10
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 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 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), pat- terning 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 self- powered 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) 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] 123 Microfluid Nanofluid DOI 10.1007/s10404-014-1454-3

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RESEARCH PAPER

Serpentine and leading-edge capillary pumps for microfluidiccapillary systems

Roozbeh Safavieh • Ali Tamayol • David Juncker

Received: 4 March 2014 / Accepted: 26 June 2014

� 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

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), pat-

terning 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 self-

powered 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)

Electronic supplementary material The online version of thisarticle (doi:10.1007/s10404-014-1454-3) contains supplementarymaterial, 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]

123

Microfluid Nanofluid

DOI 10.1007/s10404-014-1454-3

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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 con-

siderable 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 microchan-

nels 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 (Zimmer-

mann 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

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 supple-

mented with another structure in the form of a small ser-

pentine 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 func-

tionality 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 ser-

pentine 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 (Sylgard� 184, MI, USA). Curing agent and

polymer base were manually mixed at a ratio of 1:10 and

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cured for 8 h in an oven (Lindberg Blue M, Fisher Scien-

tific) 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 thick-

ness of 650 ± 10 lm were purchased from Silicon Quest

International (San Jose, USA). Shipley-1813, a positive

photoresist, was purchased from Microchem Inc. (Massa-

chusetts, 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 pun-

ched with a biopsy puncher (1 mm diameter, Ted Pella

Inc., USA). The PDMS was sealed onto the Si chip to

serve as a cover.

2.3 Characterization of the capillary pumps

CPs were imaged using optical (LV150 industrial micro-

scope, Nikon, Japan) and scanning electron microscopy (S-

3000N variable pressure SEM, Hitachi, Japan). The pro-

gression 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 inde-

pendent experiments were performed, and standard devia-

tions 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).

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

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 experi-

mental 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 direc-

tions 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 micro-

fluidic 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):

DP ¼ �ccosbþ cost

wþ coslþ cosr

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 typi-

cally *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

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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

jDP1j[ jDP2j[ jDP3j ð2Þ

Figure 2 shows the fabricated serpentine and leading-

edge CPs as well as a capillary system used in the exper-

iment, Fig. 2a–c.

The functionality of the CP depends on the balance of

capillary pressures, and the wettability of the Si structures

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

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 pres-

sure, DP, the flow resistance, R, and volumetric flow rate,

Q, of the system becomes (Juncker et al. 2002; White 2005;

Safavieh et al. 2011):

R ¼ DP

Qð3Þ

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):

Rc ¼wh3

12lL

� �1� 192h

p5wtanh

pw

2h

� �� �� ��1

Pa s m�3�

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 calcula-

tion, 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.

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 net-

work 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 ser-

pentine 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 Rh

2is one-fifth of that of Rv and thus the contri-

bution 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 col-

umns, respectively, as shown in Fig. 4b. Using Kirchhoff’s

circuit laws (Smith 1998), a conductance matrix of the

form C� P ¼ Q was constructed, where C ¼ 1

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:

R ¼ P 1; 1ð Þ � Pðj;mÞQj;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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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 fluc-

tuations 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 cos ht � DASVtþ cos hb � DASVb

þ 2� cos hr � DASVrð 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 (asso-

ciated 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 gener-

ated 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 con-

tact angles upon rapid filling (Weislogel and Lichter 1998).

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

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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 cal-

culated 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 chang-

ing 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 resis-

tance, and the contribution of the CP may be negligible.

3.4 Preprogramed serpentine CPs with varying

capillary pressure

In many applications, for example, microfluidic immuno-

assays (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

relationship between contact angle and capillary number

Ca = Ulc

� �, with U being the velocity of the filling front as:

cosha � coshd

cosha þ 1¼ 2� Ca0:5 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 cal-

culating 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 cap-

illary 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

DP34

¼ 2:0 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 Q8

Q34¼ 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

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)

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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 ser-

pentine 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 tech-

nology, 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 possi-

bilities of passive, capillary driven microfluidics for car-

rying 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 dis-

cussions. DJ acknowledges the support from Canada Research Chair.

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