Microfluidics and Nanofluidics
Supplementary Information
Experimental Methodology, Chip Fabrication, Derivation of Equations, Uncertainty
analysis and Supplementary Experimental Data
Title:
Model of droplet generation in flow focusing devices operating in the squeezing
regime
Authors:
Xiaoming Chen, Tomasz Glawdel, Naiwen Cui, Carolyn L. Ren
1
S.1 Experimental Setup
The experimental setup is shown in Fig. S1, consisting of a high precision microfluidic pressure
control system (MSFC 8C, Fluigent). This system can control up to 8 output channels at the
same time with a maximum pressure of 1bar and an accuracy of 0.5mbar. Each reservoir is
connected to the MSFC 8C via an air pressure tube, while another tube is inserted into the
bottom of the reservoir as output. Software is provided with this system to control the pressures.
The reason that a pressure control system is used instead of syringe pumps is that the response
time is very short and output is more stable than syringe pumps in terms of the flow rate.
Figure S1. Photos of experimental setup. The central photo shows the whole connection of
components.
Three in-line flow sensors are used in this study (SLG 1430, Sensirion) to measure the flow rates
of the continuous phase, dispersed phase and the main channel. The flow sensors are able to
measure water flow rates up to 40µL/min with a frequency of 100Hz and silicone oil and
glycerol/water mixtures up to 5µl/min. The calibration process can be found below.
2
Droplet formation is visualized using an inverted epifluorescence microscope system (Eclipse Ti,
Nikon) with a 20x objective and (0.5, 2.1mm) a NA. Illumination is provided by a 100W halogen
lamp for bright field applications and phase contrast and a 100W mercury halide lamp
(Intensilight C-HGFIE, Nikon) for fluorescence applications. Images are recorded on a Retiga
2000R Fast 1394 monochrome CCD camera coupled to the microscope. The digital image
quality is 12 bit with a maximum speed of 10 fps at full resolution (1600x1200). With binning
the capture speed increases up to 110 fps (8x8 binning). A custom software program that comes
with the microscope is used to control the entire system for taking images at specified times and
positions.
A high speed CMOS camera (Phantom v210, Vision Research) is connected to the microscope
on the left port using a C-mount adapter (1X DXM, Nikon). At full resolution (1280x800) the
camera can take images at 2190 fps, with a 12 bit digital image quality. By lowering the
resolution, or cropping the field of view, the camera can easily exceed 10k fps. In our
experiments, 8k fps is utilized at the resolution of 208x800. The camera continuously records
image to the on device buffer and then downloads them to the camera via firewire once the
trigger is activated.
During experiments, each chip was mounted on the microscope stage. Constant pressure flow
was used to manipulate the fluid flow using the Fluigent pressure system. Considering the small
capillary number in the squeezing regime and the flow sensor restriction as described before, the
maximum flow rate was restricted to 5µl/min and minimum was 0.5µl/min in order to get
accurate readings. The maximum/minimum applied pressure can be estimated by the
hydrodynamic resistance of the channels and the required flow rates, but the maximum value
was set to be 1050mBar limited by the Fluigent system. The minimum flow rate applied to the
continuous phase should be larger than that of the dispersed phase in order to form monodisperse
droplets. For each experiment, the estimated pressure conditions for the continuous phase, Pc ,
were divided into 3-4 levels depending on the estimated pressure range which attempted to span
the capillary number as large as possible ( for study the influence of Ca on droplet formation). At
each Pc level, a series of Pd were applied to create different flow rate ratios. After each Pd level,
the flow rate of the dispersed phase was carefully checked using Labview to ensure it was in the
applicable range (above 0. 5µl/min and working in the squeezing regime). Usually 4-8 levels of
3
Pd were applied at each Pc level. An example of a set of experimental conditions for Exp #1 is
listed in T1.
TABLE S1 Pressure conditions applied in Exp # 1 for silicone oil and 10%glyc/water with no
surfactant on a type 1 chip.
Step #
1
2
3
4
5
6
7
8
9
10
Pc (mbar)
1000
1000
1000
1000
1000
1000
1000
800
800
800
Pd (mbar)
910
900
880
860
840
820
800
750
730
710
Step #
11
12
13
14
15
16
17
18
19
20
Pc (mbar)
800
800
800
800
800
600
600
600
600
600
Pd (mbar)
700
680
660
640
610
540
520
500
480
460
S.2 Global Network Design (Chip Design)
The channel layout is illustrated in Fig. S2 which was proposed based on the design criteria
reported by T. Glawdel et al. (2012). That study showed that the global network design
influences the performance of droplet generator and reported a set of design criteria to minimize
the fluctuation of droplet production. In particular, the widths of the dispersed and continuous
phase branches were set to be the same ( w  wc  wd  100μm ) at the junction to reduce the
fluctuation in the flow rates. The width of the continuous phase channel (400 µm) was set to be
much larger than that of the dispersed phase channel (100 µm) at the inlet to satisfy the rule that
the hydrodynamic resistance of the dispersed phase channel should be much higher than that of
the continuous phase and should be close to the main channel branch. The length of the main
channel was set to be 2.5cm. If it is too short, the number of droplets in it will be too small which
will lead to large fluctuations when one droplet enters or exits the main channel. If it is too long,
the hydrodynamic resistance will be too large and the pressure control system cannot provide
enough pressure to drive the flow. The length of the dispersed phase channel was set to be 1.3cm
due to the spacing limit. Different diameters of tubing were used to match the hydrodynamic
resistance balance of the different fluids. For low viscosity fluids (10% glyc/water wt), small
tubing was connected to the inlet. For high viscosity fluids (80% glyc/water wt), a relative large
size of tubing was used.
4
Figure S2. Sketch of global network design, the widths of the dispersed phase and continuous
phase are the same at the flow focusing junction.
S.3 Flow Sensor
The flow sensor has a capillary of 480µm in the inner diameter. The sensor detects the flow rate
based on thermal anemometry principles and is thus sensitive to the physical properties of fluids.
Therefore, the sensor has to be recalibrated before use. The sensor runs in a ‘raw’ mode where
direct temperature measurements are outputted as ‘tick’ counts. The typical operational curve for
the flow sensor has a sigmoid shape. The desirable measurement should be in the linear portion
of the curve. For water, the measurement range is from -40 ~ 40 µl/min which is already
calibrated by the company. For silicone oil and glycerol/water mixtures, the linear portion is
much smaller under 5µl/min. The flow sensor is non-linear when the flow rate is above 6µl/min.
So we only used flow rates under 6µl/min during experiments. The flow sensor calibration
curves are shown in Fig. S3.
S.4 Experimental Procedure
A brief overview of the experimental procedure is listed and some of steps are described in
subsequent sections in detail.
•
Before starting the experiment, several chips with a specific height were fabricated and
prepared. The fabrication details will be described in next section.
•
Clean reservoirs, tubing and flow sensors for the first time use or switching to a new type of
oil.
5
•
Flow sensors are calibrated for each fluid following the procedure. Then they are mounted on
a piece of polycarbonate and labeled for specific fluid.
•
Connect the chip with the pressure system and pump silicone oil through the chip for at least
40 minutes so that the chip is sufficiently swelled. The chip is then checked carefully for any
debris which might block the channels.
•
If the chip is in good condition, the channel width and height are then measured (after
swelling). The protocols will be described in next section.
•
Apply pressures to the two phases and waited for several minutes until stable droplets are
generated.
•
Record the pressures and flow rates for both phases shown in the Labview program.
•
Analyze the videos and extract experimental data
Each experiment was repeated at least two times.
6
10%glyc/water
60%glyc/water
80%glyc/water
Silicone oil
Flow Rate (µl/min)
5
4
3
2
1
0
14
14.5
15
15.5
16
16.5
17
Tick Count
Figure S3 Calibration curve of flow rate vs ‘tick’ count for the SLG 1430 flow sensor. Linear
regression profiles are: -6.4804x +104.96 (silicone oil), -4.3781x + 73.142 (10%glyc/water), 4.3282x + 66.894 (60%glyc/water), -4.4637x + 69.063 (80%glyc/water) with R² > 0.997 for all
profiles.
S.5 Chip Fabrication
The chips were fabricated using polydimethylsiloxane (PDMS) via standard soft-lithography
techniques. The process for fabricating PDMS chips can be found elsewhere (Glawdel et al.,
2009). Briefly, a photo mask with the design of the microchannel layout is printed on Mylar
6
films with a 20k dpi resolution (CAD/Art services). Photoresist, SU-8 is spin coated on a silicon
wafer which has been baked at 190℃ for 10mins on a hot plate to prevent photoresist from
peeling off the substrate. The thickness of the film depends on the type of SU-8 and the spin
coating speed. Soft bake is performed at 65°C and then 95°C to evaporate the solvent in the
photoresist and harden the film. The baking time depends on the thickness of film. The silicon
wafer is placed in a UV exposure system (Newport) and illuminated with UV light (~365nm). A
hard bake at 65°C and 95°C is performed after the UV exposure and left in room temperature to
cool down. The master is immersed in a large bath of SU-8 developer to dissolve the unexposed
SU-8 regions. The wafer is then washed with clean SU-8 developer, isopropanol, deionized
water and finally is blown dry with air. A small amount of glue (Loctite 3311) is coated around
the edge of the wafer and hardened with UV light which helps to prevent photoresist from
peeling off the substrate.
After the master is fabricated, PDMS (Sylgard 184, Dow corning) is mixed with a ratio of 10:1
(base: curing agent) and degassed in a vacuum oven. Then, the PDMS is poured on the top of the
master placed in an aluminum dish and cured at 95°C for at least 3 hours. When PDMS molds
are cooled down, they are bonded to a glass slide coated with PDMS in order to create a
homogeneous microchannel using oxygen plasma. PDMS coated glass slides are fabricated by
spin coating PDMS mixture at 3000rpm for 60s and baked at 95°C for 5min. The two substrates
are exposed to oxygen plasma (PDC-001, Harrick Plasma) with power of 29.6W at 500mTorr for
10s. However, the plasma treatment changes the wetting of PDMS from original hydrophobic to
hydrophilic. Since we generate water in oil droplets, PDMS chips are heated at 190°C for 12hrs
to return back to hydrophobic. After the chip is examined and measured under a microscope, the
process of fabrication is finished.
S.6 Channel Dimension Measurement
Silicone oil is slightly soluble in PDMS which causes the dimensions (w, h) of microchannels to
shrink, therefore it is very difficult to measure the actual channel height using standard
procedures such as profilometers. Instead, measurements of the channel dimensions must be
made in situ for each chip. Since accurate channel dimensions at the junction are critical to
analyze the droplet generation process, we use flow sensor and the pressure controller to estimate
7
the equivalent hydrodynamic resistance ( Rh  P / Q ) of the dispersed channel and main channel
which are connected by the junction.
For a rectangular cross section microchannel operating under laminar flow, the hydrodynamic
resistance can be calculated by equation Error! Reference source not found. (White 1991),
Rh 
12 L
P

wh (1  0.63h / w) Q
3
(S.6-1)
Where w, h and L are the width, height and length of the channel, respectively.  is the viscosity
of the fluid flowing in the channel, P applied pressure and Q measured flow rate.
The tubing (D=1mm) and flow sensor capillary (D=480µm) have much larger diameter than the
microchannel, thus the hydrodynamic resistance caused by the connectors can be neglected.
Since the hydrodynamic resistance is inversely proportional to h3 , the estimation is very sensitive
to the measured parameters (P, Q) and hence it is an effective way of measuring the channel
height. Fig.S4 shows a schematic of the flow focusing junction and the equivalent hydrodynamic
circuit.
Figure S4. Sketch of flow focusing equivalent hydrodynamic resistance measurements for
calculating the junction height. The continuous phase is blocked and silicone is pumped from the
inlet of disperse phase to the outlet of main channel. Pressure is controlled by the Fluigent
system, and flow rate is measure by the flow sensor.
Before starting each droplet experiment, channel dimension measurements are performed after
the channel is sufficiently swollen. The junction is captured by the camera and the width can be
measured by software ImageJ. A calibrated scale is used to convert the number of pixels into
distance on the image. The length of channel can be calculated from the photo mask design.
8
After that three different levels of pressures are applied to the channel and flow rates are
recorded. Substituting these parameters into equation Error! Reference source not found., the
height of channel is calculated. The measured dimensions for each experiment are shown in
Table S2.
S.7 Droplet Speed Measurement over a Formation Cycle
In order to verify that the flow rates of both dispersed phase and continuous phase are constants,
the droplet speed over a droplet formation cycle was tracked. Fig. S5 shows one example of
droplet speed over a droplet formation cycle. Results show that the droplet speed is almost
constant within variation less than 2%, which indicates that the droplet formation process is
under quasi-steady state.
15
Droplet Velocity (mm/s)
14
13
12
11
10
9
8
7
6
5
0
5
10
15
20
25
30
35
40
Time (ms)
Figure S5. Droplet speed over a formation cycle for the case where the dispersed phase is
*
glyc/water 60% wt, height/width ratio h  0.58 , Qd  1.69μl/min , and Qc  3.59μl/min .
S.8 Compare the Model with Experiment Using Parity Plots
There are two reasons for our choice to present the comparison via parity plots: i) the nature of
our experiments prevented us from varying only one parameter at a time as discussed below in
detail; and ii) we wished to make the manuscript as succinct as possible by avoiding many lines,
plots and figures.
9
First, if the fluids are driven by a syringe pump which is supposed to output constant flow rate
regardless of the events (i.e. generating a droplet or sorting a droplet into a branch channel)
occurring in the flow network, it is possible to use the approach of varying one single parameter
at a time to study droplet microfluidics. However, using syringe pumps for droplet microfluidic
studies causes lots of problems as documented in one of our previous studies (Glawdel and Ren
2012) and other studies as well (Korczyk et al. 2011). Briefly, syringe pumps lead to short term
oscillations in the output caused by the stepper motor and long term oscillations in the output
caused by imperfections in the drive screw. Korczyk et al. 2011 demonstrated the oscillation in
the generated droplet volume when using syringe pumps and the uniformity of the generated
volume when using a pressure control system. Therefore, in this study, we used a high-precision
microfluidic pressure control system (MSFC 8C, Fluigent) to pump fluids.
When using pressure control systems to pump fluids through channel networks, only the applied
pressures at the inlets and outlets are fixed for one particular experiment which results in
challenges in decoupling flow condition. For example, when one event (such as droplet
generation or sorting) occurs in the channel network, the overall flow resistance and the flow
resistance in any channel branch would change. In addition, varying any inlet or outlet pressure
alters both flow rates simultaneously (continuous and dispersed phases). Therefore, in our
experiment, each experimental datum has a specific capillary number and flow rate ratio which is
recorded through the flow sensor and droplet frequency and volume in the video analysis. Parity
plots do verify the model’s overall performance by comparing the model predictions with the
experimental results in a concise manner as other papers have done (Malsch et al. 2010; Liu et al.
2011; Kreutzer et al. 2008; Yue et al. 2014).
*
*
S.9 Derivations of VBF* , VEF* , Vgutter
, Vd* , VBN
, VEN* (Eqns (7), (8), (10), (11), (12), (13), (14))
We adopted the method developed by van Steijn et al. (2010) to calculate the curved volume. By
assuming that the radius of the droplet curvature is half of the channel depth ( h / 2 ), which is an
accurate assumption for well controlled wetting conditions (Typically, contact angles for the oil
and PDMS must be low <60˚, and for the water and PDMS high >120˚). This method calculates
the volume by multiplying the 2D top-view area of the curved object with the height and then
subtracting the volume of the continuous phase in-between the surfaces: top and bottom walls,
10
interface perpendicular to the top/bottom walls along the perimeter of the curved object and
curved surface along the perimeter,
V  hA 
h2

(1  )l
2
4
(S.9-1)
where V is the volume of the curved object, h the height, A the 2D top-view area, and l the 2D
top-view perimeter of the curved object.
Based on this method, we derived Eqns (7), (8), (13), (14) as detailed below.

Eqn(7) VBF* :
As shown in Figure S6, first we calculate the volume without considering the curvature of the
interface:
VBF_1  A1  h
(S.9-2)
We estimated A1 as a triangle. Hence, A1 can be expressed as,
L
wLpinch
1
A1   w  pinch 
2
2
4
(S.9-3)
Second, we calculate the volume of the continuous phase in-between the curved interface.
The cross-section area of the curved interface A2 can be expressed as,
h
1  h     h2
A2   h      1  
2
2 2  4  2
2
(S.9-4)
The perimeter of the curved interface can be calculated as,
2

 w  L
l  2      pinch   w2  L2pinch
2  2 
2
(S.9-5)
Therefore, the volume of the continuous phase in-between the curved interface is,
  h
VBF_2  A2  l  1  
 4 2
2
11
w2  L2pinch
(S.9-6)
The volume of the forming droplet at the beginning of filling stage can be expressed as,
VBF  VBF_1  VBF_2
  h

wh  1  
4
 4 2
2
Lpinch
w2  L2pinch
(S.9-7)
The dimensionless form is,
*
VBF

*
2
VBF Lpinch h*   

 1   1   L*pinch 
2
wh
4
2  4
1
Lpinch
2
l
(S.9-8)
Radius h/2
h
Area A1
Area A2
(a)
(b)
Figure S6. (a) 2D top-view area of VBF* at the beginning of filling stage. (b) Cross-section area
of the curved interface A2
By using the same method we can obtain other Eqns.

Eqn(8) VEF* :
As shown in Figure S7, The 2D top-view area of forming droplet at the end of the filling
stage AEF ,
2
w
1 w  1

AEF    fill    Lfill  fill  1.5w   w  wfill 
2  2  2
2

12
(S.9-9)
The perimeter of the curved interface can be calculated as,
lEF  (
 wfill
2
wfill
 w  wfill 
2 
 1.5w) 2
  ( Lfill 
2
 2 
2
(S.9-10)
Therefore,
w
w
1 w  1
h2
 w

 w  wfill 
VEF    fill    Lfill  fill  1.5w   w  wfill  h  (1  )( fill  2 
 ( Lfill  fill  1.5w)2 )

2  2  2
2
2
4
2
2

 2 
2
2
(S.9-11)
Dimensionless form,
*
VEF

2
2
2
*
*
*
*
 h*     
  *

VEF
Wfill
1   Wfill
*
 W *  2  1  Wfill    L*  Wfill  1.5 





L


1.5
(1

W
)

1





fill
fill
fill
fill


2
4  2
2
w2 h 2   2  
 2 

 2  







(S.9-12)

*
Eqn(13) VBN
:
The 2D top-view area of the continuous phase at the junction at the beginning of the necking
stage ABN can be estimated as,
1 w  wfill
ABN  2  w
2
2
(S.9-13)
The perimeter of the continuous phase at the junction at the beginning of the necking stage,
lBN can be estimated as,
 w  wfill 
2
lBN  2  
 w
 2 
2
(S.9-14)
Therefore,
1 w  wfill h2
  w  wfill 
 2[h  w
 (1  ) 
 w2 ]

2
2
2
4  2 
2
VBN
Dimensionless form
13
(S.9-15)
2
*
VBN

*
1  Wfill*
    1  Wfill 
 h* 1   
 1
2
 4  2 
(S.9-16)
We can see that the second part volume on the right hand side of the equation is added to the
total volume, because we calculate VBN as the volume of continuous phase at the junction at
the beginning of necking stage, instead of forming droplet volume.
*
Figure S7. 2D top-view area of VEF* and VBN
at the end of filling stage (at the beginning of
necking stage)

Eqn(14) VEN* :
As shown in figure S8, the 2D top-view area of the continuous phase at the junction at the
end of the necking stage, AEN can be estimated as,
AEN  2 
Lpinch  w  wpinch 


2 
2

(S.9-17)
The perimeter of the continuous phase at the junction at the end of the necking stage, lEN can
be estimated as,
14
Lpinch 2
 w  wpinch 
 4 
)
 (
2
2


2
lEN
(S.9-18)
Therefore,
Lpinch  w  wpinch 
Lpinch 2
  w  wpinch 
2
 2
)

 h  2  h (1  ) 
 (
2 
2
4 
2
2


2
VEN
(S.9-19)
Dimensionless form
*
EN
V
L
*
pinch
*
1  Wpinch
2
 
 h* 1  
 4
1  W    L 
2
*
pinch
2
*
pinch
(S.9-20)
Figure S8. 2D top-view area of VEN* at the end of necking stage
Eqns (10) and (11) were also derived using the exactly same method based on Figure S8. The
detailed derivations of the two equations are not shown here due to simple reduplicate work.
15
S.10 Uncertainty analysis of Vd* starting from uncertainty in parameters: L*fill ( 10% ), Wfill* ,
*
( 10% ) and L*pinch ( 10% )
Wpinch
1. Uncertainty From Wfill* .
Wong et al. (1995) mentioned that the thin film thickness is proportional to the capillary number
raised to the two-thirds power when Ca  0.01, which means the thin film has a thickness on the
order of 1% to 5% of the half channel width. Therefore, we assume that the gap between the
droplet and channel wall is very thin (i.e. 0.02w) leaving to be 0.96 based on the rough
measurements. The variation of thin film thickness is in an extremely small range within 0.03
(can be treated as a constant). Therefore, the variation of the volume of droplets caused by the
*
variation of Wfill* can be ignored compared to other three parameters L*fill , L*pinch and Wpinch
.
*
2. Uncertainty From L*fill ( 10% ), L*pinch ( 10% ) and Wpinch
( 10% )
Since the variation of Wfill* is very small within 0.03 , Wfill* can be treated as a constant. Using a
*
first order Taylor expansion, uncertainty of Vd* caused by L*fill , L*pinch and Wpinch
can be expressed
as,
Vd*
1 Vd* *
1 Vd*
1 Vd*
*
*
 * * Lfill  * * Lpinch  *
Wpinch
*
*
Vd
Vd Lfill
Vd Lpinch
Vd Wpinch
(S.10-1)
Substitute Wfill*  0.96 into Eqns(7), (8), (12), (13) and (14) (from the paper manuscript), we can
obtain,
*
VBF

*
EF
V
L*pinch
4

2
2
h*   
*
*
*
*
1   1   Lpinch   0.25Lpinch  0.11h 1   Lpinch  (S.10-2)
2  4
2
2
2
*
*
*
*
 h*     
  *

 1  Wfill
  *

Wfill
Wfill
1   Wfill
*
*

  
 1.5  (1  Wfill )   1  
Wfill  2 
 1.5 
   Lfill 
   Lfill 
2  2  
2
4  2
2
 2 

 2  



 1.36  0.98 L*fill  0.11h*  3.55  2 L*fill 




(S.10-3)
16
1
 
1  Wfill*   1   h*

*
2
 4  V *  1  L*     h 1    1  ( L* ) 2    
*
Vgutter

pinch

 d 4  pinch 2  2  4 
2  
 





1   1   h*
 4
*
0.02  0.21h  *

Vd  0.25 L*pinch  0.39  0.11h* 1  ( L*pinch ) 2  1.57 

1  0.21h* 
(S.10-4)


2
*
BN
V
1  Wfill*
   1  Wfill* 
*

 h 1   
 1
2
 4  2 
(S.10-5)
 0.02  0.11h*
*
VEN
 L*pinch
 0.5L
*
pinch
*
1  Wpinch
2
 
 h*  1  
 4
1  W    L 
2
*
pinch
2
*
pinch
(S.10-6)
1  W   0.21h 1  W    L 
*
pinch
*
2
*
pinch
2
*
pinch
*
*
*
Substitute Eqns. (S.10-2) ~ (S.10-6) into Eqn. Vd*  VEF
 VBF
  VEN*  VBN*  Vgutter
  , we can
obtain,
2 
1.02



)  Vd*  1.36  0.98 L*fill  0.25 L*pinch  0.11h*  3.55  2 L*fill  1   L*pinch  
1  (1 
*
1  0.21h




(S.10-7)
2
2


*
*
 0.5L*pinch 1  Wpinch
 0.21h* 1  Wpinch
  L*pinch   (0.02  0.11h* ) 




1.02
 (1 
) 0.25 L*pinch  0.39  0.11h* 1  ( L*pinch ) 2  1.57
1  0.21h*



For a typical channel height h*  0.5 , substitute h*  0.5 into (S.10-7),
Vd* 

2 
1

1.16  0.87 L*fill  0.25 L*pinch  0.055 1   L*pinch  

1  0.14  


2
2


*
*
0.465L*pinch  0.5 L*pinchWpinch
 0.11 1  Wpinch
  L*pinch   0.0077 1  ( L*pinch ) 2  0.17 


1  0.14  

(S.10-8)
Therefore,
Vd*
0.87

*
Lfill 1  0.14 
17
(S.10-9)
Because flow rate ratio 0    1 , we can know 0.86  1  0.14  1 . Therefore,
L*fill
L*fill
1 Vd* *
0.87

L


1.01
fill
Vd* L*fill
Vd*
1  0.14  Vd*
(S.10-10)
With L*fill  10% L*fill , and L*fill much smaller than Vd* from experimental results (
L*fill
 0.5 ), we
Vd*
can estimate that,
1 Vd* *
Lfill  5%
Vd* L*fill
(S.10-11)
L*pinch
Vd*
0.25
0.055


2
L*pinch 1  0.14  1  0.14  1  L*
 pinch 

*
0.5Wpinch
0.465
0.11


1  0.14  1  0.14  1  0.14 
L*pinch
1  W    L 
2
*
pinch
2
*
pinch

L*pinch
0.0077
1  0.14  1  ( L*pinch )2
(S.10-12)
*
 0.5 , we can obtain,
Experimental results show that 2  L*pinch  4 , 0.2  Wpinch
1
L*pinch
1

 1.16 , 0 
 1,
 1.16 , 0.9 
2
1  0.14
*
1  0.14 
1   Lpinch 
L*pinch
1  W    L 
2
*
pinch
2
*
pinch
1
(S.10-13)
Therefore,
*
0.5Wpinch
Vd*
0.25
0.055
0.465
0.11
0.0077






*
Lpinch 1  0.14  1  0.14  1  0.14  1  0.14  1  0.14  1  0.14 

*
(0.58  0.5Wpinch
)  0.195
1  0.14 
 0.33
(S.10-14)
18
Then,
L*pinch
1 Vd*
*
, considering L*pinch  10% L*pinch , and L*pinch is close to Vd*

L

0.33
pinch
Vd* L*pinch
Vd*
from experimental results, we can estimate that,
L*pinch
1 Vd*
*
Lpinch  0.33
 3.3%
Vd* L*pinch
Vd*
(S.10-15)
With the same method, we can obtain,
0.5L*pinch
Vd*
0.11



*
Wpinch
1  0.14  1  0.14 
*
1  Wpinch
1  W    L 
2
*
pinch
2
*
pinch
(S.10-16)
*
0.5L*pinch
Vd*
0.11 1  Wpinch


*
Wpinch
1  0.14  1  0.14  L*pinch
*
0.5 L*pinch
Vd*
0.11 1  Wpinch


 2.76
*
Wpinch
1  0.14  1  0.14  L*pinch
Therefore, considering W
*
pinch
 10%W
*
pinch
, and
*
Wpinch
Vd*
 0.2 from experimental results, we can
obtain,
*
Wpinch
1 Vd*
*

W

2.76
 5.52%
pinch
*
Vd* Wpinch
Vd*
*
Therefore, the total error of Vd* due to the uncertainties of L*fill ( 10% ), L*pinch ( 10% ) and Wpinch
(
10% ) should be,
Vd*
 (5%  3.3%  5.52%)  13.82%
Vd*
(S.10-17)
This error considers the extreme conditions. The actual error is expected to be much smaller than
*
13.82%. In conclusion, L*fill , L*pinch and Wpinch
with errors within 10% are accurate to predict
droplet volumes.
19
TABLE S2 Measured channel dimensions after sufficiently swelling for each specific
experiment
Exp Channel
#
Type
Nominal Nominal
Disp Ph
W (µm) H(µm)
Cont. Ph
Actual Actual H
W(µm) (µm)
Length
(µm)
1
Type1
100
40
10% Glyc
Silicone oil
89
35.0
38500
2
Type1
100
40
10% Glyc
Silicone oil
89
35.5
38500
3
Type1
100
40
60% Glyc
Silicone oil
89
35.5
38500
4
Type1
100
40
60% Glyc
Silicone oil
88
35.2
38500
5
Type1
100
40
80% Glyc
Silicone oil
89
35.8
38500
6
Type1
100
40
80% Glyc
Silicone oil
89
36.8
38500
7
Type2
100
53
10% Glyc
Silicone oil
83
46.8
38500
8
Type2
100
53
10% Glyc
Silicone oil
83
49.4
38500
9
Type2
100
53
60% Glyc
Silicone oil
83
49.1
38500
10
Type2
100
53
60% Glyc
Silicone oil
83
49.2
38500
11
Type2
100
53
80% Glyc
Silicone oil
84
48.4
38500
12
Type2
100
53
80% Glyc
Silicone oil
83
47.7
38500
13
Type3
100
60
10% Glyc
Silicone oil
79
54.1
38500
14
Type3
100
60
10% Glyc
Silicone oil
80
53.5
38500
15
Type3
100
60
60% Glyc
Silicone oil
80
53.6
38500
16
Type3
100
60
60% Glyc
Silicone oil
80
53.3
38500
17
Type3
100
60
80% Glyc
Silicone oil
80
52.7
38500
18
Type3
100
60
80% Glyc
Silicone oil
80
53.7
38500
20
TABLE S3 Range of applicability of the model
1
2
3
Well designed microfluidic circuits to make sure the flow rates of both dispersed and
continuous phases are stable. The droplet generation process should be in quasi-steady
state
No surfactant should be added or the concentration of surfactant should be far more than
critical micelle concentration (CMC), so that the mass transfer of surfactant can be ignored
Well controlled wetting conditions (Typically, contact angles for the continuous phase and
wall must be low <60˚, and for the dispersed phase and wall high >120˚),
4. The droplet generation should be in the squeezing or transition regime, where the forming
droplet almost touches the walls and the shear force can be ignored compared to pressure
gradient and interfacial tension
Input parameters:
Qd Qc d c  h w
Calculate
  h* Ca
Calculate
*
VBN
by Eqn. (13)
*
*
*
VBF
[Eqn. (14)]
[Eqn. (7)] VEN
Substitute VEF* VBN
*
and Vgutter[Eqn. (12)] into [Eqn. (16)] to obtain
*
*
an Eqn. of Vd  f1  Lpinch 
Calculate
g ( ) by Eqn. (24)
Substitute Eqn. (12) into Eqn. (36) to obtain
the other Eqn. of Vd*  f 2  L*pinch 
Calculate
L*fill by Eqn. (30)
Combine Eqns.Vd*  f1  L*pinch  and Vd*  f 2  L*pinch 
to obtain Vd* and L*pinch
Calculate
*
Wpinch
by Eqn. (32)
Calculate
V by Eqn. (8)
*
Calculate f by Eqn. (1) and
*
Calculate s by Eqn. (3)
f*  f 
*
EF
Figure S9. Flow chart of the steps of the calculation
21
w2 h
Qc
TABLE S4 Experimental Data
TABLE S4-1 Experimental data with dispersed phase: glyc/water 10% by wt., and channel
height h*  0.41 .
Qc ( μl/min )
Qd ( μl/min )
L*fill
L*pinch
*
Wpinch
Vd*
2.83
2.93
2.97
3.07
3.15
3.26
3.53
3.65
2.69
2.79
2.93
3.05
2.36
2.48
2.61
2.75
2.04
2.18
2.31
1.82
2.13
1.85
1.66
1.44
1.21
1
0.83
0.66
1.66
1.36
1.08
0.83
1.35
1.06
0.8
0.59
1.01
0.76
0.55
0.58
0.865168539
0.831460674
0.797752809
0.775280899
0.730337079
0.730337079
0.707865169
0.674157303
0.831460674
0.786516854
0.730337079
0.696629213
0.842696629
0.786516854
0.730337079
0.685393258
0.820224719
0.775280899
0.741573034
0.808988764
2.820224719
2.775280899
2.797752809
2.786516854
2.797752809
2.741573034
2.797752809
2.786516854
2.876404494
2.786516854
2.752808989
2.764044944
2.808988764
2.797752809
2.786516854
2.786516854
2.831460674
2.831460674
2.865168539
2.797752809
0.269662921
0.247191011
0.280898876
0.247191011
0.247191011
0.258426966
0.258426966
0.235955056
0.235955056
0.269662921
0.280898876
0.258426966
0.247191011
0.247191011
0.269662921
0.235955056
0.247191011
0.235955056
0.224719101
0.224719101
2.660132781
2.462970774
2.292664269
2.169852828
2.045228822
1.880938568
1.745387419
1.682477783
2.400755823
2.177608282
2.009628288
1.828822968
2.476799494
2.177788858
1.940710294
1.819120735
2.345453524
2.092823263
1.927205814
2.117394956
22
TABLE S4-2 Experimental data with dispersed phase: glyc/water 10% by wt., and channel
height h*  0.58 .
Qc ( μl/min )
Qd ( μl/min )
L*fill
L*pinch
*
Wpinch
Vd*
3.58
3.65
3.74
3.84
3.9
3.98
4.06
4.15
4.23
4.34
3.14
3.24
3.32
3.41
3.49
3.58
3.67
2.6
2.68
2.78
2.89
3.19
3.03
2.87
2.72
2.53
2.37
2.21
2.04
1.88
1.66
2.42
2.25
2.09
1.92
1.77
1.61
1.45
1.89
1.71
1.58
1.44
0.987951807
0.987951807
0.963855422
0.927710843
0.927710843
0.879518072
0.843373494
0.855421687
0.819277108
0.819277108
1
0.951807229
0.915662651
0.903614458
0.879518072
0.819277108
0.807228916
1.072289157
1
0.951807229
0.891566265
2.759036145
2.734939759
2.686746988
2.722891566
2.686746988
2.734939759
2.759036145
2.710843373
2.686746988
2.65060241
2.686746988
2.710843373
2.734939759
2.746987952
2.722891566
2.662650602
2.65060241
2.746987952
2.734939759
2.771084337
2.722891566
0.385542169
0.409638554
0.385542169
0.385542169
0.385542169
0.397590361
0.397590361
0.397590361
0.385542169
0.397590361
0.385542169
0.385542169
0.409638554
0.373493976
0.409638554
0.385542169
0.385542169
0.397590361
0.397590361
0.385542169
0.385542169
2.754614093
2.679105004
2.565479265
2.501122775
2.388488719
2.312599774
2.230550117
2.146305135
2.082564503
1.990285667
2.648301481
2.561484865
2.440188152
2.35285124
2.270601396
2.171797686
2.104562686
2.754545889
2.605304052
2.462705623
2.376483213
23
TABLE S4-3 Experimental data with dispersed phase: glyc/water 10% by wt., and channel
height h*  0.66 .
Qc ( μl/min )
Qd ( μl/min )
L*fill
L*pinch
*
Wpinch
Vd*
3.43
3.59
3.7
3.81
3.96
4.11
4.29
4.46
4.63
2.69
2.8
2.91
3.08
3.23
3.4
3.61
3.79
1.99
2.09
2.21
1.65
2.7
2.34
2.08
1.83
1.59
1.35
1.12
0.9
0.7
2.04
1.75
1.51
1.27
1.06
0.87
0.66
0.48
1.36
1.13
0.92
1.06
1.202531646
1.126582278
1.063291139
0.987341772
0.936708861
0.886075949
0.835443038
0.82278481
0.810126582
1.278481013
1.202531646
1.113924051
1.075949367
0.987341772
0.924050633
0.835443038
0.784810127
1.392405063
1.316455696
1.227848101
1.481012658
2.481012658
2.53164557
2.455696203
2.430379747
2.455696203
2.443037975
2.455696203
2.430379747
2.443037975
2.53164557
2.46835443
2.481012658
2.493670886
2.46835443
2.493670886
2.455696203
2.455696203
2.569620253
2.493670886
2.53164557
2.544303797
0.430379747
0.46835443
0.455696203
0.455696203
0.443037975
0.455696203
0.443037975
0.430379747
0.46835443
0.430379747
0.443037975
0.443037975
0.455696203
0.430379747
0.430379747
0.430379747
0.430379747
0.443037975
0.430379747
0.417721519
0.405063291
2.724940716
2.531749556
2.397480245
2.264757919
2.123176649
1.999303542
1.895817451
1.772690876
1.608472707
2.923795385
2.654517434
2.504031986
2.366198391
2.157318614
2.016894415
1.832321583
1.699937798
3.087679188
2.805608839
2.579767773
3.287920491
24
TABLE S4-4 Experimental data with dispersed phase: glyc/water 60% by wt., and channel
height h*  0.41 .
Qc ( μl/min )
Qd ( μl/min )
L*fill
L*pinch
*
Wpinch
Vd*
2.54
2.66
2.76
2.9
3.03
2.29
2.39
2.52
2.63
2.72
1.96
2.05
2.16
2.25
2.36
2.47
1.73
1.86
1.83
1.71
1.57
1.44
1.32
1.78
1.62
1.5
1.37
1.23
1.67
1.53
1.4
1.25
1.1
0.99
1.35
1.17
0.806818182
0.795454545
0.75
0.772727273
0.727272727
0.829545455
0.818181818
0.795454545
0.806818182
0.75
0.852272727
0.829545455
0.806818182
0.795454545
0.715909091
0.715909091
0.875
0.806818182
3.136363636
3.056818182
3.011363636
2.920454545
2.852272727
3.113636364
3.079545455
2.920454545
2.863636364
2.863636364
3.102272727
3
2.977272727
2.931818182
2.875
2.852272727
3.090909091
2.988636364
0.295454545
0.295454545
0.306818182
0.284090909
0.284090909
0.306818182
0.295454545
0.318181818
0.306818182
0.295454545
0.284090909
0.295454545
0.284090909
0.295454545
0.295454545
0.306818182
0.295454545
0.284090909
2.197044262
2.067257046
1.930436847
1.821276779
1.734453359
2.296360068
2.100167038
1.978313606
1.851916969
1.737796087
2.431018615
2.251179083
2.093060248
1.928833138
1.767813818
1.683146597
2.376167783
2.111727605
25
TABLE S4-5 Experimental data with dispersed phase: glyc/water 60% by wt., and channel
height h*  0.58 .
Qc ( μl/min )
Qd ( μl/min )
L*fill
L*pinch
*
Wpinch
Vd*
3.21
3.31
3.47
3.63
3.81
4.02
2.81
2.99
3.18
3.36
3.59
3.84
2.38
2.57
2.78
2.98
3.21
1.83
2.02
4.24
4.43
4.62
3.04
2.97
2.79
2.6
2.4
2.2
2.53
2.31
2.09
1.88
1.69
1.49
2.01
1.77
1.55
1.35
1.15
1.57
1.36
3.58
3.32
3.07
0.987951807
0.915662651
0.891566265
0.879518072
0.855421687
0.831325301
0.987951807
0.86746988
0.86746988
0.819277108
0.78313253
0.746987952
0.963855422
0.951807229
0.86746988
0.819277108
0.795180723
1.13253012
1.024096386
0.86746988
0.831325301
0.795180723
2.963855422
2.987951807
2.939759036
2.86746988
2.807228916
2.734939759
3
2.927710843
2.879518072
2.807228916
2.78313253
2.795180723
2.963855422
2.831325301
2.831325301
2.807228916
2.710843373
3.012048193
2.951807229
2.86746988
2.831325301
2.795180723
0.385542169
0.409638554
0.385542169
0.409638554
0.397590361
0.385542169
0.385542169
0.385542169
0.385542169
0.409638554
0.385542169
0.409638554
0.373493976
0.397590361
0.373493976
0.397590361
0.385542169
0.373493976
0.373493976
0.397590361
0.373493976
0.397590361
2.545776827
2.4700761
2.321795363
2.183166375
2.050619495
1.917176323
2.553301357
2.358623771
2.173692987
2.003861234
1.863039974
1.742203523
2.571958502
2.319301032
2.097897964
1.920132659
1.772137495
2.752202777
2.414680487
2.197510853
2.065588765
1.948910137
26
TABLE S4-6 Experimental data with dispersed phase: glyc/water 60% by wt., and channel
height h*  0.66 .
Qc ( μl/min )
Qd ( μl/min )
L*fill
L*pinch
*
Wpinch
Vd*
2.58
2.81
3.09
3.36
3.69
4.02
4.37
4.75
2.3
2.59
2.95
3.31
3.7
1.86
2.07
2.33
2.59
1.64
1.93
2.2
2.4
2.13
1.87
1.59
1.32
1.05
0.81
0.55
1.63
1.32
1.04
0.78
0.52
1.6
1.33
1.11
0.91
1.15
0.96
0.74
1.2125
1.075
0.9375
0.8875
0.8625
0.8
0.7875
0.7625
1.1125
1
0.9125
0.825
0.75
1.1875
1.1375
1.05
0.975
1.25
1.0875
0.975
2.5
2.4375
2.4375
2.4875
2.3875
2.4125
2.4125
2.4
2.4375
2.4625
2.4375
2.3375
2.35
2.675
2.4625
2.425
2.4375
2.45
2.425
2.425
0.4125
0.4125
0.4125
0.4125
0.4125
0.4125
0.4
0.425
0.4
0.4
0.4125
0.4
0.425
0.3875
0.3875
0.3875
0.3875
0.4
0.4
0.4
2.708982858
2.423981423
2.207884255
1.997906566
1.820392483
1.668997619
1.561657278
1.425224167
2.590487303
2.193728943
1.940337436
1.73650985
1.553667848
2.934795576
2.536833392
2.243975991
2.017141994
2.729517815
2.400827043
2.098869595
27
TABLE S4-7 Experimental data with dispersed phase: glyc/water 80% by wt., and channel
height h*  0.41 .
Qc ( μl/min )
Qd ( μl/min )
L*fill
L*pinch
*
Wpinch
Vd*
2.08
2.03
2.11
2.2
2.3
1.73
1.81
1.88
1.98
1.33
1.41
1.5
1.57
2.41
2.51
2.1
1.12
0.95
0.85
0.82
0.8
0.77
0.78
0.75
0.71
0.68
0.74
0.7
0.66
0.63
0.78
0.74
0.66
0.62
0.715909091
0.727272727
0.704545455
0.704545455
0.659090909
0.738636364
0.727272727
0.715909091
0.681818182
0.761363636
0.75
0.727272727
0.738636364
0.704545455
0.659090909
0.681818182
0.795454545
3.761363636
3.613636364
3.556818182
3.511363636
3.420454545
3.636363636
3.556818182
3.477272727
3.443181818
3.852272727
3.693181818
3.556818182
3.443181818
3.306818182
3.295454545
3.318181818
3.806818182
0.284090909
0.295454545
0.284090909
0.295454545
0.306818182
0.306818182
0.295454545
0.295454545
0.295454545
0.295454545
0.306818182
0.295454545
0.295454545
0.295454545
0.306818182
0.295454545
0.295454545
1.551698943
1.633500586
1.553143323
1.5143926
1.464055621
1.666820784
1.606083601
1.536316337
1.481714821
1.848304867
1.75173941
1.654569885
1.585161252
1.40509907
1.351523227
1.39596618
1.906992269
28
TABLE S4-8 Experimental data with dispersed phase: glyc/water 80% by wt., and channel
height h*  0.58 .
Qc ( μl/min )
Qd ( μl/min )
L*fill
L*pinch
*
Wpinch
Vd*
2.44
2.62
2.83
3.13
3.36
3.6
3.78
2.31
2.49
2.65
2.84
3.04
3.22
3.41
1.78
1.96
2.14
2.32
2.5
2.7
1.66
1.6
1.53
1.48
1.39
1.29
1.22
1.37
1.31
1.25
1.18
1.12
1.04
0.98
1.21
1.14
1.07
0.99
0.93
0.85
0.869047619
0.869047619
0.80952381
0.821428571
0.785714286
0.785714286
0.75
0.869047619
0.833333333
0.80952381
0.797619048
0.761904762
0.773809524
0.761904762
0.964285714
0.916666667
0.857142857
0.869047619
0.785714286
0.773809524
3.30952381
3.19047619
3.095238095
2.952380952
2.928571429
2.928571429
2.857142857
3.095238095
3.071428571
3
2.94047619
2.928571429
2.904761905
2.916666667
3.095238095
3.095238095
3.023809524
2.928571429
2.857142857
2.857142857
0.380952381
0.392857143
0.380952381
0.404761905
0.392857143
0.392857143
0.392857143
0.380952381
0.404761905
0.392857143
0.392857143
0.380952381
0.380952381
0.404761905
0.380952381
0.392857143
0.369047619
0.404761905
0.392857143
0.380952381
2.055538099
1.941423504
1.822683482
1.707629865
1.6034369
1.504486262
1.437683726
1.940228718
1.825299953
1.727264302
1.631880502
1.559816103
1.468001885
1.410154668
2.176951876
1.995636748
1.840696797
1.704207611
1.611254674
1.49622061
29
TABLE S4-9 Experimental data with dispersed phase: glyc/water 80% by wt., and channel
height h*  0.66 .
Qc ( μl/min )
Qd ( μl/min )
L*fill
L*pinch
*
Wpinch
Vd*
2.29
2.48
2.68
2.9
3.11
3.34
3.57
3.77
4.02
4.26
1.89
2.13
2.38
2.62
2.88
3.15
1.5
1.76
2.04
1.28
1.31
1.24
1.16
1.07
0.98
0.9
0.81
0.72
0.64
0.54
1
0.91
0.82
0.72
0.62
0.53
0.69
0.6
0.5
0.55
0.975
0.95
0.875
0.85
0.825
0.8
0.75
0.7375
0.6625
0.6625
1.0375
0.9375
0.8875
0.85
0.825
0.775
1.0125
0.9375
0.8875
1.1125
2.725
2.675
2.675
2.6125
2.55
2.525
2.4
2.425
2.425
2.375
2.5625
2.5375
2.425
2.3875
2.3625
2.4
2.7
2.66
2.4125
2.6
0.4125
0.4125
0.4125
0.3875
0.4
0.3875
0.425
0.4125
0.4125
0.4375
0.4375
0.4125
0.425
0.425
0.4
0.4125
0.4
0.3875
0.425
0.425
2.103877311
1.97393417
1.852964672
1.733339166
1.62674308
1.552863374
1.476983647
1.41112925
1.374521119
1.306819581
2.125706331
1.939559009
1.793137592
1.65350871
1.531539535
1.45522014
2.175504977
1.951330698
1.736728865
2.2240593
30
Reference
Glawdel T, Ren CL (2012) Global network design for robust operation of microfluidic droplet
generators with pressure-driven flow. Microfluid Nanofluid 13: 469-480.
Glawdel T, Elbuken C, Lee L, Ren CL (2009) System with Integrated Electroosmotic Pumps,
Concentration Gradient Generator and Fish Cell Line (RTgill-W1)-Towards Water
Toxicity Testing. Lab Chip, 9: 3243 - 3250
Korczyk PM, Cybulski O, Makulska S, Garstecki P (2011) Effects of Unsteadiness of the Rates
of Flow on the Dynamics of Formation of Droplets in Microfluidic Systems. Lab Chip
11: 173–175.
Kreutzer MT, Gunther A, Jensen KF (2008) Sample Dispersion for Segmented Flow in
Microchannels with Rectangular Cross Section. Anal Chem 80: 1558-1567.
Liu H, Zhang Y (2011) Droplet formation in microfluidic cross-junctions. Phys Fluids 23:
082101.
Malsch D, Gleichmann N, Kielpinski M et al. (2010) Dynamics of droplet formation at T-shaped
nozzles with elastic feed lines. Microfluid Nanofluid 8: 497-507.
Yue J, Rebrov EV, Schouten JC (2014) Gas–liquid–liquid three-phase flow pattern and pressure
drop in a microfluidic chip: similarities with gas–liquid/liquid–liquid flows. Lab Chip 14:
1632-1649.
White FM (1991) Viscous fluid flow, 2nd edn. McGraw-Hill, New York
Wong H, Radke CJ, Morris S (1995) The motion of long bubbles in polygonal capillaries. Part 1.
Thin films. J Fluid Mech 292: 71-94
31