Article pubs.acs.org/Langmuir
Geometry-Induced Asymmetric Capillary Flow Dahua Shou,† Lin Ye,*,† Jintu Fan,‡,§ Kunkun Fu,† Maofei Mei,§ Hongjian Wang,† and Qing Chen§ †
Centre for Advanced Materials Technology (CAMT), School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia ‡ Department of Fiber Science & Apparel Design, College of Human Ecology, Cornell University, Ithaca, New York 14853, United States § Institute of Textiles and Clothing, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong S Supporting Information *
ABSTRACT: When capillary flow occurs in a uniform porous medium, the depth of penetration is known to increase as the square root of time. However, we demonstrate in this study that the depth of penetration in multi-section porous layers with variation in width and height against the flow time is modified from this diffusive-like response, and liquids can pass through porous systems more readily in one direction than the other. We show here in a model and an experiment that the flow time for a negative gradient of cross-sectional widths is smaller than that for a positive gradient at the given total height of porous layers. The effect of width and height of local layers on capillary flow is quantitatively analyzed, and optimal parameters are obtained to facilitate the fastest flow. are employed, the time of capillary flow was found to be independent of the total lengths of tubes with fixed wider and narrower portions.15 In 2008, Reyssat et al.16 experimentally and theoretically investigated the dynamics of capillary flow within a tube with an axial decrease in radius. They found that the penetration distance of the liquid is diffusive to time within a short time, analogous to the power law of eq 1; however, the power decreases to 0.25 for a long time and the capillary flow is slower than that of the uniform tube.16 Later, Shou et al.17 explored capillary flow in two-section circular tubes with variations in radius and height. They obtained the optimal radius and height ratios that accounted for the fastest capillary flow.17 Fu et al.18 modeled the transport time in 2D paper networks made of multiple constant-width segments used as analytical devices. However, they used the mean ratio between fluid volume and flow time as the spontaneous volumetric flux in their model. Recently, nanostructured surfaces coated with declined pins were fabricated for unidirectional spreading properties for liquids.19,20 Although capillary flow in a tube or a porous system has been extensively studied, less work has been done to investigate asymmetric transport behavior. Capillary-based asymmetric absorbents have significant applications in various areas. For instance, it was experimentally demonstrated that fibrous fabric with asymmetric absorption properties transported sweat water away more quickly from the skin side to the outer surface of the fabric, and simultaneously slowed down the invasion of water
1. INTRODUCTION Lately, there has been an upsurge in research and development motivated by the use of porous materials based on capillary flow, including plant transpiration,1 water collection of spider silk,2 textiles,3 microfluidic devices,4,5 and medical wound dressings.6 The capillary flow in terms of invasion of pores by liquids, driven by negative capillary pressure, was described by Washburn a century ago.7 Washburn7 suggested a diffusive correlation between the penetration distance of the liquid h and time t, namely, ⎛ γr cos θ ⎞0.5 h=⎜ t⎟ ⎝ 2η ⎠
(1)
where γ and η stand for the surface tension and the viscosity of the liquid, respectively, r is the tube radius, and θ is the contact angle of the liquid with the wall of tube. This law holds when the tube is placed horizontally or in the limit h ≪ he.8 Here, he is the critical height of the liquid when the capillary force is equal to the gravity of the liquid column. Washburn’s law has been widely employed to describe the process of liquid wicking in a porous medium, which was always approximated as bundles of tortuous tubes.9 The time exponent of capillary flow in these tubes was described in terms of fractal dimension.10 In actual applications, porous media were found to have different cross sections along capillary flows, affecting flow behaviors significantly.11 Numerical simulations indicated that the liquid was pinned when passing through an expansion of the tube with the actual contact angle higher than 90°.12,13 This threshold value of contact angle for a porous medium composed of beads was experimentally observed on the order of 55°, much smaller than that for a tube.14 Provided long tubes with extremely mild contractions and expansions © 2014 American Chemical Society
Received: February 4, 2014 Revised: April 17, 2014 Published: April 24, 2014 5448
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We assume h ≪ he in this study. We explore simultaneously three configurations: (1) a two-section porous medium, (2) a three-section porous medium, and (3) a trapezoidal porous medium (Figure 1). 2.1. Two-Section Medium. We begin with the capillary flow in a two-section system composed of two rectangular, homogeneous porous layers (Figure 1a). The two-section medium is globally constrained by a constant height. The total height of the system is H, and the two sections contain two different cross sections, widths a1 and a2, with corresponding heights h1 and h2, respectively. We define nij = hi /hj
(2)
mij = ai /aj
(3)
where nij and mij are ratios of height and width between the ith section and the jth section, respectively. The porous medium with uniform width is used as the control sample, and its flow time for the penetration distance H is (1/2)CH2 based on eq (S11) in the Supporting Information, where C is a constant equal to η/(γ cos θ J(s)(Kε)1/2). Normalized by (1/2)CH2, the dimensionless total flow time of the two-section medium R is given by (for details, see Supporting Information, section S1)
Figure 1. (a) Two-section porous medium, (b) three-section porous medium, and (c) trapezoidal porous medium.
from environment to skin.21 A robust framework for understanding capillary flow behavior in such architectures is therefore essential and is the subject of the present work. Here, we aim to find a general criterion for the design and optimization of porous structures composed of porous layers and trapezoidal porous media for asymmetric capillary flows.
R=
2m21n21 n212 1 + + 2 2 (1 + n21) (1 + n21) (1 + n21)2
(4)
2.2. Three-Section Medium. Next we consider capillary flow in a three-section system, which is composed of three rectangular, homogeneous porous media (Figure 1b). The three-section medium is also constrained by a constant height H. It is observed in Section 3 that the total flow time of the two-section medium increases with the increase in m21, and has a minimum value against n21 (i.e., n21 = 1) and
2. THEORETICAL MODEL This study investigates the capillary flow through a multi-section, homogeneous porous medium (with uniform porosity and pore size).
Table 1. Geometrical Parameters of the Four Groups of Porous Paper Samplesa type
first section width (a1, cm)
a b c d e f (trapezoid) g (trapezoid) h
2 2 2 2 2 2 (lower width) 1 (lower width) 1
first section height (h1, cm) second section width (a2, cm) second section height (h2, cm) 6 5 4 3 2
1 1 1 1 1 1 (upper width) 2 (upper width) 1
8
2 3 4 5 6
0
total height (H, cm) 8 8 8 8 8 8 8 8
a
Samples a−e are two-section, rectangular paper media with variation in width and height. Samples f and g are trapezoid paper media. Sample h is a uniform rectangular paper.
Figure 2. Schematics of experimental apparatus for capillary flow test. 5449
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simultaneously a maximum value when the flow occurs in the other direction. Thus, we adopt n21 = 1 in this case, as we seek to investigate the speed of a capillary flow that is as asymmetric as possible. The total flow time normalized by (1/2)CH2 is given by (for details, see Supporting Information, Section S1):
R=
n32 2 2m21 1 1 + + + 2 2 2 (2 + n32) (2 + n32) (2 + n32) (2 + n32)2 2m31n32 2m32n32 + + (2 + n32)2 (2 + n32)2 (5)
2.3. Trapezoidal Medium. Lastly, we consider liquid flow in a trapezoidal porous medium for comparison with the above multilayer cases. The trapezoidal porous system is relatively easy for processing. The trapezoidal porous medium also has a constant height H, with lower width a1 and upper width a2 (Figure 1c). The total flow time normalized by (1/2)CH2 is expressed as follows (for details, see Supporting Information, section S1): R=
⎧ ⎫ 2 ⎞⎡ ⎛ 2 ⎞ ⎤ ⎪⎛ a ⎪ a12 a2 2 ⎢ ⎥ ⎨ ⎬ ⎜ ⎟ ⎜ ⎟ ln − 1 + 1 2⎪ 2 2 ⎪ ⎢ ⎥ 2(a1 − a 2) ⎩⎝ a1 ⎠⎣ ⎝ a1 ⎠ ⎦ ⎭
(6)
3. EXPERIMENTAL SECTION 3.1. Materials. Capillary flow experiments were performed using porous absorbent paper supplied by Kimberly-Clark Corporation. We measured the flow time t against the penetration distance h for uniform paper media, two-section paper media and trapezoid paper media based on the following models, separately (for details, see Supporting Information, Section S1), (eq S10)
t=
1 2 Ch 2
(7)
(eqs S10 and S14) ⎧ 1 2 Ch 0 < h ≤ h1 ⎪ 2 ⎪ t=⎨ ⎪C a 2h1(h − h1) + 1 C(h − h )2 + 1 Ch 2 h < h ≤ H 1 1 1 ⎪ a1 2 2 ⎩
(8)
(eq S27)
H2a12C 4(a1 − a 2)2 2 ⎧ ⎫ ⎤ ⎪⎛ ⎪ h(a1 − a 2) ⎞ ⎡ ⎛ h(a1 − a 2) ⎞ ⎬ a 2ln a 1 1 ×⎨ − − − + ⎢ ⎥ ⎟ ⎜ ⎟ ⎜ 1 1 ⎪ ⎪ ⎠⎣ ⎝ ⎠ H H ⎦ ⎭ ⎩⎝ (9) We also compared our model with experiment for finding the optimal two-section paper structure accounting for the fastest or slowest flow based on eq 4. The geometrical parameters of the four groups of measured samples are summarized in Table 1. The capillary fluid used in this study is distilled water from Glendale Company, Australia. For this liquid water in a standard atmosphere of 20 ± 2 °C and 65 ± 2% relative humidity, the density is 103 kg/m3, the viscosity is η = 1.002 × 10−3 Pa/m, and the surface tension is γ = 7.275 × 10−2 N/m.23 3.2. Equipment and Penetration Test Procedure. An apparatus was designed to measure the water flow in the porous paper samples, solely driven by capillary force. In the apparatus (see Figure 2), the small liquid reservoir was used to contain liquid water, and the length scale was marked on paper samples for measuring the penetration distance of the water. Paper samples were fixed at the liquid reservoir and were placed horizontally to avoid the effect of gravity. A red hydrophobic fabric was fixed slightly below the white samples to enhance contrast. The water was quickly injected into the liquid reservoir and the fill-up is within 1 s. The immediate contact of t=
Figure 3. Variation of flow time versus flow distance in (a) uniform porous medium, (b) two-section porous medium, and (c) trapezoidal porous medium. the samples with water is complex, but it does not affect the major correlation between the flow time and the penetration distance of liquids for long durations (e.g., the total penetration time for the control sample is around 120 s), as validated in section 4. A volume equivalent of the front part (i.e., the line that is normal to the flow direction and halving the area of the wetting parts within the rectangle which exactly contains the fronts of liquids with the minimum and maximum penetration distance) was used to determine the effective 5450
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fluid front. The influence of evaporation was ignored in this experiment due to adequately fast capillary flow. The penetration distance of water at different times was continuously recorded and measured by taking snapshots periodically with the camera of a mobile digital device. The capillary flow experiments were conducted using the four groups of paper samples, including Samples a−g and the control sample h.
one flow direction is observed to be around four times that in the other flow direction with m21 = 0.25, as seen in Figure 4a and b (i.e., magnified section of Figure 4a with m21 < 1). Such asymmetries are interpreted in the sense that the capillary flow in the first section requires the same time based on eq 7, and the first section acts as a rich liquid source for the flow of the second section with m21 ≪ 1, but it becomes difficult for the first section to provide enough liquid for the second section with m21 ≫ 1. It is also demonstrated that the flow time initially increases and then decreases with increase in n21 for m21 > 1; the trend is opposite for m21 < 1; and the flow time is a constant with m21 = 1, as expected. In particular, Figure 4 indicates that both of the maximum and minimum values of flow time exist at n21 = 1. Thus, we find that the flow with m21 < 1 is much faster than m21 > 1 for the same two-section medium. We measured the capillary flow of water experimentally in two-section porous paper media with height variations. In Figure 5a, experimental results indicate that the samples of
4. RESULTS AND DISCUSSION We compare our model with experimental results in Figure 3. The experimental results in Figure 3a indicate the square root of flow time increases linearly with the flow distance and thus the validation of eq 7. By best fitting with the least-squares-fitting, straight line of experimental data, we have C = 3.7 in eq 7. With C = 3.7, we compare the flow time versus the flow distance based on eqs 8 and 9 with experimental data at different values of m21 and n21 = 1, for two-section porous medium and trapezoidal porous medium in Figure 3b and c, respectively. It is evident that our model agrees well with experimental results. With eq 4, we can easily plot the values of the normalized flow time versus the height ratio of two-section porous layers at different width ratios, as shown in Figure 4. The figure clearly illustrates that the flow time increases with increasing m21 at a fixed n21. For the same sample, the flow time with m21 = 4 in
Figure 5. Normalized flow time of the two-section porous medium versus height ratio with (a) m21 in a full range and (b) m21 < 1.
contraction with identical local heights approximately have the fastest capillary flow, which validates the prediction that n21 = 1 accounts for the minimum flow time required for a given total height in Figure 4b. On the other hand, in Figure 5b it seems that the slowest capillary flow exists in the samples of extraction
Figure 4. Normalized flow time of the two-section porous medium versus height ratio with (a) m21 in a full range and (b) m21 < 1. 5451
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Figure 6. Fluid fronts in horizontally placed samples (group 1) of both contraction (left) and extraction (right) (a−e) and the control sample (h) at 5, 10, 40, 80, and 120 s.
when m21 is very small, the flow time remains around a constant 0.5, which is equal to twice of flow time of the first section. This is expected that the first section becomes a full liquid supplier when the width of second extremely small comparing to the former and thus the capillary flow process in the second section reoccurs as that of the first section based on eq 4. On the contrary, when m21 ≫ 1, the first section is difficult to provide adequate liquid for flow to the second section with great width expansion and thus the capillary flow becomes very slow. We next investigated the capillary flow process in the threesection porous medium. To avoid most of the liquid absorption being concentrated in one side, we set m31 = 0.5. The normalized flow time R versus n31 on the basis of eq 5 is shown in Figure 8 with different values of m23. It is demonstrated that different values of minimum flow time correspond to different width ratios of m23. The minimum normalized values of R are in a narrow range close to 0.75, demonstrating that the minimum flow time is just slightly sensitive to the variation of m23.
with n21 = 1, as shown in Figure 4a. As well, by comparing Figure 4a and b, it is evident that the liquid flow is faster in samples of contraction but is slower in samples of extraction than in the control samples with uniform width. Comparison of the flow propagation between the samples of contraction and extraction and the control sample in Table 1 is illustrated based on a series of snapshots in Figure 6. Series of snapshots for the rest samples are provided in section S2 of the Supporting Information. It is demonstrated that the paper samples with similar heights of the two sections tend to have the fastest and the slowest flow, as illustrated at t = 80 s and t = 120 s in Figure 6, respectively. It is noted that the flow time generally scales with the square of penetration distance of liquids, so the flow becomes much slower for long durations. As such, although the fluid fronts look close between different samples, the variations in the total flow time for the penetrations distance H are very large, as indicated by Figures 4 and 5. Figure 7 demonstrates the effect of width ratio on the flow time of two-section porous medium with n21 = 1. It is seen that 5452
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In Figure 9, we compare the normalized flow time versus the normalized penetration distance in the trapezoidal medium, the two-section medium and the uniform medium at m21 = 0.5. It is clear in Figure 9 that the capillary time against the penetration distance of liquids in the nonuniform medium does not follow the Washburn equation. Liquid impregnation in the twosection medium is faster for capillary flow from a1 to a2 than from a2 to a1, with R = 0.8068 and R = 1.2726, respectively. Figure 9 suggests that the trapezoidal porous medium is also asymmetric to capillary flow. As well, it is noted that the fastest capillary flow occurs in the two-section medium from the wider section to the narrow section. The next fastest time is found in the trapezoidal porous medium from a1 to a2, and the slowest time is in the uniform medium. The trend is reversed when the capillary flow occurs in the other direction. Figure 7. Normalized flow time of the two-section porous medium versus width ratio.
5. CONCLUSION In this work, we have developed a general framework for the design of porous architectures for capillary flow behaviors with asymmetries. The asymmetric transport due to geometrical asymmetry is facilitated in the two-section, the three-section and the trapezoidal homogeneous porous medium. The models obtained are expressed as a function of height and width ratio. Experimental measurements of capillary flow in two-section rectangular paper networks and trapezoidal paper networks were conducted to illustrate our findings. The capillary flow of the porous medium with uniform width is found to follow Washburn’s equation, but all flow processes in the nonuniform media are different from Washburn’s description. The capillary flow in multi-section media is faster from the wider section to the narrower section than that from the opposite flow direction. This asymmetric transport trend also holds for the trapezoidal porous medium. It is particularly interesting that an optimal height ratio exists, accounting for the fastest/slowest flow in the multi-section porous media, as readily verified by the experiment. Moreover, an increase in the width ratio increases the flow time of the two-section porous medium under the given total height. The flow time of the trapezoidal porous medium is found to be close to the minimum flow time of the multi-section porous medium, indicating that the trapezoidal porous medium is a good candidate for asymmetric transport due to its simple geometry. This study provides an initial exploration of asymmetric capillary flow in simple structures, and insights gained from this work offer new opportunities to tailor more complex and actual systems to achieve active control of asymmetric transport under different constraints.
Figure 8. Normalized flow time of the three-section porous medium versus height ratio.
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ASSOCIATED CONTENT
S Supporting Information *
Model generation and fluid fronts in different samples. This material is available free of charge via the Internet at http:// pubs.acs.org/
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AUTHOR INFORMATION
Corresponding Author
Figure 9. Comparison of the normalized flow time between the trapezoidal medium, the two-section medium, and the uniform medium.
*Tel.: +61 02 9351 4798. Fax: +61 02 9351 3760. E-mail:
[email protected].
When m23 = 1 or m23 = 2, the normalized flow time becomes 0.75 and the three-section medium becomes two-sectional.
Notes
The authors declare no competing financial interest. 5453
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ACKNOWLEDGMENTS L.Y. is grateful for an ARC Discovery Project grant that supports this work.
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NOTATIONS
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REFERENCES
a, cross-section width (cm) A, width of the medium (cm) C, constant (ss/cm2) h, penetration distance of the liquid (cm) he, critical height of the liquid (cm) H, total height (cm) i, the ith section j, the jth section J, Leverett J-function K, permeability tensor of the medium (cm2) m, width ratio n, height ratio p, pressure (Pa) Q, flow rate (cm3/s) R, normalized flow time or flow time ratio s, saturation t, flow time (s) u, flow velocity (cm/s) ε, porosity γ, surface tension (N/cm) r, tube radius (cm) η, liquid viscosity (Pa·cm) θ, contact angle
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