A Microfluidic Device for Producing Controlled

Collisions Between Two Soft Particles

by

DINESH KUMAR

A thesis submitted in conformity with the requirements

for the degree of Master of Applied Science

Department of Chemical Engineering and Applied Chemistry

University of Toronto

© Copyright by Dinesh Kumar 2016

ii

A Microfluidic Device for Producing Controlled Collisions

Between Two Soft Particles

Dinesh Kumar

Masters in Applied Science

Department of Chemical Engineering and Applied Chemistry

University of Toronto

2016

Abstract

We report the design and operating characteristics of a computer-controlled microfluidic device

for the confinement and manipulation of two particles in a viscous fluid undergoing a two-

dimensional extensional flow. Using this device, two particles are trapped at two stagnation points

in a circular slot having three alternating fluid inlets and outlets which are connected to a pressure-

controlled liquid reservoir. By employing a control algorithm based on theoretical flow pattern

and model predictive control, two particles can be steered to arbitrary target positions within the

slot by adjusting the fluid flow rates into and out of the slot. Thus, two soft particles can be trapped

indefinitely at two different points, and be manipulated along arbitrary, independent trajectories in

the slot. This device is suitable for investigation of coalescence between drops, and adhesion

between vesicles by achieving collisions between two particles at precise contacting forces and

glancing angles.

iii

To those who were always there when I needed them the most

iv

Acknowledgements

First and foremost, I would like to express my sincerest and deepest gratitude to my advisor,

Professor Arun Ramachandran, for his constant guidance, support and funding over the course of

my work. His trust in my potential as a graduate student and continuous encouragement, has helped

me develop personally and professionally. With his enthusiasm, passion and dedication for his

profession, he has inspired me to become a better researcher. It is a remarkable honor for me to

have worked with him. I am grateful to Professor Eugenia Kumacheva for granting me permission

to use plasma cleaner equipment in her lab. I also thank the Center of Microfluidic Systems (CMS)

at University of Toronto for microchannel fabrication facilities and Syncrude Canada & NSERC

for funding.

Secondly, I would like to express my sincerest appreciation to Suraj Borkar for his kind help during

the initial phase of my experiments. My special thanks are due to Sachin Goel- my roommate and

labmate for being very helpful and supportive in and out of the lab. I have been extremely fortunate

to be in the company of excellent researchers and friends at the laboratory of complex fluids. I

would like to thank - Ghata, Rajesh, Jayant and Shadman for making my stay here, fun and

memorable. I am also thankful to Jenny Wei who established the platform for my research work

prior to my arrival.

-Dinesh Kumar

v

Table of Contents

List of Figures ................................................................................................................................ ix

List of Appendices ....................................................................................................................... xiv

Chapter 1 ....................................................................................................................................... 1

Introduction ................................................................................................................................... 1

Chapter 2 ....................................................................................................................................... 3

Literature Review ......................................................................................................................... 3

Chapter 3 ....................................................................................................................................... 9

3. Design, Working Principle, Experimental Methods and Materials ..................................... 9

3.1 Geometry of the microfluidic particle steering device ......................................................... 9

3.2 Operating mechanism for particle control .......................................................................... 13

3.3 Microfluidic circuit ............................................................................................................. 15

3.4 Device fabrication ............................................................................................................... 18

3.5 Experimental setup.............................................................................................................. 20

3.6 Control scheme using the analytical solution of flow field ................................................ 22

3.7 Failure of Proportional Control and Proportional Derivative Control .................................... 24

3.8 Control scheme using the Model Predictive Control (MPC) for drops .................................. 28

3.8 Materials and methods ........................................................................................................ 34

3.8.1 Calibration experiments ............................................................................................... 35

vi

3.8.2 Soft particle experiments ............................................................................................. 35

Chapter 4 ..................................................................................................................................... 37

4. Device Calibration and Algorithm Testing........................................................................... 37

4.1 Determination of the device center ..................................................................................... 37

4.2 Stagnation point calibration ............................................................................................ 38

4.3 Particle trapping at two stagnation points using analytical solution ............................... 41

4.4 Particle pair manipulation along predefined arbitrary paths using analytical solution ... 45

4.5 Particle trapping at arbitrary stagnation points and manipulation along a predefined path

using Model Predictive Control (MPC) ................................................................................ 47

4.6 Trapping of a pair of water drops using Model Predictive Control (MPC) .................... 49

Chapter 5 ..................................................................................................................................... 52

5. Avenues for improvement of the device and future work ................................................... 52

5.1 Improvements for the automated trap ................................................................................. 52

5.1.1 Better experimental procedure ..................................................................................... 52

5.1.2 Geometry of the device ................................................................................................ 53

5.1.3 Control loop time ......................................................................................................... 54

5.1.4 Material of the microfluidic device .............................................................................. 54

5.1.5 Limitation on the range of strain rates for particle trapping ........................................ 55

5.1.6 Scaling of pressure vector in each feedback loop ........................................................ 57

5.2 Future Work ............................................................................................................................ 59

vii

5.2.1. Controlled coalescence of water drops using the circular microfluidic flow device .......... 59

5.2.2 Theory and experimental results of hydrodynamic drainage time for coalescence ......... 60

5.2.3 Collisions of vesicles to measure adhesion rates ......................................................... 62

Bibliography ................................................................................................................................ 65

Appendix A .................................................................................................................................. 71

Appendix B .................................................................................................................................. 75

Appendix C .................................................................................................................................. 76

Appendix D .................................................................................................................................. 78

viii

List of Tables

Table 5.1 Range of strain-rates available in two-particle control experiments with the current set-

up

ix

List of Figures

Figure 2.1 Schematic of cross-slot device employed by Schroeder et. al. [36]

Figure 2.2 Schematic of diamond-slot device employed by Motagamwala [37]

Figure 3.1 Schematic of the 3-D shape of microfluidic extensional flow device.

Figure 3.2. A streamline plot of a circular microfluidic extensional flow device simulated in

MATLAB. The dimensionless flow rates in each port is [3, 1.4, 3, 1, 2.4, 1]. The flow rates are

rendered dimensionless by ᅀP/R where ᅀP is the pressure-drop and R is the hydrodynamic

resistance.

Figure 3.3. A streamline plot of a circular microfluidic extensional flow device simulated in

MATLAB. The dimensionless flow rates in each port is [3, 1.4, 3, 1, 2.4, 1]. The flow rates are

rendered dimensionless by ᅀP/R where ᅀP is the pressure-drop and R is the hydrodynamic

resistance.

Figure 3.4. Dimensionless strain rate contours in the microfluidic device for equal flow rates

through each port. The strain rates are rendered non-dimensional by 2/Q bR where Q is flow rate

in each port, b is half-depth of channel and R is the radius of circular geometry.

Figure 3.5. Particle steering towards their target point. (a) Snapshot of flow-field at t=0. (b)

Snapshot of flow-field at t=t1. (c) Snapshot of flow-field at t=t2 (t2>t1) (d) Snapshot of flow-field

at t=t3 (t3>t2). The snapshots at various times shows that how current stagnation points adjusts

themselves to steer the particles towards their target positions. See Video1 for more details.

Figure 3.6 Schematic of the flow device and connection to the fluid reservoirs. R1, R2, R3, R4, R5

and R6 are the hydrodynamic resistances of each arm of the circular slot and R is the hydrodynamic

resistance of the external circular tubing.

x

Figure 3.7. Electrical representation of microfluidic channel. P1, P2, P3, P4, P5 and P6 are the

pressure in the reservoirs. Q1, Q2, Q3, Q4, Q5 and Q6 are the flow rates through each arm leading to

the circular slot.

Figure 3.8 Image of the PDMS microfluidic extensional flow device showing three inlets and

outlets (a) with no side ports (b) with two extra side ports for introducing water drops during

collision experiments.

Figure 3.9. Top-view of the schematic of the complete experimental setup of microfluidic

extensional flow device.

Figure 3.10. Block diagram for feedback control scheme using the analytical solution.

Figure 3.11. Snapshot of slot showing the initial particle positions and target stagnation points.

Figure 3.12. Offset error for two particles for Proportional controller simulation.

Figure 3.13. Offset error for two particles using PD controller simulation.

Figure 3.14. MATLAB code to define the dynamic model of the flow field in the device.

Figure 3.15. MATLAB code to export the integrators and solvers for solving the OCP in

equation (3.13) and (3.14).

Figure 3.16. MATLAB code for solving the OCP in real time

Figure. 3.17 Trajectory of two particles MPC scheme. Two particles are made to move along a

predefined path, a square using MPC. See ‘Video2’ for more details.

Figure 4.1 Image of a representative device showing the edges of the circle (see equation in

Appendix II), the device center is at [786.29.45 m, 633.15m] with respect to the origin at the

top left corner and the length of the device edge is 486.4 m.

Figure 4.2 Stagnation point position with different inlet and outlet pressure combinations. In each

subfigure, experimental streamlines are shown on left, and streamlines simulated in MATLAB for

the same pressure combinations are shown on the right.

xi

Figure 4.3 Plot showing the error between the stagnation point position obtained by experiments

and theory. Red and black markers denote the positions of theoretical stagnation points and

experimental stagnation points. Similar markers represent the pair of stagnation points above and

below y=0. For example, red square above and below the line y=0 represent a theoretical pair of

stagnation for any arbitrarily applied pressure. The corresponding black squares represent the

experimental stagnation points.

Figure 4.4. Snapshots of two particles being controlled (a-d) at different time instants using the

analytical solution control scheme. The green squares represent the target stagnation positions and

green circles represent the current position of both particles. The pressure for this experiment were

[3.8 3 4 3 4 3] psi. See Video3 for more details.

Figure 4.5. Separation between the particle positions and target points with respect to iteration

number (time)

Figure 4.6. Snapshots of two particles being controlled (a-d) using the analytical solution control

scheme. The green squares represent the target stagnation positions and green circles represent

the current position of both particles. The pressure for this experiment at the final positions of the

particles were [3.8 1 4 1 4 1] psi. See Video4 for more details.

Figure 4.7. Manipulation of two surfactant free water drops of 50 μm diameter using the analytical

solution. Snapshots of initial and final positions of both drops are shown. Red and blue square

denotes the target position of drop 1 and drop 2 respectively. Red and blue asterisk represents the

current positions of both drops. The pressure for this experiment at the final positions of the

particles were [1.8 1 2 1 2 1] psi. See Video5 for more details.

Figure 4.8. Manipulation of two fluorescent beads of 1 μm diameter to trace a square. Snapshots

(a-d) of both particles at different time instants. Green stars denote the desired trajectory of

particles while green circles represent the current position of particles. The pressures in this

experiment were [3.8 3 4 3 4 3] psi. See Video6 for more details.

Figure 4.9. Snapshots of two particles being controlled (a-d) using MPC scheme at various time

instants. Green square denotes the target position of particles while green circle represents the

xii

current positions of both particles. This final pressures in the experiment were [3.8 1 4 1 4 1] psi.

See Video7 for more details.

Figure 4.10. Manipulation of two fluorescent beads of 1 μm diameter to trace a square.

Snapshots (a-d) of both particles at different time instants are shown. Green square denotes the

target position of particles while green circle represents the current positions of both particles.

This final pressures in the experiment were [3.8 1 4 1 4 1] psi. See Video8 for more details.

Figure 4.11. Trapping of two drops using MPC algorithm. Snapshots(a-d) of both drops at

different time instants are shown. Green square denotes the target position of drops while red

asterisk represents the current positions of both drops. See Video9 for more details.

Figure 4.12. Circumambulation of drop 1 over stationary drop 2 using MPC algorithm. Snapshots

of positions of drop 1 at different time instants are shown. Red asterisk represents the current

positions of both drops. See Video10 for more details.

Figure 5.1. The optimum channel shape for Hele-Shaw microfluidic device with two stagnation

points

Figure 5.2: Flowchart representing the scaling of pressure vector required in each control loop

Figure 5.3 Three stages of coalescence between two drops pushed by a constant force

Figure 5.4. Controlled Coalescence of water drops at Pressure = [4.3, 1.45, 4.5, 1.45, 4.5, 1.45]

Snapshot of both drops at various time instants are shown. Red asterisk denotes the current

position of both drops. See Video11 for more details.

Figure 5.5. Snapshots showing controlled collision of two vesicles at different time instants. The

suspending medium is 70% glycerol-water. See Video12 for details.

Fig. A.1. Schematic of the circular slot with the boundary conditions on stream function

Figure C.1 The GUI interface used to trap and manipulate particles using analytical solution.

Consol highlighted by orange box is used to change the reservoir pressures manually. Drop down

xiii

menu highlighted by red box is used to toggle between 'manual control', 'Particle trapping' and

'Particle manipulating'.

Figure D.1. Two Hele-Shaw drops pushed against each other by a constant force in an

extensional flow

xiv

List of Appendices

Appendix A Analytical solution for hydrodynamic flow field in the circular slot

Appendix B Determination of device center and radius

Appendix C GUI description

1

Chapter 1

Introduction

Suspensions of soft particles are prevalent in nature and industrial applications, including blood

[1], physiological fluids [2], emulsions in petroleum extraction industry [3], contrast agent and

drug delivery vehicles in the pharmaceutical industry, pastes, dressings and dairy products in the

food industry, microbial suspensions in wastewater treatment plants, liquid fabric softeners [4],

[5], and polymer blends [6]. The soft particles in the above examples can be broadly classified into

capsules, microgels, vesicles and drops, and these have received a fair amount of attention in

literature [7][9][49][12]. Capsules consist of a bag of an incompressible fluid surrounded by a thin

elastic skin. They are often used as model systems for the study of red blood cells (RBCs) [11].

As an example, the aggregation of human RBCs to form rouleaux and the breakup of these

aggregates under shear are problems of considerable interest in biology [46] [47] [48]. Microgels

are colloidal suspensions of gel particles consisting of an intramolecular cross-linked polymeric

network [49][50][51]. When immersed in a solvent, these polymeric networks can exist in swollen

or deswollen states, depending on external parameters such as temperature, pH and ionic strength.

Thus, the chemical and physical properties of microgels can be controllably modified, and this

makes them a suitable carrier for targeted drug delivery. The design of gel particles for this purpose

requires the detailed understanding of interaction between microgels and oppositely charged

proteins [52] [53].

Another type of soft particle is a vesicle, which is also a closed object enclosed by a special

membrane called the lipid bilayer. The bilayer is a thin membrane which is generally a self-directed

arrangement of amphiphilic molecules when introduced into an aqueous medium [7]. Vesicle

suspensions have widespread applications ranging from food products and pharmaceuticals to

cosmetics [8][10][11]. The macroscopic properties of concentrated vesicle suspensions are

strongly related to the thermodynamic and hydrodynamic interactions between vesicles. The

intervesicle interactions are, in fact, tuned in some applications to achieve a desired function. For

example, vesicle-based hair lotions rely on vesicle adhesion to hair followed by rupture to deliver

the therapeutic [8]. In other cases, vesicle adhesion is detrimental to the application (e.g. in fabric

2

enhancers, liquid soaps etc.), wherein, gravitationally-induced settling strongly diminishes the

shelf lives of these suspensions [8]. Emulsions, which represent another class of soft particles, are

extremely important in the oil industry, food industry, pharmaceutical industry and polymer

blending. The soft particle in the emulsion system is a drop of one fluid immersed in a second,

immiscible fluid. The mechanics of the interface between the two fluids has an extremely simple

mathematical description: an isotropic interfacial tension [54]. Coalescence in emulsions is

desirable when separation of the dispersed and suspending phases is required, and is undesirable

when a long shelf life of the emulsion product is required. Unfortunately, in spite of the simplicity

of the interface, and the importance of the phenomenon, coalescence remains poorly understood.

In all the examples discussed above, a common link, and often a critical determinant of the

macroscopic behaviour of a suspension, is the collision of one soft object with another under the

influence of an external force, often induced by hydrodynamics. A facility that could perform

such collisions in a controlled manner would enable a deeper understanding of this process. In

particular, a wealth of insights can be obtained by studying the flow-induced coalescence,

adhesion, or fusion of drops, vesicles, capsules and microgels.

This motivated us to build a microfluidic platform that implements hydrodynamically-induced

interactions between two particles at specific strain rates and specific glancing angles by using

hydrodynamic trapping methods. This thesis is organized as follows: Chapter 2 provides a

background on previous trapping methods used for confining and manipulating small particles,

and an overview of hydrodynamic trapping. Chapter 3 discusses the design, operating principle

and experimental materials of the microfluidic extensional flow device. The calibration of the

device and experimental testing of control algorithm based on theoretical flow and Model

Predictive Control (MPC) is discussed in Chapter 4. Finally, in Chapter 5, we discuss the

improvements in the device and directions for future research, as well as our preliminary results

from experiments involving inter-droplet and inter-vesicle collisions.

3

Chapter 2

Literature Review

There are many circumstances in which it is useful to control the dynamics of small particles/drops

in a linear flow. A significant amount of effort has been devoted in the literature to develop the

means to trap and manipulate single molecules and particles in solution. For example, optical traps

have been used to apply controlled forces on living cells [16], to assemble microstructures from

dielectric microparticles [17] and to probe the intrinsic properties of DNA and RNA [26] [27].

Several other researchers have used magnetic [13] [14] [15], acoustic [18] [19] [20] and

hydrodynamic [21] [22] [23] [24] [25] force fields to confine and manipulate particles for use in a

variety of fields, including biology, chemistry, material science and medical science [14] [19] [24].

Optical traps use a highly focused laser beam to hold and move dielectric particles with very high

precision. The optical force required for restricting the motion of a particle to a particular location

in a suspending medium is inversely proportional to particle size, which potentially leads to

damage of the particle, and hence, optical tweezers are not very useful for trapping of objects

smaller than 100 nm [34]. For particles larger than few tens of microns, the force exerted by an

optical trap is on the order of nanoNewtons or smaller, which limits the trapping of such particles

to very slow flows [56]. Another disadvantage is that optical traps can be used only for very dilute

suspensions. This is because the laser beams generate a local region of potential minimum in the

fluid where, the object of interest falls and gets trapped. With a non-dilute dispersion of micron-

sized particles, it would be very difficult to predict the number of particles trapped in the potential

well. While the use of magnetic trap solves the problem of local temperature increase associated

with laser traps, they are limited to only magnetically active particles whose magnetic

susceptibility is lower than the suspending fluid medium [15][59]. Electrophoretic methods are

strictly dependent on particle polarizability and suspending medium conductivity and utilize

electrical forces that may adversely affect the particle structure due to electric field interaction [35]

while acoustic trapping requires a high frequency sound wave (>20 MHz) to create a pressure node

where particles are trapped. For stable acoustic trapping, the limitation is that acoustic impedance

of particles should be lower than the propagation medium [60]. In this regard, hydrodynamic

4

force-field based trapping methods offers an inherent advantage over other trapping techniques,

simply because it does not pose any restriction on the physical or chemical properties of the

particle, and hence, can be used to trap/manipulate particles of any size and kind, as long as we

can optically image the particles [28].

Particle trapping using hydrodynamic forces is fundamentally different from other techniques, as

it does not utilize an additional external field such as a magnetic or electric field for particle

manipulation; the flow in the separation device used to supply and remove the particles can be

manipulated to trap particles. The first attempt in the literature that made use of hydrodynamic

forces to trap particles was in 1934, when G. I. Taylor developed a four-roll mill apparatus to

generate mixed flows that can be varied from purely rotational to shear to purely extensional

through the appropriate choice of speed and direction of rotation of four cylindrical rollers [21].

The four-roll mill is characterized by the presence of a stagnation point located centrally which

serves a trapping site for particles/drops. Taylor used this four-roll mill to study drop deformation

in a purely straining flow by manually controlling a single drop at the center stagnation point at

low flow rates. But, since the stagnation point position is not a steady state position for a drop, a

continuous control of the drop position is actually required to maintain it at the target location for

prolonged times. Thus, the absence of automatic control over drop position limited the study of

drop dynamics done by Taylor to weak velocity gradients.

In 1985, Bentley and Leal developed a computer-controlled four-roll mill which allowed them to

control the drop position at the stagnation point of the linear flow for extended periods of time at

sufficiently higher flow rates. This automated four-roll mill also enabled the study of drop

dynamics for mixed flows that were not possible with manual control. Therefore, the deformation

characteristics of viscous drops in mixed flows were investigated, and experimental data was

obtained for a wide range of viscosity ratio [22]. Apart from single drop deformation studies, the

computer-controlled four-roll mill was also used to study the head-on and glancing collisions of

Newtonian droplets in a purely extensional flow [29] [30]. However, the computer-controlled four

roll mill has certain drawbacks. For example, the movement of four large rollers in the immediate

neighborhood of millimeter-sized drop, interferes with the drop imaging. Also, the rollers in the

device (10 cm x 10 cm x 10 cm) sit in a deep fluid tank, resulting in a non-planar flow, which

5

causes the drops to drift up or down and therefore, the trapped drop may drift out of focus during

the experiments. Additionally, the suspending fluid has to be an extremely viscous medium to

prevent the settling of non-neutrally buoyant particles, and this places a very strong restriction on

the types of fluids that can be employed in the experiments. The Bentley/Leal four-roll mill is also

not suitable for confinement of small micron-sized particles due to a very high feedback loop time

of the system (~5 seconds). The miniaturized version of this four-roll mill developed by Hsu in

2000, provides a relatively lower feedback time and has been used to perform coalescence

experiments with drops of 60 μm diameter [61].

With the advent of microfluidics, several researchers have made attempts to build a microfluidic

version of the four-roll mill so that particle trapping can be achieved at shorter time scales.

However, it is very challenging to mimic the four-roll mill concept at micron-length scales just by

changing the flow rates in each channel. Hudson and co-workers [31] suggested a design of

microfluidic four roll mill which consists of six intersecting channels with asymmetric baffles

between channels in a chiral arrangement. By varying the flow rates in each channel, they could

change the flow type but they were unable to realize purely rotational flow due to asymmetry in

the design. Muller and co-workers further developed a symmetric microfluidic four-roll mill with

a cross-slot geometry capable of producing extensional flows and rotational flows [24]. These

devices lack a continuous computer control, and therefore, long time measurements of particle

dynamics at the stagnation point were not achievable.

The first proposal to use feedback controls in microfluidic platforms was made by Shapiro and

coworkers [32], who used a combination of electro-osmotic actuation and feedback control theory

to steer multiple particles along independent trajectories in a microfluidic chamber. Using this

method, they experimentally controlled a single fluorescent nanocrystal in a viscous solution to a

remarkable precision of ∼50 nm. Cohen et al. [33] [34] [35], have developed a feedback control

mechanism based on electric fields in a quadrupole configuration, that induces a drift to steer the

particle to a desired position. They used the Anti-Brownian Electrokinetic (ABEL) trap to study

the equilibrium motion and fluctuation of single λ-DNA molecules for extended periods of time

without perturbing internal dynamics, and observed evidence of internal hydrodynamic

interactions between molecules. The ABEL trap is guaranteed to trap only a single object which

6

offers unique opportunity to study single particle dynamics, but the effectiveness of the trap highly

depends on the surface properties of the channel. For instance, the presence of ions in the

suspending fluid medium to create the counterions is essential for electrosmotic flows and hence,

the correct implementation of control algorithm of ABEL trap. This restricts the applicability of

the electrokinetic trap to only electroosmotic flows, and hence, it cannot be used to study the

dynamics of water drops in oil.

In 2010, Schroeder et. al. designed a cross-slot device having two inlets and outlets (see Fig. 2.1)

that employs hydrodynamic force-field to trap and manipulate micro and nanoscale particles using

the sole action of fluid flow [23] [36]. They employed a linear feedback proportional controller to

demonstrate the confinement of a 500-nm-diameter particle to within ~0.18 μm of the target

stagnation point. In this pressure-driven hydrodynamic trap, the stagnation point position is

adjusted by using a pneumatic, Polydimethylsiloxane (PDMS)-based valve, thereby altering the

flow rate through the inlet and outlet of the device. They achieved a feedback time of ~130 ms

with their set-up. Later, they also employed a proportional-integral-derivative (PID) control to

Inlet 1

Outlet 1

Inlet 2

Outlet 2

Trapped particle

Figure 2.1 Schematic of cross-slot device employed by Schroeder et. al. [36].

7

improve the trap efficiency and addressed the effects of gain constants, choice of control scheme

on the stability of the trap in the cross-slot design [57].

Recently, Motagamwala [37] built a diamond shaped microfluidic extensional flow device (shown

in Fig. 2.2) which has a higher region of constant strain rate at the center compared to the cross-

slot device. The device has two alternating inlets and outlets which are connected to a liquid

reservoir. The liquid reservoir is pressurized, and the pressure difference between the reservoirs

causes the flow through the device. The flow rate through the inlets and the outlets and, in turn,

the stagnation point position in the slot can be changed by altering the pressures maintained in the

reservoirs. This device was used to study flow-induced deformation of Hele-Shaw drops, and

measure ultra-low interfacial tensions in emulsion systems.

psi

Dim

ensionless Length

Dim

ensi

onless

Len

gth

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Trapped particle

Inlet 1 Inlet 2

Outlet 1

Outlet 2

Figure 2.2 Schematic of diamond-slot device employed by Motagamwala et. al. [37].

8

All of these microfluidic devices [36] [37] are strictly limited for trapping and confining only a

single particle/drop in a free solution. Due to limited degrees of freedom, it is not possible to

achieve a controlled coalescence/adhesion of two drops/vesicles. From this perspective, there is a

strong need to develop a microfluidic platform which allows confinement and manipulation of

multiple particles in solution using the sole action of fluid flow. Schneider and co-workers

performed a computational study to demonstrate an algorithm for controlling multiple particles in

Hele-Shaw geometry but no experiments were performed [63]. Control over multiple particles

using a hydrodynamic flow field will allow us to perform detailed studies of soft particle

interactions including vesicle adhesion and drop-coalescence. In this work, we have built an

automated microfluidic trap based on two different feedback control strategies, namely, the use of

theoretical flow pattern and Model Predictive Control (MPC) to independently steer and

manipulate two particles in a microfluidic circular slot.

Concurrent to our work on the building of two particle trap based on hydrodynamic forces, the

group of Charles Schroeder at University of Illinois at Urbana Champaign independently

developed the idea of combining fluidics and control theory to manipulate and confine multiple

particles in a free solution [58]. Their device operates on a similar principle (MPC as a feedback

control) but they are restricted to only using MPC for trapping particles, which is computationally

costly. They mainly focused on developing a system to trap multiple particles and treated it as only

a control problem, rather than understanding the hydrodynamics of flow. In contrast, our in-depth

understanding of the flow-field characteristics also allowed us to build the device using the

analytical solution as a feedback control (in addition to MPC scheme) which only requires solving

four simultaneous linear equations and hence, is relatively less computationally intensive. We have

also performed drop-coalescence and vesicle-collisions experiments which they haven’t done yet.

9

Chapter 3

3. Design, Working Principle, Experimental Methods and Materials

3.1 Geometry of the microfluidic particle steering device

The schematic of the microfluidic particle steering device used in this study, is shown in Figure

3.1. It is a circular-shaped slot with three alternating inlets and outlets. The choice of circular

shape was dictated by the availability of an analytical solution of the flow-field, which is discussed

in Appendix A. Each port is connected to a liquid reservoir, which is pressurized using a pressure

controller. The pressure difference between the reservoirs causes the flow through the device with

three incoming and three outgoing streams of fluid. The width and depth of the device channels

are 200 µm and 100 µm respectively as shown in the 3D schematic of device in Fig. 3.1.

The flow-field characteristics in the device are related to the flow rates through each inlet and

outlet, and in turn, the gauge pressures maintained in the reservoirs connected to the ports. A given

combination of pressure differences applied to the liquid reservoirs produces a steady two-

Figure 3.1 Schematic of the 3-D shape of the microfluidic extensional flow device

Inlet 1

Inlet 2

Inlet 3

Outlet 1

Outlet 2

Outlet 3

10

dimensional flow field in the circular slot that has, in general, two stagnation points (co-ordinates

with zero velocity vectors), as shown in the example in Fig. 3.2. The exception is the case of equal

pressure differences, where there is only one stagnation point at the center of device (see Fig. 3.3).

The streamlines shown in the two figures have been drawn using the analytical expression for the

streamfunction of the flow field derived in Appendix A, and were generated in MATLAB.

Figure 3.2. A streamline plot of a circular microfluidic extensional flow device simulated

in MATLAB. The dimensionless flow rates in each port is [3, 1.4, 3, 1, 2.4, 1]. The flow

rates are rendered dimensionless by ᅀP/R where ᅀP is the pressure-drop and R is the

hydrodynamic resistance.

Dimensionless X

Dim

en

sio

nle

ss Y

Streamlines

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

0

0.5

1

1.5

11

The strain rate at any point in the device is related to the velocity gradient and hence, the flow rates

into/out of each channel. We used the analytical solution to plot the contours of strain rates in the

circular geometry for equal flow rates in the ports (see Fig. 3.4). It can be observed from Fig. 3.3

and Fig. 3.4 that when the stagnation points merge to one at the device center, this location

becomes a degenerate critical point where both, the local velocities and the velocity gradients are

identically zero. Thus, the velocity field in the vicinity of the center has a quadratic nature. In

contrast, both stagnation points in Fig. 3.2 are saddle critical points where two streamlines enter

the stationary point (called the compressional axis) and two issue from it (extensional axis),

producing a pure, Hele-Shaw extensional flow at these points. The velocity field in the vicinity of

such saddle stagnation points is linear. It can also be inferred that a particle is attracted toward the

Dimensionless X

Dim

en

sio

nle

ss Y

Streamlines

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

0

0.05

0.1

0.15

0.2

0.25

Figure 3.3. A streamline plot of a circular microfluidic extensional flow device

simulated in MATLAB. The dimensionless flow rates in each port is [3, 1.4, 3, 1, 2.4, 1].

The flow rates are rendered dimensionless by ᅀP/R where ᅀP is the pressure-drop and

R is the hydrodynamic resistance.

12

stagnation point along the compressional axis, but eventually always repelled from the stagnation

point along the extensional axis for infinitesimal perturbations away from the compressional axis.

A prominent attribute of our device geometry is that it permits an analytical solution of the flow

field (Appendix A), and can, thus, allow the direct use of theoretical flow-field as a feedback

controller to steer two particles towards their respective target stagnation points. For the purpose

of this thesis, we have experimented two different control schemes. The first one is based on the

direct use of analytical solution of the flow field while the second one is the Model Predictive

Control (MPC) method.

Figure 3.4. Dimensionless strain rate contours in the microfluidic device for equal flow

rates through each port. The strain rates are rendered non-dimensional by where Q

is flow rate in each port, b is half-depth of channel and R is the radius of circular geometry

13

3.2 Operating mechanism for particle control

The control task is to steer two particles independently in a hydrodynamically-driven microfluidic

device by creating an underlying fluid flow that will carry the particles along any two pairs of

arbitrary trajectories. A common-sense strategy is to modify the underlying fluid flow and place

the stagnation points around the particles in such a way that the velocity vector of two particles in

the slot always points towards their respective target positions. For example, consider two particles

placed at two arbitrary points in the circular slot as shown in Fig. 3.5. If the two particles (initial

position is denoted by blue asterisk A and B in Fig. 3.5(a) while the current position at intermediate

time is denoted by two red ‘+’ symbols in Fig. 3.5(b)) are required to be moved towards the target

stagnation points (denoted by blue circles C and D) as shown in Fig. 3.5, then both the stagnation

points need to be shifted continuously (denoted by red asterisk) in such a way so that the net

instantaneous velocity vector of both particles is always directed towards C and D respectively.

Hence, by appropriately positioning the stagnation points continuously, both particles can be made

to move along an arbitrary path in the slot. This strategy is also used to bring two soft particles

close to each other to achieve a controlled collision which may result into adhesion/coalescence.

As discussed earlier, the stagnation point represents a steady state position for a particle along the

compressional axis, but an unstable position along the extensional axis (see fig 3.3). In other words,

any disturbance in the particle positions along the compressional axis decay, but such disturbances

along the extensional axis grow exponentially in time, and will eventually cause the particles to

drift away from the stagnation positions. Hence, once the particles reach their target positions,

another challenge is to maintain the particle at their respective target positions for longer time

scales [21, 22]. An active feedback control is required to achieve this by manipulating the

stagnation points as discussed in the previous paragraph and the particles can be maintained at

their target positions for indefinite times over a certain range of pressure drops. This is

demonstrated in chapter 4 by controlling two fluorescent particles at their respective target

stagnation points for up to 10 min.

14

Please select their stagnation points respectively

Dimensionless Length

Dim

en

sio

nle

ss L

en

gth

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

psi

Dimensionless Length

Dim

en

sio

nle

ss L

en

gth

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

psi

Dimensionless Length

Dim

en

sio

nle

ss L

en

gth

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

psi

Dimensionless Length

Dim

en

sio

nle

ss L

en

gth

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 3.5. Particle steering towards their target point. (a) Snapshot of flow-field at t=0. (b)

Snapshot of flow-field at t=t1. (c) Snapshot of flow-field at t=t2 (t2>t1) (d) Snapshot of flow-field at

t=t3 (t3>t2). The snapshots at various times shows that how current stagnation points adjusts

themselves to steer the particles towards their target positions. See Video1 for more details

A

B

C

D

a b

c d

15

3.3 Microfluidic circuit

To adjust the position of the stagnation points in the circular slot, the flow rates of the fluid in the

six ports/channels need to be manipulated, and this is achieved using six pressure-controlled

reservoirs. For conversion of flow-rates to pressure values, we utilize the flow circuit in Fig. 3.6.

The small length scales and low flow rates in the microfluidic device imply laminar, low Reynolds

number flows, and this allows us to define a linear relationship between the pressure drop, P ,

and the flow rate, Q , as:

.P RQ (3.1)

Figure 3.6 Schematic of the flow device and connection to the fluid reservoirs. R1,

R2, R3, R4, R5 and R6 are the hydrodynamic resistances of each arm of the circular slot

and R is the hydrodynamic resistance of the external circular tubing.

R2 R

3

R1 R

4

R5 R

6

16

In Eq. 3.1, R is the hydrodynamic resistance. The inlets and outlets channels in the microfluidic

device have rectangular cross-sections, and the resistance of each channel is denoted by iR , where

1 2, 3 4 5 6, , , ,R R R R R R denotes a particular inlet or outlet. The microfluidic circuit employed in this

work is shown in Fig. 3.7. The length of the tubing used for connecting the microfluidic device

to the fluid reservoirs is same for each inlet and outlet. Let us call the resistance of each tubing as

R . If the average pressure in the slot is 0P , the flow rate in each arm is

0 .i

i

i

P PQ

R R

(3.2)

Figure 3.7. Electrical representation of microfluidic channel. P1, P2, P3, P4, P5

and P6 are the pressure in the reservoirs. Q1, Q2, Q3, Q4, Q5 and Q6 are the flow

rates through each arm leading to the circular slot.

P3 P

2

P1 P

4

P5 P

6

R+R1

R+R2

R+R3

R+R4

R+R5

R+R6

P0

Q1

Q2

Q3

Q6

Q5

Q4

17

The pressure 0P is obtained by the requirement of incompressibility of the liquid,

6

1

0,i

i

Q

(3.4)

Which yields

6 6

0

1 1

1/ .i

i ii i

PP

R R R R

(3.5)

The length of the circular tubing (PEEK tubing of ID 127 μm) is chosen so that the tubing

resistances are much higher than the channel resistances, typically by an order of magnitude. This

choice is dictated by the following reasons: The control algorithm that manipulates particles in the

slot is based on flow rates, which are calculated assuming equal resistances in each port of the

microfluidic device. Hence, if the resistances are unequal in any port, flow rate, iQ , in that arm

could be altered. This can interfere with the implementation of the control algorithm and result in

inefficient trapping. However, if R is chosen such that iR R , the flow rate in each arm is

6

1

1 1.

6i i i

i

Q P PR

(3.6)

Thus, iQ becomes independent of the individual resistances, iR , within the microfluidic device and

depends only on the control variables, iP . The choice of tubing with iR R also suppresses

the effect of fabrication-induced differences in the resistances of the inlet and outlet arms on the

control process.

18

3.4 Device fabrication

Soft lithography was used to fabricate the microfluidic extensional flow device in PDMS. This

method has been extensively reviewed in the literature [38]. Our methods are discussed below

briefly.

The microfluidic channel was designed using AutoCAD, a computer-aided design and drafting

software. The design was then printed at a high resolution of 20,000 DPI on a transparency mask

at the Center of Microfluidic Systems (CMS) at University of Toronto. A 100 m layer of negative

photo-resist (SU-8 50) was spin-coated onto a 3" diameter silicon wafer. The wafer was then pre-

baked at 95oC for 15 min. The transparency was used as a photo-mask in contact photolithography

using a mask aligner with 16 mJ.cm-2.s-1 lamp power. The spin-coated SU-8 50 was exposed to

UV light (365 nm) for 35 seconds. The unexposed photo-resist was removed by dissolving in SU8

developer, yielding a silicon wafer with a positive low-relief of photoresist that served as a casting

mold for PDMS.

PDMS elastomer was mixed with the pre-cursor in the ratio of 10:1, degassed and poured over the

cast. The cast was then cured at 65oC for 4 hours. Cured PDMS was cut and carefully peeled off

the master. Access holes were punched using an 18-gauge syringe needle. The device was cleaned

by sonicating in isopropyl alcohol (IPA) for 2 minutes and subsequently dried with nitrogen. We

use Corning glass slides of dimensions, 75x50x1mm, purchased from Scientific Product and

Equipment as a base for the PDMS device. Finally, the PDMS block with casted micro-channels

was bonded to the glass slide by oxygen plasma etching. IPA, Acetone and DI water are used to

clean the glass slides to remove the contaminants and undesired dust-particles. Fig. 3.8 shows an

image of a bonded microfluidic extensional flow device.

19

Figure 3.8 Image of the PDMS microfluidic extensional flow device showing three inlets

and outlets (a) with no side ports (b) with two extra side ports for introducing water drops

during collision experiments

a b

20

3.5 Experimental setup

The top-view of the complete experimental setup is shown in Fig. 3.9. The circular microfluidic

flow device was mounted on an inverted microscope (Nikon TI-Eclipse). A 16-bit monochrome

camera (Retiga 2000R, Q-imaging) was used as the imaging device to capture images of the flow

field in real time. The resolution of the image obtained from the camera is approximately 1

m/pixel at a magnification of 10X. Nikon’s NIS software was used to record the videos during

the calibration of microfluidic device (see section 4.1). The Image acquisition and Image

processing toolboxes of MATLAB® were used to extract the particle/drop positions for trapping

experiments. The sequence of getting a frame of information and finding the center of particle/drop

comprised the measurement portion of the feedback control process which is discussed in section

3.6. The fluid was delivered to the device using rigid PEEK tubing (0.005 in. inner diameter 0.0625

in outer diameter). One end of the tubing was inserted into the inlet/outlet of the device and the

other end was submerged under the liquid surface in the reservoir. Four 100 ml, GL-45 screw cap

glass bottles (Fisherbrand) were used as liquid reservoirs. GL45 2-ported ¼-28, PTFE insert, blue

Polyethylene Collar bottle caps (IDEX-Health & Science) with two access holes were employed

to shut these reservoirs. The two access holes are required, one for the PFA tubing, and the other

for pressurizing the bottle contents which connects to the supply pressure.

The reservoirs are pressurized using Type 2000 pressure transducer (Marsh Bellofram) which

regulates an incoming supply pressure down to a precise output that is directly proportional to an

electrical control signal. The electrical input to the pressure controller based on a proportional

control algorithm is provided through NI-USB 6351 DAQ card.

21

NI D

A

Q Ca

rd

Camera

Figure 3.9. Top-view of the schematic of the complete experimental setup of microfluidic

extensional flow device

22

3.6 Control scheme using the analytical solution of flow field

Our objective in the design of the control algorithm is to steer both the particles as close as possible

to their respective target stagnation points. When a particle drifts away from its target point,

alterations to the flow field must be such that the new flow tends to return the particles to the target.

It is clear that for the particles to tend to return to their target stagnation points (say, A & B), they

must be at a point in the new flow where their net velocity vector is towards A and B respectively,

and this can be achieved by placing each stagnation point suitably around each particle. The

availability of the analytical solution to the flow field that provides the pressures imposed in each

reservoir to place the stagnation points at desired locations in the slot facilitates this control

approach. The control scheme is especially effective for small particles (~ 60 μm), because the

inherent assumption in the scheme that the particle velocity is equivalent to that of a fluid element

placed at its center of mass in the undisturbed flow, is exact for point particles. The error in this

assumption is a weak quadratic correction for asymptotically small but finite-sized particles

according to Faxen’s law [62]. The sequence of events in one control loop are as follows:

Step 1. Capture the image of the circular slot (fig. 3.1) using the camera, and read the image into

MATLAB®.

Step 2. Use the regionprops function in the Image Processing toolbox of MATLAB to determine

the positions of the particles ( x ) in the image. Save the image for future analysis, and

specify the target points ( 0x ).

Step3. Determine the velocity vector desired at the current particle positions using the equation

0v K x x (3.7)

where K is a constant or a tuning parameter.

Step 4. Given the velocities in Step 3, invert the equations (A.9) and (A.10) from Appendix A to

obtain the updated flow rates ( iQ ).

23

Step 5. The flow rates were converted to pressures using the relation:

6

1

1

6i i i

i

P P RQ

(3.8)

where Pi is the pressure in each port and R is the hydrodynamic resistance. These updated pressures

iP can either be smaller or greater than the average pressure 0P at the center of the device as shown

in Fig. 3.7. This allows any port to become an inlet or outlet during the feedback loop which may

be requirement to steer the particles towards their target locations.

Step5. Alter the pressures in the reservoirs to the new values via instructions issued to the NI-

DAQ card.

Figure 3.10. Block diagram for feedback control scheme using the analytical solution

In the first step of the control loop, a background image of the flow-field is subtracted from the

current image. This background subtraction is useful because it removes the particles stuck on the

device surface and hence, aids the regionprops command to track the particles of interest

efficiently. To maximize the speed of control code, two small sub-images (about 50 x 50 pixels),

for which we call the cropping algorithm imcrop, are extracted from the original image captured

by the camera. In the first frame, the origin of each imcrop image is chosen as the initial position

Current

Positions Updated

pressure

∆𝑑ሱሮ

∆𝑡 Eq. 3.8 𝑓൫𝑣Ԧ, 𝑋Ԧ𝐶൯

+

− 𝑋Ԧ𝐶

Distance

∆𝑑

(Error) Velocities

V՜

Flow

rates

𝑄ሬԦ 𝑃ሬԦ𝑛𝑒𝑤

𝑓൫𝑣Ԧ, 𝑋Ԧ𝐶൯

Updated

Positions

DX

24

of both particles chosen by the user. Each of these imcrop images is chosen to be small enough to

contain only one particle, but large enough so that if the particle is in the center of the imcrop, it

does not leave the imcrop by the time camera delivers the next frame. For the subsequent frames,

imcrop is centered on the centroid of particles calculated for preceding frame. In case, we have

more than one particle in each imcrop, we select the particle which is closest to the centroid of the

particles in the preceding frame. Our control algorithm also accepts inputs from the user, in the

form of mouse clicks indicating which two particles need to be controlled or the change in the

target stagnation points. In each control loop, we also save the images, the coordinates of both

particles and the set of applied flow-rates.

3.7 Failure of Proportional Control and Proportional Derivative Control

The control scheme based on analytical solution is not suitable for channel spanning water drops

because the presence of large drops in the circular slot significantly modifies the flow field and

interferes with the control scheme. Hence, a conventional proportional feedback control scheme

was tested to achieve control of drops in the circular slot. Since drops have a tendency to drift

apart through the Outlet 1 (or Outlet 3) and Outlet 2, we tested the success of simple proportional

control using a computer simulation described by Eq. 3.9 and 3.10.

25

Figure 3.11. Snapshot of slot showing the initial particle positions and target stagnation points

For y1>0, the pressures of Outlet 1 and Outlet 2 were updated by the following relation (Kp11, Kp12,

Kp21, Kp22 are proportional gain constants):

1 1

2 2

11 12 2 21

21 22 1 11

O Op p si i

O Op p si i

K K x i xP P

K K y i yP P

(3.9a)

For y1<0, a similar equation is used where the pressure of outlet 3 is changed instead of outlet 2:

1 1

3 3

11 12 2 21

21 22 1 11

O Op p si i

O Op p si i

K K x i xP P

K K y i yP P

(3.9b)

These simulations were performed by starting with an initial particle positions, updating the

pressures using (3.9) and (3.10), and calculating the new particle and stagnation points based on

the new flow field. The proportional control scheme was not successful because it failed to

correctly account for the closed loop interaction between flow rates in the ports and two stagnation

26

points or two particle positions. A series of computer simulations showed that if the initial particle

positions are very close to their target points, the control scheme exhibited an oscillatory behavior.

However, if the initial particle positions were far away from their target positions, the scheme is

unstable and particles diverge. The offset error for both particles over time for a simulation with

Proportional control scheme with an oscillatory behavior is shown in Fig. 3.12

Figure 3.12. Offset error for two particles using Proportional controller simulation

Similar simulations were performed using proportional-derivative (PD) control where the

pressures in each control loop were updated by the relation (Kd11, Kd12, Kd21, Kd22 are derivative

gain constants and loopt is the time taken to complete one closed loop cycle:

1 1

2 2

11 12 2 21

21 22 1 11

2 2 2 211 12

21 22 1 1 1 1

11

1

O Op p si i

O Op p si i

s sd d

d dloop s s

K K x i xP P

K K y i yP P

x i x x i xK K

K Kt y i y y i y

(3.10)

0 50 100 150 200 250 300 350 400 450 5000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Iteration #

Off

set

err

or

for

bo

th p

art

icle

s

Error for particle 1

Error for particle 2

27

We deduced that the PD controller suppresses the fluctuations around the target positions. This

happens because PD controller is based on the rate of change of error and hence, it can modulate

the control output based on the current velocity of the particles. The offset error for both particles

over time for a simulation with PD control (see Fig. 3.12) shows that the fluctuation of particles

around their respective target points are damped, but monotonic convergence was not achieved.

Figure 3.13. Offset error for two particles using PD controller simulation

The initial values of gain constants in Eq. 3.9a, 3.9b and 3.10 were calculated using the relations:

1,3 1,3 2 2

2 21 1

11 11 12 12 21 21 22 22, , ,O O O O

d p d p d p d p

x yx y

P P P PK K K K K K K K

y x y x

The typical values of gain constants for the simulation results presented in Fig. 3.13 were:

11 12 21 220.81, 0.63, 0.51, 0.43p p p pK K K K

Several experiments showed that a simple PD controller was difficult or impossible to successfully

implement. One possible reason for the failure of conventional PD control was that these feedback

28

controllers have a low robustness for multiple inlet multiple outlet (MIMO) system due to closed

loop interaction between control input (flow rates in the device) and output state (two stagnation

points or two particle positions). Regulating a single loop introduces a change in the other loop

which PD control is not able to robustly account for. If the interaction is strong, it may create

instability in the control process. We quantified the interaction between loops by calculation of

relative gain arrays [44] using the dynamic model of the process (Appendix A) and found that the

interaction is very large. The strong coupling between both state vectors (stagnation point

positions) tends to make the controller tuning more difficult because of large number of gain

constants (see Eq. 3.10) in the PD control. One way of improving the performance of PD controller

is to convert the multivariable system into multiple single loops by identifying the interaction and

control loops, and then use any existing single input single output (SISO) PD controller tuning

approach to tune each loop independently. However due to the non-linearity of the model and

constraints involved in the problem, decoupling the system is a challenging problem. Another way

to account for the non-linearity of the system in PD control is to schedule all the gain constants

depending on the state variables (current particle positions) in each feedback loop using frequency

response analysis. However, the task of finding an appropriate non-linear function to schedule all

the gains constants in (3.9) and (3.10) is complicated and highly involved. Therefore, we

proceeded with an alternative feedback controller, the details of which are discussed now.

3.8 Control scheme using the Model Predictive Control (MPC) for drops

We mainly followed the idea of Brenner [63] to obtain optimized trajectories for the movement of

two particles in the slot. A more robust control technique, namely the Model Predictive Control

(MPC), was used for controlling two particle positions in the circular device. MPC relies on the

prediction model of the process (Eq. 3.11, 3.12, Appendix A) and allows the current time slot to

be optimized, while keeping future timeslots in account. This is achieved by optimizing a finite

time-horizon, but only implementing the control inputs in the current time slot. Hence, MPC is

capable of anticipating future events and can account for model mismatch or external disturbances

in the system. The dynamic model of the flow field is written as follows (see Fig. 3.11 & Appendix

A for details):

29

1,2 1

1,2 1,2 1,2 1,2 1,2 1,2 1,2

1

sin cos , ,n

n n

n

dxnr A n B n f x y q

dt

(3.11)

1,2 1

1,2 1,2 1,2 1,2 1,2 1,2 1,2

1

cos sin , ,n

n n

n

dynr A n B n g x y q

dt

(3.12)

Here, 𝑥1, 𝑦1, 𝑥2, 𝑦2 are state variables and q = [𝑞1, 𝑞2, 𝑞3, 𝑞4, 𝑞5, 𝑞6] are control variables.

MPC repeatedly calculates the control input (flow rates) that optimizes the trajectory of the two

particles over a finite time horizon in the optimal control problem (OCP) which is defined in the

continuous time format as follows:

22 2

mini

i

t T

s s

i i i i

t

F x t x q t dt x t T x

(3.13)

subjected to constraints:

0

,

1

ix t x

dx th x t q t

dt

x t

(3.14)

Here, t is the current time instant, x t is the state variable, sx is the reference state variable, q t

is the control variable at time t and T is the length of the prediction horizon. The objective function

for non-linear model predictive control (NMPC) problem is defined by (3.13) while the path and

terminal constraints are denoted by (3.14). The term 2

sx t x in the objective function is called

stage cost, 2

q t is termed control cost and 2

s

ix t T x is called the terminal cost. The

weighting coefficient i represents the deviation of particles from the predicted trajectory while

i ensures that the particles at the end of time horizon do not deviate significantly from the target

30

stagnation positions. The weighting coefficient i ensures that the flow rates do not change by an

order of magnitude in two consecutive time slots. For example, a lower value of i means that the

optimized trajectory joining initial and desired positions is achieved by small changes in flow rates.

The non-linear dynamics in the second equation of (3.14) are described explicitly by a system of

ordinary differential equations (ODEs) in (3.12) and (3.13). The OCP depends on the positions of

the two particles at time ti, ix t through the initial value constraint in (3.14). Hence, the control

trajectory obtained by solving (3.13) and (3.14) provides a feedback strategy ,new i iq t x t which

depends on the current state variable and time. In practice, the OCP was solved by discretizing the

problem into M number of equal intervals in a finite time horizon [ti, ti+T] using multiple shooting

method to obtain a structured Nonlinear problem (NLP) which was non-convex. Each equal

interval is called sampling time. A generalized Gauss-Newtonian method based on real time

iteration algorithm is capable of finding the locally optimal solution of (3.13) by solving the

Karush-Kuhn-Tucker (KKT) conditions. The finite time horizon is chosen to be 30 ms and the

sampling interval to be 0.3 ms based on empirical testing following the ideas of Diehl [64] [65] .

At each sampling interval in the time horizon, we solve the OCP on the fly using the current system

state as initial value which provides an optimal control output (flow rates). The optimized flow

rates are calculated for each sampling interval in the finite time horizon but only the first set of

flow rates are implemented. Each flow rate is bounded such that -10< newq <10 allowing any of the

port to become inlet or outlet. The first set of optimized flow rates and initial states are used to

find the future position of particles by integration of dynamic model using implicit RK method in

simulations while the subsequent particle positions are measured by an image processing algorithm

in the experiments. This procedure is repeated continuously until convergence is achieved. Hence,

MPC is a feedback control strategy which solves the open loop optimization problem in real time.

One way of solving the OCP in (3.13) is to model ODEs represented by (3.11) and (3.12) using

MATLAB Simulink and integrate it with Predictive control toolbox to obtain the control variables

in real time. However, we require the feedback at very high sampling rates (typically ~150 ms).

So, MATLAB is not a good tool for online solving of the MPC problem in each sampling time.

We use an open source toolkit called Automatic Control and Dynamic Optimization (ACADO)

31

[43] which solves the OCP by using the Real Time Iteration (RTI) algorithm. Using the Code

Generation feature of ACADO toolkit, we export the RTI scheme from ACADO toolkit in a

MATLAB environment. The following four crucial steps can be identified for solving the

Nonlinear MPC problem in MATLAB interface:

1. Define the model equations (3.11) and (3.12) as shown in the Fig. 3.14. Another MATLAB

program communicates the real time particle locations x1, y1, x2, y2 to ACADO. The

variables used are explained in the analytical solution presented in Appendix A.

Figure 3.14. MATLAB code to define the dynamic model of the flow field in the device

32

2. SIM export: An integrator is generated to accurately predict the future states of particles at

each sampling interval.

Figure 3.15. MATLAB code to export the integrators and solvers for solving the OCP in

equation (3.13) and (3.14)

3. NMPC export: THE OCP is formulated in ACADO syntax including the dynamic model,

objective function, constraints and corresponding solvers are exported to solve the OCP

33

Figure 3.16. MATLAB code for solving the OCP in real time

4. Solving NMPC: The exported solvers are used to find the updated flow rates 1 0,newq t x

for an optimized trajectory. These updated flow rates are fed to the ‘main control code’

which converts them to pressures and the pressure controllers are appropriately actuated

by NI USB DAQ cards.

34

Simulations of this control scheme showed satisfactory results and both the particles moved

monotonically to their respective target points without any oscillations. Using computer

simulations, we also demonstrated that it was possible to manipulate both particles along

independent trajectories (Fig. 3.17) using the MPC feedback strategy.

Figure. 3.17 Trajectory of two particles MPC scheme. Two particles (blue and red markers) are

made to move along a predefined path, a square using MPC. See ‘Video2’ for more details.

3.8 Materials and methods

The materials and methods for the calibration experiments and the particle manipulation

experiments with emulsion systems are described below.

35

3.8.1 Calibration experiments

Fluorescent beads (Polyscience Inc.), 1 m in diameter, suspended in glycerol/water (80 wt%

glycerol) mixture are used for the device calibration and control algorithm testing. During the

calibration experiments, the stagnation point position is obtained by observing the streamlines

formed by the fluorescent particles by taking long exposure images of the flow field. A high

concentration (~1000 particles/l) of particles is required for a reasonable exposure time.

Fluorescent beads, 1 m in diameter, were also used to demonstrate the ability of the device to

control and steer two particles along independent pre-defined arbitrary paths. While performing

the particle trapping experiments, we used a matched-density suspending fluid (glycerol and water)

so that the density difference between the particles and the suspending medium was small to avoid

the gravity-induced settling of the particles in the device.

3.8.2 Soft particle experiments

In order to demonstrate the ability of the device to study the collisions of soft particles, we choose

a water-in-oil emulsion as a model system because its interfacial properties are well-characterized.

Light mineral oil is used as continuous phase and Span® 80 (Sorbitane monooleate), a non-ionic

surfactant was added to mineral oil at 0.5% (w/w) to prevent spontaneous drop coalescence. The

dispersed phase is 1mM SDS (Sodium dodecyl sulfate) solution. All chemicals were purchased

from Sigma-Aldrich and were used without further purification.

The two drop collision experiments were performed to study the coalescence of drops, and these

preliminary experiments were compared with the scaling theory developed to estimate the

hydrodynamic drainage time.

The water-in-oil emulsion was generated using a T-junction at two inlets (I2 and I3) of the circular

region of the device. The flow rate of dispersed phase (water) is kept at 1 l/hr using a syringe

pump. The suspending phase (oil) is flowed at a pressure differential of 3 psi which was sufficient

enough to create short slugs of the dispersed phase. The pressure differential of 3 psi leads to flow

rats of ~ 20 μl/hr at which the, interfacial forces dominate the hydrodynamic stresses [40], and

36

hence, the droplet completely spans the channel width. In this regime, the size of the discrete fluid

segment is given by the following relationship.

1s water oilL w Q Q (3.15)

Here, sL is the length of the slug, w is the channel width, waterQ and oilQ , are the flow rates of

water and oil, respectively, and is a constant which depends on the geometry of the T-junction.

These drops spanned the channel depth and thus no gravity effects are observed. When a few

droplets are formed at the junction, the water flow is cut off and the last two slugs traversing the

channels enter the circular slot, and are subsequently brought to their respective target stagnation

points for drop collision experiments.

37

Chapter 4

4. Device Calibration and Algorithm Testing

In this chapter, we describe the calibration procedure that establishes the relationships between the

applied pressure levels in the inlet and outlet reservoirs, and the stagnation points. The

characterization allows us to invoke the correct pressures in the reservoirs to manipulate the

positions of particles, and impose known strain rates at target stagnation points for droplet-droplet

collision experiments.

4.1 Determination of the device center

Figure 4.1 Image of a representative device showing the edges of the circle (see equation in

Appendix II), the device center is at [786.29.45 m, 633.15m] with respect to the origin at the

top left corner and the length of the device edge is 486.4 m.

x

y

38

Before the device calibration and particle trapping experiments can be performed, it is necessary

to determine the device center using equations in Appendix II. To achieve this, we take a bright

field image of the device under stagnant conditions. A MATLAB® code then computes the center

of the device and the radius of the device. Fig. 4.1 shows the device center, radius of a

representative extensional flow device.

4.2 Stagnation point calibration

The fluorescent beads suspended in 80% glycerol-water solution were illuminated by an external

light source (mercury lamp, illumination wavelength 500 nm) which is connected to the Nikon

microscope and imaged through the microscope objective (10X) onto the Q-imaging camera at a

frame rate of 7.5 fps and exposure time of about 900 ms. For calibration experiments, the

microfluidic device used has no extra side ports [see Fig. 3.8 (a)]. Before starting the experiments,

the microfluidic device was flushed with 1% w/w Bovine serum albumin (BSA) in Phosphate-

buffered saline (PBS) in order to inhibit the fluorescent beads from sticking onto the PDMS

microchannel walls. These long exposure images of fluorescent beads, which act as tracer

particles, are required to determine the streaklines of particles and hence, the two stagnation points

for a series of pressure combinations. All of the streamline images in these experiments were

recorded at the mid-plane of the microchannel. The location of central plane was found by shifting

the fine focus by half-depth of the channel starting from either the top or bottom plane of the

channel. Fig. 4.2 shows such exposure images, where the inlet and the outlet pressures are shown.

Also shown in each subfigure is the flow profile expected from simulations in MATLAB. The

position of stagnation points in the experimental image is determined by the intersection point of

extensional and compressional axis. One can see the agreement between experiment and theory is

good, but not perfect. Fig. 4.3 provides a more detailed comparison between the stagnation point

position obtained from the analytical solution and the experiments. The maximum error in

stagnation point position is 36.8 m, which is 3.68% of the device side length. The error in the

stagnation point position at the device center is 1.2 m.

39

The mismatch between the analytical solution and the experiments may be due to errors in

fabrication of the device and external disturbances during the experiments. For example, there

could be a mismatch in the lengths of the arms/ports device, non-uniformities in the channel depth,

and/or unequal resistances in the tubes leading to the device. Fortunately, for the experiments

performed in this work, these are acceptable errors; there is no significant deviation between the

experimental behavior and theoretical flow-field, and the errors do not interfere significantly with

our control procedure.

The drops and particles can thus be trapped at two arbitrary stagnation points for extended periods

of time (e.g. see Fig. 4.6). As we will see later in the thesis, disturbances in the flow-field become

significant when two extra ports are introduced to form water drops at the T-junction. For drop-

collision experiments, we have implemented a model-predictive control scheme that can account

for disturbances and generate control sequences to achieve prolonged trapping.

Figure 4.2 Stagnation point position with different inlet and outlet pressure combinations. In

each subfigure, experimental streamlines are shown on left, and streamlines simulated in

MATLAB for the same pressure combinations are shown on the right.

Dimensionless Length

Dim

en

sio

nle

ss L

en

gth

(0.000, 0.000)(0.000, 0.000)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

0.5

1

1.5

Dimensionless Length

Dim

en

sio

nle

ss L

en

gth

(-0.324, 0.560)

(0.176, -0.305)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Pressure = [4, 1, 4, 1, 4 , 1] psi Pressure = [3, 0, 1.5, 0, 3 , 0] psi

Dimensionless Length

Dim

en

sio

nle

ss L

en

gth

(0.431, 0.290)

(-0.466, -0.228)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Dimensionless Length

Dim

en

sio

nle

ss L

en

gth

(0.612, 0.354)

(-0.612, -0.354)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

0.5

1

1.5

Pressure = [2, 1, 3, 1, 2, 0] psi Pressure = [2.5, 1, 3, 1, 2.5 , 0] psi

40

-600 -400 -200 0 200 400 600-500

-400

-300

-200

-100

0

100

200

300

400

500

x(in microns)

y(i

n m

icro

ns

)

Experimental

Analytical

Figure 4.3 Plot showing the error between the stagnation point position obtained by

experiments and theory. Red and black markers denote the positions of theoretical

stagnation points and experimental stagnation points. Similar markers represent the

pair of stagnation points above and below y=0. For example, red square above and

below the line y=0 represent a theoretical pair of stagnation for any arbitrarily

applied pressure. The corresponding black squares represent the experimental

stagnation points.

41

4.3 Particle trapping at two stagnation points using analytical solution

After the stagnation point calibration, trapping of two fluorescent beads suspended in 80%

glycerol-water was performed in the circular slot at two stagnation points on the x-axis. In the

initial trapping experiments, we introduced the particles from two side ports [Fig. 3.8 (b)].

Although the magnitude of flow rates in the side ports was one order lower than the flow rates in

the main channel, we observed that they strongly interfere with the theoretical flow-pattern and

our experiments failed repeatedly. Therefore, the two extra ports in the microfluidic device [Fig.

3.8 (b)] were completely closed and the fluorescent particles were introduced in the slot from the

Inlet 2 liquid reservoir (see Fig. 3.1). In order to control the particles, fluid flow was started by

actuating the pressure controllers through MATLAB GUI interface (Appendix III). The pressure

differential was chosen so that the flow velocities are not high because at high velocities, if the

particles are caught along the extensional axis, they are swept away rapidly due to the

exponentially increasing separation from the stagnation point with time. The time available for

controlling the particles is limited by the control loop time which is discussed in Chapter 5. Once,

sufficient number of particles are spotted in the circular slot, the control was shifted to automatic

control scheme (discussed in Chapter 3) based on analytical solution (Appendix III). The GUI

allows the user to choose the two particles that he/she wants to control. The fluctuation of both

particles around their respective target stagnation points is shown in Fig. 4.5. In the initial

demonstration of the two-particle control [see Figure. 4.4], one iteration was equal to 500 ms but

we were able to reduce it to ~150 ms in recent experiments (see section 5.2). We maintained the

particles at their respective stagnation points for 9 minutes after which the experiment was stopped.

We also performed experiments where the two particles were trapped at two different positions on

the y-axis; the results are shown in Fig. 4.6. We trapped the particles for about 17 minutes at their

respective stagnation points. Note that we have performed experiments where particles were

trapped for as long as 30 min, and we believe that there is no reason why this cannot be extended

to much longer times.

42

Figure 4.4. Snapshots of two particles being controlled (a-d) at different time instants

using the analytical solution control scheme. The green squares represent the target

stagnation positions and green circles represent the current position of both particles.

The pressure for this experiment at the final positions of the particles were [3.8 3 4 3 4

3] psi. See Video3 for more details.

Particle 2

Particle 1

a b

Snapshot of flow field at t=0 sec Snapshot of flow field at t=60 sec

d c

Snapshot of flow field at t=120 sec Snapshot of flow field at t=230 sec

43

Figure 4.5. Separation between the particle positions and target points with respect to

iteration number (time)

We also used the analytical solution to manipulate water drops of diameter 50 μm (Fig. 4.7).

suspended in light mineral oil. Since this control scheme works only when the two extra ports are

not present, we used a water in oil emulsion in one of the liquid reservoirs to introduce the drops

in the circular slot. After trapping the drops, we maintained the drops at their target stagnation

points for 15 minutes using the control scheme. For experiments with drops, the constant K in (3.7)

was increased slightly above the values used in fluorescent bead control, which means that a higher

pressure difference in each control loop was required to achieve convergence of drops. This is

discussed in detail in the Appendix C. As mentioned in section 3.6, the analytical solution control

scheme gives accurate results for particles not more than ~60 μm diameter because in this limit,

the particles do not adversely affect the theoretical flow-field in the slot. The updated pressures

corresponding to the new flow rates in each feedback loop are scaled to the range (0-5 psi) of the

pressure controllers used in the experiments. The scaling of pressures relies on the fact that the

flow-field in the slot remains same on multiplying all the pressures by a numeric constant or adding

a constant to all the pressures. We have observed that scaling of pressures is required only when

100 200 300 400 500 600 7000

50

100

150

200

250

300

Iteration #

Off

set

err

or

for

bo

th p

art

icle

s(i

n m

icro

ns)

Error for particle1

Error for particle 2

44

particles are travelling to their desired locations because the flow-field changes considerably

during the steering phase. Once particles are trapped, the changes in pressures required to hold

them still, at their target points is very small and no scaling is required. In simpler words, the

particles can be maintained at their target locations by only small changes in the flow-field,

thereby, not significantly affecting the strain rate in the flow. This is particularly useful for drop-

collision experiments where we want to perform experiments at nearly constant strain-rates.

Figure 4.6. Snapshots of two particles being controlled (a-d) using the analytical

solution control scheme. The green squares represent the target stagnation

positions and green circles represent the current position of both particles. The

pressure for this experiment at the final positions of the particles were [3.8 1 4 1 4

1] psi. See Video4 for more details.

Snapshot of flow field at t=0 sec Snapshot of flow field at t=30 sec

Snapshot of flow field at t=50 sec Snapshot of flow field at t=80 sec

Particle 1

Particle 2

a b

c d

45

4.4 Particle pair manipulation along predefined arbitrary paths using

analytical solution

With the ability to trap two particles at any arbitrary positions in the circular slot, we demonstrate

two-dimensional manipulation of two particles along independent trajectories. Two particles are

initially trapped at arbitrary stagnation points. Each independent trajectory of respective particles

is divided into a fixed number of points which serve as the successive target points for the particles.

The two particles are first steered towards the starting points of their trajectories using the control

scheme. Once the distance between the particle position and target stagnation point (fixed points

in the trajectory) is below a threshold, the trapping position is updated to the next point along the

Initial position of two drops Final position of two drops

Figure 4.7. Manipulation of two surfactant free water drops of 50 μm diameter using the

analytical solution. Snapshots of initial and final positions of both drops are shown. Red

and blue square denotes the target position of drop 1 and drop 2 respectively. Red and blue

asterisk represents the current positions of both drops. The pressure for this experiment at

the final positions of the particles were [1.8 1 2 1 2 1] psi. See Video5 for more details.

46

predefined path. This sequence is continued for all points on the two independent paths. Fig. 3.8

shows the trajectory taken by both particles when the predefined path is a square. At some points

in the trajectory, the particles deviate significantly from their predefined path which can be

attributed to the external disturbances in the system or model mismatch. However, the control code

was strong enough to bring the particles back and the predefined trajectories were traced with an

acceptable error.

a b

c d

Snapshot of flow field at t=0 sec Snapshot of flow field at t=140 sec

Figure 4.8. Manipulation of two fluorescent beads of 1 μm diameter to trace a square.

Snapshots (a-d) of both particles at different time instants. Green stars denote the desired

trajectory of particles while green circles represent the current position of particles. The

pressures in this experiment were [3.8 3 4 3 4 3] psi. See Video6 for more details.

Snapshot of flow field at t=280 sec Snapshot of flow field at t=450 sec

47

4.5 Particle trapping at arbitrary stagnation points and manipulation

along a predefined path using Model Predictive Control (MPC)

Since the computer simulations of MPC scheme showed satisfactory results (see section 3.8), it

was also implemented in the experiments. To test the performance of MPC, we manipulated two

1 μm-diameter fluorescent particles suspended in 80% glycerol water solution. Once the control

scheme parameters were properly selected (the scheme was sensitive to the choice of weighting

matrices in the objective function in equation 3.13 which is discussed in Appendix C), the particles

could be maintained within few microns of their respective target stagnation points for flow rates

up to 20 μl/hr.

a b

c d

t=0 sec t=8.2 sec

t=18 sec t=24 sec

Figure 4.9. Snapshots of two particles being controlled (a-d) using MPC scheme at

various time instants. Green square denotes the target position of particles while green

circle represents the current positions of both particles. This final pressures in the

experiment were [3.8 1 4 1 4 1] psi. See Video7 for more details.

48

t=0 sec t=100 sec

t=200 sec t=300 sec

a b

c d

Figure 4.10. Manipulation of two fluorescent beads of 1 μm diameter to trace a square.

Snapshots (a-d) of both particles at different time instants are shown. Green square

denotes the target position of particles while green circle represents the current

positions of both particles. This final pressures in the experiment were [3.8 1 4 1 4 1]

psi. See Video8 for more details.

49

As shown in Fig. 4.9, there is an offset between the steady state position of the particles and the

target stagnation points. The control code maintained both the particles at their steady state

positions for a sufficiently long time. To reduce this offset, we need to add an integral control in

the objective function of equation (3.13) because it is capable of eliminating the steady-state errors.

However, since the RTI algorithm used for solving the OCP in MPC works well for path tracking

problems i.e. when the initial state vectors are closer enough to the target stagnation points, we

also demonstrated the manipulation of particles along two separate squares with the same objective

function as in (3.13). The terminal constraint 1x t in (3.13), is omitted during the

experiments because inequality constraints make the solution time of OCP relatively slower.

4.6 Trapping of a pair of water drops using Model Predictive Control

(MPC)

We also demonstrate the trapping of water drops at two arbitrary stagnation points using the MPC

algorithm (see Fig. 4.11). Again, since the order of offset error is much smaller than the size of

trapped drops, the trapping was achieved successfully. Larger drops are much easier to control

than fluorescent beads of 1 μm, because they do not drift apart due to small fluctuations in the

pressures through the ports.

50

a b

c d

Figure 4.11. Trapping of two drops using MPC algorithm. Snapshots(a-d) of both drops

at different time instants are shown. Green square denotes the target position of drops

while red asterisk represents the current positions of both drops. See Video9 for more

details.

51

Next, we use the MPC algorithm to circumambulate one drop over another stationary drop. At

time t=0, the two drops are trapped at two separate positions. The stagnation point of drop 2 (see

Fig. 4.12) is kept same while the target stagnation point of drop 1 is gradually changed such that

the distance between both target stagnation points is always equal to the center to center distance

of both the drops.

Figure 4.12. Circumambulation of drop 1 over stationary drop 2 using MPC

algorithm. Snapshots of positions of drop 1 at different time instants are shown.

Red asterisk represents the current positions of both drops. See Video10 for more

details.

Drop 2

Drop 1

a b

c d

52

Chapter 5

5. Avenues for improvement of the device and future work

Over the last two years, we have redesigned the geometry of the two particle trapping and collision

device, and tested six different control schemes, before finally converging to a control scheme that

is robust for small and large particles. However, there are many potential avenues for improving

the device remaining to be explored. We highlight these in the last chapter of this thesis, along

with the future work that will be undertaken with the device.

5.1 Improvements for the automated trap

5.1.1 Better experimental procedure

We have used the two-particle control trap to demonstrate collision of Hele-Shaw drops leading

to coalescence. As discussed earlier, water drops are generated at T-junctions formed by the

intersection of Inlet 2 (or Inlet 3) and an extra side port (Fig. 3.4) connected to a ‘Kd Scientific’

Syringe pump which unfortunately, involves a large dead volume in the syringe and tubing. As a

result, slugs of water drops keep flowing into the circular slot even if the syringe pump supply is

cut-off. This means that the user has to spend long periods of time staring at the computer screen,

waiting to have only two drops in the circular slot so that he/she can start the collision experiments.

In some experiments, there is only one drop in the circular slot and by the time, the second drop

floats by, the first one exits the slot through one of the outlets. Occasionally, the accommodation

time before the user generates two droplets at the junction is very long. Therefore, it is really

difficult to perform a vast number of drop-collision experiments in one sitting. A good experiment

should have the facility of on-demand drop generation so that there no unwanted drops in the slot.

This problem will be solved in the future by using a solenoid valve to generate drops on demand.

53

5.1.2 Geometry of the device

The choice of circular shape of the device in our thesis was governed by the availability of an

analytical solution for the flow field and stagnation points for arbitrary combination of inlet and

outlet flow rates. In this device, there is some redundant flow near the circumference of circle

which limits the highest possible strain rate at each point in the slot. Another geometry with three

alternating inlets and outlets producing two stagnation points is a regular hexagon (see Fig. 5.1).

Since the MPC algorithm is robust and works on the online optimization of a suitably chosen

objective function, we believe that it will work even if there are inaccuracies in the dynamic model

of the system, as has been verified in other control problems in the literature [66] [67] [68]. Hence,

the dynamic model of the theoretical flow field for circular slot can be used for the hexagonal slot

Figure 5.1. The optimum channel shape for microfluidic device with two stagnation points

also. In future work, we can perform collision of drops in this optimized geometry.

54

5.1.3 Control loop time

The most critical design feature in the automated trap is achieving a short feedback time. If the

control loop time is short, particles can be trapped at relatively higher strain rates. Our first

demonstration of particle trapping had a feedback time of ~500 ms and over time, we improved

our algorithm to achieve a feedback time of 120-150 ms. The sequence of events in our control

scheme along with their typical times in milliseconds are (1) Capture of the image of flow-field

into MATLAB (10 ms), (2) Image analysis to yield the center of the particles (25-70 ms) (3)

Calculation of the pressures required to be imposed on six reservoirs (~ 10 ms) for analytical

solution control scheme and ~30 ms for MPC scheme (4) Instructing the pressure controllers

through the DAQ card to impose the pressures ( 50-70 ms). Hence, the total time consumed in

one loop of the iteration is, thus, about 120-150 ms for analytical solution control scheme and

about 150-180 ms for MPC scheme.

Currently, we acquire the entire image of the flow-field (1600 x 1200) and then, divide it into two

sub-images. Instead of capturing the entire image, if we make the control code acquire a specific

region of interest (500 x 500) that contains both particles, the resultant image will be shorter and

the time required to detect the position of particles can be reduced which will speed up step (2) of

the control loop. Alternatively, MATLAB can be used to communicate with an external

microcontroller that can calculate the particle positions in nanoseconds. Another scope of

improvement is to reduce the time taken in step (4) by employing pressure transducers with smaller

response times. But it has to be noted that the lowest achievable times are limited by the time

scales corresponding to the speed of sound, which governs the time between the application of the

pressure to the controllers, and the translation of this pressure to flow within the microchannels.

After the adjustments made in step (2) and step (4), we believe that the control loop time can be

reduced to ~30 ms which is five times faster than the current loop time.

5.1.4 Material of the microfluidic device

All our experiments were performed using PDMS-based microfluidic channels, because these are

cheaper and easy to fabricate. One future application of the device is to investigate the coalescence

of Hele-Shaw drops in bitumen, which is important in the oil industry. Unfortunately, PDMS

presents compatibility issues when it is used bitumen as it contains Toluene which can cause

55

swelling of the PDMS device [43]. Since we have tested the device with a simpler material, we

are ready to move to a more stable platform i.e. glass. Although glass-based microfluidic channels

are more expensive, they are compatible with a much wider range of liquids. They can be subjected

to a variety of treatments to render them hydrophilic or hydrophobic Unlike PDMS channels, they

do not show bowing or sagging even for high aspect ratios, a requirement of the Hele-Shaw

configuration.

5.1.5 Limitation on the range of strain rates for particle trapping

To induce flow in the circular slot, we use pressure-controlled liquid reservoirs. The maximum

rating of the pressure controllers is max 5P psi which limits the flow rates and hence, the strain

rates at which the particles can be trapped in the slot. Another factor that limits the maximum

available strain rates is the control loop time. As discussed in section 3.6, the cropping algorithm

imcrop is set in such a way that the particle does not leave it after one control loop time, tc. For a

cropping window of size W x W, the maximum possible velocity, maxv , is

max .

c

Wv

t (5.1)

The flow rates, Q, therefore, must be maintained such that,

max .c

c

c

R bWQ R bv

t (5.2)

Equivalently, the characteristic pressure differences should satisfy

0

maxmin , ,

.

c

c

R R bWP P

t

(5.3)

where we have used Eq. (3.1),

56

,P

QR

(5.4)

with

0 ,R R (5.5)

and 0 4

128.

LR

D (5.6)

We are also limited by the least count of the pressure transducer, minP , which provides a lower

bound on the flow rate. Thus,

0min maxmin , ,

.

c

c

R R bWP P P

t

(5.7)

The constraints on the pressure difference can be converted to restrictions on the strain rate,

knowing that the strain rate in the device obeys the relationship

* *

2 2

0c c

Q PG G G

bR R bR

(5.8)

where *G is the non-dimensional strain rate, and depends on the position in the device. Therefore,

** *

maxmin

2 2

0 0

min , ,

.

c c c c

G PG P G WG

R bR R t R bR

(5.9)

The typical experimental parameters are:

16 39.5 10 / , 500μm, 100μm, 100 μm, t 150o c cR Pa s m R b W ms

57

Using Eq. (5.4), (5.5) in Eq. (5.7) at an arbitrary target location , 0.35,0x y where

* 19.542G deduced from analytical solution, we present the range of strain rates available in our

experiments for different ranges of pressure controllers, suspending medium viscosities and

whether our control scheme (with control loop time ~150 ms) can successfully control the particles

or not:

minP maxP G Success of

control

scheme

0.1 psi 5 psi 1 cp 5.67 / 26 /s G s Unsuccessful

0.1 psi 5 psi 30 cp 0.19 / 9.45 /s G s Successful

0.1 psi 60 psi 1 cp 5.67 / 26 /s G s Unsuccessful

0.1 psi 60 psi 30 cp 0.19 / 26.0 /s G s Unsuccessful

Table 5.1 Range of strain-rates available in control experiments with the current set-up

If the upper bound on strain rate G in Eq. (5.9), is dictated by G*W/Rc tc , we cannot trap the particles

successfully with the current feedback time of ~150 ms. As evident from the table, we have

demonstrated successful trapping of particles suspended in light mineral oil (viscosity: 30 cp).

5.1.6 Scaling of pressure vector in each feedback loop

The updated pressure vector corresponding to the new flow rates in each feedback loop are scaled

to the range (0.1-5 psi) of the pressure controllers used in the experiments. The scaling of pressure

vector relies on the fact that the flow-field in the slot remains same on multiplying all the pressures

by a numeric constant or adding a constant to all the pressures. We have observed that scaling of

pressure vector is required only when particles are travelling to their desired locations because the

flow-field changes considerably during the steering phase. Once particles are trapped, the change

in pressure vector in subsequent time loops, required to hold them still, at their target points is very

58

small and hence, no scaling is required because the pressures remain within the bounds of 0.1-5

psi. In other words, the particles can be maintained at their target locations by only small changes

in the flow-field, thereby, not significantly affecting the strain rate in the flow. This is particularly

useful for drop-collision experiments where we want to perform experiments at nearly constant

strain-rates. A flow-chart representing the methodology to scale the pressure vector is shown in

Fig. 5.2. Each element of the updated pressure vector, iP , can either be smaller or greater than the

average pressure, 0P , at the center of the device as shown in Fig. 3.7. This allows any port to

become an inlet or outlet during the feedback loop which may be a requirement to steer the

particles towards their target locations.

Obtain the pressure vector from control

scheme P= [P1, P2, P3, P4, P5, P6]

Does all the elements of

vector P satisfy the

inequality 0.1<P<5.0?

Scale the pressure vector using the relation:

P_scaled=P/max(P)*5.0

P_scaled= P-min(P)+0.1

No

Scaling of vector P is not

required.

P_scaled=P

Scaling of pressure

vector is completed.

Yes

Figure 5.2: Flowchart representing the scaling of pressure vector required in each

control loop

59

5.2 Future Work

5.2.1. Controlled coalescence of water drops using the circular microfluidic flow

device

To demonstrate the capabilities of the circular microfluidic device to study soft particle collision,

we show that the device can be used to study the controlled coalescence of Hele-Shaw drops at

specific strain rates and glancing angles.

Figure 5.3 Three stages of coalescence between two drops pushed by a constant force

As a motivation for this work reserved for future, recall that, as mentioned in the introduction,

coalescence is still a poorly understood phenomenon. Surprisingly, even the canonical problem of

coalescence between two identical, Newtonian drops colliding in a compressional flow is yet to be

understood completely [54] [55]. This is due to the inherent multistage and nonlinear nature of the

phenomenon. As shown in Fig. 5.2, a typical coalescence process starts with two Newtonian drops

approaching each other, due to a constant hydrodynamic force from the external fluid, leading to

collision. As the drops approach each other, the thin film of ambient fluid trapped between the

F

F

Drop I

Drop 2

Drop I

Drop 2

Coalescence

Step 1: Collision under constant force F

Step 2: Thin film drainage

Step 3: Coalescence after film rupture

60

contact zones of two drops begins to drain. Finally, the film thickness reduces to a critical value,

below which the non-hydrodynamic forces ruptures the film completely facilitating coalescence.

The time required to completely drain the thin film of ambient fluid is called hydrodynamic film

drainage time, and this time dominates the total time to coalescence when the suspending medium

is highly viscous.

We present here some preliminary theory and experiments that elucidate the hydrodynamic

drainage time of coalescence. While prior studies have been performed with 3-D drops in

unbounded flows [29] [30], we perform our experiments for drops under confinement, i.e., Hele-

Shaw drops for which the undeformed drop diameter is significantly greater than the depth of the

microfluidic channel confining the drop.

5.2.2 Theory and experimental results of hydrodynamic drainage time for

coalescence

The theory relating the drainage time Dt , strain rate G, radius of drops R , and the interfacial

tension is developed for Hele-Shaw drops in an extensional flow in Appendix D. We assume

that the drop is cylindrical such that the interface between the drop and suspending medium is flat

with zero curvature along the depth of channel. The scaling analysis presented in Appendix D

yields the hydrodynamic drainage time as

2

3~ Cad

c

bt G

h

(5.1)

where, Ca, is the Capillary number defined as:

3

2Ca

GR

b

(5.2)

is the viscosity of suspending phase, G is the strain rate, ch is the critical film thickness and

b is the half-depth of channel.

61

During collision experiments, two water drops of 1mM SDS solution in mineral oil (0.5 % Span-

80) are created in inlet up streams of the circular slot of the device. The drops are trapped at two

separate target stagnation points using the MPC control algorithm. Next, the target stagnation

positions for both drops are gradually moved towards each other so that drops come in contact.

The GUI in the control code allows the user to change the weighting coefficients in the objective

function (see Eq. 3.12) continuously during the experiments. We observe that larger weighting

coefficients are needed to control the drops as the size of the drops increases. The drainage time

from experiments was found to be 30 minutes. The drainage time calculated from the scaling

analysis in Eq. (5.1) is 47 minutes. We have done some preliminary experiments of drop-collision

with increasing strain rates but a comprehensive comparison of experimental data with the

theoretical model requires a series of drop-collision experiments at different strain rates and

different drop sizes which is a part of future work. It should be noted that the theoretical model of

film drainage for Hele-Shaw drops only provides order of magnitude estimates. This will

obviously have an effect on comparing the experimental observations and theoretical predictions.

62

5.2.3 Collision of vesicles to measure adhesion rates

Vesicles are soft particles in which a finite volume of liquid is enclosed by a membrane called the

bilayer [42]. Although the dynamics of vesicles in different linear, unbounded flows have been

studied extensively [45], there have been no studies for vesicle-vesicle interactions in linear flows

under confined conditions. Our device has the ability to study hydrodynamic interactions between

two vesicles. We have done some preliminary experiments using the control scheme based on

Figure 5.4. Controlled Coalescence of water drops at Pressure = [4.3, 1.45, 4.5, 1.45, 4.5,

1.45] Snapshot of both drops at various time instants are shown. Red asterisk denotes the

current position of both drops. See Video11, Video12 and Video14 for more details.

a b

c d

t=0 sec t=30 sec

t=2 min t=30 min

63

analytical solution to achieve collision between two vesicles (see Fig. 5.4), but obviously, more

work needs to be done in this area. Two SOPC vesicles grown with small mole % of fluorescently

labelled NBD suspended in 70% glycerol-water was used for vesicle-collision experiments.

Figure 5.5. Snapshots showing controlled collision of two vesicles at different time

instants. The suspending medium is 70% glycerol-water. See Video13 for details

a b

c d

64

5.3 Conclusion

Our experiments have demonstrated the feasibility of controlling the position of two

particles/drops in a circular microfluidic device based on two different feedback control strategies,

the analytical solution and non-linear Model Predictive Control (MPC) scheme. In this work, we

have presented the construction and calibration of a microfluidic device for studying the interaction

between soft particles. We have also demonstrated the capabilities of the device by performing

controlled coalescence experiments between two water drops. The microfluidic platform we have

created will open new doors for the studies of vesicle fusion, droplet coalescence and a wide

variety of other soft particle interactions.

65

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71

Appendix A

Analytical solution for hydrodynamic flow field in the circular slot

The schematic of the circular slot is shown in the figure A.1. The inlets and outlets are assumed to

be point inlets and outlets. Since the flow between two parallel plates separated by an infinitely

small gap is a Hele-Shaw flow, the governing equation for stream function (A.1) and the boundary

conditions (A.2) can be written as:

Fig. A.1. Schematic of the circular slot with the boundary conditions on stream function

Q1

Q2 Q

3

Q4

Q6

Q5

𝜓 = 0

𝜓 = 𝜓1

𝜓 = 𝜓2

𝜓 = 𝜓3

𝜓 = 𝜓4

𝜓 = 𝜓5

(0, 0) x

y

72

2 2

2 2 2

1 10

r rr r

(A.1)

6

11

1

3mr

m

mQ H

(A.2)

The general solution of Laplace equation (.1) approximated by Fourier series is given as follows:

0

1

cos sinn

n n

n

A r A n B n

(A.3)

To satisfy the boundary condition (.2), we must write:

6

011 1

1cos sin

3n n mr

n m

mA A n B n Q H

(A.4)

where H represents the Heaviside function. Using orthogonality condition on (.4) and mass

balance equation

6

1

0m

m

Q

, we can write:

2 26 6

0

1 10 0

6 6 5 5

1 1 1 1

1 11 1

2 3 2 3

11 1 1 12 7 7 1 6

2 3 6 6 6

i m

m m

m m m m

m m m m

m mA Q H d Q H d

mQ Q m Q m Q m

(A.5)

2 26 6

1 10 0

226 6 6

1 1 1( 1) /3 ( 1) /3

1 11 1cos cos

3 3

sin 11 1 1cos sin

3

11 5sin sin

3 3

n i m

m m

m m m

m m mm m

m

m

m mA n Q H d Q n H d

n n mQ n d Q Q

n n

n mnQ

n

5

1

5

1

6 42sin cos

6 6m

m

n m n mQ

n

(A.6)

73

(A.7)

Hence, the final solution of flow-field in the circular slot is given by:

0

1

cos sinn

n n

n

A r A n B n

where 5

0

1

16

6m

m

A Q m

5

1

6 42sin cos

6 6n m

m

n m n mA Q

n

(A.8)

5

1

6 42sin sin

6 6n m

m

n m n mB Q

n

The velocity field in the circular slot is obtained as follows:

1

1 1sin cosn

r n n

n

v nr A n B nr r

(A.9)

1

1

sin cosn

n n

n

v nr A n B nr

(A.10)

74

At the two stagnation points 1 1,r and 2 2,r ,

1 1 2 2, , 0r rv r v r and 1 1 2 2, , 0v r v r (A.11)

Hence, the stagnation point positions in the slot are obtained as a function of flow rates in the six

ports.

75

Appendix B

Determination of center and radius of the circular device

If 1 1,x y , 2 2,x y and 3 3,x y are three points on the circumference of circle as shown in Fig.

4.1, the center of the circle ,c cx y can be written as follows:

2 22 2

2 31 21 3

1 2 2 3

3 21 2

1 2 3 2

2

c

x xx xy y

y y y yx

x xx x

y y y y

(B.1)

2 2

1 21 2

1 21 2

1 22c c

x xy y

x xy yy x

y y

(B.2)

The radius of the circle ‘r’ is obtained as:

1

1

c

c

x xr

y y

(B.3)

76

Appendix C

GUI description

A MATLAB Graphical User Interface (GUI) is created to facilitate the trapping of a particle/drop.

Fig. C.1 shows the user interface used to trap the particle. The left half of the user interface is used

to define the pressure of the inlets and the outlets. When the program is running in manual mode,

the 'Manual Control' mode is active and the pressure of all the reservoirs can be varied

independently. This mode is used to ensure that we have at least two particles.

Figure C.1 The GUI interface used to trap and manipulate particles using analytical solution.

Consol highlighted by orange box is used to change the reservoir pressures manually. Drop down

menu highlighted by red box is used to toggle between 'manual control', 'Particle trapping' and

'Particle manipulating'. Drop down menu highlighted by blue box is used to toggle between ‘Click

mode OFF’ and ‘click mode ON’ which allows the user to choose the two particles which they

want to trap/manipulate. Consol highlighted by brown box can used to change exposure time,

while the one highlighted by green box is used to change the gain constant.

77

The device can be accessed live in the manual control mode (highlighted by red box) to observe

the fluid flow. Once at least two particles are present in the circular slot, the ‘Particle trapping’

mode is selected from the drop down menu (red box) and the “Click mode ON” is selected from

the drop down menu (highlighted by blue box) which allows the user to click on any two particles

which he/she wants to control. The gain constant can be changed by changing the value of ‘K11’

(green box). Decreasing the gain decreases the rate at which both particles approaches their

respective target stagnation points. After some trial and error, we found that the gain of 0.8 was

optimal for controlling fluorescent beads by analytical solution control scheme. For controlling

water drops by analytical solution, we increased the gain constant upto 2.5 to achieve efficient

trapping. The user can also change the exposure time in order to view the particle more clearly by

changing the value of 'Exposure' (brown box). We found that the exposure time of 0.120 sec was

good for fluorescent particles trapping and 0.010 sec was optimal for drop control.

For steering particles along a predefined path, the ‘Particle manipulation’ mode is selected from

the drop down menu (red box) and the position of target stagnation points is successively changed

by checking the ‘Click to define new Xs’ option (black box). For controlling fluorescent beads by

MPC scheme, the parameters used were 1 and 0.01 [64]. While controlling the drops,

we have to increase the parameters and to 10.

78

Appendix D

Scaling theory for hydrodynamic drainage time

Consider two drops of viscosity ̂ suspended in a fluid of viscosity , and squeezed between

two plates separated by a constant distance, 2b (see Fig. D.1). The interfacial tension between the

two fluids is .

Figure D.1. Two Hele-Shaw drops pushed against each other by a constant force in an

extensional flow

We assume that the shape of drops is cylindrical with radius R and when the drops deform under

the action of constant hydrodynamic force from the suspending flow medium, the resulting

geometry is a thin disk-like film with a film thickness, h . If the lateral extent of the disk-like film

is a , the quasi-steady force balance between viscous force pushing the drops together and surface

tension force is established as:

2

2~

GRbR ba

b R

(D.1)

Hence, the lateral dimension of the thin film scales as:

Contact zone

(Strain rate)G

R

R

79

3

2~ Ca

a GR

R b

(D.2)

where Ca, the capillary number, is the ratio of the externally imposed hydrodynamic forces (that

tend to deform the drops) to the surface tension forces (that tend to keep the drops spherical). The

rate of change of the film thickness, /dh dt , can be related to the total radial velocity, u , within

the film by means of an overall volume balance, as:

~dh

ba ubhdt

(D.3)

The total radial velocity in the thin film region can we written as p tu u u , where pu is the

parabolic portion of the velocity profile driven by the capillary pressure gradient, / R , while tu

is the uniform portion of the velocity profile due to the slip at the interfaces between the thin film

and the drops. For clean interfaces without surfactant, the shear-stress continuity equation on the

interface may be written as:

ˆ~

1~

pt

t p

uu

b h

bu u

h

(D.4)

where viscosity ratio, , is the ratio of the viscosity of the drop to the viscosity of the suspending

fluid. For b>>h, the tangential component dominates the total flow

1

~ ~t p

dh bu h u

dt a a

(D.5)

The flow in the film can be approximated as that between two squeezing disk, with a resultant

velocity magnitude pu equal to

2 /~p

h Ru

a

(D.6)

80

Incorporating (5.6), the scaling (5.5) may be rewritten as:

2

3

/ /~

d h b h b

d tG Ca

(D.7)

If hc is the critical film thickness at which the film ruptures because of non-hydrodynamic

forces. This integrates out to 3

0

d

c

t Gb b

h h Ca

(D.8)

and for hc<<h0,

3~ Cad

c

bt G

h

(D.9)

Let the Hamaker’s constant be HA . At the inception of the instability of the thin film,

3

~H

c

A

h a

,

1/3

1/3

3~ Cac H

h A R

b b

(D.10)

Therefore, the scaling for dimensionless drainage time is as follows:

-1/3

8/3

3~ Ca H

d

A Rt G

b

(D.11)

If the interface is immobile as is the case for surfactant coated interfaces,

~ p

dhba u bh

dt

(D.12)

And this subsequently leads to

81

3

3

/ 1~

Ca

d h b h

d tG b

(D.13)

The dimensionless drainage time for immobile interfaces is as follows:

2

3~ Cad

c

bt G

h

(D.14)

Eq. (D.14) was used to compare the preliminary experiments in section 5.2