Modeling and simulation of droplet dynamics for microfluidic applications

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Ph.D. Thesis

Modeling and simulation of droplet dynamics for
microfluidic applications

Graduate School of Yeungnam University
Department of Mechanical Engineering
Major in Mechanical Engineering

Van Thanh Hoang

Advisor: Professor Jang Min Park, Ph.D.

August 2019



Ph.D. Thesis

Modeling and simulation of droplet dynamics for
microfluidic applications

Advisor: Professor Jang Min Park, Ph.D.

Presented as Ph.D. Thesis

August 2019

Graduate School of Yeungnam University
Department of Mechanical Engineering
Major in Mechanical Engineering

Van Thanh Hoang

ACKNOWLEDGMENTS
I would like to dedicate this thesis for my late father who highly encouraged me to
pursue a master and a doctoral program when he left this world almost nine years
ago. The thesis also is dedicated to the author’s mother who is seventy six years
old and living far from me now.
I really would like to express my deepest gratitude to my thesis advisor, Professor
Jang Min Park for dedicated help, valuable and devoted instructions, and
everything he has done for me in academic direction and in my life as well over the
last three years of my doctoral program.
I am so grateful to the committee members, Prof. Jiseok Lim, Prof. Jungwook Choi,
Prof. Kisoo Yoo, and Prof. Kyoung Duck Seo for attending my presentation as well
as providing pieces of advice for my doctoral thesis completion.
During my doctoral program, I wish to express my thanks to the Yeungnam
University for supporting the scholarship and providing an excellent academic
environment.
I also thank all of Lab. members, Mr. Gong Yao, Mr. Liu Wankun, Mr. Wu Yue,
Mr. Heeseung Lee, Mr. Seung-Yeop Lee, always gave me encouragement and
support during my doctoral program. Finally, I would like to thank my family,
especially my wife for their constant support and encouragement.
Date: May 15th, 2019
Van Thanh Hoang (호앙반탄)
Multiphase Materials Processing Lab., ME/YU

I


ABSTRACT
Design of microchannel geometry plays a key role for transport and
manipulation of liquid droplets and contraction microchannel has been widely used

for many applications in droplet-based microfluidic systems. This study first aims
to investigate droplet dynamics in contraction microchannel for more details and
then to propose a simplified model used for microfluidic systems to describe
droplet dynamics. In particular, for contraction microchannel, three regimes of
droplet dynamics, including trap, squeeze and breakup are characterized, which
depends on capillary number (Ca) and contraction ratio (C). Theoretical models
have been also proposed to describe transitions from one to another regime as a
function of capillary number and contraction ratio. The critical capillary number of
transition from trap to squeeze has been found as a function of contraction ratio
expressed as CaIc=a(CM-1), whereas critical capillary number CaIIc = c1C-1 depicts
the transition from squeeze to breakup. Additionally, the deformation, retraction
and breakup along downstream of the contraction microchannel have been
explored for more details.
To describe dynamics of droplet in microfluidic system, one-dimensional
model based a Taylor analogy has been proposed to predict droplet deformation at
steady state and transient behavior accurately. The characteristic time for droplet
reaching steady state is dependent on viscosity ratio and the droplet deformation at
steady state is significantly influenced by viscosity ratio of which the order of
II


magnitude ranges from -1 to 1. Finally, theoretical estimation of condition for
droplet breakup was also proposed in the present study, which shows a good
agreement with experimental result in the literature.

Keywords:

Droplet

dynamics,

Microfluidics,

Numerical simulation, Taylor analogy model.

III

Contraction

microchannel,


TABLE OF CONTENTS

ACKNOWLEDGMENTS .......................................................................................I
ABSTRACT ........................................................................................................... II
TABLE OF CONTENTS ...................................................................................... IV
LIST OF FIGURES .............................................................................................. VI
NOMENCLATURES ........................................................................................ VIII
CHAPTER 1. INTRODUCTION ........................................................................... 1
1.1. Droplet-based microfluidic system .............................................................. 1
1.2. Contraction microchannel in microfluidic system ....................................... 2
1.3. Dynamics of droplet in contraction microchannel ....................................... 2
1.4. Droplet dynamics in extensional flow ......................................................... 4
1.5. Problem statement ........................................................................................ 5
1.6. Dissertation overview .................................................................................. 7
CHAPTER 2. PROBLEM DESCRIPTION ........................................................... 8
2.1. Problem description of contraction microchannel ....................................... 8
2.2. Problem description for proposed model ..................................................... 9
2.3. Dimensionless numbers ............................................................................. 11

CHAPTER 3. TAYLOR ANALOGY MODELING ............................................ 12
3.1. Damped spring-mass model ....................................................................... 12
3.2. Taylor analogy breakup (TAB) model ....................................................... 13
3.3. Proposed model .......................................................................................... 15
3.4. Condition for droplet breakup .................................................................... 17
CHAPTER 4. COMPUTATIONAL MODEL AND VALIDATION .................. 18
4.1. Computational model and methods ........................................................... 18
4.2. Computational domain of contraction microchannel ................................. 19
4.3. Computational domain for the proposed model......................................... 22
4.4. Validation of simulation results in planar extensional flow ...................... 25
CHAPTER 5. RESULTS AND DISCUSSIONS.................................................. 27
5.1. Droplet dynamics in the contraction microchannel ................................... 27
5.1.1. Three regimes of the droplet dynamics ............................................... 27
IV


5.1.2. Droplet dynamics along downstream of contraction microchannel ... 34
5.2. Performance of the proposed model .......................................................... 41
5.2.1. Steady behavior of droplet deformation ............................................. 42
5.2.2. Transient behavior of droplet deformation ......................................... 44
5.2.3. Critical capillary number for droplet breakup .................................... 45
CHAPTER 6. CONSCLUSIONS AND RECOMMENDATIONS ...................... 47
6.1. Conclusions ................................................................................................ 47
6.2. Recommendations ...................................................................................... 48
REFERENCES ..................................................................................................... 50
요약....................................................................................................................... 59
CURRICULUM VITAE ....................................................................................... 61

V

LIST OF FIGURES
Fig. 1.1. Overview of the dissertation ..................................................................... 7
Fig. 2.1. Geometry of the contraction microchannel: (a) a full geometry and
symmetric domain for computational model, which is illustrated by the grey color
and (b) a view from top of the contraction microchannel [52]. .............................. 9
Fig. 2.2. Illustration of (a) a droplet suspending in a planar extensional flow and (b)
a description of the droplet magnified at XY plane [53]. ...................................... 10
Fig. 4.1. Schematic diagram of T-junction used in validation: (a) a full geometry
and (b) side view of the geometry. Dimensions unit is in micrometer. Wc and Wd
are the inlet widths for the continuous phase and dispersed phase, respectively (WT
= Wc = Wd) and LT is droplet length in the downstream. ...................................... 20
Fig. 4.2. Three regimes of droplet generation. (1) Experimental results from Li et
al., (2012) and (2) present simulation. (a) vct=0.83mm/s, vd=0.69mm/s, (b)
vct=3.47mm/s, vd=0.28mm/s, (c) vct=10.0mm/s, vd=5.0mm/s, (d) vct=20.0mm/s,
vd=10.0mm/s, where vct and vd represent the continuous phase inlet velocity and the
dispersed phase inlet velocity, respectively. ......................................................... 21
Fig. 4.3. Dimensionless droplet length as a function of Ca for two different flow
rates (8.06μL/h and 20μL/h) of the dispersed phase. Li et al.’s experiment 1 and
present simulation 1 applied the disperse phase flow rate of 8.06μL/h, and Li et
al.’s experiment 2 and present simulation 2 applied the disperse phase flow rate of
20μL/h. .................................................................................................................. 22
Fig. 4.4. A one-eighth of the full model used for the computational domain in planar
extensional flow. [53]. .......................................................................................... 23
Fig. 4.5. Mesh convergence test for λ=1 and Ca=0.067; (a) steady state, (b)
transient behavior of the droplet deformation. ...................................................... 24
Fig. 4.6. Comparison of droplet deformations at steady state [53] between
computational simulation and experiments [37,40]. ............................................. 26
Fig. 4.7. Comparison of droplet deformation at transient behaviors between
computational simulation and experiments [40] when the viscosity ratio of unity

[53]. ....................................................................................................................... 26
Fig. 5.1. Droplet cross-section snapshots at the symmetric plane viewed from top:
(a) trap behavior, (b) squeeze behavior, and (c) breakup behavior [52]. .............. 28
Fig. 5.2. Droplet dynamics described by a map of three regimes depending on C
and CaI, CaII: (a) trap and squeeze regimes and (b) squeeze and breakup regimes.
The results of trap (■), squeeze (○) and breakup (▲) are plotted by symbols, and
transition models of trap-to-squeeze (―) and squeeze-to-breakup (---) are plotted
by fitted curves [52]. ............................................................................................. 30
Fig. 5.3. Description of trap mechanism of the droplet at the early stage of the
contraction microchannel [52]. ............................................................................. 32
VI


Fig. 5.4. Two types of droplet breakup: (a) the first type of droplet breakup as CaII
> 2.4, tail formation (1), initial breakup (2), breakup into small pieces (3), the entire
droplet breakup (4) and (b) the second type of breakup as CaII < 2.4, back-interface
collapse and neck formation (1), initial tearing (2), tearing growth (3) and breakup
(4)-(5) [52]. ........................................................................................................... 35
Fig. 5.5. Positions of front (Zf) and back (Zb) interfaces and length (Ld) of the
droplet in contraction microchannel [52]. ............................................................. 36
Fig. 5.6. The relationship between normalized droplet length (Ld/D) and
normalized droplet position (Zd/D) for various values of C and CaII [52]. ........... 36
Fig. 5.7. Droplet position at steady state as a function of contraction C and capillary
CaII. Simulation results are plotted by the symbols, CaII = 0.1 (■); CaII = 0.3 (○);
CaII = 0.5 (▲), and predicted model is plotted by curves, CaII = 0.1 (―); CaII = 0.3
(---); CaII = 0.5 (-·-) [52]. ...................................................................................... 40
Fig. 5.8. Initial topology change position (Zd/D) of the droplet in the contraction
microchannel as a function of CaII [52]. ............................................................... 41
Fig. 5.9. Droplet deformations at steady state depending on viscosity ratio and
capillary number [53]. ........................................................................................... 43

Fig. 5.10. Steady behavior of droplet deformations obtained by the numerical
simulation and the proposed model for various viscosity ratios [53]. .................. 43
Fig. 5.11. The droplet deformation at transient behaviors for different viscosity
ratios as Ca=0.085 [53]. ........................................................................................ 44
Fig. 5.12. The verification of the proposed model by the numerical simulation for
the droplet deformation at transient behavior when the viscosity ratio of unity [53].
............................................................................................................................... 45
Fig. 5.13. Critical capillary number for droplet breakup as a function of viscosity
ratio ....................................................................................................................... 46

VII


NOMENCLATURES
Ca

Capillary number

CaI

Capillary number at large microchannel

CaIc

Critical capillary number for transition from trap to squeeze

CaII

Capillary number at contraction microchannel

CaIIc

Critical capillary number for transition from squeeze to breakup

Cac

Critical capillary number for droplet breakup

Re

Reynolds number

We

Weber number

B

The half breadth of droplet in Y-direction

C

Contraction ratio

Cf

Friction factor

C1, Ck, CF, Cd, Q

Dimensionless constants

c1, c2, M, N

Dimensionless constants

D

Diameter of droplet

Df

Parameter of droplet deformation

Ds

Droplet deformation at steady behavior

Δph

Hydrostatic pressure

Δpb

Laplace pressure at back interface of droplet

Δpf

Laplace pressure at front interface of droplet
VIII

Δp

Net Laplace pressure

μm

Medium viscosity

μd

Droplet viscosity

ρm

Medium density

ρd

Droplet density

σ

Surface tension between medium and droplet phases

𝜀̇

Extensional rate

λ

Viscosity ratio

к

Density ratio

ξ

Damping ratio

v

Characteristic velocity

vi

Velocity in large microchannel

vc

Velocity in contraction microchannel

vX

Velocity component in X direction

vY

Velocity component in Y direction

vZ

Velocity component in Z direction

d

Damping coefficient

F

External force

IX


Fc

Capillary force

Fd

Viscous drag force

k

Spring coefficient

L

The half length of droplet in X-direction

Ld

Droplet length in microchannel

Li

Large microchannel length

m

Mass of droplet

R

Radius of droplet

Rb

Back interface radius of droplet

Rf

Front interface radius of droplet

x

Dimensional displacement of droplet equator

y

Dimensionless displacement of droplet equator

ys

Dimensionless displacement at steady state of droplet equator

Zb

Position of back interface of droplet in microchannel

Zf

Position of front interface of droplet in microchannel

Zd

Position of droplet in microchannel

Zs

Steady state position of droplet in microchannel

W

Width of contraction microchannel

X

CHAPTER 1. INTRODUCTION
In the first chapter, introduction to microfluidics and droplet-based microfluidic
systems will be presented first. Then applications and droplet dynamics in
contraction microchannel, and extensional flow are reviewed in details. Finally,
problem statements which include objectives of the research will be stated.

1.1. Droplet-based microfluidic system
Microfluidics is a terminology divided into Micro and Fluidics, where fluid
and/or gas are introduced into small size on the order of microliters or nanoliters.
Microfluidics is well known for simple high-performance biochemical analysis.
There has many advantages such as precise control and manipulation, fast
processing, small amounts of samples and reagents, and so forth. Microfluidics has
been employed for development of Lab-on-a-Chip and micro total analysis systems
for applications in pharmaceutical, biomedical, chemistry and life science [1]. The
flow regime in a microfluidic system is described by laminar flow and with
assumption of no slip boundary condition, so it can control the flow for
manipulating chemicals and reagents precisely [2]. Microfluidic system is first
employed to produce droplets for materials processing applications. Later, dropletbased microfluidic devices have been applied for development of chemical and
biological analysis, and droplets can be considered as micro-reactors with small
volume [3-6].

1


1.2. Contraction microchannel in microfluidic system
In droplet-based microfluidic devices, contraction microchannel is typically
employed to generate extensional flow with high strain rates [7]. This configuration
microchannel has many applications. For instance, Anna et al. [8] used contraction

microchannel to generate water droplets which are suspended in continuous phase
of oil.

Zhu et al. [9] experimentally studied droplet breakup in expansion-

contraction microchannel. Large DNA molecules are controlled and stretched
continuously for optical detection and genes analysis by using a hyperbolic
contraction microchannel [10]. In addition, rheological properties of polymeric
materials can be measured by employing contraction microchannel [11].

1.3. Dynamics of droplet in contraction microchannel
Droplet dynamics in contraction microchannel has been investigated recently
via numerical methods, experiments and some theoretical analysis. In relation to
numerical studies is concerned, nearly previous researches were carried out with a
two-dimensional case. For example, the effects of viscoelasticity on drop and
medium were explored in 5:1:5 planar contraction-expansion microchannel via
applying a finite element method [12-14]. Entrance effects of contraction geometry
and rheology on the droplet behavior were studied by using numerical method [15].
The impact of shear and elongation on the droplet deformation was also
numerically and experimentally examined by using a hyperbolic convergent-

2


divergent microchannel [16]. Christafakis and Tsangaris [17] investigated the
effects of capillary number (Ca), Reynolds number (Re), Weber number (We) and
viscosity ratio (λ) on the droplet dynamics in a two-dimensional contraction
microchannel. Harvie et al., [18-20] studied the influence of Reynolds number,
capillary number, and viscosity ratio on droplet dynamics in an axisymmetric 4:1
microfluidic contraction. In regard to three-dimensional numerical studies of

droplet dynamics in contraction microchannel, there are only few studies in the
literature. Zhang et al. [21,22] performed three-dimensional numerical
investigations on the deformation of droplet in different three-dimensional
contractions. In the meantime, as far as experimental study is concerned, there are
not many researches in the literature. Droplet deformation and breakup in a planar
hyperbolic contraction microchannel were experimentally examined by Mulligan
and Rothstein [23,24]. Chio et al., [25] studied influence of transient pressure,
bubble deformation and bubble length on clogging pressure in microchannel
contraction. Faustino et al., [26] explored the deformability of red blood cells
undergoing extensional and shear flow generated in hyperbolic microchannel with
low aspect ratio. Carvalho et al. [27] proposed an aqueous fluid containing GUVs
to mimic the rheological behavior of blood by using hyperbolic extensional flow.
Regarding theoretical studies, Jensen et al., [28] provided a theoretical and
numerical research of large wetting bubbles in contraction microchannel for
minimizing the clogging pressure.

3


1.4. Droplet dynamics in extensional flow
Dynamics of droplet in microfluidic systems is controlled by the strength of the
flow type which is extension or shear [23]. Planar extension is a typical flow
selected to describe droplet dynamics in microfluidic systems. Planar extensional
flow is well known for several practical applications in materials processing and
microfluidics in this study. The planar extensional flow is first well known for
charactering emulsions, polymers and obtaining droplet viscosity by measuring the
drop deformation [29]. In droplet-based microfluidic systems, extensional flow is
usually used for generation, trap, mixing and manipulation of liquid droplets with
small volume [2]. There are also several investigations about cells or vesicles which
undergo the extensional flow. For example, planar extensional flow is employed to

trap and manipulate cells for long time scales [30], and to measure cellular
mechanical behavior [31]. A microfluidic cross-slot device generating planar
extensional flow is used to study dynamics of vesicles [32]. The mechanical
damage of cells in bioreactors was quantitatively assessed via planar extensional
flow [33].
There are several experimental, numerical and theoretical investigations
reported previously. The deformation and breakup under shear and extensional
flow were first presented by Taylor [34]. Similar to the Taylor’s research, the
experimental studies on the droplet dynamics were carried out for a wide range of
flow conditions [34-37] and for the details of three-dimensional droplet shapes at

4


steady and transient states [38]. In addition, non-Newtonian impact of the droplet
and medium were explored due to the complication of rheological properties in
polymer processing [39,40]. Later, the effects of rheological properties of the
droplet and/or medium on the droplet dynamics of deformation and breakup were
studied [41-43]. Also, droplet position and difference of flow rates in axisymmetric
extensional flow were taken into account for study on the asymmetric breakup [44].
In order to explore droplet deformation in extensional flow, there have been
some theories studied. A theory of small deformation was suggested by Taylor [34]
for prediction of the droplet deformation at steady state at low capillary number
flow. Later, theoretical models for transient behavior of droplet were investigated
by Cox [45] and Barthès-Biesel and Acrivos [46]. Another approximate theory was
developed to depict the breakup of a slender droplet at large deformation [47].
Droplet deformation in three-dimensional shape for arbitrary flow was described
by a phenomenological model [48].

1.5. Problem statement

It is noted that according to fabricated methods, the practical microfluidic
systems have microchannel geometry of rectangular cross-section. It is necessary
to have a clearer understanding about droplet dynamics in such microchannel.
However the previous studies on droplet behavior in contraction microchannel are
limited to two-dimensional model. Therefore, for a detailed analysis droplet

5


dynamics in geometry of contraction microchannel should be studied by threedimensional model. Up to our review, it has not been found a full guideline in the
literature for designing a contraction microchannel for droplet manipulation. In this
regard, the first objective of this study is to investigate droplet dynamics in details
in contraction microchannel through three-dimensional numerical simulation and
theoretical modeling.
In droplet-based microfluidic systems, dynamics of droplet in microfluidic
systems is determined by the strength of the flow type which is extension or shear
[23]. In practical cases, it is impossible to perform a three-dimensional simulation
for the whole microfluidic systems due to high computational cost. Therefore,
theoretical models for prediction of droplet deformation should be encouraged in
this case. As far as the previous research are concerned, the theoretical models are
quite complicated [45-47]. In the present study, the second objective is to propose
a simplified model for description of droplet deformation in the microfluidic
systems. The approach is based on an analogy between a droplet dynamics and a
damped spring-mass system. Particularly, the external force and damping force are
developed for investigating droplet dynamics in low Reynolds number and
capillary number. The proposed model has been examined the accuracy relying
upon an extensive computational simulation. Additionally, theoretical estimation
of critical capillary number for droplet breakup has been proposed in this study.

6

1.6. Dissertation overview
This dissertation has been divided into 6 chapters. The research framework is
shown in Fig. 1.1. Chapter 1 introduces the related works, motivations and
objectives of this study. Chapter 2 presents a problem description of the study. In
Chapter 3, the Taylor analogy is briefly presented, and details of the proposed
model are described. Chapter 4 is a presentation of three-dimensional
computational model and validation also. The results and discussions are provided
in Chapter 5. Finally, conclusions and recommendations are given in Chapter 6.

Fig. 1.1. Overview of the dissertation

7


CHAPTER 2. PROBLEM DESCRIPTION
2.1. Problem description of contraction microchannel
A contraction microchannel geometry is shown in Fig. 2.1, where a droplet of
diameter D suspended in a medium fluid. The droplet is initially located in large
microchannel which has a length and width of Li and 2D, respectively. The droplet
then transports into contraction microchannel which has a length of 15D and width
of W. To completely capture droplet dynamics and to eliminate effect of outlet
boundary, the length of the contraction microchannel of 15D was used in this study.
The depth of the whole microchannel of 2.5D was utilized so that effect of walls at
top and bottom of the microchannel on droplet dynamics can be neglected [49-51].
A dimensionless number of contraction level is defined as C=D/W for studying
effect of the width of the microchannel on the droplet dynamics. The initial droplet
diameter D is always larger than the microchannel width W, and the contraction
value C ranges from 1.11 to 2.5. For saving computation time, a symmetric model

which corresponds to a quarter of the full three-dimensional geometry was used as
shown by the grey color in Fig. 2.1 (a).

8


Fig. 2.1. Geometry of the contraction microchannel: (a) a full geometry and
symmetric domain for computational model, which is illustrated by the grey color
and (b) a view from top of the contraction microchannel [52].

2.2. Problem description for proposed model
In this research, planar extensional flow was selected to study droplet dynamics
in microfluidic systems relying on a proposed theoretical model and simulation
data. Fig. 2.2 illustrates a droplet having radius R suspending in a medium fluid
undergoing planar extensional flow. The velocity field used to describe the planar

9


extensional flow is expressed by Equation (2.1) where 𝜀̇ is extension rate and
velocity components in X, Y and Z directions are termed VX, VY and VZ, respectively.
𝑣𝑋 = 𝜀̇𝑋, 𝑣𝑌 = −𝜀̇𝑌, 𝑣𝑍 = 0

(2.1)

Fig. 2.2 (a) is an illustration of the droplet at XY plane, and a magnification of
the droplet is shown in Fig. 2.2 (b). The parameters of droplet deformation are
defined in terms of L and B, where L is the half length of droplet in X-direction and
B is the half breadth of droplet in Y-direction. The displacement of droplet equator
in X-direction is defined as x = L – B. It is assumed that droplet deformation is

ellipsoidal in all times, so droplet deformation parameter is commonly defined by
a dimensionless parameter D as Equation (2.2) [37,48]:
𝐷𝑓 = (𝐿 − 𝐵)/(𝐿 + 𝐵)

(2.2)

Fig. 2.2. Illustration of (a) a droplet suspending in a planar extensional flow and
(b) a description of the droplet magnified at XY plane [53].

10