Article pubs.acs.org/Langmuir
Capillary Flow Resistors: Local and Global Resistors Jean Berthier,*,†,‡ David Gosselin,†,‡ Andrew Pham,§ Guillaume Delapierre,†,‡ Naceur Belgacem,∥ and Didier Chaussy∥ †
University Grenoble Alpes, F-38000 Grenoble, France Department of Biotechnology, CEA, LETI, MINATEC Campus, F-38054 Grenoble, France § Cain Department of Chemical Engineering, Louisiana State University, South Stadium Road, Baton Rouge, Louisiana 70803, United States ∥ LGP2, Grenoble-INP Pagora, University of Grenoble, 38402 Saint-Martin d’Hères, France ‡
S Supporting Information *
ABSTRACT: The use of capillary systems in space and biotechnology applications requires the regulation of the capillary flow velocity. It has been observed that constricted sections act as flow resistors. In this work, we also show that enlarged sections temporarily reduce the velocity of the flow. In this work, the theory of the dynamics of capillary flows passing through a constricted or an enlarged channel section is presented. It is demonstrated that the physics of a capillary flow in a channel with a constriction or an enlargement is different and that a constriction acts as a global flow resistor and an enlargement as a local flow resistor. The theoretical results are checked against experimental approaches.
dimensions of the cross section of the channel.18 The geometry of open channel has not yet been considered. In a uniform cross-section channel, the capillary flow velocity decreases as a function of the inverse of the square root of time. We show here that a well-dimensioned constriction can force the flow velocity to a nearly constant value downstream of the contraction. We also show that a well-dimensioned enlargement can produce the same result in the enlarged section. To determine the cross-sectional changes needed to adjust the flow velocity to the requirements of capillary point-of-care systems, we focus in this work on the dynamics of a capillary flow passing through a constriction or an enlargement of the channel. We derive closed-form expressions for the velocity and travel distance of a spontaneous capillary flow (SCF) in an open, rectangular cross-section channel with a sudden constriction (Figure 1) or a sudden enlargement (Figure 2). As there are sharp 90° corners at the bottom of the channel, the contact angle is assumed to be in the range 45°−90° in order to avoid precursor corner filamentsoften called Concus−Finn filaments.19−23 The analytical expressions are favorably checked by an experimental approach using open, rectangular channels milled in plastic. It is found that a constriction acts as a flow resistor, i.e., reduces the velocity of the flow once the interface has passed the constriction. It has a “global” effect downstream of the
1. INTRODUCTION The developments of capillary-driven microsystemsfor space and biotechnology applicationsrequire a precise passive control and regulation of the flow velocity. However, because space devices must be very lightweight and biotechnological devices targeting point-of-care and home care must be portable and stand alone, auxiliary aids such as actuated valves or regulated pumps are not the solution. In the absence of active methods, a passive control and regulation of the velocity must be performed by an adapted geometry. A first example is that of Suk and Cho, who proposed the use of hydrophobic obstacles to slow down a capillary flow.1 In this work, we show how simple changes of cross-sectional dimensions can be used to regulate the velocity of the capillary flow. In fact, dynamics of capillary flow has been widely investigated for uniform-section, closed channels.2−10 However, in the case of nonuniform cross section, the reported approaches are mostly experimental in the geometry of closed channels.11−15 For example, it has been experimentally shown that the introduction of a section of very small dimensions acts as flow resistor, i.e., reduces the capillary flow velocity. From a theoretical standpoint, Yang and colleagues proposed a modification of the Laplace pressure expression to take a sudden decrease of the cross section of a capillary channel.16 Elizalde et al. have theoretically investigated the effect of continuous, progressive change of cross section of closed channels.17 Finally, Erickson and coworkers numerically analyzed the effect of change of the © XXXX American Chemical Society
Received: June 8, 2015 Revised: December 15, 2015
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DOI: 10.1021/acs.langmuir.5b02090 Langmuir XXXX, XXX, XXX−XXX
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Multiplying both sides of eq 3 by the cross-sectional area S = hw, one finds eq 1. On the other hand, the friction force is the sum of the friction in all the segments preceding the interface. An approximation of the friction is Fdrag, k =
⎡ 2h w ⎤ Fdrag, k − 1 + 6μVk(z − zk − 1)⎢ + k⎥ h⎦ ⎣ wk i = 1, k − 1
∑
(4)
Figure 1. Sketch of a capillary channel with a sudden constriction.
where Vk is the velocity in the section kwhich is not constant but decreases from the beginning to the end of section k. This approximation is based on quadratic profiles at the wall. Let us decompose the drag force into two parts: the one on the vertical walls and the one on the bottom plate. Assuming quadratic velocity profile along a horizontal plane and semiquadratic velocity profile along a vertical plane, the wall friction can be approximated by V Fsides ≈ 6μ (2h)z w V Fbottom ≈ 6μ wz h
Relation 5 immediately leads to expression 4 for the drag force. A more rigorous approach based on spatial Fourier series has been proposed by Ouali et al., Yang et al., and Baret et al.9,16,30 The two expressions produce very similar results (see Supporting Information). Let us denote r = w2/w1, the “resistor” ratio. For convenience, let us note
Figure 2. Sketch of a capillary channel with a sudden enlargement.
constriction. On the other hand, an enlargement acts locally as a reducer of the capillary forcewhen the interface passes through the enlargementand can be seen as a “local” flow resistor.
2. THEORY We consider open, rectangular U-grooves and assume a piecewise constant cross-section microchannel, such as that of Figures 1 and 2. We follow the approach pioneered by Lucas, Washburn, and Rideal in the 1920s for cylindrical channels2−4 and by Rye and co-workers for open V-grooves.5−8 The dynamics of the capillary flow is deduced by the balance between the capillary force at the tip of the flow and the friction force on the solid walls. Inertia forces can be neglected at the microscale when the hydraulic diameter is sufficiently small, as in the case of biotechnology, and in a zero-gravity environment, as in the case of space.23,24 In each segment of uniform cross section k (k = 1,3), the capillary force is25,26 Fcap, k = (wk + 2h)γ cos θ − wkγ = γhfk
w 2h 2 + 1 = + e1 w1 h e1
C2 =
w 2h 2 + 2 = + e2 w2 h e2
(6)
Fdrag,1 = 6μV1zC1
(7)
When the interface is located in the second segment, the drag force is
(1)
Fdrag,2 = 6μV1L1C1 + 6μV2(z − L1)C2
(8)
For an interface located in the third segment, after the resistor, Fdrag,3 = 6μV1L1C1 + 6μV2L 2C2 + 6μV3[z − (L1 + L 2)]C1
(2)
(9)
where e1 and e2 are, respectively, the aspect ratios e1 = w1/h and e2 = w2/h. Note that, in the geometry of Figure 1 and considered here, we have w1 = w3. Expression 1 can also be derived from the Laplace pressure in closed rectangular channels proposed by Delamarche et al.27 and verified by Safavieh and colleagues.28,29 Extrapolating to the case of open channel, and using for the free surface a contact angle value of π, the Laplace pressure is ⎡ cos θ − 1 2 cos θ ⎤ |PC| = γ ⎢ + ⎥ ⎣ h w ⎦
C1 =
The constants C1 and C2 are nondimensional and represent the friction force. For an interface in the entrance region
where h is the depth of the channel, θ is the contact angle, γ is the surface tension, and f k is a nondimensional capillary characteristic number defined by fk = (ek + 2)cos θ − ek
(5)
In eqs 7, 8, and 9, the velocities V1, V2, and V3 are related by the mass conservation equation V1w1 = V2w2 = V3w1
(10)
Upon substitution in eqs 8 and 9, we obtain Fdrag,2 = 6μV2[rL1C1 + (z − L1)C2] ⎤ ⎡L C Fdrag,3 = 6μV3⎢ 2 2 + (z − L 2)C1⎥ ⎦ ⎣ r
(3) B
(11)
DOI: 10.1021/acs.langmuir.5b02090 Langmuir XXXX, XXX, XXX−XXX
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if r ≪1. In such a case, relation 15 yields a velocity jump J2,3 ≈ −(γ/6μ)(2h cos θ/L2C2). Note that the magnitude of the positive jump at the constriction entrance is much higher than that of the exit: if r ≪1, the ratio J1,2/J2,3 is given by
Neglecting inertia, we equate the capillary and drag forces to obtain the velocities V1, V2, and V3 as functions of the penetration distance z ⎡ γ ⎤ (w + 2h) cos θ − w1 V1 = ⎢ ⎥ 1 ⎣ 6μ ⎦ zC1
J1,2 J2,3
⎡ γ ⎤ (w + 2h) cos θ − w2 V2 = ⎢ ⎥ 2 ⎣ 6μ ⎦ [rL1C1 + (z − L1)C2] ⎡ γ ⎤ (w + 2h) cos θ − w1 V3 = ⎢ ⎥ L1C ⎣ 6μ ⎦ ⎡ 2 2 + (z − L )C ⎤ 2 1⎦ ⎣ r
=−
L 2C 2 1 L1C1 r
(16)
This ratio varies as 1/r. Conversely, for an enlargement, r > 1, relation 14 predicts a velocity decrease at the entrance of the enlarged segment, and relation 15 predicts a velocity increase when the advancing interface leaves the enlarged section. If r ≫1, the negative jump at the entrance of the enlarged channel is J1,2 ≈ −(γ/6μ)(2h cos θ/ L1C1), and the positive jump at the exit of the enlarged channel is J2,3 ≈ −(γ/6μ)(2h cos θ/L1C1) = −J1,2. In such a case the velocity is restored to its value before the enlarged section. Figure 3 illustrates the effects of constriction and enlargement.
(12)
The volumetric flow rate can easily be derived from relation 12, by remarking that Qk = Vkhwk. Introducing the nondimensional capillary force (eq 2) in eq 12 yields ⎡ γ ⎤ f V1 = ⎢ ⎥h 1 ⎣ 6μ ⎦ zC1 ⎡ γ ⎤ f2 V2 = ⎢ ⎥h ⎣ 6μ ⎦ [rL1C1 + (z − L1)C2] ⎡ γ ⎤ f1 V3 = ⎢ ⎥h L C ⎡ ⎤ 2 2 ⎣ 6μ ⎦ ⎣ r + (z − L 2)C1⎦
(13)
At the changing of cross section, there is a velocity jump, i.e., a sudden change in the magnitude of the average velocity. Rigorously such a sudden change should imply inertial effects.31−33 However, it can be shown that in our case the time constant for the inertial effects is τ ≈ 2ρS/μ ≈ 1/100 s, where ρ is the density of the fluid and S is the cross-sectional area of the channel. In view of the axial dimensions of the constriction or enlargement (a few millimeters), and at the time scale of our experiences (2−10 s), the inertial effect associated with a sudden change of cross section is negligible. Hence, we can calculate the velocity jump from relation 13. This velocity jump is given by calculating the difference between the velocities at the entrance of the section and at the exit of the preceding section:
Figure 3. Sketch of the average velocity V of the capillary flow crossing a constriction (red curve) or an enlargement (green curve) compared to the velocity in the straight channel (orange curve).
In this problem, the variables are the velocity, the penetration distance, and the time. Expressions 12 and 13 relate velocity and distance. An expression for the distance as a function of time can be derived from eq 12 using V = dz/dt. The travel distance z is the solution of the differential equation. Integration of eq 12 using the initial conditions z = 0, z = L1, and z = L1 + L2 for t = 0, t = t1, t = t1 + t2, respectively, yields
⎡ γ ⎤ 2h cos θ J1,2 = V2(z = L1) − V1(z = L1) = ⎢ ⎥h ⎣ 6μ ⎦ L1C1 ⎛1 − r ⎞ ⎟ ⎝ r ⎠
z=
⎜
(14)
C1
ht , for z < L1
rL C z = L1 − 1 1 + C2
J2,3 = V3(z = L1 + L 2) − V2(z = L1 + L 2)
r
f1
⎛ rL1C1 ⎞2 γ f2 (t − t1) , ⎟ + ⎜ 3μ C2 ⎝ C2 ⎠ for L1 + L 2 > z > L1
and ⎡ γ ⎤ 2h cos θ ⎛ r − 1 ⎞ ⎜ ⎟ =⎢ ⎥ ⎣ 6μ ⎦ L1C1 + L2C2 ⎝ r ⎠
γ 3μ
⎛ LC ⎞ z = L1 + L 2 − ⎜L1 + 2 2 ⎟ rC1 ⎠ ⎝ 2 ⎛ LC ⎞ γ f1 (t − t1 − t 2) , + ⎜L1 + 2 2 ⎟ + 3μ C1 rC1 ⎠ ⎝ for z > L1 + L 2
(15)
In the case of the constriction, r < 1 and the velocity jump is positive. Relation 14 indicates that the velocity increases when the liquid front penetrates a reduced cross-section channel. This is in agreement with the literature.2−6 In addition, if r ≪1, the positive velocity jump is proportional to 1/r. When the advancing liquid interface regains the nominal dimension channel after the constriction, relation 15 predicts a negative velocity jump. This negative velocity jump does not depend on r
(17)
Finally, the average velocity as a function of time V(t) is obtained by a simple time derivation of eq 17. C
DOI: 10.1021/acs.langmuir.5b02090 Langmuir XXXX, XXX, XXX−XXX
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3. NUMERICAL RESULTS AND DISCUSSION Expressions 12 and 17 have been programmed with MATLAB, for an open, rectangular nominal channel of 500 μm depth and 200 μm width. The fluid physical properties are that of water (μ = 10−3 Pa·s, γ = 72 mN/m). The contact angle is set to 60°. 3.1. Straight Channel. A straight channel is obtained by setting w = w1 = w2. In that case, the average velocity determined by eq 12 is ⎡ γ ⎤h f V=⎢ ⎥ ⎣ 6μ ⎦ z C (18) For a given depth h, the velocity is proportional to f/C, which represents the balance between the nondimensional capillary force and the nondimensional friction force. In Figure 4 the values of f/C as a function of the aspect ratio e are plotted. For a contact angle of 60° (cos θ = 1/2), there is a
Figure 5. Average velocity in an open, rectangular channel with a constriction: the velocity jumps positively at the passage of the constriction (green curve) and negatively at its exit. The three (light to dark) green curves correspond to constriction widths of 150, 100, and 50 μm. The orange curve corresponds to the channel without constriction.
⎛ 2 cos θ cos θ ⎞⎟ ΔP = −γ ⎜ + ⎝ w h ⎠
(20)
The pressure jump at the constriction is then ⎛1 1⎞ ΔP2 − ΔP1 = 2γ cos θ ⎜ − ⎟ w1 ⎠ ⎝ w2
However, when the liquid penetrates the constriction, the fluid friction on the walls increases, due to the small dimensions of the constricted channel. The flow velocity decreases sharply at the constriction exit, according to the negative jump predicted by eq 15. At the exit of the constriction, the Laplace pressure is set back to its nominal value and the wall friction is increased by the presence of the constriction. Hence, the velocity at the exit of the constriction is smaller than that of the uniform nominal channel: in Figure 5, the green curves drop below the orange curve. Downstream, the constriction acts as a resistor, similarly to what it would do in a pressure-actuated flow. This property is associated with the decreased value f 2/C2 = 0.05 shown in Figure 4. Figure 6 shows the penetration distance with time. After the 50
Figure 4. Variation of the ratio f/C with the channel aspect ratio. The ratio f/C has a maximum for e = √3−1 (for θ = 60°); the red dot corresponds to the nominal ratio, and the two green dots correspond to the constriction (50 μm) and the enlargement (800 μm). In the present case, the depth h = 500 μm.
maximum for the ratio f/C for a value of the aspect ratio e = √3 − 1 ≈ 0.73. We obtain a first interesting result: for a given etching depth, there is an optimal width providing the maximum average velocity in an open, rectangular groove. A similar observation has already been established for a suspended channel.34 The value of the maximum velocity is ⎡ γ ⎤h ⎛ (e + 2) cos θ − e ⎞ ⎟ Vmax = ⎢ ⎥ max⎜ ⎝ ⎠ ⎣ 6μ ⎦ z e + 2/e
(21)
(19)
Hence, one can expect that just a change of channel widthfrom the nominal width to the optimal width, for examplewill have consequences on the flow velocity. 3.2. Channel with a Constricted Section. Three constriction widths have been considered: 150, 100, and 50 μm. Figure 5 shows the relation between average velocity of the liquid and the penetration distance. The positive velocity jump predicted by eq 14 at the entrance of a constricted channel corresponds to a sharp and sudden acceleration of the fluid velocity. In fact, this acceleration is due to a sudden increase of the driving Laplace pressure, conjointly to the effect of mass conservation. According to the formulation of Delamarche et al.27 for a closed rectangular channel of dimensions w and h, and after extrapolation to an open, rectangular channel, the Laplace pressure can be approximated by
Figure 6. Penetration distance vs time in the channel with a constriction: the penetration velocity is considerably reduced by the 50 μm constriction (dark green curve). The three green curves correspond to constriction widths of 150, 100, and 50 μm. The orange curve corresponds to the channel without constrictions.
μm constriction, the advancing interface velocity decreases substantially, and the constriction acts as a “global” flow resistor. 3.3. Channel with an Enlarged Section. Three enlargement widths have been considered: 366, 600, and 800 μm. Note that the width 366 μm corresponds to the maximum of the value f 2/C2 in Figure 4. Figure 7 shows the relation between average velocity of the liquid and the penetration distance. D
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Figure 7. Average velocity in an open, rectangular channel with an enlargement: the velocity jumps negatively at the passage of the constriction (green curve) and positively at its exit. The three green curves correspond to enlargement widths of 400, 600, and 800 μm. The orange curve corresponds to the channel without enlargement.
Figure 8. Penetration distance vs time in the channel with an enlargement: the penetration velocity is considerably reduced by the 800 μm enlarged section (dark green curve). The three green curves correspond to enlarged widths of 366, 600, and 800 μm. The orange curve corresponds to the channel without enlargement. Note that the light green curve, corresponding to a width of 366 μm, and to the maximum of f/C, provides a slightly faster penetration.
The flow velocity considerably decelerates in the enlarged channel due to a decrease of the magnitude of the Laplace pressure. The velocity jump is given by relation 14. In fact, if the enlargement is too wide, the capillary flow will stop. The condition for the stopping of the flow is22 w2 > cos θ w2 + 2h (22)
4.1. Constriction. The device with the constriction is shown in Figure 9. The constriction has a width of 200 μm and a length of 3.6 cm.
which is identical to f 2 < 0. Such a negative value indicates that the capillary force is negative and that the flow recedes. At the exit of the enlarged section, the flow velocity increases again, because the Laplace pressure is restored. The positive velocity jump is given by relation 15. After the advancing interface has passed the enlarged section, the fluid friction on the walls is reduced due to the relatively lower wall friction in the enlarged section. On the other hand, the driving Laplace pressure is restored to the nominal value, while the wall friction is less due to the enlarged section; hence the average flow velocity is higher than that in the channel without enlargement. If the width of the enlarged cross section is sufficient (r ≫1), the wall friction is negligible in the large section, and the velocity is restored to its value at the entrance of the enlargement, as shown in Figure 7. Figure 8 shows the penetration distance as a function of time. The penetration velocity is decreased by the 800 μm enlargement but is not decreased by the 366 μm enlargement, which corresponds to the maximum of the ratio f/C.
Figure 9. Left: top view of the channel with a constriction just after the inlet port; right: progression of the tinted IPA in the open channel. Because of the milling limitations, and the etching depth of 500 μm, small widths (50−100 μm) are not feasible. Hence, the constriction will have a limited effect. Figure 10 compares the theoretical and experimental results for the penetration distance as a function of time. The effect of the constriction is to delay the penetration by slowing down the SCF downstream of the constriction.
4. EXPERIMENTAL RESULTS Open, rectangular channels with a constricted section and with an enlarged section have each been fabricated, by milling a PMMA (polymethyl methacrylate acetyl) substrate using a Charly4U milling machine (Mecanumeric, France). The liquid used is a 20% solution of IPA (isopropyl alcohol). The advantage of IPA is that it does not require a preliminary treatment of the PMMA walls. As there is minimal food coloring added in order to color the IPA, the characteristics of the solution are only slightly different from that found in the literature.35 We have then considered a surface tension γ = 31 mN/m and a viscosity μ = 2.15 cP. The contact angle of the 20% IPA solution with the PMMA substrate has been measured in the range 61−64° with a Krüss device (Krüss GMBH, Germany). We found that the value of 62.7° produces the best fit with the experimental results. The milling depth is 500 μm, and the nominal width is 200 μm for the device with the enlargement and 400 μm for the device with the constriction. After placing the device on millimetric paper aligned with the channels, the IPA solution is introduced promptly into the inlet port with a pipet and the advancing interface is simply monitored by photographs at periodic times (every 2 s).
Figure 10. Comparison between theoretical and experimental results: penetration distance versus time. The experimental data are the green dots, the theoretical prediction is the red line, and the orange line corresponds to the penetration in the same channel without a constriction. The slight experimental oscillations are caused by careful refill of the inlet port with the pipet to maintain a zero pressure at inlet. E
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Langmuir 4.2. Enlargement. The device with the enlargement is shown in Figure 11. The enlargement has a width of 800 μm and a length of 7.7
biochemical reactions (polymerase chain reaction (PCR) chambers, for example, as shown in Figure 13). It can also be used to determine the velocity of the flow in a series of successive microchambers separated by constrictions.
Figure 11. Left: top view of the channel with an enlargement just after the inlet port; right: progression of the tinted IPA in the open channel. Figure 13. Sketch of a network designed to perform PCRs in parallel.36
■
APPENDIX: FRICTION IN A RECTANGULAR OPEN CHANNEL Following Ouali and colleagues,9 the average velocity in an open, rectangular channel is obtained by a development in the Fourier
Figure 12. Comparison between theoretical and experimental results: penetration distance versus time. The experimental data are the green dots, the data from the model are shown in red, and the orange line corresponds to the penetration in the same channel without enlargement. mm. Figure 12 compares the theoretical and experimental results for the penetration distance as a function of time. The experimental results confirm the theoretical analysis. The SCF velocity decreases at the passage of the enlargement and then is restored to a value close to that of a channel without enlargement (the two orange and red curves of Figure 12 are parallel early on).
Figure 14. Velocity comparison between spatial Fourier series (eq 27) and approximate model (eq 28).
5. CONCLUSION In this work, an analysis of the effect of a cross-section reduction or enlargement is performed. It illustrates how the velocity of a spontaneous capillary flow is dependent on two forces: the capillary force and the wall friction. Capillary-driven microflows differ from pressure- or flow-ratedriven ones by the fact that the engine of the motion is placed at the forefront of the liquid. Hence, the strength of the engine, i.e., the capillary force, depends on the geometry found by the advancing interface. On the one hand, the friction force depends on all the geometrical features placed upstream from the advancing interface. In this respect, a geometrical constriction of the flow channel acts as a resistor: once the liquid interface has passed the constriction, the total wall friction reduces the flow velocity. On the other hand, a geometrical enlargement of the flow channel acts as a resistor: as long as the advancing interface is localized in the enlargement, the capillary force is reduced and the flow velocity is diminished. However, when the leading interface has passed the enlarged section, the flow velocity approximately regains its nominal value. The model developed in this work can be used to predict the capillary filling of networks aimed to performing multiparallel
series of the channel perimeter. In such a case, the expression of the average velocity is V=−
∞ ⎡ tanh(αi) ⎤ 1 ∂P ⎛ 8w 2 ⎞ − 1⎥ ⎜ 4⎟ ∑ 4⎢ ∂z ⎝ μπ ⎠ i = 0 (2i + 1) ⎣ αi ⎦
(23)
where αi = (2i + 1)
π e
(24)
Defining an aspect ratio function by ⎡ ⎛ 24e 2 ⎞ ∞ tanh(αi) ⎤ 1 1− ζ0(e) = ⎜ 4 ⎟ ∑ ⎥ 4⎢ αi ⎦ ⎝ π ⎠ i − 0 (2i + 1) ⎣
(25)
the drag force is given by fdrag = −
3μezV ζ0(e)
(26)
Equating the drag and capillary forces yields V= F
γ 3μ
⎤ ⎞ h ⎡⎜⎛ 2 ⎢⎝ + 1⎟⎠ cos θ − 1⎥ζ0(e) ⎣ ⎦ z e
(27)
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(15) Safavieh, R.; Juncker, D. Capillarics: pre-programmed, selfpowered microfluidic circuits built from capillary elements. Lab Chip 2013, 13, 4180. (16) Yang, D.; Krasowska, M.; Priest, C.; Popescu, M. N.; Ralston, J. Dynamics of Capillary-Driven Flow in Open Microchannels. J. Phys. Chem. C 2011, 115, 18761−18769. (17) Elizalde, E.; Urteaga, R.; Koropecki, R. R.; Berli, C. L. A. Inverse Problem of Capillary Filling. Phys. Rev. Lett. 2014, 112, 134502. (18) Erickson, D.; Li, D.; Park, C. B. Numerical simulations of capillary-driven flows in non-uniform cross-sectional capillaries. J. Colloid Interface Sci. 2002, 250, 422−430. (19) Concus, P.; Finn, R. On the behavior of a capillary surface in a wedge. Proc. Natl. Acad. Sci. U. S. A. 1969, 63 (2), 292−299. (20) Concus, P.; Finn, R. Capillary surfaces in a wedge: Differing contact angles. Microgravity Sci. Technol. 1994, 7, 152−155. (21) Jokinen, V.; Franssila, S. Capillarity in microfluidic channels with hydrophilic and hydrophobic walls. Microfluid. Nanofluid. 2008, 5, 443− 448. (22) Berthier, J.; Brakke, K. A.; Gosselin, D.; Huet, M.; Berthier, E. Metastable capillary filaments in rectangular cross-section open microchannels. AIMS Biophys. 2014, 1 (1), 31−48. (23) Berthier, J.; Brakke, K. A.; Furlani, E. P.; Karampelas, I. H.; Poher, V.; Gosselin, D.; Cubizolles, M.; Pouteau, P. Whole blood spontaneous capillary flow in narrow V-groove microchannels. Sens. Actuators, B 2015, 206, 258−267. (24) Weislogel, M. M.; Chen, Y.; Bolleddula, D. A better nondimensionalization scheme for slender laminar flows: The Laplacian operator scaling method. Phys. Fluids 2008, 20, 093602. (25) Berthier, J.; Brakke, K. A. The physics of microdrops; ScrivenerWiley Publishing: 2012.10.1002/9781118401323 (26) Berthier, J.; Brakke, K. A.; Berthier, E. A general condition for spontaneous capillary flow in uniform cross-section microchannels. Microfluid. Nanofluid. 2014, 16 (4), 779−785. (27) Delamarche, E.; Bernard, A.; Schmid, H.; Bietsch, A.; Michel, B.; Biebuyck, H. Microfluidic networks for chemical patterning of substrate: design and application to bioassays. J. Am. Chem. Soc. 1998, 120, 500− 508. (28) Juncker, D.; Schmid, H.; Drechsler, U.; Wolf, H.; Wolf, M.; Michel, B.; de Rooij, N.; Delamarche, E. Autonomous microfluidic capillary system. Anal. Chem. 2002, 74, 6139−6144. (29) Safavieh, R.; Tamayol, A.; Juncker, D. Serpentine and leadingedge capillary pumps for microfluidic capillary systems. Microfluid. Nanofluid. 2015, 18, 357−366. (30) Baret, J.-C.; Decré, M. M. J.; Herminghaus, S.; Seemann. Transport dynamics in open microfluidic grooves. Langmuir 2007, 23, 5200−5204. (31) Yang, D.; Krasowska, M.; Priest, C.; Popescu, M. N.; Ralston, J. Dynamics of capillary-driven flow in open microchannels. J. Phys. Chem. C 2011, 115, 18761−18769. (32) Quéré, D. Inertial capillarity. Europhys. Lett. 1997, 39 (5), 533− 538. (33) Stange, M.; Dreyer, M. E.; Rath, H. J. Capillary driven flow in circular cylindrical tubes. Phys. Fluids 2003, 15 (9), 2587−2601. (34) Berthier, J.; Brakke, K. A.; Gosselin, D.; Bourdat, A.-G.; Nonglaton, G.; Villard, N.; Laffite, G.; Boizot, F.; Costa, G.; Delapierre, G. Suspended microflows between vertical parallel walls. Microfluid. Nanofluid. 2015, 18, 919−929. (35) Park, J.-G.; Lee, S.-H.; Ryu, J.-S.; Hong, Y.-K.; Kim, T.-G.; Busnaina, A. A. Interfacial and electrokinetic characterization of IPA solutions related to semiconductor wafer drying and cleaning. J. Electrochem. Soc. 2006, 153 (9), G811−G814. (36) Diakite, M. L.; Rollin, J.; Jary, D.; Berthier, J.; Mourton-Gilles, C.; Sauvaire, D.; Philippe, C.; Delapierre, G.; Gidrol, X. Point-of-care diagnostics for ricin exposure. Lab Chip 2015, 15, 2308−2317.
On the other hand, the approximate relation 5 yields V* =
γ 3μ
⎤ ⎞ h ⎡⎜⎛ 2 1 ⎢⎝ + 1⎟⎠ cos θ − 1⎥ ⎦2 1 + z⎣ e
(
2 e2
)
(28)
The two relations are compared in Figure 14, for e = 0.3, h = 1 mm, γ = 72 mN/m, and μ = 1 mPa·s. The two expressions produce very similar results.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.5b02090. Two methods to calculate spontaneous capillary flow (PDF)
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AUTHOR INFORMATION
Corresponding Author
*
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank François Boizot for the milling of the plastic devices. The work was funded by the CEA-Leti, in the frame of the program on capillary devices for biotechnology at the Laboratory for Biology and Microfluidic Architecture.
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REFERENCES
(1) Suk, J. W.; Cho, J.-H. Capillary flow control using hydrophobic patterns. J. Micromech. Microeng. 2007, 17, N11−N15. (2) Lucas, R. Ueber das Zeitgesetz des Kapillaren Aufstiegs von Flussigkeiten. Colloid Polym. Sci. 1918, 23, 15. (3) Washburn, E. W. The dynamics of capillary flow. Phys. Rev. 1921, 17, 273−283. (4) Rideal, E. K. CVIII. On the flow of liquids under capillary pressure. Philos. Mag. Ser. 6 1922, 44, 1152−1159. (5) Mann, J. J. A.; Romero, L.; Rye, R. R.; Yost, F. G. Flow of simple liquids down narrow V-grooves. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1995, 52, 3967. (6) Rye, R. R.; Yost, F. G.; Mann, J. A., Jr. Wetting Kinetics in Surface Capillary Grooves. Langmuir 1996, 12, 4625−4627. (7) Romero, L. A.; Yost, F. G. Flow in an open channel capillary. J. Fluid Mech. 1996, 322, 109−129. (8) Yost, F. G.; Rye, R. R.; Mann, J. A. Solder wetting kinetics in narrow V- grooves. Acta Mater. 1997, 45, 5337−5345. (9) Ouali, F. F.; McHale, G.; Javed, H.; Trabi, C.; Shirtcliffe, N. J.; Newton, M. I. Wetting considerations in capillary rise and imbibition in closed square tubes and open rectangular cross-section channels. Microfluid. Nanofluid. 2013, 15, 309−326. (10) Conrath, M.; Canfield, P. J.; Bronowicki, P. M.; Dreyer, M. E.; Weislogel, M. M.; Grah, A. Capillary channel flow experiments aboard the International Space Station. Phys. Rev. E 2013, 88 (6), 063009. (11) Zimmermann, M.; Schmid, H.; Hunziker, P.; Delamarche, E. Capillary pumps for autonomous capillary systems. Lab Chip 2007, 7, 119−125. (12) Zimmermann, M.; Hunziker, P.; Delamarche, E. Valves for autonomous capillary systems. Microfluid. Nanofluid. 2008, 5, 395−402. (13) Gervais, L.; Delamarche, E. Toward one-step point-of-care immunodiagnostics using capillary-driven microfluidics and PDMS substrates. Lab Chip 2009, 9, 3330−3337. (14) Gervais, L.; de Rooij, N.; Delamarche, E. Microfluidic chips for point-of-care immunodiagnostics. Adv. Mater. 2011, 23 (24), H151− H176. G
DOI: 10.1021/acs.langmuir.5b02090 Langmuir XXXX, XXX, XXX−XXX
