Electro-kinetic Transport with Biochemical Reactions

7/29/2019 Electro-kinetic Transport with Biochemical Reactions

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Electrokinetic Transport with Biochemical Reactions

A dissertation submitted to the University of Manchester for the

degree of Master of Science in the Faculty of Engineering and

Physical Sciences

2012

Anupam Kumar Pilli

School of Mechanical, Aerospace and Civil Engineering


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CONTENTS

List of Tables & Figures 4

List of Symbols 6

Abstract 8

Declaration 9

Copyright 9

The Author & Acknowledgements 10

Chapter 1: INTRODUCTION

1.1 Introduction to Microfluidics 11

1.1.1 Micro-Electro Mechanical Systems (MEMS) & Microfluidics 11

1.1.2 Microfluidics, History and Various Microfluidic Systems 11

1.2 Introduction to Electrokinetics 12

1.3 Introduction to Lab-on a-Chip, LOC devices 13

1.4 Objectives and Structure of this Dissertation 14

1.4.1 Objectives 14

1.4.2 Structure of this Dissertation 15

CHAPTER 2: LITERATURE SURVEY

2.1 Basics of Electrokinetic phenomena 16

2.1.1 Electrical Double Layer (EDL) 16

2.1.2 Electro-osmosis 18

2.1.3 Electrophoresis 19

2.2 Relevant research to the topic of this dissertation 20


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2.2.1 Biochemical analysis and characterization of flows

in microfluidic systems 20

2.2.2 Mathematical model for non-Newtonian

fluids under electrokinetic microchannel flows 22

2.2.3 Rheology and Hemodynamics of Blood 30

2.2.4 Slip (or) Wall-depletion effects 32

2.2.5 Mathematical model for visco-elastic fluids under electro-osmotic

flow in Microchannel 34

CHAPTER 3: MATHEMATICAL MODELLING

3.1 Microchannel and its parameters 39

3.2 Governing equations 40

3.3 Analytical solution 42

3.4 Non-dimensional velocity and length scales 44

CHAPTER 4: PRESENTATION AND DISCUSSION OF RESULTS

4.1 Comparison of the present model with Das & Chakraborty's model 46

4.2 Effect of relative viscosity and haematocrit on the flow with depletion effects 47

4.3 Effect of Debye-Huckel factor, Ion charge density and EDL thickness 50

4.4 Effect of the factor De2 54

Chapter 5: Conclusions

5.1 Conclusions 56

5.2 Future Work 57

References 58

FINAL WORD COUNT = 11,184


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LIST OF TABLES AND FIGURES:

Tables: page no

Table 1: Physical properties and problem data used by Das & Chakraborty (2006). -30

Figures:

Figure 1: Schematic drawing of the micro-electrophoresis device. In mm 12

Figure 2: Schematic representation of microfluidic components in a LOC device. 13

Figure 3: Illustrative diagram showing EDL and various layers of liquid. 17

Figure 4: Illustration of an electrical double layer potential field for a flat surface in

contact with an aqueous solution. 18

Figure 5: Velocity profiles for (a) Electro-osmotic flow; (b) pressure-driven flow 22

Figure 6: Schematic diagram of the microchannel used for the model. 23

Figure 7: Velocity profiles obtained for various haematocrit fractions 29

Figure 8: Shear-rate VS Viscosity for normal blood. 31

Figure 9: Effect of haematocrit fraction on blood viscosity. 32

Figure 10: Wall depletion of RBCs in a micro channel. 33

Figure 11: Schematic diagram of flow involving wall depletion. are thickness of

electric double layer and thickness of depletion layer, respectively. 34

Figure 12: Schematic of the flow in a microchannel 34

Figure 13: Electro-osmotic velocity profiles for several Debye-Huckel factors,

Figure 14: A schematic depicting the problem domain 39

Figure 15: Comparison of present model with Das & Chakrabortys model at h =0.45

and for N = 1 and N at 370C 47

Figure 16: Velocity profiles for different relative viscosities N. 48


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Figure 17: Effect of haematocrit on relative viscosity and velocity (bottom pic is an

enhanced view of near wall region) 49

Figure 18: Effect of Debye-Huckel parameter on velocity profiles. 50

Figure 19: Velocity profile for Different EDL thicknesses. 51

Figure 20: Relationship between EDL thickness and Ion density. 52

Figure 21: Relationship between EDL thickness and Debye-Huckel Factor 52

Figure 22: Relationship between Debye-Huckel factor and Ion density 53

Figure 23: Effect of potential applied on the net charge density. 53

Figure 24: Velocity profile for various values ofDe2 at N=1 54

Figure 25: Velocity profile for various values of De2 at N=5 55


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NOMENCLATURE:

Electric field potential.

Zeta potential

e Charge density,

Dielectric constantsZ valence

no ion density

ui,j,k Velocity component

p, px Pressure

Stress tensore Charge on electron

e Specific internal energy

qi , q Heat flux

Cv, Cp Specific heat constant at constant volume and pressure respectively

kth Thermal conductivity of fluid

T Absolute temperature.

sol Viscosity of fluid,

Permittivity of fluidt Time

kb Boltzmann constant

EDL , L Thickness of EDL and depletion layer, respectivelyC Concentration


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H Half width of microchannel

Debye-Huckel parameterN Relative viscosity

DNA De-oxy ribose Nucleic Acid

RBC Red Blood Cells

LOC Lab-on a-Chip

UCM Upper Convected Maxwell


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ABSTRACT

Biological macromolecules are often handled through microfluidic systems, in which

these molecules can transport and react. A common driving force behind such

microfluidic transport processes is the electrokinetic force, which originates as aconsequence of interaction between the electrical double layer potential distribution

and the applied electric field. The first part of the study is focused on understanding

the electrokinetic phenomena and their importance in the field of microfluidics. Then

an analytical solution is derived for the velocity of a non-Newtonian bio-fluid under

the influence of electro-osmosis. Here the non-Newtonian fluid is assumed to undergo

wall depletion. The fluid within the depletion layer is assumed to show Newtonian

characteristics and the fluid outside the depletion layer is assumed to show visco-

elastic characteristics and is modelled using Phan-Thein-Tanner model adopted from

Pinho et al. (2009). As a case study, the flow behaviour of a blood sample is analysed

and compared to the findings of Das & Chakraborty (2006). And the solution is

investigated for various factors effecting the generated velocity profiles of the blood

sample, such as haematocrit fraction, relative viscosity and EDL thickness.


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DECLARATION

No portion of the work referred to in the dissertation has been submitted in support ofan application for another degree or qualification of this or any other university or

other institute of learning.

COPYRIGHT

i. The author of this dissertation (including any appendices and/or schedules to this

dissertation) owns certain copyright or related rights in it (the Copyright) and s/he

has given The University of Manchester certain rights to use such Copyright,

including for administrative purposes.

ii. Copies of this dissertation, either in full or in extracts and whether in hard or

electronic copy, may be made only in accordance with the Copyright, Designs and

Patents Act 1988 (as amended) and regulations issued under it or, where appropriate,

in accordance with licensing agreements which the University has entered into. This

page must form part of any such copies made.

iii. The ownership of certain Copyright, patents, designs, trade marks and other

intellectual property (the Intellectual Property) and any reproductions of copyright

works in the dissertation, for example graphs may not be owned by the author and may

be owned by third parties. Such Intellectual Property and Reproductions cannot and

must not be made available for use without the prior written permission of the

owner(s) of the relevant Intellectual Property and/or Reproductions.

iv. Further information on the conditions under which disclosure, publication and

commercialisation of this dissertation, the Copyright and any Intellectual Property

and/or Reproductions described in it may take place is available in the University IP

Policy (see http://documents.manchester.ac.uk/display.aspx?DocID=487), in any

relevant Dissertation restriction declarations deposited in the University Library, The

University Librarys regulationsand in The Universitys Guidance for the Presentation

of Dissertations. (see http://www.manchester.ac.uk/library/aboutus/regulations)


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THE AUTHOR

I am Anupam Kumar Pilli, from India. I am a Mechanical Engineering graduate,

pursuing MSc. Thermal Power & Fluid Engineering. So far, this work has been my

first piece of research on an international platform. The present topic of dissertation

Electrokinetics with biochemical reaction is a totally a new branch of physics for

me to work with as, neither my first degree nor my current program deals with

Electrokineticsand biochemical reactions.But Ive been always keen to explore new

fields and this work is an example of my interest in inter-disciplinary research. I am

extremely thankful to my school and my dissertation guide for giving me thisopportunity and the resources to complete my research successfully.

ACKNOWLEDGEMENTS

I would like to take this opportunity to thank my parents for giving me this

opportunity to go far away from home to pursue my dream, without whose support

this wouldnt have been possible.

I would like to express my sincerest gratitude to my personal tutor and dissertation

guide Prof. Ali Turan for presenting me with this challenging topic for my dissertation

and for guiding me throughout the process of research undertaken and for giving his

valuable advices that made possible this dissertation today.

Finally, I would like to thank all my friends and family for supporting me throughout

my career and helping me go through ups and downs of life.

- Anupam Kumar Pilli


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Chapter 1

INTRODUCTION

1.1 Introduction to Microfluidics1.1.1 Micro-Electro Mechanical Systems (MEMS) & Microfluidics:

The world is running towards miniaturization and is getting smaller and smaller from

day to day. The computers which once occupied a whole room now can fit in ones

palm. Over the past two decades a lot of research is being done on miniaturization of

various applications in day to day life to applications that employ various mechanical,

fluidic, electromechanical and thermal systems. This led to the development of MEMSwhich provided a platform for the development of revolutionary devices for various

chemical, biological and biomedical applications. MEMS employ various types of

fluid flows which paved the way for research into fluid flows at micro and nano scale.

1.1.2 Microfluidics, History and Various Microfluidic Systems:

Microfluidics is the study of fluid flows, simple and complex, single and multi-phased

those occur in micro-scale systems. The first device that employed microfluidic

phenomenon was developed around 1975. It was a gas chromatography system that

circulated gas by electromagnetic injection through micro canals etched in silicon. But

the science community for its own reasons didnt welcome the development of such

technologies at that point of time (Reyes. et al 2003).

It was only after 1990 the world and scientific community concentrated on

miniaturization, and different varieties of microfluidic systems. Since then, lots of

microfluidic systems are being developed. Some examples of the microfluidic systemsare chemical micro-reactors, micro-mixers, electro-phoretic separation systems,

electro-osmotic pumping systems, diffusive separation systems, DNA amplifiers,

cytometers and the list goes on.

Microfluidic systems are very useful in biomedical and biochemical applications. The

development of micro and nano fluidic devices has provided the possibility to examine

and study biological processes on a length scale where most of the biological

processes takes place, like DNA sequencing, DNA hybridization etc.,. These devices


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are called Lab-On a- Chips (LOCs). The LOCs can accommodate and perform various

functions of the different devices that are necessary for the study of biological

processes on a single chip of the size of a few square centimetres.

Figure 1: Schematic drawing of the micro-electrophoresis device. Dimensions in mm.

The above device was developed at the Forschungszentrum Karlsruhe, Germany.

Barz, (2008) developed a comprehensive model of electrokinetic flow and migration

in micro channels with conductivity gradients that could explain various phenomena

occurring in the above device.

1.2 Introduction to Electrokinetics:Electrokinetic phenomena are usually characterized by the tangential motion of liquid

with respect to an adjacent charged surface, (Lyklema 1992). Reuss. (1809), first

observed that clay particles, dispersed in aqueous media, migrate under the influence

of an applied electric field. This was example of the electrokinetic phenomenon called

electrophoresis. In this example the surface was that of a clay particle and it moved

with respect to the stationary liquid. Conversely, the particle may be stationary and the

liquid might move with respect to the particle, this is called electro-osmosis. A

detailed explanation of these electrokinetic phenomena is provided in chapter 2.

The main idea of the LOC devices is to perform various functions such as separation,

concentration and detection systems onto a single device. Thus, enabling these devices


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to perform a wide variety of tasks for various applications in the different fields.

Generally, a biochemical process involves manipulation, concentration and/or

separation of different types of bio-particles/molecules. And these operations are

usually done by employing various techniques (Morgan & Green 2003) such as optical

tweezers, ultrasound, magnetic sorting (MACS), fluorescence (FACS), filtration,

centrifugation and electric-field approaches. With the advent of MEMS and

microfluidics, many of these techniques are miniaturized onto LOCs with the help of

electrokinetic phenomena. The most used electrokinetic phenomena are electro-

osmosis and electrophoresis. Electric-field based manipulation and separation methods

are so successful that many of the devices are now commercially available.

Pohl (1978), had been a pioneer in this field and had written a classic text on

Dielectrophoresis.

1.3 Introduction to Lab-on a-Chip, LOC devices:A LOC is a micro-scale chemical or biological laboratory built on a thin glass or a

plastic plate with a set of micro-channel networks, electrodes, sensors and electronic

circuits (Li 2004a). The flow of fluid and other operations are controlled by applying

electric fields through electrodes. As the LOC devices are miniature versions of theconventional lab equipment which normally occupy huge space and require relatively

huge amounts of reagents and samples, they reduce the amount of samples and

reagents required and the analysis time. They also give high throughput and provide

portability.

Figure 2: Schematic representation of microfluidic components in a LOC device.


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The above figure shows the sketch of a common LOC device. (Li 2004a) A LOC

usually consists of a number of components such as pump, mixer, reactor, dispenser

and a separator as shown in Figure 2. This allows the performance of multiple

operations on a single chip. (Li 2004a) One can be using a simple LOC similar to the

one in Figure 2 for DNA hybridization. The reagent loading wells are filled with a

known single stranded DNA tagged with a fluorescent dye and an unknown single

stranded DNA solutions and then pumped into a mixer by applying electric fields

through respective electrodes. This mixed solution with then be pumped into a reactor

where the unknown DNA fragments will react with the dye tagged DNA fragments at

a specific temperature. The matched DNA samples will bind with the fluorescent

DNA fragments. The reaction products will be pumped to a dispenser where they will

be subjected to another switching electric field which causes a plug of DNA fragments

to migrate into a separation channel where they are separated according to the charge

to mass ratio by electrophoresis. Then a laser is imposed on the separated DNA, the

one that tagged along with the known DNA probe molecules will give out

fluorescence, the larger the separated fragment the stronger the fluorescence. The

detected light intensities are fed to a computer to provide the sample analysis.

These LOC devices are reducing the cost of health care which is why the scientificcommunity around the world is keen about developing more LOC technology. As it is

known that most of the important media in biomedical analysis and diagnostics are

fluids, like whole blood samples, proteins, cell suspensions, antibody solutions etc., it

is required to know the quantitative control of fluid flow and mass transport processes

in the micro-channels, which are attributed to the fields of microfluidics and

electrokinetics.

This is a budding field in scientific research and there is not a lot of research done on

these topics and an understanding of complex electrokinetic phenomena in

microchannels is necessary for the design and development of better, durable and

reliable LOC devices.

1.4 Objectives and Structure of this Dissertation:

1.4.1 Objectives:

The objectives of this dissertation are:


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1. To gain familiarity and understand the concepts in the fields of electrokineticsand microfluidics related to biochemical processes and LOC devices.

2. To find an analytical solution for the velocity of a non-Newtonian bio-fluid(blood) in a pure electro-osmotic micro-channel flow.

3. To compare the results with the solution given by Das & Chakraborty (2006).4. To observe various factors affecting the velocity profile and limitations if any.

1.4.2 Structure of this Dissertation:

As it can be observed from the above introduction, the fields of electrokinetics and

microfluidics are relatively new topics in scientific research. This dissertation is

structured in such a way to provide the reader with all the necessary information to

understand the basic concepts of electrokinetics and microfluidics to enable the reader

to perceive the analytical solution and analyse the results obtained in the end of this

dissertation. Chapter 2 provides all the basic concepts of electrokinetics and equations

that describe various electro-kinetic phenomena. It will also throws light on the

relevant research that has been done concerning the present topic, i.e. Electrokinetic

transport with biochemical reactions. Chapter 3 presents the reader with the

theoretical model and the analytical solution for velocity of a pure electro-osmotic

blood flow in micro-slit channel. Chapter 4 presents the reader with the results

obtained from the model and compares them with Das & Chakraborty (2006)s model.

Finally the dissertation concludes with the findings, contribution of this dissertation to

the fields under discussion and giving directions and scope for further research.


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Chapter 2

LITERATURE SURVEY

The objective of this literature survey is to know the basic electrokinetic phenomena

that are involved in microfluidics relevant to topic of this dissertation and also to gain

the knowledge from previous research relevant to the topic under discussion.

This literature starts with a biochemical analysis with microfluidic systems then the

characterization of liquid flows within microfluidic systems. Then it continues giving

an analytical solution for velocity, concentration and temperature fields of a non-

Newtonian bio-fluid (in this case blood) for electro-osmotic flow within micro-

channels. As the bio-fluid that will be used for validating the present model with Das

& Chakraborty (2006)s model is blood, this literature also include the rheology and

hemodynamics of blood and describes different aspects of blood as a non-Newtonian

fluid, also explains the dependence of blood viscosity on haematocrit fraction of

blood. Then it explains the importance of slip/wall depletion condition. And then it

throws light on some visco-elastic effects on electrokinetic flow in micro-channels and

gives a basic model for a visco-elastic fluid in electrokinetic flow through

microchannels. Finally it concludes describing the work that is to be carried out for the

successful completion of this dissertation.

2.1 Basics of Electrokinetic phenomena:

2.1.1 Electrical Double Layer (EDL):

Most microfluidic applications employing electrokinetics use di-electric materials, itsuseful to know few characteristics of di-electric media. Some examples of di-electric

media are plastics, organic /bio fluids, water, electrolyte solutions and gases. The

molecules of di-electric material are permanently polarized because of their

asymmetrical molecular structure (Li 2004b). When such materials are subjected to an

electric field its molecules align to form di-poles i.e. two equal and opposite charges

separated by a distance.


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And the electric potential in such a medium is given by following equation called

Poisson equation,

Where, = the electrical field potential, e = charge density, 0 and are the dielectric

constants in the medium and vacuum, respectively.

When a solid surface comes in contact with an aqueous solution it acquires some

surface charge or surface potential. Generally, the aqueous solution is electrically

neutral with equal number of positively and negatively charged ions. As the surface in

contact with the aqueous solution is charged, it tries to attract the counter ions towards

the surface, and the population of the counter ions at the solid liquid interface

increases i.e. the concentration of counter-ions at the surface is higher than that of in

the bulk solution. And the concentration of co-ions at the surface is lower when

compared to that of the bulk solution far away from the solid surface, due to electric

repulsion. This creates a net charge close to the surface and this net charge should

balance the charge at the solid surface (Li 2004b).

The solid surface and the layer of the liquid containing this balance charge is EDL.The layer of the liquid immediately in contact with the immobile ions is called

compact layerand the rest is called diffuse layer. The plane of contact of the immobile

compact layer and mobile diffuse layer is called shear plane.

Figure 3: Illustrative diagram showing EDL and various layers of liquid.


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The net charge density gradually goes to zero from compact layer to the electrically

neutral bulk liquid. The ion and potential distributions in the diffuse layer are given by

an equation called Poisson-Boltzmann equation, which will be discussed later in the

literature. The electric-potential and solid liquid interface is difficult to measure, but

the potential at the shear plane can be measure experimentally and is called zeta

potential () which is approximated to give the surface potential (Li 2004b), the

electric potential due to ions falls off exponentially with distance from wall and the

distance at which the potential falls of to 1/e of its maximum value is called Debye-

length (e charge on electron) .

Figure 4: Illustration of an electrical double layer potential field for a flat surface in

contact with an aqueous solution.

Li (2004b), in his book presented a theoretical model for analysis of EDL,

This equation is called Poisson-Boltzmann equation, Where, is the ion density (inmolar units), e is the charge on electron, Z is the valence, is the Boltzmannconstant, T is the absolute temperature

2.1.2 Electro-osmosis:

If an electric field is applied across such a configuration as shown in Figure 3, theexcess counter-ions in the diffuse layer of the EDL will move under the applied


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electrical force. This is called the electro-osmosis. (Li 2004c) As the ions move, they

drag the surrounding liquid molecules to move with them due to the viscous effect,

resulting in a bulk liquid motion. Such a liquid motion is called the electro-osmotic

flow. In the literature many a studies have been done on the steady state electro-

osmotic flow in microchannels of various shapes, Tsao(2000) & Kang et al. (2001)

presented model on annulus shaped channels, Koh & Anderson (1975) presented

model for elliptical microchannels. Models for rectangular microchannels are

presented by Arulanandam & D. Li (2000) and the T and Y shaped microchannels by

Patankar & Hu (1998); Harrison et al. (1999); Bianchi et al. (2000). Electro-osmotic

flow is a very important phenomenon for the design LOC devices as it is employed for

transport of liquids and mixing different solutions through micro-channel network.

2.1.3 Electrophoresis:

When a charged particle in a suspension is subjected to an electric field, it experiences

some force due to the surface charge present on the particle. As a result the particle

tends to move in a certain direction depending on the net force on the particle. This

movement of the particle is called Electrophoresis. When the particle is subjected to a

non-uniform electric field the resulting motion of the particle is called di-

electrophoresis.

Electro-kinetic flow involves various processes including fluid flow, electrostatic

interaction, species diffusion, and sometimes energy transfer. And it is considered one

of the most typical multi-physical transport phenomena because of its presence in

almost all electrolyte solutions in engineering applications (Li 2004d) (Masliyah

2006). The mass transport in ion channel cells can be understood in a much better way

with an extensive knowledge on micro/nano scale electro-kinetic flows (Doyle et al.

1998) (Coalson & Kurnikova 2005). Accurate predictions of electro-osmotic flow in

microfluidic devices may help in producing optimal designs of bio-macromolecules

diagnostics (Sharp & Honig 1990), (Wong et al. 2004), (Stone et al. 2004) & (Squires

& Quake 2005).

The present work is about electro-osmotic flows of non-Newtonian bio-fluids in

particular and the literature that follows was compiled to understand the basic models

that are available already and various aspects of non-Newtonian fluids that are ofimportance in microfluidic flows.


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2.2 Relevant research to the topic of this dissertation:

2.2.1 Biochemical analysis and characterization of flows in microfluidic

systems:

A qualitative analysis of microfluidic systems for biochemical applications was given

by Bilitewski et al. (2003). Their work presents the reader with importance of

microfluidic phenomena in biochemical assays such as analysis of nucleic acids,

enzymes and immunoassays. They laid down the principles of microfluidics systems

and gave justification for the miniaturization of analytical systems to microfluidic

systems and also they have given some classic examples of application of micro-chips

to biochemical analysis which will be discussed below. They also noted that the

development of microsystems was accelerated by improvement of fabrication

techniques and they noted the interface between the micro-ship and the macro-world,

i.e. the sample to be analysed slows down widespread application of the micro-chips

because, even though the sample taking systems like pipettes are replaced by more

efficient systems, the sample must be introduced manually, thus losing the benefit of

high throughput analysis.

It was presented by Bilitewski et al. (2003) the development of miniaturized analyticalsystems to microsystems is not just the transfer of analytical assays to microsystems

but also because, in a micro-channel the surface to volume ratio is larger than in the

normal equipment used for analytical assays, thus the chemical nature of the surfaces

are important, because the present techniques of electrokinetics doesnt need any extra

components such as pumps or valves instead a couple of electrodes and application of

electric field is all that is needed for electrokinetic flows (Pyell 2003; Bousse et al.

2000). Also because the flows resulting from the capillary tubes mostly tend to be

laminar, also detection becomes easy with reduced dimensions that is why the

fluorescence detectors are often used in microsystems (Vandaveer et al. 2002; Lacher

et al. 2001).

In their paper, Bayraktar & Pidugu (2006) presented different types of liquid flows in

microfluidic systems, materials used for microfluidic systems, various cross-sections

employed in microchannels and entrance effects, effects of pressure and friction, flow

control techniques in microchannels and mixing in microchannels, which will bediscussed below.


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The governing equations of a normal fluid flow are given by Navies-Stokes equations:

()

Where, uirepresents the flow velocity; is the density; p is the pressure; is the stress

tensor; e is the specific internal energy; F is the body force; and q is the heat flux. The

repeated indices in any single term indicate a summation following a standard

summation convention. The above equations need to be modified to include forces by

electric field in electrokinetic flows.

Electrokinetic flows are mostly characterized by low Reynolds number and typical

electric fields of about 100 V/cm (Stroock & Whitesides 2003). Though pressure

driven flows can be used in microfluidic systems, electrokinetic flows are preferred

due to their uniform velocity profiles across the channel, i.e. the velocity across the

channels is almost the same except in the region close to the wall provided the charge

on the channel walls is uniformly distributed. Bayraktar & Pidugu (2006) mainly

concentrated on electrophoresis and electro-osmosis. The equations of motion for an

incompressible, Newtonian, isotropic fluid in the presence of an external electric field

are given as:

Where, is the viscosity; e is the electric charge density, E is the external electric

field, Cv is the specific heat at constant volume, kth is the thermal conductivity. The


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term in momentum equation is the Lorentz body force and the term inenergy equation is the corresponding work term.

For one-dimensional, fully developed electro-osmotic flow, Eq.(2.1.5) reduces to

Where, is the permittivity and is the electric potential.

Figure 5: Velocity profiles for (a) Electro-osmotic flow; (b) pressure-driven flow.

The flow profiles of a pressure driven flow and a flow under the influence of electric

field is given in Figure 5. The advantages of electrokinetic flow are it is very useful in

flows where separation of mixtures is important, no necessity of moving components,

it is also advantageous in flows with branched channels as such flows doesnt require

any valves and can be controlled by varying voltage across the channels, the

disadvantages of electrokinetic flows are they are limited to a certain solvents only and

they require high electric and are highly sensitive to surface contamination.

2.2.2 Mathematical model for non-Newtonian fluids under electrokinetic

microchannel flows:

Most bio-fluids show non-Newtonian characteristics, and with the usage of

electrokinetics for transport of bio-fluids through LOC devices, it is important to know

mathematical characteristics of flow of such fluids. Das & Chakraborty (2006) has


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provided analytical solutions for velocity, concentration and temperature fields for a

non-Newtonian bio fluid under electro-osmotic microchannel flow. And the model

was used to study the flow characteristics of blood, in which the flow characteristics

are modelled as functions of haematocrit fraction (will be discussed later sections).

It was known from their research that, transfer of momentum, heat and solute in many

application involving bio-fluids within LOCs are not adequately explained by the

generic electrokinetic models for Newtonian fluids. This is because the constitutive

equations for most of bio-fluids are nonlinear and strain-rate dependent.

In their model, Das & Chakraborty (2006) considered that the non-Newtonian fluid

behaves as a power-law fluid under electro-osmotic forces. It is assumed that flow is

fully developed, incompressible and fluid properties are unaltered by temperature and

has no external pressure gradient imposed. The temperature field is steady and fully

developed; the flow is under constant pressure gradient. The charge density is

calculated on the basis of average temperature, and Debye-Huckel linearization

principle remains valid.

Figure 6: Schematic diagram of the microchannel used for the model.

Mathematical model(Das & Chakraborty 2006):

Microfluidic transport of a non-Newtonian fluid though a parallel slit channel of

height 2H, length L0 and width w, w >> 2H as shown in Figure 6. Bottom half of the

channel is designated asH and upper half as H. A potential gradient is applied along

the axis of the channel that provides the necessary driving force for the electro-osmotic

flow.


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Continuity equation:

Where, is the density of the fluid and Vis the flow velocity.

Linear momentum equation:

Where, is the stress tensor, is the net electric charge density and is the appliedelectric field.

Poisson-Boltzmann equation for potential distribution within EDL:

Where, denotes the EDL potential and is permittivity of the fluid. is given by:

Where, is the ion density (in molar units), e is the charge on electron, Z is thevalence, is the Boltzmann constant, T is the absolute temperature. The relationshipbetween the net charge density and Debye length is given by:

Thermal energy conservation equation:

( ) Where, is the strain-rate tensor, is the thermal conductivity of the fluid and isthe heat generation per unit volume due to joule heating, given by


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Where, is the electrical conductivity of the fluid.Species conservation equation for the transported solute

Where, C denotes the instantaneous concentration of the solute being transported (the

solutes are assumed to be uncharged) and D is the diffusion co-efficient of the solute

in the fluid. The constitute relation for stress tensor for a power law fluid is given by:

After taking the assumptions into account, the governing equations become,

Where, T/x = dTm/dx = dTW/ dx = MT which is a constant for thermally fully

developed flow with constant wall heat flux boundary condition. Even though the

velocity and temperature fields are fully developed, the concentration field is transient.

Such kind of behaviour is typical to many biotechnological applications of relevance.

Hence, the transient term of the concentration field is retained and is given by,

Because of symmetry only bottom half of the channel is considered for the

mathematical analysis. The corresponding boundary conditions are taken as follows:


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Where, (t) describes the reaction taking place at the channel surface and can be

expressed as, (t) = - E1 Cwall and Cwall is the instantaneous wall concentration of the

solute at a given axial location and E1 is a constant. And Cwall is given by Cwall =

E2exp(-Ft), where E2 and F are time-dependent constants. Therefore, (t) =Eexp(-Ft).

where, E and F are some time-dependent constants, and (t) = 0 for non=reactive

channel walls.. The initial condition for unsteady concentration field is taken as: C = 0

and t = 0.

Combining Eq. (2.2.4) & (2.2.11) gives,

Using, Debye-Huckel linearization principle (sinhX ~ X),

The above equation can be solved by using the boundary condition in Eq. (2.2.15)

Where, and using, , the velocity is obtained from Eq(2.2.10) & using Eq. (2.2.14),

[ { } { }]


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[

]

In a similar approach the temperature and concentration fields are deduced but they are

beyond the scope of the present study and only their final expressions are presented

below,


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Other terms appearing the above equations are defined as,


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Das & Chakraborty (2006) applied their model for blood flow in a microchannel, as

blood shows non-Newtonian behaviour with a non-Newtonian behavioural index (n)

lying between 0 and 1. For this model it is taken as a power law fluid, with the non-

Newtonian indices based on the haematocrit percentage (discussed in later section) the

blood sample. And they obtained the velocity profile as shown in Figure 7. They

employed the following equations to determine the kand n indices, the non-Newtonian

indices of the blood based on the haematocrit fraction of blood,

Where, h is the haematocrit fraction and C1, C2, C3 are characteristic coefficients. These

coefficients depends on various factors like plasma globulin, plasma protein,

temperature, (Das & Chakraborty 2006). The physical properties Das & Chakraborty

(2006) used for the following velocity profile is given in Table 1.

Figure 7: Velocity profiles obtained for various haematocrit fractions


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Table 1: Physical properties and problem data used by Das & Chakraborty (2006).

This model serves as a basic model for a non-Newtonian bio-fluid and helps in

understanding the basic theoretical modelling necessary for electro-osmotic flow a

bio-fluid in microchannels.

2.2.3 Rheology and Hemodynamics of Blood:

Blood is made up of different cells such as red blood cells (RBC); white blood cells

(WBC) and platelets, suspended in an aqueous solution made up of proteins and salts

called plasma. Hence, blood can be considered as a two-phase suspension. And blood

shows non-Newtonian fluid characteristics as its apparent viscosity depends on shear

forces acting upon it. Apparent viscosity of blood is determined by various factors

like, haematocrit fraction, plasma viscosity, RBC aggregation and the mechanical

properties of RBCs.

Baskurt & Meiselman (2003) gave an excellent piece of literature about the rheology

and hemodynamics of blood explaining the manner in which blood viscosity is

affected by haematocrit, shear rate and RBC aggregation.


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Blood viscosity:

As blood shows non-Newtonian fluid characteristics, its viscosity cannot be described

by a single value of viscosity. (Baskurt & Meiselman 2003) The apparent viscosity of

a non-Newtonian fluid may decrease (shear thinning behaviour) or increase (shear-

thickening behaviour). Human blood shows shear thinning behaviour, the apparent

viscosity of blood decreases with increasing the shear see Figure 8, at low shear rates

the apparent viscosity is high.

Figure 8: Shear-rate VS Viscosity for normal blood.

Plasma viscosity:

Plasma is an aqueous suspension in which different cells of the blood are suspended;

therefore any change in its viscosity will affect the whole blood viscosity. Plasma

exhibits Newtonian fluid characteristics in general.

Haematocrit value:

It is defined as the %volume of RBCs present in a blood sample. The presence of the

cells, dominantly RBCs is the main reason for the viscosity of blood being higher than

the plasma viscosity. As the number/concentration of cells increase in the blood the

viscosity of the blood increases. And the ratio of blood viscosity to that of plasma is

called the relative viscosity of blood (N). The relation between haematocrit fractionand blood viscosity is shown in Figure 9 below.


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Figure 9: Effect of haematocrit fraction on blood viscosity.

It can be observed from the above figure that viscosity varies exponentially with

haematocrit fraction. Pries et al. (2012) in their work described the dependence of

blood viscosity on haematocrit factor; in their work they derived the following

empirical relation for relative viscosity of blood:

Where, N= relative viscosity of blood, Nrel 0.45 = relative viscosity of blood when

haematocrit % = 0.45, HCTD = Discharge haematocrit, which in experiments can be

takes as the volume concentration of RBC entering or leaving the channel, C describes

the curvature of the relation between apparent blood viscosity and haematocrit, it is

equal to unity if the relation is linear and it is less than unity when the convex shape of

the relationship is towards the abscissa. (From Figure 9 it can be known that, C


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phase close to the solid surface. The typical thickness of a depletion layer would be

around 0.1 to 10 micrometres in geometries of quite larger magnitude.

He noted that, smooth walls and geometries with high shear-rate gradients,

concentrated solutions of high molecular weight polymers with suspensions of large

particles and emulsions of large droplet size are susceptible for wall depletion

phenomenon.

Wall depletion occurs when a two/multi-phase liquid comes in contact with a smooth

solid surface. The suspended particles in the liquid cannot penetrate through the wall,

hence the local micro structure of the liquid is affected, and this can be occurred even

without a flow, i.e. no-flow slip/wall depletion. Apart from this, the local isotropy of

Brownian motion of very small particles close to the wall is destroyed causing

additional slip, Delime & Moan (1991) noted that the collision of dynamic particles is

also altered in near wall regions. It also occurs due to the repulsion of particles

adjacent to the wall due to different physico-chemical forces between particles and the

walls, like electro-static forces.

Barnes (1995) also gave the following condition that may lead to large and significant

wall depletion effects:

1. In fluids with large particles as disperse phase;2. In fluids with large dependence of viscosity on the concentration of the

dispersed phase, smooth walls and small flow dimensions;

3. Usually in low speed flows;4. In situations with walls and particles carrying like electrostatic charges and the

continuous phase is electrically conductive.

A flow with slip or wall depletion effects can be imagined as flow with a very thin

layer at the boundary, and the bulk flow with the original concentration.

Figure 10: Wall depletion of RBCs in a micro channel.


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Figure 11: Schematic diagram of flow involving wall depletion. are thicknessof electric double layer and thickness of depletion layer, respectively.

2.2.5 Mathematical model for visco-elastic fluids under electro-osmotic flow in

microchannels:

Afonso et al. (2009) presented a model for the flow of visco-elastic fluids in

microchannels, namely, parallel slit and pipes under the influence of electrokinetic and

pressure forces. They used a simplified Phan-Thein-Tanner (sPTT), (Bird et al. 1987)

and FENE-P model, but only PTT model is explained in this literature as it seems

more relevant to visco-elastic fluids. The flow characteristics of non-Newtonian fluids

are fairly known when their rheological descriptions are inelastic and rely on simple

models like power law as shown in Das & Chakraborty (2006)s model.

Figure 12: Schematic of the flow in a microchannel.


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Similar to Das & Chakraborty's (2006) model presented in section 2.2.3, Afonso et al.

(2009) used a channel as shown in Figure 12 and considered only half of channel for

modelling because of the symmetry. And gave the governing equations as follows:

Continuity:

Momentum equation:

Where, is the polymeric extra-stress contribution.Constitutive equations for sPTT model:

From network theory arguments (Bird et al. 1987) who derived the following

equations:

Where, D = is the rate of deformation tensor, is the relaxation time ofthe fluid, is the polymeric viscosity coefficient and represents the upper convectedderivative of, defined as follows: The stress coefficient function, is given by the linear form (Bird et al. 1987)

Where, represents the trace of extra-stress tensor and is parameter that imposesthe upper limit of elongational viscosity.

Poisson-Boltzmann equation:

Where, is the di-electric constant of the solution, as seen in previous model by Das& Chakraborty (2006) it can be integrated to get,


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Boundary conditions and assumptions:

Only one half (0 ) is considered for the analysis. The flow is considered to befully developed and the velocity and stress fields depend on the transverse coordinate

y. It is assumed that the ionic charge distribution is low such that the EDL formed is

thin with a weak electric field. For this EDLs Eq.(2.6.7) can be approximated as

follows, this equation is called Debye-Huckel approximation. And the rest of the

boundary conditions are:

Following the similar approach for solving the above equations, as present in above

model by (Das & Chakraborty 2006), the electro-osmotic velocity was obtained as,

Solution:

As the flow is considered one dimensional,

The momentum equation becomes,

Integrating the Eq. (2.2.20) is obtained and substituted in Eq.(2.6.10) to get Using the constitutive equations, the normal stress component was obtained. (in detail

derivation will be presented in next chapter)


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Using the above two equations, velocity gradient across the half channel is found and

integrating with the suitable boundary conditions the electro-osmotic velocity is

obtained as U = uE

+ uP

+ uEP

.. Where,

Where, .U

E= velocity due to electro-osmosis, u

P= velocity due to pressure gradient, u

EP=

velocity due to combined effect.

Figure 13: Electro-osmotic velocity profiles for several Debye-Huckel factors, .


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It was known from the various results obtained from this model that the analysis is

restricted to very small EDL thicknesses. In pure electro-osmotic flows, the velocity

profiles show plug like characteristics as shown in Figure 13. And the velocities

increase significantly with high values of

and thin double layers.

From the above literature, a good understanding of the various electrokinetic

phenomena is accomplished, and it is known that blood functions as a non-Newtonian

liquid with shear thinning behaviour. It can be inferred from Barnes (1995) work that

blood is susceptible to wall depletion in microfluidic phenomena as blood flow in

microfluidic phenomena exhibits the characteristics of a fluid that is susceptible for

wall depletion as mentioned in Barnes (1995) work. The literature on rheology and

hemodynamics on blood gave a lot of important information about the behaviour of

blood flows and a good account of the factors that affect blood viscosity. Models

presented by Das & Chakraborty(2006; Afonso et al. (2009) helped in understanding

the governing equations of the electrokinetic flows and various constitutive equations

that can be employed in different flow situations. Their models are trivial yet

important as they serve as a template to carry on further research in developing

complex models.

Further in this dissertation, a mathematical model will be presented to predict the flow

of a non-Newtonian bio-fluid under the influence of electro-osmosis in microchannels

including the wall depletion phenomenon.

The model is based on Afonso et al. (2009) adoption of sPTT model with additional

boundary conditions for wall depletion. And the model will be validated against Das &

Chakrabortys (2006) model by applying the model for the flow of blood in same

conditions as modelled by Das & Chakraborty(2006).


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Chapter 3

MATHEMATICAL MODELLING

In this chapter, an analytical solution will be obtained for the electro-osmotic flow of a

non-Newtonian bio-fluid in a microchannel. The fluid is to have wall depletion effects

and was modelled accordingly. Most bio-fluids are non-Newtonian and are

two/multiphased. As discussed in literature, such fluids are susceptible to wall

depletion. The phase of the fluid in depletion layer is assumed to show Newtonian

behaviour hence a linear stress strain constitutive equation is used. But the bulk fluid

is assumed to visco-elastic behaviour and the constitutive equations used by (Pinho et

al. 2009) and proposed by (Bird et al. 1987) are used. And model is validated by

applying the model to blood flow, and presented as a case study with the findings in

next chapter.

3.1 Microchannel and its parameters:

A parallel plate microchannel is considered such that its length (L 0) is much greater

than its width (2H) i.e. L0 >> 2H. See the figure below, the thickness of eletrical

double layer is denoted by EDL and the depletion layer formed is considered to be at a

distanve ofL. As the flow is symmetric over geometry, only the half isconsidered.

Figure 14: A schematic depicting the problem domain.


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3.2 Governing equations:

Continuity:

Where, is the density ofthe fluid and V is the flow velocity.

Linear momentum equation:

Where, is the stress tensor,

eis the net electric charge density and E is the applied

electric field.

Poisson-Botzmann equation:

Where, denotes the EDL potential and is permittivity of the fluid. e is given by:

Where, n0 is the ion density (in molar units), e is the charge on electron, Z is the

valence, kB is the Boltzmann constant, T is the absolute temperature.

Constitutive equations:

Layer 1: Newtonian solute depleted to depletion layer ( ):As the dispered phase of the fluid (solvent phase) to depletion layer is assumed to

show newtonian characteristics, the shear stress is given by a linear strees strain

relation:

Where, xy is the shear stress of the solvent, sol = coefficient of viscosity of the solvent


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Layer 2: Bulk fluid that is assumed to show visco-elastic characteristics

( ):Afonso et al. (2009) have used the constitutive equations of sPTT model laid down by

Bird et al. (1987) which are based on Network theory arguments (Thien & Tanner

1977). Same relations are employed to describe the stress strain relations of the bulk

fluid, i.e., visco-elastic fluid.

Where, D = (uT+u)/2 is the rate of deformation tensor, is the relaxation time of thefluid, is the polymeric viscosity coefficient and represents the upper convectedderivative of , defined as follows:

The stress coefficient function, f(kk) is given by the linear form (Bird et al. 1987)

Where, kkis a trace of extra stress tensor and ' is a parameter that imposes an upper

limit to elongational visocsity and is the coefficient of viscosity of the bulk fluid.

Assumptions:

. It is assumed that flow is fully developed, incompressible and fluid properties are

unaltered by temperature and has no external pressure gradient imposed. The

temperature field is steady and fully developed. The charge density is calculated on the

basis of average temperature, and Debye-Huckel linearization principle remains valid.

Boundary conditions:


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3.3 Analytical solution:

The charge density field can be calculated by combining Eq. (3.2.3) & (3.2.4) which

gives:

Using, Debye-Huckel linearization principle (sinhX ~ X),

Employing the boundary condition in Eq.(3.2.11) in above equation and integrating it

twice gives the solution for distribution of electric potential as follows:

Where,

called Debye-Huckel parameter and is related to EDL thicknessas Substituting Eq. (3.3.3) in Eq. (3.2.4) gives

Substituting the above equation in momentum Eq.(3.2.2) equation and intergrating

applying the boundary condition Eq (3.2.11) gives,

The integration constant goes to zero because

at y = 0.


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Solution for velocity profiles:

Within depletion layer (layer 1):

Using Eq. (3.2.5) and Eq. (3.3.5),

Integrating above equation with boundary condition in Eq. (3.2.9) gives,

Outside the depletion layer (layer 2):

The constitutive equation for sPTT model is obtained from equations Eq(3.2.6)-(3.6.8)

considering a fully developed flow u = (u(y),0,0) as follows,

Where, yy = xx + yy is the trace of extra-stress tensor, du/dy and

Then dividing the Eq (3.3.7) by EQ (3.3.9) the specific function f(xx) cancels out and

a relation between normal and shear stresses is obtained as

Substituting the expression for xy, Eq (3.3.5) in the above equation gives,

The velocity profile for bulk fluid is obtained by combining the equations (3.3.9)-

(3.3.11), which gives,


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Upon substituting the corresponding expressions in the above equation it gives,

Integrating it for u with the boundary condition in Eq. (3.2.10) gives,

3.4 Non-dimensional velocity and length scales:

Velocity profiles obtained are normalized by Deborah number(De) based on EDL

thickness and on the Helmholtz-Smoluchowski electro-osmotic velocity (uhs), which

are defined as (Park & Lee 2008)

/H andRelative viscosity of the non-Newtonian fluid (N) =

.

Employing the above to expressions to normalize the velocity profiles result in U*,

Non-dimensional velocity profile for fluid within depletion layer:


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Non-dimensional velocity profile for fluid within depletion layer:

Length scale is obtained by dividing the distance from centre of the channel by half

channel widht, Y*

= y / H.


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Chapter 4

PRESENTATION AND DISCUSSION OF RESULTS

As discussed earlier, the model was used to study the flow of blood under electro-

osmotic microchannel flow. The physical parameters used for the study were the same

as that of Das & Chakraborty (2006), refer table 1, section 2.2.3, except for the

average temperature, which for this study was taken as 370C. As studied in literature

the blood is a two phased liquid with different kinds of cells suspended in a aqueous

suspension, its viscosity depends on various factors. For the present study, the plasma

viscosity and apparent blood viscosity were taken at 370C, they are 1.5 mPa.sec and 4

mPa.sec respectively and at this temperature the haematocrit fraction of blood wouldbe 0.45. In order to see the effect of depletion layer on the flow, the depletion layer

was set as 0.1 m as Barnes (1995) noted that generally the thickness of depletion

layer would be around 0.1 10 m. And for all the cases unless specifies, istaken as zero according to the Upper convected Maxwells model(UCM) (Pinho et al.

2009).

4.1 Comparison of the present model with Das & Chakraborty'smodel:Non-dimensional velocity profile is obtained for Das & Chakraborty's model with a

constant haematocrit fraction of 0.45. And non-dimensional velocity profiles were

obtained for the present model with the relative viscosity(N) of 1 and relative viscosity

of blood at 370C. It can be observed from Figure 15, that the non-dimensional profile

for N = 1 matches exactly with profile obtained by Das et al,. But for N at 370C the

present model gave a higher velocity. The first case, i.e. N = 1, means that the blood

is considered as a homogenous and single phased fluid, meaning the effect of

depletion was not taken into account. In the second case N is taken at 370C, the value

of N at 370C is about 2.67, the plasma that is supposed to get depleted of RBCs within

the depletion layer is modelled separately as a Newtonian fluid and the rest of fluid is

modelled separately as a visco-elastic fluid. The velocity profile obtained in the second

case gave a higher velocity output because, within the depletion layer, the fluid in the

first case with N = 1, having not considered the RBC depletion was modelled as a

more viscous fluid than the fluid (blood plasma) in the second case, which has taken


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into account the RBC depletion from near wall region. Thus, giving a higher velocity

than in first case.

Therefore, Das & Chakraborty (2006) model when used for modelling a flow that

involves wall depletion, which most bio-fluids are susceptible to, underestimates the

velocities as it only considers the fluid as single-phased power law fluid. The present

model, on the other hand takes into account the wall depletion effects while modelling

a bio-fluid. This model can be used for predicting flow of blood in various LOC

devices which are used for plasma separation techniques. This can also be used

predicting the flow in LOCs used as separators.

Figure 15: Comparison of present model with Das & Chakrabortys model at h =0.45

and for N = 1 and N at 370C

4.2 Effect of relative viscosity and haematocrit on the flow with depletion

effects:

The study of the effects of varying relative viscosity N on the flow is important as it

impacts the velocity of the flow significantly, can be observed from the Figure 16 and

this is important as N indicates the level of depletion i.e., the higher the value of N the


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higher is the effect of depletion. Modelling with different N helps to study the

behaviour of flow at various levels of depletion. Here the velocity profiles are obtained

for different values of N and for the same value of . Within the depletion layer, thevelocity is directly proportional to N, hence the apparent increase in velocity with

increase in N.

.

Figure 16: Velocity profiles for different relative viscosities N.

Generally in the flow of a bio-fluid in microchannel, the higher value of N indicates

higher depletion and so the thickness of depletion layer is higher relative to flows with

lower relative viscosity.

As it was studied during the literature survey, the blood viscosity is a function of

haematocrit fraction. The viscosity of the blood varies exponentially with haematocrit

fraction. Therefore, the effect of haematocrit on relative viscosity is studied; velocity

profiles were obtained for various haematocrit factors as shown in Figure 17. The

relative viscosity values for different haematocrit factors was calculated by the

empirical relation formed by Pries et al. (2012) and Baskurt & Meiselman (2003), Eq

2.3.1. Different velocity profiles were obtained with different haematocrit and their

corresponding N values and were plotted. It can be observed from Figure 17 that the


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present model gives fairly consistent profiles, whereas Das & Chakraborthys model

performs poorly at higher haematocrit factors as its velocity profile at higher

haematocrit factor of 0.7 .

Figure 17: Effect of haematocrit on relative viscosity and velocity (bottom pic is an

enhanced view of near wall region)


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4.3 Effect of Debye-Huckel factor, Ion charge density and EDL thickness:

It is important to study the effects of above three parameters because, they are inter-

related and varying one parameter, affects another and hence affects the flow as a

whole. Velocity profiles were obtained for different values of as shown Figure 18

Figure 18: Effect of Debye-Huckel parameter on velocity profiles.

It can be observed that, as the values decrease the velocity profiles are fairly

consistent, except for the velocity profile for = 1e6. This is because, at this value of

, the EDL overlaps with electrical double layer. Therefore, the effect of double layer

is felt both inside and outside of the depletion layer and it can also be observed that forthe rest of the profiles, the velocity became constant well before the depletion layer

interface, but for the profile = 1e6 the velocity havent become constant at the

interface. In such cases, the sudden increase in viscosity and sudden reduction in shear

rate with a fairly constant shear stress causes the electro-osmotic velocity to lower.

As decreases, EDL thickness increases and so is its influence on the both sides of

the depletion layer. Different velocity profiles were plotted with increasing EDL

thickness to observe its effect on velocity profiles. See Figure 19 below.


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Figure 19: Velocity profile for Different EDL thicknesses.

As found earlier in the case of decreasing Debye-Huckel parameter, with the increase

in EDL thickness, its effect on velocity on both sides of the depletion layer increased.

And it can be observed that the model serves better when the EDL thickness is lower

than about 0.3 micrometres.

The reason for increase in EDL influence on both sides as its thickness increases is, as

its thickness increases the ionic density (no.of ions/cubic meters) decreases and hence

the influence of electro-osmosis reduces across the interface.

As EDL thickness increases, ionic density decreases and as ionic density decreases the

Debye-Huckel parameter decreases and vice versa. The following figures give their

relationship plots. Figures, 20, 21 and 22.


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Figure 20: Relationship between EDL thickness and Ion density.

Figure 21: Relationship between EDL thickness and Debye-Huckel Factor.


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Figure 22: Relationship between Debye-Huckel factor and Ion density.

Figure 23: Effect of potential applied on the net charge density.


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Charge density is also affected by potential applied, the above Figure 23, shows the

relation between potential applied and charge density, as the potential applied

increases, the charge density across the channel also increases. It is quite evident

because, as the potential increases, more number of particles are polarised and charged

according to the applied potential.

4.4 Effect of the factor :The quantity imposes the upper limit to elongational viscosity, thus it is important

to see its effects on velocity profiles.. Two different plots were made, each for N = 1

and N = 5 with different for different Debye-Huckle parameters as shown infigures below. It can be observed that the model is giving recirculation flow within the

channel at high and low Debye-Huckel parameter i.e. as decreases, EDLthickness tends to increase to depletion layer thickness. Therefore, as discussed earlier,

there will be a sudden jump in viscosity and the shear thinning effect comes into

action, hence recirculation should not be occurring and should be treated as an

unphysical phenomenon. For, high Debye-Huckel parameter and very low values, the model gave good results. It is expected to give completely physical results

for velocity profiles with high relative viscosity.

Figure 24: Velocity profile for various values at N=1.


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Figure 25: Velocity profiles for various values at N =5.


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Chapter 5

CONCLUSIONS

5.1 Conclusions:

A thorough understanding of the concepts in the fields of electrokinetics and

microfluidics related to biochemical processes and LOC devices is achieved. A

mathematical model is employed to find the analytical solution for the velocity of a

non-Newtonian bio-fluid in a pure electro-osmotic microchannel flow considering the

wall depletion effects. And the results of a case study with blood as the bio-fluid are

compared with the results obtained by Das & Chakraborthy. It was found that for

relative viscosity of unity, both the models agree with each other. But, most of the bio-

fluids show wall depletion characteristics and the present model is giving more

sensible velocity profiles that are consistent with the characteristics of a bio-fluid as

the model considers the bio-fluid as two-phased fluid, the one within depletion layer

being Newtonian and the one being outside the depletion layer as a visco-elastic fluid.

The effect of various factors on the model is observed and it is found that the relative

viscosity and haematocrit fraction have a huge effect on velocity profiles and it is also

observed that Das & Chakraborthys model deviate from electro-osmotic velocity

characteristics at higher haematocrit fractions. The present model produced velocity

profiles that fairly consistent.

The effect of other factors like Debye-Huckel parameter, ion charge density, thickness

of EDL are also studied and it is found that their effects on the velocity profiles are

inter-related. As EDL thickness increases, ionic density decreases and as ionic density

decreases the Debye-Huckel parameter decreases and vice versa. It is found that themodel serves better when the EDL thickness is lower than about 0.3 micrometres.

Other parameter that was studied to see its effects on velocity profiles is , itwas found that the model is giving recirculation at low Debye-Huckel parameters

corresponding to the EDL thickness approaching the depletion layer thickness and at

low relative viscosities (N ~ < 10), because in such flows the shear thinning becomes

dominant and the EDL approaching the thickness of depletion layer will bring in a


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sudden change in the viscosity of the fluid. Good results can be obtained for high

Debye-Huckel parameter and high relative viscosity.

It is important to appreciate that the present model works good for modelling the flow

of bio-fluids with UCM (Upper Convected Maxwells) condition i.e. andaccounts for the wall depletion effect that is to be expected from most of the bio-fluids

under microfluidic phenomena. It even works just fine for all values of withflows having high Debye-Huckel parameter and high relative viscosity. Hence this

model can be employed for predicting flows in application such as plasma separators,

for application is trapping DNA using wall depletion, it will also be useful in

predicting the blood coagulation point etc.

5.2 Future Work:

Microfluidics being a relatively young science, there is a not much literature available

and there is plenty of scope of research in multiple dimensions. There are branches of

electrokinetics like Non-linear electrokinetics, AC electrokinetics, etc.

Basing on the work done in this dissertation, one can find solutions for rectangular andcircular channels, with wall depletion. And a numerical analysis can also be performed

for these models. Defining various non-Dimensional parameters to explain their

effects on the flow field will come in handy. For the case study presented in this

dissertation, blood haematocrit is predominantly influenced by RBC and that is what

considered for the analysis, but apart from RBC, there are other cells in blood like

WBC and platelets which also affect the blood viscosity and haematocrit, therefore

further investigation in that direction will enhance this model.

It will be interesting to see the microfluidic simulations in Smoothed Particle

Hydrodynamics (SPH), particularly, Lattice-Boltzmann methods are quite relevant to

Microfluidic phenomenon and SPH codes can handle Lattice-Boltzmann methods

effectively.


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References:

Arulanandam, S. & Li, D., 2000. Colloids Surfaces A. , 161, p.89.

Barnes, H.A., 1995. A review of the slip (wall depletion) of polymer solutions,

emulsions and particle suspensions in viscometers: its cause, character, and cure.Journal of Non-Newtonian Fluid Mechanics, 56(3), pp.221251. Available at:http://www.sciencedirect.com/science/article/pii/037702579401282M.

Barz, D.P.J., 2008. Comprehensive model of electrokinetic flow and migration in

microchannels with conductivity gradients.Microfluidics and Nanofluidics, 7(2),

pp.249265. Available at: http://www.springerlink.com/index/10.1007/s10404-008-0382-5 [Accessed August 20, 2012].

Baskurt, O.K. & Meiselman, H.J., 2003. Blood rheology and hemodynamics.

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