International Journal of Multiphase Flow 93 (2017) 130–141

Contents lists available at ScienceDirect

International Journal of Multiphase Flow

journal homepage: www.elsevier.com/locate/ijmulflow

Three dimensional phase-field investigation of droplet formation in

microfluidic flow focusing devices with experimental validation

Feng Bai a , b , Xiaoming He

c , Xiaofeng Yang

d , Ran Zhou

a , Cheng Wang

a , ∗

a Department of Mechanical and Aerospace Engineering, Missouri University of Science and Technology, 400 W. 13th St., Rolla, Missouri, 65409, USA b School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China c Department of Mathematics and Statistics, Missouri University of Science and Technology, 400 W. 12th St., Rolla, Missouri, 65409, USA d Department of Mathematics, University of South Carolina, 1523 Greene Street, Columbia, South Carolina, 29208, USA

a r t i c l e i n f o

Article history:

Received 7 October 2016

Revised 14 April 2017

Accepted 14 April 2017

Available online 15 April 2017

Keywords:

Phase-field modeling

Droplet formation

Microfluidic flow focusing

Droplet breakup

a b s t r a c t

In this paper, the droplet formation process at a low capillary number in a flow focusing micro-channel

is studied by performing a three-dimensional phase field benchmark based on the Cahn–Hilliard Navier–

Stokes equations and the finite element method. Dynamic moving contact line and wetting condition are

considered, and generalized Navier boundary condition (GNBC) is utilized to demonstrate the dynamic

motion of the interface on wall surface. It is found that the mobility parameter plays a very critical role

in the squeezing and breakup process to control the shape and size of droplets. We define the character-

istic mobility M c to represent the correct relaxation time of the interface. We also demonstrate that the

characteristic mobility is associated with the physical process and should be kept as a constant as the

product of the mobility tuning parameter χ and the square of interfacial thickness ε2 . This criterion is

applied for different interfacial thicknesses to correctly capture the physical process of droplet formation.

Moreover, the size of the droplet, the velocity of the droplet along the downstream, and the period of

droplet formation are compared between the numerical and experimental results which agree with each

other both qualitatively and quantitatively. The presented model and criterion can be used to predict the

dynamic behavior and movement of multiphase flows.

© 2017 Elsevier Ltd. All rights reserved.

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1. Introduction

Emulsions (or micro-droplets) have a wide range of applica-

tions in food industry ( Muschiolik, 2007 ), cosmetics ( Gallarate

et al., 1999 ), drug delivery ( Yamaguchi et al., 2002 ), and chemi-

cal synthesis ( Odian, 2004 ). Traditional methods of emulsion pro-

duction, e.g., direct agitation of immiscible fluids, often produces

a broad size distributions. Droplet microfluidic technology has

shown great potential for production of highly mono-dispersed and

micron-sized emulsions ( Shah et al., 2008; Teh et al., 2008 ). Ex-

perimental study of small vesicles generation at a low Reynolds

number in micro-devices was performed firstly ( Nisisako et al.,

2002; Thorsen et al., 2001 ). Systematic experimental investiga-

tion clarifies the mechanism of droplet breakup and formation

process in three regimes (squeezing, dripping and jetting) in T-

junction geometries ( Garstecki et al., 2006 ). By experimental ob-

servations the capillary number, flow rate ratio and geometry are

concluded as the major factors from squeezing to dripping regimes

∗ Corresponding author.

E-mail address: wancheng@mst.edu (C. Wang).

n

a

http://dx.doi.org/10.1016/j.ijmultiphaseflow.2017.04.008

0301-9322/© 2017 Elsevier Ltd. All rights reserved.

Christopher et al., 2008; Xu et al., 2008 ). Besides the T-shape

esign, the cross-junction, co-flowing and flow focusing devices

re also common configurations for producing uniform emulsions

Christopher and Anna, 2007 ). The scaling and mechanism of emul-

ification in cross-junction configuration are discussed to control

he monodisperse emulsification process ( Tan et al., 2008 ). A flow

ocusing design with unbounded downstream geometry is reported

o investigate the droplet size in W/O emulsions by varying flow

ates ( Anna et al., 2003 ). The role of geometry and fluid proper-

ies in planar flow focusing devices is studied systematically using

caling methods to optimize the control of emulsification process

Lee et al., 2009 ). The emulsions in a confined flow focusing de-

ices are reported and investigated for identifying the mechanism

f breakup process ( Garstecki et al., 2005 ) and the effects of wet-

ing energy on boundaries ( Li et al., 2007 ). Adding surfactant in

uids alters the surface tension and wetting conditions, and pro-

uces both monodisperse O/W (oil in water) or W/O (water in oil)

mulsions ( Xu et al., 2006a; 2006b ).

On the other hand, numerical simulation is a powerful means

ot only for understanding the complex physical processes, but

lso for predicting and guiding the practical designs of experi-

F. Bai et al. / International Journal of Multiphase Flow 93 (2017) 130–141 131

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ents ( Wörner, 2012 ). Numerical studies of multiphase flows in

icrofluidic devices are introduced both in discrete and continu-

us approaches ( Wörner, 2012 ). As a typical discrete model, the

attice Boltzmann (LB) method is suitable and efficient for mod-

ling micro-scale binary viscous fluids by simulating the interac-

ions among a number of lattice particles. The first three dimen-

ional LB model of multiphase flow was formulated twenty years

go ( Martys and Chen, 1996 ). The LB benchmarks using the free

nergy theory and wetting boundary conditions were implemented

o study droplet formation in different configurations ( Gong et al.,

010; Graaf et al., 2006; Gupta and Kumar, 2010; Liu and Zhang,

009 ), to describe the flow regimes and the transitions in cross-

unction structure ( Liu and Zhang, 2011 ) and to investigate the dy-

amic characteristics of water droplet on a hydrophobic gas dif-

usion layer surface ( Hao and Cheng, 2009 ). In contrast to dis-

rete methods, continuous methods are usually utilized to cap-

ure the dynamic behaviors of the interface between multiphase

uids, including the volume of fluid method (VOF), the level set

ethod (L-S) and the phase field method (P-F). Numerical simu-

ation of droplet formation in a T-junction configuration using the

OF method is validated by comparing the experimental visual re-

ults ( Sivasamy et al., 2011 ). The effects of uniform magnetic field

n emulsification process in a flow focusing micro-channel are

tudied numerically using a refined L-S method ( Liu et al., 2011 ).

The idea of P-F approach can be dated back to the ancient

tudies a century ago ( Rayleigh, 1892; van der Waals, 1893 ). The

ree moving interface between multiple material components is

onsidered as a continuous, but steep change of some physical

aterial properties, for instances, density or viscosity, thus a

ontinuous phase field variable is introduced and the interface

s represented by a thin but smooth transition layer. One major

dvantage of this method is that the free interface can be auto-

atically tracked without imposing any mathematical conditions

e.g. Young–Laplace junction condition ( Edwards et al., 1991; Kro-

ov and Rusanov, 1999; Probstein, 1994 ) in other sharp interface

odel) on the moving interface, thus it provides an easy treatment

f topological variations at the interface.

The commonly used phase field model for two phase fluids sys-

em couples an advection-diffusion equation, which represents the

volution of the phase variable, with the Navier–Stokes equations

y introducing an extra interfacial stress term induced from the

hemical potential. The evolution equation for the phase variable

s derived from the energetic variational of the action functional of

he Landau-Ginzburg free energy which uses the flux of chemical

otential to demonstrate the non-uniform phase system of binary

uids ( Cahn, 1959; Cahn and Hilliard, 1958 ). One great advantage

f the phase-field model is that it leads to well-posed nonlinearly

oupled systems that satisfies thermodynamically consistent en-

rgy dissipation laws due to the energy-based variational formal-

sm. Recently, it has been successfully employed in many fields of

cience and engineering and become one of the major modeling

nd computational tools for the study of microfluidic interfacial

henomena (cf. Boyer et al., 2010; Cahn and Hilliard, 1958; Chen

nd Wang, 1996; Chen et al., 2015; Du et al., 20 04; Feng, 20 06;

eng et al., 2007; 2016; Feng and Wise, 2012; Han and Wang, 2016;

acqmin, 1999; Kim, 2012; Kim and Lowengrub, 2005; Liu and

hen, 2003; Lowengrub et al., 2009; Miehe et al., 2010; Nochetto

t al., 2014; Shen and Yang, 2009; 2010; 2014; 2015; Spatschek

t al., 2010; Wang and Wise, 2011; Wise, 2010; Yang et al., 2013b;

ue et al., 2004; Zhao et al., 2016a; 2016b; 2016c ).

When the fluid-fluid interface touches a solid wall, it leads to

he well known mathematical singularity if no-slip condition is

sed ( Cox, 1986 ). The P-F model can overcome this singularity

y introducing a diffuse-interface and demonstrate the fluid-wall

nteractions by using chemical energy diffusion instead of shear

tress ( Jacqmin, 20 0 0 ). Hydrodynamic slip boundary condition is

iscussed and several theories are outlined in Blake (2006) . The

oving contact line problem is studied using phase field equa-

ions by considering sharp interface limit ( Yue et al., 2010 ) and

nergy relaxation between fluid and solid wall surface ( Yue and

eng, 2011 ). The generalized Navier boundary condition (GNBC)

or dynamic moving contact line is introduced mathematically and

s verified by comparing with molecular dynamics (MD) simula-

ions ( Qian, 2006; Qian et al., 20 03; 20 04; 20 06 ). It is also ap-

lied to study the moving contact line on heterogeneous surfaces

Qian et al., 2005; Wang et al., 2008 ). A similar general form

s utilized to study dynamic moving wetting line ( Carlson et al.,

009 ). Another type of slip boundary condition for moving con-

act line is formulated by analyzing the balance of physical fric-

ion forces on triple-phase contact line and is also validated by

D simulations ( Ren and E., 2007; 2011; Ren et al., 2010 ). Nu-

erical schemes are updated to improve the stability of moving

ontact line models ( Gao and Wang, 2012; 2014; Shen et al., 2015 ).

he phase-field method has been employed to study various mul-

iphase problems, including droplet impact on homogeneous sur-

aces ( Khatavkar et al., 20 07a; 20 07b ), droplet spreading on par-

ially wetting substrate ( Gao and Feng, 2011 ), impingement and

preading process of a micro-droplet ( Lim and Lam, 2014 ), electro-

ydrodynamic multiphase flow ( Lin et al., 2012; Yang et al., 2013a ),

s well as droplet formation process in a T-junction configuration

De Menech, 2006; De Menech et al., 2008 ).

Most of the existing works on phase field models are based on

heoretical and mathematical study, which is important and can

e further validated by experimental results. In this paper, we use

he P-F model with the finite element method to study the droplet

ormation process in a 3D microfluidic flow focusing device and

ompare the results with the corresponding lab experiment results.

ur aim is to clarify the physical process of droplet formation, and

dentify the role of energy diffusion and the relationship between

ime relaxation parameter and diffuse interface based on physical

nd mathematical scaling. Through systematic numerical studies,

e investigate the effect and physical meaning of several charac-

eristic model terms and parameters on the droplet and formation

rocess, and derive a criterion for the parameter selection in order

o consistently and correctly capture the physical phenomenon.

The rest of the paper is organized as follows. In §2 , we describe

he model system and illustrate the phase field equations coupled

ith the generalized Navier boundary condition (GNBC). Then we

escribe the experimental details, including the configuration of

icrodevice, and fluids, in §3 . The implementation and verification

f the P-F model will be presented, and important findings based

n the systematic numerical investigation in comparison with the

xperimental data will be emphasized in §4 .

. Mathematical formulation and description of the model

ystem

.1. Description of the model system

The three dimensional geometry of a flow focusing configura-

ion in micro-device is shown in Fig. 1 . The width of this micro-

hannel system is defined as L which is also considered as the

haracteristic length of this system. There are one main inlet and

wo side inlets. Each of inlets has a square cross-section with the

idth L . In the downstream there is only one outlet and its height

s 1.6 L . Two immiscible fluids are injected into the microfluidic sys-

em. The disperse phase is injected into the channel through the

ain inlet, and the continuous phase is injected through the two

ide inlets. The flows at the inlets are assumed as fully developed

aminar flows with the flow rates of disperse phase Q d and con-

inuous phase Q c . We assume the continuous phase has a perfect

etting condition on the channel wall and the disperse phase has

132 F. Bai et al. / International Journal of Multiphase Flow 93 (2017) 130–141

Fig. 1. 3D schematic of the flow focusing microfluidic channel. The characteristic

length is use as L to show the geometric size of the model. There are one main

inlet for the disperse phase ( Q d ) and two side inlets for the continuous phase ( Q c ).

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non-wetting condition on the channel wall. These boundary con-

ditions match the wetting properties used in our experiments and

will be discussed in §2.3 . The relevant physical properties of the

two immiscible fluids are: viscosities μc and μd , densities ρc and

ρd and surface tension σ . The droplet is formed by the squeez-

ing force on the disperse phase from the continuous phase. The

normalized droplet volume V ∗ = V d /L 3 is a physical dimensionless

parameter and depends on the following dimensionless numbers:

the Reynolds number Re = ρc Q c /Lμc , and the Capillary number

a = μc Q c /L 2 σ . The flow rate ratio between the two binary flu-

ids Q = Q d /Q c is also an important factor in this droplet formation

regime.

In our study, we use two specific immiscible fluids with given

physical properties to numerically investigate the emulsification

process. These two immiscible fluids will also be used in our ex-

periments. Our goal is not to discuss the flow transition from shear

stress dominated regime to surface tension dominated regime, but

to investigate the mechanism of droplet formation in the squeez-

ing regime by using phase field equations. In our model, the value

of Capillary number is small enough ( Ca ≈ 0.016) so that the flow

regime can be confirmed as surface tension dominated. The Cap-

illary number can be considered as a constant since the flow rate

of continuous phase is fixed. Only the flow rate of disperse phase

is varied in order to control the size of droplet. Thus the effect of

capillary number variation on the droplet size can be neglected.

2.2. Mathematical formulation of phase field equations

In the phase field models ( Kim, 2012 ), the physical sharp limit

interface is replaced by a diffuse-interface which has a non-trivial

thickness in which the physical properties and the interfacial ten-

sion force are smoothly and continuously distributed. The govern-

ing equation we choose for the phase variable is known as the

Cahn–Hilliard equation which can be seen as a kind of advection-

convection equation ( Jacqmin, 1999 ):

∂φ

∂t + u · ∇ φ = M∇

2 G,

G = −K∇

2 φ + f ′ (φ) ,

(1)

where φ is the phase variable and its value could be varying on

[ −1 , 1] . The phase variable means the pure continuous phase when

φ = 1 and the pure disperse phase when φ = −1 . The nonlinear

term in the chemical potential is f ′ (φ) = uφ3 − rφ. M is called the

phenomenological mobility parameter. r, u are the computational

parameters obtained from MD simulations and have a relationship

of r = u =

K ε 2

, where K , ε are the mixing energy density and inter-

facial thickness, respectively ( Qian et al., 2003 ). G is the chemical

potential representing the change rate of free energy at the inter-

ace and it is calculated as functional derivative of the free energy

= δF /δφ, where F ( φ) is the free energy. The first term in G rep-

esents the interfacial energy, and the second term represents the

ulk free energy. Then the free energy can be expressed as:

(φ) =

∫

[ 1

2

K(∇φ) 2 + f (φ) ]

d r . (2)

The mixing energy density K is related to the surface tension

oefficient σ and the interfacial thickness ε as follows,

=

2

√

2

3

K

ε . (3)

his formula clearly shows that the surface tension force is pro-

ortional to the energy density at the interface between two im-

iscible fluids. It also indicates the connection between the energy

ensity and the sharp interface limit. As the interfacial thickness εhrinks to zero, the energy density K should decrease to zero in

rder to keep the surface tension as a constant.

Then the Cahn–Hilliard equation transforms into:

∂φ

∂t + u · ∇ φ =

MK

ε 2 ∇

2 ,

= −ε 2 ∇

2 φ + φ(φ2 − 1

),

(4)

here is an auxiliary variable to separate the fourth order equa-

ion into two second order equations. The mobility parameter M

etermines the relaxation time of interface and the timescale of

iffusion in Cahn–Hilliard equation ( Yue et al., 2004 ). This param-

ter is proportional to square of the interfacial thickness M = χε 2 ,here χ is the mobility tuning parameter ( Lim and Lam, 2014 ).

The fluid flow is modeled using the Navier–Stokes equation and

he equation of continuity: [∂ u

∂t + ( u · ∇ ) u

]= −∇p + ∇ · μ

(∇ u + ∇ u

T )

+ G ∇φ + f ext ,

(5)

· u = 0 , (6)

here u is the velocity vector, p is the pressure and f ext is the ex-

ernal body force exerting on the fluids such as gravity or buoy-

ncy. In our study the two-phase flows are incompressible in the

icrofluidic channel and there is no other external force field ap-

lied in this system, thus the external body force can be neglected

afely. The fluid properties of the two phases are different and vary

ith the phase variable φ especially at the sharp limit interface.

he density in disperse phase is ρd corresponding to φ = −1 and

he density in continuous phase is ρc corresponding to φ = 1 . The

iscosity in disperse phase is μd corresponding to φ = −1 and the

iscosity in continuous phase is μc corresponding to φ = 1 . There-

ore the density we used in our computation can be expressed as

=

1 2 [(1 − φ) · ρd + (1 + φ) · ρc ] and the viscosity in our compu-

ation is expressed as μ =

1 2 [(1 − φ) · μd + (1 + φ) · μc ] . The term

· μ(∇ u + ∇ u

T )

represents the effect of viscosity; and G ∇φ is a

ody force transformed by using the divergence theorem to repre-

ent the surface tension force. Other forms of representations have

een used to represent the surface tension force using the free

nergy theory ( Kim, 2012 ). However, in our study the most com-

only used expression G ∇φ is selected to calculate the interfacial

ension force. The chemical potential G is expressed by the surface

ension coefficient σ and the interfacial thickness ε:

=

3 εσ

2

√

2

(

−∇

2 φ +

φ(φ2 − 1

)ε 2

)

. (7)

It can be seen that the chemical potential of the fluids varies

irectly with the interfacial thickness. The interfacial thickness is

F. Bai et al. / International Journal of Multiphase Flow 93 (2017) 130–141 133

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sually determined to satisfy the sharp interface limit and energy

quilibrium ( Jacqmin, 1999; 20 0 0 ). Therefore the surface tension

ody force does not have a specific constant value and this varia-

ion should be balanced by the phenomenological mobility in the

ahn–Hilliard equation as a timescale relaxation parameter to con-

rol the interface movement according to the transport of binary

uids and the interfacial diffusion. In order to fully understand

he theory, these governing equations are non-dimensionalized to

dentify four dimensionless numbers. The width of micro-channel

ystem L is chosen as the characteristic length. The other non-

imensional variables are u

∗ = u L 2 /Q c , t ∗ = tQ c /L 3 , p ∗ = pL 4 /ρc Q

2 c ,

∗ = ρ/ρc , μ∗ = μ/μc and ∇

∗ = L ∇ . Then the non-dimensional

quations are

∂φ

∂t ∗+ u

∗ · ∇

∗φ =

3

2

√

2

1

P e ∇

∗2 [−Cn

2 ∇

∗2 φ + φ(φ2 − 1

)], (8)

∗ · u

∗ = 0 , (9)

ρ∗[∂ u

∗

∂t ∗+ u

∗ · ∇

∗u

∗]

= −∇

∗ p ∗ +

1

Re ∇

∗ · μ∗(∇

∗u

∗ + ∇

∗u

∗T )

+

3

2

√

2

1

C nC aRe

[−Cn

2 ∇

∗2 φ + φ(φ2 − 1

)]∇

∗φ. (10)

The four dimensionless numbers in the governing equations

re:

e =

ρc Q c

Lμc , Cn =

ε

L , Ca =

μc Q c

L 2 σ, P e =

εQ c

MLσ, (11)

here Q c and μc are the flow rate and viscosity of the continu-

us phase. The Reynolds number Re describes the ratio of inertial

orce to viscous force. The Cahn number Cn is defined as the ratio

f interfacial thickness between the binary fluids to the character-

stic length L , thus it is also known as the dimensionless interfa-

ial thickness. The Capillary number Ca shows the ratio of viscous

orce to surface tension force. The Péclet number Pe represents the

atio of advection to diffusion in the two-phase flows. And the Pé-

let number Pe in our model differs from the usual Peclet num-

er for mass transfer since it is specifically for the phase field

odel. Because the physical properties and geometric parameters

re chosen to be constant, Ca and Re in our model have specific

alues Re ≈ 0.01, Ca ≈ 0.016. The Péclet number and Cahn number

re both functions of interfacial thickness ε which is used to ap-

roximate the physical sharp limit interface. The phenomenologi-

al mobility M in Pe is an artificial parameter introduced into the

ahn–Hilliard equation by using interfacial energy theory ( Cahn

nd Hilliard, 1958 ).

.3. Discussion of boundary conditions

To save the computational time, we only simulate one quar-

er of the whole domain since both of the micro-channel geom-

try and the flow pattern are symmetric. Therefore the symmetric

oundary conditions are used for all of the symmetric surfaces in

he system:

· n = 0 , δn − ( δn · n ) n = 0 , (12)

here δ is the stress tensor matrix and n is the unit normal vector.

There are three inlets in the micro-channel: the main inlet is for

he disperse phase and the two side inlets are for the continuous

hase. Fluid flow in the model belongs to laminar flow ( Re < 1),

hus the boundary conditions for all the three inlets are defined as

he fully developed inlet flow rate. The mean flow rates are Q d for

he disperse phase and 0.5 Q c for the continuous phase. All the inlet

elocity distributions of the fully developed laminar flow satisfy

he parabolic profile and the maximum velocities in each inlet are

ocated on the center point of the 2D inlet cross section domains:

t · u = 0 , (13)

ent ∇ t · [ −p · I + δt ] = −p ent n , (14)

here L ent is the length of entrance before the inlet and ∇ t , δt rep-

esent the tangential operators. Non-viscous, pressure constraint

utflow is applied as the boundary condition of the outlet, i.e., the

ressure on the outlet is fixed as zero

n = 0 , p = 0 . (15)

ll of the above boundary conditions can be found and performed

n the phase-field module using Comsol-Multiphysics.

However, the boundary conditions related to the moving con-

act line or fluid-wall interactions, have to be implemented based

n theory of free energy ( Qian et al., 2006 ). The total free energy

f the fluid-fluid-wall triple contact system can be described as

=

∫

f m

( φ, ∇φ) d +

∫ ∂

f w

( φ) d S, (16)

here is the domain of two-phase fluids and ∂ is the area

f solid wall surface. f m

is the fluid-fluid mixing energy and this

xpression can be written as

f m

( φ, ∇φ) =

K

2

| ∇φ | 2 +

K

4 ε 2

(φ2 − 1

)2 . (17)

The wall relaxation energy f w

is a function of phase field vari-

ble. The energy equilibrium of static contact angle can be de-

cribed by no-slip condition at the wall and the corresponding

oundary conditions to demonstrate the stationary energy balance

t solid wall are called the natural boundary conditions ( Yue et al.,

004 ). However, the dynamic behavior of the moving contact line

t solid wall should be stated by the minimum energy dissipation

rinciple. To demonstrate the energy dissipation of moving contact

ine on the solid surface, the generalized Navier boundary condi-

ion (GNBC) is introduced ( Qian et al., 2003 ). The GNBC is formu-

ated by stating the Navier slip which is proportional to the to-

al tangential stress on the solid surface. The total tangential stress

onsists of two parts: the viscous surface stress and the uncom-

ensated Young stress. The wall relaxation energy f w

is formulated

s Jacqmin (20 0 0)

f w

( φ) = −1

2

σ cos θs sin

(π

2

φ). (18)

ince no slip boundary condition is not adequate to describe the

hree phase contact line at the wall surface ( Qian et al., 2004;

005 ), the partial slip ( Zhu and Granick, 2002 ) at wall surface in

angential direction is used in GNBC as the Navier slip and the

quilibrium contact angle θ s of the binary fluids can be deter-

ined by Young–Dupré’s equation which demonstrates the balance

f surface energy near the triple phase contact line:

os (θs ) =

σda cos (θds ) − σca cos (θcs )

σdc

, (19)

here σ da , σ ca and σ dc are the disperse phase-air, continuous

hase-air and disperse phase-continuous phase surface tension re-

pectively. And θds and θ cs are the static contact angle between

isperse phase and air and the angle between continuous phase

nd air on solid surface, respectively. Then the GNBC can be writ-

en as a combination of the viscous component and the uncom-

ensated stress component. Since there is no mass flux across wall

oundary according to the Eq. (12) and then the diffusion across

all is also equal to zero. Therefore the expression of GNBC with

he impermeability conditions u · n = 0 and ∂ G/∂ n = 0 is

( φ) u τ = −δ · n + L w

( φ) ∇φ, (20)

134 F. Bai et al. / International Journal of Multiphase Flow 93 (2017) 130–141

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w

4

4

l

b

o

a

t

s

f

i

i

c

C

w

c

S

r

p

t

d

c

s

t

s

t

t

m

s

C

t

r

f

t

i

c

a

s

s

c

t

d

i

s

t

where β( φ) is the slip parameter and this value depends on the

contact characteristics of wall surface. u τ is the small slip veloc-

ity on the surface in tangential direction. δ is the viscous stress

tensor. The first term in the right side of this equation is the vis-

cous part and the second term is the uncompensated Young stress

with L w

( φ) = K∂ n φ + ∂ f w

( φ) /∂ φ as the surface chemical poten-

tial. Here the parameter K is the mixing energy density which is

the same as K used in the free energy. Then we can obtain the

transport boundary condition with the wall relaxation dynamics of

φ

∂φ

∂t + u · ∇φ = −�L w

( φ) , (21)

where � is a positive phenomenological parameter which repre-

sents the wall relaxation in the transport process ( Yue and Feng,

2011 ). Then the wall boundary conditions are coupled with the

GNBC and the wall transport equation using Navier slip and wall

relaxation energy.

3. Experimental setup and method

The schematic of the microfluidic channel is depicted in Fig. 1 .

The geometry used in experiments is the same as that used in the

computational work. The characteristic length of this geometry is L ,

which is designed as 70 μm. All of the main inlet and the two side

inlets have the cross-square section with the characteristic length

L and the height of outlet is about 1.6 L . The width of this micro-

system is the characteristic length L in the z direction. Due to the

limitations of the fabrication technique ( Zhang et al., 2015 ), there

are some fillets in the corners of the micro-device. The radius of

fillets at the cross intersection of inlets is measured to be R =17.5 μm, and the radius of fillets at the expansion of downstream

is measured to be r = 10.5 μm, as shown in Fig. 1 .

This microfluidic device was fabricated in PDMS using a soft

lithography technique, and the Master molds were manufactured

in a dry film photoresist (MM540, 35 μm thick, DuPont) by litho-

graphic patterning. The details of fabrication process can be found

in earlier experimental studies ( Zhang et al., 2015; Zhou and Wang,

2016 ). To study the droplet formation process, the two fluids, min-

eral oil (Sigma M5904, Sigma-Aldrich) with 2 % surfactant Span 80

(Sigma-Aldrich) and distilled water, were delivered into the inlets

separately using two precision syringe pumps (NE-300, New Era

and KDS 200, KDS Scientific). Since the material of PDMS is hy-

drophobic, mineral oil was chosen as the continuous phase and

distilled water was the disperse phase. The physical properties of

the two immiscible fluids were measured experimentally. The min-

eral oil has a density of ρc = 840 kg/m

3 and a viscosity of μc =23.8 mPa ·s. The distilled water has a density of ρd = 10 0 0 kg/m

3

and a viscosity of μd = 1 mPa ·s. The surface tension coefficient be-

tween the binary fluids is about σ= 5 mN/m.

The equilibrium contact angle θ s of the two immiscible flu-

ids we used in our experiments can be calculated using Young–

Dupré’s equation ( Li et al., 2007 ). The difference of the surface

energy between the two immiscible fluids on PDMS surface is

σ ·cos ( θ s ). Based on the calculation, the value of cos ( θ s ) is −6 . 2

and it is less than −1 , which means that the distilled water is the

de-wetting phase and does not touch the PDMS wall surface, and

the oil mineral is a completely wetting phase. This energy differ-

ence at the PDMS surface could be expressed by the generalized

Navier boundary condition (GNBC) in numerical simulations.

In the droplet formation experiments, the flow rate of contin-

uous phase (mineral oil) was fixed at Q c = 1 μL/min while the

flow rate of disperse phase (distilled water) was varied from Q d =0.1 μL/min to 1 μL/min. To maintain stability of the two phase

flow, gas tight glass syringes were used to reduce the effect of mo-

tor’s step motion. Droplet formation process in different flow ratio

as monitored with an inverted microscope (IX73, Olympus) and

high-speed camera (Phantom Miro M310, Vision Research). We

sed ImageJ to analyze the experimental results, and extracted the

mages to compare them with the numerical results.

In the experiment, the droplet velocity was analyzed and deter-

ined by using software ImageJ and MATLAB. First, the area center

osition ( x c , y c ) of a droplet at different image frames (i.e., time)

ere extracted by using ImageJ. The time-position data were then

inearly fitted with MATLAB built-in curve-fitting function. The

lope of the best fitted curve gave the corresponding droplet ve-

ocity. The experiment images were taken after five minutes when

he flow rates were changed, so that the flow conditions were sta-

le. And for each experiment condition, a total of 70–80 droplets

ere analyzed to determine the average droplet velocity.

. Results and discussions

.1. Numerical implementation of the phase field model

Ideally as the thickness of the diffuse interface ( ε) decreases,

ess energy dissipation will occur, and better approximation will

e achieved in phase field models. However, the proper selection

f thickness of diffuse interface needs to take computational cost

nd capability into consideration to achieve accurate solution of

he phase variable φ. The principal criterion is that numerical re-

ults of φ should be smooth enough to express the diffuse inter-

ace and the simulations can lead to numerical convergence both

n the bulk of fluids and on the wall surface. An important criterion

s proposed for the sharp interface limit when the diffuse-interface

ontacts a wall surface ( Yue et al., 2010 )

n c = 4 S, S =

√

Mμe

L , μe =

√

μc μd , (22)

here μe is the equivalent viscosity for the binary fluids. Cn c is the

ritical Cahn number for the convergence of sharp interface limit,

represents the bulk diffusion of the two-phase fluids and

√

Mμe

eflects the diffusion length at the contact line. This criterion is

roposed according to the stability of numerical convergence using

he Galerkin finite element method. Thus Cn varies with the bulk

iffusion for a sharp-interface limit and this sharp-interface limit

an be approached by reducing Cn while keeping S constant.

On the other hand, the value of interfacial thickness must be

elected based on the mesh size. For 3D cubic mesh, the interfacial

hickness should be equal to or larger than half of the mesh size

o that the phase function φ is described smoothly. In our model,

he cubic mesh is utilized and the mesh size is 0.05 L . Therefore

he thickness ε should be equal or greater than 0.025 L . In our

odel, four different thicknesses 0.025 L , 0.03 L , 0.035 L , 0.04 L are

elected for numerical analysis and the corresponding values are

n = 0 . 025 , Cn = 0 . 03 , Cn = 0 . 035 , Cn = 0 . 04 . According to the cri-

erion of numerical convergence for the sharp-interface limit, this

elationship ( Eq. (22) ) is determined by two parameters: the inter-

acial thickness ( ε) and tuning mobility ( χ ). Thus for ε = 0 . 025 L

he value of minimum mobility is determined by the correspond-

ng tuning parameter χ = 12 . 8 m ·s/kg. It should be noted that this

riterion only determines the lower limit of mobility. The reason-

ble value of χ in numerical simulations depends on real physical

ystems, and can be found by comparison with experimental re-

ults. Once determined, the same value will be valid for other flow

onditions provided that the same fluids are used in channels of

he same surface properties.

The generalized Navier boundary condition (GNBC) is used to

emonstrate the dynamic interaction between wall surface and flu-

ds. In GNBC the Navier slip term is equal to the sum of viscous

tress term and the uncompensated Young stress. The Navier slip

erm β( φ) u τ is the product of the Navier slip velocity u τ and the

F. Bai et al. / International Journal of Multiphase Flow 93 (2017) 130–141 135

Fig. 2. The 2D cross section schematic of droplets formation for numerical results in four different values of mobility ( M = χε 2 ). The 2D cross section is in the middle of the

channel, which will be clarified in Fig. 3 . The thickness of interface is fixed: ε = 0 . 025 L . The tuning mobility parameter are χ = 12 . 8(a ) , 204 . 8(b) , 820(c) , 1280(d) (unit: m ·s / kg )

respectively. The flow rate of the disperse phase (blue) and the continuous phase (red) are 0.5μL/min and 1μL/min respectively. The length of arrows indicates the velocity

of fluids. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

b

o

u

v

t

a

t

(

l

v

r

p

l

r

t

t

i

v

s

s

a

S

t

q

c

b

t

t

e

l

4

b

D

f

b

o

e

e

o

(

t

a

b

t

fi

e

t

fi

s

e

i

a

h

a

i

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M

o

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u

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w

t

t

oundary friction coefficient β( φ). This friction coefficient depends

n the fluid-wall interaction at the contact line and has the same

nit as viscosity ( Ren et al., 2010 ). The value of this coefficient

aries from 0.05 to 4 for various strength of fluid-wall interac-

ion. In our simulation we estimate the value of friction coefficient

s β = 2 in the LJ unit(Lennard-Jones potential unit) according to

he expression of contact line velocity acquired by MD simulation

Blake, 2006; Qian et al., 2004; Ren and E., 2007 ). The phenomeno-

ogical parameter � in the boundary transport equation of phase

ariable φ indicates the energy dissipation and represents the cor-

esponding relaxation time on wall surface. This wall relaxation

arameter can be determined by dimensional analysis using the re-

ation of [ M] = [�][ Length ] 3 ( Qian et al., 2006 ), where the [ Length ]

epresents the scaled length in the phase field system and links

he two parameters M and �. The order of this length is propor-

ional to the interfacial thickness. In our computation, this length

s chosen to be the same as the interfacial thickness ε since the

ariation of φ is located in the diffuse interface.

To obtain accurate numerical results, the mesh size should be

mall enough and the interfacial thickness is selected to satisfy the

harp-interface limit criterion. Considering the computational us-

ge in the 3D phase field model the mesh size is chosen as 0.05 L .

ince we use the symmetry of micro-channel to compute one quar-

er of the geometry, there are 10 × 10 cubic mesh cells in the one

uarter square section of the main inlet and 20 × 10 cubic mesh

ells in one half section of each side inlet. The total number of cu-

ic mesh elements in one quarter geometry is over 16,0 0 0 and all

he fillets are mapped by non-cubic mesh. In this study we use

he quadratic finite elements for velocity field and linear finite el-

ments for pressure. The time step size is 0.0 0 01 s and the simu-

ation stops after 50 0 0 steps.

.2. The phenomenological mobility in Cahn–Hilliard equation

Similar phase field models in T-shape micro-channels have

een reported using finite difference method ( De Menech, 2006;

e Menech et al., 2008 ). These studies focus on the transition

rom squeezing to dripping regime by varying the capillary num-

er Ca . A recent study of rising bubbles suggested the importance

f the mobility parameter on affecting the bubble rise velocity( Cai

t al., 2016 ). However, the role of the diffuse term in Cahn–Hilliard

quation remains unclear. Here, we keep the flow rate of continu-

us phase constant, which means the capillary number is constant

Ca = 0 . 016 ), and investigate the effect of mobility in the diffusion

erm, M ∇

2 G . The Péclet number Pe =

εQ c MLσ represents the ratio of

dvection to diffusion. According to the expression of Pe , the mo-

ility and the interfacial thickness are important variables to reflect

his ratio. First the effect of mobility will be tested in the phase

eld model.

Fig. 2 shows the comparison of emulsification with four differ-

nt values of mobility M ( ε = 0 . 025 L ), and all of the results satisfy

he criterion of numerical convergence ( χ ≥ 12.8 m ·s/kg). In this

gure we use 10 × 10 cubic mesh cells in one quarter of square

ection in the main inlet and 20 × 10 cubic mesh cells in half of

ach side inlet as the original mesh case to be mapped for the

nitial numerical tests. The color legend bars of phase variable φre shown in Fig. 2 . The undershoot of φ is found in the results,

owever, the lower bound is relatively small (the lower bound is

round φ = −1 . 044 ). We can observe that as the value of mobil-

ty increases the size of droplet in micro-channel becomes larger.

his means that as the relaxation time of interface, the mobility

plays a very critical role in the two-phase flow: a higher value

f mobility may cause the increase of relaxation time and the dif-

usion will be stronger. This value of M also reflects the effect of

nterfacial energy on the fluid flow. Thus the mobility should be

elected such that it is neither too low for satisfying the numerical

onvergence and nor not too large for ensuring the diffusion not to

verly damp the flow ( Yue et al., 2004 ). There is no clear criterion

or the selection of mobility to justify the correctness of simula-

ions. However, we can use the experimental results to determine

his value.

Fig. 3 is the comparison between the numerical and experimen-

al results of emulsification with the flow rates 0.5 μL/min for dis-

erse phase and 1 μL/min for continuous phase. In Fig. 3 we try

o use 20 × 20 cubic mesh cells in square section of the main in-

et and 40 × 20 in half section of each side inlet. The size of the

roplet in experiments matches the numerical results well. The in-

erfacial thickness in this case is ε = 0 . 025 L . The radius of droplet

s about R e = 51.5 μm in the experiments, and R n = 51.2 μm

n numerical simulations. Thus we identify this value of mobil-

ty ( M c = χε 2 , χ c ≈ 820 m ·s/kg) as the characteristic value that

an correctly reflect the real physical emulsification process in our

odel using distilled water and mineral oil as disperse and contin-

ous phase respectively. When M < M c , the droplet size in numer-

cal simulations will be smaller than the experimental result; and

hen M > M c , the droplet size will be larger. It should be noted

hat this value of mobility is an approximation for the characteris-

ic mobility, and we find this value by matching experimental re-

136 F. Bai et al. / International Journal of Multiphase Flow 93 (2017) 130–141

Fig. 3. The comparison between numerical and experimental results in the charac-

teristic mobility ( M c = χε 2 , χ = 820 m ·s/kg). (a) The experimental result, (b) the

numerical result in 2D cross section and (c) the 3D numerical result. The 2D cross

section is in the middle of the channel, which can be observed in (c). T 0 in (b) rep-

resents the experimental time span before the current process in this figure. In this

figure we use refined mesh cases with 20 × 20 cubic mesh cells in square section

of the main inlet and 40 × 20 cubic mesh cells in each side inlet. The thickness of

interface is fixed: ε = 0 . 025 L . The flow rate of the disperse phase and the continuous

phase are 0.5μL/min and 1μL/min respectively.

Fig. 4. The process of droplet formation in the characteristic mobility ( M c = χε 2 ,

χ = 820 m ·s/kg). (a) The numerical results; (b) the experimental results in 2D cross

section. The 2D cross section is in the middle of the channel. The thickness of in-

terface is fixed: ε = 0 . 025 L . T 0 in (a) represents the experimental time span before

the current process in this figure. The flow rate of the disperse phase and the con-

tinuous phase are 0.5μL/min and 1μL/min respectively.

t

d

4

a

b

p

G

i

o

t

t

a

s

f

t

C

t

f

o

c

s

m

a

χ

r

t

b

sults. The corresponding Péclet number satisfying the characteristic

mobility with the interfacial thickness ε = 0 . 025 L is Pe c ≈ 33.25. In

this case the droplet diameter is 2 R d = 103 μm and is larger than

the width of micro-channel system L = 70 μm. However, since the

disperse phase is completely de-wetting, this phase can not con-

tact the wall surface. Therefore the droplet is not a sphere but is

squeezed in the width direction so that the interface of droplet

does not touch the wall surface and is parallel to the wall (See

Fig. 3 (c)).

The whole emulsification process is demonstrated in Fig. 4 . This

comparison shows excellent match between numerical and exper-

imental results, including the size of droplet and the movement of

droplet in the downstream. The value of mobility is experimentally

justified as M c = χε 2 , where χ = 820 m ·s/kg and ε = 0 . 025 L . The

whole duration, from the disperse phase entering into the throat

of micro-channel to the exiting of droplet near the computational

outlet is about 70 ms. It is found that the breakup location of dis-

perse phase is near the entrance of throat and the breakup process

experiences a very short time (less than 1 ms). The breakup pro-

cess indicates the dominant capillary effect over the shear stress

in squeezing regime. The velocity of droplet in the downstream is

constant, which is confirmed in both the experiments and numer-

ical simulations.

Fig. 5 shows the comparison of droplet volume in initial shape

and in downstream respectively. Four different values of mobil-

ity are used to investigate the effect of interfacial diffusion on

the droplet movement process in the micro-channel. In phase field

method, the interfacial diffusion(the Gibbs–Thomson effect) is in-

evitable, since the Cahn-Hilliard equation utilizes the diffusion

term in the right hand side of the Eq. (8) . The Péclet number Pe =εQ c MLσ represents the ratio of advection to diffusion, which means

that large mobility will cause strong interfacial diffusion. The

droplet volume nearly keeps constant in these four different values

of mobility. In Fig. 5 , the largest mobility is χ = 1280 (unit: m ·s / kg )

where ε = 0 . 025 L . The diffusion effect in our model is so small that

he distance between the leading edge and the rear edge of droplet

ecreases less than 1%.

.3. The effect of interfacial thickness on droplet formation process

nd breakup of the disperse phase

The interfacial thickness is an artificial diffuse layer introduced

y phase field method and plays a very important role in two-

hase flow models. In the Navier-Stokes equation, the force term

∇φ represents the surface tension force acting on the binary flu-

ds as a body force. This force is not fixed because it is a function

f the chemical potential G , which in turn is a function of the in-

erfacial thickness. Thus the surface tension body force varies with

he interfacial thickness. Besides, the phenomenological mobility is

lso a function of interfacial thickness M = χε 2 . Therefore, we can

ee that the interfacial thickness both affect the surface tension

orce and the relaxation time of interface. In our numerical simula-

ions, we define the thickness as four different values: Cn = 0 . 025 ,

n = 0 . 03 , Cn = 0 . 035 , Cn = 0 . 04 to investigate the effect of ε on

he emulsification process.

In Fig. 6 , it can be seen that the size of droplet varies with

our different interfacial thicknesses when the tuning parameter

f mobility is fixed as χ = 820 m ·s/kg. The size of droplets in-

reases with the value of ε and the shape of droplet changes from

phere to plug as the volume becomes larger and the droplet for-

ation process is delayed. This indicates that the original energy

nd force balance is broken and the value of mobility M = χε 2 ,= 820 m ·s/kg is not the characteristic mobility if ε � = 0.025 L . The

elaxation time changes with the mobility, thereby greatly affects

he droplet formation process. Thus we should find a relationship

etween the tuning mobility parameter χ and interfacial thickness

F. Bai et al. / International Journal of Multiphase Flow 93 (2017) 130–141 137

Fig. 5. The comparison between initial droplets and droplets in downstream with different values of mobility (where ε = 0 . 025 L ). The tuning mobility parameter are χ =

12.8(a), 204.8(b), 820(c), 1280(d) (unit:m ·s/kg) respectively. The left column shows the initial droplets and the right column shows the droplets in downstream. The 2D cross

section is in the middle of the channel. The flow rates of the disperse phase and the continuous phase are 0.5μL/min and 1μL/min respectively.

Fig. 6. The 2D cross section comparison of droplet using four different interfacial thicknesses. The 2D cross section is in the middle of the channel. The value of mobility is

M = χε 2 , where χ = 820 m ·s/kg. (a) ε = 0 . 025 L ; (b) ε = 0 . 03 L ; (c) ε = 0 . 035 L ; (d) ε = 0 . 04 L . The flow rates of the disperse phase and the continuous phase are 0.5 μL/min and

1 μL/min respectively.

εv

p

s

w

f

c

t

I

i

o

f

t

i

o

i

c

ε

r

t

r

t

r

g

s

t

to keep the relaxation time and the diffusion term as reasonable

alues so that the phase field model can match the experimental

henomena.

The relaxation time of interface should be determined as con-

tant to keep the energy stable and the Péclet number would vary

ith the control of mobility and interfacial thickness so that inter-

acial diffusion is calculated to be reasonable. The Péclet number

orresponding to the characteristic mobility is the exact ratio be-

ween advection and diffusion to reflect the real physical process.

n Fig. 7 we use four different thicknesses and the correspond-

ng reasonable tuning mobilities to examine the breakup process

f disperse phase and the dynamic shape of droplet. In all of the

our groups (a,b,c,d), the relation between tuning mobility and in-

erfacial thickness is fixed to keep the phenomenological mobil-

ty constant M = χε 2 = constant, which means the relaxation time

f interface is also kept to be constant. For ε = 0 . 025 L the tun-

ng mobility is χ = 820 m ·s/kg as the characteristic value, then the

orresponding characteristic values for ε = 0 . 03 L, ε = 0 . 035 L and

= 0 . 04 L are χ = 570 m ·s/kg, χ = 418 m ·s/kg and χ = 320 m ·s/kg

espectively. The Péclet number Pe =

εQ c MLσ in these four different

hickness are Pe = 33 . 25 , Pe = 39 . 9 , Pe = 46 . 55 and Pe = 53 . 2 cor-

espondingly. It means that the Péclet number is linearly propor-

ional to the interfacial thickness since the mobility and other pa-

ameters are kept to be constant. The numerical results of the four

roups show excellent agreements in the breakup process and the

ize of droplet.

The interfacial body force and the interfacial energy dominate

he breakup process. The time span of breakup process is very

138 F. Bai et al. / International Journal of Multiphase Flow 93 (2017) 130–141

Fig. 7. The comparison of breakup process using four different interfacial thicknesses. The 2D cross section is in the middle of the channel. The characteristic mobility is

M c = χε 2 : (a1-a2) ε = 0 . 025 L , χ = 820 m ·s/kg; (b1-b2) ε = 0 . 03 L, χ = 570 m ·s/kg; (c1-c2) ε = 0 . 035 L, χ = 418 m ·s/kg; (d1-d2) ε = 0 . 04 L, χ = 320 m ·s/kg. The flow rate of the

disperse phase and the continuous phase are 0.5 μL/min and 1 μL/min respectively.

4

fi

u

m

d

r

t

i

m

w

t

m

t

s

0

f

s

a

=

t

b

i

c

c

d

o

t

I

d

c

f

n

t

a

c

short (less than 1 ms) and there is an instability of the two-phase

flow which causes the minimization of interfacial energy after the

breakup. From the enlarged picture at the entrance of throat, we

can see that there are symmetric circular flows in the bulk with

bullet shape, which indicates the instability of interfacial energy

and the momentum equilibrium in horizontal direction.

4.4. The effect of flow rate on the size of droplet

The capillary number is calculated using the physical properties

and flow rate of the continuous phase Ca = μc Q c /L 2 σ . In our nu-

merical study, however, the value of Ca is fixed to 0.016 since the

physical properties and the flow rate of the continuous phase are

constant. Therefore we only investigate the effect of flow rate of

the disperse phase on the size of droplet.

Fig. 8 clearly shows the comparison of droplet volume between

experimental and numerical results. The experimental and numer-

ical results match very well. The size of droplet increases with

the flow rate of disperse phase. This phenomenon satisfies the

mass conservation law. However, the droplet volume is not lin-

early proportional to the flow rate of disperse phase because the

droplet formation process is also affected by the surface tension

force and the dynamic energy equilibrium (including both interfa-

cial energy ∫

1 2 ε|∇φ| 2 d and bulk free energy

∫

(φ2 −1) 2

4 ε d) in

the two phase fluids system. Experimental results of droplet radius

in the x − y plane are 40 μm, 42.2 μm and 51.5 μm for the flow

rate of disperse phase at 0.1 μL/min, 0.2 μL/min and 0.5 μL/min re-

spectively. The shape of droplet at 1 μL/min is not a sphere but a

plug and the 3D the numerical results show that the interface of

this dispersed plug does not touch the wall surface. The length of

this plug flow is about 128 μm in the x direction. It can be noticed

that the droplet volume obtained from the experiments fluctuates

because the unsteady flow due to the micro-pumps. The numer-

ical results of droplet radius are 41.5 μm, 45.2 μm and 51.2 μm

correspondingly and the length of plug is 140.2 μm. The discrep-

ancy between the experimental and numerical results is less than

10% and is caused by the relatively large instability induced by the

micro-pump when operating at a low flow rate.

.5. The velocity of droplet and period of droplet formation process

The velocity of droplet in the downstream channel after emulsi-

cation process is studied in both experiments and numerical sim-

lations. From the experiments we can see that the droplet or plug

oves along the micro-channel at a constant speed. The velocity of

roplet in P-F model is also acquired by analyzing the numerical

esults.

Different from the experimental measurements, the computa-

ional cases only include 0.5 s in time and only 2–4 droplets forms

n the whole process. Therefore we cannot use the time average

ethod to measure the velocity of around 100 droplets. Instead,

e use the position of leading edge of interface in disperse droplet

o measure the droplet velocity. The measurement starts from the

oment when the droplet is completely formed and ends when

he leading interface gets close to the outlet. The velocity we mea-

ured in numerical cases is also not stable for each frame (per

.0 02–0.0 05 s), thus we used the average velocity for the whole

rames to express the droplet velocity in the downstream.

Numerical results of the velocity are higher than the corre-

ponding experimental values when Q d = 0.1 μL/min, 0.2 μL/min

nd 0.5 μL/min, but lower than the experimental results when Q d

1 μL/min. However, the discrepancy is less than 15%. Besides

he instability caused by the micro-pumps, there are other possi-

le reasons which can contribute to this discrepancy. These factors

nclude the variation of physical properties caused by temperature

hange and the wall surface roughness which can lead to the in-

rease of resistance at wall surface and change the boundary con-

ition at solid wall. From Fig. 9 , it clearly shows that the velocity

f droplet increases with the flow rate of disperse phase and this

endency can be found in both experimental and numerical results.

t indicates that the numerical results are reasonably accurate in

escribing the transitional movement process in the downstream

hannel.

The periods of droplet formation process are obtained in dif-

erent flow rates of the disperse phase both in experiments and

umerical models. To account the effect of unsteady flow due to

he micro-pump, we analyze a very long time span (over 10 s) to

cquire the variation of period. We carefully observe the emulsifi-

ation process in the experiments and acquire the periods in four

F. Bai et al. / International Journal of Multiphase Flow 93 (2017) 130–141 139

Fig. 8. The comparison of droplet size in four different disperse flow rates (the left column shows the 3D results, the middle column shows the 2D crosssection results

and the right column shows the experimental results.). The 2D cross section is in the middle of the channel. The characteristic mobility is M c = χε 2 , where ε = 0 . 025 L ,

χ = 820 m ·s/kg. The flow rate of disperse phase varies from 0.1 μL/min to 1 μL/min. (a) Q d = 0.1 μL/min; (b) Q d = 0.2 μL/min; (c) Q d = 0.5 μL/min; (d) Q d = 1 μL/min. The

flow rate of continuous phase is fixed as 1 μL/min.

Fig. 9. The comparison of droplet velocity between experimental and numerical re-

sults in four different flow rates of disperse phase. The characteristic mobility is

M c = χε 2 , where ε = 0 . 025 L , χ = 820 m ·s/kg. The flow rate of continuous phase is

fixed as 1 μL/min.

d

A

p

=

Q

w

u

c

i

a

t

s

v

t

p

W

l

Fig. 10. (a) The evolution of pressure in central point at the entrance of throat and

(b) the location of fluid pressure in 2D cross section. The characteristic mobility is

M c = χε 2 , where ε = 0 . 025 L , χ = 820 m ·s/kg. The flow rates are 0.2 μL/min and

1 μL/min for disperse and continuous phase respectively.

o

r

f

4

e

w

o

m

n

o

ifferent flow rates (0.1 μL/min, 0.2 μL/min, 0.5 μL/min, 1 μL/min).

ccording to the experimental data, the periods of emulsification

rocess are 116 ∼ 170 ms for Q d = 0.1 μL/min, 81 ∼ 89 ms for Q d

0.2 μL/min, 54 ∼ 63 ms and Q d = 0.5 μL/min and 38 ∼ 46 ms for

d = 1 μL/min. It can be seen that the span of period is very large

hen the flow rate of disperse phase is small, which indicates the

nsteadiness effect is obvious at smaller flow rates.

We also obtain the periods in the numerical simulations. We

hoose the time step size to be 0.0 0 02 s and perform 30 0 0 steps

n time evolution. The period of droplet formation process can be

ccurately determined from the evolution of pressure. The evolu-

ion of pressure in the central point at the entrance of throat is

hown in Fig. 10 , with Q d = 0.2 μL/min.

The numerical solutions of pressure at the entrance of throat

ary periodically with time in Fig. 10 and there are about 3 fluc-

uations from 0.2 s to 0.5 s. The pressure increases when the dis-

erse phase is squeezed by the continuous phase to form a droplet.

hen the droplet breaks up, the pressure suddenly drops to a very

ow value. It clearly indicates that there are three sudden drops

f pressure at T ≈ 0.26 s, 0.31 s and 0.41 s. We obtain the pe-

iods for four given flow rates: the values of period are 179 ms

or 0.1 μL/min, 99 ms for 0.2 μL/min, 61 ms for 0.5 μL/min and

1 ms for 1 μL/min. By comparing the numerical solutions with the

xperimental periods we can see that the numerical periods are

ithin the experimental spans when the flow rate is 0.5 μL/min

r 1 μL/min. It indicates the numerical and experimental periods

atch very well when the disperse flow rate is relatively high. The

umerical periods are slightly higher than the experimental peri-

ds when the flow rate is small (0.1 μL/min and 0.2 μL/min). The

140 F. Bai et al. / International Journal of Multiphase Flow 93 (2017) 130–141

B

B

C

C

C

C

C

C

D

F

F

F

G

G

G

G

G

G

G

H

H

J

unsteady flow may lead to this mismatch, and it is difficult to keep

flow steady in the small flow rates.

5. Conclusions

A 3D phase field model, the Cahn–Hilliard–Navier–Stokes model

with generalized Navier boundary condition, was used with finite

element method to simulate the droplet formation process in a

flow-focusing device. To justify this phase field numerical model

and investigate the selection criterion of several critical parame-

ters, experiments in the same configuration was set up. The fluid

flow was dominated by interfacial force since the capillary num-

ber was fixed as a small value ( Ca ≈ 0.016). We found the phe-

nomenological mobility in the Cahn–Hilliard equation is a criti-

cal parameter for the emulsification process. This parameter de-

termines the diffusion term in the Cahn–Hilliard equation and the

relaxation time of the interface in order to control the droplet vol-

ume and shape. We defined the characteristic mobility by a com-

parison between numerical and experimental results. This charac-

teristic mobility is associated with the chemical energy in the dif-

fusion term of the Cahn–Hilliard equation, and thus the mobility

should be affected by the interfacial tension between given spe-

cific fluids. In this paper the two-phase fluids consisted of water

for the disperse phase and mineral oil for the continuous phase.

Therefore the mobility in this water-mineral oil system was found

by comparing with the experimental results. The mobility of other

different two-phase fluid systems should also be investigated in

our work in future. In our model the numerical convergence of

sharp interface limit was guaranteed with the properly selected

interfacial thickness. The interfacial thickness was also found to

play a key role in the phase field model. The variation of thick-

ness could lead to the change of energy dynamic equilibrium in

the two-phase system and also cause the variation of characteris-

tic mobility. To ensure the numerical results capturing the phys-

ical process correctly, i.e., the experimental data, we found that

the multiplication between the tuning mobility and the square of

interfacial thickness needed to be kept constant ( χε 2 = Constant)

to guarantee the mobility ( M = χε 2 ) as a constant. This criterion

also guarantees that Péclet number ( Pe =

εQ c MLσ ) represents the rea-

sonable ratio between advection and diffusion. The Péclet number

Pe is proportional to the interfacial thickness when the mobility

and other physical properties are fixed. The effect of flow rate of

the disperse phase was also studied and a good agreement be-

tween numerical results with the characteristic mobility and ex-

perimental results was achieved. The droplet velocity in the down-

stream and the period of droplet formation process obtained in

the numerical model generally matched the experimental results.

By comparing the numerical results with experiments, we justified

the correctness of our phase field model, thus this model can be

used for our further study to investigate more complex two-phase

flow systems.

Acknowledgments

This work is supported in part by the University of Missouri

Research Board and is partly supported by the National Natu-

ral Science Foundation of China through Grant Nos. 51376129

and 51036005 . X Yang’s research is partly supported by the U.S.

National Science Foundation under grant numbers DMS-1200487

and DMS-1418898 .

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