any additional fitting parameter. Results are shown for charging up to t = 14 (equivalent to about 2 h) followed by discharging up to t = 28 and for four external voltages of 0.4, 0.6, 0.8, and 1 V. Time is normalized by the diffusion time scale across the electrode thickness, L2e /De , which is order 500 s based on Table 4.2. Figure 4.5 shows a fairly rapid drop in the outlet concentration at around t = 0.5 − 1 followed by a slowly increasing concentration phase. The initial rapid phase of charging process corresponds to fast (RC time scale) adoption of ions originally inside macropores. The adsorption process is then slowed down by diffusion-limited uptake of new ions from the gap, which in turn, results in gradual increase of outlet salt concentration. We discuss this adsorption process in more detail in Section 4.5.2. The discharging process shows a similar trend: an initial fast desorption followed by a gradual decrease of concentration. This two-phase adsorption/desorption is typical of fbCDI systems and has been observed experimentally and computationally. [34, 36] In this figure, the model follows the overall trend of experiments, however, it only approximately matches the measured data. We attribute the discrepancies to two main sources. First is our assumption of constant micropore capacitance is not sufficiently precise.


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Figure 4.5: (a) Measured and (b) model predicted normalized effluent salt concentration vs. time (normalized by diffusion time L2e /De ) for constant voltage operation at 0.4, 0.6, 0.8, and 1 V and constant flow rate of 0.42 mL/min. Feedwater concentration is 20 mM KCl, and time is normalized by diffusion time scale across the electrode thickness. As we mentioned in Section 4.2.1, we hypothesize that the model can be improved by (explicitly) relating the micropore capacitance to the micropore charge state. Second, the method of model calibration using mainly the equilibrium measurements, which is commonly used in CDI, [34, 174] neglects exact dynamic behavior of the system. We hypothesize that additional ex-situ characterization may help better characterize the electrode material. For example, electrochemical impedance spectroscopy [112, 189, 190] and cyclic voltammetry, [138, 191, 192] maybe employed to extract a more accurate and voltage-dependent micropore capacitance. Further, in Figure 4.6, we present time evolution of cumulative adsorbed salt Γ (t) and cumulative transferred electric charge Σ (t) during charging for voltages similar to those in Figure 4.5. We again see that the model follows the observed trends fairly well, but it is not in perfect agreement with the experiments.

4.5

Spatiotemporal results

In Section 4.4, we discussed the process by which we evaluated the free parameters of our unsteady, 2D model. We here focus on this model’s predictions of two-dimensional spatiotemporal concentration and electric potentials during charging and discharging.


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Figure 4.6: (a) Cumulative stored salt Γ (t) (mg/g) and (b) electric charge Σ (t) (C/g) vs. normalized time for 2 h of charging for constant external voltage operation at 1 V and flow rate of 0.42 mL/min. Each is reported per electrode dry mass.

4.5.1

Macropore concentration

Figure 4.7 shows normalized concentration profiles in the gap (0 < y < Ls /2Le ) and electrode region (Ls /2Le < y < 1 + Ls /2Le ) of the cell (collectively the top symmetric half of the cell) at selected times for 0.42 mL/min flow rate. In the electrode region, we plot the concentration of ions in the (net neutral) macropores. The figure shows color maps and contours of concentration during charging at Vext = 1 V in the top row, and for discharging at Vext = 0 V in the bottom row. The dashed line at y = Ls /2Le represents the electrode-solution interface. The less interesting inlet and outlet sections (see Figure 4.2b) are not shown here. Time is normalized by the diffusion time scale across the electrode thickness, L2e /De , which is order 500 s for the parameters listed in Table 4.2. In this section we describe simulation cases for 20 mM KCl inlet concentrations. We summarize a simulation at 200 mM in the next section. The cell is charged to t = 14 (equivalent to about 2 h) at 1 V and then discharged at 0 V, all with total plotted run time of t = 19. The figure shows concentration fields at charging times t = 0.3, 0.6, 4, and 8, and discharge times 14.08, 15, 17, and 19. We also provide videos of the full spatiotemporal concentration fields as addendum in Section 4.7. Figure 4.7a-I shows that, shortly after application of external voltage, a near-gap depletion zone forms within the electrode. This low concentration region grows and its boundary propagates toward the top of the electrode at y = 1 + Ls /2Le . The salt adsorption capacity of


93

our cell at 1 V is approximately 4.5 times greater than the initial salt contained within the total cell volume. So, macropore concentration decreases to order c ≈ 0.001 shortly after charging at 1 V (within about t ≈ 1, or about 8 min after charging starts). We refer to this phenomenon as electrode starvation, where the simulated local macropore concentration approaches zero. This rapid ion-depletion phase, however, does not lead to fully charged micropores throughout the electrode. For example, at t = 1 only 15% of micropores are more than 80% charged, while 50% of them have a charge state less than 30% of their maximum (see Figure 4.13 for a plot of micropore counter-ion concentration). We discuss this phenomenon in more detail in Section 4.5.3. After t = 1, the micropores continue to charge and deionization continues. This relatively slow phase of the charging is shown in Figures 4.7a-III and IV, and is characterized by slow, diffusionlimited transport of ions from the gap into the electrode. As mentioned earlier, this diffusion-limited ionic transport is a characteristic of fbCDI systems. [34, 36] At times t = 1 and 2, the electric field at the electrode-gap interface is reduced to respectively 1.5% and 1% of its initial value; favoring diffusive interchange of ions over electromigration (see Figure 4.15a). The angled contour lines within the electrode for times t = 4 to 8 subsequently show how this process is limited by the influx of ions entering the cell and then diffusing into the electrode, predominantly entering the electrode at the electrode surfaces nearest the inlet. Note the near vertical portions of the concentration contours at t = 8 indicating the importance and persistence of 2D transport in the charging of this system. Figure 4.7b-I shows concentration fields immediately after the cell is short circuited at t = 14. The color bar range is here from 1 to 2 to aid in visual representation (the maximum non-dimensional concentration in the discharge process is about 3). We see a rapidly formed desorption region in the near-gap region of the electrode. This high-concentration region expands and its boundary propagates to the top of the electrode. In a manner analogous to the charging process, the desorption covers the entire electrode within a period of about ∆t = 1. This rapid propagation is then followed by a low-electric field, slow, diffusion-limited transport of ions out of the electrode. See electric potential in the cell for selected times in Figure 4.15. In Section 4.7 (Figure 4.11), we present line plots (along x = 1/2) associated with the results of Figure 4.7.


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Figure 4.7: Normalized ion concentration fields in the gap and macropores for the symmetric top half of the cell (not-to-scale) for constant voltage (a) charging (1 V) and (b) discharging (0 V) (as per Table 4.2). The dashed lines indicate the electrode-solution interface. Time is normalized by the diffusion time scale associated with the electrode thickness. In charging, a depletion region forms near and propagates upward from the electrode-solution interface. This rapid charging is followed by a slow, diffusion-limited uptake of new ions from the gap. At the beginning of discharge, formerly adsorbed ions (now have same sign of charge as electrode) are quickly electrostatically repelled from micropores in the near-gap region. This forms a near-gap depletion region whose boundary then propagates upward. Discharge then slows down significantly as the cell proceeds into a diffusionlimited transport of ions out of the electrode.

4.5.2

Macropore concentration for high inlet concentration

We here present additional results for high inlet concentration (200 mM) and compare these to the low-concentration simulations of the previous section. For 200 mM, the salt adsorption capacity is smaller than the salt initially within the total cell volume. The ratio of salt capacity to initial salt in this case is about 0.45. Figure 4.8 shows the gap and electrode concentration at selected times similar to those of 4.7. Note that the normalized concentrations through the cell and for the entire charging and discharging processes are now near unity. This is expected from the smaller ratio of adsorption capacity to initial salt. Despite the mild changes in concentration, the simulations show many of the multidimensional transport features discussed in the previous section. These include the fast and slow dynamics as well as the high-angle contours associated with trade-offs between


95

advection and diffusion. In Figure 4.12, we present the line plots associated with Figure 4.8 along the line x = 1/2. The results show that the time scales for fast adsorption phase of charging process as well as discharge process are similar to those of the 20 mM case. For example, the minimum concentration (at the end of fast adsorption phase) occurs at around t = 1. Also, the time scales associated with the discharge process are similar to those of the low concentration case. Interestingly, the qualitative shape of normalized concentration curves in discharge are also very similar to those of the higher concentration change (see Figures 4.11b and 4.12b). However, the results here are different from the low concentration case in two ways. First, no electrode starvation is observed as the amount of salt initially in the system is less than total salt capacity of the cell even for 1 V applied potential. Second, the charging process is now significantly faster. For example, the concentration field contour lines at t = 4 for 200 mM (Figure 4.8a-III) are similar to t = 8 for 20 mM (Figure 4.8a-IV). During starvation (in times between t ≈ 1 to 5 in 20 mM case), some portion of electrode has negligible concentration, and hence, no ion adsorption is possible in those regions. The electrode is then forced to rely only on diffused ions for adsorption. In contrast, for the high concentration case, the electrode can adsorb due to both locally available supply of ions and due to the diffusion of ions. Hence, charging is eventually dominated by diffusive transport of ions from bulk solution in both low and high concentration cases explored here. But the main difference between these is that, in the low concentration case, starvation more strongly retards the charging process. We hypothesize that diffusive transport will have increasingly negligible effect on charging time as initial salt supply becomes progressively higher than the electrode capacity.

4.5.3

Micropore concentration and evidence of a concentration shock

As we mentioned above, we characterize the end of the initial rapid phases of charging and discharging (at roughly t = 0 to 1 for charging and t = 14 to 15 for discharging) as a transition to diffusion-dominated transport. However, we again stress this transition does not necessarily imply fully charged micropores. We demonstrate this effect in Figure 4.9, where we plot normalized micropore counter-ion concentration (a measure of micropores charging level) throughout the upperelectrode in charging process for 20 mM inlet salt concentration. We also present line plots associated


96

Figure 4.8: Ion concentration fields in gap and macropores normalized by a 200 mM inlet concentration. Shown is the not-to-scale top half of the cell for constant voltage (a) charging (1 V) and (b) discharging (0 V). All other parameters are similar to those of Figure 4.7. In a similar manner to Figure 4.7, a depletion region rapidly forms inside the electrodes and charging rate is then limited by diffusion. However, the diffusion-limited charging process occur faster than 20 mM case. We hypothesize this difference is associated with electrode starvation, which is only observed in the low concentration (20 mM) case. Discharge process shows an analogous trend, i.e. rapid but brief discharge followed by slow diffusion. with Figure 4.9 along x = 1/2 in Figure 4.13 during charging and discharging. Figure 4.9 shows that the near-interface micropores (lower edge of squares) are rapidly charged upon application of external voltage, while the rest of micropores remain partially-charged. The ion diffusive transport from bulk solution coupled with local adsorption into the partially-charged micropores results in a steepening of the micropore concentration profile within the electrode. As shown in Figure 8, the interplay of diffusive transport and local adsorption continuously sharpens the low- and high-charge state micropores and results in what appears to be a propagating shock wave in ion concentration. This shockwave propagation was observed in the 1D mD model of Porada et al. [16]; although Porada presents no discussion of its physical origin. We here present a hypothesis ∂c for formation of such interface. In Figure 4.14, we show directional diffusion flux − ∂y and local

(instantaneous) salt adsorption rate

∂ wm ∂t

along x = 1/2 within the electrode. The diffusive flux

distribution quickly forms a step-like function, with a rapid drop in flux within the electrode. The


97

step function in flux coincides with a spatial peak in adsorption rate. This is an evidence that the starved micropores are forced to “wait” for diffusion. We refer the reader to Section 4.7.4 for more discussion. We further discuss the results for the high salt concentration case here. Our simulations for the charging process for 200 mM salt concentration (not presented here) show no concentration shock within the electrodes. For 200 mM concentrations, micropores are almost uniformly charged upon application of voltage. As we mentioned earlier, or 200 mM, the salt initially in the gap and electrodes is abundant, and this implies no starvation. We hypothesize the shock wave is the result of local electrode starvation (where at least some region of the electrode is starved), and this is only observed in the low concentration case.

Figure 4.9: Normalized micropore concentration profile in the upper electrode. Note that plots are not-to-scale. Micropores in the regions near solution-electrode interface (lower edge of squares) are almost completely charged quickly, while the rest of micropores remain partially-charged. As time goes on, the interface between low and high concentration regions becomes thinner and propagates toward the collector boundary (upper edge of squares) until all the micropores are completely charged.

In the Section 4.7.5 we further discuss the micropore potential distribution and its correlation to micropore charge state. This includes plots of micropore potential along the line x = 1/2 at selected times during charging and discharging process.


4.6

98

Conclusions

In this Chapter, we formulated and solved what is to our knowledge the first 2D model of a CDI cell. This model included coupled solutions of external electrical networks, charge conservation, and species conservation in bimodal pore structure electrodes. We coupled the desalination cell to an arbitrary voltage- or current-control external power source (with a purely resistive element). The model includes three free parameters µatt , Cm , and pm which respectively characterize non-electrostatic adsorption of ions into micropores, volume-normalized adsorption capacitance of micropores, and porosity of micropores. We applied the model to a flow-between architecture desalination cell (with a water-filled gap between a single pair of porous electrodes). We built a meso-scale fbCDI cell and performed a series of experiments at constant external voltage and constant flow rate in an open loop configuration (i.e., constant inlet species concentrations). We chose low influent concentration (20 mM) as test case in our experiments to investigate strong ion removal and study diffusion-limited desalination. We calibrated the model by using dynamics effluent salt and near-equilibrium salt and charge data. The extracted free parameters are fixed throughout the whole work (both parts). We presented effluent salt concentration as well as cumulative and equilibrium adsorbed salt and transferred charge as a function of time. We showed a reasonable agreement between model results and experimental data. The measured effluent concentration suggested that desalination process in fbCDI cells has two distinct phases: a fast adsorption step at the beginning of charging followed by a slow salt removal step. We performed numerical simulations to investigate the adsorption/desorption dynamics in capacitive deionization cell operated under continuous flow rate. We performed simulations at low (20 mM KCl) and high (200 mM KCl) inlet salt concentrations. We presented predictions for the spatiotemporal evolution of potential as well as ion concentrations in the gap, macropores, and micropores. We showed that low inlet concentration can lead to electrode starvation (near zero macropore concentration). The coupling of diffusion-limited transport and adsorption can result in a sharpening of the gradient between high and low micropore charge concentration. This sharpening apparently results in an ion concentration shock wave in micropore charge concentration propagating through the electrode (toward the current collector). We hypothesize this shock wave is caused by


99

the coupling of diffusion and adsorption in the low-concentration regime where adsorption can lead to local electrode starvation.


4.7

100

Addendum to: A Two-Dimensional Porous Electrode Model for CDI

4.7.1

Effect of electrode compression on contact resistance

As mentioned in Chapter 4, in addition to the effect of conductive epoxy on the system resistance, we observed that electrode compression can significantly improve the electrode-to-current-collector contact. We quantified the effect of compression force on this contact resistance using ex situ voltagecurrent readings and comparison of these to pressure force exerted on the stack of electrode-current collector by fasteners. We found 3 to 4 times reduction in contact resistance at ∼ 100 kPa pressure. A recent study [111] suggests that this is because compression increases the number and quality of microscopic point contacts. We show three-dimensional isometric drawings of the two major components of our desalination cell flow structure in Figure 4.10. Our design included four compression slot features (see Figure 4.10a) which we use to improve the contact between the electrodes and current collectors. As shown in the figure, these slot features are implemented together with a set of plastic shim features (labeled “plastic pieces” in Figure 4.10c). Upon sealing of the two major halves of the cell, these shims create four regions of high compression (they “pinch” the “wings” of the electrodes within the slot features). We found that, upon assembly, these four pinched regions decrease the contact resistance by a factor of 3 to 4. We hypothesize that these four peripheral compression regions leave the large majority of the central electrode regions unchanged.

4.7.2

Macropore concentration plots for low and high inlet concentration

Figures 4.11 and 4.12 show line plots of macropore concentration for 20 and 200 mM inlet concentration cases. Figures 4.11a/4.12a and 4.11b/4.12b respectively present ion concentrations in macropores along the line x = 1/2 during charging and discharging processes. In Figure 4.11, as charging progresses, counter-ions are adsorbed to micropores while co-ions are expelled. In the early phase of charging (gray contours), counter-ions are transported within the electrode via both electromigration and diffusion. In this phase, nonlinear coupling of ion adsorption, diffusive transport,


101

Figure 4.10: (a) Three-dimensional drawing of CDI cell clamshell structure. We sandwiched a pair of activated carbon electrodes inside the upper and lower clamshells and sealed the desalination cell using gasket and fasteners. Electrodes have dimension of 100 × 20 mm with 0.68 mm thickness. Our design had compression slots we used to compress outer regions of the electrode upon assembly. This technique led to 3- to 4-fold decrease in contact resistance. (b) Cross-section schematic of stack of activated carbon electrodes between stainless steel current collectors. Current collectors were bonded to backside of the electrodes by silver epoxy. We placed thin frames of plastic between the electrodes to avoid the electrodes touching each other. (c) Cross sectional view of the assembled cell. and electromigration mechanisms here creates a local concentration minimum near the interface. This local minimum propagates through the electrode until ions in the macropore are largely depleted by around t = 1 (electrode starvation). Charging continues subsequent to t = 1 but at a rate which is progressively more limited by diffusion. This results in linear profiles near the y = Ls /2Le boundary of the electrode as primarily diffusive transport supplies new ions to the electrode volume. The initial rapid (t = 14 to 15) and subsequent diffusion-limited transport phases of discharge are shown in the gray and black contours of Figure 4.11b, respectively. The combined electromigrationdiffusion-accumulation dynamics here result in a local concentration maximum near the gap which propagates into the electrode. This is followed by a long-duration, diffusive transport of ions out of the electrode. On the other hand, electrode starvation is not observed for the high concentration case (see Figure 4.12). However, time scales associated with the early charging as well as discharging phase are similar to low concentration case. Additionally, the concentration profiles in discharging process are quite similar to 20 mM case.


102

Figure 4.11: Normalized ion concentration in gap and macropores along the line x = 1/2 in constant voltage (a) charging and (b) discharging under conditions similar to those of Figure 4.7 of Chapter 4. Vertical dashed line denoted the electrode interface and results are shown for upper-half of the system. Gray lines represent concentration profiles shortly after starting charging or discharging, and black lines show concentration evolution at longer times. Each line is labeled with corresponding non-dimensional time (scaled by L2e /De ).

Figure 4.12: Ion concentration in gap and macropores normalized by 200 mM inlet concentration along x = 1/2 for constant voltage (a) charging and (b) discharging under conditions similar to those of Figure 4.8 of Chapter 4. The label for each curve corresponds to time non-dimensionalized by L2e /De . In contrast to Figure S3a, electrode starvation is not observed. However, early charging as well as discharging have time scales analogous to 20 mM case. Moreover, the concentration lines for the discharge process are quite similar to 20 mM case.


4.7.3

103

Micropore concentration plots and evidence of shockwave

Normalized micropore counter-ion concentration throughout the upper-electrode in charging process for 20 mM inlet salt concentration is presented in Figure 4.13. Figure 4.13a shows that the counterion concentration within near-interface micropores increases to c+ m ≈ 30 within about t = 1 (compare to the fully charged state of c+ m,max ≈ 35). During this time, the counter-ion concentration at the collector boundary (y = 1 + Ls /2Le ) increases to c+ m ≈ 10 but then remains approximately “pinned” near this value from about t = 1 to 4, indicating only partially charged micropores throughout much of the near-collector region of the electrode. The location of the steep-gradient between + this near-interface value of c+ m ≈ 30 and the near-collector value of cm,max ≈ 10 then propagates

toward the collector as ions diffuse into this region and adsorb. The interplay between diffusion and concentration-dependent adsorption rate results in a sharpening of the concentration interface. We later discuss this phenomenon in more detail (c.f. Figure 4.14 for a plot of diffusive transport vs. adsorption rate). This sharpening results in what appears to be a propagating micropore ion concentration shock wave which propagates through the electrode at these conditions. The micropore counter-ion concentration of the discharge process is shown in Figure 4.13b. As described above, diffusion and electromigration mechanisms result in an initially rapid depletion of counter-ion concentration. This is then followed by a slow diffusion-limited expulsion of ions.

4.7.4

Diffusion flux and salt adsorption rate

We here further discuss diffusion limitations of charge transport. Figures 4.14a and 4.14b respectively ∂c show normalized directional diffusion flux − ∂y and local (instantaneous) salt adsorption rate

∂ wm ∂t

along x = 1/2 within the electrode at selected times during charging. Figure 4.14a shows how the diffusive flux distribution quickly (within about t of unity) forms a step-like function, with a rapid drop in flux within the electrode. This high gradient flux feature then propagates toward the current collector. Comparison of Figures4.14a and 4.14b show that this step function in flux coincides with a spatial peak in adsorption rate. This is expected as the “sink function” of adsorption results in a rapid change in flux. To the left of this adsorption peak, the micropores approach saturation. To the right, micropores are starved and are forced to “wait” for the propagating front. Interestingly,


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Figure 4.13: Normalized micropore counter-ion concentration along x = 1/2 for constant voltage (a) charging and (b) discharging (under the charging conditions of Figure 4.3). The degree of charge of micropores near the interface quickly increases to 85% (corresponding to c+ m ≈ 30). Counter-ion diffusion (from bulk) and adsorption to partially-charged micropores sharpens gradient between the low- and high-charge state zones, and apparently forming a moving shock which slowly propagates toward the collector. The process continues until micropores are filled. Discharging shows an initially rapid expulsion of ions, followed by a slow, diffusive expulsion. the diffusion and adsorption dynamics couple to create what is apparently an ion concentration shockwave (see Section 4.5.3 of Chapter 4).

4.7.5

Macropore potential and micropore charge state

As follows from discussion in Chapter 4, the local macropores potential can be indicator of micropores charge state. This is because a higher potential drop from electrode surface to macropores implies greater the stored charge in micropores (c.f. Equation 4.16 of Chapter 4). In Figures 4.15a and 4.15b, we present the gap and macropore potential along x = 1/2 in charging and discharging processes, respectively (reference potential located at y = 0). Figure 4.15a shows that upon application of 1 V to the system (0.5 V and −0.5 V potentials at the two electrodes), potential of near-collector macropores quickly drops to ∼ 7VT and remains approximately pinned at this value from t ≈ 1 to 4. So, from a total potential of ∼ 20VT dropped from each electrode to centerline (the 0.5 V value) about 65% is taken by combination of Donnan and micropore potential drops, ∆φD + ∆φm (c.f. Section 4.2.1 of Chapter 4 for more information). This means that the micropores are capable of


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Figure 4.14: Normalized (a) diffusion flux in y-direction and (b) local salt adsorption rate along the line x = 1/2 during charging process. (a) Shows the formation of a step-like distribution of diffusion flux (after time t = 1), and this feature propagates to the right. Coincident with this step feature is a corresponding local peak in adsorption rate. This peak in adsorption rate occurs at the left boundary of the shrinking starvation region. adsorbing additional ions, but the near-collector regions of electrode are starved during this time (see Figure 4.13a). As micropores are being slowly charged, the macropore potential approaches zero and hence ∆φD +∆φm approaches 20VT . Note, in the absence of Faradaic reactions (as Figures 4.15a and 4.15b show) the equilibrium condition for either charging or discharging is uniformly zero macropores potential.

4.7.6

Note on potential boundary condition

We here discuss our choice of boundary conditions (BCs) for electric potential. We show our model is insensitive to the choice of BC at the inlet and outlet regions. This is an intended and direct result of our use and implementation of extended inlet and outlet flow regions of the domain. In most experimental CDI cells, there are standoff regions in the form of inlet and outlet tubing upstream and downstream of the cell. The distance between external ground nodes (e.g., metal connection touching electrical ground) are typically made sufficiently far from the cell so as to not interfere with the cell’s operation. We capture this characteristic in our model by adding extended inlet and outlet flow regions to our solution domain (see Figure 4.2 of Chapter 4). The potential in the inlet, outlet, and spacer regions of our cell is governed by a nonlinear Laplace equation of the form ∇ · (c∇φ) = 0 (since conductivity is not uniform), as discussed in Section 4.2.3


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Figure 4.15: Gap and Macropore potential (normalized by thermal voltage) along x = 1/2 during (a) charging and (b) discharging processes (under the charging conditions of Figure 4.11). In charging, initial macropore potential at the near-collector region quickly drops to ∼ 7VT and remains approximately pinned near this value from t = 1 to 4 (consistent with partially charged micropores). Potential drops to zero throughout the electrode as micropores fully charge (at about t = 8). During discharge, macropore potential quickly increases and is positive as net charge is released. Macropore then relaxes back to zero as the electrode is discharged. of Chapter 4. Unique solutions to this require at least one Dirichlet boundary condition (BC). One Dirichlet BC for potential in our model is the symmetry line (which is extended along the lower edge of computational domain). This Dirichlet BC is sufficient to specify the unique solution of the potential. By using extended inlet/outlet sections, we make our model insensitive to the specific choice (or values) of BCs at the inlet (x = −L/8) and outlet (x = L + L/8). Hence, our BC at inlet/outlet can be either φ = 0 or ∂φ/∂x = 0. We chose φ = 0 at these boundaries as a simple, convenient choice. To validate our choice, we solved our model for all four possible cases (φ = 0 or ∂φ/∂x = 0 at either/both inlet or outlet) and confirmed that our solutions are insensitive to this. We further validated this idea by evaluating various lengths of inlet and outlet domains. The latter work showed negligible change in the solution for increases of inlet and outlet domain lengths beyond our chosen values. We illustrate this point using the Figure 4.16 below. Here we show the potential along y = Ls /4 evaluated at time t = 20s. Parameter settings are similar to those in Figure 4.7 of Chapter 4. Figure 4.16 shows overlap of all solutions for the four choices of boundary conditions at the extreme


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Figure 4.16: Normalized potential in the inlet, outlet, and spacer regions along y = Ls /4 at time t = 20 s for four possible choice of BCs at inlet and outlet. Results show overlap of all solutions. Parameter settings are similar to those in Figure 4.7 of Chapter 4. boundaries of our inlet and outlet conditions.

Chapter 5

Equilibria model for pH variations and ion adsorption in CDI electrodes Sections of this Chapter are based on article published in the journal of Water Research, [66] and are reproduced here with minor modifications.

5.1

Introduction

CDI is normally achieved by flowing a salt solution through or near porous carbon electrodes, where an applied bias induces ion migration and salt adsorption within the porous structures. [108, 132, 155, 193, 194] Often, the carbon pores are modeled as having a bimodal distribution of so-called macropores and micropores. The solution within macropores is assumed to be net neutral, and macropores are dominantly responsible for transport of ions. Ions are assumed to be stored in the electric double layers within (overlapping) micropores which dominate charge storage. [95, 162] The nature of micropore adsorption and equilibrium between micropores and macropores is thus at the heart of much of CDI research. An important micropore property that can dramatically impact this

108

CHAPTER 5. EQUILIBRIA MODEL FOR PH VARIATIONS ...

109

coupling is chemical surface charge functionalization. [103, 195–199] Studies on surface functional groups and the potential or pH at the point of zero charge (PZC) of activated carbon is a longstanding field of study. [168, 200–205] In CDI applications, however, the role of surface functional groups on enhancement of performance has been of significant interest only recently. [28, 198, 199, 206] Current CDI-based models treat chemical surface charge by considering a simple fixed (constant and permanent) charge density (of a certain sign and magnitude in Coulombs per volume) residing within the micropore volume. [103] This simple treatment of net surface charge has the effect of modifying the equilibrium charge composition in the micropores. An important implication of surface charge is that the porous material can electrostatically adsorb salt ions from the solution within macropores, sustaining macropore electroneutrality by altering the pH. Notably, this adsorption can occur with no externally applied electric potential. This fixed charge changes the degree of charging of the electric double layers and so fundamentally changes the CDI cell’s PZC and the relationship between charge efficiency and applied potential. For example, applied electrical potentials of a sign opposite to the fixed charge first expels ions of charge opposite to the fixed charge before reaching the PZC. Only after crossing the PZC, can electric charges effectively accumulate net charge of the same sign as the fixed charge. [28, 37, 103] Recently, Gao et al. [28] and Wu et al. [198] experimentally demonstrated increases in the charge efficiency, salt-absorption capacity, and operating voltage window of CDI devices with positive surface charge functionalization (treated with ethylenediamine or quaternized poly (4-vinylpyridine)) and negative surface charge (treated with nitric acid) functionalization. While the results presented by Gao et al. [28] are in general agreement with a simple fixed-charge amphoteric model for surface charge, [207] a detailed model based on basic physicochemical equilibria which includes pH and an arbitrary composition of acidic and basic surface charge sites within the micropores has yet to be developed. In this work, we aim to expand upon current models for understanding surface charge effects in porous carbon materials for CDI by coupling the modified Donnan (mD) model [97] with weak electrolyte acid-base equilibria theory. [208] Namely, we propose a treatment wherein micropores have a certain analytical concentration of acidic and basic groups with their own equilibrium constants (pK). We propose a model wherein the degree of dissociation of (immobile) functional groups within the micropores is governed by their acid dissociation constants, an equilibrium between


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micropores and macropores, and the pH of the external solution (e.g. in macropore). To validate our model, we perform well controlled titration experiments for porous carbons. The model assumes that dry electrodes are initially electroneutral, and then upon submersion into an aqueous solution develop a net surface charge due to dissociation or adsorption of protons according the micropore-tomacropore equilibrium. We describe the model and present experimental data demonstrating very good agreement between predictions of initial and final pH of a solution after adding the activated carbon sample, and for final background concentrations of individual ionic species.

5.2 5.2.1

Theory Multi-equilibria surface charge model

In recent work, Biesheuvel et al. [207] developed a model they termed amphoteric Donnan (amphD) model which assumes carbon surfaces with a fixed (constant) value for positive and negative surface charges. Their model is based on the assumptions that (1) the concentrations of positive and negative surface charges are each fixed and independent of pH, and (2) each group has its own (independent) Donnan potential and micropore-to-macropore Boltzmann distribution. The result of this formulation is that surfaces are assumed to always adsorb both positive and negative ions from solutions at equal rates independent of solution pH. In general, we recommend that models with constant surface charge (e.g. those that neglect the effects of pH on the degree of dissociation of surface charges) are not physical and have limited use in predicting adsorption capabilities across variations in pH and ionic strength. In the current work, we propose a model of surface groups based on first principles from weak electrolyte theory. We assume that the surfaces of micropores contain acid and base groups which can protonate or deprotonate according to their pK values and in equilibrium with the solution within micropores. We then assume a standard Boltzmann type equilibrium between the solution in the micropores and the local solution outside of micropores. In this way, the charge density of positive surface groups and negative surface groups is a function of solution pH and ionic strength. Our formulation is general and can be used to study the adsorption properties of activated carbon with arbitrary number of immobile (acidic or basic) surface functional groups.


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A schematic of our model is shown in Figure 5.1. We assume nX acidic surface groups and nY basic surface groups. Throughout this work, subscripts X and Y respectively correspond to acidic and basic surface functional groups. The degree of dissociation of functional groups (hence net chemical surface charge) is determined by the micropore pH environment and can be modeled with acid-base dissociation reactions for the form

Xi H Yj H+

KXi

KYj

+ X− i + Hm

Yj + H + m

(i = 1, ..., nX )

(5.1)

(j = 1, ..., nY )

(5.2)

+ are where Xi H is the i-th acidic functional group and Yj is the j-th basic group. X− i and Yj H

the associated conjugate base and conjugate acid, respectively. H+ m represents hydronium ion in the micropores. KXi and KYj are acid dissociation constants for the weak electrolytes and are defined as KXi = cX− · cm,H /cXi H

(5.3)

KYj = cYj · cm,H /cYj H+

(5.4)

i

where cX− , cXi H , cYj , and cYj H+ are volume-average concentration of functional groups (moles per i

micropore volume), and cm,H is hydronium micropore concentration. We follow Persat et al. [208] and use only the acid dissociation constants Ka (for both weak acid and weak base) and avoid using a base dissociation constant Kb (the subscript “a” is then dropped for this reason). For reference, Ka and Kb are related through Ka Kb = Kw , where Kw is the water autoprotolysis constant (10−14 M2 at 25◦C). The micropore volume is vm = pm ve , where pm is micropore porosity and ve is (macroscopic) electrode volume. For amount (moles) of each site (X− i , Yj , etc), one should multiply the concentrations shown here by the pore volume vm . We further follow the convention of Persat et al. [208] in which one only keeps track of hydronium (not hydroxide) ions in the formulations. Hydroxide concentration can, of course, be determined from the relation for water autoprotolysis in the bulk liquid, cOH = Kw /cH , and in the micropores, cm,OH = Kw /cm,H . The analytical concentration of each acidic or basic group (cXi ,0 and cYj ,0 ) is a fixed value for the sample and is equal to sum of the


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concentrations of each ionization state of the functional group (see Persat et al. [208]). Hence, we write cXi ,0 = NXi ,0 /vm = cXi H + cX−

(5.5)

cYj ,0 = NYj ,0 /vm = cYj H+ + cYj

(5.6)

i

Note, the concentrations here are per micropore volume. NXi ,0 and NYj ,0 are amount (moles) of acidic and basic groups of the sample, respectively. Chemical charge density of carbon (in units of Coulombs per micropore volume) is then

σchem =

X

σX− +

X

i

i

σYj H+ = −F

X

j

i

!−1 −1 X KYj cm,H cXi ,0 1 + +F cYj ,0 1 + KXi cm,H j

(5.7)

where σX− = −F cX− and σYj H+ = F cYj H+ are surface charges associated with i-th acidic and j-th i

i

basic functional group, respectively, and F is Faraday’s constant.

5.2.2

Electric double-layer model

We use an mD model in which the electric double layers (EDLs) in micropore volume are treated as strongly overlapping. [97, 136] This model thus assumes uniform ion concentration within the micropores. Concentration of protons in micropores can then be related to that of macropores assuming a micropore-to-macropore equilibrium following a Boltzmann distribution of the form

cm,H = cH exp −∆φD /VT ,

(5.8)

where ∆φD is Donnan potential drop between the micropores and the adjacent (local) electrolyte solution and VT is thermal voltage. By taking logarithm of both sides and multiplying by negative one, Equation 5.8 can be transformed to

pHm = pH +

∆φD ln(10)VT

(5.9)

where pHm and pH are respectively the pH value at micropores and the (local) net neutral electrolyte within macropores. Equation 5.9 shows that micropore pH value is linked with electrolyte pH


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through Donnan potential. Note that we explicitly assume that the micropore pH (and not the electrolyte pH) directly determines the degree of dissociation of functional groups, although all quantities are coupled according to Equation 5.9. Also, note the Donnan potential is only negligible in the high ionic strength limit. So, for a finite ionic strength electrolyte, the measurable pH (electrolyte pH) does not by itself determine the degree of dissociation of functional groups. We discuss this in more detail in Section 5.4.1. Throughout this chapter, we assume a binary and fully dissociated salt such that

cm,± = c± exp −z± ∆φD /VT

(5.10)

where subscripts + and − correspond to cationic and anionic species, respectively. So, cm,+ , for example, is the micropore concentration of the salt cation, and c+ is electrolyte concentration of salt cation (outside the micropores) with valence z+ . The formulation here can easily be extended to arbitrary (non-binary) weak electrolytes in solution by taking into account the dissociation constant of dissolved species (similar to Equations 5.3 and 5.3). Ionic charge density in the micropores (in units of Coulombs per micropore volume) can then be written as

σionic = F

X

zk cm,k

(5.11)

k

where the summation is over all ionic species (k = H+ , OH− , +, −). The third form of charge (besides the chemical σchem and ionic σionic ) is the electronic charge. In the simplest form, electronic charge in carbon matrix adjacent to acidic or basic sites is a linear function of micropore potential drop ∆φm (potential difference between electrode surface and center of micropore) as

σelec = Cm ∆φm ,

(5.12)

where Cm is volumetric capacitance of micropores. Multiplying Cm by vm , we arrive at total capacitance of the carbon electrode. The chemical, ionic, and electronic charges add up to zero (charge compensation) as in σelec + σchem + σionic = 0

(5.13)


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Figure 5.1: Schematic of our acid-base equilibria model for activated carbon with bimodal (micropore and macropore) pores. Here, the carbon is electrically floating (not connected to an electrical source) and in equilibrium with the solution in a beaker. Electrolyte (outside carbon) and macropores (pathways for ion transport) are electroneutral. Micropores, however, are ion adsorbing regions wherein electronic, ionic, and surface charges (due to acid-base equilibrium) are in balance with zero net charge. We here depict a special case of one acidic and one basic surface functional group in the micropores. Note the global net neutrality of surface charges, charge in solution, and net surface electrons within the micropore. The potential difference between carbon matrix and electrolyte solution (external voltage Vext ) is related to micropore potential and Donnan potential as

Vext = ∆φm + ∆φD

(5.14)

In the case of a single floating electrode (one electrode with no external voltage applied), electronic charge vanishes and we have σelec = ∆φm = 0. The voltage drop across electrode and electrolyte solution is then simply Donnan potential ∆φD .

5.2.3

Mass conservation

The model developed above is applicable in the case of negligible Faradaic reactions, i.e. carbon at zero or sufficiently low applied voltage. In this work, we consider ion adsorption behavior of activated carbon electrodes disconnected from any power source and in equilibrium with a local solution as shown in Figure 5.1. The sample is in contact with a (binary) electrolyte solution with initial cation concentration of c+,0 and anion concentration of c−,0 (see Figure 5.1). We then seek a prediction of the finial concentration of species (c+ and c− ), salt adsorption, micropore and electrolyte pH, and degree of dissociation of functional groups.


115

To begin, electroneutrality holds in the electrolyte in contact with the porous material such that

z+ c+ + z− c− + cH − Kw /cH = 0

(5.15)

Next is the mass conservation for individual ionic species (c+ and c− here). Upon the contact between carbon and electrolyte, the ions redistribute between the electrolyte and micropore volumes such that vsol c±,0 = (vsol − vm )c± + vm cm,±

(5.16)

0 where vsol is total volume of the electrolyte. For example, in a titration experiment, vsol = vsol + 0 vtitrant , where vsol is initial volume of electrolyte and vtitrant is volume of added titrant. The left-

hand side of Equation 5.16 16 is total amount (moles) of each ion. The first and second term in the right-hand side are moles of each ion present respectively in the electrolyte and in the acidic/basic micropores. Note that for binary salt, we can define micropore cationic and anionic salt adsorption as Γ± = vm (cm,± − c±,0 ). So, Γ+ and Γ− are respectively amount (in moles) of adsorbed cations and anions. The set of equations in Sections 5.2.1 to 5.2.3 fully describe the steady state solution of the problem with cH , c+ , c− , ∆φD , and ∆φm as unknowns. For our case of floating electrode, we drop Equation 5.14 and set σelec = 0 and ∆φm = 0. Also, note that for the special case of ideal solution p (infinite dilution) limit, c+ , c− → 0, and so cH = Kw (pH = 7) in the electrolyte according to electroneutrality condition. The set of Equations 5.13 and 5.14 in ideal solution limit then result in non-trivial (non-zero) solutions for ∆φD , and ∆φm . Refer to Section 5.6.1 for further discussion.

5.3 5.3.1

Materials and methods Materials

We used a commercially available activated carbon cloth (ACC) (Zorflex FM50K, Calgon Carbon Corp., PA, USA) with 0.26 g cm−3 volumetric density. We modified pristine ACC with nitric acid (70% w/w, Sigma-Aldrich) solution to increase the surface acidity and to remove residual surface impurities as much as possible. To this end, we immersed ACC into nitric acid at room temperature


116

for 2 h. We then rinsed ACC thoroughly with deionized water (DI) multiple times. The electrodes were soaked in DI overnight and dried at 150◦C in a convection oven. We used this nitric acid treated activated carbon cloth (N-ACC) in all experiments in this work.

5.3.2

Experimental setup

We prepared several solutions of NaCl with a wide pH range using 1 M HCl and NaOH stock solutions. To this end, we filled two sets of 14 containers (28 in total) with 20 mM NaCl solution (pH∼7, purged with nitrogen gas for 1 h) and added volumes of HCl or NaOH ranging from ∼ 3 µL to ∼ 1 mL to each container. The final volume of solution in each container was 25 mL. Our pH meter was an Omega PHE-3700 probe connected to a PHH-37 pH meter (Omega Engineering, CT, USA). The measured pH values of solutions were 1.76, 2.01, 2.25, 2.69, 3.01, 3.87, 5.62, 9.33, 10.83, 11.36, 11.61, 11.92, 12.16, and 12.41. We added 0.75 g of N-ACC to each container of the first set (sample solutions). No N-ACC was added to the second set (control solutions). Control and sample solutions were sealed and gently agitated for 14 h. We measured pH value and individual ion content of the solutions after this equilibration. Sodium and chloride ion contents were measured respectively by inductively coupled plasma mass spectrometry (ICP-MS X Series II, Thermo Scientific, MA, USA) and ion chromatography (DIONEX DX 500, DIONEX, CA, USA) at Stanford ICP-MS/TIMS Facility.

5.4 5.4.1

Results and discussion Model results and physical insights

In this section, we focus on model predictions in an attempt to gain insight into salt adsorption and surface charge trends for carbon under various pH environments.

5.4.1.1

Effect of analytical concentration and pK ’s of surface groups on titration curves

First, we note that our model has 2(nX + nY ) free parameters (cXi ,0 , cYj ,0 , KXi , and KYj for i = 1, ..., nX and j = 1, ..., nY ). To study the effect of each individual parameter, we model titration


117

of carbon with a strong acid and base (i.e. HCl and NaOH) and compare it to blank titration (in absence of carbon). Although our model allows for arbitrary number of weak electrolyte groups, for simplicity, we use nX = 1 and nY = 1, and consider dissociation reactions below

XH

YH+

KX

KY

X− + H + m

(5.17)

Y + H+ m

(5.18)

Figure 5.2 shows titration curves for four types of carbon, each with a single surface functional group with a fixed pKX or pKY . Two carbon samples have each a single acidic surface group with pKX = 4 or 10 (Figures 5.2a and 5.2b). The two others have each a basic surface group with pKY = 4 or 10 (Figures 5.2c and 5.2d). For each case, we show the results for three values of cX,0 or cY,0 , namely, 0.1, 0.5, and 1 M (solid lines). Dashed lines show blank titration curves. We here used an initial salt concentration c0 = 20 mM, final solution volume vsol = 25 mL, and titrant concentration of cstock = 1 M (same for acid and base titrants). Other parameters used to generate these plots are listed in Table 5.1. Refer to Section 5.6.2 for details of this titration model. Figure 5.2 summarizes basic features of model predictions which are very useful in interpreting titration experiments. In particular, the qualitative shape of titration curves help us individually identify if a sample has primarily weak acid groups, primarily weak basic groups, multiple weak acid, multiple weak bases, or combination of weak acids or weak bases. For example, when titrated with base, the acidic carbon (Figures 5.2a and 5.2b) result in a negative surface charge. The deprotonation of the surface charge, through macropore-to-micropore equilibrium, drives a net recruitment of cations from solution and tend to lower the final pH of the solution (pHf ). Titrating an acidic surface sample with strong acid, results in negligible effect on pH curve of the titration as expected. We also note that higher values of the analytical concentration of the weak acid cX,0 result in a stronger decrease in pH compared to blank titration. Comparison of Figures 5.2a and 5.2b, highlights the effect of the absolute value of pK of the acidic group. Note how a more acidic (lower pK) surface group results in a more profound departure of the titration curve from the blank at mid-range pH values. The insets of Figures 5.2a and 5.2b show change in concentration of univalent anions and cations in the solution outside of the micropores (∆c± = c± − c±,0 ) versus pHf . Here, c±,0 is initial


118

Table 5.1: Model parameters used in titration model and Figure 5.2 Parameter vm me 0 vsol cstock z±

Description Micropore volume Electrode mass Initial volume of solution Titrant (HCl, NaOH) stock concentration Cation and anion valence

Value 0.5 0.75 25 1 ±1

Units mL g mL M

concentration of species (right after addition of titrant), and c± is their concentration after carbon and electrolyte are equilibrated. As expected, we observe preferential adsorption of cations at pHf > pKX and negligible expulsion of anions throughout the titration. Moreover, the cation adsorption increases with analytical concentration of acidic groups cX,0 (see arrows). Titration of basic carbon (Figures 5.2c and 5.2d) with strong acid, on the other hand, results in pHf values higher than blank curve and a net positive surface charge. This departure from blank is pronounced for pHf < pKY . We again stress that KY and pKY are the acidic K and pK of weak base Y (see Equation 5.4 for definition). For example, a stronger weak base Y corresponds to lower KY and higher pKY value. The insets of Figures 5.2c and 5.2d show adsorption of anions at pHf < pKY and negligible expulsion of cations throughout the titration. The anion adsorption here is primarily due to net positive surface charge of basic carbon. The preferential adsorption of cations and anions is directly due to the ability of acidic and basic functional groups to protonate or deprotonate according to their pK value. As an example, the model developed by Biesheuvel et al., [103] would predict constant and equal values of anion versus cation adsorption (and thus constant net salt adsorption) independent of pH. We discuss the ion adsorption in more detail in the next section.

5.4.1.2

Effect of ionic strength on macropore-to-micropore equilibrium

We here explore the effect of ionic strength on pH variations and surface chemical charge in the current model. Figure 5.3a shows final pH of micropore (pHm,f ) versus final pH of solution (pHf ) for initial salt concentrations in the range of 0 to 500 mM (solid lines). We assume titration of a


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Figure 5.2: Summary of characteristic shapes of predicted titration curves for samples with weak acids and weak bases for porous carbon samples. (a) and (b) each consider a single acidic functional group, and respective pKX are 4 and 10, while (c) and (d) each consider a single basic functional group with respective pKY of 4 and 10. Titrants are 1 M HCl and 1 M NaOH. We show results for cX,0 , cY,0 = 0.1, 0.5, and 1 M (solid lines). Dashed lines indicate blank titrations (cX,0 = cY,0 = 0). Insets show change in concentration of anion/cation (∆c± = c± − c±,0 ) as a function of pHf . Initial salt concentration and solution volume are c0 = 20 M and vs ol = 25 mL. Other parameters are listed in Table 5.1.


120

carbon with one acidic (cX,0 = 1 M, pKX = 4) and one basic (cY,0 = 1 M, pKY = 10) surface group. Other parameters used are identical to those of Figure 5.2. Dashed line shows where pHm,f = pHf . Note that intersection of solid and dashed lines is actually the pH of the PZC (pHPZC ), which also corresponds to a pH where ∆φD = 0 and σchem = 0. Results show that pHm,f is lower (higher) than pHf above (below) pHPZC . The surface functional groups of micropore (which act as weak electrolyte) oppose variations in pHm,f compared to pHf . Further, we note that pHm,f approaches pHf only at high ionic strengths. This is a feature not observed in simple buffers and is strictly resulting from the macropore-to-micropore equilibrium. As we will discuss later in this section, this dependence on ionic strength is indicative of a shift in the adsorption dynamics from net salt adsorption to ion swapping. Also, note that as per Equation 5.8, the Donnan potential ∆φD approaches zero in the limit of high ionic strength. The difference between pHm,f and pHf implies that the measured pH of electrolyte (pHf ) may not accurately represent micropore environment, and consequently, surface pK and surface charge. We illustrate this in Figures 5.3b and 5.3c, where we plot concentration of charged acidic and basic groups (cX− and cYH+ ) as a function of pHf and pHm,f . When plotted against pHm,f , cX− and cYH+ artificially appear to be strong functions of initial salt concentration c0 (see Figure 5.3b). Note that the observed pK’s (half-dissociation points) in this case approach the actual values (pKX = 4 and pKY = 10) only in the limit of c0 → ∞. On the other hand, Figure 5.3c shows that all cX− and cYH+ curves collapse when plotted against pHm,f on which they depend strongly. We note that dissociation constants are here assumed to be constant. Ideally, dissociation constants are corrected for ionic strength, for example using a Davies-type equation. [208] Further, dissociation constants might be functions of carbon electric potential as well. We chose to not to adopt such corrections in this work in order to simplify the presentation and because (1) our current focus is the effect of ionic strength on differences in electrolyte and micropore environments through adsorption process, (2) the applicability of the Davies equation for correction of pK values within a micropore (and overlapped double layers) is an open question. We hope to study this further in future work. Throughout this section, for simplicity of presentation, we considered a symmetric case for carbon with cX,0 = cY,0 , pKX = 4, and pKY = 10. Hence, the results are symmetric around pHPZC = 7.


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Figure 5.3: Ionic strength affects the equilibrium between carbon micropores and the electrolyte (outside the micropore). (a) Final micropore pH (pHm,f ) versus final electrolyte pH (pHf ) for titration model with c0 = 0, 1, 20, 100, and 500 mM. Surface parameters are cX,0 = 1 M, pKX = 4, cY,0 = 1 M, and pKY = 10, and other parameters are similar to those of Figure 5.2. pHm,f is lower (higher) than pHf above (below) PZC (pHPZC = 7 here), and pHm,f approaches pHf only at high ionic strengths. (b) Concentration of charged acidic and basic groups (cX− and cYH+ ) versus pHf and (c) versus pHm,f . Curves for cX− and cYH+ collapse onto a single curve when plotted against pHm,f , as per our formulation. In Section 5.6.3, we present the results for an asymmetric carbon with cX,0 = cY,0 and pKX = 4 (as before), but with pKY = 8. 5.4.1.3

Determination of point of zero charge

As mentioned earlier, PZC corresponds to σchem = 0 and ∆φD = 0. We here discuss the effect of dissociation constant of surface groups and their concentrations as well as ionic strength on pHPZC for a simple case of one acidic and one basic surface functional group. We also provide insights for the general case. PZC in our simple case requires

−

cX,0 cY,0 + =0 1 + cH /KX 1 + KY /cH

(5.19)

which can be recast as a quadratic equation in cH with a unique real positive root (from Descartes’ rule of signs). The first term in Equation 5.19 is the (negative) surface charge associated with acidic group (σX− ), and the second term is (positive) charge due to basic group (σYH+ ). By assuming fixed dissociation constants, Equation 5.19 and hence pHPZC are independent of ionic strength (see


122

Figure 5.3a). For a general case of nX acidic and nY basic groups, the resulting equation is a polynomial of degree nX + nY . We solve Equation 5.19 for cX,0 > cY,0 and cX,0 < cY,0 , take the logarithm of the solution, and arrive at

pHPZC =

  pKX − log α1 − log ε

cX,0 > cY.0

 pKY + log α2 + log ε

cX,0 < cY,0

(5.20)

where 0 < α1 = cX,0 /cY,0 − 1 and 0 < α2 = cY,0 /cX,0 − 1, and ε can be written as

ε=

p 1 1 + 1 + 4(α + 1)/α2 · KY /KX , 2

(5.21)

where α is either α1 or α2 (both result in the same value of ε). Equation 5.20 generalizes the effect of individual pK’s and analytical concentrations (cX,0 and cY,0 ) on PZC. In special case of cX,0 = cY,0 (i.e. in the limit of α1 , α2 → 0), we arrive at the simple result that pHPZC = (pKX + pKY )/2 (which may be familiar to readers experienced in buffer calculations). In Figure 5.9, we show pHPZC as a function of α1 and α2 for pKX = 4 and pKY = 10. We show that pHPZC decreases with α1 (where cX,0 > cY,0 and increases with α2 (where cX,0 < cY,0 ), and further demonstrate that, for α1 α0 (where α0 is some threshold that depends solely on pKX and pKY ), ε approaches unity and according to Equation 20, pHPZC is independent of pKY . In the case where α2 α0 , ε approaches unity but then pHPZC is independent of pKX . Refer to Section 5.6.4 for more information. These observations help in quick estimates of pHPZC given some knowledge of surface pK values. 5.4.1.4

Chemical charge efficiency: a parameter for salt removal efficiency due to surface charge

Lastly, we discuss the effect of pH and ionic strength on adsorption mechanisms and define parameters to identify them. Figure 5.4a shows net salt adsorbed Γchem as a function of pHm,f for a carbon with one acidic (cX,0 = 1 M, pKX = 4) and one basic (cY,0 = 1 M, pKY = 10) surface groups at various initial salt concentration c0 in the range of 0 to 500 mM. Other parameters used are identical


123

to those of Figure 5.2. Salt removal Γ (in units of moles) for a univalent electrolyte is calculated as

Γ = Γ+ + Γ− = vm (cm,+ − c+,0 ) + (cm,− − c−,0 )

(5.22)

where Γ+ and Γ− are cationic and anionic salt adsorption (sodium and chloride ions, for instance). We stress that c+,0 and c−,0 differ from initial salt concentration c0 . These are theoretical cation and anion concentrations the solution would obtain without carbon-electrolyte equilibration, while c0 is an initial salt concentration in the absence of added titrant. Results show considerable salt adsorption at pHm,f < pKX and pHm,f > pKY and much lower adsorption otherwise. This expected trend is in accordance with higher net negative (positive) micropore surface charge at pH values lower (higher) than pKX (pKY ). In Section 5.4.2, we show experimental results which support this observation. Figure 5.4a also shows that at a fixed pH environment (fixed pHm,f ), salt adsorption Γ decreases with increase in c0 (or equivalently, initial ionic strength). For example, at pHm,f = 3, salt adsorption is almost halved when initial salt concentration increases from 20 to 500 mM. This result can be attributed to a higher degree of expulsion of co-ions (ions with like charge relative to that of surface groups) at higher initial concentrations. At lower salt concentrations, co-ion expulsion is less important than counter-ion recruitment into the micropore. The two insets illustrate this point. The insets show individual cationic and anionic salt adsorption Γ+ and Γ− for c0 = 20 and 500 mM cases. At low ionic strengths, counter-ion adsorption (i.e. adsorption of cations at high pHm,f and anions at low pHm,f ) dominates relative to co-ion expulsion. For example, Γ+ |Γ− | at pHm,f = 11 and |Γ+ | Γ− at pHm,f = 3. Conversely, at high ionic strengths corresponding to small Donnan potentials, counter-ion adsorption and co-ion expulsion are of the same order (|Γ+ |≈ |Γ− |). In other words, low ionic strength can be associated with counter-ion recruitment and salt adsorption, while high ionic strength leads to increased ion swapping. We here propose a characterization of this effect by defining a signed “chemical charge efficiency” parameter as

Λchem =

FΓ F (Γ+ + Γ− ) = , Σchem vm σchem

(5.23)

where Σchem is net micropore surface charge in units of Coulombs. Figure 5.4b shows chemical charge efficiency as a function of pHm,f for parameters identical to those of Figure 5.4a. Results show that


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at pHm,f = pHPZC , Λchem = 0, and associated zero anion or cation adsorption. At pHm,f > pHPZC , cation adsorption is greater than anion expulsion (Γ+ > |Γ− —), and the surface charge is positive (Σchem > 0), and hence |Λchem |> 0. At pHm,f < pHPZC , however, cation adsorption is lower than anion expulsion (|Γ+ |< Γ− ), surface charge is negative (Σchem < 0), and so |Λchem |< 0. Further, chemical charge efficiency approaches unity (in magnitude) at low ionic strengths, since counter-ion adsorption is much greater than co-ion expulsion. For pHm,f values very close to pHPZC , however, chemical charge density and adsorbed salt both vanish. At this limit, adsorbed salt Γ approaches zero faster than surface charge Σchem (in magnitude) such that chemical charge efficiency (FΓchem /Σchem ratio) approaches zero (see Figure 5.4.). We note that negative chemical charge efficiency here does not mean ion expulsion dominates ion adsorption (note, Γ > 0 for all ionic strengths after all), nor does it imply that electrolyte ionic strength (outside carbon micropores) increases to a higher value. A negative chemical charge efficiency simply indicates that net micropore surface charge Σchem (or equivalently, surface chemical charge density œchem ) is negative.

Figure 5.4: Effect of ionic strength and micropore pH environment on salt adsorption dynamics. (a) Net adsorbed salt Γ = Γ+ + Γ− as a function of pHm,f for a carbon with one acidic (cX,0 = 1 M, pKX = 4) and one basic (cY,0 = 1 M, pKY = 10) surface groups for c0 = 0, 20, 100, and 500 mM. Other parameters are identical to those of Figure 5.2. Salt adsorption is considerable at pHm,f < pHX and pHm,f > pHY . Further, the magnitude of salt adsorption decreases with increasing ionic strength (or equivalently with increasing c0 ). The insets show individual cationic and anionic salt adsorption (Γ+ and Γ− ) for c0 =20 and 500 mM. (b) Chemical charge efficiency versus pHm,f for the same c0 values as in (a). At low ionic strengths, counter-ion adsorption (cations at high pHm,f and anions at low pHm,f ) dominates co-ion expulsion and so |Λchem |→ 1. At high ionic strengths, however, counter-ion adsorption and co-ion expulsion are of the same order (ion swapping) and hence |Λchem |→ 0.


5.4.2

125

Experimental results and model verification

As described in Section 5.3.2, we prepared 28 containers of NaCl solutions with a wide pH range by adding 1 M HCl or NaOH to 20 mM NaCl electrolyte. Measured pH values were in the range of 1.76 to 12.41 and final volume of each container was 25 mL. We added 0.75 g of N-ACC samples to half of the containers. The rest of containers were used as control (no N-ACC added). We measured final pH value and individual ion concentration of the solutions after 14 h agitation. Figure 5.5a shows final pH of electrolyte (pHf ) as a function of initial pH of solution (pH0 ) for both experiments and model. Circle and square markers show the results for N-ACC samples and control experiments, respectively. Dashed line represents pHf = pH0 . Control experiments (square markers) very closely follow the dashed line, indicating trace dissolved carbon dioxide results in a minimal drift in pH (all samples purged with nitrogen gas and carefully sealed). Solutions with N-ACC samples (circle markers), however, noticeably deviate from dashed line. pHf is higher than pH0 for pH values below ∼ 5, and is lower than pH0 for pH values above ∼ 5. This suggests that N-ACC has pHPZC ≈ 5 , as it behaves as a base at pH0 < 5 and behaves as acid at pH0 > 5. Figure 5.5b shows change in concentration of sodium and chloride ions versus pHf for the same experiments as in Figure 5.5a. Results show strong adsorption of chloride ions at low pH (pHf < 5) and strong adsorption of sodium ions otherwise. Based the experimental measurements above, and for simplicity, we modeled N-ACC surface as an ion adsorbent with one acidic and one basic group. Solid lines in Figures 5.5a and 5.5b show the model predictions using cX,0 = 1.17 M, cY,0 = 1.1 M, pKX = 4, and pKY = 8 as free parameters. We found values of free parameters by minimizing the absolute value of the error between experiments and model. We list the parameters and other constants used in Table 5.2. Results show a good agreement between experiments and model and suggest that a simple two pK model can predict pH shifts and ion adsorption characteristics of N-ACC samples. Goodness of fit to the model is evaluated using coefficient of determination (R2 ). R2 is 0.95 and 0.998 for sample and control experiments (Figure 5.5a) and is 0.94 and 0.93 for cation and anion concentration change (Figure 5.5b).


126

Figure 5.5: Model and experimental results of pH shift and ion adsorption. (a) Final electrolyte pH (pHf ) versus initial pH (pH0 ) for N-ACC samples (circles), control samples (squares), and fitted model (solid line). Dashed line shows pHf = pH0 . pHf for N-ACC samples is higher than pH0 for pH0 < 5, and is lower than pH0 for pH0 > 5. N-ACC therefore is an ion adsorbent with pHPZC ≈ 5. (b) Change in concentration of sodium and chloride ions versus pHf for experiments and fitted model. Results show specific adsorption of chloride ions at pHf < pHPZC and strong adsorption of sodium ions otherwise. We modeled N-ACC with cX,0 = 1.17 M, cY,0 = 1.1 M, pKX = 4, and pKY = 8. Other parameters used in the model are listed in Table 5.2.

Table 5.2: Parameters used in model for N-ACC samples Parameter cX,0 cY,0 pKX pKY vm ve vsol cstock z± c0

Description Analytical concentration of acidic group Analytical concentration of basic group Acid dissociation constant Base dissociation constant Micropore volume Electrode volume Final volume of solution Titrant (HCl, NaOH) stock concentration Cation and anion valence Initial salt concentration

Value 1.17 1.1 4 8 0.5 2.9 25 1 ±1 20

Units M M

mL mL mL M mM


5.4.3

127

Prediction of N-ACC surface charge

We used the fitted model above to estimate the surface charge of N-ACC samples. Figure 5.6a shows surface charge density (σchem ) as well as its associated acidic and basic surface charges (σX− and σYH+ ) as a function of micropore pH (pHm,f ). We here used c0 = 20 mM (similar to experiments). Other c0 values result in identical curves. We again stress that c0 is initial dissolved salt concentration in the absence of added titrant. Figure 5.6b shows σX− , σYH+ , and σchem as a function of final electrolyte pH (pHf ) for c0 values in the range of 0 to 500 mM. This figure shows that when plotted against pHf , surface charges (σX− , σYH+ , and σchem ) artificially appear to be strong functions of initial salt concentration c0 . Left and right axes show surface charge densities in units of C cm−3 and M, respectively. Note that σchem = σX− + σYH+ according to Equation 7. The magnitudes of σX− and σYH+ each strictly decrease with pHf and result in a net positive surface charge for pHf < 5 and net negative charge for pHf > 5. This is in accordance with specific adsorption of anions (cations) at pHf values lower (higher) than ∼ 5 observed in experiments. Other parameters used in calculation of charge densities are listed in Table 5.2. We note that in CDI operation, the electrodes usually experience voltage windows of around −1.2 to 1.2 V. This applied voltage range has been shown [139, 140] to profoundly change pH of effluent solution (up to 5 units). Therefore, since surface charge density σchem is pH dependent, it inevitably varies in a CDI cycle. This highlights the importance of CDI models where surface charge is allowed to vary based on surface dissociation constants.

5.4.4


We here present a proposed generalization of the often cited and reported charge efficiency parameter for CDI systems. The charge efficiency is typically defined as ratio of adsorbed salt to electronic charge as in

FΓ Σchem ,

[136, 188, 196, 197] where Σchem = vm σelec is electronic charge in units of

Coulombs. However, the common expressions for charge efficiency are only function of external voltage, initial salt concentration, and electrode capacitance, and thus do not account for surface charge effects. Hence, we here propose the following generalization for charge efficiency including functional groups with surface dissociation constants (pK). For simplicity, we assume that the electrolyte volume is large enough such that (1) initial and final electrolyte salt concentrations


128

Figure 5.6: Prediction of net, acidic, and basic surface charge densities (σchem , σX− , and σYH+ ) of N-ACC samples in units of C cm−3 (left axis) and M (right axis) versus (a) micropore pH (pHm,f ) and (b) final electrolyte pH (pHf ). Net surface charge density is sum of acidic and basic surface densities. pHPZC (where σchem = 0) is ∼ 5 for our N-ACC. Parameters used in calculation of charge densities are listed in Table 5.2. are approximately equal to each other and constant (c+ = c+,0 and c− = c−,0 ), (2) initial and final pH of electrolyte are equal, and (3) pH is moderate (4 to 10) [208] such that salt anion (cation) concentration is considerably larger than hydronium (hydroxide) concentrations (e.g., salt concentration in CDI is typically 10−100 mM whereas cH = 0.1 mM at pH = 4). We then substitute Equations 5.7, 5.11, and 5.12 into Equation 5.13 and solve for ∆φD . Salt concentration Γ can now be written as 2c0 vm (cosh ∆φD /VT − 1) (see Equation 5.22). Refer to Section 5.5 for more detailed derivations. We then define the generalized charge efficiency as F (Γ − ΓPZC ) 2F c0 vm cosh ∆φD /VT − cosh VPZC /VT Λ= = · Σelec Cm Vext − ∆φD

(5.24)

where ΓPZC is salt adsorbed at potential of zero charge VPZC (i.e. potential at which Σelec = 0) and can be written as 2c0 vm (cosh VPZC /VT − 1). Note that charge efficiency is now a function of external voltage, initial pH, ionic strength (or c0 ), and surface properties (i.e. cX,0 , cY,0 , pKX , and pKY ). We also note that due to presence of surface charge, it is possible for Γ to have a non-zero value when Σelec = 0 (i.e. at point of zero charge). Hence, Equation 5.24 defines generalized charge efficiency in terms of the salt adsorbed relative to a base value of ΓPZC . Hence, we formulate in terms of the difference Γ −ΓPZC in the numerator (this also ensures that the quantity Λ is bounded). In Section 5.6.5, we show plots of adsorbed salt and electronic charge as well as generalized charge


129

efficiency vs Vext in the range of −0.5 to 0.5 V. We present these results and formulation as a purely theoretical description of generalized charge efficiency which we hope to experimentally validate as part of future work, including experiments of full CDI cells with electrodes with significant surface charge.

5.5

Conclusions

We here expanded the current CDI-based surface charge models by coupling the mD model with weak electrolyte acid-base equilibria theory. Our model is based on a more physical treatment of the surface functional groups wherein the surface groups can protonate or deprotonate based on their individual pK values and micropore pH environment. Our formulation is applicable to arbitrary number of immobile (acidic or basic) surface functional groups. We applied our model to titration of activated carbon and showed (1) specific adsorption of cations and expulsion of anions at electrolyte pH values higher than pK of acidic groups, and (2) specific adsorption of anion and expulsion of cations at pH values lower than pK of basic groups. We studied equilibrium between micropores and macropores and proposed a model wherein micropore pH (and not directly the electrolyte pH) directly determines the degree of dissociation of functional groups. Hence, the electrolyte pH alone does not reflect ionization state of functional groups. This feature is strictly resulting from the macropore-to-micropore equilibrium and is not observed in simple buffers calculations. We further showed that this equilibrium is ionic strength dependent and is indicative of a shift in the adsorption dynamics from net salt adsorption to ion swapping. We introduced a new parameter we call hemical charge efficiency and showed that (1) it approaches unity (in magnitude) at low ionic strengths, where counter-ion adsorption dominates co-ion expulsion, and (2) it vanishes at high ionic strengths, where counter-ion adsorption and co-ion expulsion are of the same order (ion swapping). We validated our model by comparing predictions to experimental measurements from well controlled titration experiments of N-ACC samples. We measured initial and final pH of electrolyte after adding N-ACC samples as well as initial and final concentrations of individual ionic species (sodium and chloride) inductively coupled plasma mass spectrometry (ICP-MS) and ion chromatography (IC). Our fitted model with one acidic and one basic surface group showed a very good agreement with the experiments (with cX,0 = 1.17 M, cY,0 = 1 M, pKX = 4, and pKY = 8). Results showed strong


130

adsorption of chloride ions at pHf < 5 and strong adsorption of sodium ions otherwise. Finally, we used our model to estimate the surface charge of N-ACC samples and showed that net surface charge is negative (acidic) at and is positive (basic) otherwise.


5.6

131

Addendum to: Equilibria model for pH variations and ion adsorption in CDI electrodes

5.6.1

Ideal solution limit

We here discuss adsorption and surface charge dynamics in the ideal solution limit (infinite dilution where c+ = c− = 0) with and without external voltage. Electroneutrality condition in the ideal p solution limit requires pH of electrolyte after equilibration with carbon to be 7 (cH = Kw ). So, the micropore concentration of hydronium is

p cm,H = Kw exp −∆φD /VT ,

(5.25)

and ionic charge density in the micropores can be written as

p σionic = F (cm,H − Kw /cm,H ) = −2F Kw sinh ∆φD /VT .

(5.26)

For the case where external voltage Vext is applied, charge compensation requires σchem + σionic + σelec = 0. So, we substitute Equations 5.25 and 5.26 into charge compensation equation and arrive at a single equation below for Donnan potential

−

nX X

cXi ,0

+

p

nY X

cYi ,0 KY i

K exp ∆φD /VT j=1 1 + p 1 + K w exp −∆φD /VT Kw Xi p Cm − 2 Kw sinh ∆φD /VT + (Vext − ∆φD ) = 0. F

i =1

(5.27)

Note, the left-hand side of Equation 5.27 strictly decreases with increasing ∆φD . So, this equation has at most one real solution for any Vext . In the absence of applied voltage (floating electrode) electronic charge σelec vanishes and thus micropore ionic charge and surface chemical charge compensate each other (i.e., σchem + σionic = 0). Equation 5.27 can then be simplified to

−

nX X i=1

n

Y X cYj ,0 cXi ,0 + + cm,H − Kw /cm,H = 0. 1 + cm,H /KXi j=1 1 + KYj /cm,H

(5.28)

Equation 5.28 can be recast as a polynomial of degree nX + nY + 2. Similarly, note each term in


132

Equation 5.28 is strictly increasing in cm,H and thus this equation has at most one real solution.

5.6.2

Titration model

In Section 5.4.1 of Chapter 5, we showed results of titration model for carbon with acidic and/or basic surface charges. We here discuss titration model in more detail. We assume electrolyte solution 0 has initial volume of vsol (before addition of any titrant) and initial salt concentration of c0 . We then

add either strong acid or base titrant of concentration cstock and volume vtitrant to the electrolyte solution. We define the pH value and hydronium concentration at this point (after addition of titrant but prior to the addition of carbon to the electrolyte) pH0 and cH,0 , respectively. Assuming the added titrant completely dissociates, electroneutrality requires that

c+,0 − c−,0 + cH,0 − Kw /cH,0 = 0,

(5.29)

where c+,0 and c−,0 are cation and anion concentrations right after addition of titrant and before carbon-electrolyte equilibration. Note, c0 is initial salt concentration with no titrant added. In case of titration with strong acid, c+,0 and c−,0 can be written as

c+,0 =

0 c0 vsol , 0 +v vsol titrant

(5.30a)

c−,0 =

0 c0 vsol + cstock vtitrant , 0 +v vsol titrant

(5.30b)

and in case of titration with strong base, as

c+,0 = c−,0 =

0 c0 vsol + cstock vtitrant , 0 +v vsol titrant 0 vsol

0 c0 vsol , + vtitrant

(5.31a) (5.31b)

Substituting Equations 5.30 and 5.31 back into Equation 5.29, we arrive at cH,0 − Kw /cH,0 v0 , cstock − (cH,0 − Kw /cH,0 ) sol cH,0 − Kw /cH,0 = v0 , −cstock − (cH,0 − Kw /cH,0 ) sol

vtitrant = vtitrant

(5.32a) (5.32b)


133

for acid and base titration, respectively. Now, for known pH0 values, we calculate c+,0 , c−,0 , and vtitrant using Equations 5.30 to 5.32. We then use these parameters as inputs to our multi-equilibria surface charge model (see Section 5.2 of Chapter 5). The electrolyte volume before addition of carbon 0 is vsol +vtitrant . The final electrolyte and micropore pH (pHf and pHm,f ) are then determined using

the governing equations discussed in Section 5.2 of Chapter 5.

5.6.3

Salt adsorption and micro-to-macropore equilibrium for asymmetric carbon

In model results section of Chapter 5 (Section 5.4.1), we discussed micropore and macropore electrostatic environments as well as salt adsorption and surface charge efficiency for symmetric case where cX,0 = cY,0 and pHPZC = (pKX + pKY )/2 = 7. We here show an asymmetric case where we set cX,0 = cY,0 = 1 M (as before), pKX = 4, and pKY = 8. Note, (pKX + pKY )/2 6= 7 here. For other parameters, refer to Table 5.1 of Chapter 5. Figure 5.7 shows pHm,f (final pH of micropore) versus pHf (final pH of solution) for initial salt concentrations in the range of 0 to 500 mM (solid lines). Dashed line shows pHm,f = pHf . Note that all curves coincide at pHPZC = 6 as expected, however, the plot is not symmetric around pHPZC . Moreover, similar to symmetric case, pHm,f approaches pHf only at high ionic strengths. Figures 5.7b and 5.7c show concentration of charged acidic and basic groups (cX− and cYH+ ) as a function of pHf and pHm,f . Results again show that cX− and cYH+ curves all collapse when plotted versus pHm,f (i.e. they are not functions of ionic strength). Additionally, note that cX− and cYH+ are not symmetric around pHPZC = 6 when plotted versus pHf , but are symmetric around pHPZC when plotted versus pHm,f . Figure 5.8a shows net salt adsorbed Γ as a function of pHm,f at different initial salt concentrations c0 in the range of 0 to 500 mM. Other parameters are identical to those of Figure 5.7. According to Figure 5.7c, micropore surface charge is more negative (positive) at pH values lower (higher) than pKX (pKY ). This explains the observation in Figure 5.8a that salt adsorption is considerable at pHm,f < pKX and pHm,f > pKY and is much lower otherwise. Moreover, Γ decreases with ionic strength (or with c0 ). These trends are similar to the symmetric case discussed in Chapter 5. However, there are


134

Figure 5.7: Effect of ionic strength on micropore-to-macropore equilibrium for asymmetric carbon. (a) pHm,f versus pHf for titration of an asymmetric carbon with cX,0 = cY,0 = 1 M, pKX = 4, and pKY = 8 at initial concentrations c0 = 0, 1, 20, 100, and 500 mM. pHm,f is lower (higher) than pHf above (below) pHPZC = 6. (b), (c) Concentration of charged acidic and basic groups (cX− and cYH+ ) versus pHf and versus pHm,f . Similar to symmetric carbon, each cX− and cYH+ collapses on a single curve when plotted against pHm,f . the following differences:(1) the minimum adsorption occurs at pHm,f = pHPZC = 6 rather than at pHm,f = 7, and (2) salt adsorption saturates at pHm,f several units larger than pKY (i.e. pHm,f > pKY + 2). The latter is because micropore surface charge saturates at pHm,f several units higher (lower) than pKY (pKX ). Figures 5.8b and 5.8c show individual cationic and anionic salt adsorption (Γ+ and Γ− ) for c0 = 20 and 500 mM cases, respectively. We again see that (1) counterion adsorption (i.e. adsorption of cations at high pHm,f and anions at low pHm,f ) dominates co-ion expulsion at low ionic strength, and (2) counter-ion adsorption and co-ion expulsion are of the same order at high ionic strengths. Chemical charge efficiency Λchem (see Figure 5.8d) shows the trend similar to symmetric case with the following differences: (1) PZC is moved to pHPZC = 6, and (2) Λchem is not symmetric around PZC.

5.6.4

Determination of point of zero charge

As discussed in Section 5.4.1, PZC corresponds to σchem = 0 and ∆φD = 0. This condition can be written as

nX X

σX− +

nY X

i

i=1

j=1

σYj H+ = −

nX X i=1

n

Y X cYj ,0 cXi ,0 + = 0. 1 + cH /KXi j=1 1 + KYj /cH

(5.33)


135

Figure 5.8: Effect of ionic strength and micropore pH environment on salt adsorption dynamics for carbon with asymmetric surface chemistry parameters of cX,0 = cY,0 = 1 M, pKX = 4, and pKY = 8. (a) Net adsorbed salt Γ = Γ+ + Γ− as a function of pHm,f for c0 = 0, 20, 100, and 500 mM. Salt adsorption is considerable at pHm,f < pKX and pHm,f > pKY and is much lower otherwise. Moreover, decreases with ionic strength (or with c0 ). (b), (c) Cationic and anionic salt adsorption (Γ+ and Γ− ) for c0 = 20 and 500 mM cases. (d) Chemical charge efficiency Λchem versus pHm,f . General trend of Λchem is similar to the case discussed in Chapter 5, however, Λchem is not symmetric about PZC (pHPZC = 6) here. where σX− and σYj H+ are surface charges associated with i-th acidic and j-th basic functional group. i

The first term is sum of the (negative) surface charges associated with acidic groups, and the second term is sum of (positive) charges due to basic groups. Note, Equation 5.33 is strictly increasing (since each term is strictly increasing) and so has at most one real root. Equation 5.33 can also be recast as a polynomial of degree nX + nY in cH . Exact value of PZC in general case then requires finding this root. PZC for nX = nY = 1, on the other hand, requires finding the real positive root of the quadratic below and is straightforward.

(cY,0 /KX )c2H + (cY,0 − cX,0 )cH − cX,0 KY = 0.

(5.34)

For a better representation of the results, we split the solution of this quadratic to two cases, namely, cX,0 > cY,0 and cX,0 < cY,0 . We define α1 = cX,0 /cY,0 − 1 for when cX,0 > cY,0 , and α2 = cY,0 /cX,0 − 1 for when cX,0 < cY,0 . So, α1 and α2 are both positive quantities. These two parameters are a measure of asymmetry in concentration of acidic and basic groups. For example, α1 = 0 and α2 = 0 correspond to cX,0 = cY,0 , and α1 > 0 and α2 > 0 correspond to cX,0 > cY,0 and


136

cX,0 < cY,0 , respectively. The solution can be written as

cH =

  KX α1 ε

cX,0 > cY.0

  KY /(α2 ε)

cX,0 < cY,0

(5.35)

where ε is ε=

p 1 1 + 1 + 4(α + 1)/α2 · KY /KX 2

(5.36)

and α is either α1 or α2 (they both result in the same ε). Taking logarithm of Equation 5.35, pHPZC can be written as pHPZC =

  pKX − log α1 − log ε

cX,0 > cY.0

 pKY + log α2 + log ε

cX,0 < cY,0

(5.37)

Figure 5.9 shows pHPZC versus α1 and α2 with pKX = 4 and pKY = 10. The results show that pHPZC decreases with α1 and increases with α2 . Taking the limit of Equation 5.37 as α1 → 0 (i.e. p cX,0 = cY,0 ), ε approaches KX /KY /α1 and we arrive at familiar equation pHPZC = (pKX +pKY )/2. On the other hand, according to Equation 5.36, ε approaches unity for large enough α (α1 or α2 ) and we have pHPZC =

  pKX − log α1

cX,0 > cY.0

 pKY + log α2

cX,0 < cY,0

(5.38)

For Equation 5.38 to be valid, we need (α + 1)/α2 · KY /KX 1, or equivalently,

α α0 =

1 KY p 1 + 1 + 4KX /KY 2 KX

(5.39)

So, for α1 α0 , ε approaches unity and pHPZC is independent of pKY . For α2 α0 , ε approaches unity and pHPZC becomes independent of pKX . The dashed lines in Figure 5.9 show extrapolation of pHPZC for α1 α0 and α2 α0 (as described by Equation 5.38). These derivations can help in quick estimates of pHPZC given some knowledge of surface pK values and analytical concentrations.


137

Figure 5.9: pHPZC as a function of α1 and α2 for pKX = 4 and pKY = 10 (solid lines). pHPZC decreases with α1 and increases with α2 . At small values of α1 and α2 (i.e., cX,0 = cY,0 ), pHPZC approaches (pKX + pKY )/2. At high values of α1 , on the other hand, pHPZC varies almost linearly with log (α1 ) as pHPZC = pKX − log (α1 ) and so pHPZC is independent of pKY . At high values of α2 , pHPZC is independent of pKX and varies linearly with log(α2 ) as pHPZC = log (α2 ) − pKY . Dashed lines are extrapolation of pHPZC for cases where α1 α0 and α2 α0 .

5.6.5


We introduced a generalized charge efficiency for CDI systems with functionalized surface charges in Section 5.4.4 of Chapter 5. We here present details of derivation. As mentioned in Chapter 5, we assume electrolyte volume is large enough and thus (1) initial and final electrolyte salt ion concentrations are equal to each other and constant (c+ = c+,0 and c− = c−,0 ), (2) initial and final pH of electrolyte are equal, and (3) pH is moderate such that salt concentration is considerably larger than hydronium or hydroxide. The latter can be justified by the fact that salt concentration in CDI is typically in 10-100 mM range, while even at pH = 4, cH = 0.1 mM. So, cH is at least two


138

orders of magnitude less than salt concentration. Thus, we can write Kw σionic ≈ F −2c0 sinh (∆φD ) + cH exp (−∆φD ) − exp (∆φD ) , cH

(5.40)

−1 −1 # KY cH + cY,0 1 + , exp (−∆φD ) exp (∆φD ) = F −cX,0 1 + KX cH

(5.41)

σelec = Cm (Vext − VT ∆φD ).

(5.42)

"

σchem

Substituting Equations 5.40 to 5.42 into the charge compensation equation σionic +σchem +σelec = 0, results in a transcendental equation for ∆φD . We then calculate salt adsorption as

Γ = vm (cm,+ − c+,0 ) + (cm,− − c−,0 ) ≈ 2c0 vm (cosh (∆φD ) − 1).

(5.43)

Moreover, we express salt adsorbed at potential of zero charge VPZC (i.e. Vext at which σelec = 0) as

ΓPZC ≈ 2c0 vm (cosh (VPZC /VT ) − 1).

(5.44)

The charge efficiency can then be defined as

Λ=

Γ − ΓPZC . Σchem /F

(5.45)

In this definition, Γ − ΓPZC is the extra salt adsorbed associated with applied potential Vext (note, Γ can be non-zero at potential of zero charge VPZC ). Note that charge efficiency here is a function of external voltage, surface properties (i.e. cX,0 , cY,0 , pKX , and pKY for nX = nY = 1), salt concentration c0 , and initial pH. In Figure 5.10, we show adsorbed salt and electronic charge as well as charge efficiency as a function of Vext for various values of those parameters. Unless otherwise noted, other parameters used are listed in Table 5.3.


139

Figure 5.10: Volumetric adsorbed salt and electronic charge (in units of M) vs external voltage Vext at various (a) analytical concentration of acidic and basic surface charges, (c) dissociation constant of acidic functional groups, (e) salt concentrations, and (g) electrolyte pH values. (b), (d), (f), (h) Charge efficiency vs Vext for parameters as in (a), (c), (e), and (g). Other parameters used are listed in Table 5.3.


140

Table 5.3: Parameters used in Figure 5.10 Parameter cX,0 cY,0 pKX pKY Cm z± c0

Description Analytical concentration of acidic group Analytical concentration of basic group Acid dissociation constant Base dissociation constant Volumetric micropore capacitance Cation and anion valence Initial salt concentration

Value 1 1 4 10 145 ±1 20

Units M M

F cm−3 mM

Chapter 6

Conclusions In this dissertation, we presented our efforts in development of experimental setups and modeling frameworks for quantification of energy costs as well as mass/charge transport mechanisms and surface chemistry in capacitive deionization. Here, we list the main contribution of this dissertation and provide recommendations for future work on capacitive deionization.

6.1

Contributions

6.1.1

Identification and quantification of energy loss mechanisms

1. Presented a general top-down approach to calculate minimum energy requirement of electrosorptive ion separation using variational form of free energy formulation, applicable to most known EDLs irrespective of EDL geometry and thickness. 2. Fabricated a low series resistance CDI cell by embedding titanium mesh into the activated carbon electrodes, operated the cell at various current and flow rates, and demonstrated lowenergy desalination with unprecedented 9% thermodynamic efficiency and only 4.6 kT energy requirement per removed ion (NaCl solution). 3. Designed and fabricated a radial fbCDI cell to quantify individual loss mechanisms during charging and discharging and to characterize their dependence on charging current and cell

141

CHAPTER 6. CONCLUSIONS

142

voltage. 4. Identified resistive and Faradaic (parasitic) losses as two main energy loss mechanisms, and showed the two categories of loss favor different charging rates: resistive losses are minimized at low charging currents and Faradaic losses are minimized at high charging rates (as the cell spends shorter time at high voltage).

6.1.2

Charge and mass transport in two-dimensions

1. Formulated and solved the first two-dimensional model of a CDI cell coupling external electrical network, charge conservation, and mass conservation in bimodal pore structure electrodes. The model was further applied to a fbCDI cell. 2. Fabricated a meso-scale fbCDI cell and performed a series of experiments to calibrate the model by using dynamic effluent salt and near-equilibrium salt and charge data. Results showed a good agreement between model results and experimental data. 3. Demonstrated that desalination process in fbCDI cells has two distinct phases: a fast adsorption step at the beginning of charging followed by a slow salt removal step. Results showed that the coupling of diffusion-limited transport and adsorption can result in a sharpening of the gradient between high and low micropore charge concentration. This sharpening resulted in ion concentration shock wave in micropore concentration propagating through the electrode (toward the current collector).

6.1.3

Effect of surface functional groups on salt adsorption

1. Expanded the current CDI-based surface charge models by coupling the modified Donnan model with weak electrolyte acid-base equilibria theory. Model was further validated by comparing predictions to experimental measurements from well-controlled titration experiments of electrode samples. The fitted model with one acidic and one basic surface group showed a very good agreement with the experiments. 2. Applied the model to titration of activated carbon and showed (1) specific adsorption of cations


143

and expulsion of anions at electrolyte pH values higher than pK of acidic groups, and (2) specific adsorption of anion and expulsion of cations at pH values lower than pK of basic groups.

6.2

Recommendations for future work

We here list our recommendations for extending the current work.

6.2.1

Long term performance

Currently, there has been only a few studies on long term performance of CDI and stability of CDI cells. To be a competitive technology, carbon electrodes used in CDI cells should be stable (both physical integrity and chemical stability) for an extended period of time. In addition to need for studies on stability of carbon electrodes during CDI operation, more work is needed on storage of electrodes in contact with various salt solutions. Studies on methods to reduce electrode oxidation and fouling can significantly increase the viability of CDI technology.

6.2.2

Temperature effects

There has been little to no work on effects of feed solution temperature on performance of CDI desalination. Temperature can affect the electrolyte conductivity as well as EDL structure formed inside the electrode pores, and thus can affect (or adversely affect) desalination performance.

6.2.3

Surface conduction effects

Surface conduction is transport of charged species tangential to the charged surface (surface of electrode pores). This effect opens a new pathway for charge and salt transport and thus can enhance electrosorptive desalination. However, almost all of the current charge and mass transport models intended for CDI applications neglect surface transport effects. The work by Shocron and Suss [209] and Mirzadeh et al. [99] are among the few that investigate this effect.


6.2.4

144

Species-specific adsorption and selectivity

Species-specific electrosorption with CDI is highly desired in both point-of-use and industrial applications. Research studies on methods for selective adsorption via CDI can open opportunities in applications such as ionic contaminant removal, rare metal ion extraction, and trace ion detection. In recent years, studies on selective ion removal via CDI has been an area of research, however, a general approach for user-specific ion selectivity is lacking. These work are intended for a limited set of ions, for example, lithium, nitrate, and fluoride. Recently, Oyarzun et al. [210] has shown selectivity of nitrate ions (versus chloride ions) by treating the carbon electrodes with positively charged and negatively charged surfactants. We thus recommend a comprehensive study on wide range of surfactants currently used on ion-exchange resins.

Appendix A

Notes on fabrication and test of CDI cells A.1 A.1.1

Cell design and fabrication Sources for materials

The main components of a CDI cell are activated carbon electrodes, non-conducting spacers, current collectors, gaskets, and casing. Carbon electrodes can be custom made or purchased commercially. Particle adsorbent sheets from Material Methods LLC, particularly PACMM 203, as well as Zorflex activated carbon cloth are two options for the electrodes. PACMM 203 electrodes are cut to 6×6 cm area with average uncompressed thickness of 270 µm with unnoticeable batch-to-batch variability. Although, under continuous operation, the carbon cloth tends to oxidize in a faster rate compared to PACMM 203 activated carbon sheets. A variety of materials can be used as spacers. Filter papers, coffee filters, and woven plastic meshes are a few options. 30 to 300 µm thick woven plastic meshes are recommended because they are less prone to clogging and provide higher permeability and so less pressure drop. Current collectors need to be conductive, chemical-resistant, and nonreactive. Titanium foils and graphite sheets are two common choices. Finally, the stack of CDI (electrodes, current collectors, and spacers) are placed between two end plates and sealed with

145

APPENDIX A. NOTES ON FABRICATION AND TEST OF CDI CELLS

146

Table A.1: Recommended materials and providers for fabrication of CDI cells. Item Electrode Spacer Current collector End plates Gaskets

Material PACMMTM 203 activated carbon electrode Plastic mesh (Cat. number 9218T91, 9218T83, 9275T41, or similar) 50 µm grade 2 titanium foil Acrylic sheets O-rings or PDMS gaskets

Provider Material Methods, LLC McMaster-Carr Solution Materials, LLC McMaster-Carr McMaster-Carr

gaskets. Poly(methyl methacrylate) (PMMA, also known as acrylic) and polycarbonate (PC) are used frequently. These materials can be CNC-cut as well for more precise designs. Another option is 3D printing using thermoplastic materials. In Table A.1, we list recommended list of raw materials and providers for fabrication of CDI cells.

A.1.2

fbCDI cell designs

Fabrication of a prototype CDI cell is relatively easy and straightforward. More precise fabrication using CNC is more involved. For a quick cell fabrication, one can hand- or machine-cut acrylic endplate sheets, electrodes, spacers, gaskets, and current collectors similar to those shown in Figure A.1. Input and output ports are cut on acrylic end-plates and are threaded with thread tap for Luer-lock external connections. Spacers are cut slightly larger than electrode, and current collectors are cut smaller than the two to prevent shorting. Current collectors have tabs for external connections. The stack can then be pressed by C-clamps or other fasteners. Alternatively, end-plates can be cut precisely using CNC machine. A 3D CAD design and photo of a CDI cell fabricated by this method is shown in Figure A.2.

A.2

Test of CDI cells

A.2.1

Experimental setup

As mentioned in the main text, our CDI setup consisted of a Watson Marlow 120U/DV peristaltic pump, eDAQ conductivity sensors and isoPods (which interfaces the sensors with the computer


Figure A.1: Schematic and a photo of a CDI cell for quick fabrication.

Figure A.2: 3D design and a photo of a CDI cell made by CNC machine.

147


148

Figure A.3: Schematic of our experimental setup consisting a solution reservoir, a peristaltic pump, eDAQ conductivity and pH sensors, Keithley sourcemeter, and a CDI cell. via USB connection), and Keithley sourcemeter. We controlled the CDI process and recorded the voltage, current, and concentration using serial connection between computer and measuring devices. A schematic of the experimental setup is shown in Figure A.3.

A.2.2

Pre-testing and troubleshooting

• Short circuit: short circuit is a common problem and is caused mainly by electrical short between opposite current collectors or opposite electrodes. In the case of short circuit, the cell series resistance appears to be very low (order 1 Ω or lower). To diagnose, (i) perform a constant voltage test and monitor the electric current. If there is a short, initial current jump is unusually high and the current does not decay to near-zero current. Similarly, (ii) perform a constant current test and monitor the cell voltage. If there is a short, voltage profile tends to plateau (instead of almost-linear increase). Alternatively, (iii) perform a cyclic voltammetry test and construct the CV plot. If the plot is linear with voltage (rather than leaf-shaped or rectangular) for moderate and low scan rates, short circuit is probable. • Faradaic reactions: to reduce Faradaic reactions, de-gas the solution with inert gasses (such as nitrogen or argon) prior to or during the experiments. Additionally, reducing the high voltage


149

limit (for example to 0.8 V), or use of ion exchange membranes (as in MCDI) can significantly reduce Faradaic reactions. • Priming new cells: due to small pore size of activated carbons, fully wetting the electrodes with solution is critical for performance of the cell. The cell shows high series resistance if electrodes are partially wet. To wet the electrodes, immerse the electrodes in the solution and use vacuum (desiccator or vacuum chamber) for 10 min. Solution will impregnates the electrodes and trapped are leaves the pores. Additionally, to further wet the electrodes after cell assembly, prime the cell by flowing solution for at least 1 h. • Quick test of CDI cells: Cycling voltammtry and electrochemical impedance spectroscopy (see Appendix B) are two useful methods for measuring capacitive and resistive components of CDI cells. These methods, although reliable and robust, can take several hours. For a quick and reasonable measurement of capacitance and resistance, one can use a simple constant current (galvanostatic) charging of the cell instead. Through this experiment, series resistance can be roughly approximated as R ≈ ∆V /I with I and ∆V being the applied current and initial voltage jump upon application of the current. Additionally, capacitance C can be approximated by calculating the slope of voltage profile versus time as in C ≈ I/(dV /dt).

A.2.3

Serial communication with devices

An example Python code for serial communication with pump, sensors, and sourcemeter is shown below. Note, we use pySerial module for communication. This example uses two conductivity sensors (at inlet and outlet) and operates the CDI cell as the following: (i) constant voltage charging up to Vmax = 1 V under no flow condition, (ii) open circuit (zero current) flush with flow, (iii) zero voltage discharge under no flow condition, and (iv) a second open circuit flush. The code is modular and can readily be modified under various other current/voltage and flow conditions.

1 2 3 4

import import import import

serial time datetime sys

5 6

import numpy as np


7 8 9

import matplotlib.pyplot as plt from drawnow import drawnow from matplotlib import style

10 11 12 13

timestamp = ('run_{:%Y%m%d_%H%M%S}'.format(datetime.datetime.now())) filename = sys.argv[1]+"-"+timestamp+".txt" fw = open(filename,"w+")

14 15 16 17 18 19 20 21

timeArr = [] voltArr = [] currArr = [] condArr = [] cond2Arr = [] plt.ion() cond_range = 20

# maximum conductivity value [mS/cm]

22 23 24 25 26 27 28 29 30 31 32 33 34

ch_time = 3600 # charging time [s] flush_time = 1200 # flush time [s] cycles = 9 # number of cycles datafreq = 40 # frequency of data for plot Vflow = 0.175; # voltage for 2 rpm [V] Vnoflow = 0.0; # Voltage for no flow [V] vmax=1.0 vmin=0.4 tcc = 5*60 # an arbitrary high number Vflowlist = [ 0.5315, #Volt. for 2.0 mL/min 0.1915, #Volt. for 0.5 mL/min 0.3065 ] #Volt. for 1.0 mL/min

35 36 37 38 39

#### function definitions def writeDataToFile(data): fw.write(str(data)+'\n') fw.flush()

40 41 42 43 44 45 46 47 48 49

def updatePlot(): plt.subplot(2,2,1) plt.plot(timeArr,voltArr) plt.subplot(2,2,2) plt.plot(timeArr,currArr) plt.subplot(2,2,3) plt.plot(timeArr,condArr) plt.subplot(2,2,4) plt.plot(timeArr,cond2Arr)

50 51 52 53 54 55 56 57

def voltcommand(vlevel, device): if device == 'vi_cv': com = [':OUTP OFF',':SOUR:FUNC VOLT',':SOUR:VOLT:MODE FIXED', \ ':SOUR:VOLT:RANG 1E+1',':SOUR:VOLT:LEV '+str(vlevel), \ ':SENS:FUNC "CURR"',':SENS:CURR:RANG 1',':SENS:CURR:PROT 1', ':OUTP ON'] elif device == 'vi_cc': com = [':OUTP OFF',':SOUR:FUNC CURR',':SOUR:CURR:MODE FIXED', \

150


58 59 60 61 62 63 64

':SOUR:CURR:RANG 1',':SOUR:CURR:LEV '+str(vlevel), \ ':SENS:FUNC "VOLT"',':SENS:VOLT:RANG 5E+0',':SENS:VOLT:PROT 10',':OUTP ON'] elif device == "pump": com = ['*RST',':SOUR:FUNC VOLT',':SOUR:VOLT:MODE FIXED', \ ':SOUR:VOLT:RANG 1E+1',':SOUR:VOLT:LEV '+str(vlevel), \ ':SENS:FUNC "CURR"',':SENS:CURR:RANG 0.025',':SENS:CURR:PROT 1', ':OUTP ON'] return com

65 66 67 68 69

def pump_update(p0,state): if state=="flow": for i in voltcommand(Vflow,'pump'): p0.write(i+"\n")

70 71 72 73

elif state=="noflow": for i in voltcommand(Vnoflow,'pump'): p0.write(i+"\n")

74 75 76

def srcmtrcom(v0, p0,c0,c1, cmdlst, run_time, t0,V_thresh, tsleep,mode,state): # Define Empty Arrays

77 78 79 80 81

v0.flushOutput() p0.flushOutput() c0.flushOutput() c1.flushOutput()

82 83 84 85

# send commands to keithley for V-I for i in cmdlst: v0.write(i+"\n")

86 87 88

# update pump pump_update(p0,state)

89 90 91

tstart = time.time() t1 = time.time()

92 93 94 95 96

cntr = 1; while t1-tstart < run_time: c0.write('v\r') c1.write('v\r')

97 98 99 100

t1 = time.time() v0.write(':READ?\n') p0.write(':READ?\n')

101 102 103 104

inv0 = [ i.lstrip("+") for i in v0.readline().rstrip().split(",") ] outl1 = ",".join( [str(t1-t0),] + inv0) outl2=outl1.split(',')

105 106 107 108

tt = float(outl2[0]) voltage = float(outl2[1]) current = float(outl2[2])

151


109 110 111 112 113 114

cond_str = c0.readlines() cond = float(cond_str[1]) cond2_str = c1.readlines() cond2 = float(cond2_str[1]) datastr = str(tt)+'\t'+str(voltage)+'\t'+str(current)+'\t'+ \ str(cond)+'\t'+str(cond2)+'\t'+str(Vflow)

115 116 117

print datastr writeDataToFile(datastr)

118 119 120 121 122 123 124

#Assign values to plot timeArr.append(tt) voltArr.append(voltage) currArr.append(current) condArr.append(cond) cond2Arr.append(cond2)

125 126 127 128

if cntr%datafreq==0: drawnow(updatePlot) # draw and refresh graph plt.pause(0.000001)

129 130 131 132 133 134 135 136

v0.flushOutput() p0.flushOutput() c0.flushOutput() c1.flushOutput() sys.stdout.flush() time.sleep(tsleep) cntr = cntr+1

137 138 139 140 141

## CC charge if mode=='cc' : if current>0 and voltage>=V_thresh: return t1-tstart

142 143 144

if current Cyclic Voltammetry). 6. In the pop-up window, enter the output file name and CV setting values as follows. Select 0 V as initial and final voltage. High and low voltage limit can be for example 0.5 and −0.5 V, as shown in Figure B.5. Select at least three to five cycles to ensure reaching dynamic steady state. 7. Scan rate (s) is an important parameter that can affect the capacitance measurement. A good value for s is between 0.1 to 2 mV/s. Low scan rate s is desirable for accurate capacitance measurement. Relatively high scan rate changes the CV curve into a leaf-shape (as opposed to a rectangular shape) and thus results in inaccurate capacitance measurement. To elucidate this issue, we show in Figure B.4 examples of CV measurements for high (s = 5 mV/s), medium (s = 1 mV/s), and low (s = 0.2 mV/s) scane rates. The voltage limits are ±0.5 V.

B.2

Electrochemical impedance spectroscopy

EIS is an electrochemical method that can be primarily used to determine resistive components of a CDI cell. Refer to Ref. [111] for more information on these various components. We here show the steps to perform EIS test on a CDI cell using a potentiostat. 1. In contrast to CV test, EIS does not deplete the electrolyte, as the applied voltage is significantly lower (on the order of mV). So, EIS under stopped flow condition is possible.

APPENDIX B. ELECTROCHEMICAL MEASUREMENTS OF CDI CELLS

158

Figure B.3: An example of Gamry Instruments Framework software settings for cycling voltammetry measurement. 2. Turn on the potentiostat, wait for about 15 s, and open Gamry Instruments Framework software. 3. Connect working electrode lead to the positive electrode and counter electrode lead to the negative electrode. 4. For a two-point measurement, connect reference lead to the counter electrode lead. For threepoint measurement, connect the reference lead to the reference electrode. 5. Navigate through Gamry Instruments Framework software and select Potentiostatic EIS (select Experiments > Electrochemical Impedance > Potentiostatic EIS). 6. In the pop-up window, enter the file name and other values associated with EIS test, such as


159

Figure B.4: An example of CV measurement for high to low scan rates at s = 5, 1, and 0.2 mV/s. Scan rate of 5 mV/s is too high (CV curve is leaf-shaped) and should not be used to approximate the capacitance. high and low frequency limits. 700 kHz to 1 mHz are good starting points. Enter AC (RMS value) and DC voltage values as well. 7.07 mV RMS and 0 V are typical values. 7. Start the test.

The potentiostat will start from high to low frequency and records the

impedance values at each frequency. The resulting EIS data is typically negative of imaginary part of impedance versus real part of impedance.


160

Figure B.5: An example of Gamry Instruments Framework software settings to perform electrochemcial impedance spectroscopy test.

Appendix C

Use of CNC machine for fabrication CDI cells C.1

Preparation

The instructions given here are based on Roland MDX-540 Milling Machine. We will introduce a workflow of fabricating a CDI cell using this CNC-machine, and present details and suggestions. The list of materials and equipment required for CNC-machining process is as follows: Endmills of desired diameter and length, workpiece (e.g. acrylic or polycarbonate), wipers, isopropyl alcohol or methanol, and double sided tape. Additionally, create your 3D model with any design software and export as “stl” file format. Remember dimensions of the designed model.

C.2

Hardware setup

Mounting the workpiece Cut your workpiece to desired size and secure it to the CNC stage using double sided tape. However, to prevent damaging the stage, use a sacrificial layer (a piece of acrylic etc.) underneath the workpiece. To this end, clean all the surfaces with wipers and isopropyl alcohol, secure the sacrificial

161

APPENDIX C. USE OF CNC MACHINE FOR FABRICATION CDI CELLS

162

Figure C.1: Photo of spindle of the CNC machine with endmill mounted. layer on the stage, and secure your workpiece on top of them. Make extra effort to make sure all pieces are placed completely flat.

Mounting the endmill Open the spindle cover and use two wrenches to unscrew the bolt located at the bottom of the spindle, as shown in Figure C.1. Place the endmill inside the spindle and while holding it (to prevent from falling), first fasten the bolt by hand and then continue fastening by use of the the two wrenches. Close the spindle cover. To unmount the endmill, follow the instructions in reverse and make sure to prevent the endmill from falling and hitting the stage.

Setting the origin of coordinate system Use the hand-held controller and click the “Coord. System” button a few times such that “USER” appears on top left of the device’s screen. Now, click either of X, Y, or Z buttons and rotate the knob on the hand-held controller to move the spindle in X, Y, or Z directions. Once facing the machine, X direction is left-to-right, Y direction in front-to-back, and Z direction is vertical direction. To set the origin of the device’s coordinate system, move the spindle to approximately one of the workpiece’s corners (move the spindle down in Z direction for better placement of spindle). Click


163

Figure C.2: Photo of hand-held controller (left), used for setting the Z-coordinate origin. The Zsensor is placed on top of work place and endmill is lowered to about 1 cm above the sensor (top right). Make sure that the sensor is properly connected to the machine front panel (bottom right) before pressing Z0 SENSE button. on X button and then hold “Origin” for a few seconds. Do the same with Y button. Click on X again and move the spindle by approximately half of the designed piece (refer to 3D design for your model). Click and hold “Origin” again. Do the same with Y axis. At this point, origin of X and Y coordinates are set to be in the middle of the designed 3D model. Set origin for the Z coordinate using as instructed in the following (see Figure C.2). Carefully clean the endmill and Z-sensor with isopropyl alcohol to remove any oil on their surface. This will ensure electric current can pass through the endmill to the Z-sensor as soon as they touch during the process. Move the spindle down to about 1 cm above the top surface of the Z-sensor. Make sure Z-sensor is plugged into the CNC machine. Place the Z-sensor beneath the endmill and press “Z-origin”. The spindle slowly moves downward to touch the top surface of Z-sensor. Spindle will move upward as soon as endmill touches Z-sensor. Be ready to press “Emergency Stop” is endmill penetrates the Z-sensor’s top surface.


C.3

164

Software settings

Open “SRP Player” software and load the “stl” file from File menu. Model will appear in the main window. Use the radio-buttons around the green figure on the right hand side of the software to change the orientation of the model (Located under “Model Size and Orientation tab”). For faster cutting, make sure to align the shortest dimension of the model with Z-axis. Then navigate to Options menu and click Add/Remove Tool. Find the endmill placed in the spindle earlier in the list and select it. If not listed, Add the endmill to the list of tools. Hit Ok. Navigate to step 2 and click on “Type of Milling”. For faster cut, select “Faster cutting time”. If the piece has mainly flat surfaces rather than round surfaces, select “Model with many flat surfaces”. Select “Cut top only”. Navigate to step 3 (“Create tool path”) and select type of material and enter dimension of the model. Note, these dimensions take into account the endmill diameter as well. Suggested values (based on endmill diameter are shown to the left of the text boxes). A green box enclosing the 3D model appears on the main window. Next, in order to set the cut settings, click on “Edit”. To reduce the job time, remove “Finishing” form the list. Finishing is not required for cutting plastic pieces such as acrylic. Click on “Roughing” and then “Cutting parameters”. The default cutting parameters are set based on the type of endmill and material to be cut. Do not increase the parameters more than 20% to 30% of their nominal values, otherwise, the endmill might break during the cutting process. Make sure to set “Finish margin” value to zero. Click “Apply” and then “Create Tool Path”. The software calculates the cutting paths and cutting time (postprocessing). Next, click on step 4 (“Preview Results”) to see the estimated time of cutting. Review the final results by clicking on “Preview Cutting”. It is very important to visually confirm the final product. As an example, CNC machine will skip the features that are smaller than the endmill diameter. Next, click on the last step (“Perform Cutting”). Make sure no object is left inside the machine and on top of the stage. Then hit “Start Cutting” and perform the cut.

C.4

Notes on operation of CNC machine

• At any point, press the “Emergency Stop” to immediately stop the operation of CNC machine.


165

• Before performing the cut, make sure no object is left inside the cutting area and the spindle can freely move. • It is always a good practice to thoroughly clean the machine and surrounding areas from debris after the job is finished.

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