i

ELECTROKINETIC TRANSPORT WITH TEMPERATURE AND SOLUTAL GRADIENT

Thesis submitted in partial fulfillment of the requirements for the award of the degree

of

Master of Technology in

Mechanical Engineering

(Thermal Sciences and Engineering)

by

Arka Prabha Roy Roll no. 07ME3203

Under the guidance of

Prof. Suman Chakraborty

DEPARTMENT OF MECHANICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR

MAY 2012

ii

CERTIFICATE

This is to certify that the thesis entitled Electrokinetic Transport with Temperature

and Solutal Gradient, submitted by Arka Prabha Roy (Roll No. 07ME3203) in partial

fulfillment of the requirements for the degree of Master of Technology in Thermal Sciences

and Engineering in the Department of Mechanical Engineering, is a bona fide research work

carried out by him under my supervision and guidance during the year 2011-2012.

Prof. Suman Chakraborty

Department of Mechanical Engineering

Indian Institute of Technology, Kharagpur

iii

DECLARATION

I hereby declare that:

a. The work contained in this thesis is original and has been done by me under the

guidance of my supervisor.

b. The work has not been submitted to any other institute for any degree or diploma.

c. Whenever I have used materials (data, theoretical analysis figures, text) from other

sources, I have given due credit to them by citing them in the text of the thesis and

giving their details in the references.

Arka Prabha Roy Department of Mechanical Engineering

Indian Institute of Technology, Kharagpur

iv

ACKNOWLEDGEMENT

I would sincerely thank my project supervisor- Professor Suman Chakraborty for

giving me opportunity to work in this fascinating topic and for his continuous motivation.

I would also like to thank my senior Mr. Jeevanjyoti Chakraborty for guiding me in

the analytical work pertaining to computer simulations.

I pay my sincere regards to my father Mr. Nirmal Kumar Roy and my mother Mrs.

Sujata Roy, who have encouraged me and helped me to remain motivated throughout my life.

I would also like to take this opportunity to thank Pratiti and my brother, Anish for their

unfailing faith in me and for their support throughout the duration of the work.

v

ABSTRACT

Electrokinetic transport has been utilized extensively in the micro-/nano-fluidics

community and the fundamental understanding behind such transport has also been

investigated for long in the colloidal science literature. Yet, most traditional studies have

focused rather simplistically on phenomena that are intrinsically coupled in reality through

expeditious assumptions leading to modeling paradigms that are, at best, piecemeal

representations of such real problems. As such, looming questions in electrokinetic transport

coupled intricately with ancillary flow effects still do remain unanswered; these have

potentially far-reaching consequences in micro-/nano-fluidics. The present thesis addresses

two such coupled phenomena with the critical role of osmotic pressure in establishing the

concerted interplay (between the fluid dynamics and the temperature and/or the

concentration) acting as the common refrain in both.

In the first example problem, a non isothermal electro-osmotic fluid flow has been

considered for a micro-confinement filled with a binary electrolyte. We have incorporated

the effect of the induced thermo electric field due to the temperature gradient with the

traditional electrokinetic flow in a microchannel to study the changes in the ionic species

distribution and the velocity profile. In the process we have shown that the deposition of the

ionic species on the charged surface is altered in an uneven way. We have found that the

velocity profile for a non-uniform temperature distribution significantly varies from the

isothermal condition. The discrepancy also depends heavily on the characteristics of the ions

and the zeta potential of the wall surfaces.

In the second example problem, a novel transport mechanism which we term

“induced  diffusioosmosis”  is  brought  about  through  imposition  of  a  zeta  potential  gradient  as  

a clear departure from the established norm of actuating diffusioosmotic flow through

imposed concentration gradients. Since under conditions of electric double layer (EDL)

overlap, the channel centre-line concentration is a function of the zeta potential, it is the

gradient of the zeta potential which induces a gradient in concentration implicitly. It is this

latter induced concentration gradient which results in the diffusioosmotic flow. The induced

diffusioosmotic flow is computed as a function of the gradients of zeta potential at the walls

vi

of the channel, the concentration of the electrolyte solution and reservoir concentration. It is

revealed that for thick EDLs, the fluid velocity increases with increasing magnitudes of the

zeta potential and increasing reservoir concentration. As a concrete proof of concept we also

show that the diffusioosmotic flow vanishes when the gradient in the zeta potential is set to

zero.

vii

LIST OF SYMBOLS

Zeta Potential

Debye Length

Induced Potential in medium

u

u

v

E

T1(x)

T2(x)

0T

e

Applied Potential

Velocity vector

Horizontal velocity component

Vertical velocity component

Electric field acting in the system

Temperature at the top wall surface

Temperature at the bottom wall surface

Reference temperature at the inlet

Charge density

i

E

Electrochemical potential of i-th species

Electronic charge

z

Valency of positive / negative species

n

n

J

D

Number density of positive / negative species

Total number density of both ions

Current of the positive/ negative species

Einstein diffusion coefficient for positive/ negative species

viii

Q

Bk

T

r

0

P

F

0P

Pc

k

H

L

w

U

To

0

Ionic heat of transport of positive/negative species

Boltzmann constant

Temperature

Dielectric permittivity of water

Relative dielectric permittivity

Permittivity of vacuum

Pressure

Effective body force

Osmotic pressure

Density of fluid

Specific heat of fluid

Thermal conductivity of fluid

Viscosity of fluid

Viscous dissipation

Electrical conductivity of the fluid

Microchannel height

x– extent of the microchannel

Microchannel width

Entry velocity

Ambient temperature

Applied voltage at entry

ix

Re

x

y

n

T

u

v

P

n

n0

1A

2A

3A

4A

B

1C

2C

3C

4C

Reynolds number

Dimensionless x co-ordinate

Dimensionless y co-ordinate

Dimensionless positive/ negative ion density number

Dimensionless induced potential

Dimensionless potential

Dimensionless temperature

Dimensionless horizontal velocity component

Dimensionless vertical velocity component

Dimensionless pressure

Normal unit vector

Ion concentration at both entry and exit of microchannel

0/ Bez k T

0/ BQ k T

0 0/ Bez k T

/UH D

20 /ezn H

1/ Re

20 0 /Bn k T U

20 /zen U

20 0 /zen U

x

1D

2D

3D

a

b

C

0p

h

L

2H

LR

C

x

Pe

m

F

R

Vref

/ Pk C UH

0/ PU C HT

20/ PC UHT

Temperature gradient at the top wall surface

Temperature gradient at the bottom wall surface

Ratio of the ionic heat of transports

Centreline potential

Centreline pressure

Nanochannel half height

Length of the nanochannel

Reservoir height

Reservoir length

Dimensionless centreline potential

Dimensionless axial electric field

x-derivative

Chacarteristic diffusion coefficient

Peclet number

Ionic mobility of positive/ negative ion

Faraday constant

Universal gas constant

Reference velocity

xi

A

S

ind

/ refzFC h V

/refV h D

/zF RT

2 /zFCh

Surface Area

Dimensionless induced potential at left reservoir boundary

Slope of zeta potential on the top wall surface

xii

LIST OF FIGURES

Fig. 1.1 Formation of EDL close to the charged surface and the variation of the potential field within the same.

Fig. 3.1 Schematic representation of the Temperature Gradient problem

Fig. 4.1 Schematic Diagram of the slit nanochannel

Fig. 5.1.1(a) Ion Number Distribution along the cross section at x = 0.7 for = 10,

0T T = 0.12, = -50mV and 0 = 25mV

Fig. 5.1.1(b) Induced Potential Distribution along the cross section at x = 0.7 for = 10,

0T T = 0.12, = -50mV and 0 = 25mV

Fig. 5.1.2 Horizontal Velocity profile ( u ) at x = 0.7 for = 10, 0T T = 0.12, = -

50mV and 0 = 25mV

Fig 5.1.3 Variation of Ion Number Density Differences 0( )wn n on the wall with

0T T

Fig 5.1.4 Profiles of axial velocity ( u ) for = 1, 10 and 25 for constant 0T T = 0.12

Fig. 5.1.5 Variation of axial velocity ( u ) with 0T T for = 10 and 0 = 25m

Fig. 5.1.6 Variation of axial velocity ( u ) with 0T T for = 10 and 0 = 50mV

Fig. 5.2.1(a) Concentration of the positive (c+ ) and negative (c-− ) ions along the x- axis for diffusioosmotic problem

Fig. 5.2.1(b) Induced electric potential along the x-axis for diffusioosmotic problem

Fig. 5.2.2 Variation of the horizontal velocity component along the axial direction of the nanochannel with different gradients ( 0 , 0.6 and 1.2 ) of wall surface zeta potential with fixed ion concentrations ( / 1c C ) at the reservoirs

Fig. 5.2.3 Variation of the horizontal velocity component along the axial direction of the nanochannel with different reservoir concentrations ( / 1c C , / 1.5c C and

xiii

/ 1c C ) for constant zeta potential gradient ( 1.2 )

Fig. 5.2.4 Variation of the horizontal velocity component along the axial direction of the

nanochannel with different for the channel height to debye length

( / 1.6h , / 3.2h and / 4.8h ) for constant zeta potential gradient

( 1.2 )

xiv

CONTENTS

TITLE PAGE i

CERTIFICATE ii

DECLARATION iii

ACKNOWLEDGEMENT iv

ABSTRACT v

LIST OF SYMBOLS vii

LIST OF FIGURES xii

CONTENTS xiv

CHAPTER 1: INTRODUCTION 1

1.1 Electrical Double Layer 2

1.2 Overview of Electrokinetic Effects 3

1.3 Thermal Effects on Electroosmosis 4

1.4 Diffusioosmotic Flow 6

1.5 Objective and Outline 7

CHAPTER 2: THEORETICAL MODEL FOR THE COUPLED TRANSPORT OF IONIC

SPECIES, MOMENTUM AND HEAT 9

2.1 Concentration Distribution 9

2.2 Velocity Profile 11

2.3 Temperature Distribution 11

CHAPTER 3: INFLUENCE OF TEMPERATURE GRADIENT ON ELECTROSMOTIC

FLOW IN MICROCHANNEL 13

3.1 Mathematical Description 14

3.2 Non Dimensional Formulation 16

3.3 Boundary Conditions 17

3.4 Numerical Analysis 18

xv

CHAPTER 4: DIFFUSIOOSMOTIC FLOW INDUCED BY ZETA POTENTIAL

GRADIENT 19

4.1 Preliminaries 19

4.1.1 Semi Analytical Method 20

4.2 Fully-Coupled Model Formulation 23

4.3 Non Dimensional Formulation 25

4.4 Boundary Conditions 26

4.5 Code Validation 27

CHAPTER 5: RESULTS AND DISCUSSIONS 28

5.1 PART A: Influence of the Temperature Gradient on Electro-osmotic Flow in

Microchannel 28

5.1.1 Variation in Ion Number Density 31

5.1.2 Effect on Velocity Profile 32

5.2 PART B: Diffusioosmotic Flow Induced by Zeta Potential Gradient 35

CHAPTER 6: CONCLUSIONS AND FUTURE WORK 40

6.1 PART A: Influence of the Temperature Gradient on Electro-osmotic Flow in

Microchannel 41

6.2 PART B: Diffusioosmotic Flow Induced by Zeta Potential Gradient 42

REFERENCES 44

1

CHAPTER 1

INTRODUCTION In recent years, many researchers have focused their research on developing

micro-/nano devices to study small biological species, such as DNA, proteins, and other

molecules for bio-chemical analysis (Berg & Wessling, 2007; De Leebeeck & Sinton,

2006; Eijkel & Berg, 2005; Mukhopadhyay, 2006; Austin, 2006). In such devices one of

the primary challenges is to propel and control the fluid movement along with fluid

mixing and separation. This challenge stems from the fact that the routinely utilized

pressure-gradient- or gravity- driven mechanism in macroscopic flows cannot be trivially

scaled down to the micro-/nanoscopic regimes. The reason is simple: as dimensions scale

down, hydraulic resistances to volumetric actuation mechanisms (like pressure-gradient

or gravity) scale as the third power of the characteristic micro- or nano- dimension; thus,

necessitating impractically high values of pressure-gradient and a total uselessness of

gravity for all practical purposes. This is apparently a paradox. For, while lower

dimensions offer the practical benefits of handling lower volumes of fluid (with the

attendant benefits of lower times in fluidic assays and lower quantities of sample

required), these very same small dimensions make the traditional actuation mechanisms

largely impractical. Interestingly, the answer to this paradox lies in the scaling argument

itself: since volumetric actuation mechanisms face cubic orders of hydraulic resistance, it

is the surface actuation mechanisms which face only quadratic orders of hydraulic

resistance that might be usefully exploited with the scaling down of the characteristic

dimensions of the fluidic devices. Two very important surface effects which

fundamentally lay behind many different flavours of these surface actuation mechanisms

are surface tension and electrokinetics. Again, of all the flow manipulation techniques

which exploit surface effects, electrokinetic effects are, arguably, the most popular ones

because these offer numerous controls over the fluid flow as a consequence of the

intricate coupling between the fluid dynamics and the physico-chemical characteristics

both of the surface and of the fluid. Before launching into any discussion of electrokinetic

2

effects and of harnessing its advantages for fluid flow actuation at the micro-/nano-scale,

it is important to realize that the understanding of all electrokinetic phenomena is

contingent on an understanding of the electrical double layer. As such, we first describe

the genesis of all electrokinetic phenomena: the electrical double layer (EDL, in short) in

the following.

1.1 ELECTRICAL DOUBLE LAYER Electrolyte solutions invariably have dissolved ions (e.g., from dissolved salts or

dissociated water groups) present in them. The ions which are charged oppositely to the

charged surface are called counter-ions, and the ions which have the same charge polarity

as the surface are called co-ions. The charged surface, naturally, attracts the counter-ions

and repels the co-ions. If there were no thermal motion of the ions, the charged surface

would be perfectly shielded by a layer of counter-ions stacked against the surface.

However, ions have non-zero absolute temperature and the concomitant random thermal

motion of the ions precludes such a physical picture. What happens, then, is that a balance

is established between the electrostatic forces and the thermal interactions so that, at

equilibrium, a certain charge distribution prevails adjacent to the surface (of course, with

the predominant presence of the counter-ions in the vicinity of the surface). This charge

distribution with the predominant distribution of counter-ions together with the surface

charge is called the electrical double layer (EDL).

A schematic depicting the charge and the potential distribution associated with the

EDL is shown in Fig. 1.1. A layer of immobile counter-ions is present just next to the

charged surface. This layer is known as the compact layer or the Stern layer or the

Helmholtz layer. The thickness of this layer is about a few Angstroms and, hence, the

potential distribution within it may be assumed to be linear. From this Stern layer to the

electrically neutral bulk liquid, the ions are mobile. This layer of mobile ions beyond the

Stern layer is called the Gouy–Chapman layer or the diffuse layer of the EDL (Hunter,

1981). Besides this, there is a plane called the shear plane or surface which is considered

to be the plane at which the mobile portion of the EDL can flow past the charged surface.

The potential at the shear plane is called the zeta potential (ζ). The zeta potential is an

extremely significant parameter that may eventually dictate the necessary coupling

between electrostatics and hydrodynamics. One may characterize the span of the EDL as

3

the distance from the shear plane over which the EDL potential reduces to (1/e) times of

ζ,  which  is  also  known  as  the  Debye  length  (λ).  

Figure 1.1: Formation of EDL close to the charged surface and the variation of the

potential field within the same.

1.2 OVERVIEW OF ELECTROKINETIC EFFECTS

The four primary electrokinetic effects are as follows:

1. Electroosmosis: It refers to the relative movement of liquid over a stationary

charged surface, with an external electric field acting as the actuator.

2. Streaming Potential: It refers to the electric potential that is induced when a

liquid, containing ions, is driven by a pressure gradient to flow along a stationary

charged surface.

3. Electrophoresis: It refers to the movement of a charged surface (for example, that

of a charged particle) relative to a stationary liquid due to the application of an

external electric field.

4. Sedimentation Potential: It refers to the potential that is induced when a charged

particle moves relative to a stationary liquid.

It is clear from the preceding discussion that the primary electrokinetic

phenomena may be divided between those which are driven by an externally applied field

(electroosmosis and electrophoresis) and those which are driven by a flow (streaming

potential and sedimentation potential). Close in spirit with these primary electrokinetic

phenomena, as far as the fundamental dependence on the EDL goes, are:

4

Diffusioosmosis, Diffusioophoresis, Thermoosmosis, Thermophoresis and so on. While

attention to these important effects in the micro- and nano-fluidics community have only

recently started to be given (Daiguji, 2010; Fayolle, Bickel, & Würger, 2008; Keh & Ma,

2007a; Qian, Das, & Luo, 2007; Würger, 2007), the fundamentals of these have been

studied for long in the context of classical colloid science. Yet, inspite of the presence of

the extensive literature on the fundamental aspects of these various phenomena, certain

important issues pertaining to these very fundamentals still remain unresolved. For

instance, what is the effect of thermal effects on electroosmotic flow which have

traditionally been modelled under the simplifying assumption of isothermal flow

conditions? How far does the imposition of temperature gradients affect electroosmotic

flow through intricate couplings of the latter with thermoosmotic flow which may not be

negligible under these conditions? Is it not possible to establish diffusioosmotic flows in

situations where such concentration gradients are not externally imposed (as done

traditionally) but are, in turn, established as a consequence of other effects? Another

extremely important point on which the electrokinetic effects in micro- and particularly

so, in nano-fluidics depart from the classical developments of the colloidal science

literature is the breakdown of the thin-Debye-layer limit routinely invoked in the latter.

The use of the thin-Debye-layer limit relegates the hydrodynamics within the EDL to a

thin boundary layer region near the substrate, with the hydrodynamics in the outside

region influenced by the electrokinetic effects only through effective boundary conditions

stemming from the solutions of the inner boundary layer region. While the use of such

limits have successfully captured many colloidal phenomena (because, first, while

colloidal particles themselves may be small macroscopically they are still orders of

magnitude larger than typical Debye-layer thickness; second, the physical domains of

interest confining such colloidal phenomena are indeed macroscopic. It is these latter

conditions especially that are necessarily violated in micro- and nano-fludic domains. As

such, the resolution of these questions and issues has far reaching consequences in the

micro- and nano-fluidics domain; the current thesis is a humble attempt towards that

broad aim. However, before attempting to investigate these apparently sophisticated

issues it is necessary to first present the backdrops against which such profound questions

arise. In what follows, we focus, for the sake of the discussions in the ensuing chapters of

this thesis, on only a couple of these aspects.

5

1.3 THERMAL EFFECTS ON ELECTROOSMOSIS The theory behind electroosmosis is well-established, and is successfully used to

model and predict practical applications. It is important to realize that this theory is,

nevertheless, based on many simplifying assumptions – the biggest among which is the

assumption of isothermal flow conditions. While such an assumption does indeed

expedite the mathematical development necessary for achieving a decent understanding

of the physical picture, it does remain far from complete. Also, these ideal isothermal

conditions are extremely hard to realize in practice. Pertinently, however, there might

indeed be advantages to actually impose external thermal gradients to bring about fine-

tuned control over analyte distributions that often accompany electroosmotic flow. In

these latter cases, isothermal idealizations of electroosmotic flow are fundamentally

wrong, and are essentially incapable of capturing the entire physical picture. There have

been some efforts to address these issues in recent times: foremost, among which has

been the through the incorporation of Joule heating effects which invariably accompany

electro-osmotic flow because of the passage of an electric current under the application of

an electric field (Xuan, 2008; Xuan, Xu, Sinton, & Li, 2004). Considerable research has

also been done to study the heat transfer characteristics on various forms of electrokinetic

flow. However, most of these models are again not devoid of simplifications and

assumptions. The most striking simplifying assumption is that the electrokinetic flow

velocity, which is inherently dependent on the ionic distribution, is independent of the

temperature, when this ionic distribution itself is actually dependent on the temperature.

Moreover, the temperature gradients generated as a result of Joule heating may also

induce local gradients in conductivity, permittivity, density and viscosity. These are

together categorized under electrothermal effects which may lead to net forces acting on

the liquid. For example, conductivity gradients produce free volumetric charges and

Coulombic forces, while gradients in the permittivity lead to dielectric forces

(Chakraborty, 2008). Closely associated with the fluid forces originating due to

electrothermal effects is the phenomenon of thermoosmosis. It refers to the actuation of

fluid motion due to gradients in osmotic pressure which are induced due to gradients in

temperature. Thermoosmosis will take place irrespective of the mechanism in which the

thermal gradients are generated. Thus, in the absence of any other actuator like pressure-

gradient or electrical field, the sole application of a temperature gradient may also lead to

the actuation of fluid motion through the phenomenon of thermoosmosis.

6

Another phenomenon which is a direct consequence of the presence of thermal

gradients in a solution containing ions is the generation of a thermoelectric field (Würger,

2008). Indeed, when a temperature gradient is imposed on a solution containing free ions,

these migrate (Soret Effect)(Fayolle et al., 2008) so that in the stationary state, a

concentration gradient is built up. The variable degree and direction of migration of the

ions ultimately lead to the generation of an electric field. Although described, here, in a

step-by-step way, this stationary state is achieved through a transient equilibrating

mechanism where all the three effects of temperature gradient, Soret effect migration and

the inducement of an electric field develop synchronously. The electric field thus

developed is fundamentally a consequence of the imposed temperature gradient and is,

hence, referred to as the thermoelectric field.

1.4 DIFFUSIOOSMOTIC FLOW

Instead of externally applying an electric field, an electrolyte solution in a micro-

/nano-channel can also be driven by means of diffusioosmosis through the application of

gradients of solute concentration. Like the well-known electroosmosis phenomenon,

diffusioosmosis originates from electrostatic interaction between the electrolyte and the

charged solid surface that is in contact with the electrolyte solution. The concept of

diffusioosmosis is mainly based on difference in ion concentrations along the channel

axial direction. Due to the presence of the concentration gradient, electrolyte ions diffuse

in the nanochannel, accompanied by a net diffusive flux of charge when the mobilities of

the anion and cation are not equal. As a result, an electric field is induced that

compensates for the net diffusive flux of charge across the nanochannel. The induced

electric field, through its action on the counter-ions accumulated in the EDL, creates a

body force that, in turn, induces fluid motion. In contrast to the electroosmotic flow

driven by an externally applied electric field, the fluid motion due to the electroosmotic

effect in diffusioosmosis phenomenon is driven by the induced electric field in the

absence of an externally applied electric field. Therefore, it is obvious to note that both

electroosmosis and diffusioosmosis fall into the same category of surface-driven

phenomena that take advantage of the increase of surface to volume ratio (Ajdari &

Bocquet, 2006).

Previously researchers have studied diffusioosmotic flows near plane surfaces and

capillary tubes with uniform zeta potential along the wall surfaces (Keh & Ma, 2007a). In

7

addition, most analysis of diffusioosmotic flows has been subjected to several restrictions,

such as a thin EDL (Lowell, 1984; Prieve, 1952), low zeta potential consideration along

the wall (Keh & Wu, 2001a), and neglected effects of the ionic concentration

distributions and the induced local electric field .

1.5 OBJECTIVE AND OUTLINE Traditionally electrokinetic transport in microfluidics is studied under isothermal

conditions and with no concentration gradient in the flow direction. The corresponding

implications in electroosmotic flow and flows mediated by streaming potential are well

studied and documented in the literature (Andrade, 1980; Chakraborty & Das, 2008; Das

& Chakraborty, 2009; Yariv, Schnitzer, & Frankel, 2011). There also exists some related

literature on heat transfer effects on electrokinetic transport. However, in those works, the

electrochemical transport is decoupled from the energy equation and averaged

temperature in the Poisson-Boltzmann formalism is employed to represent the

consequences of ionic species transport. This approximation, although practical for

several problems, suffers from some fundamental shortcomings. The primary

shortcoming stems from the fact that gradients in temperature introduce additional

gradients in the number density of the ionic species leading to an EDL potential profile

that may significantly deviate from a Poisson-Boltzmann picture hypothesized in terms of

an equivalent temperature. In addition, the gradients in the system are likely to introduce

osmotic pressure gradients which can influence the flow characteristics. Thus, the

scenario can be more appropriately assessed by a coupled consideration of Poisson-

Nernst-Planck-Navier-Stokes-Energy equations (PNPNST) which is a consideration yet

to be invoked in the literature.

Interestingly, electrokinetic transport may also be assessed with concentration

gradients along the intended flow direction. Such concentration gradients imposed

externally may be trivial. However, in several circumstances the concentration gradients

may be implicitly induced in the system as a combined consequence of surface charge

patterning and EDL overlap. Surface charge patterning or axial variation in zeta potential

over the surfaces is a more realistic condition than uniform wall potential and essentially

mimic the surface defects or inhomogeneity of the platform (Ghosal, 2004). Another

important point is that surface charge modulation induces longitudinal gradient in the

axial velocity, due to which a transverse velocity component is introduced. Now,

8

satisfying the flow continuity some re-circulating velocity components are generated

which forms local vortices that can enhance mixing of adjacent streams and cross stream

migration. Motivated by the observation of mixing in narrow confinements, various

researchers have considered axial variation of the interfacial potential in a controlled way

to achieve their desired functionalities (Chang & Yang, 2008; Chen & Cho, 2007, 2008;

Meisel & Ehrhard, 2006; Zhang, He, & Liu, 2006).

Importantly, subjected to the combined effect of axial gradients of zeta potential

and EDL overlap, axial gradients in the concentration of the ionic species may be

spontaneously induced in a fluidic channel. This, in turn, may give rise to a

diffusioosmotic transport, as a combined consequence of osmotic pressure gradients and

species diffusion. Although a plethora of literature has been reported on diffusioosmosis

in micro- and nano-fluidic confinements, this issue has by far been overlooked.

Aim of the present thesis, accordingly, is to assess the implications of temperature

and concentration gradients on electrokinetic transport in narrow confinements. In an

effort to delve deeper into the underlying consequences, two separate yet related

problems are investigated. The first problem analyzes electroosmosis in a narrow

confinement with imposed temperature gradients on the system. The second problem

addresses diffusioosmosis in narrow confinements as a combined consequence of axially

patterned zeta potential and EDL overlap phenomenon. The organization of the remaining

part of the thesis is as follows. In Chapter 2, a theoretical model is described to address

coupled transport of ionic species, momentum and heat. In Chapter 3, the same model is

utilized to assess the implications of heat transfer on electroosmotic transport in narrow

confinements. In Chapter 4, mathematical modelling on diffusioosmosis in presence of

implicitly induced concentration gradient due to non uniform zeta potential at the wall

surfaces is presented. In Chapter 5, some key results are presented and discussed. In

Chapter 6, conclusions are drawn from the present work and further scopes of work are

identified.

9

CHAPTER 2

THEORETICAL MODEL FOR THE COUPLED

TRANSPORT OF IONIC SPECIES, MOMENTUM AND HEAT

In this chapter we state the governing equations for fluid motion, concentration

distribution and temperature distribution. The basic equation system for the fluid motion

consists of the Continuity equation and the Navier-Stokes equation. Similarly, the

concentration distribution inside any fluidic confinement is governed by the Nernst-

Planck equation and the Poisson equation. In order to study the thermal profile we have

taken the Energy equation as well. These equations forms a coupled system that

determines the basic variables related to an electrolytic fluid motion inside a micro-

/nanochannel.

2.1 IONIC SPECIES TRANSPORT

The transport of ions in an electrolyte, in a general continuum model, is given by

the species conservation equation:

·ii

n Jt

. . . (2.1)

where in is the local concentration of the i-th ionic species, and iJ

is the flux, assuming

that there is no source term due to generation or consumption of ions by any bulk reaction

within the electrolyte. In a moderately dilute solution, the flux can be expressed in terms

of the gradient of the electrochemical potential i as

i i i i iuJ m n n . . . (2.2)

where im is the mobility and u is the mean fluid velocity. It is important to understand

that while the first term on the RHS subsumes within it ion transport due to both diffusion

and electro-migration it is not a truly generally representation of concentration multi-

10

component transport because it does not include any inter-species interactions other than

those between a particular diffusing species and the solvent. In the dilute solution limit,

however, the electrochemical potential i may be conveniently decomposed as,

lni B i ik T n z e . . . (2.3)

where is the electric potential, iz is the ionic valence and e is the electronic charge. The

Nernst-Einstein relation i B iD k Tm expresses the diffusivity, iD , in terms of the mobility.

In this limit, the traditional expression of the flux popularly used to address

electrochemical transport problems is recovered:

ii i i i i

B

z eJ D n n n uk T

. . . (2.4)

Considering, henceforth, the electrolyte to be a binary one with symmetric valencies of

the cations (+) and anions (-), the flux expression becomes

B

z eJ D n n n uk T

. . . (2.5)

Now this expression is valid only when there is no thermal inhomogeneity in the system.

But if there is a thermal gradient present, then because of the enhanced solute-solvent

interaction the particle velocity modifies and becomes a function of the temperature

gradient (Fayolle et al., 2008; Rasuli & Golestanian, 2008; Würger, 2007, 2008).

Previously it has been studied for charged colloids that, how the particle velocity is

affected due to the manifestation of salinity gradient caused by the Soret effect of the

mobile ions in the system (Würger, 2008). In order to incorporate the alteration of the

concentration distribution due to the temperature variation, the current of each mobile ion

is taken as directly proportional to the local thermal gradient. So, to consider the effect of

the diffusion and advection in a non isothermal condition the flux per unit area of an ionic

species is taken as a function of the potential gradient as well as the temperature gradient

and local fluid velocity. Thus, the current of the charged species can be written

respectively as,

2B B

Q z eJ D n n T n n uk T k T

. . . (2.6)

11

where, first term in the R.H.S refers to the normal diffusion with Einstein Coefficient D ,

second term comprises the thermal diffusion with the Ionic Heat of Transport *Q , third

term denotes the drift caused by total potential gradient and the last term is due to the

advection by the fluid flow.

Assuming steady state, the modified Nernst-Planck equation for the charged

species is given by

0J

. . . (2.7)

According to the theory of electrostatics, due to the variation of the charged

species number density the potential distribution is governed by the Poisson Equation

(near the wall surface), written as

.( ) e . . . (2.8)

where, ( )e ez n n is the charge density assuming z z z , consists of the

relative dielectric permittivity ( r ) and permittivity of vacuum ( 0 ) such that o r .

2.2 VELOCITY PROFILE

The fluid velocity and pressure are governed by the continuity and momentum

equation for incompressible fluids,

0u

. . . (2.9) 2u u P u F

. . . (2.10)

where, P is the hydrostatic pressure of the fluid and F

is the body force density that

consists contributions from the osmotic pressure ( 0 BP nk T ) and the potential gradient.

Thus F

can be represented as:

B eF k n T T n

. . . (2.11)

where n is the total number density of the mobile ions and is denoted by n n n .

2.3 TEMPERATURE DISTRIBUTION As the developed thermo-electric field induces the thermo-osmotic flow in the

entire two dimensional space, the electric current passing through results in Joule Heating

(Ruckenstein, 1981). So, the general Energy equation in steady condition takes the form:

2

Pu T k T Ec . . . (2.12)

12

where, Pc and k are the specific heat and thermal conductivity of the fluid, and are

the fluid density and viscosity, respectively, all are assumed to be constant. The last term

is due to the joule heating. is the viscous dissipation term and expressed as, 2 22

2 2u v u vy x x y

. . . (2.13)

It is important to note that the presented system of equations is intricately coupled

and non-linear in nature. There does exist previous instances in the literature where such

coupled Poisson-Nernst-Planck (PNP) equations together with the Navier-Stokes (N-S)

equations have been fully considered; however, none of them have considered the

thermoelectric effect as done here. Additionally, the explicit consideration of the osmotic

pressure contribution to the body force term in the momentum equation imparts a hitherto

unaddressed coupling among the N-S, the PNP equations and the energy equation. While

such an intricate set of equations precludes any analytical tractability, it does free up the

analysis, albeit numerical, from any restrictive regimes ensuing from the simplifying

assumptions that have to be otherwise invoked. For the sake of nomenclature, we call this

the coupled Poisson-Nernst-Planck-Navier-Stokes-Energy (PNPNST) system. The

numerical resolution of this PNPNST system is achieved through the commercial Finite

Element package COMSOL Multiphysics.

13

CHAPTER 3

INFLUENCE OF TEMPERATURE GRADIENT ON ELECTROOSMOTIC FLOW IN MICROCHANNEL

In the present chapter we want to study the effect that the application of an

imposed temperature gradient has on the flow physics (inside a micro-channel) of a fluid

which is basically an aqueous solution of ions, together with the consideration of the

surface charging effects leading to the establishment of an Electric Double Layer (EDL),

which is driven by a combined pressure and potential gradient. The application of a

temperature gradient leads to fluid flow as described earlier following electrothermal

effects and the phenomenon of thermoosmosis. Concomitantly, the surface charging

effects due to the interaction of the confining surface with the aqueous solution of ions

lead to the establishment of an EDL through the equilibration of Coulombic and entropic

interactions. But, contrary to simple isothermal condition such EDL distribution may not

be simply described through the Boltzmann distribution. Indeed, in our case, gradients of

temperature (and, hence, of thermo-physical quantities) exist. But for simplicity, the

gradients of thermo-physical quantities are neglected. Simultaneously, the imposition of a

temperature gradient tends to establish a thermoelectric field. The establishment of the

EDL, or more generally, a concentration gradient depends on this developed

thermoelectric field. Conversely, the thermoelectric field itself is dependent on the

concentration field. Thus, in this situation, we have a rich and coupled interplay between

the imposed temperature gradient, the induced gradients of ionic distribution, and the

induced thermoelectric field. The temperature gradient and concentration gradient

changes the osmotic pressure in the fluid flow. The developed thermoelectric field

determines the effective body force in the fluid flow. So, we can see that the coupled

parameters have a direct impact on determining the fluid flow.

14

3.1 MATHEMATICAL DESCRIPTION As a physical system we consider a parallel plate microchannel of height H, length

L and width w, with w>>H, L. In this problem, E

is the total electric field developed

due to the combined influence of the external electric field, the thermoelectric field and

the EDL distribution (so that, E

). The two parallel plate surfaces possess

unequal temperature profile with different temperature gradients. The temperature profile

at the bottom surface and top surfaces are taken as T1(x) and T2(x) respectively. It is

assumed that the left end of the domain is at constant temperature, such

that 1 2 0(0) (0)T T T . In Fig. 3.1, the schematic diagram is shown; origin is taken at the

bottom surface of the left boundary.

Figure 3.1: Schematic representation of the present problem

As explained earlier the distribution of these ions in the stationary state will be a

consequence of the coupled equilibrating influences of the EDL, induced thermoelectric

field and applied electric field. This is like a two way coupling process where the

temperature driven migration of the ions originates a thermoelectric field which

eventually again re-distributes the ion distribution in the non-uniformly charged fluidic

system. In this case, the Poisson equation (for induced potential) and Laplace equation

(for applied electric field) should be solved simultaneously with the species continuity

and the energy equation to get the ionic species distribution and the overall potential

distribution. The Navier-Stokes momentum equations and continuity equation are solved

with the above mentioned equations in a coupled way to determine the velocity field.

15

Here, the theoretical concept from the previous chapter is used to analyze the

intricate electro osmotic flow. Due to the axially applied electric field is modifications are

done in the terms consisting of the potential gradients. Now the current of the charged

specie can be expressed as,

2B B

Q z eJ D n n T n n uk T k T

. . . (3.1)

where, the third term denotes the drift caused by total electric field.

Assuming steady state, the modified Nernst Planck equation for the charged

species is given by

0J

. . . (3.2)

According to the theory of electrostatics, due to the variation of the charged

species number density the potential distribution is governed by the Poisson Equation

(near the wall surface), written as

.( )o r e . . . (3.3)

The fluid velocity and pressure are governed by the continuity and momentum

equation for incompressible fluids,

0u

. . . (3.4) 2u u P u F

. . . (3.5)

where, F

is the body force density that consists effects of the osmotic pressure

( 0 BP nk T ) and the total potential gradient. Thus F

can be represented as

( )B eF k n T T n

. . . (3.6)

As the developed thermo-electric field induces the thermo-osmotic flow in the

entire two dimensional domain, the electric current passing through results in Joule

Heating. So, the general Energy equation in steady condition takes the form

2

Pu T k T Ec

. . . (3.7)

where, is the viscous dissipation term and the last term is due to the joule heating.

As one new variable is introduced to the coupled system, so to close the system one

more equation is needed. The potential developed, due to the externally applied electric

field, inside the channel satisfies the Laplace Equation, given by 2 0 . . . (3.8)

16

3.2 NON DIMENSIONAL FORMULATION The governing equations are rewritten in dimensionless form using the

characteristic parameters of the system: the channel height H , the inlet ionic number

density 0n , the wall zeta potential , the applied voltage 0 , the ambient temperature 0T

and the entry velocityU . The hydrostatic pressure is non-dimensionalised by 2U . Thus

the new dimensionless variables become:

xxH

, yyH

, 0

nnn

,

, 0

, 0

TTT

, uuU

, vvU

, 2

PPU

Thus the coupled non linear equation system takes the following form.

Nernst Planck Equation:

1 2 3 42. 0Tn A A A A u nT T T

. . . (3.9)

1 2 3 42. 0Tn A A A A u nT T T

. . . (3.10)

where, 10B

ezAk T

, 20B

QAk T

, 03

0B

ezAk T

, 4UHAD

Poisson Equation:

2 B n n . . . (3.11)

where,2

0ezn HB

Laplace Equation: 2 0 . . . (3.12)

Continuity Equation:

. 0u

. . . (3.13)

Navier-Stokes Equation:

21 2 3 4u u P C u C n T T n C C

. . . (3.14)

where, 11

ReC

UH

, 0 0

2 2Bn k TC

U , 0

3 2

zenCU

, 0 0

4 2

zenCU

17

Energy Equation: 2

21 2 3u T D T D D E

. . . (3.15)

where, 1P

kDC UH

, 20P

UDC HT

,

2

30P

DC UHT

3.3 BOUNDARY CONDITIONS We consider the left boundary of the domain as an equi-potential and an

isothermal surface with the 0 and 0T as the respective values. As boundary conditions on

the entry side we take both the ion number densities as constant and equal to 0n . There is

no induced potential on the left and right side of the channel. We have previously used

the inlet horizontal velocity as the reference velocity. Thus, at 0x we have, 1 , 1n , 1u , 0v , 1T , 0 . . . (3.16)

On the outlet boundary the pressure and the potential is taken as zero. We used the

convection boundary condition for the temperature and concentration field. Thus at

Lx LH

we get,

0 , n 0J

, 0P , 0Tx

, 0 . . . (3.17)

where, n is the unit vector along the normal to the surface. At all channel wall surfaces no

slip and no penetration boundary condition is applied for the flow field. It is considered

that both the positive and negative ions on reaching the solid wall surface do not give up

their electric charge or in any way do not react with the surface so that the ion flux normal

to the surface (relative to the surface) must be zero (Cox, 1997). Both the walls are

assumed to be at fixed zeta potential. There is no potential gradient along the normal to

the wall surface for the applied electric field. Moreover, the temperature gradient on the

top surface is considered to be less compare to the bottom wall surface. So, on the top

surface ( 1y ) we have,

0y

, n 0J n u

, 0u , 1 1T T ax , 1 . . . (3.18)

and on the bottom surface ( 0y ) the boundary conditions look like,

18

0y

, n 0J n u

, 0u , 2 1T T bx , 1 . . . (3.19)

where, a b .

3.4 NUMERICAL ANALYSIS

The numerical simulation is done by the commercial finite-element-method

software COMSOL Multiphysics 3.5 and the post-processing is performed by writing

scripts in Matlab 2007. The most significant changes of the variable parameters occur

near both the wall surfaces. So to study the variation of the flow field with respect to the

characteristics variable it is necessary to have a fine discretization mesh near the channel

wall. The size ratio of the triangular mesh used near the wall boundary and subdomain

space is almost five times smaller.

At first the numerical simulation is done using some practical values for the

variable parameters considering the non isothermal condition. Then the variation of the

ion number density and velocity profile is studied for various temperature gradient and

combinations of ionic heat of transports.

19

CHAPTER 4

DIFFUSIOOSMOTIC FLOW INDUCED BY ZETA

POTENTIAL GRADIENT

The basic premise for bringing about diffusioosmosis is the existence of an axial

concentration gradient. While traditional implementations of diffusioosmosis have been

solely realized through externally imposed concentration gradients it is important to

understand that there is no fundamental requirement on how such a gradient is

established; thus, diffusioosmotic flow might very well be actuated through an induced

gradient. Here in this chapter, we have described such a method for inducing the axial

concentration gradient which, in turn, induces a diffusioosmotic flow. Generally, the zeta

potential at the channel walls is taken as constant as the surface charge at the solid walls

is uniform. But, if we consider a gradient in zeta potential along the channel walls in the

axial direction, then due to the non uniform electrostatic interaction, a concentration

gradient is created in the axial direction for both the co-ions and counter ions.

Importantly, this is of significance only when the EDL penetrates sufficiently into the

bulk, that is, under conditions in which EDL overlap exist whence the channel centre-line

concentration is different from the bulk reservoir values. Notably, under these conditions,

the channel centre-line potential gets manifested as function of the surface charging

condition (expressed through the zeta potential). Due to the presence of the concentration

gradient, electrolyte ions diffuse in the nanochannel, accompanied by a net diffusive flux

of charge when the mobilities of the anions and cations are not equal. As a result, an

electric field is induced that compensates for the net diffusive flux of charge across the

nanochannel. The induced electric field, through its action on the counter-ions

accumulated in the EDL, creates a body force that, in turn, induces fluid motion. 4.1 PRELIMINARIES

Diffusioosmosis has been extensively studied through simplified mathematical

models by Keh and co-workers (Hsu & Keh, 2009; Keh & Hsu, 2007, 2009; Keh & Ma,

20

2004, 2005, 2007b, 2008; Keh & Wei, 2002a, 2002b; Keh & Wu, 2001b; Ma & Keh,

2005, 2006, 2007; Wei & Keh, 2003a, 2003b; Wu & Keh, 2003). In these models, the

ionic concentration of each species is described by the Boltzmann distribution and the

electrical potential is then described by the commonly used Poisson–Boltzmann model.

Consequently, the electrostatics and hydrodynamics are decoupled. However, the

Boltzmann distribution is strictly valid under the following assumptions: (i) the system is

in equilibrium (i.e., no convection and diffusion); (ii) the channel wall has a

homogeneous surface charge; and (iii) the charged surface is in contact with an infinitely

large liquid medium where the potential is zero and the ionic concentration is the same as

that of the bulk solution (Li, 2004). Inspite of these assumptions, these simplified models

are useful in obtaining physical insights into the problems. As such, in what follows, we

first develop a similar simplified model based, particularly on the work of Keh and Ma

(2006), discuss a possible algorithm to resolve the one-way coupled equations and then

move on to a model based on the fully-coupled set of equations, following Qian et al.

(2007).

4.1.1 Semi Analytical Method We consider diffusioosmostic flow induced by the interplay of zeta potential

gradient and EDL overlap in a channel of height 2h and length L connecting two

reservoirs having electrolyte with concentration n . Here, a simplified semi-analytical

model for the determination of the axial velocity is described. The major assumption

behind this model is that Poisson-Boltzmann equation is valid and 1D flow is considered.

This scheme considers an iterative way of solving the momentum equations to predict the

velocity distribution. In the first step the pressure distribution is calculated from the y

momentum equation in a 1D approach and then using the pressure we find the axial

velocity profile from the x momentum equation. In the following discussion, number

density ( n and n ) is considered instead of the molar concentration to specify the

concentration of the ionic specie. Here the centreline potential is taken as C and

centreline potential as 0p . Here the non dimensional parameters are taken in a slightly

different manner as, , ,B

ez x yx yk T L H

Now writing the y momentum equation, neglecting the transverse velocity, we get

21

0dze n ny dp

y

. . . (4.1)

From the literature we get the local number density as a function of the reservoir ionic

concentration ( n ) and the absolute temperature (T ) (Baldessari & Santiago, 2008). So

the above equation becomes

0Bdn k T e edy

py

. . . (4.2)

Solving the equation by integrating and using the centreline condition

as, 0y , 0 , Cp p , we get the final expression of pressure as,

0 2 cosh cosh CBp p n k T . . . (4.3)

Now let us look at the x momentum equation with an electrical body force based on the

axial electric field ( E )

2

2

u p ez n n Ey x

. . . (4.4)

Differentiating the pressure term, obtained from the previous expression, with respect to x

and substituting it, we get

2

2 2 sinh sinh 2 sinhCC

du dn ez n ez Ey dx dx

. . . (4.5)

Integrating the above equation, using the symmetric boundary condition ( 0 at 0u yy

)

at channel centreline and no slip boundary condition ( 0 at u y H ) at the wall, we

obtain

0

2 sinh sinh sinhy y C

CH

ddu n ez E dydydx dx

. . . (4.6)

Now we have expressed the axial velocity as an integral function of some non

dimensionalised parameter, as following,

2

1 0

2 sinh sinh sinhy y CB

Cref

dk Tn H du dydyV L dx dx

. . . (4.7)

where, denotes the non dimensional axial electric field, such that, B

ezELk T

.

We have obtained the functional value of from the local zero current condition, i.e, it is

considered that the current density of the positive and negative ions are same at every

22

point inside the domain. This is a necessary simplification which was very much needed

to solve . So considering J J and writing the axial component only, we get

x x x xB B

B B

ez ezD n D n D n D n n n uk T k T

ez ezD n E D n Ek T k T

. . . (4.8)

Substituting the values for u , and n n in the above equation the expression for E is

found.

2 sinh 11 1

ref BV k TuE

D eze e

. . . (4.9)

where D DD D

.

Finally the expression for is obtained by using the above expression,

2 sinh1 1

uPe

e e

. . . (4.10)

where

2

refV LPe

D D

This scheme has some major simplifications, but it can be used to determine the

nature of an approximate velocity profile. At first, a guess value of zero is taken for

and using that value, the distribution of u is determined from equation (4.5). Then from

equation (4.10), the induced axial electric field is predicted and it is modified from the

initial guess, which is further used to determine the u profile. In this iterative manner the

procedure is done until the difference of consecutive iteration values exceeds a certain

minimum tolerance value. This is a good approach to study the velocity profile in an

analytical way considering 1D approach and Poisson Boltzmann approximation.

However, as already mentioned, the Boltzmann distribution is not valid in our case. The

flow field affects the mass transport due to convection. On the other hand, the mass

transport, in turn, affects the flow field through the induced electric field. The model

requires one to simultaneously solve the coupled equations including the Navier-Stokes

equations, the Nernst–Planck equation, and the Poisson equation. This intricately coupled

model is described in detail in the subsequent section.

23

4.2 FULLY-COUPLED MODEL FORMULATION Let us consider a charged slit nanochannel with length L and height 2h connecting

two identical reservoirs on either side. The schematic is depicted in Fig. 4.1(Qian et al.,

2007). The length and height of the reservoir are respectively LR and 2H. Utilizing the

symmetry of the geometry, a two dimensional Cartesian co-ordinate system (x, y) with

origin located at the centre of the nanochannel is represented. The x and y coordinates are,

respectively, parallel and perpendicular to the axis of the nanochannel. The symmetrical

model geometry is represented by the region bounded by the outer boundary and the line

of symmetry, AB. The dashed line segments, BC, CD, GH, and HA, represent the regions

in the reservoirs. The length LR and height 2H of the reservoir are sufficiently large to

ensure that the electrochemical properties at the locations of BC, CD, GH, and HA are not

influenced by the nanochannel.

Figure 4.1: Schematic Diagram of the slit nanochannel

We consider that the walls of the two reservoirs (line segments DE and FG) are

electrically neutral surfaces and the channel wall surfaces have a horizontally varying zeta

potential. The left and right reservoirs are filled with two identical electrolyte solutions

with same bulk concentration, C, so that there is no imposed concentration gradient along

the x-direction. We also assume that there is no externally applied pressure gradient and

horizontal electric field across the two reservoirs. In the following sections, we present

dimensional mathematical models and the non dimensional formulation for the fluid

motion and ionic concentration distribution through the nanochannel.

A

Virat Kohli is always involved in wonderful chases.

B

H G

F E

D C

h

L

H

LR LR

x y

24

We assume a binary, symmetric electrolyte solution enclosed in the nanometer

channel between the two reservoirs. Due to the low Reynolds number of the fluid flow,

we neglect the inertial forces in the Navier-Stokes equation. So, the motion of the

incompressible solution due to the pressure gradient and electrical body force created as

the effect of the induced concentration gradient is described by the modified Stokes

equation,

u 0 . . . (4.11) 2u ( ) 0p F z c z c . . . (4.12)

In the above, ˆ ˆu i ju v is the fluid velocity, where i and j are respectively the unit

vectors in the x- and y-directions; u and v are respectively the velocity components in the

x- and y-directions; p is the pressure; is the electric potential in the electrolyte

solution; c and c are, respectively, the molar concentrations of the positive and negative

ions in the electrolyte solution; z and z are respectively the valences of the positive and

negative ions satisfying the relation z z z ; F is the Faraday Constant; and is the

dynamic viscosity. The last term of the left hand side describes the electrostatic force

through the net charge density and induced electric field. The concentration distribution of the solution inside the nanochannel can be found

out by solving the bi-ion mass transport model that includes the Nernst–Planck equation

for the concentration of each ionic species and the Poisson equation for the electric

potential in the electrolyte solution. The flux density of each aqueous species is given by

(Qian et al., 2007),

N uc D c z m Fc . . . (4.13)

In the above, c denotes the respective ion molar concentration; D is the respective ion

Diffusion coefficient; z is the respective ion valence; and m is the respective ion

mobility. The three terms in the right hand side of the equation denotes the convective,

diffusive and migratory fluxes respectively. The mobility m is expressed in terms of the

diffusivity D , the universal gas constant R , and the absolute temperatureT using the

Nernst–Einstein relation as following,

DmRT

. . . (4.14)

25

Thus the ionic concentration distribution, at the steady state, can be solved using

the Nernst Planck equation,

N 0 . . . (4.15)

In the above equation there are three unknowns; the +ve and –ve ion concentrations

c and c respectively and the potential . We can find the potential using the Poisson

equation: 2 ( )Fz c c . . . (4.16)

where, is the permittivity of the electrolyte solution.

4.3 NON DIMENSIONAL FORMULATION

The governing equations are rewritten in dimensionless form using the

characteristic parameters of the system: the nanochannel height h , the ion concentration at

the reservoirsC and the wall zeta potential .The velocity components are non-

dimensionalised by a reference velocity refV and the pressure by refVh

. So the new

dimensionless variables become:

xxh

, yyh

, ccC

, ccC

,

, ref

uuV

, ref

vvV

, ref

ppVh

Thus the coupled non linear equation system takes the following form:

Continuity Equation:

.u 0 . . . (4.17)

Navier Stokes equation:

2u 0p c c . . . (4.18)

Nernst Planck equation:

. u 0c c . . . (4.19)

. u 0c c . . . (4.20)

Poisson equation:

2 c c . . . (4.21)

26

In the above equations, some non dimensional coefficients appeared which depend on the

characteristic physical values. These are as following:

A=ref

zFC hV

, refV hD

, refV hD

, zFRT ,

2zFCh

4.4 BOUNDARY CONDITIONS

A no-slip boundary condition (i.e., 0u v ) is used at the wall surfaces of the

nanochannel and the reservoirs (line segments DE, EF and FG in Fig. 2). At the planes

BC and AH of the reservoirs, normal pressure equals to zero is specified, since there is no

externally applied pressure gradient across the two reservoirs. Slip boundary conditions

are used on the segments CD and GH, since they are far away from the entrances of the

nanochannel. Finally, a symmetric boundary condition is used along the line of

symmetry, AB.

In the plane BC and AH, the concentrations of the positive and negative ions are

the same as the bulk concentration C of the electrolyte solution present in the two

reservoirs. At the walls of the reservoirs and the wall of the nanochannel (line segments

DE, EF and FG in Fig.1), the net ion fluxes normal to the rigid walls are zero as the wall

surfaces are impervious to ions. Zero normal flux is used for the segments CD and GH, as

these surfaces are in the bulk electrolyte reservoirs. Along the segment AB, symmetric

boundary condition is used. Thus the boundary conditions for the Nernst–Planck

equations are as following:

1c c in the plane BC and AH . . . (4.22)

ˆ ˆn N n u 0c c . . . (4.23)

in the plane AB, CD, DE, EF, FG and GH

ˆ ˆn N n u 0c c . . . (4.24)

in the plane AB, CD, DE, EF, FG and GH; where, n is the outward unit normal vector.

Now, the boundary condition for the Poisson equation along the plane HA is

specified by ind , which is determined from the zero current condition (Qian et al.,

2007),

+ ˆN N nS

Fz dS

. . . (4.25)

27

Here, S is the surface area of the plane HA and the current density is denoted by,

+N Ni Fz . The condition of non-zero ionic current is valid for any cross section of

the reservoir and the nanochannel. So, in our non-dimensional scheme, the boundary

condition for the plane HA can be written as,

ind . . . (4.26)

A symmetric boundary condition is used along the plane AB. As the walls of the

reservoirs (planes DE and FG) do not carry a fixed charge and since the surfaces CD and

GH are far away from the nanochannel, no charge boundary condition is used for these

four surfaces, i.e.,

n 0 . . . (4.27)

Along the plane BC, the potential is taken as zero, i.e.,

0 . . . (4.28)

A linearly varying zeta potential is used as the boundary condition on the wall

surface EF. Thus, the potential in the plane EF looks like,

1 *( 15) / 30x . . . (4.29)

where is an arbitrary constant.

4.5 CODE VALIDATION

We solved the strongly coupled system with the commercial finite element

package COMSOL Multiphysics 3.5. The computational domain bounded by

ABCDEFGH in Fig. 4.1 was discretized into quadrilateral non-uniform elements. We

compared the solutions obtained for different mesh sizes to ensure that the numerical

solutions are convergent, independent of the size of the finite elements, and satisfy the

various conservation laws. At first we simulated the ionic mass transport in nanofluidic

channels without considering the convection (i.e., u=0), solving the set of Nernst–Planck

equations and the Poisson equation and our numerical results agree well with those

obtained from previous literature (Daiguji, 2010). Then, the numerical scheme is

validated with the results for previous study on diffusioosmotic flows in slit nanochannel

by Qian et al., 2007. As compared to their work, our model shows that a non-zero

velocity can be achieved without the externally imposed concentration gradient, just by a

linear variation in surface zeta potential value.

28

CHAPTER 5

RESULTS AND DISCUSSION

In this chapter, some key results are mentioned from the numerical simulations of

the two example problems described in the previous chapters. In Part A, the variation in

the ion number density and the effect on axial velocity profile due to the applied

temperature gradient on the wall surfaces is discussed. In Part B, the induced

diffusioosmotic flow inside a nanochannel due to the gradient in wall surface zeta

potential is explained.

5.1 PART A: Influence of the Temperature Gradient on Electro-osmotic Flow in Microchannel

The geometrical domain for the numerical calculation is taken as a rectangular

channel, with height ( H ) equals to 1 micron and length ( L ) ten times the height. Other

parameters with constant values are taken as the following: = -0.05 V, 0 = 25 mV, U =

0.01 mm/sec, 0n = 1×1021 per cubic meter and 0T = 300 Kelvin. The variables are non-

dimensionalised using these characteristics value and the constant physical parameter

values of , , k , PC and . Generally the ionic heat of transport of positive ions is

greater than the negative ions. Here we have defined a new non dimensional parameter

such that*

*

QQ

, considering ionic heat of transport for negative ions ( *Q ) as 8.8×10-21

Joules per ion. The temperature gradients on the wall surfaces are so chosen that the

temperature difference along the entrance and exit for the bottom wall surface is 60

degree Kelvin and for the top wall surface is 10 degree Kelvin. The focus of our analysis

is at the cross section which is at 7x . Thus the temperature difference ( T ) between

the two wall surfaces at that cross section is 35 degree Kelvin which implies 0/T T =

0.12.

29

Fig. 5.1.1(a) shows the distribution of both the opposite ions along that cross

section. In our model both the wall surfaces are considered as negatively charged, so the

positive ion density is maximum at the channel walls. This is due to the fact that

positively charged ions possess a strong affinity towards negative surface. Here the

electric double layer is thick enough to merge for the two walls. Thus in the centerline of

the channel the bulk concentration condition is not achieved. The number densities for the

opposite ions at the centre of a cross section, differs by a certain amount which implies a

non zero induced potential. It is important to note that due to the temperature gradient

created vertically along the cross section, the ion distributions are asymmetric with

respect to the centerline; the minimum ion density for the positive ions appears below the

centerline and maximum ion density for the negative ions occurs above the centerline.

The difference in temperature distribution also affects the ion number densities on the

wall surfaces. The positive ion concentration is less on the hot surface in comparison to

cold surface, although the difference of ion densities at the walls for negative ions is less

than the difference for the positive ions, which shows that the temperature field has a

larger   effect   on   determining   the   positive   ions’   distribution   than   negative   ions’  

distribution.

(a)

Figure 5.1.1: (a) Ion Number Distribution along the cross section at x = 0.7 for = 10,

0T T = 0.12, = -50mV and 0 = 25mV

30

(b)

Figure 5.1.1: (b) Induced Potential Distribution along the cross section at x = 0.7

for = 10, 0T T = 0.12, = -50mV and 0 = 25mV

The combined induced potential, due to the electric double layer and thermo-

electric effect, is plotted against the height of the micro-channel in Fig. 5.1.1(b). The

induced potential has a maximum value of unity at both the walls and the minimum value

is at the middle of the channel where the difference in number density is minimum for the

oppositely charged ions.

In Fig. 5.1.2, x-velocity profile is plotted along the cross section. The magnitude

of the horizontal velocity near the wall surfaces are greater compared to the value at the

middle of the channel. This can be explained by the effective electrical body force which

is huge near the wall as the positive ion concentration is more than the negative ion

concentration in this region. But, in the middle of the channel the difference between the

negative ion concentration and positive ion concentration decreases, so the effect of

osmotic pressure dominates over the electrical body force. As the osmotic pressure acts as

a positive pressure gradient for a fluid flow with charged species, thus in this case mainly

due to the influence of osmotic pressure the velocity profile goes backward and gradually

decreases and attains a local minimum near the centre of the channel. Similar to the

31

potential distribution, the cross sectional velocity profile also shows asymmetry with

respect to the centerline, which is studied in a greater detail in the later sections.

Figure 5.1.2: Horizontal Velocity profile (u ) at x = 0.7 for

= 10, 0T T = 0.12, = -50mV and 0 = 25mV

5.1.1 Variation in Ion Number Density

Now we have seen that the concentration distributions for opposite ions vary for

the same temperature gradient, which can be demonstrated by the effects of wall surface

potential and the different ionic heat of transports. The term ionic heat of transport has a

significant relevance on determining the ionic charge distribution inside the medium

because this term along with the temperature gradient incorporates the deviation from

isothermal condition. The non dimensional parameter signifies the comparison of the

effect of temperature gradient on positive ion over negative ion. If we consider the

isothermal condition on both the negatively charged wall surfaces then there will be no

difference in corresponding ion densities on the walls at a certain cross section; i.e., ion

density for positive ion on top surface and bottom surfaces are same. But for different

temperature on the wall surfaces, the ion number densities of corresponding ions vary.

Let the difference of ion number density on the wall surfaces for corresponding ions are

denoted by wn and it is non-dimensionalised by 0n . In Fig. 5.1.3, the non dimensional

32

wn (for both the ions) is plotted against the non dimensional temperature differences for

different . From the graph it is seen that 0wn n for negative ions is invariant with the

temperature variation and is almost zero, i.e. the negative ion number densities are same

on both the channel walls. But on the other hand, wn for positive ions increases (almost

linearly) with the increasing wall temperature differences; under normal conditions 0.12

is considered as the maximum achievable value of 0T T for the cross section at x = 0.7.

It is also observed that with the increment of , 0wn n increases with a steeper slope,

which reflects the greater association of positive ions with the temperature variation.

Figure 5.1.3: Variation of Ion Number Density Differences 0( )wn n on the wall with

0T T

5.1.2 Effect on Velocity Profile

The horizontal velocity component highly depends on the effective body force term

appearing in the momentum equation, which comprises of the osmotic pressure as well as

the effects of the potential gradients. Now as the ionic distribution of positive ions varies

33

with different and varying temperature difference ( T ), so it is easily understood that

the velocity profile has a direct connection with these two parameters.

Figure 5.1.4: Profiles of axial velocity ( u ) for = 1, 10 and 25

for constant 0T T = 0.12

In Fig. 5.1.4, the non dimensionalised velocity profile is plotted along the cross

section for 3 different values of , considering 0T T being 0.12. For = 1, the velocity

profile is almost symmetric with respect to the centerline. But for = 10 and = 25, we

can see the asymmetric nature of the velocity profile. The maximum velocity occurs at

y = 0.81 and its value is 1.37 (for = 25) and 1.28 (for = 10). On the other hand near

the bottom surface the velocity increases but it is less than the velocity reported near the

top surface. The non dimensional velocity magnitude at the middle of the channel is less

than unity. As increases, the positive ion concentration increases near the cold surface

than the warm surface. Considering the negative ion concentration as to be almost similar

near to the both walls, we can say that the effective body force due to the applied

potential is more near the colder surface. This account for an increasing trend of velocity

profile near the cold surface (top wall) compare to the hot surface (bottom wall).

34

Figure 5.1.5: Variation of axial velocity ( u ) with 0T T for = 10 and 0 = 25mV

In Fig. 5.1.5, the velocity profile is plotted against the channel height for 3

different values of temperature difference along the cross section. is taken as constant

and equals to 10. It shows that with increasing temperature difference the magnitude of

maximum velocity increases near the colder surface. This can be explained by the less

accumulation of positive ions towards the warm surface for an increased temperature

difference. But the local minimum for the velocity profile near the center line appears

below the centerline. One important factor determining the velocity profile is the applied

potential; if we double the value of 0 from 25 mV to 50 mV then the asymmetric nature

of velocity profile becomes more prominent. We have plotted the velocity profile

considering as 10 and 0 as 50 mV in Fig 5.1.6. In this case, minimum non

dimensional velocity in the middle of the channel goes below 0.5, which is less than half

of the value attained for 0 = 25 mV. The maximum velocity also increases and for

0T T = 0.12, u reaches almost 1.8. This shows that the channel surfaces with high zeta

potential and uneven temperature gradients can heavily influence the cross sectional

velocity profile inside a microchannel.

35

Fig. 5.1.6 Variation of axial velocity ( u ) with 0T T for = 10 and 0 = 50mV

5.2 PART B: Diffusioosmotic Flow Induced by Zeta Potential Gradient

In this section, we present some selected numerical results to highlight the key

trends of the induced diffusioosmotic flow in a slit nanochannel. In order to have a

concrete basis for comparison of our simulation results, we follow Qian et al. [ref] and

consider the following values for the various geometrical and physic-chemical

parameters, 5h nm, 150L nm, 20H nm, 30RL nm, 10C mM, 0.050 V, and

1.2 . The temperature ( 0T ) of the electrolyte solution in the nanochannel is taken as

300K. The diffusion coefficients of the ions K+ ( c ) and Cl- ( c ) are respectively,

91.95 10 m2/s and 92.03 10 m2/s.

36

(a)

(b)

Figure 5.2.1: (a) Concentration of the positive (c+ ) and negative (c-− ) ions along

the x- axis (b) Induced electric potential along the x-axis

Fig. 5.2.1(a) depicts the concentration of the positive and the negative ions along

the x-direction of the nanochannel when 1.2 . The concentration distributions are

plotted for the symmetry plane AB. Due to the influence of the negative surface potential,

the concentration of the positive (K+) ions is enhanced and the concentration of the

37

negative (Cl-) ions is reduced near the wall surface. Here the electric double layer is thick

enough so that a condition of EDL overlap is established throughout the channel length.

Thus in the centerline of the channel the concentration is different from the bulk value

(i.e. equal to that in the reservoir) that would, otherwise, have been established had there

been no EDL overlap. Furthermore, it is expected that the concentration of the positive

ions is greater than the concentration of the negative ions inside the nanochannel, which

is clearly seen in the figure. Another point to be noted is that at the steady state, along the

channel, the concentration of positive ion increases but the concentration of negative ion

decreases. It is fundamentally the establishment of these gradients, brought about solely

(and implicitly) through a gradient of the zeta potential, and without externally imposing

any differences in the concentrations, that lies at the heart of the novel transport

mechanism we address here. It is also seen, that a non-zero potential is manifested self-

consistently with the EDL overlap scenario. Fig. 5.2.1(b) shows the distribution of the

induced potential. The potential ind obtained from our numerical simulations, considering

the ionic current to be zero at any cross section, is non zero and have a finite positive

value. The non-dimensional potential increases along the +ve x-direction inside the

channel. The potential is almost zero at the two reservoirs since is no effect of the varying

zeta potential.

Figure 5.2.2: Variation of the horizontal velocity component along the axial direction of

the nanochannel with different gradients ( 0 , 0.6 and 1.2 ) of wall surface zeta

potential with fixed ion concentrations ( / 1c C ) at the reservoirs

38

Fig. 5.2.2 depicts the variation of centre-line velocity with three different values.

It is extremely important to note the case 0 denoting a constant value of the zeta

potential along the wall surface. Even with EDL overlap, the absence the zeta potential

gradient precludes the establishment of any gradient in the concentration along the

channel centre-line; as such, no (induced) diffusioosmosis is possible. In this scenario the

fluid velocity is expected to be zero which is indeed seen (bold line) in the figure. This

important case acts a concrete proof of concept of the hypothesized novel transport

mechanism. Of course, as the surface potential is varied linearly, an axial velocity does

get induced. We can see from Fig. 5.2.3, in the middle of the nanochannel, for 0.6

there is a non-dimensional axial velocity of 0.12, and which increases with the zeta

potential gradient, along the channel. For 1.2 , the enhancement in the centre-line

velocity is more pronounced; the centre-line velocity increases almost one and a half

times as the fluid moves from left reservoir to right reservoir.

Figure 5.2.3: Variation of the horizontal velocity component along the axial direction

of the nanochannel with different reservoir concentrations ( / 1c C , / 1.5c C

and / 1c C ) for constant zeta potential gradient ( 1.2 )

These studies are done for a fixed reservoir concentration of 10mM i.e.,

1c c . Since for thick EDLs the positive and negative ions possesses different

39

centerline concentration, so there is a concentration difference along the channel. For this

reason, there is always a potential gradient induced along the centre-line. It is also

noticeable that with increasing the reservoir concentration of both ions, for a fixed

gradient of zeta potential ( 1.2 ) the axial velocity increases in a non-linear way.

Now the effect of zeta potential gradient on concentration and velocity

distribution is highly dependent on the thickness of the debye length ( ). As the ratio of

height is to debye length increases, the concentration difference at the centreline

decreases and as a result the effective induced electric field is lowered. So, with

increasing channel height, for a fixed reservoir concentration and constant zeta potential

gradient, the magnitude of the axial velocity at the centre of the channel decreases. In the

previous results the debye length is 3.15 nm; obtained from the theoretical expression of

debye length, as 02 22

Bk Tn e z . From Fig. 5.2.4, we can observe that as the value of

/h increases from 1.6 to 3.2 and 4.8, the velocity magnitude gets decreased; if the

channel height increases further, the axial velocity might get lowered and eventually

becomes zero.

Figure 5.2.4: Variation of the horizontal velocity component along the axial direction of

the nanochannel with different for the channel height to debye length

( / 1.6h , / 3.2h and / 4.8h ) for constant zeta potential gradient ( 1.2 )

40

CHAPTER 6

CONCLUSIONS AND FUTURE WORK

In the present study, we have set up and investigated two model problems with the

intention of addressing some fundamental issues in electrokinetic transport especially as it

pertains to the micro-/nano-fluidics domain. As mentioned in the prefatory remarks of

this work, the motivation behind this is to contribute towards a deeper insight into

intrinsically coupled phenomena for which traditional modeling paradigms are, at best,

incomplete representations. In the two model problems, we study interesting flow

phenomena originated out of interactions between externally imposed gradients (such as

temperature gradients and zeta potential gradient) and concentration distribution together

coupled with electrical potential gradients in narrow confinements. Perhaps the most

important point to note is the critical role played by osmotic pressure gradients in

establishing the concerted interplay between the fluid dynamics influenced by electrical

potential gradients and the gradients of temperature and concentration. Interestingly, it is

the deceptively simple dependence of the osmotic pressure on temperature and

concentration that ultimately gives rise to the rich non-linear coupling, and, indeed, it this

gradient of the osmotic pressure which is the common refrain throughout both the

problems addressed. Another thematic commonality stems from the clear departure in the

models of both problems from the thin-Debye-layer limit routinely invoked in the

colloidal science literature (and to a great extent even in the microfluidics literature). The

use of such limits is predicated on the assumption that the characteristic thickness of the

EDL is orders of magnitude smaller than the characteristic physical dimensions of the

system. This justifies the relegation of the flow dynamics mediated by the physico-

chemical factors to be relegated to thin boundary layer regions. Such an artifice is perfect

from the point of view of analytical tractability achieved through the asymptotic matching

formalism. It also simplifies the analysis to a considerable extent by the representation of

the entire physics within those thin boundary layers as effective boundary conditions for

the rest of the so-called   “outer   region”.   However,   in   the   nano-fluidic domain, such

simplifying assumptions routinely do break down because the characteristic physical

41

dimensions indeed match the orders of the characteristic thickness of the EDL.

Appreciating this, we have, thus, kept our formulations more general. While such

relaxations come at the price of analytical tractability and necessitate the use of full-scale

numerical simulations as done in the current work, they do, however, allow the set-up of a

framework that is not constrained to work only within narrow, restrictive regimes of

interest. In the following sections, the summary of the trends obtained from the results of

the two problems based on the afore-mentioned broad modeling paradigms and

additionally incorporating full-fledged non-linear couplings are delineated. Even beyond

such sophisticated investigations, the scope of future works that may provide further

insights into certain interesting niche areas is identified.

6.1 PART A: Influence of the Temperature Gradient on Electro-osmotic Flow in Microchannel

In the first problem, the temperature dependency of the ionic species distribution

is studied in the case of fluid flow taking place inside a narrow confinement. The outcome

confirms that the accumulation of the ions on the channel surfaces depends heavily on the

temperature gradient of the wall surface along with the zeta potential. From the results

obtained for this study, considering the fact that the driving forces of the fluid flow

(external pressure gradient and electric field) are kept constant, it can be inferred that the

fluid velocity profile deviates significantly due to temperature variation inside the

domain. With increasing temperature gradient on the wall surfaces the deviation of the

cross-sectional velocity from the isothermal case is enhanced. It is seen that by applying

different temperature gradient on each wall surface, the velocity profile becomes

asymmetric with respect to the channel centerline as well. The maximum value of the

velocity profile is then identified near the less warm channel surface. It is also observed

that the deviation of the velocity profile from the isothermal case increases with high wall

surface zeta potential and high ratio of the characteristic ionic heat of transports for the

ionic species. The minimum value of the velocity is being attained at centerline, which is

almost halved when the wall surface zeta potential is doubled.

The work presented here is focused primarily on the effect of a linear temperature

gradient on the ionic distribution and velocity profile of an electro-osmotic flow. The

effect of patterned temperature distribution and non linear thermal gradient on wall

42

surfaces, on the flow physics of an electroosmotic fluid transport can be studied in the

future. Furthermore, the non isothermal electroosmotic flow for altering zeta potential can

also be investigated. This might give some idea on controlling and counter balancing the

effect of thermal gradients on transport characteristics by the imposed surface potential.

6.2 PART B: Diffusioosmotic Flow Induced by Zeta Potential Gradient

In the second problem, we test our proposed hypothesis of the novel transport

mechanism   “induced   diffusioosmosis”   brought   about   through   the interplay of zeta

potential gradient and EDL overlapping. The most notable feature is that despite the

absence of any of the traditional external fields, namely pressure gradient or electric field

or a concentration gradient a flow is established due to the induced concentration gradient

along the channel, in the thick EDL regime. In particular, the manifested flow field is

strongly dependent on the zeta potential gradient as well as the magnitude of the bulk

concentration in the reservoirs; the axial velocity shows an increasing trend with increase

in the zeta potential gradient and also with increase in the reservoir concentration. As it is

expected from the consideration of the basic assumption of EDL overlapping, the results

showed a decreasing magnitude of axial velocity with increasing channel height.

The comparison of normal diffusioosmotic flow due to the axially applied

concentration gradient with the induced diffusioosmotic flow due to the imposed gradient

in surface potential, on the flow parameters can be investigated. As a future assignment,

the study of induced diffusioosmotic flow for a non isothermal case can be done, which

might have more significance related to nanofluidics based experiments.

Another important point to note is that the current investigation of the

diffusioosmotic problem serves primarily as a proof of concept of the hypothesized novel

induced transport mechanism. The smooth linearly varying zeta potential utilized to

induce such diffusioosmotic transport is but only an idealization, and is extremely

difficult to realize in practice. What is practically realizable, however, is to have discrete

patches of different zeta potentials. It is intuitive to expect that when arranged in

increasing (or, decreasing) steps these patches may also induce such diffusioosmotic

transport. This problem may be addressed, in the future, as a more practical flavour of the

current one. Such investigations may be of immediate interest to experimentalists because

43

these can provide a generalized framework for design of experiments. Even from the

point of view of fundamental flow physics, this proposed future work may offer

sophisticated avenues to delve into richer fluid dynamical phenomena (with possibilities

of vortical structures) through an expected interplay between the different length scales-

with an important extra contribution being from the length scale of the zeta potential

patches.

44

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