The Impact of Electrostatic Correlations on the Electrokinetics

UNLV Theses, Dissertations, Professional Papers, and Capstones

5-1-2020

Elaheh Alidoosti

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This Dissertation has been accepted for inclusion in UNLV Theses, Dissertations, Professional Papers, and Capstones by an authorized administrator of Digital Scholarship@UNLV. For more information, please contact [email protected]. THE IMPACT OF ELECTROSTATIC CORRELATIONS ON THE ELECTROKINETICS

Elaheh Alidoosti

Bachelor of Science-Geoscience University of Shahid Beheshti 2003

Master of Science in Electrical Engineering University of Nevada, Las Vegas 2014

A dissertation submitted in partial fulfillment of the requirements for the

Doctor of Philosophy in Mechanical Engineering

Department of Mechanical Engineering Howard R. Hughes College of Engineering The Graduate College

May 2020

Dissertation Approval

The Graduate College The University of Nevada, Las Vegas

April 21, 2020

This dissertation prepared by

Elaheh Alidoosti entitled

The Impact of Electrostatic Correlations on the Electrokinetics is approved in partial fulfillment of the requirements for the degree of

Doctor of Philosophy Mechanical Engineering Department of Mechanical Engineering

Hui Zhao, Ph.D. Kathryn Hausbeck Korgan, Ph.D. Examination Committee Chair Graduate College Dean

Yi-Tung Chen, Ph.D. Examination Committee Member

Kwang Kim, Ph.D. Examination Committee Member

Yingtao Jiang, Ph.D. Graduate College Faculty Representative

Pengtao Sun, Ph.D. Graduate College Faculty Representative

Abstract

The Impact of Electrostatic Correlations on the Electrokinetics

Elaheh Alidoosti

Advisor: Dr. Hui Zhao

In this project we are interested in electrokinetics phenomenon (i.e., study of electrical charges in a liquid matter) and focus on three families of this phenomenon, dielectrophoresis, electro-osmosis and streaming current. For the first work, at concentrated electrolytes, the ion-ion electrostatic correlations effect is considered as an important factor in electrokinetics. In this project, we compute, in theory and simulation, the dipole moment for a spherical particle (charged, dielectric) under the presence of an alternating electric field using the modified continuum Poisson-Nernst-

Planck (PNP) model by Bazant et al. (Phys. Rev. Lett. 106, 2011) [25]. We investigate the dependency of the dipole moment in terms of the frequency and its variation with such quantities such as zeta potential, electrostatic correlation length, and double layer thickness. With thin electric double layers, we develop simple models through performing an asymptotic analysis to the modified PNP model. We also present numerical results for an arbitrary Debye screening length and electrostatic correlation length. From the results, we find a complicated impact of electrostatic correlations on the dipole moment. For instance, with increasing the electrostatic correlation length, the dipole moment decreases and reaches a minimum, and then it goes up. This is because of initially decreasing of surface conduction and it s finally increasing due to the impact of ion- show that in

iii contrast to the standard PNP model, the modified PNP model can qualitatively explain the data from the experimental results in multivalent electrolytes.

For the second study, Ion-ion electrostatic correlations are recognized to play a significant role in the presence of concentrated multivalent electrolytes. To account for their impact on ionic current rectification phenomenon in conical nanopores, we use the modified continuum Poisson-Nernst-

Planck (PNP) equations by Bazant et al. [Phys. Rev. Lett. 106, 046102 (2011)]. Coupled with the

Stokes equations, the effects of the electroosmotic flow (EOF) are also included. We thoroughly investigate the dependence of the ionic current rectification ratios as a function of the double layer thickness and the electrostatic correlation length. By considering the electrostatic correlations, the modified PNP model successfully captures the ionic current rectification reversal in nanopores filled with lanthanum chloride LaCl3. This finding qualitatively agrees with the experimental observations that cannot be explained by the standard PNP model, suggesting that ion-ion electrostatic correlations are responsible for this reversal behavior. The modified PNP model not only can be used to explain the experiments, but also go beyond to provide a design tool for nanopore applications involving multivalent electrolytes.

The third part of this project deals with applying modified PNP equations to compute streaming current which is generated by a pressure-driven liquid flow in nanochannels. Streaming currents are the results of the transport of counterions in the electric double layer, when a pressure gradient is applied. The accumulated counterions at the end of the reservoir are inducing the streaming potential. Streaming current can provide an effective tool to convert hydrostatic pressure differences into electrical energy. In this project, by using modified PNP, we compute streaming current and potential in a silica nanochannel. The efficiency of nanochannel is increasing as the correlation length is getting higher. As the correlation length is increasing the negative ions are

iv depleted inside the nanochannel and the ionic current is mainly consisting of cations. So, at higher correlation length the smaller the thickness of the EDL, the higher the efficiency is.

Acknowledgments

I would like to thank my valuable advisor and chair Dr. Hui Zhao for his support and recommendations. I really had the privilege to work with him on his groundbreaking projects and ideas. I would also like to thank to my committee members, Dr. Yingtao Jiang, Dr. Yi-Tung Chen,

Dr. Kwang Kim, and Dr. Pengtao Sun for their assistance and advice. I also like to thank Dr.

Yingtao Jiang, and Dr. Yi-Tung Chen for their support and guidance. Finally, I would like to thank my family for their encouragement of pursuing this project.

Research reported in this publication was supported, in part, by the National Institute of

General Medical Sciences of the National Institutes of Health under Award Number

R15GM116039 and National Science Foundation under Grant No. ECCS-1509866. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health and National Science Foundation.

Table of Contents

Abstract ...... iii

Acknowledgments...... vi

List of Figures ...... x

Chapter 1 ...... 1

Introduction ...... 1

1.1 Backgrounds ...... 1

1.2 Electrokinetics ...... 3

1.2.1 Electroosmosis, Electrophoresis, and Streaming Potential ...... 3

1.2.2 Zeta Potential, Diploe Moment, and Dielectrophoresis ...... 5

1.2.3 Over screening and Charge Inversion ...... 6

1.3 Modified Poisson-Nernst-Plank Model ...... 8

1.4 Applications ...... 11

1.5 Objectives and Significance of This Thesis ...... 13

Chapter 2 ...... 15

On the Impact of Electrostatic Correlations on the Double-Layer Polarization of a Spherical

Particle in an Alternating Current Field ...... 15

2.1 Introduction ...... 15

2.2 The Modified Poisson-Nernst-Planck Model ...... 19

2.3 Perturbation Expansion ...... 22

vii

2.3.1 The Zeroth Order Approach ...... 23

2.3.2 The First Order Equation ...... 23

2.4 PNP Models...... 25

2.4.1 The Maxwell-Wagner- Model ...... 25

2.4.2 The Dukhin-Shilov (DS) Model ...... 26

2.5 Results and Discussion ...... 28

2.6 Conclusion ...... 36

Chapter 3 ...... 38

The Effects of Electrostatic Correlations on the Ionic Current Rectification in Conical Nanopores

...... 38

3.1 Introduction ...... 38

3.2 Mathematical Model ...... 41

3.3 Results and Discussions ...... 46

3.4 Conclusions ...... 53

Chapter 4 ...... 54

The Impact of Electrostatic Correlation Length on Streaming Current and Streaming Potential in

Nanochannels ...... 54

4.1 Introduction ...... 54

4.2 Mathematical Model ...... 55

4.2.1 Governing Equations ...... 55

viii

4.2.2 Computational Domain...... 57

4.2.3 Boundary conditions ...... 60

4.2.4 Energy Conversion Efficiency ...... 61

4.2.5 Validation ...... 63

4.3 Results and Discussion ...... 64

4.4 Conclusion ...... 71

Chapter 5 ...... 72

Conclusions and Future Work ...... 72

5.1 Conclusions ...... 72

5.2 Future Work ...... 73

Appendix A ...... 75

Appendix B ...... 77

References ...... 78

Curriculum Vitae ...... 89

List of Figures

Figure 1. The schematic of Electrical Double layer (EDL) ...... 5

Figure 2. a) Over screening (happens at smaller voltage). b) Crowding (happens at higher

voltage)...... 7

Figure 3. The schematics of the coordinates for a submerged particle in an electrolyte...... 19

Figure 4. The dipole moment Re f with respect to frequency for 7 and C 0.1 for

various D . The solid lines represent MWO model; the dash-dotted lines represent the DS

model; and the circles, squares, and triangles represent the modified PNP model...... 28

Figure 5. The dipole moment Re f with respect to frequency for D 0.01 , and c 1. For

different . The solid lines represent MWO model; the dash-dotted lines represent the DS

model; and the circles, squares, and triangles represent the modified PNP model...... 29

Figure 6. The dipole moment Re f with respect to frequency for different c when 5 ,

D 0.01 and the solid lines represent MWO model; the dash-dotted lines represent the DS

model; and the circles, squares, and triangles represent the modified PNP model...... 31

Figure 7. The Dukhin Number with respect to the correlation length: The solid line represents

the Du number, the dashed line represents the contribution from the convection (Duc), and

the dash-dotted line represents the contribution from the migration (Dum) when 5 and

D 0.01 ...... 31

Figure 8. The dipole moment Re f versus frequency for different c when 7 and D 0.3 .

...... 32

Figure 9. The dipole moment Re f with respect to frequency with the surface charge density

2 0.16 mC/m , a 20 nm andC0 5 mM . The solid line represents the dipole moment with

monovalent ions, the dash line represents the dipole moment with divalent ions ( lc 0 ),

and the dash-dotted line represents the dipole moment with divalent ions ( lc 1 nm )...... 34

Figure 10. The dipole moment Re f with respect to frequency for various D when 7 and

lc 0.2 ...... 35

Figure 11. A schematic of the simulated system and the cylindrical coordinates ...... 41

Figure 12. The I V curves for a monovalent solution KCl in a conical nanopore. a) D 1 , b)

D 0.17 , c) D 0.1 . The different lines stand for the PNP model from [44] and the

3 different symbols represent the modified PNP model with lc 10 . The solid line (circles),

dashed line (squares), and dash-dotted line (triangles) are for surface chargeies 2.73

2 ( 10 mC m2 ), 13.66 ( 50 mC m ) and 27.32 ( 100 mC m2 )...... 45

Figure 13. Current rectification ratio IR as a function of the electrostatic correlation length lc

when D 0.2 , 40 andV0 100 ...... 46

Figure 14. The averaged cross-sectional ionic concentrations in which the solid line and the

dashed line correspond, respectively, to anion Cl- and cation La3+ (the top) and the

3 averaged electrical potentials (the bottom) for D 0.2 and 40 whenlc 2 10 (a);

lc 0.02 (b); an lc 0.6 (c). The left, the middle, and the right column represent,

respectively, the applied voltage ofV0 100 ,V0 0 andV0 100 ...... 48

Figure 15. The current rectification ratio IR as a function of 1/ D when lc 0.6 andV0 100 .

The line with stars and that with circles represent, respectively, to 20 and 40 . ... 50

Figure 16. The flow rate as a function of lc when D 0.2 ,V0 100 and 40 ...... 51

3 Figure 17. The flow field near the tip of the conical nanopore for (a) lc 2 10 and (b) lc 0.6

when D 0.2 , V0 100 and 40 ...... 51

Figure 18. Distribution of the concentration of cation (La3+) near the tip region of the conical

3 nanopore for (a) lc 2 10 and (b) lc 0.6 when D 0.2 ,V0 100 and 40 ...... 52

Figure 19. A schematic of simulated system ...... 59

Figure 20. The efficiency in terms of the bulk concentration for monovalent solution KCL. The

dashed line represents for the PNP model and the symbol represents for the Modified PNP

3 2 model at 1mM KCL solution for lc 10 Here surface charge density is 1mC m , and the

size of the nanochannel is 2h=60nm...... 63

Figure 21. Streaming current and streaming potential in terms of pressure for a 5 nm channel

2 height and Sigma=-10 mC m . Part A & B are for l c =0.001 with no charge inversion. Part C

& D are for l c =0.3 with charge inversion happening at higher D ...... 64

Figure 22. Efficiency in terms of bulk concentration (mM) for sigma=-10 mC m2 and h=30nm.

...... 65

Figure 23. The efficiency in terms of bulk concentration for C0=10.47 (mM) and 20(mM), ...... 66

xii

Figure 24. Cation and anion cross sectional averaged for sigma=-10 mC m2 , h=30nm,...... 68

Figure 25. Concentration of Cations (La3+) inside the nanochannel near the inlet for ...... 70

Figure 26. Distribution of electric potential inside the nanochannel near the outlet for C0=1mM

and sigma=-10 mC m2 . a) lc=0.001, b) lc=0.3 ...... 70

Figure A1. The simulation domain including the particle (domain 1) and the electrolyte

(domain 2) 75

Figure A2. Mesh-independence studies: the dipole moment with respect to the frequency for

3 4 4 The mesh size 10 ,5 10 , and 10 when 7, D 0.01, and c 0.1 ...75

2 Figure A3. The dipole moment as a function of frequency D for 3

and D 1, D 2, D 3, and D 4 . The solid line and the symbols are respectively the

ones from the standard PNP model and those from the modified PNP model accounting

for electrostatic correlations. Here c 0.01 ... .76

xiii

Chapter 1

Introduction

1.1 Backgrounds

Electrokinetics define as the motion of the fluid itself or the located particle inside the fluid with an applied electric filed or the pressure difference inside the fluid with having a charged surface. Nowadays it is applicable in biological subjects due to the ability of manipulation process in the microfluidics or nanofluidics area. As a result, the electrokinetics open a new door for biology and technology to influence the molecular degree of the substances. For this to be happen, scientists are trying to understand the basic characteristics and the restrictions of the existing forces in this phenomenon.

The different forces are including the forces on the fluid with applying AC voltage or without applying the voltage which are the electro osmotic and streaming potential, respectively. The forces that are applying on the particle inside the fluid are electrophoresis and dielectrophoresis

(DEP) for the charged and dielectric particles in the existence of the electric field, respectively.

Electrophoresis and electro osmosis discovered in 1808 by F.F. Reuss a German scientist in Russia

[1]. In Berlin in 1859, Georg Quincke not only worked on electro osmosis but also discovered streaming potential [2]. Other findings from him, was the linearity of streaming potential in terms of pressure and observation of achieving lower streaming potential by having higher concentration of KCL. His innovations led to the electrical double layer (EDL) discovery. As a result, a quantitative theory of EDL was presented by Hermann Helmholtz in 1879. Next step in

1 electrokinetics was taken by Ernst Dorn in 1880 in which he measured the potential different between two electrodes in sedimentation process which is called Dorn effect or sedimentation potential [3]. By this time all the electrokinetic phenomena completely discovered in the classical form.

Electrokinetic phenomena has four categories. The first two of them are happening when we have applied voltage. In which if a liquid works against a solid we have electro osmosis. Otherwise if we have solid particles that are working against a liquid then the electrophoresis exists. For the next two categories, there is no applied voltage, but the only force is potential difference. In this case, we have streaming potential and sedimentation potential. In streaming potential, the liquid is moving parallel to the solid surface. In sedimentation potential the solid object is moving parallel to the liquid.

Microfabrication technology helped the useful telecommunications for today devices. The development of this technology resulted in improving MEMS fabrication (Micro-Electrical-

Mechanical Systems) in which the technology is based on the Nano/Micro scaled electrode integration. Also, lab-on-a-chip (LOC) devices are using a Microchip to integrate the lab results with a fast performance.

Nano/Micro liter devise first miniaturized total chemical analysis systems TAS which was the branch of the total chemical analysis system

TAS . Developing from this new system, then Micro/Nano fluidics and Lab-On-a-Chip system were invented. After that the Microelectromechanical systems (MEMS) was obtained by integrating microelectronics and micromechanics. It was applying the capillary electrophoresis in

2 a large number of connected microchannels to come by the very first chip that was created in early

19th with two methods including fluorescent dyes and amino acids [4-6].

The scale of the electrokinetics is in micro and nano scale in the same rage of biological objects which makes that a suitable technology in biomedical. Today the diagnostics happening with the minimum number of samples by using the micro/nano devices. Electrokinetics are the best match of the length scale for biological subjects like DNA, Proteins and other cell tissues.

One the best advantages of nanofluidics and nano scale systems is having a high surface to volume ratio which makes the study of near wall (solid-liquid) surface more important. Also, that makes these kinds of fluids suitable for some properties that other fluids do not have like having higher heat transfer and better thermal conductivity.

One of the disadvantages of using nanoparticles that can be used for future work is that they are making the nanofluid more viscous. So, scientists are trying to produce the nanofluids in the best synthesis procedure to achieve the efficient heat transfer with having less viscosity.

1.2 Electrokinetics

1.2.1 Electroosmosis, Electrophoresis, and Streaming Potential

Electrokinetic phenomena includes electro-osmosis, streaming potential, and electrophoresis.

It happens when there is a surface charge distribution on the surface of the particle or on the wall of the nanopore. When a surface is positively or negatively charged, it can attract anions or cations respectively which have the opposite charges. In this case the attracted charges are named counterions. The other charge that is the same as the surface charge is called co-ions. There is an

3 electric double layer between the fluid and the solid surface (EDL). As it is shown in Fig.1, the electric double layer consists of two parallel layers. The first layer is Stern layer which is exists because of the chemical interaction close to the solid surface in which the ions are located in compact by adsorption. The second layer is called diffuse layer which is created because of the

Coulomb force. This layer is compensating the net charge. The EDL is in contact with a fluid. The

thickness of the EDL is called D which has an inverse relationship with the bulk concentration of the ions in the fluid. The higher the ion concentration is the thinner the EDL thickness is. That is because the ions are not going to diffuse too much, and the thickness remains thin. The charged solid surface is absorbing the counter-ions which have the opposite charge to the surface. The next layer absorbs co-ions which are the same charge sign as the surface. And so on. Electro-osmosis happens when a solution experiences an electric field and a stationary solid charge surface exists.

However, when there is no applied electric potential, the fluid is transferring through the micro/nano channel due to the applied pressure gradient with the existence of the surface charge.

This is called the streaming current. Then relative to this streaming current, there is an electrostatic potential which can be measured as streaming potential. This streaming potential flow is coming from the potential difference on the two sides of the channel. In the diffused part of the EDL, the co-ions and the counterions are cancelling each other. The remaining charge density is moving by streaming potential to the opposite direction of the pressure driven flow like the one by the streaming current. Electrophoresis is the movement of the charged solid particles (counterions) when there is an applied electric field in a stationary fluid.

Figure 1. The schematic of Electrical Double layer (EDL)

1.2.2 Zeta Potential, Diploe Moment, and Dielectrophoresis

The potential difference between the solid surface of the dispersed particle in the conducting liquid like water and the bulk of the liquid is called the zeta potential. In ion transportation, when there is an applied electric field, it causes the excess counterions inside the EDL to migrate. As the counterions are migrating, the water is moving with those ions. So, the electro osmotic flow is forming. The molecules that are polarized are showing a positive and a negative side. The difference of this polarity is called dipole moment. The dipole moment can be charged with migration and convection of the ions which polarizes the EDL. And also, the applied electric filed causes the electrophoretic motion and the dipole moments will be modified. By having an

5 electrolyte solution with a charged and dielectric nanoparticle, the EDL can be formed close to the surface of the particle [7, 8]. Now by applying an electric field the counterions inside the EDL are repositioning which leaves the EDL to be polarized. When the EDL is polarizing, the charges of the particle is changing in a way that if we look far away from the particle the dipole moment exists. When the applied electric field is not uniform (AC electric field), then the particle is subjected to the sum of the forces on its poles and can migrate to the higher electric filed which is called dielectrophoresis (DEP). In DEP the particles are dielectric and not charged. If the particles are charged, it is called electrophoresis. DEP application is to separate, or assemble the nanoparticles [9, 10]. The biomolecules like DNA also can be manipulate with DEP [11, 12].

1.2.3 Over screening and Charge Inversion

In Fig. 2 over screening is compared with crowding. When we are using the multivalent ion systems and the solution has a high concentration of ions, then over screening happens. In this situation the first layer starts to have more counterions, and the second layer compensate that. It continues until the ions are compensating each other and neutrality reaches.

Figure 2. a) Over screening (happens at smaller voltage). b) Crowding (happens at higher voltage).

When a charged particle is in an electrolyte or when a solid charged surface is in contact with the electrolyte, the counterions are attracted to the charged surface. If the attracted counterions are large and more than the number that should be neutralized by the surface charge, then the solid surface charges appears to be reversed. This is called Charge Inversion. Most of the time this inversion of the charge happens due to the ion-ion correlations. In this case, no interaction happens between the surface charge and the ions. But the ions which are located outside of the surface are interacting together.

Although charge inversion is happening by ion-ion correlation, but sometimes the chemical similarity of counterions to the surface is responsible for that [13]. That is because the ion-ion correlations happen when other ions are distributed around one ion. And the ion-surface correlations happen when other ions are distributed around a surface.

If the number of the attracted counterions are even more so that they form some condensed layers of counterions close to the charged surface, then crowding happens. It is happening by increasing the voltage. In this case, the quality of the solutions permittivity is getting poor.

1.3 Modified Poisson-Nernst-Plank Model

When the ions are at the equilibrium that means there is no external electric field. So cations

and anions are located uniformly. In this case, assuming the ions concentration is low, the chemical potential of a single ion can be calculated by [7, 8]

ln C (1.1)

This equation is dimensionless. Here C relates to ions concentrations, and is the electric potential. We assume the particle is charged negatively which is located in a symmetric electrolyte

1 1 .

The Boltzmann distribution can be obtained by integration of Eq. 1.1, when 0 as:

C e (1.2)

Then for the Poisson equation also called the Poisson-Boltzmann (PB) eq.:

2 C C Sinh 2 2 (1.3) 2 D D

Here also the variables are dimensionless. D is the Debye Screening length which can be

* * * * *2 * 1 2 * defined as, D 1 a 1 R T 2F C0 and it a .

From the classical Boltzmann distribution in Eq. 1.2, the concentration of the counterions close to the surface is not real, especially when the zeta potential is high. This is happening in the PNP model. But in the modified PNP model this error can be corrected. Also, some phenomena like the ionic depletion is not going to be recognized by the classical Poisson Boltzmann theory. In the modified PNP model finite ion sizes and water molecules occupy a square cell with a uniform grid

* size ai [14-18]. This new model is based on the excluded volume effects. Then for electro chemical potential the modified form would be:

ln C ln 1 v 2 C C (1.4)

* *3 The final phrase demonstrates the excluded volume effects in which v 2C0 ai defined as the effective volume fraction of the bulk ions. Based on the excluded volume effects, the counterion concentrations inside the double layer cannot be more than 2 v . This constrain shows the existence of the chemical potentials in the electric double layer. The Boltzmann distribution for the modified

PNP model in the equilibrium in which 0 can be obtained by:

e C (1.5) 1 2vsinh 2 2

Eq. 1.5 considered to be saturated at a concentration of 2 v which is happening by choosing large electric potential. Then the Poisson equation defines as:

2 sinh 2 2 (1.6) D 1 2vsinh 2

To solve this equation numerically, the boundary conditions should be used as:

At r 1, (1.7)

d At r 1, (1.8) dr

At r , 0 (1.9)

By using Eq. 1.7 and 1.8 as the boundary conditions the electric potential close to the surface of the particle can be define as:

r 2 1 1 2 2 2 (1.10) 6 D v r 3 D vr 2 D v

By using boundary condition in Eq.1.7, the zeta potential is high and the counterion concentration close to the surface of the particle is maximum 2 v . Since the applied charge on the

d thickness can be calculated by equation 1.10 as 0 [19]: dr

2 1 3 lc 1 3 Dv 1 (1.11)

2 In the thin EDL or when D 0 then lc Dv . Since by Gauss law, lc , then lc

1 2 growth can be calculated by [14].

Short range forces are decreasing with the distance. For example, at a point the electric potential will be zero. Whereas, long range potentials are unlimited in distance. The electrostatic forces are considered the long-range forces. And since they decay very slowly with the distance, they are still very important in dilute electrolytes too. The minimum boundary of the correlation

2 a , and the maximum bounded distance is the z lB in which

2 lB e 4 kBT is called the Bjerrum length. If the system is deeply charged, the length scale

which the Coulomb energy between the two ions is related to the thermal fluctuation energy. For

water lB equals to 0.7nm. The ions distance determines if they are experiencing a mean field

(screened) electrostatics (their distance is more than lc ), or Columbic electrostatic correlations

(their distance is less than lc ). The modified Poisson equation is then defined as

2 4 2 lc in which the electric potential is defined as , and the ionic charged density is defined as e z n z n , where z and z are the valences of cation and anion. Also n and n are the number densities of cation and anion respectively.

1.4 Applications

Electrokinetics by application in Micro/ Nano scale is considered a very ambitious field nowadays. Its popularity in different scientific field comes from bringing the importance of some physical phenomena that are not well known in Macro scale devices.

The energy conversion efficiency which is the result of the differential pressure and inducing the streaming potential is a useful method for converting the mechanical energy into electricity.

To achieve this aim, an integration method is introduced (Lab-On-a-Chip) to have thousands of

Micro/Nano channels with the size of about 100nm on a small chip. This way, the small electronic devices can be powered efficiently. This application helps to improve rechargeable batteries on the portable devices and reduce building and discarding the replaceable batteries. The future work of the surface property for investigating of energy conversion efficiency is in high demand.

The separation technic that is performing on DNA is happening in a very good quality in

Micro/Nano fluidics by employing electrophoresis. Also, a micro total analysis system (mTAS) is characterizing DNA fragments in a very small amount of time by PCR, separation and detection techniques on a micro-scale. Other application of electrophoresis in such systems are separation in capillary electrophoresis, Separation and analysis of protein, lipids and fatty acids, DNA, RNA and Gene [20, 21].

Another electrokinetic technique is dielectrophoresis, in which the electrodes are responsible for applying an AC current on a nanofluidic system. This process can influence the nanoparticles like DNA, Protein, Cells and Viruses to be polarized by the nonuniform applied electric field and assembling them. Diselectrophoresis is an effective way of finding infection in one cell among 1e5 healthy cells [22].

Electro osmosis application is mainly in the networks of interplay capillaries like MEMS devices. Electro-osmosis is able to control the flow patterns by surface propertied like the applied electric field or surface charged. The future of the micro-scale mixing is depending on the electro- osmosis phenomena.

Nanofluids are using for removing heat from systems. So, they can be used in the cars and in any electronic devices. They have optical and biomedical applications. Nanofluids can be used as

12 the friction reduction in mechanical systems. And they have a lot of more potential applications like using in aerospace or petroleum refinery. Some of their applications are named below.

Nanofluids can be used to remove heat from electronic devices. These devices should be designed in a way that the heat can be removed easily. Scientists also try to increase the heat capacity. Nanofluids are very good solution to this problem. Nanofluids in Microchannel heat sink is already designed [23].

As nanofluids are capable to remove the heat from the systems, they can be used in automobiles

or any other engines. The size of the cooling system can be reduced. Then by using less

materials and weights, the cars are more economic and efficient. The studies in this case were

performed on the Al 2 O 3 as the nanoparticles [24]. Some more potential applications are using

in aerospace or petroleum refinery. The Micro/Nano fluidics will be changing the world like

the microelectronics did nowadays.

1.5 Objectives and Significance of This Thesis

Unlike the Poisson equation the modified Poisson equation (MPE) is considering a correlation length which makes the equation a fourth order one. This correlation length is very essential in electrokinnetics phenomena like streaming potential, streaming current, electrophoresis, dielectrophoresis and electro-osmosis. The results were the same when the classical PNP model was compared to a very small correlation length of the modified PNP model. But as correlation length was increasing for the trivalent solution, the charge inversion was happening due to overscreening of counter-ions near the charged surface. The MPE prediction of flow reversal is as a result of the ion-ion electrostatic correlation length which is defined in the new modified Poisson

13 equation as lc . This dissertation is organized as following. In chapter 2, the impact of the electrostatic correlation for a charged particle in the dielectrophoresis phenomena was investigated. Then in chapter 3, the effect of the electrostatic correlation on the ionic current rectification in a conical nanopore by using electro-osmosis phenomena was studied. And in chapter 4, the influence of the electrostatic correlation on the streaming current and streaming potential in a nanochannel was examined. Finally, some future work and conclusion is organized in chapter 5 and 6.

Chapter 2

On the Impact of Electrostatic Correlations on the Double-Layer

Polarization of a Spherical Particle in an Alternating Current Field

2.1 Introduction

Dielectrophoresis (DEP) becomes a promising technique for managing and manipulating particles in particle separation, particle assembly, and characterizing biomolecules [26-30]. DEP can also be used to measure the magnitude and , 32]. The DEP device cost is low, making it convenient for lab-on-a-chip systems. DEP finds important applications in different areas of the medical field or water management [33-39].

When a nonuniform electric field exists, a polarized particle tends to move so that it reaches to the base of the extreme electric field, which defines the DEP [31, 32]. The dipole moment can specify the direction and magnitude of the motion. When the ratio between the particle size and the characteristic length of the electric field is small, the dipole moment can estimate the DEP force. In electrolyte, the charged surface of the particle results in an electric double layer (EDL) with excess counterions. The external electric field in the electrolyte can affect the counterions as well [40, 41], causing them to migrate and induce an electro-osmotic flow. The double layer can be polarized with two kinds of mechanisms: migration and convection. This polarization can, in turn, redistribute the charge along the particle, resulting in the high-frequency dispersion. This charge redistribution inside the double layer can be approximated by a dipole moment. The migration and convection also create a bulk concentration gradient by repelling ions from the

15 double layer and attracting ions into the double layer at various sides of the particle. At a frequency lower than the diffusion frequency, ions diffuse back in a direction opposite to the migration and/or convection. The dipole moment can be modified by the bulk diffusion as well, which leads to the low-frequency dispersion.

Basically, the dipole moment can be obtained from the classical Poisson-Nernst-Planck (PNP) equations. These equations deal with migration, convection, and diffusion and the dipole moment has been calculated for spherical [42-53], cylindrical [54, 55], soft [56], and porous particles [57].

The standard PNP model can capture different frequency dispersions. A favorable agreement of the experimental results on the double-strand DNA molecules polarization in dilute solutions with the theoretical results from the PNP model has been obtained [55, 57].

In addition, under the thin EDL assumption, the surface conduction approximates the migration and convection. The PNP model can be reduced to a simple Maxwell-Wagner- i (MWO) model [58]. In this model diffusion does not take into consideration. Hence, it can only predict the high-frequency dispersion. To account for the diffusion, within the thin quasi-equilibrium EDL assumption, another simple theory named the Dukhin-Shilov (DS) model can be deduced from the

PNP model and successfully captured the low-frequency dispersion [59, 60]. In general, when the

EDL is assumed to be thin, the combination of the MWO and DS models can adequately describe the polarization process.

Although the estimations from the standard PNP model are in fairly agreements with experiments, often, they underestimated the dipole moment amplitude [61-64]. For example, recent experiments on estimating the tension inside actin filaments induced by the DEP suggest that this tension force is higher than the one obtained by the PNP model in a condensed electrolyte

[30]. In addition, in the existence of multivalent ions, the standard PNP model even cannot

16 qualitatively predict the experimental results. For example, with divalent counterions (Mg2+), experiments showed that -DNA experienced negative DEP instead of positive DEP as with monovalent ions as estimated by the standard PNP model.41 In another set of experiments, in the presence of Ca2+, the DNA orientation anisotropy under the operation of the electric field, related to the polarizability, increased and then decreased as the concentration of Ca2+ increases [66].

Again, this nonmonotonic relationship cannot be achieved through the standard PNP model.

Similarly, experiments on electrophoresis of colloidal particles also observed mobility reversal at high salts and in multivalent electrolytes that again cannot be even qualitatively captured by the standard model [67-70].

It is recognized that a phenomenon called overscreening exists in concentrated electrolytes and is prominent in the existence of multivalent ions [71]. For overscreening, the first layer with excess counterions close the charged surface excessively compensates the charge of surface and results in a new layer with excess coions, which leads to one more separate layer of charge. This process continues until the charge is totally neutralized. This charge oscillation inside EDL is attributed to Coulomb short-range electrostatic correlations [72, 73]. Recently, to capture the electrostatic correlations, Bazant et al proposed a simple continuum model, Landau-Ginzburg-type

[25]. The impact of the electrostatic correlations is included into the free energy using Cahn-

Hilliard-gradient set expansions. By letting the free energy being minimal, a fourth-order modified

Poisson equation was derived that can capture overscreening. Due to the simplicity of this model, it can be implemented numerically for electrokinetics. Indeed, this model was able to predict the electro-osmotic flow reversal near a flat surface [74] and electrophoretic mobility reversal [75].

Consider the significant discrepancies of the dipole moment between the predictions and experimental results in concentrated multivalent electrolytes and highly charged particles, where

17 overscreening is most likely to happen. It is worthwhile to investigate the impact of electrostatic correlations on the dipole moment. Besides, if we assume a thin double layer thickness, we can also extend the standard MWO and DS models to account for the electrostatic correlations. These models can provide a simpler theoretical tool to understand and design experiments.

The outline of this article is mentioned here. The modified PNP model considering for electrostatic correlations as well as a perturbation theory are introduced first, respectively. Then the dipole moments for both high-frequency and low-frequency regimes, assuming that the double layer thickness is small, are derived, followed by the results and discussion. Finally, we conclude the paper.

2.2 The Modified Poisson-Nernst-Planck Model

* 1 e j t Ee er

* 2 ez

Figure 3. The schematics of the coordinates for a submerged particle in an electrolyte.

Fig. 3 depicts a spherical particle that is submerged into an electrolyte surrounded by a uniform

AC electric field. The particle is dielectric, and it is charged uniformly. a* is the radius of particle.

* * 2 s dielectric permittivity. 1 The particle

i t undergoes an electrophoretic motion with a velocityU t U 0e . We will obtain the velocity of particle, U as a part of the solution process. To solve this problem, as shown in Fig. 3, we consider spherical coordinates r, , where the origin of the coordinates is in the center of the particle and

the angle between two vectors er and ez is defined as .

The counterions within the EDL react with the applied electric field to induce the electro- osmotic flow. Moreover, the electrophoretic motion of particle causes neighboring fluid to move

the small Reynolds number, one can use the

Stokes equation:

1 2 p2 z C z C1 u 0. (2.1) 2 D

For the incompressible condition we have

u 0 , (2.2) where p and C are the pressure, cation and anion concentration; E defines the electric field; subscript 1 defines the liquid and subscript 2 denotes the particle; and for the dimensionless

* * * 1 1 R T * * Debye screening length D * *2 * , a C0 a 2F C0

* * cation bulk concentration; R is related to the ideal gas constant; F is related to the Faraday

* constant; T defines the temperature. Variables with the superscript * show the dimensional form and those without the superscript * denote the dimensionless form. Here, length scale equals to

RT** *R *2 T *2 a* ; electric potential scale equals to ; velocity scale equals to 1 ; concentration scale F * *F *2 a *

*R *2 T *2 equals to C * ; time scale is a*2 / D* ; pressure scale equals to 1 ; and electric charge scale 0 F*2 a *2

*** R T D* equals to 1 . The dimensionless molecular diffusivities define as D (i.e. D 1). To F** a D* make the calculati D D .

To describe the ion-ion electrostatic correlations, we consider the modified Poisson equation for inside the electrolyte as presented by Bazant et al. [25]:

2 4 2 1 lc 1 1 2 zCzC , (2.3) 2 D

Where lc denotes the electrostatic correlation length. To facilitate the simulations, Eq. (2.3) can be split into two second-order equations:

2 2 1 lc gg2 zCzC , (2.4) 2 D and

2 1 g . (2.5)

The electric potential the Laplace equation:

2 2 0. (2.6)

N D C z D C 1 mC u (2.7)

Obeys the Nernst-Planck equations:

C N 0 . (2.8) t

* R*2T *2 Here m 1 is the mobility and * denotes the dynamic viscosity of solvent. * D* F *2

In a long distance from the particle,

1 rcos , g 0, Czz / ,

i t u U0e ez r , (2.9)

Where defines the correlation between the external elec equilibrium EDL.

1 2 g , 0 , 0 , n r n n 1 2

n N 0 ( r 1). (2.10)

* * In the above, the relative permittivity is defined as r 2 1 , and n is defined as the outer normal to the surface.

2.3 Perturbation Expansion

This study is considering a non-conducting particle with a spherical shape that is surrounded

* * by an applied low-intensity electric field. We assume that 2 1 , so that the electric field of the equilibrium EDL is slightly perturbed by the external electric field. Considering these assumptions, the variables can be expressed via a regular perturbation expansion:

(0) (1) 1 1 C C (0) Re( C (1) ei t ) O( 2 ) (2.11) u 0 u (1)

Here Re shows the real part and i denotes the unit complex value. To compute the force related to the particle.

2.3.1 The Zeroth Order Approach

The zeroth order concentrations C(0) can be obtained from the Boltzmann distribution as,

z (0) C(0) zze/ 1 (2.12)

(0) The zeroth order electric potential 1 obeys the modified Poisson-Boltzmann equation, which is axisymmetric, only as a function of radial coordinate (r):

d2(0) g2 dg (0) zC(0) zC (0) 2 (0) , (2.13) lc 2 g 2 dr r dr 2 D

And d2(0)2 d (0) 1 1 g (0) . (2.14) dr2 r dr

The corresponding boundary conditions are:

(0) (0) dg (1) (1) , (0) ( ) 0 , g(0) ( ) 0 and 0 . (2.15) 1 1 dr

2.3.2 The First Order Equation

The imposed electric field just affects a little the EDL at equilibrium. The linearity of the first order equations of the imposed quantities makes the variables proportional to the forcing frequency.

By plugging series (2.11) into Eqs. (2.1) (2.8) and using i to substitute the time derivative, we get:

11 (0) (0) 1 (1) (1) 0 2 1 p2 z C z C1 z C z C 1 u 0 , (2.16) 2 D

u 1 0, (2.17)

2 2 (1) (1)1 (1) (1) lgc g2 zC zC , (2.18) 2 D

2 (1) (1) 1 g , (2.19)

2 1 2 0, (2.20) and

1 1 0 1 1 0 0 1 i C C z C 1 C 1 mC u 0 . (2.21)

It is possible to further reduce Eqs. 2.16-2.21 to ordinary differential equations due to axisymmetry, which then is solvable by the commercial finite element software COSMOL 5.2®

(Comsol is a product of ComsolTM, Boston). The computing geometry contains an interval

4 between R 0 and R 10 (Figure A1 in Appendix A). We increased R and found out that there is little variation in the results, indicating that R is large enough to lead to R-independent computational results. To capture the details of the EDL, an uneven mesh was selected and more condensed meshes were implemented close to the particle surface and inside the EDL. The mesh was refined to make sure to get the mesh-independent results (Figure A2 in Appendix A). The detailed solution procedures are similar to the one in [2], except that we add two new Eq. (2.13) and Eq. (2.18) which are the second-order equations with the standard boundary condition and readily implemented into Comsol.

The electric field is slightly perturbed by the particle and the EDL. Going away from the particle, the generated field behaves the same as the one caused by a dipole, which can be expressed as

f ( r ) cos . The dipole moment coefficient is determined by the real part of f. This means r 2 that the dipole coefficient can be obtained by the electric potential from the first-order equations in the far field.

When lc 0 , Eq. (2.18) becomes the standard PNP model. Here we let lc 1 and estimated the dipole coefficient f for a variety of double layer thicknesses and zeta potentials. The comparisons between our modified model and the standard model are excellent (Figure A3 in

Appendix A). In addition, dipole moments calculated this way agree with those from the simple models under the thin double layer assumption (see below), which again validated the computational algorithm.

2.4 PNP Models

2.4.1 The Maxwell-Wagner- Model

Under the thin double layer, the spherical following equ

* * 2 1 f * * . (2.22) 2 2 1

* * * i * * Here i i i * , where 1 defines the complex permittivity of the electrolyte and 2 is the one

z F*2 DC * * for the particle. The conductivity of electrolyte is obtained from16 k* (1 ) a 0 1 z RT**

(DL)* and t ffective conductivity is * (i)* 2 s 2 2 a*

(i)* (DL)* where 2 is the intrinsic conductivity of the particle and s is the surface conductivity of the

EDL.

For thin EDL ( D 1), we have [40],

()*DL * (0) (0) (0) (0) (0) * s D zC zC2 mzC zC dr 1 . (2.23) 1

* * 2 1 2Du 1 (DL)* * * When 0 , Eq. (22) becomes f * * . Here Du s /(a 1 ) is the 2 2 1 2Du 2

Dukhin number [40].

2.4.2 The Dukhin-Shilov (DS) Model

When the frequency is high, migration and convection dominate the diffusion. The MWO model is able to adequately estimate the dipole moment. At frequencies around D* /a*2 , the surface conduction can create concentration polarization, inducing the diffusion process. The diffusion can impact both the electric current and dipole coefficient. Since the diffusion is performed through the opposite direction to the migration and convection, it reduces the dipole moment, resulting in low-frequency dispersion. We notice that since the diffusion does not play a role in the MWO model, it cannot yield the dipole moment at low-frequency ranges.

For the case of thin EDL, to investigate the influence of the diffusion on the dipole coefficient, an asymptotic analysis was presented by Dukhin and Shilov [52, 30, 59, 60]. In short, outside the double layer, the respective and the bulk concentration satisfy the Laplace and diffusion equations. In terms of boundary conditions, one can assume that the chemical potential is not dependent on the (r) coordinate (the EDL is in local equilibrium). Selecting the proper effective boundary conditions, one can find the resulted dipole moment obtained from the electric potential in the far field as,

1Wj 1 WR 2 2 R 2 1 WWjUR 2 1 UR 2 1 f . (2.24) 2 1Wj 1 WRR 2 1 WWjUR 1 UR 1

In the above,

RGm(0) zzCdr/ (0) , 1

(2.25)

r U G m r gsds() gssds () z / z C(0) dr , (2.26) 1 r 0 and

W / 2 .

(2.27)

(0) Here, G(0)( C (0) zzdr /) and g(s) cosh . The detailed derivation can be 1 found in reference [4].

At high frequencies ( 1), another form of Eq. (2.24) is,

2Du 1 f f . (2.28) 2Du 2

In the above, to simplify the equation, we use the identity R R 2Du . Hence, the high- frequency limit of the DS model converges to the low-frequency limit of the MWO model.

2.5 Results and Discussion

=0.1 d 0.5

Re(f) =0.03 d

=0.01 d

-0.5 0 5 10 10

Figure 4. The dipole moment Re f with respect to frequency for 7 and

C 0.1 for various D . The solid lines represent MWO model; the dash-dotted lines represent the DS model; and the circles, squares, and triangles represent the modified PNP model.

For typical dimensional units, the thickness of the double layer varies from nm to hundred nm.

The electrostatic correlation length is in order of nm. The zeta potential is around hundred mV.

Fig. 4 plots the dipole moment coefficient Re f with respect to frequency when 7 and

0.1 = -7

Re(f) = -6

-0.2

= -5

-0.4 = -3

-3 0 5 10 10 10

Figure 5. The dipole moment Re f with respect to frequency for D 0.01 , and

c 1. For different . The solid lines represent MWO model; the dash-dotted lines represent the DS model; and the circles, squares, and triangles represent the modified PNP model.

C lc / D 0.1for various D . Here C characterizes the relative importance of the electrostatic correlations. The solid lines represent MWO model. The dash-dotted lines represent the DS model.

The symbols represent the modified PNP model. As anticipated, the MWO model is not able to

s at low frequencies in which the diffusion is important. Also, the DS model cannot predict the high-frequency dispersion where the EDL is not in local equilibrium. Moreover, the dipole coefficients obtained by these two simple models are in good agreements with the results from the modified PNP model at high and low frequencies. In

addition, with increasing D , the predictions from the simple models deviate from those by the modified PNP model since the simple models are derived under the thin double layer assumption.

Fig. 5 plots the dipole moment Re(f) with respect to the frequency for various when

D 0.01 and c 1. For large zeta potentials, the estimations from the DS model do not match the ones predicted by the modified PNP model. Interestingly, the MWO model goes along with the modified PNP model at higher zeta potentials. The cause for this disagreement remains in the fact that the derivation of the DS model is under the local chemical equilibrium assumption for the double layer. This local equilibrium assumption was justified using the rigorous singular

2 / 2 perturbation analysis, if De 1[76]. Clearly, the DS model does not work at higher zeta potentials which break the justification.

Fig. 4 and Fig. 5 are consistent with those from the standard MWO, DS, and PNP models though there are quantitative differences. The discussions on the frequency dependence have been documented in our previous works in details [3]. Here, we do not repeat them. Instead, we focus on electrostatic correlations. In order to study the performance of electrostatic correlations in the particle polarization, Fig. 6 generates the dipole moment Re(f) with respect to the frequency for

various C when 5 and D 0.01. Interestingly, by increasing the correlation length, a non- monotonous relationship happens and causes a decrease in the dipole moment and an increase after that.

=2 c -0.2

=1.5 c

-0.3 =0.1 c Re(f) =1 c -0.4

-3 0 5 10 10 10

Figure 6. The dipole moment Re f with respect to frequency for different c when 5 , D 0.01 and the solid lines represent MWO model; the dash-dotted lines represent the DS model; and the circles, squares, and triangles represent the modified PNP model.

1 10

0 10 Du c Du Du m

-1 10

-2 10 0 2 4 6 8 10 c

Figure 7. The Dukhin Number with respect to the correlation length: The solid line represents the Du number, the dashed line represents the contribution from the convection (Duc), and the dash-dotted line represents the contribution from the migration (Dum) when 5 and D 0.01

Fig. 6 suggests a complicated influence of electrostatic correlations on the dipole moment.

Inside the double layer, charge oscillation may decrease migration and convection due to the cancellation from different charges. On the other hand, electrostatic correlations can effectively

extend the double layer thickness beyond the Debye screening length D . In particular, at large C ,

53 the double layer thickness is characterized by c D instead of D . The increase of the double layer thickness can, in turn, increase the migration and convection. Indeed, studies show that the electro-osmotic mobility decreases, changes the sign (charge inversion), and then monotonously

increases as the correlation length C increases [74].

To quantitatively characterize the importance of electrostatic correlations on the surface

conduction, Fig. 7 plots the Dukhin number as a function of C when 5 and D 0.01.

=2 c

2 =1 c

=0.01 Re(f) c 1 =0.1 c

=0.5 c

-3 0 5 10 10 10

Figure 8. The dipole moment versus frequency for different when and Re f c 7

D 0.3 .

Du rzC2 (0) zC (0)2 mrzC 2 (0) zC (0) (0) dr (2.29) 1

The surface conduction has both migration and convection component. The first term in Eq.

(2.29) refers contribution (Dum). The second term is the input from the convection (Duc). In Fig. 7, the solid line represents the Du number, the dashed line represents the contribution from the convection (Duc), and the dash-dotted line represents the contribution from the migration (Dum). Initially, the surface conduction decreases due to charge oscillation. Also, the

migration is dominant over convection. When C increases, charge inversion occurs, and the surface conduction begins to increase. Interestingly, convection contribution becomes larger than migration contribution. Eventually, the surface conduction is determined by the convection. The

electro-osmotic mobility is proportional to C when C 1[74]. Hence, it is not surprising that

the surface conduction linearly increases at large C as shown in Fig. 7.

For an arbitrary double layer thickness and correlation length, we can solve them numerically.

Fig. 8 is showing various C for the dipole coefficient Re(f) in terms of the frequency for the

condition of 7 and D 0.3 . Similar to Fig. 6, the dipole moment decreases and then rises

when the correlation length C , the trend of the impact of electrostatic correlations on the dipole moment qualitatively remains the same: the charge oscillation initially reduces the surface conduction inside the double layer until the occurrence of charge inversion. Once charge inversion is initiated, a further increase of electrostatic correlations begins to enhance convection prominently, leading to an increase of the dipole moment.

0.8

0.4

Re(f)

-0.4

-2 2 6 10 10 10

Figure 9. The dipole moment with respect to frequency with the surface Re f 2 charge density 0.16 mC/m , a 20 nm andC0 5 mM . The solid line represents the dipole moment with monovalent ions, the dash line represents the dipole moment with divalent ions ( lc 0 ), and the dash-dotted line represents the dipole moment with divalent ions ( l 1 nm ). c

Next, we examine whether the modified PNP model accounting for electrostatic correlations can adequately explain experimental observations which are unable to be estimated by the standard

PNP model. Recently, Gan [65] measured the polarizability of -DNA using the insulated-based dielectrophoresis and found out that in the divalent buffer (Mg2+) (5 mM), at low frequencies

(hundreds Hz), the -DNA experienced negative DEP. In contrast, in the monovalent buffer, the

-DNA only showed positive DEP. Fig. 9 is showing the dipole moment coefficient as a function of the frequency when the surface charge density 0.16 mC/m2 (the DNA surface charge density), the particle radius is 20 nm, and the bulk concentration is 5 mM, the same as the experimental results. In Fig. 9, the solid line corresponds to the dipole moment with monovalent

34 ions, the dash line corresponds the one with divalent ions in the absence of electrostatic correlations

(lc 0 ), and the dash-dotted line represents divalent ions when electrostatic correlations exist

(0) (lc 1 nm ). A constant surface charge is specified as 1 / n in equation (2.15). The rest of equations and the boundary conditions do not change. Clearly, it is not possible to estimate the negative dipole moment of low frequencies when using the standard PNP model. In contrast, the modified model with electrostatic correlations can successfully predict the negative DEP, suggesting the importance of the electrostatic correlations in multivalent electrolytes.

Finally, we fix the electrostatic correlation length lc and change the double layer thickness.

Fig. 9 is plotting the dipole moment coefficient in terms of frequency for various D when 7

and lc 0.2. Fig. 10 suggests that the dipole moment increases, and then decreases by increasing

EDL as a result of the electrostatic correlations. Interestingly, recent experiments on the DNA orientation anisotropy in the presence of divalent electrolytes showed a similar trend [66] which

=0.3 2 d

=0.1 d Re(f) 1 =0.5 d

-3 0 5 10 10 10

Figure 10. The dipole moment Re f with respect to frequency for various D

when 7and lc 0.2 .

35 again cannot be captured by the standard PNP model. Interestingly, our modified PNP model accounting for electrostatic correlations is in qualitative agreements.

2.6 Conclusion

Under the action of an external AC electric field, the impact of electrostatic correlations on the

EDL polarization for a spherical particle was thoroughly investigated numerically by using a modified Poisson-Nernst-Planck equation unfolded by Bazant et al. [25] In the condition of thin double layer thickness and small electrostatic correlation lengths, we investigated the Maxwell-

Wagner- model for high frequency as well as the Dukhin-Shilov model for low frequency to account for electrostatic correlations. The computed dipole moments from these simple models were in favorable agreements with the numerical predictions obtained by the modified PNP model. Consistent with the standard DS model, at higher zeta potentials, the modified DS model deviated from the full modified PNP model, limiting its application to moderately charged particles.

the thickness of EDL, and electrostatic correlation length. The impacts of zeta potential and EDL thickness on the dipole moment under the influence of electrostatic correlations are similar to those without considering ion-ion electrostatic correlations. Electrostatic correlations are more prominent in concentrated multivalent electrolytes or near highly charged surfaces. Electrostatic correlations generally induce charge oscillation. The increase of the electrostatic correlation length can eventually lead to charge inversion. This phenomenon makes the dependence on electrostatic correlations complicated. Initially, due to charge oscillation, the surface conduction decreases as both migration and convection are reduced. However, once charge inversion occurs, the electro-

36 osmotic mobility reverses and starts to increase as the electrostatic correlation length increases.

Eventually, the contribution from convection to the surface conduction becomes dominant, which

continuously increases with C . Such enhancement leads to a high-level dipole coefficient over a broad spectrum of frequency.

Comparisons performed with the experimental data in multivalent electrolytes show that the modified PNP model accounting for electrostatic correlations can qualitatively explain the experimental results that the standard PNP model fails to predict, suggesting the importance of electrostatic correlations in multivalent electrolytes.

The electrostatic correlation length is around nanometers. For an electrolyte with a

concentration larger than mM, C is above 0.1 and the impact of electrostatic correlations becomes important. When the salt concentration further increases, the influence of electrostatic correlations is more prominent. However, when the salt concentration increases to a point that the double layer

thickness is below the ion hydration, corresponding to the larger C , i.e. C 10 , ion steric effects

(finite ion size) need to be taken into consideration. Additional improvement by including steric effects [52] may further extend the applicability of the modified PNP model to even more highly concentrated electrolytes.

Chapter 3

The Effects of Electrostatic Correlations on the Ionic Current

Rectification in Conical Nanopores

3.1 Introduction

Ionic channels are important in living cells. Synthetic nanopores are important model systems for studying ionic channels [79-85]. Indeed, one of the best ways to investigate alive organ activities is to explore transportation of ions in synthetic nanopores [86-88]. Besides the implications in living cells, due to the unique phenomena occurred at nanoscale, synthetic nanopores themselves currently attract great interests as well [89]. For example, electrical measurements of nanopores show interesting properties of ion transport such as ion permselectivity [90], ion enrichment and depletion [91-93] and ion current rectification [94]. In addition, DNA and proteins can be identified or studied by synthetic nanopores via monitoring the ionic current since they change the current magnitude when translocating through the nanopore

[80, 82, 84, 95 100]. Nanopores have numerous applications in biotechnology since they can be used in many separations and sensing processes [94 96, 101 107].

The current-voltage curves through the conical nanopore pose the asymmetric behavior generally. The ionic current magnitude depends on the direction of the applied voltage across the nanopore, leading to the ionic current rectification [108]. In ionic current rectification, the magnitude of the ionic current with the electric field direction from the tip side to the base side is different from the one with the opposite electric field. The ionic current rectification can be used to make a gate inside the pore that is related to voltage fluctuation [109], transfer ions in the conical nanopore in different electrochemical characteristics [110, 111], enhance and exhaust ions [108].

Even more interestingly, in the presence of multivalent electrolytes i.e. lanthanum chloride

LaCl3, the ionic current rectification can even reverse in comparison to that in the monovalent electrolyte i.e. KCl [112], where the magnitude of the ionic current with the electric field from the tip side to the base side is smaller than that with the electric field from the base side to the tip side.

The reversal was attributed to charge inversion. Charge inversion is an inclusive incident which attracts a lot of scientific researchers from theory to applications [79, 102, 103]. It happens when there is deep interference between ions in a solution so that interfacial charges absorb ions with different charges much more than their existing charges [79]. This incident happens a lot in biological process like DNA and proteins. Charge inversion is also considered for large biological channels like Lysenin and OmpF channels [81 83]. For these channels, to act like a diode, using various pH solutions is helpful [84 86]. Although charge inversion is used to explain the ionic current rectification reversal, no detailed numerical simulations without many assumptions were carried out to thoroughly understand the underlying physics.

Overscreening or charge oscillation where the layer of excess counterions is immediately compensated by a charged layer with excess co-ions has been used to explain charge inversion

[78]. The charge oscillation inside the double layer is due to Coulomb short-range electrostatic correlations which are prominent in concentrated multivalent electrolytes [113 115]. Recently,

Bazant et al. incorporated the electrostatic correlations into a Landau-Ginzburg-typed free energy by using Cahn-Hilliard gradient-based expansions and derived a modified Poisson-Nernst-Planck

(PNP) model [78]. This model not only successfully predicted the overscreening but also explained the electro-osmotic flow reversal [116], electrophoretic mobility reversal [117], and dielectrophoretic polarization reversal [118]. The modified PNP model has also been implemented to study electro-convective instability, nano electro-osmotic flow, and electro-convective flow on

39 a curved surface [119]. However, it has not been used to study the ionic current in a conical nanopore where there is a clear qualitative discrepancy between the experimental results and the predictions from the standard PNP model. To bridge this qualitative discrepancy, it is worthwhile to quantitatively investigate the impact of electrostatic correlations on ionic currents in conical nanopores and examine if electrostatic correlations alone can explain the ionic current rectification reversal without invoking other assumptions. The main contribution of our work is to use a simple continuum model and demonstrate that the model can adequately capture essential physics and explain the reversal of the ionic current rectification in the multivalent electrolyte LaCl3. By capturing the underlying physics in multivalent electrolytes, this continuum model can also be used to study other electrokinetic phenomena related with nanopores. Although more complex models have been proposed to study electrostatic correlations [120], these models are usually non- local and involve coupled integral equations. Therefore, they are typically intractable for complex dynamic problems or complicated geometries like here. In contrast, the modified PNP model used here is simple enough to be directly applied to dynamic electrokinetic problems.

In this manuscript, we will employ the modified Poisson-Nernst-Planck (PNP) model accounting for ion-ion electrostatic correlations with the Stokes equation since recent studies suggest that the electro- be negligible at high imposed voltages and large surface charges [121].

3.2 Mathematical Model

H G Reservoir

Rb E F

0 r L

D C

Rt LR A B

Figure 11. A schematic of the simulated system and the cylindrical coordinates

In this study, we consider a conical nanopore with a length of L* (Fig. 11). The nanopore has

* * * a narrow tip radius Rt and wider base radius Rb . Two large reservoirs with a size of LR are connected to the end sides of the nanopore. The reservoirs and the nanopore are filled with an electrolyte solution with density * , viscosity * , and permittivity * . The reservoirs are set to be so large that the concentration far from the nanopore remains a constant. The nanopore has a uniform negative surface charge * charged nanopore attracts counterions and repels co-ions, forming the electric double layer. An applied electric field across the nanopore can induce an ionic current consisting of migration,

41 convection, and diffusion through the nanopore. Consider the axis-symmetry of the system. We use the cylindrical coordinate. The radial r is at a right angle, and the axial z is collateral with the axis of the nanopore. The origin of the coordinate lies at the center of the nanopore.

Next, we present the mathematical model in the dimensionless form. Here, we use the superscript * to denote the dimensional forms. The variables without superscript * are dimensionless. Due to the nature of the low Reynolds number, we employ the Stokes equation to describe the flow motion,

1 2 p2 z C z C u 0 . (3.1) 2 D

For incompressible fluid:

u 0 . (3.2)

In the above, u is the velocity vector in which ur and uz are the r and z velocity components; p is the pressure; is the electric potential; C and C are cation and anion concentration,

1 ***RT respectively; z and z are the valences of cation and anion; and D * *2 * is the Rt 2 FC0

* * dimensionless double layer thickness where C0 is the cation bulk concentration; R is the ideal gas constant; F* is the Faraday constant; and T* is the temperature.

* * * * For nondimensionalization, we chose Rt , the tip radius, as the length scale; R T /F as the

* *2 *2 * *2 * * electrical potential scale; RT/ ( FRt ) as the velocity scale; C0 as the concentration scale;

* *2 *2 *2 *2 ***** RT/ ( FRt ) as the pressure scale; RT/ ( FRt ) as the surface charge scale.

We apply the symmetric condition on the boundary AH, the non-slip boundary conditions on

specify p 0.

To account for the impact of ion-ion electrostatic correlations, we employ the modified PNP model developed by Bazant et al. [78]:

2 4 2 1 lc 2 zCzC (3.3) 2 D

Here lc is the electrostatic correlation length.

on, and migration, we have:

NCu CzC (3.4)

* * *2 *2 D R T * * Here * in which D0 * *2 , D and D are the molecular diffusivity of cation and D0 F anion.

The Nernst-

N 0 (3.5)

In terms of boundary conditions, we apply the axisymmetric conditions along the boundary

DE, the segment CD, EF, and reserv n N 0 (3.6)

For the boundary AB and GH, the concentrations are equal to the bulk concentration:

C 1andC z . (3.7)

To compute the ionic current rectification ratio IR, we need to apply the electric fields both from the tip to the base and from the base to the tip. Here, the boundary condition on the GH is

0 , (3.8) and the boundary condition on the AB is

V0 , (3.9) depending on the direction of the electric field.

. (3.10) n

Here for simplicity, we do not consider the charge regulation and their impact in the multivalent ions. For the remaining boundaries, we apply the insulated condition for the electrical potential.

The resulting ionic current through the nanopore can be obtained by integrating the current density consisting of migration, convection, and diffusion along any cross section in the computational domain. Here we integrate the current density along AB. We also integrated along other cross section areas and found out that the difference is negligible.

I zN zN nds (3.11)

The following values are used in our simulations: *10 3 kg/m 3 , *10 3 Pa s ,

T * 298 K , the diffusivities of La3+, K+, Cl- are, respectively, 6.16 109 m 2 s , 1.95 109 m 2 s,

9 2 * * and 2.03 10 m s , and Rt 5 nm . Here throughout the manuscript, the tip radius Rt is fixed to be 5 nm. The coupled nonlinear equations (Eqs. 3.1-3.5) are numerically solved by the

200 60 (A) 80 (B) (C) 0

I 40 30 -200

0 -400 0

-600 -40 -30 -40 -20 0 20 40 -40 -20 0 20 40 -40 -20 0 20 40 V

Figure 12. The I V curves for a monovalent solution KCl in a conical nanopore. a)

D 1 , b) D 0.17 , c) D 0.1 . The different lines stand for the PNP model from [44]

3 and the different symbols represent the modified PNP model with lc 10 . The solid line

(circles), dashed line (squares), and dash-dotted line (triangles) are for surface chargeies

2 2.73 ( 10 mC m2 ), 13.66 ( 50 mC m ) and 27.32 ( 100 mC m2 ). commercial finite element software COSMOL 5.2® (Comsol is a product of ComsolTM, Boston).

The computational domain is given in Fig. 11. Here, Rb6, L 200, L R 40. Further increasing

L does not change the results, indicating the computational domain is sufficiently large. Quadratic triangular elements are used for discretization. Nonuniform elements are employed adjacent to the

etails of the double layer. We refined the meshes a few times to make sure that the results are mesh independent.

Furthermore, when lc 0 , Eq. (3.3) turns to be the standard PNP model. By letting lc 1 and z 1, we can compare the predictions from the modified PNP model with the ones from the standard PNP model for monovalent electrolyte KCl for various double layer thicknesses and surface charges [121]. The I-V curves of three different double layer thicknesses and three different surface charge densities are plotted in Fig. 12. There are good agreements between the PNP model and the modified PNP model for KCl, validating the computational algorithm. In addition, it is seen that the negative and positive parts of the I V curve are not symmetric and behave like a diode, consistent with experimental observations in conical nanopores.

3.3 Results and Discussions

0 -4 -2 0 10 10 10 lc

Figure 13. Current rectification ratio IR as a function of the electrostatic correlation length

lc when D 0.2 , 40 andV0 100 .

To characterize the ionic current rectification, we can define an ionic current rectification ratio

I( V ) IR 0 . First, we examine the impact of electrostatic correlations on the ionic current I( V0 ) rectification ratio IR. Fig. 13 shows the ionic current rectification ratio as a function of the

46 electrostatic correlation length lc , when D 0.2 , corresponding to the bulk concentration

* C0 100 mM , 40 andV0 100 . Interestingly, the IR first increases, reaches the maximum, and then decreases below 1, indicating that the ionic current rectification reversal occurs.

To understand this behavior, Fig. 14 plots the averaged cross-sectional ionic concentrations of

3+ - 3 La and Cl and the electric potentials cross the nanopore for different lc and V0. Whenlc 2 10

, the ionic distributions and the electrical potential are similar to those in the absence of electrostatic correlations. Due to the asymmetry of the conical nanopore, the relative double layer thickness at the tip side is larger than that of the base side. The variation of the relative double

I V 100 II V 0 III V 100

12 (A)

0 0 100

-50 50

-100 -0.12 0

-100 0 100 -100 0 100 -100 0 100

(B) 12

0 0 100

-50 50

-100 -0.12 0

-100 0 100 -100 0 100 -100 0 100

0 0.25 100

-50 50

0 -100 0

-100 0 100 -100 0 100 -100 0 100 Z

Figure 14. The averaged cross-sectional ionic concentrations in which the solid line and the dashed line correspond, respectively, to anion Cl- and cation La3+ (the top) and 3 the averaged electrical potentials (the bottom) for D 0.2 and 40 whenlc 2 10 (a); lc 0.02 (b); an lc 0.6 (c). The left, the middle, and the right column represent, respectively, the applied voltage ofV0 100 ,V0 0 andV0 100 .

layer thickness creates a cation trap near the tip side in the absence of the external electric field

[94, 122] (Fig. 14a II).

When V0 is positive, the applied electric field drives positive ions into the nanopore from the tip side and these ions are trapped inside the nanopore, leading to a higher concentration or a higher ionic current. In contrast, when V0 is negative, the electric field is from the base to the tip. Positive ions are pulled out from the tip side of the nanopore and enter from the base side. Since the electric field at the tip side is larger than the one at the base side, fewer ions are present inside the nanopore, leading to a lower ionic current. Accordingly, the ionic current rectification ratio is larger than 1.

When lc 0.6 , the impact of electrostatic correlations is prominent, causing the charge to oscillate. Without the external electric field, the charge oscillation induced by electrostatic correlations results in anion trap instead of cation trap. This anion trap leads to a larger ionic current when V0<0. Hence, the ionic current rectification ratio becomes less than 1 or the ionic current rectification reversal occurs, which is successfully predicted by the modified PNP model.

Interestingly, Fig. 13 also suggests that when lc 0.02 , the ionic current rectification ratio increases slightly compared to the smaller lc. It can be understood due to the charge oscillation induced by ion-ion electrostatic correlations, more negative ions are stored inside the nanopore.

At the same time, since lc is still small orlc/ D 0.1, cation concentration does not change significantly. As a consequence, the net ionic current increases slightly, resulting in an increase of the current rectification ratio. Recent experiments showed that by adding LaCl3 into the KCl solution, the rectification ratio slightly increases and then decreases to less than 1 [113]. Our theoretical predictions are qualitatively consistent with the experimental observations: small ion- ion electrostatic correlations can slightly increase the anion or Cl- concentration without

49 significantly decreasing the cation concentration, leading to a slight increase of ionic current rectification ratio.

We examine the impact of the surface charge and double layer thickness on the ionic current rectification in the presence of the electrostatic correlations. Fig. 15 plots the ionic current

rectification ratio as a function of 1/ D for both 40 and 20 when lc 0.6 andV0 100 .

0.9

0.8 IR 0.7

0.6

0.5 2 1/ 3 4 5 d

Figure 15. The current rectification ratio IR as a function of 1/ D when lc 0.6

andV0 100 . The line with stars and that with circles represent, respectively, to 20 and 40 .

Both rectification ratios are less than 1, suggesting that electrostatic correlations are important to induce the anion trap near the tip region of the nanopore. In addition, the rectification ratio of

40 is generally smaller than that of 20 since the impact of the electrostatic correlations is more significant at a larger surface charge. It is not surprising that the ratio is smaller. In addition, as the relative double layer thickness increases, the ionic rectification ratio decreases. This trend is consistent with the one without the electrostatic correlations. This is attributed to the overlapping of the double layers [121].

6000

Q 2000

0 0.1 0.2 0.3 0.4 lc

Figure 16. The flow rate as a function of lc when D 0.2 ,V0 100 and 40 .

A 2.5 B 40 35 2 30 1.5 25 20 1 15 10 0.5 5

3 Figure 17. The flow field near the tip of the conical nanopore for (a) lc 2 10 and (b) lc 0.6 when D 0.2 , V0 100 and 40 .

Next, we examine the impact of the electrostatic correlations on the induced electroosmotic flow. Fig. 16 plots the flow rate Q as a function of the electrostatic correlation length lc when

D 0.2 , V0 100 , and 40 . Consistent with the impact of electrostatic correlations on the electro-osmotic flow near a flat surface [116], a flow reversal is observed with the increase of lc and the flow rate is proportional to lc when lc is large. Fig. 17 shows the flow field for different lc.

A flow reversal is clearly demonstrated.

Figure 18. Distribution of the concentration of cation (La3+) near the tip region of 3 the conical nanopore for (a) lc 2 10 and (b) lc 0.6 when D 0.2 ,V0 100 and 40 .

3 Finally, Fig. 18 plots the cation concentration distribution near the tip regions for lc 2 10

3 lc 0.6 when D 0.2 ,V0 100 and 40 . Interestingly, atlc 2 10 , the

concentration is depleted outside the tip region. In contrast, when lc 0.6 , the concentration is enriched outside the tip region. In other words, at large electrostatic correlations, the concentration polarization is reversed as well, similar to the flow reversal.

3.4 Conclusions

By considering the effects of ion-ion electrostatic correlations, we investigated the ionic current rectification in a conical nanopore in the presence of multivalent electrolytes by using the modified continuum Poisson-Nernst-Planck (PNP) equations presented by Bazant et al. [78]. Here we chose the trivalent cation solutions consisting of lanthanum chloride as the electrolyte solution and focused on the high salt concentration (~100 mM), where the double layer thickness is smaller than the nanopore radius. It is recognized that ion-ion electrostatic correlations are prominent in concentrated multivalent electrolytes and can lead to the charge oscillation. Our numerical results showed that at larger electrostatic correlation lengths, the ionic current rectification reverses, leading to the ratio less than 1. By examining the electrical potential distribution in the absence of external electric fields, we showed that the cation trap at the tip side with a smaller electrostatic correlation length turns to be the anion trap with a larger electrostatic correlation length, explaining the ionic current rectification reversal. Our studies concluded that ion-ion electrostatic correlations inducing the charge oscillation are responsible for the ionic current rectification reversal. In addition, our studies also suggested flow reversal and concentration polarization reversal induced by ion-ion electrostatic correlations. Our studies can help to better understand experimental results and the modified continuum PNP model can also be used to design various nanopore based devices with multivalent electrolytes.

Chapter 4

The Impact of Electrostatic Correlation Length on Streaming Current and Streaming Potential in Nanochannels

4.1 Introduction

Micro/Nano fluidics has got a lot of improvements in electrokinetics (e.g. electrophoresis, electro-osmosis, streaming potential and streaming current) [123, 124]. When an electrolyte with the polarizability effect is in contact with a hard surface that makes the surface to be charged [125].

The counter-ions are absorbed to the surface charge. As a result, the Electrical Double Layer (EDL) is forming adjacent to that surface which is consist of immobile ions (stern layer) and mobile ions

(diffusion layer) [126, 127]. The thickness of EDL is named the Debye length [128]. The importance of the surface charged in nanochannels are resulting from the high surface-to-volume ratio [129, 130]. When there is no applied electric field, the only driving force is the pressure difference at the nanochannel. Due to this pressure driven flow, the fluid moves the charges inside the EDL toward downstream. The resulting electrical current is called the streaming current. The charges that obtained in this way, can be measured as the electric potential difference at the outlet, which is called the streaming potential [131]. Having streaming current and potential establishments give us the opportunity to achieve electrical power from mechanical energy [132-

141]. By knowing the amount of pressure between inlet and outlet and the volume flow rate, the input power can be achieved. The calculated streaming potential and streaming current at the outlet leading to the calculation of the output power. The hydrodynamic input power is later turning into

54 the electrical output power [134]. The efficiency then can be calculated by knowing the input and output power. Nanoscale channels have the ability to represent the overlapped EDL in which the highest efficiency is happening [142-147]. Heyden et al. [148] performed experiments for measuring the streaming currents in a silica rectangular nanochannel. The streaming current had a linear relationship with the pressure difference and the sign of the nanochannel was responsible to change the direction of the streaming current. The purpose of this study is to measure the streaming current and streaming potential and the efficiency inside the nanochannel by performing the modified PNP model presented by Bazan et al. [25]. The solution consists of Lanthanum Chloride

(LaCl3) in the water solvent. Lanthanum is a three-valence white and soft metal. It is called a rare earth metal which can be found in other sediments. Lanthanum has a lower electrical conductivity compare to some other three valences cations. The counterions with lower electrical conductivity generate more efficiency due to having less power dissipation.

In experiments the maximum efficiency of a nanochannel by using KCL as the solution was just about 3% for a silica nanochannel with the 75nm height [149]

4.2 Mathematical Model

4.2.1 Governing Equations

Fig.19 shows a schematic of simulated system in which a two-dimensional geometry is considered in symmetry. The Cartesian coordinates (x, y, and z) are employed. The rectangular silica nanochannels are connected to two big reservoirs.

To solve this model we introduce the mathematical part in which we assume: (i) an equilibrium

(chemically and dynamically) condition; (ii) the system is solved

55 molecules are much shorter than the size of the channel; (iv) the stern

ppropriate

Poisson Boltzmann (PB) is considered (the electrical double layer thickness is one tenth of the channel height) [150]. (vii) Chemical reaction is considered to be zero. (viii) The viscosity is always higher in nanochannel than the reservoir which makes all the pressure drop inside the nanochannel and it is negligible in the reservoirs. Then we can define the governing equations as following:

The flow inside the nanochannel is considered to be creepy flow so the motion of this flow is defined by Stokes equation as

1 2 p2 z C z C u 0 . (4.1) 2 D

Also the fluid is incompressible so we have:

u 0 . (4.2)

In which the velocity vector is defined as u ; pressure is defined as p; electric potential is defined as ; cation and anion concentration are defined as C and C respectively; also valences of cation and anion are defined as z and z ; and the dimensionless double layer thickness is

* * * 1 R T * * * defined as D *2 * where C0 is the bulk concentration; R is the ideal gas constant; F is a 2F C0 the Faraday constant; and T* is the temperature.

Here the superscript * denotes the dimensional forms otherwise it is considered nondimensional.

All the equations are nondimensionalized by using the following scales. The size of the nanochannel is calling a* , which can be used as the length scale; R*T*/F* as the electrical potential

* *2 *2 * *2 * * * *2 *2 *2 *2 scale; R T F a as the velocity scale; C0 as the concentration scale; R T F a as the pressure scale; *R*T * F*a* as the surface charge scale. The modified PNP model which is developed by Bazant et al. [25] is using for the impact of ion-ion electrostatic correlations as:

2 4 2 1 lc 2 zCzC . (4.3) 2 D

in which lc is defined as the electrostatic correlation length.

The flux density is normalizing by U0C0 as:

NCu CzC . (4.4)

* * *2 *2 D R T * * In which * also D0 * *2 , D and D represent the molecular diffusivity of D0 F positive and negative charges.

The Nernst-Planck equations for ation:

N 0 (4.5)

4.2.2 Computational Domain

The geometry is designed in two dimensional in which a nanochannel is connected to two big reservoirs. Fig. 19 is showing the geometry of simulated system. The length of the nanochannel

57 defined as L. The width and height are equal to a=h=5nm. At the end of the nanochannel, there are

two reservoirs with the size of LR . The concentration remains constant far from the nanochannel due to the larger sizes of the reservoirs. The large reservoirs cause all the pressure difference to be happening inside the nanochannel. The negative surface charge * is just applying on the wall of the nanochannel and other walls are remained unchanged.

After the nondimensionalization by sionless

lengths as: LR 25, L 50, a 1.

H G Reservoir LR

E F

e y n n

a L

h 0 c x o n a N

D C a

A B

Figure 19. A schematic of simulated system

The parameters are defined as: *10 3 kg/m 3 , *10 3 Pa s , T * 298 K , the diffusivities of La3+, K+, Cl- are, respectively, 6.16 109 m 2 s , 1.95 109 m 2 s, and 2.03 109 m 2 s . In order to solve these equations (Eqs. 4.1-4.5), the commercial finite element software COSMOL 5.4®

(Comsol is a product of ComsolTM, Boston) was employed. The main domain is meshed with

Quadratic triangular elements. The wall of the nanochannel uses Nonuniform elements along with

60 elements of boundary layer to capture the effect of the EDL thickness. The mesh is checked and refined many times so that the results do not depend on mesh anymore.

4.2.3 Boundary conditions

Nanofluidics boundary condition should be choosing carefully. We are defining a silica surface for the nanochannel in which the boundary condition is nonslip. Some research shows the improvement in energy conversion efficiency by using the slip boundary condition for nanochannel walls [151-153]. For channel heights size of 2 nm or higher, the nonslip wall is acceptable [154-158].

The boundary conditions for Navier-Stokes (N_S equation) are defined for the pressure at the

AB and GH, and also, we specify the slip and no-slip walls in N_S equation. As a result, the pressure for inlet and outlet equals to P1=1, 2, 3, 4 and P2=0 respectively. Slip wall condition applied for the reservoir walls BC and FG. While a nonslip boundary condition is used for the segment ED which is the nanochannel wall. Axisymmetric condition is applied on the boundary

AH.

The boundary conditions for Poisson Equation are designated for the surface charge density and the gradient of the potential. The boundary condition on the wall DE is described as the applied surface charge density as,

. (4.6) n

To get the results of the simulation for streaming current both inlet (HG) and outlet (AB) are assigned a zero potential so that the only driving force is Pressure:

0 , (4.7)

For simulating the streaming potential, the electric potential will stay zero at the inlet and it is going to be calculated at the outlet.

The boundary conditions for Nernst Plank Equation explain the concentration far from the nanochannel at the inlet and outlet with a no-flux boundary for the walls.

The boundary conditions on AB and GH for concentration are:

C 1andC z . (4.8)

We have a no flux condition for all the walls:

n N 0 (4.9)

4.2.4 Energy Conversion Efficiency

The streaming current can be obtained by integrating the Ionic current at the two ends of the

2 reservoirs. And the dimensionless ionic current can be normalized by FU0C0a as:

I zN zN nds (4.10)

Also, the streaming current can be integrated on any cross section inside the nanochannel by using the integration of:

2a I str C C udy (4.11) 0

The accumulated ions at the end of the nanochannel induces an electrical potential which is called streaming potential. This streaming potential is inducing the conduction current. The direction of this current is opposite of the streaming current. So, the total ionic current should be

equal to zero as Iionic Icond I str 0. This phenomenon called the electro viscous effect. Which is quite different in soft and rigid nanochannels [159].

Then the Maximum power generation (P) can be calculate by using half of the Streaming

Current (I), and Streaming potential (V) as:

IV P out 4

And the efficiency can be obtained by:

P IV out , (4.12) Pin 4Q( p)

Here the flow rate (Q) can be calculated by integrating the velocity profile. The pressure difference p , causes the the ions are accumulating at the end of the channel. That is why the channel is in generation mode like a battery. If the channel is in the pumping mode in which the voltage is decreasing as the pressure increasing toward the outlet, then the efficiency calculation is power input divided by power output. So, the energy conversion is happened inside the nanochannel in two ways which are called Pumping and generation modes. The pumping mode is related to electro-osmosis in which electrical energy is converted into the mechanical energy. While the generation mode is related to the streaming potential in which mechanical energy is converted into electrical energy.

4.2.5 Validation

Figure 20. The efficiency in terms of the bulk concentration for monovalent solution KCL. The dashed line represents for the PNP model and the symbol represents for the Modified PNP model at 1mM KCL solution for Here surface charge

density is , and the size of the nanochannel is 2h=60nm.

In order to realize the verification of the results, the potassium chloride (KCL) solution is

solved for the modified PNP model with the correlation length lc 1. The calculated efficiency in terms of the bulk concentration of the KCL is plotted in Fig. 20 in which the modified PNP model and the classical PNP model at the bulk concentration of 1mM are compared. So the computational method is verified due to this comparison.

4.3 Results and Discussion

10-3 4 C

=0.3 d 2

=0.1 d

=0.01 d 0 =0.5 d

-2 0 1 2 3 4 Pressure

Figure 21. Streaming current and streaming potential in terms of pressure for a 5 nm channel height and Sigma=-10 . Part A & B are for l =0.001 with no charge inversion. Part C & D are for l =0.3 with charge inversion happening at higher .

In Fig. 21, the streaming current and streaming potential in terms of the pressure are showing linear relationship. By increasing the EDL thickness the higher correlation length demonstrates a reversing direction which is the indication of charge inversion for the three valences ions. This

charge inversion can be seen in Fig. 21 at l c =0.3. The simulation was performed for a channel height of (h= 5 nm) and an applied sigma of (-10 mC m2 ).

Figure 22. Efficiency in terms of bulk concentration (mM) for sigma=-10 and h=30nm.

Figure 23. The efficiency in terms of bulk concentration for C0=10.47 (mM) and 20(mM), l =0.001, 0.1, 0.3, 0.5

To compute the efficiency, different correlation lengths were simulated for a fixed Debye length. The efficiency was calculated from the nondimensional parameter in the result section by using Eq.4.12. First the streaming current was simulated separately by choosing the inlet and outlet to be in zero voltage so that the only effective force is the pressure difference. In this case, the calculated streaming current at both sides of the reservoir was the same and obtained by Eq.4.10.

Then for streaming potential the voltage at the inlet was set to be zero and at post processing the streaming potential was calculated at the outlet. Fig. 22 shows that for smaller bulk concentration,

66 the calculated efficiency is decreasing by increasing bulk concentration when the correlation length is very small. While the result is reverse as correlation length is increasing which is captured by modified PNP model.

In Fig. 22, at a very small correlation length at part A, the efficiency is like the classical PNP model in which the efficiency is decreasing by increasing the bulk concentration. And as correlation length is increasing at first there are some oscillations (part B) then the charge inversion is happening at higher correlation length (part C&D) and as a result the efficiency is increasing by increasing bulk concentration. Also, overall efficiency is increasing by increasing the correlation length. In Fig 23. The correlation length increased to 20 and as one can see the charge inversion

is happening at higher correlation length (l c =0.5).

Figure 24. Cation and anion cross sectional averaged for sigma=-10 , h=30nm,

A) C0=1mM and lc=0.001, B) C0=10.47mM, lc=0.001, C) C0=1mM, lc=0.3, D) C0=10.47mM, lc=0.3.

In Fig. 24, the averaged cross-sectional ionic concentration is plotted for the cation (c1) and anion (c2) concentration of LaCl3. EDL thickness of two bulk concentrations and two correlation

68 lengths are compared. In which for smaller correlation length both species show enrichment inside the nanochannel, but at higher lc, cations are enriched, and anions are depleted. That also explains the reason of having higher efficiency in Fig. 22 and Fig. 23, by having higher correlation length.

there exist both anions and cations inside the channel. Although the applied surface charge is negative, so the cations are the majority and their concentration are higher than the anions.

For l c =0.001, the depletion happens at the inlet while the ions are accumulating at the outlet in which streaming potential is inducing a conduction current. This current is in the reverse route of the streaming current and is responsible for the zero counterionic flux and no concentration polarization effect for streaming potential case.

For both, the classical PNP model and the modified PNP model with very small correlation length, the maximum efficiency is happening when the EDL is overlapping. But by using the modified PNP model for higher correlation length, the lower EDL has the higher efficiency for a fix correlation length. That means as correlation length is getting bigger the maximum power is happening when the EDL is smaller in which the bulk concentration is dominated by the cations.

To make this clearer, Fig. 25, shows the concentration of cations close to the inlet at the streaming potential mode. For streaming potential, the inlet electrostatic potential is set to zero.

And at the outlet the electrostatic potential is going to be obtain in the post processing.

3+ Figure 25. Concentration of Cations (La ) inside the nanochannel near the inlet for Sigma=-10 and C0=1mM. a) lc=0.001 b) lc=0.5.

Figure 26. Distribution of electric potential inside the nanochannel near the outlet for C0=1mM and sigma=-10 . a) lc=0.001, b) lc=0.3

is plotted in Fig. 25 for lc=0.001 and lc=0.5. The concentration of the cations is getting higher as the correlation length is getting bigger.

In Fig. 26, the electric potential was shown in nanochannel close to the wall. For the same bulk concentration there is a higher measured electrical potential within the nanochannel with a higher correlation length.

At part A, the electric potential is getting higher toward the centerline while at part B, the higher electric potential was moved closer to the wall with the higher value. That is the reason of having higher maximum power and maximum efficiency as the correlation length is increasing.

4.4 Conclusion

The modified continuum Poisson-Nernst-Planck (PNP) equations presented by Bazant et al.

[25] is used in this study to explain the ion-ion electrostatic correlation length effect on streaming current and streaming potential in a nanochannel. The working fluid was chosen as a three-valence medium called lanthanum chloride. Different bulk concentrations which lead to various EDL thickness were examined. Both streaming current and streaming potential show a charge reversal inside the nanochannel by having a higher correlation length and smaller bulk concentration (larger

EDL thickness). The efficiency is improving by considering the electrostatic correlation length.

By using the Correlation length used in modified PNP model we were able to capture the depletion and enrichment of anions and cations. The cations concentration near the nanochannel wall are smaller for the smaller correlation length and higher concentration expected for higher correlation length.

Chapter 5

Conclusions and Future Work

5.1 Conclusions

The application of Electrokinetics in nanofluidics regime was investigated in this study by using Modified Poisson-Nernst-Planck model. Dielectrophoresis, electro osmosis, and streaming potential was considered in three different studies.

The dielectric particle that experiences an AC electric field, tends to move toward the downstream. This particle senses an electric field which creates a velocity and can be measured at the end of the calculation. The diploe moment which is the net charge on the two sides of the particle can be measured by classical and modified PNP model. In this study, the modified PNP model was successfully presented the measurement of these results more accurately and efficiently. In conclusion the modified PNP model was able to predict a wide range of frequency for dipole moment. Also, the effect of different correlation length for measuring the dipole moment was investigated and a charge reversal for higher correlation length in multivalent electrolyte was observed.

In another study that was performed in the conical nanopore in the action of an AC electric field, electro osmosis in three valence electrolytes was able to capture the charge inversion by considering the electrostatic correlation length in modified PNP model. The flow field was plotted and a charge reversal in ionic current was obtained for the correlation length more than 0.6.

Finally, the streaming potential showed a charge inversion in calculating the efficiency for different bulk concentrations by using different correlation lengths. For very small correlation

72 length, the efficiency is decreasing as the concentration is increasing. This is the same as the classical PNP model. The three-valence ionic current was considered in this study and it was seen that the efficiency is increasing by considering correlation length. As the correlation length is increasing, first oscillation is happened and then the charge inversion is happened. The depletion and enrichment of anions and cations which are happening at higher correlation length are the reason of having higher efficiency in nanochannel by existing ion-ion correlation length.

In conclusion, these results may not be captured by the standard PNP model which shows that the ion-ion electrostatic correlation is important as the surface to volume ratio is increasing.

5.2 Future Work

The nanofabrication is still in developing stage. The main application of that is in biochemistry.

For example, the production of lab-on-a-chip systems is limited to a handful of products nowadays.

There are some great ideas, commercialize them yet. The fundamentals are well established but the real sample tests still needs more work due to its complexity. Also, the materials that the micro pipes are made up should be examined for different purposes. The preparation process of nanoparticles needs to be studied. The synthesis process is expensive. And nanoparticles tend to aggregate, and sedimentation happens [160, 161]. The leaking issue of Micro/Nano devices should be taken into account. The bubble formation during separation process is problematic.

Another challenge is that having the large surface to volume ratio, increases the evaporation rate for microsystems. This issue should be overcome. This area of study needs a level of funding for industrial research.

Moreover, to overcome the designing issue for micro/nanofluidics, the computer model should be able to simulate the micro fabrication design fast and robust for the coming future products.

Some of the future works of the electrokinetics with the application on Micro/Nanofluidics are the ability to scaling the devices down along with the cooperation of fluid flows and forces along the boundaries with the ionic interactions.

Lab-On-a-Chip devices right now are at the laboratory testing but soon, everybody will experience having their own diagnostic devices just like a personal computer that we are using today [162].

Appendix A

Figure A1. The simulation domain including the particle (domain 1) and the electrolyte (domain 2).

Figure A2. Mesh-independence studies: the dipole moment with respect to the frequency for the mesh size10 3 ,5 10 4 , and 10 4 when 7, 0.01, and 0.1. They show excellent D c agreement.

Figure A3. The dipole moment as a function of 2 frequency D for 3 and D 1, D 2, D 3, and

D 4 . The solid line and the symbols are respectively the ones from the standard PNP model and those from the modified PNP model accounting for electrostatic correlations. Here c 0.01.

Appendix B

The Model Setup in Comsol

Here we used the general form in Comsol to set up the zeroth order and the first-order perturbation equations. In Comsol, the general form has a format:

d F (B1) dr

In the above, and F can be specified as an arbitrary function. For example, for equation (2.13),

2 0 0 0 lc d 2 dg 0 z C z C 2 r g 2 (B2) r dr dr 2 D

Eq. (B2) can be rewritten as

0 0 0 d 2 2 dg 2 0 2 z C z C lc r r g r 2 (B3) dr dr 2 D

We can implement Eq. (B3) into Comsol by assigning

0 0 2 2 0 2 0 2 z C z C lc r gr and F r g r 2 (B4) 2 D

In this way, Eq. (2.13) is setup in Comsol. Similar procedures were applied to other equations.

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Curriculum Vitae

Graduate College University of Nevada Las Vegas

Elaheh Alidoosti Email: [email protected]

Research Interests Computational fluid Dynamic including Finite Element, Finite Volume Electrokinetics, designs and applications Design and Multiphysics analysis Nanotechnology and Applications Micro/Nanofluidic devices

Education University of Nevada Las Vegas (UNLV); (2016-2020) Department of Mechanical Engineering PhD in Mechanical Engineering (GPA 3.69/4) Advisor: Dr. Hui Zhao

University of Nevada Las Vegas (UNLV); (2011-2014) Department of Electrical and Computer Engineering M.S. in Electrical Engineering (GPA 3.77/4)

Thesis: Simulation of Thermal Transit-Time Flow Meter for High Temperature, Corrosive and Irradiation Environment Advisor: Dr. Yingtao Jiang

Research Experience Dept. of Mechanical Engineering, University of Nevada Las Vegas, (2016-2020) Research Assistant Calculation of Streaming Currents and streaming potential in a Nanochannel Evaluating the Effects of Electrostatic Correlations on the Ionic Current Rectification in Conical Nanopores Evaluating the Impact of Electrostatic Correlations on the Electrokinetics Design and modeling the Impact of Electrostatic Correlations on the Double-Layer Polarization of a Spherical Particle in an Alternating Current Field

Dept. of Electrical and Computer Engineering, University of Nevada Las Vegas, (2011-2014) Research Assistant Studied numerically and analytically measurement of the flow in a pipe and proposing a new technique for flow measurement analysis. Designed and simulated a new flow measurement method based on correlated thermal signals using ANSYS software package Teaching Experience Dept. of Mechanical Engineering, University of Nevada Las Vegas, (2016-2020) Teaching Assistant Analysis of Dynamic Systems (ME 330) Spring 2018 Undergraduate level course; helped the instructor. Graded the homework. Fluid Dynamics Laboratory (ME 380L) Spring Fall 2018 Undergraduate level course; helped the instructor. Worked in the lab. Helped students for final reports. Graded the lab reports. Engineering Thermodynamics I (ME 311) Spring 2019

Undergraduate level course; helped the instructor. Graded the homework. Fluid Dynamics for Mechanical Engineers (ME 380) Spring Fall 2019 Undergraduate level course; helped the instructor. Graded the homework. Introduction to Heat Transfer (ME 314) Fall 2019 Undergraduate level course; helped the instructor. Graded the homework.

Publications [1] E. Alidoosti and H. Zhao, Calculation of Streaming Currents in a Nanochannel preparation. [2] E. Alidoosti and H. Zhao, The Effects of Electrostatic Correlations on the Ionic Current Rectification in Conical N Electrophoresis, 2019, 00, 1-7. [3] E. Alidoosti ct of Electrostatic Correlations on the Double-Layer Langmuir, 2018, 34(19), 5592 5599. [4] E. Alidoosti and H. Zhao, The impact of electrostatic correlations on Dielectrophoresis of Non- conducting Particles, The 70th Annual Meeting of the APS Division of Fluid Dynamics, vol. 62, No.14. (2017), Denver, CO. [5] E. Alidoosti, J. Ma, Y. Jiang and T. - Time Flow Meter for High Temperature, Co ASME 2013 Heat Transfer Summer Conference, pp. V004T19A006; 6 pages, July 2013.

Awards Graduate Fellowship, University of Nevada Las Vegas (Spring 2020) Patent: Inventors: Yingtao JIANG, Elaheh ALIDOOSTI, Taleb MOAZZENI, Efficient data transmission using orthogonal pulse shapes, WO2017027469A1, 8 August 2016

Skills Extensive experience with commercial finite element and finite volume software COMSOL, MATLAB, and CFD ANSYS Fluent Proficiency in Excel, Word, Strong in modeling nanofluids, and thermal heat transfer Strong in numerical simulation Strong interpersonal skills

Independently learner and motivated

References 1. Dr. Hui Zhao Associate Professor, Department of Mechanical Engineering University of Nevada Las Vegas, Las Vegas, NV 89154 Phone: 702-895-1463, E-mail: [email protected]

2. Dr. Yingtao Jiang Professor, Department of Electrical and Computer Engineering University of Nevada Las Vegas, Las Vegas, NV 89154 Phone: 702-895-2533, E-mail: [email protected]

3. Dr. Yi-Tung Chen Professor, Department of Mechanical Engineering University of Nevada Las Vegas, Las Vegas, NV 89154 Phone: 702-895-1202, E-mail: [email protected]

4. Dr. Taleb Moazzeni Senior Lecturer, Department of Electronics and Telecommunications Global College of Engineering and Technology, Muscat, Oman Phone: +9 687-192-8276, E-mail: [email protected]