Microfluid Nanofluid (2015) 18:357–366
DOI 10.1007/s10404-014-1454-3
RESEARCH PAPER
Serpentine and leading-edge capillary pumps for microfluidic
capillary systems
Roozbeh Safavieh • Ali Tamayol • David Juncker
Received: 4 March 2014 / Accepted: 26 June 2014 / Published online: 17 July 2014
Ó Springer-Verlag Berlin Heidelberg 2014
Abstract Microfluidic capillary systems operate using
capillary forces only and require capillary pumps (CPs) to
transport, regulate, and meter the flow of small quantity of
liquids. Common CP architectures include arrays of posts
or tree-like branched conduits that offer many parallel flow
paths. However, both designs are susceptible to bubble
entrapment and consequently variation in the metered
volume. Here, we present two novel CP architectures that
deterministically guide the filling front along rows while
preventing the entrapment of bubbles using variable gap
sizes between posts. The first CP (serpentine pump) guides
the filling front following a serpentine path, and the second
one (leading-edge pump) directs the liquid along one edge
of the CP from where liquid fills the pump row-by-row. We
Electronic supplementary material The online version of this
article (doi:10.1007/s10404-014-1454-3) contains supplementary
material, which is available to authorized users.
R. Safavieh A. Tamayol D. Juncker
McGill University and Genome Quebec Innovation Centre,
McGill University, 740 Dr. Penfield Avenue, Montreal,
QC H3A 0G1, Canada
R. Safavieh A. Tamayol D. Juncker
Biomedical Engineering Department, McGill University,
3775 University Avenue, Montreal, QC H3A 2B4, Canada
A. Tamayol
Harvard-MIT Division of Health Sciences and Technology,
Cambridge, MA, USA
A. Tamayol
Harvard Medical School, Boston, MA, USA
D. Juncker (&)
Department of Neurology and Neurosurgery, McGill University,
3801 University Street, Montreal, QC H3A 2B4, Canada
e-mail: [email protected]
varied the angle of the rows with respect to the pump
perimeter and observed filling with minimal variation in
pumping pressure and volumetric flow rate, confirming the
robustness of this design. In the case of serpentine CP, the
flow resistance for filling the first row is high and then
decreases as subsequent rows are filled and many parallel
flow paths are formed. In the leading-edge pump, parallel
flow paths are formed almost immediately, and hence, no
spike in flow resistance is observed; however, there is a
higher tendency of the liquid for skipping a row, requiring
more stringent fabrication tolerances. The flow rate within
a single CP was adjusted by tuning the gap size between
the microposts within the CP so as to create a gradient in
pressure and flow rate. In summary, CP with deterministic
guidance of the filling front enables precise and robust
control of flow rate and volume.
Keywords Capillary flow Passive pumping Microfluidics Pore network modeling Point-of-care
1 Introduction
Capillary systems are widely used for manipulating liquids
in heat pipes (Groll et al. 1998), regulating fluid flow in
low-gravity environment in space (Weislogel 2003), patterning biomolecules on surfaces (Delamarche et al. 1997,
2005; Lovchik et al. 2010), immunoassays (Cesaro-Tadic
et al. 2004), and diagnostic applications (Juncker et al.
2002; Gervais and Delamarche 2009; Gervais et al. 2011a;
Safavieh and Juncker 2013). Capillary systems are selfpowered and self-regulated by capillary effects and are thus
self-contained as there is no need for external pumps and
tubes. Capillary pumps (CPs) are essential components of
many capillary systems and provide a negative (suction)
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pressure for manipulating liquids. CPs are notably used in
assays to flow accurate volumes of sample (i.e., 1–10 lL)
over the assay surface to reach a high sensitivity (Gervais
et al. 2011b; Safavieh and Juncker 2013). The volume of
the CP serves as a metering device—the pump’s capacity
sets the volume of the liquid drawn in—and as a waste
reservoir for reagents (Juncker et al. 2002; Safavieh and
Juncker 2013). Indeed, since capillary systems operate by
applying a negative pressure, CPs are placed at the end of
the flow path and thus hold the liquid after it passes through
the reaction zones. For immunoassays, it is often desired to
have CPs that can provide a desired flow rate for a considerable amount of time (i.e., 1–30 min) during the
incubation step. CPs that generate a particular capillary
pressure and that do not significantly increase the overall
flow resistance as they are being filled are thus needed.
Successful designs include CPs with parallel microchannels arranged in an arborescent pattern (Juncker et al.
2002) or arrays of posts (Gervais et al. 2011b), or meshes
(Mirzaei et al. 2010). Postarrays and meshes are more
robust than branching channels, because they are less prone
to clogging or flow stopping as the liquid can progress
along many parallel, connected paths. In practice, pumps
are often covered to avoid evaporation and contamination
of the sample. As a consequence, bubbles can easily get
trapped, if the filling front of the liquid progresses faster
along one side of the CP and reaches the outlet prior to
filling the CP entirely. Trapped bubbles reduce the liquid
drawn into the CP and thus result in unreliable metering
and inaccurate incubation time during immunoassays. An
occurrence of bubble trapping is illustrated in the Movie S1
in the electronic supplementary information (ESI). The
issue of bubble trapping becomes even more pronounced
when multiple pumps are connected in series (Zimmermann et al. 2007).
A strategy to avoid bubble trapping in microfluidic
systems is to control the progression of the filling front.
This has been widely implemented in hydrophobic systems
where external pumps, pressure, vacuum, or centrifugal
forces are used to drive liquids, and where the progression
is controlled and preprogramed by adding hydrophobic
obstacles to the flow path (McNeely et al. 1999; Vulto et al.
2011). The control of the liquid front is more difficult in
self-filling, hydrophilic capillary systems because of the
constraints of maintaining self-filling. Zimmerman et al.
(2007, 2008) designed CPs with posts to enable filling and
imposing specific angles on the filling front-indented
sidewalls that slow down the progression along the edges.
However, their CP did not provide deterministic flow
control, and the filling front was shown to jump rows, and
in angled pump, the front could progress more rapidly on
one side than the other. Hence, trapping of bubbles could
not be excluded, and as a result, they introduced outlets
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made by a series of arch-shaped, hierarchical ‘‘bridges’’
with increasing size that fill in predetermined order. The
outlets further help minimize bubble trapping, but at the
expense of a weaker capillary pressure that reduces further
as the cross section of the bridging channels increases, thus
making the system susceptible to contaminations and
stopping of the flow. Moreover, this structure was supplemented with another structure in the form of a small serpentine side channel, which acted as a delay valve
(Zimmermann et al. 2008). This channel however comes at
a cost of reduced metering accuracy, and ultimately, each
new external element added to the CP affects the functionality of the circuit and may itself become a source of
failure.
Here, we introduce CPs that deterministically guide the
progression of the filling front via the geometry of the
microstructures of the CPs. These CPs comprise arrays of
microposts that are spaced differently along the X (parallel
to the bulk flow) and Y (perpendicular to the bulk flow)
axes so as to form rows along the Y axis. The liquid
entering the pump flows into the first row and fills the
narrow gaps between the posts on the downstream side of
the row, which acts as a temporary stop valve; the valving
effect is robust because of the negative pressure of the
filling front in the row (Zimmermann et al. 2007). The
liquid front is thus guided and fills the row entirely, and
only at the edge of the CP does it progress into the next
row. Two CP designs and strategies were developed that
allow a precise control over the advancement of the filling
front: first, a serpentine CP, where the front is guided in
an alternate direction in each row in a serpentine pattern;
second, a leading-edge CP, where the liquid initially is
guided to fill one of the edges of the CP and then fills
each row starting from this ‘‘leading-edge’’. The filling
front advances perpendicularly to the bulk flow along
each row, and as a result, both follow different paths. We
analyze and compare the two CPs and demonstrate the
robustness of both designs by changing the angle of the
rows. Next, we develop an electrical network analog for
the serpentine pump and use it to calculate the flow
resistance of the CP as the liquid fills it. Finally, a serpentine CP with continuously varying gap between the
rows was designed, and the theoretical and experimental
filling speeds were compared.
2 Materials and methods
2.1 Chemicals and materials
Polydimethylsiloxane (PDMS) was purchased from Dow
corning (SylgardÒ 184, MI, USA). Curing agent and
polymer base were manually mixed at a ratio of 1:10 and
Microfluid Nanofluid (2015) 18:357–366
cured for 8 h in an oven (Lindberg Blue M, Fisher Scientific) at 60 °C. A 1 mg/mL solution of Fluorescein sodium
salt (C20H10Na2O5) (fluorescein dye, Sigma-Aldrich, USA)
in water (Milli-Q purified water, Millipore, USA) was used
for characterizing the pumps; 600 Si wafers with the thickness of 650 ± 10 lm were purchased from Silicon Quest
International (San Jose, USA). Shipley-1813, a positive
photoresist, was purchased from Microchem Inc. (Massachusetts, USA).
2.2 CPs fabrication
We designed CPs using a layout editor software (CleWin
5, WieWeb software, Netherlands) and obtained 700 9 700
chrome mask with 65,000 dots per inch resolution from
Thin Metal Parts Inc. (Colorado Springs, USA). The CPs
were fabricated by first coating a Si wafer with a 2.8 lm
thick photoresist layer followed by photolithography and
deep reactive ion etching (DRIE, Tegal SDE 110, USA).
Next, we diced the wafers into individual chips using a
diamond scriber (Ted Pella Inc., USA). The Si chips were
rendered hydrophilic by flaming them for *45 s until
they became glowing red using a microtorch (Butane fuel
microtorch MT-51, Master Appliance, USA). A flat
2-mm-thick PDMS was cut to size, and vents were punched with a biopsy puncher (1 mm diameter, Ted Pella
Inc., USA). The PDMS was sealed onto the Si chip to
serve as a cover.
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3 Results and discussion
3.1 Design of the capillary pumps
The CPs comprise an array of 40 9 80 lm2 posts with
rounded edges in a staggered arrangement. The gap
between adjacent posts in two separate rows, W1, is larger
than the distance between two columns, W2, Fig. 1a. By
making the surface hydrophilic, the liquid spontaneously
fills the CP owing to the capillary pressure generated inside
the CP, and first enters the large space forming the first row
(t0), but then immediately fills the narrow gaps between
adjacent posts (t1). However, the liquid does not flow into
the next row, because the geometry of the posts acts as a
temporary capillary stop valve (Zimmermann et al. 2008).
The liquid front thus progresses along the next post and
2.3 Characterization of the capillary pumps
CPs were imaged using optical (LV150 industrial microscope, Nikon, Japan) and scanning electron microscopy (S3000N variable pressure SEM, Hitachi, Japan). The progression of the filling front in the CPs was visualized by
capturing the flow of fluorescein in deionized water. For
each experiment, 2.5 lL of solution was delivered to the
loading port and using a stereomicroscope (Leica MZ10f,
Leica Microsystems, Germany). Fluorescence videos were
recorded at a rate of 30 frames per second (infinity 3 CCD
camera, USA). Videos were converted into images using
VirtualDubMod Surround 1.6.0 (Lee 2010) and the velocity
of the filling front was established by measuring the change
in position between individual frames. At least three independent experiments were performed, and standard deviations were calculated. The flow resistance of the CP as a
function of the position of the filling front was calculated by
modeling the pump as a nodal network and using an in
house code programed in Matlab (v7.13, MathWorks Inc.,
USA). The advancing and static contact angles between DI
water and a flat Si surface, which was rendered hydrophilic,
were measured using a contact angle analysis equipment
(VCA optima, AST products Inc., USA).
Fig. 1 Schematics of serpentine and leading-edge CPs with arrays of
posts. a Schematic illustrating the progression of capillary flow
between the posts from time t0 to t2. The liquid fills the rows by
capillary pressure, as well as the gap between the posts, but the liquid
cannot jump into the next row as each pair of posts forms a capillary
stop valve, and thus, the liquid progresses along the row until it
reaches the edge of the pump and proceeds to fill the next row.
b Serpentine CPs comprise guiding structures at the end of each
alternating row to guide the liquid filling front in a serpentine pattern.
c Leading-edge CPs draw the liquid into the narrow edge with a
capillary pressure (DP2), and then sequentially into each of the rows,
which have the highest capillary pressure (DP1), starting with the first
one. Finally, the wide gap (DP3) is filled up to the exit. The path of the
filling front is indicated by the red arrows and the bulk flow by the
white hashed arrows; note the difference in the paths. d A capillary
system with a loading port, a CRV, a flow resistor, and a vent is used
to test the CPs (color figure online)
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further along the row (t2), until it fills the next narrow gap,
and so on. Oval shaped posts were chosen to both perform
as a temporary stop valve, preventing the liquid to leak
from one row to the other, and avoid trapping bubbles
while filling the posts one after the other in the same row.
It may, however, also be possible to use rectangular posts,
but the flow resistance for the same gap would be higher
and the fabrication becomes more difficult. The experimental data confirmed the effectiveness of the oval shaped
posts in temporarily stopping of liquid. The filling front
thus fills the entire row, and only after reaching the edge
of the CP, advances to the next row. The CP is thus filled
row-by-row.
Two distinct CPs were designed that guide the filling
front in two different ways. The first is called a serpentine
pump, because it guides the liquid front along a serpentine
path, Fig. 1b. As it reaches the edge of the CP, the liquid
front is guided to the next row while guiding structures are
included to prevent premature leakage into the subsequent
row. The second CP is called leading-edge pump, because
the liquid front is led along one edge of the CP and starts
sequentially filling each row from this edge, Fig. 1c. The
liquid front is guided by reducing the gap between the
‘‘leading’’ edge and the posts compared with the other
edges—hence creating a higher capillary pressure at the
leading edge—and thus forcing the liquid to fill each row
from that edge. In both CPs, the angle of the rows is varied
and tilted between 08 to 458 relative to the edge of the CP.
Manipulation of the filling front along the required directions allows filling of CPs with various geometries and thus
could save the footprint of the microfluidic circuits. Also,
by varying the filling front direction, the number of posts
per unit area of the CP and its capacity can be adjusted.
To test the two CP designs, simple capillary systems
consisting of a loading port, a capillary retention valve
(CRV), an external flow resistor, and the respective CP
were built, Fig. 1d (Juncker et al. 2002; Hitzbleck et al.
2011). PDMS is widely used in the fabrication of microfluidic systems but it is hydrophobic natively, and although
it can be hydrophilized by plasma treatment, it reverts to a
hydrophobic state within hours. To avoid complications
owing the changes in surface wettability of PDMS, the
chips were made out of Si. Si is capped by a native SiO2
layer, which can be hydrophilized readily, for example, by
plasma treatment, and remains hydrophilic over extended
periods of time. This enables us to preserve the chips over
for few weeks and test them, when they are needed. The
system was then sealed with a hydrophobic PDMS cover
that contained a loading port to fill samples, and one
opening for venting.
The capillary pressure, DP, in the CPs was estimated
using the pressure equation for rectangular cross sections
(Delamarche et al. 1998):
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DP ¼ c
cos þ cos cos þ cos b
t
l
r
þ
w
h
ð1Þ
where c is the surface tension of the liquid, i = b, t, l, r are
the contact angles of the liquid–air meniscus on the bottom,
top, left, and right walls, respectively, and w and h are the
width and height of the rows formed between the pillars.
The actual pressure is expected to be lower owing to the
fact that the channel is not continuous, but lined by posts
with narrow gaps.
The narrow gaps, W2, between the pillars were constant
and equal to 15 lm for all CPs except the one with the
variable capillary pressure, which was equal to 25 lm, Fig.
S1A in ESI. The width of the rows, W1, was varied from 34
to 130 lm, depending on the CP. The depth of the pumps
was fixed during the DRIE fabrication step and was typically *100 lm. The spiral shaped external resistor had a
width of 30 lm and an overall length of 3,400 lm, Fig.
S1C in ESI. The footprints of the CPs are from 4.2 9 4 to
4.2 9 4.5 mm2 with a porosity ranging from 47 to 73 %
and thus can accumulate between 1.02 and 1.65 lL. To
have CPs with higher volumes, one could either increase
the height of the structures or instead expand the footprint
Fig. 2 Micrographs of CPs and the capillary system. a A serpentine
CP with a 15° angle with ridges at the edge. b A leading-edge CP with
15° showing the leading-edge gap and the trailing-edge gap. Note that
the posts are cut at the edges here. c Overview of the capillary system
used to test the CPs. The PDMS cover with a vent was aligned to the
chip manually using an stereomicroscope
Microfluid Nanofluid (2015) 18:357–366
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of the CPs. The deeper the structure, the higher the aspect
ratio of the posts, which is difficult to extend beyond 10 in
Si. Increasing the area of the CPs leads to an increase in the
overall footprint of each device, and an increased cost per
unit. For the leading-edge CPs, Fig. 1c, we adjusted the
gaps between rows that generate DP1, the gap of the
leading edge (left and top) that results in DP2, and the gap
of the trailing edge (bottom and right) that creates DP3, so
that the absolute value of the capillary pressures was
jDP1 j [ jDP2 j [ jDP3 j
ð2Þ
and the PDMS cover. The gap of W2 acts as a temporary
stop valve and prevents the filling front from jumping from
one row to the next, until the entire row is filled. The
robustness of these temporary stop valves can be increased
by adjusting 3 different parameters: (i) by decreasing the
ratio of W2/W1; (ii) by making the surface of the chip less
hydrophilic; and (iii) by increasing the capillary pressure
within a row to create a strong negative pressure.
3.2 CPs with rows at different angles
Figure 2 shows the fabricated serpentine and leadingedge CPs as well as a capillary system used in the experiment, Fig. 2a–c.
The functionality of the CP depends on the balance of
capillary pressures, and the wettability of the Si structures
The deterministic control over the filling path allows
designing serpentine and leading-edge CPs with angled
rows, Fig. 3. We studied the effect of the angle on the flow
rate of the serpentine CPs. The three serpentine CPs with
row angles relative to the edges of a = 0°, 15°, and 45° are
Fig. 3 Time lapse imaging showing the progression of the filling
front. Three serpentine CPs with a 0°, b15°, c 45° angle are shown
and d a leading-edge CP with a 15° angle. Dilute solution of
fluorescein in distilled water was used to perform all the tests. The red
arrows show the direction of the liquid filling front, whereas the white
arrows demonstrate the direction of the flow. Real-time movies of the
filling as shown in the ESI. The scale bar is 1 mm (color figure
online)
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shown in Fig. 3a–c. In general, no significant difference
was observed between the flow rates through the pumps.
We also fabricated a leading-edge CP with an angle of
a = 15° shown in Fig. 3d. Real-time movies S2-6 in (ESI)
illustrate the filling of different CPs. Angled rows afford
greater flexibility in directing the filling front, and a 45°
angle further minimizes the risk of bubble trapping at the
outlet as the final rows are very short and row jumping less
likely to occur.
3.3 Flow model of the CPs
The flow rates of the system used here are few tens of
nanoliters per second only; thus, the flow is laminar and
corresponds to what is known as creeping flow. Under
these conditions, the relationship between capillary pressure, DP, the flow resistance, R, and volumetric flow rate,
Q, of the system becomes (Juncker et al. 2002; White 2005;
Safavieh et al. 2011):
R¼
DP
Q
ð3Þ
This is similar to the electrical relationship between
voltage, resistance, and current; however, in this case, the
flow resistance of the CP continuously changes as the filling
front progresses, while flow front and bulk flow follow
different paths. Both of these factors need to be considered
when calculating the flow resistance and flow rate.
The flow resistance of rectangular microchannels is
defined as (White 2005; Akbari et al. 2009):
3 pw1 wh
192h
Rc ¼
Pa s m3
1 5 tanh
pw
2h
12lL
ð4Þ
where L, w, and h are the length, width, and height of the
microchannel, respectively, and l is the viscosity of the
liquid. The CP can be considered as an array of unit cells
that allow for flow along the row and across the row
through the small gaps, Fig. 4a. To simplify the calculation, the gap between the posts can be treated as straight
conduits. The resistance of each unit cell in a row can be
lumped into a single resistance Rh, and the resistance along
the narrow gap between posts as Rv, Fig. 4b. Tamayol et al.
(2013) have shown that the suitable length scale for
determining the pressure drop between adjacent cylinders
is the minimum gap size. Thus, to simplify the analysis, we
neglected the effects of the rounded edges in the post and
assumed that the posts are completely rectangular shape
with 40 9 80 lm2 and that the effective gaps to calculate
Rh and Rv are W1 = 35 lm and W2 = 15 lm, respectively.
The values of Rh and Rv can then be obtained using Eq. 4.
More details are provided in ESI.
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The regular structure of the CP lends itself to a pore
network modeling approach, common in the analysis of
flow through porous media, which is an equivalent to network approach as used in electronics (Ho et al. 1975;
Chapuis and Prat 2007; Prat 2011). In this approach, the
pump can be modeled as a network of nodes interconnected
by Rh and Rv resistors, Fig. S2 and S3. For a typical serpentine CP, Rh = 3.99 9 1011 Pa s m-3 and Rv = 9.90 9
1011 Pa s m-3, ESI. The staggered architecture of the CP
with a = 0°, Fig. 4b, was modeled as an in-line array of
pillars, so that all the four resistances connect to a single
node, thus simplifying the calculations. This approximation
is justified by the fact that the shift in the structure of the
resistance R2h is one-fifth of that of Rv and thus the contribution of the lateral flow resistance between two rows is
relatively negligible to the resistance of the narrow gap.
To calculate the equivalent resistance of the network
model, each node was assigned a coordinate from 1 to
(j 9 m), where j and m are the number of rows and columns, respectively, as shown in Fig. 4b. Using Kirchhoff’s
circuit laws (Smith 1998), a conductance matrix of the
1
form C P ¼ Q was constructed, where C ¼ R
is the
conductance matrix and has the dimension of
(j 9 m) 9 (j 9 m), and once is multiplied by the matrix of
nodal pressures, P, the summation of the flow rates flowing
in and out of each node is obtained with P and Q having the
dimensions of (j 9 m). Assigning an arbitrary value for the
Qj,m, and the same negative value for Q1,1, we calculated
the associated values for Pj,m, and P1,1 by solving the
matrix. Finally, the resistance between the node 1 and
j 9 m of the pump can be calculated as:
R¼
Pð1; 1Þ Pðj; mÞ
Qj;m
ð5Þ
The detailed algorithm is provided in the ESI. The total
flow resistance of the circuit is the summation of the
resistance associated with the CP and the connecting
channel.
Figure 4c illustrates the resistance of the CP as the
function or position of the filling front. For a serpentine CP
with a = 0°, the resistance increases rapidly, as the filling
front fills the first row, because here, the bulk flow and filling
front follow the same path, and the resistance reaches a local
maximum when the liquid reaches the edge. Indeed, as the
liquid travels back along the second row, the two rows
become connected by an increasing number of narrow gaps,
which all serve as parallel flow paths, and hence, the flow
resistance diminishes as the liquid front advances. In fact,
the flow resistance of the CP shown here diminishes between
the first and seventh row, from R = 6.98 9 1012 to
R = 1.87 9 1012 Pa s m-3, but then starts to increase
continuously until it is filled entirely as the vertical
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Fig. 4 Flow resistance of a
serpentine CP. a Schematic
showing the liquid flow of a CP.
b Lumped resistance model of
the CP. c Average flow
resistance for the filling front at
each row based on nodal
analysis of the electrical circuit
equivalent, see ESI. The
calculated flow resistance
decreases sharply as the liquid
fills the first few rows of the CP,
and starts to increase linearly
after the seventh row.
d Comparison of theoretical and
experimental flow rates using a
capillary pressure 1,210 Pa for
fitting. The error bars in the
experimental data are the RMS
values of three independent
experiments
resistance of the narrow gaps dominate the overall flow
resistance. Our analysis showed that for after a spike in
resistance in the first row that has no parallel flow path, the
resistance diminishes in subsequent rows as many parallel
flow paths exist, and then gradually increases in a linear
manner up to R = 7.53 9 1012 Pa s m-3, Fig. 4c. The
overall flow resistance of the CP remains small compared
with microchannels, and in a full capillary circuit, the fluctuations would be further minimized owing to by upstream
resistances that may be much larger than the one of the CP.
Alternatively, using a CP at a 45° angle, or an inlet placed
closer to an edge, the large jump in flow resistance can be
avoided as the initial resistance increase is due to the filling
of the first, long row in the design shown in Fig. 4.
The normalized calculated flow rates of the CP obtained
from the computed flow resistance using the capillary
pressure as a free parameter set to P = 1,210 Pa matches
and reproduces the flow rates observed experimentally,
Fig. 4d.
The average capillary pressure for each unit cell of the
typical serpentine and leading-edge CPs (shown in Fig. 3)
is constant and can be calculated by dividing the change in
the Gibbs free energy of the unit cell (before and after
filling by the liquid) over the volume of the liquid, which is
transported into the cell. The equation can be formulated as
(Leverett 1941; Hassanizadeh and Gray 1993):
DP ¼ cLV ðcos ht DASVt þ cos hb DASVb þ 2 cos hr DASVr Þ
Vl
ð6Þ
where cLV is the surface tension of the liquid (i.e., water),
hi=t,b,r are the contact angle of the liquid–air meniscus in
the top, bottom, and right surfaces, DASVi¼t;b;r are the
respective changes of interfacial areas between solid, and
air before and after filling of the cell for the top, bottom,
and right surfaces, and Vl is the volume of the liquid
transferred in the unit cell (see ESI for more details). From
experiments shown in Figs. S6 and S7, we observe that the
advancing contact angle of a freshly treated flat Si (associated with the right and bottom surfaces) is 26.4° ± 3.3°,
and the advancing contact angle of a layer of PDMS
(associated with the top surface of the CP) is
118.7° ± 1.7°; thus, the average capillary pressure generated calculated from Eq. (6) equals to DP = 3,375 Pa (see
the ESI for the details of the calculation). Equation 6,
however, only constitutes an approximation as it does
neither include corner flow effects nor the change in contact angles upon rapid filling (Weislogel and Lichter 1998).
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We measured the flow rates for the pump and repeated
the experiments at least three times. We observed the
variation in the flow rates in the order of 20 % as shown in
the error bars of the experiment, Fig. 4d. Our experiments
allowed determining the relative change of the flow rate
over time, and the capillary pressure was used as a free
parameter to correlate flow rate with calculated flow
resistance and was DP = 1,210 Pa. This value corresponds
to an equivalent of a capillary pressure generated from a
glass tube with the diameter of 80 lm (which can be calculated analytically), which is commensurate with the
dimensions of the CP used here with a depth of *100 lm
and a distance of *75 lm between the rows.
The flow rate of the CP is governed by the capillary
pressure and the flow resistance, which are both dominated
by the geometry of the CP. The capillary pressure is
dominated by the cross section of the rows, and for deep
pumps, mainly by W1, and is independent on the overall
dimension. The flow resistance is mostly dominated by W2
and the number of posts and gaps between them. The flow
resistance of CPs with nonrectangular geometries can be
calculated using the electrical network analogy by changing the number of nodes and resistors on each line to match
the width of the pump. But as mentioned, in most cases, the
external resistance will dominate the overall flow resistance, and the contribution of the CP may be negligible.
3.4 Preprogramed serpentine CPs with varying
capillary pressure
In many applications, for example, microfluidic immunoassays (Parsa et al. 2008), it is desirable to adjust the flow
rate for various steps of the assay. The control over the
filling front enables to design CPs with variable capillary
pressure. We designed CPs with a gradual decrease in the
capillary pressure by increasing the width of the rows from
34 to 130 lm with incremental steps of 3 lm, Fig. 5a and
Movie S7 in ESI.
Figure 5b illustrates the variation in the average flow
rates from one row of the CP to the other. Although the
capillary pressure decreases continuously, initially, the
speed of the filling front increases rapidly from 3 mm s-1
in the 1st row to 5 mm s-1 in the 8th row of the CP. This
value corresponds to a growth from 33 to 62 nl s-1 in the
flow rate of the CP. This increase is due to the decrease in
the flow resistance, as explained in the section describing
on the flow model of the CP. Subsequently, the speed of the
filling front diminishes continuously to 2 mm s-1 in the
last row, which corresponds to a flow rate 23 nl s-1.
It is known that the contact angle of a liquid with a
surface increases for increasing flow velocity (Hoffman
1982; Seebergh and Berg 1992; Weislogel and Lichter
1998). Bracke et al. (1989) empirically established a
123
Microfluid Nanofluid (2015) 18:357–366
Fig. 5 Preprogramed serpentine pumps with variable capillary
pressure. a Fluorescence image of a CP filled with a green fluorescent
solution revealing the increasing width of the rows that lead to a
capillary pressure gradient. b Flow rates in each of the rows measured
by video microscopy. The error bars are standard deviations of seven
experiments. The scale bar is 1 mm (color figure online)
relationship between contact angle and capillary number
Ca = Ul
c , with U being the velocity of the filling front as:
cosha coshd
¼ 2 Ca0:5
cosha þ 1
ð7Þ
With ha being the static advancing contact angle and hd
the dynamic advancing contact angle. The static contact
angles of Si and PDMS surfaces are ha = 26.4° and
ha = 118.7°, respectively, and thus, the dynamic contact
angles of these surfaces are expected to increase in the 8th
and last row to 27.3° and 29° for the Si, and to 118.8° and
119.1° for PDMS, respectively, Table S1. Thus, when calculating the capillary pressure at different rows, one needs
not only to consider the change in geometry, the increase in
flow resistance, but also the change in contact angle.
If we neglect the increasing flow resistance of the capillary pump and other dissipative energies, in theory, the
average ratio of the generated capillary pressure between
the 34th row and the 8th row of the pump is expected to be:
DP8
¼ 2:0
DP34
ð8Þ
where DP8 and DP34 are the average capillary pressure
generated in the 8th and 34th row, respectively. This ratio
is orderly consistent with our experimental data QQ348 ¼ 2:7,
where Q8 and Q34 are the average flow rates in the 8th and
34th rows, respectively, and the difference may be in part
due to the approximations that were made. The detail of the
calculation for the ratio of the capillary pressures can be
found in ESI.
4 Conclusions
We introduced novel designs for CPs that deterministically
guides the filling front along a predetermined path along
rows and at the edges. We implemented two novel
Microfluid Nanofluid (2015) 18:357–366
architectures, namely a serpentine and leading-edge filling
path, respectively. The flexibility and robustness of the
design was highlighted using CPs with different filling
angles and gradients of pressures. A pore network analysis
model was developed to calculate the flow resistance and
was used to predict the pressure and flow properties of the
different CPs. In comparison with previous CPs, our serpentine and leading-edge CPs can control the progression
of the filling front row-by-row deterministically; thus, they
can (i) robustly avoid trapping bubbles and be used as
capillary driven metering elements, and (ii) they can be
used to accurately change the capillary pressure of the CPs
at each row. Moreover, it is expected that with this technology, the shape of the CP may be adjusted as well, and
various geometry could be made, for example, to modulate
the flow resistance as the CP is being filled.
The developed CPs are robust and can further expand
the functionality of capillary systems. For example, by
tailoring the capillary pressure of the CP and the flow
resistance of the circuit, the filling time may be tuned from
a few seconds to tens of minutes, while accurately metering
the liquid. CPs with gradients of capillary pressure may
allow varying the liquid flow rate during the incubation
steps of bioassays to minimize the time to saturation (Parsa
et al. 2008). Also, it will be possible to make CPs that
produce a constant flow rate, by compensating the change
in resistance with an increase in capillary pressure.
We believe programmable CPs will expand the possibilities of passive, capillary driven microfluidics for carrying out advanced fluidic operations autonomously, and
this may open new opportunities for the use of capillary
systems in immunoassays, point-of-care diagnostics and
other areas.
Acknowledgments We would like to acknowledge funding from
NSERC, CIHR and CFI, and the assistance of the McGill Nanotools
and Microfab Laboratory (funded by CFI, NSERC and Nanoquebec).
We also acknowledge Prof. Elizabeth Jones for allowing us to use the
facilities in her lab. We also acknowledge Arash Kashi, Mohammad
Ameen Qasaimeh, and Mohammadali Safavieh for their helpful discussions. DJ acknowledges the support from Canada Research Chair.
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