Supplemental Materials for “High Frequency Microbubble-switched
Oscillations Modulated by Microfluidic Transistors”
Fanghao Yang, Xianming Dai, and Chen Lia)
Department of Mechanical Engineering, University of South Carolina, Columbia, SC, 29208
USA
a) Electronic mail: li01@cec.sc.edu
Microfabrication Methods: Double side polished 100 mm <100> 500 m-thick P-type wafers
were coated with 0.5 m thickness thermal oxides on both sides. Additional 1 m-thick silicon
oxide was deposited on the front side by plasma-enhanced chemical vapor deposition (PECVD)
as etching mask for the deep reactive-ion etching (DRIE) in the next step. 10 nm-thick titanium
and 200 nm-thick aluminum thin-film layers were deposited onto the backside of the wafers by
DC sputtering. Photolithography and wet-etching were used to form a built-in thin-film
micro-heater (10 mm-long and 2 mm-wide) on the backside of the test chips (Figure S1b). This
thin-film micro-heater would power four parallel microfluidic devices in one test chip. For each
device, the effective area of the heater is (10 mm x 500 m) Deep micro trenches (250 m depth)
were fabricated by DRIE on the front side of the wafers. Through-holes were etched on the
backside of wafer as flow inlet, outlet and pressure transducer ports for a microfluidic testing
chip in this study. Finally, a 500 m thick Pyrex glass wafer was bonded onto the top of the
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wafers as a transparent window for visualization study. Both sides of the microfluidic testing
chip are shown in Figure S1.
Measuring Methods: Two-phase microbubble-switched oscillations were tested and visualized in
an established two-phase test platform. The visualization system consists of a high-speed camera
(Phantom V 7.3) with 256 x 256 pixels at approximate 40,000 FPS (frames per second) and an
Olympus microscope (BX-51) with 400X amplifications. Pressure drop data were collected by
two Omega PX01-C1 pressure transducers with ±0.05% linearity and repeatability. All original
points of transducer output were calibrated under 1 atmosphere prior to measurements. The
sampling rate of transient pressure drop data was 1k Hz. Uniform input heat flux was supplied by
a digital power supply through a built-in thin-film heater on the backside of a microfluidic test
chip. Pressurized nitrogen gas served a constant-pressure pump to supply DI water through a
needle valve into a test chip. The mass and heat fluxes were precisely controlled in the test
platform. Mass fluxes were measured by Sensiron ASL1600 flowmeter with 0.03 kg m-2 s-1
resolution. All pressure drop data, mass and heat flux data were automatically recorded by a
real-time digital data acquisition system.
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FIG. S1. Top-view of two sides of a microfluidics test chip. (a) The gray area in the front side of
the test chip is the solid wall or bonding area. The light blue indicates wet area. Thermal isolation
gaps are used to reduce heat loss. (b) The green area in the back side of the test chip is a built-in
aluminum thin-film heater. Two contact pads are designed to achieve great electric contact with
pogo-pins.
Testing System: The front side and the back side of testing chip are shown in Figure S1. And a
testing package module is shown in Figure S2b and S2c. This testing package module could
provide reliable hydraulic and electrical connections to microfluidic chips. So that oscillation of
pressure drop can be precisely measured in real-time. Several pogo-pins are used to ensure
perfect electric contact between pins and the metal pads on the backside of microfluidic chips.
Hydraulic ports are sealed by micro rubber O-rings. The schematic map of testing platform as
shown in Figure S2 illustrates functions and connections of major components in the system.
Blue wires show the hydraulic connections through stainless steel tubing. The red lines denote
gas tubing. Green wires represent electrical wires/connections. The yellow line indicates the
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optical path for visualization study.
FIG. S2. The two-phase testing platform. (a) The picture of the testing platform. (b) An exploded
3D model of testing package module, which provide hydraulic and electrical connections. (c) A
picture of the testing package module. (d) Schematic map of the testing platform.
Data Reduction: The electrical power of microheater, P, is calculated as P  V  I , where, V is
 , is defined in
the voltage on the micro-heater, and I is the current. The effective heat flux, qeff
  ( P  Qloss ) / Ae . Here Qloss
 is the heat loss and Ae is the effective area of
our study as: qeff
heating. The heat loss Qloss includes the natural convection heat transfer between testing system
and the ambient environment. The heat loss is a function of temperature of the micro-heater as
shown in Figure S3.
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FIG. S3. Heat losses as a linear function of temperature of the heater. Correlation coefficient is
larger than 0.99.
The transient pressure drop oscillation posc in Figure 4 is derived by Eq. (2),
posc (t )  p(t )  p
(1)
,where  p is transient pressure drop as a function of time t and p is the averaged pressure
drop in relative long time (4 minutes).
Detailed flow resistance modeling during bubble growing/collapsing (BGC) processes is
described in this section. The high frequency periodic flow oscillation is primarily governed by
the non-linear bubble dynamic process in this study. A model of two-phase flow resistance has
been established to estimate the magnitude and frequency of HF-BGC in terms of flow resistance.
Each microchannel is modeled as an individual unit. A system-level modeling is coupled through
the cross-junction from two side-channels and a main-channel. Transient flow resistance during a
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period of flow oscillation inside a confined microchannel has been decomposed into several
states in a time domain. As shown in Figure 2, flow structures have been characterized to five
regimes, each of which has its own flow resistance model for individual regime.
Before bubble nucleation, the flow in microchannel is single-phase flow. For single-phase flow
inside microchannels, the Reynolds number, Re, is 2300, in this study. The Hagen-Poiseuille law
is applicable and adopted to estimate the hydraulic resistance, R1h  8 L / rh2 Ac , where is the
dynamic viscosity, L is the length of a microchannel, rh is the hydraulic radius and Ac is the
cross-section area. Overall, for a constant volumetric flow rate Q, the pressure drop Δp is
calculated as p  Q  R , where, R is the overall flow resistance of a microchannel. Flow
oscillation resistances are resulted from BGC process inside microchannels and can be
categorized into several typical flow regimes. Without considering the flow acceleration, the
overall hydraulic resistance of a single channel R is simplified to three components: the hydraulic
flow resistance in single phase flow R1h , the additional hydraulic flow resistance caused by
bubbles R2h and the capillary pressure induced by confined bubbles when attaching to walls
R3h . To simplify this complicated problem, we use a single-bubble model as shwn in Eq. (2),
which assumes that only one confined bubble in a microchannel. This assumption has been
validated by the visualization study: there is only one confined bubble in side-channels and
main-channels during the HF oscillation.
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 R1h , t  t0
 R  R (t ), t  t  t
 1h
2h
0
1
R(t )  
 R1h  R3h (t ), t1  t  t2
 R1h  R3h (t2 ), t2  t  t3
(2)
R is a non-linear function of time t in a period of BGC. The period is estimated by t  t3  t0 . A
confined bubble grows and reaches the walls in t1  t1  t0 during bubble growing period. Then,
the liquid film formed between vapor bubble and solid wall is evaporating and drying out in time
t2  t2  t1 . After that, the bubble lasts a short time t3  t3  t2 until the confined bubble
collapses due to the direct condensation. The time of ultrafast condensation, which is
approximately 0.2 ms and only a small fraction of a total period at approximately 5 ms, is
neglected because current model seeks to capture the transient flow resistance during the BGC
process.
According to Choi et al’s modified model1 , the hydraulic resistance of confined bubble is
estimated by Eq. (3) as following
R2 h (t1 )  (0.07  106.8 / Re) l v 2 / Q
(3)
where Reynolds number is defined as Re  Dh l v /  , and Dh is the hydraulic diameter,  l is
the mass density of liquid phase, v is the superficial velocity of fluid. This empirical equation fits
most cases of different Re from 50 to 700. The flow resistance resulted from capillary pressure or
surface tension force on the two ends of a bubble attaching on the walls is derived in Eq. (4)
R3h (t2 )  4 / (QDh )
(4)
where γ is the surface tension. The time of bubble growing prior to directly attaching on solid
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walls can be calculated from several proposed models2 as described in Eq. (5)
t1  rh2 / ( l Ja 2Cb 2 )
(5)
where,  l is the liquid thermal diffusivity. Jacob number is Ja  l clTsat / v h fg and Cb is a
constant with value from
4 /  to
12 /  .
An additional critical time during the thin liquid film evaporation is calculated as
t2  d film l h fg / q
(6)
where, q is the heat flux. h fg is the latent heat. The thickness of liquid film d film is estimated
in Eq. (7) as below
d film  0.67 DhCa2/3 / (1  3.35Ca2/3 )
(7)
Eq. (7) is called Taylor’s law from fitted experimental data3. Ca is the Capillary number and
estimated by Ca  0.445 /  . Finally, t3  t  t3 , which is primarily governed by the critical
size of bubble collapsing and has not been well studied in a micro-domain, is estimated from the
visualization study due to the complex physics. According to high speed camera videos,
t  N f / N FPS , here N f is the average number of frames in an oscillation period and N FPS is
number of Frames Per Second (FPS).
A numerically fitted equation is developed to estimate the transient flow resistance such that it
can be conveniently calculated during a period. As a switched oscillator, a logistic function is
adopted. The least square curve fitting method has been employed to calculate coefficients.
R(t )  R1h  R3h (t2 ) / (1  ae  bt )
(8)
where, a and b are coefficients, which are case sensitive. Eventually, this equation was used to
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generate curves in the Figure 4c. It shall be noted that this is one component channel
flow-resistance function within a period t during HF-TPOs.
The transient flow resistance is described by Eq. (8). The lumped model at system level
modeling is illustrated in Figure 4a. The high-aspect-ratio restrictor is designed to against the
overall resistance of main channel and side channel. As a result, the high resistance resulting
from a restrictor, R1 , is nearly constant during the two-phase oscillations4. Except for R1 , each
“flow resistor” in Figure 4c is treated as a periodic function during a BGC process. Moreover,
these functions are non-synchronous so that there are several measured time constants between
each individual BGC process in side-channels. R21 (t ) , R22 (t ) and R3 (t ) are transient flow
resistances in side-channels and main channel, respectively. Resistances are treated as periodic
functions as following: R21 (t )  R2 (t  N / f 2 ) and R22 (t )  R21 (t  td 2 ) as shown in Figure 4b,
where, N is a natural number. Oscillation frequency in a side channel is measured by high speed
camera as: f 2  N FPS / N f . Two side channels are non-synchronous with time difference, td 2 .
According to observations on bubble dynamics as illustrated in Figure 2 and Figure S4, the
bubbles inside the main-channels collapse as a result of jetting flow induced by the bubbles
collapsing in side-channels. A delay time, td 1  l / vw , is estimated as length l divided by
rewetting velocity, vw , which was measured by high speed camera.
Once component flow resistance in individual channels has been modeled as a function of time, a
microfluidic transistor model can be developed to explain the interactions between each
independent component in this system. As transistor-like function at the cross-junction, a
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corresponding negative correlation between the flow resistance R3 (t ) from “source” to “drain”
and the local pressure
p gate
at “gate”, which is proposed as a logistic function,
R3 (t )  R1h  R3h (t2 ) exp( t ( pbubble  pgate )) / (1  aebt ) , according to video research at Figure 2 and
Figure 3; where, R (t ) is the function as Eq. (7) with specific coefficient, pbubble is the local
pressure inside the confined bubble and  is a positive constant. This model indicates that the
bubble collapsing process in main channels due to the direct condensation always leads a sharp
local pressure reduction and consequently, results in accelerated fluid flow through the "gate" on
the walls. In this simplified modeling, is a large positive value (e.g. 1000) to indicate the
accelerated process of bubble collapse in main channel. This problem is simplified because of the
complicated mechanisms of phase change and critical diameter for bubble collapsing, however,
the time period of bubble collapsing is much shorter than the total period time and thereof is
negligible. It shall be noted that this problem is not a critical part of the model. The pressure drop
due to the bubble collapsing process is roughly estimated by Bernoulli’s equation:
pgate  pbubble   (v2j  vs2 ) / 2 , where is the liquid density, v j is the velocity of fluid jetting and
vs is average superficial velocity.
Finally, this model is programmed in MATLAB. The system-level pressure drop oscillation
function of time is plotted in Figure 4d and compared with experimental data. Great agreement
has been achieved.
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FIG. S4. Bubbles grow, attach to walls and collapse in side-channels. Arrows show the local flow
directions. The white rectangle highlights a confined bubble, which is dark black shadow in a
side channel under optical microscope. From 0 ms to 1.7 ms, two confined bubbles rapidly grow
in side-channels and induce reversal flows into sub-cooled upstream. The direct contact between
the sub-cooled water and bubbles results in condensation and hence the collapse of the confined
bubbles at 4.2 ms. The additional pressure drop established by collapsed bubbles accelerates the
sub-cooled liquid, which is pumped into the main-channel through the cross-junction. (Refer to
Movie S3)
Movies S1
A close look at the HF bubble growth/collapse processes in the main-channel near the
cross-junction. Flow oscillations (pulses) in side-channels cause the bubble collapse in the
main-channel. The working condition: mass flux at 224 kg m-2 s-1 and heat flux on wall at 111.7
W cm-2. This video are recorded at FPS 14035 and replayed at FPS 5.
Movies S2
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The interactions of high frequency bubble growth/collapse processes in the side-channels and
main-channel. Bubbles nucleate and grow in the side-channel until they are confined by walls.
Due to guidance of the converging gate, a confined bubble in side-channels expands only
towards upstream, where the bubble contacts with sub-cooled fluids at a velocity and rapidly
collapses due to the direct and promoted condensation. The jetting flow, which is pumped by the
pressure gradient established by the bubble collapse in side-channels, introduces the direct
condensation on the large confined bubble and causes the bubble collapse in the main-channel.
The working condition: mass flux at 150 kg m-2 s-1 and heat flux on wall at 184 W cm-2. This
video are recorded at FPS 14035 and replayed at FPS 5.
Movies S3
Fluid flows in the inlet area. The reverse flows due to bubble expansion in the side-channels are
recorded. Rapidly growing bubbles expand into the sub-cooled upstream, directly condense and
eventually collapse. The resulted pressure gradient pumps sub-cooled fluid into main-channels.
Due to light refractive index difference of two phases (liquid and gas), moving dark areas are
confined bubbles in this video while bright areas are liquid. The working condition: mass flux at
298 kg m-2 s-1 and heat flux on wall at 150 W cm-2. This video are recorded at FPS 14035 and
replayed at FPS 5.
References
1
C. W. Choi, D. I. Yu, and M. H. Kim, Int. J. Heat Mass Transf. 53 (23-24), 5242 (2010).
2
E. Ory, H. Yuan, A. Prosperetti, S. Popinet, and S. Zaleski, Phys. Fluids 12 (6), 1268
(2000); D. B. R. Kenning, D. S. Wen, K. S. Das, and S. K. Wilson, Int. J. Heat Mass
Transf. 49 (23-24), 4653 (2006).
3
P. Aussillous and D. Quere, Phys. Fluids 12 (10), 2367 (2000).
4
A. Kosar, C. J. Kuo, and Y. Peles, J. Heat Transf.-Trans. ASME 128 (3), 251 (2006).
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