Article pubs.acs.org/Langmuir
Optimal Design of Porous Structures for the Fastest Liquid Absorption Dahua Shou,† Lin Ye,*,† Jintu Fan,‡ and Kunkun Fu†,§ †
Centre for Advanced Materials Technology (CAMT), School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, New South Wales 2006, Australia ‡ Department of Fiber Science and Apparel Design, College of Human Ecology, Cornell University, Ithaca, New York 14850, United States § College of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 20092, People's Republic of China S Supporting Information *
ABSTRACT: Porous materials engineered for rapid liquid absorption are useful in many applications, including oil recovery, spacecraft life-support systems, moisture management fabrics, medical wound dressings, and microfluidic devices. Dynamic absorption in capillary tubes and porous media is driven by the capillary pressure, which is inversely proportional to the pore size. On the other hand, the permeability of porous materials scales with the square of the pore size. The dynamic competition between these two superimposed mechanisms for liquid absorption through a heterogeneous porous structure may lead to an overall minimum absorption time. In this work, we explore liquid absorption in two different heterogeneous porous structures [three-dimensional (3D) circular tubes and porous layers], which are composed of two sections with variations in radius/porosity and height. The absorption time to fill the voids of porous constructs is expressed as a function of radius/porosity and height of local sections, and the absorption process does not follow the classic Washburn’s law. Under given height and void volume, these two-section structures with a negative gradient of radius/porosity against the absorption direction are shown to have faster absorption rates than control samples with uniform radius/porosity. In particular, optimal structural parameters, including radius/porosity and height, are found that account for the minimum absorption time. The liquid absorption in the optimized porous structure is up to 38% faster than in a control sample. The results obtained can be used a priori for the design of porous structures with excellent liquid management property in various fields. where D is the diffusion coefficient of the liquid, γ and η stand for the (liquid−vapor) surface tension and the viscosity of the liquid, respectively, r is the tube radius, and θ is the contact angle characterizing the wetting of the liquid on the wall of the tube. This law holds in the limit z ≪ ze13 or when the tube is placed horizontally, where ze is the final height of the liquid rise with the capillary force simultaneously being equal to the gravity of the liquid column. Recently, Washburn’s law was found to be valid down to nanoscale pore sizes.14 Washburn’s law was also extended to characterize the process of water absorption in a porous medium, which was approximated as bundles of tortuous tubes.10 In reality, different cross-sections in tortuous tubes along the absorption direction have been found to affect the liquid absorption rate significantly.15 Numerical simulations showed that the liquid meniscus suffered from complex deformation when moving through varying cross-sectional widths; in particular, the liquid passing through an expansion was pinned near a high-curvature point when the actual contact angle exceeded 90°.16,17 By introducing a sole factor, effective capillary radius, Masoodi et al.18 obtained a
1. INTRODUCTION Liquid absorption in capillary tubes or porous media is a rich phenomenon in nature that attracts wide attention in industrial applications, such as water collection of artificial silk,1 spacecraft life-support systems,2 micropatterns,3 liquid composite molding,4,5 textiles,6,7 paper-based analytical devices,8 medical wound dressings,9 oil recovery,10 and fuel cells.11 Although the study of liquid absorption by means of capillary flow in a hollow tube or a porous medium is a relatively mature topic of fundamental and technological interest, little attention has been paid to the design and optimization of engineered porous architectures for the fastest liquid absorption. A century ago, Washburn12 quantitatively described the quasi-steady dynamics of capillary flow in a tube as a result of the negative capillary pressure caused by the meniscus curvature of the liquid. Washburn12 applied Poiseuille’s law to describe the slow flow motion in a smooth circular tube and suggested a diffusive correlation between the distance of liquid movement z and the time t, viz. z = (Dt )0.5
with
D=
γr cos θ 2η
© XXXX American Chemical Society
Received: September 3, 2013 Revised: October 6, 2013
(1) A
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structure is monotonously dependent upon the pore size based on the balance of the two mechanisms and Darcy’s law of eq 2. However, the minimum absorption time may be found on the basis of the dynamic competition between the above two mechanisms superimposed, with different sensitivities to the pore size through varied sections contained in the heterogeneous porous structures. From this perspective, we expect to find a general criterion for the design of porous structures for the fastest absorption of liquid, by tailoring the evolution of heterogeneous sections to meet desirable configurations.
diffusive model of water absorption in a homogeneous layer of porous wicks, similar to the classic result of eq 1. A sharp wet front analysis of the propagating liquid was used to realize the absorption of water into a multilayer composite of porous media.19 The proposed model was validated by experimental data, which also revealed no evident hydraulic contact resistance at the interface between layers.19 The studies discussed above were based on tubes or porous systems with uniform or locally uniform cross-sections of pores, but less work has dealt with liquid absorption in non-uniform and heterogeneous porous matrixes. Reyssat et al.20 investigated water absorption within a conical tube with axial variation in the radius. They found that, for short durations, the propagation height of the liquid in a circular tube of conical shape increases as the square root of time, with a power of 1/2 as that of Washburn’s law; for long durations, however, a power of 1/4 responses.20 Washburn’s law was also modified by a power of 1/3 for radial liquid absorption in semi-infinite porous media.21 Later, Reyssat et al.22 explored the absorption rate of liquid into layers of packed beads of various sizes (see Figure 1), observing that the capillary flow for a negative gradient of
2. THEORETICAL MODEL We investigate here liquid absorption through heterogeneous porous structures constrained by fixed void volumes and height. The assumption of z ≪ ze or horizontal liquid advancement is adopted in this work. Two configurations are explored simultaneously, namely, (1) a two-section three-dimensional (3D) circular tube and (2) twosection porous layers. The two sections vary in radius/porosity and height. The absorption time of the heterogeneous porous construct is normalized by that of a control sample of a uniform structure, which has a constant absorption time at given constraints. 2.1. Three-Dimensional Circular Tubes. We begin with the liquid absorption in 3D circular tubes (Figure 2). The two tubes have
Figure 1. Sketch of a two-layer system composed of solid beads of different sizes.
bead or pore sizes was faster than for a positive gradient. Recently, Bal et al.23 created various absorption patterns based on different combinations of fabric layers of varying porosity and pore size distribution. Although the principle of liquid absorption in homogeneous porous media has been extensively studied and pioneering research has been conducted on heterogeneous systems, little work has focused on the optimization of porous structures with the minimum absorption time at a fixed constraint. Such porous absorbents have significant applications in various fields. For instance, the optimized spacecraft life-support system having minimum liquid absorption time can accelerate circulation and separation of water with microgravity in space. As well, fabrics with fast liquid absorption provide control of the movement of body water (e.g., sweat) in such a way that they are transported quickly from the skin to the outer surface of the fabric, where they can evaporate quickly. Garments produced from such fabrics keep the skin dry and provide maximum comfort to the wearer. In the creeping regime, Darcy’s law is commonly used to describe viscously dominated capillary flow24 K u = − ∇p η (2)
Figure 2. (a) Uniform 3D tube and (b) heterogeneous two-section 3D tube. identical volumes and height. The uniform tube in Figure 2a has height H and radius R, whereas the two-section tube in Figure 2b contains two different cross-sections, radii r1 and r2, with corresponding heights h1 and h2, respectively. Because the total absorption time (i.e., the time required for filling up the voids of porous materials with liquid) is a constant for the tube with the uniform cross-section (Figure 2a), that tube is used as the control sample for the heterogeneous two-section tube (Figure 2b). The height of the tube is usually much larger than the radius in experiments.27 As well, the effect of the liquid meniscus at the transition of different cross-sections is commonly neglected on the basis of long tubes with extremely mild contractions and expansions.28 This assumption is also adopted in the present work. On the basis of the constraint, we have the following equations:
h1 + h2 = H
(tube)
πr12h1 + πr2 2h2 = πR2H
(3) (4)
and we define
h 2 = nh1
(tube)
(5)
r2 = mr1
where K is the permeability tensor of the medium, ∇p is the pressure gradient, and u is the flow velocity. It is evident that the capillary pressure driving the liquid absorption increases with a decrease in the pore size.25 Conversely, the permeability of porous materials is proportional to the square of the pore size.26 The liquid absorption speed in a uniform porous
(6)
where n and m are the ratios of height and radius, respectively, between the upper (second) and lower (first) sections of the twosection tube. Substituting eqs 5 and 6 into eqs 3 and 4, we have
h1 = B
H 1+n
(tube)
(7)
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nH 1+n
(tube)
t3D,2 = 4C
(8)
H
1
(1 + m2n)0.5 m3n (1 + m2n)0.5 n2 ⎤ H + ⎢2 ⎥ A⎣ m(1 + n)2.5 ⎦ (1 + n)2.5
(9)
(10)
t3D =
t3D,1 + t3D,2 T3D
The capillary pressure in the 3D tube in Figure 2a is given by
=
2γ cos θ Δp = − R
(11)
πR2K ∂p η ∂z
u=
with
∂z ∂t
and
K=
R2 8 (12)
Thus, the hydrostatic pressure drop accounting for the viscous flow is obtained, viz. z
dx ∂z ⎛ 8z ⎞ = −η ⎜ 2 ⎟ 2 ∂t ⎝ R ⎠ πR K
(1 + m2n)0.5 (1 + m2n)0.5 m3n (1 + m2n)0.5 n2 +2 + 2.5 2.5 (1 + n) (1 + n) m(1 + n)2.5
(21) It is noted that, when m is equal to 0 (or infinite), the radius of the second (or the first) section of the 3D tube becomes 0. Thus, the liquid cannot flow through the second (or the first) section, which is completely blocked, and the total corresponding absorption time approaches infinity by comparison to the control sample. To avoid this, we define 0 < m < +∞ in this work. 2.2. Porous Layers. Then, we investigate the general and practical case of liquid absorption in two-layer porous media. A porous medium with uniform porosity (Figure 3a) is the control sample, which is used
and the flow rate is obtained on the basis of Darcy’s law.29
∫0
(20)
The normalized absorption time of the two-section circular tube in Figure 2b in terms of m and n is given as
⎛ 1 + n ⎞0.5 ⎟ mR r2 = ⎜ ⎝ 1 + m2n ⎠
Q = πR2u = −
⎡ r 3h z − h1 ⎤ ⎢ 2 41 + ⎥dz = 2C r2 ⎦ ⎣ r1
2⎡
⎛ 1 + n ⎞0.5 ⎟ R r1 = ⎜ ⎝ 1 + m2n ⎠
Δp = − ηQ
∫h
(13)
Substituting eq 11 into eq 13 leads to γ cos θ ∂z ⎛⎜ 4z ⎞⎟ = η ∂t ⎝ R ⎠
(14)
Therefore, the absorption time of the uniform tube T3D is derived by integrating eq 14, viz.
T3D =
4η γ cos θ
∫0
H
Figure 3. (a) Uniform porous medium and (b) heterogeneous twolayer porous media.
z dz R
(15) for comparison to a two-layer porous structure with a porosity gradient (Figure 3b). A previous study has indicated that the hydraulic contact resistance between two porous layers may be negligible.19 The two porous structures in Figure 3 have identical void volumes, particle sizes, cross-sectional areas, and height. The porous medium in Figure 3a has height H and porosity E, whereas the two-section porous layers in Figure 3b have two different porosities ε1 and ε2, with corresponding heights h1 and h2, respectively. Thus, we have the following equations:
which can be simplified as
T3D = 2C
H2 R
(16)
where we define a constant C = η/(γ cos θ). Then, we calculate the absorption time required to fill the two sections with liquid in Figure 2b, by adding the absorption time for the two local sections (or tubes) with different cross-sections. Initially, the liquid moves to the lower section with a uniform radius. The absorption time of the lower section t3D,1 is obtained analogous to that of the control sample in Figure 2a, viz.
t3D,1 = 2C
h12 r1
2
= 2C
2
h1 + h2 = H
Sε1h1 + Sε2h2 = SEH
(17)
h 2 = nh1
When the liquid moves to the upper section (with height z > h1), the total pressure drop is the sum of the two local hydrostatic pressure drops in the lower and upper sections, respectively, viz.
Δp = − ηQ
∫0
h1
dx − ηQ πr12K1
∫h
z
1
dx πr2 2K 2
with
K1 =
and
2γ cos θ r2
(23)
(24) (25)
where n and m are the ratios of height and porosity, respectively, between the upper (second) and lower (first) layers in Figure 3b. Substituting eqs 24 and 25 into eqs 22 and 23, we have
r12 8
h1 =
H 1+n
(porous media)
h2 =
nH 1+n
(porous media)
ε1 =
1+n E 1 + mn
(28)
ε2 =
1+n mE 1 + mn
(29)
(18)
which is equal to the capillary pressure in the upper section
Δp = −
(porous media)
ε2 = mε1
2
r K2 = 2 8
(22)
where S is the cross-sectional area of the porous structures. Then, we define
0.5
H (1 + m n) A (1 + n)2.5
(porous media)
(19)
Thus, the absorption time of the upper section t3D,2 is obtained by integration C
(26) (27)
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The pressure drop in the porous medium in Figure 3a is given by11 ⎛ E ⎞0.5 Δp = − γ cos θ ⎜ ⎟ ⎝K ⎠
K=
with
tp =
ra 2E3 2
4(1 − E)
with
+
∫0
z
(31)
∂z ⎛ z ⎞ dx = −η ⎜ ⎟ ∂t ⎝ K ⎠ SK
3. RESULTS AND DISCUSSION We first compare our model of liquid absorption in 3D tubes to numerical results of Erickson et al.,28 who studied the capillarydriven flow in the convergent tube (Figure 4) using finite
(32)
On the basis of eqs 30 and 32, we have ⎛ E ⎞0.5 ∂z ⎛⎜ z ⎞⎟ γ cos θ ⎜ ⎟ = η ⎝K ⎠ ∂t ⎝ K ⎠
(40)
When m is equal to 0 (or infinite), the porosity of the second (or the first) layer of the porous media becomes 0. As such, the liquid cannot penetrate the second (or the first) layer, which is blocked, and the total corresponding absorption time approaches infinity by comparison to the control sample. Here, we define 0 < m < +∞ to avoid this phenomenon.
Thus, the pressure drop driving the viscous flow is expressed as follows:
Δp = − ηQ
1 ⎧ mn(1 + mn)(1 + mn − E − nE)2 ⎨2 1 − E ⎩ (1 + n)4 (1 + mn − mE − mnE)
n (1 + mn)(1 + mn − mE − mnE) m2(1 + n)4 (1 + mn)(1 + mn − E − nE) ⎫ ⎬ + (1 + n)4 ⎭
(30)
∂z u= ∂t
Tp
=
2
where ra is the mean particle radius, ε is the porosity of the porous medium, and K is the Kozeny−Carmon permeability of the porous medium.30 In this porous medium, the flow rate is obtained on the basis of Darcy’s law
SK ∂p Q = Su = − η ∂z
t p,1 + t p,2
(33)
Therefore, the absorption time of the control sample Tp is obtained by integration, viz. Tp =
η γ cos θ
∫0
H
z dz E 0.5K 0.5
(34)
which is simplified as
Tp = C
H2 1 − E ra E2
(35)
Then, we present the absorption time for filling the porous structures with liquid in Figure 3b, by calculating the absorption time for the two layers, respectively. The absorption time of the lower layer tp,1 is obtained analogous to that of the control sample in Figure 3a, viz.
t p,1 = C
h12 1 − ε1 H2 (1 + mn)(1 + mn − E − nE) C = ra ε12 (1 + n)4 raE2
Figure 4. Comparison of absorption time versus capillary rise height in the 3D convergent circular tube between the present model and numerical results.
(36) The pressure drop when the liquid penetrates the upper layer (with height z > h1) is the sum of the Darcy pressure drops of the two local layers, viz.
∫0
Δp = − ηQ and
K2 =
h1
dx − ηQ SK1
∫h
z
1
dx SK 2
with
K1 =
element numerical simulations. The tube is circular and contains two subsections connected by an extremely mild contraction. The tube has diameter D1 = 100 μm and D2 = 50 μm and height H1 = 20 mm and H2 = 2.86 mm.28 The liquid fluid properties are γ = 0.03 N/m, η = 0.001 kg s−1 m−1, and θ = 30°.28 In Figure 4, we plot the curve lines of absorption time versus the capillary rise height based on eqs 17 and 21, with h1 = H1 and h1 = H1 + H2, respectively. It is shown that both curve lines agree closely with numerical results of Erickson et al.,28 indicating the accuracy of our model and the negligible effect of the mild contraction. With the validated model of eq 21, it is easy to plot the normalized absorption time of the heterogeneous 3D tube versus the radius ratio m and the height ratio n in panels a and b of Figure 5, respectively. The result is that t3D in Figure 5a first decreases and then increases dramatically with the increase in m at three different n values (i.e., n = 0.5, 1, and 2), and the three curves converge at m = 1 because all two-section tubes become identical to the control sample in Figure 2a. It is clear that the distance of spontaneous liquid movement in the upper section is not proportional to the root of time when m ≠ 1 and then t3D ≠ 1, because the normalizing term based on eq 16 follows
ra 2ε13 4(1 − ε1)2
ra 2ε2 3 4(1 − ε2)2
(37)
which is equal to the capillary pressure in the upper layer, viz. ⎛ ε ⎞0.5 Δp = − γ cos θ ⎜ 2 ⎟ ⎝ K2 ⎠
(38)
Thus, the absorption time of the upper layer tp,2 is derived
t p,2 = C
∫h
H
1
⎛ K 2 ⎞0.5⎡ h1 (z − h1) ⎤ + ⎥ dz = C ⎜ ⎟ ⎢ K2 ⎦ ⎝ ε2 ⎠ ⎣ K1
H ⎡ nm(1 + mn)(1 + mn − E − nE)2 ⎢2 raE2 ⎣ (1 + n)4 (1 + mn − mE − mnE) 2
+
n2(1 + mn)(1 + mn − mE − mnE) ⎤ ⎥ ⎦ m2(1 + n)4
(39)
The normalized absorption time tp in terms of m and n is as follows: D
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Figure 5. Normalized absorption time in the heterogeneous 3D tubes versus (a) radius ratio (m) at different height ratios (n) and (b) height ratio (n) at different radius ratios (m).
important application in advanced fabrics, as mentioned in the Introduction. It is expected that the minimum t3D in terms of m and n can simultaneously be found by analyzing Figure 5. With the help of MATLAB, the minimum t3D is found, viz., 0.6186, with m = 0.5141 and n = 0.6513 (the calculation process is shown in the Supporting Information). This result indicates that the liquid absorption time can be decreased by 38% compared to the 3D uniform tube at a fixed height and volume. The minimum normalized absorption time tp of heterogeneous two-layer porous media with different system porosities E versus the porosity ratio m and the height ratio n is clearly visible in Figures 6 and 7, respectively. Note that the red dotted
Washburn’s law. Similar phenomena are found later in porous layers, indicating that liquid absorption in these heterogeneous porous structures is not diffusive. Moreover, the concave curves in Figure 5a indicate the optimal m between 0 and 1 that results in the minimum absorption time for the heterogeneous tube, and the corresponding optimal values of m vary with n. This is expected in this moderate area of m. The lower section of the tube with a larger radius has greater permeability and requires less absorption time (i.e., t3D,1) based on eq 17, whereas the upper section with a smaller radius creates higher capillary pressure to drive the overall liquid movement. When m reaches the higher limit (m ≫ 1), however, the radius is extremely small in the lower section and the flow resistance rises markedly, significantly slowing the process of liquid absorption. On the other hand, when m decreases to nearly 0, the permeability of the upper section, which is proportional to the square of the radius, is significantly reduced. The extremely low permeability suppresses the effect of the capillary pressure, driving all of the liquid in two sections. In the two-section heterogeneous 3D tube, therefore, the minimum absorption time against the radius results from the two superimposed mechanisms, capillary pressure and permeability. Figure 5b presents the normalized absorption time t3D for the 3D tube (Figure 2b) versus n at different values of m (m = 0.5, 1, and 2). It is evident that t3D is not sensitive to n with m = 1, as expected. When m = 0.5 and n approaches 0, the height of the upper tube is close to 0 based on eq 8 and the radius and height of the lower section are close to those of the uniform tube in Figure 2a. Thus, t3D tends to 1. Similarly, when m = 0.5 and n approaches infinity, the upper tube becomes very close to the uniform tube of Figure 2a and they have almost the same absorption time. For moderate n, radii of both the upper and lower sections vary with n. Accordingly, the minimum t3D occurs by adjusting the contribution of liquid absorption in the two local sections based on the phenomena that the lower, wider section promotes higher permeability and simultaneously the upper, narrower tube causes higher driving pressure for the liquid. It is interesting to observe that the maximum t3D exists with m = 2, which is ascribed to the physics being opposite to the above case (m = 0.5) for the minimum t3D. Therefore, the slowest liquid absorption simultaneously accompanied by the fastest liquid absorption from the inverse flow direction leads to a structure with asymmetric absorption behavior, which has
Figure 6. Normalized absorption time in the heterogeneous porous layers versus porosity ratio (m) at different height ratios (n) and system porosities (E).
line in Figure 7b is anomalous. This is ascribed to the artifact of eq 29 as a result of ε2 > 1, which is obtained when n approaches 0 based on eq 29. It is necessary, therefore, to restrict ((1 + n)/ (1 + mn))E < 1 and ((1 + n)/(1 + mn))mE < 1 for eqs 28 and 29, respectively. Moreover, the optimal parameters of porosity and height for the fastest absorption are found to be sensitive to the system porosity. It is also demonstrated in Figure 7 that the capillary flow for a negative gradient of pore sizes (as a result of E
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Figure 7. Normalized absorption time in the heterogeneous porous layers versus height ratio (n) at different system porosities (E) and local porosity ratios, i.e., (a) m = 0.5 and 1 and (b) m = 1 and 2.
the porosity gradient) is faster than the capillary flow for a positive gradient, which is consistent with experimental results of Reyssat et al.22 It is noted that we find the optimal porosity ratios between the two porous layers by taking the mean particle radii identical in both porous layers, because porous layers made up of particles with identical sizes are widely found in practical applications (e.g., layers of packed, uniform beads).22 In fact, this is an example of optimizing the structure of porous materials by fixing the radius of particles, and many extensions of this work can also be considered. The present analytical method used to design the porous structures to promote the fastest liquid absorption is very general and can be easily extended to find the optimal structures based on other structural parameters. For instance, we can find the optimal value of the particle radius ratio by fixing the porosity. As well, the porous structures are not limited by sets of particles; they can be widely used fiber- or network-based porous assemblies with different permeabilities.31−33 The minimum tp against m and n is simultaneously also obtained using MATLAB, viz., tp = 0.7122, with m = 0.5980 and n = 0.4579 for E = 0.4; tp = 0.6928, with m = 0.6173 and n = 0.4617 for E = 0.5; and tp = 0.6680, with m = 0.6403 and n = 0.4621 for E = 0.6 (the calculation process is shown in the Supporting Information). The results reveal that the speed of liquid absorption can be increased by around 30% over that of the uniform porous medium at fixed height and system porosity.
We observe that the absorption time of the heterogeneous system is less than that of the control sample for a negative radius/porosity gradient against the absorption direction. The hydrodynamics of the flow front advance in these heterogeneous porous materials is not diffusive and causes a deviation from Washburn’s law. It is demonstrated that optimal parameters exist, including height and radius/porosity radio of local sections, for the fastest liquid absorption. As well, the values of the absorption time of optimized porous structures are found to reduce by around 30% compared to the control samples. This study provides an initial theoretical exploration of the fastest liquid absorption in common structures, but the proposed robust framework can be extended to more complex systems under different constraints. The present analysis encourages us to design novel porous structures with asymmetrical absorption behaviors and to explain some of the underlying mechanisms behind the natural phenomena, such as the effect of radius/length variation in different branching hierarchies of trees on their liquid absorption behavior.
■
ASSOCIATED CONTENT
S Supporting Information *
Calculation of optimal values of m and n. This material is available free of charge via the Internet at http://pubs.acs.org.
■
AUTHOR INFORMATION
Corresponding Author
4. CONCLUSION A quantitative model has been developed to find the fastest liquid absorption in two-section heterogeneous porous architectures. The heterogeneous arrangement can accelerate or decelerate liquid absorption in different sections. As a whole, the optimized design and adjustment of the sections leads to the minimum absorption time for the total porous system. The proposed model of time-dependent capillary rise in the heterogeneous circular 3D tube is verified by collected numerical results. As well, the model of capillary flow in the heterogeneous porous layers is consistent with experimental results in the literature. The absorption time of heterogeneous structures is compared to that of a uniform medium, which is employed as a control sample at fixed height and void volume.
*Telephone: +61-02-9351-4798. Fax: +61-02-9351-3760. Email:
[email protected]. Notes
The authors declare no competing financial interest.
■ ■
ACKNOWLEDGMENTS Lin Ye is grateful for an Australian Research Council (ARC) Discovery Project grant that supports this work. REFERENCES
(1) Zheng, Y.; Bai, H.; Huang, Z.; Tian, X.; Nie, F.-Q.; Zhao, Y.; Zhai, J.; Jiang, L. Directional water collection on wetted spider silk. Nature 2010, 463 (7281), 640−643.
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dx.doi.org/10.1021/la4034063 | Langmuir XXXX, XXX, XXX−XXX
