Langmuir 2006, 22, 8431-8434
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Relation between Contact Angle and the Cross-Sectional Area of Small, Sessile Liquid Drops C. W. Extrand* Entegris Inc., 3500 Lyman BouleVard, Chaska, Minnesota 55318 ReceiVed May 10, 2006. In Final Form: July 11, 2006 In this theoretical study, the cross-sectional areas of small, sessile drops were calculated for solid surfaces with a wide range of wettability. These areas were then used to estimate obstruction of rectangular gas flow channels by sessile liquid drops. Our findings suggest that even a small improvement in wettability (i.e., a lower contact angle) will lead to a substantial decrease in blockage. This work has implications for small channels that contain two-phase flow, such as those found in fuel cells.
Introduction Fuel cells hold great promise as a more environmentally friendly energy source. For fuel cells to become ubiquitous, a number of technical issues must be overcome.1,2 One such issue is flooding.3,4 In a low temperature ( 150°). When θ ) 90°, A/A0 is 0.79. At θ ) 45°, A/A0 declines slightly more to 0.61, and then
Figure 2. Normalized meridian cross-sectional area, A/A0, of a small, sessile liquid drop versus contact angle, θ, for constant radius of curvature, R, constant volume, V, and constant contact diameter, D, eqs 8-10. For constants R and V, A0 is the cross-sectional area of the small spherical drop prior to deposition, θ ) 180°. For constant D, A0 is the cross-sectional area of a small hemispherical sessile drop, θ ) 90°.
falls steeply toward zero for θ < 5°. The variation in A/A0 assuming a constant contact diameter is also shown in Figure 2 for θ values between 90° and 0°. With increased wettability, A/A0 decreases in a nearly linear fashion toward zero. Obstruction of a Rectangular Channel by a Sessile Liquid Drop. If the gas phase of a fuel cell is saturated with water vapor, improvement in performance due to greater hydrophilicity occurs as a result of a reduction in flow channel obstruction. (On the other hand, if the gas phase in the flow channel is not saturated, then rendering the surface of the flow channels hydrophilic may increase the gas-liquid interfacial area, which may increase evaporation and improve water removal. Relative gas-liquid interfacial areas of small drops are discussed in the Appendix.) Consider the case of water drops inside the small, saturated, gas flow channel of a fuel cell. The cell operates at steady state, and the volume of liquid water in its gas flow channels is constant. The liquid water inside a nonwetting (θ > 0°) channel would likely be present as a multitude of drops confined by the walls of the channel. The smaller the drop volume and the lower the contact angle, the less obstruction there is of the channel. This can be demonstrated by the following example. Figure 3 shows the side view of a rectangular channel that is partially blocked by a small liquid drop. The channel is twice as wide as it is high (w/h ) 2). The cross-sectional area of the channel, Ac, is
Ac ) wh ) w2/2
(11)
Contact Angle and Cross-Section of Drops
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Figure 3. Side view of a rectangular channel that contains a small liquid drop. The channel is twice as wide as it is high (w/h ) 2). The drop partially blocks the channel. The contact diameter of the drop is equal to the width of the channel (D ) w). The fraction of obstruction varies with contact angle: (a) a drop with θ ≈ 90° and (b) another drop with θ ≈ 45°.
Figure 4. Fraction of obstruction, f, versus contact angle, θ, for a small, rectangular channel with width/depth ) 2, where contact diameter ) channel width, as depicted in Figure 4. Values were calculated according to eq 12.
A large drop could completely wet the perimeter, creating a slug of liquid that would totally block the channel.15 However, shy of complete obstruction, the worst case would be a drop with a contact diameter equal to the width of the channel (D ) w) and a large θ. In Figure 3a, the channel is hydrophobic (or lyophobic) with θ ≈ 90°. Therefore, most of the channel is blocked by the drop cross-section. In Figure 3b, the channel is more hydrophilic, θ ≈ 45°, and obstruction is less. For this particular example, the fraction of the rectangular channel that is blocked by the meridianal cross-section of the drop, f, can be calculated from eqs 4 and 11 by assuming D remains constant,
f ) A/Ac ) (1/2)(θ - cos θ sin θ)/sin2 θ
(12)
Figure 4 shows the fraction of obstruction, f, versus contact angle, θ, according to eq 12. Because the contact diameter of the drop spans the entire width of the channel, for θ ) 90°, the rectangular channel is largely obstructed, f ) 0.78. Significant gains in flow can be expected from improved wettability. The largest incremental reductions in obstruction come from the reducing the contact angle of more hydrophobic surfaces. For example, if θ is reduced from 90° to 60°, then channel obstruction is cut nearly in half. Further reduction of θ to 45° shrinks the obstruction to f ) 0.28. As the cross-section decreases, the liquid must go somewhere. In a real system, either the number of drops would increase or drops with D ) w would elongate along the channel. Blockage would be less if the volume of liquid in a system were divided into a greater number of drops. For a given drop volume, drops with lower contact angles can merge more easily than drops on hydrophobic surfaces having larger contact angles. In principle, a perfectly wettable channel (θ ) 0°) would completely eliminate obstruction due to liquid drops; however, in practice it is difficult to maintain near-zero contact angles on (15) A sufficiently large drop can form a slug that will completely block a hydrophobic flow channel. If this occurs, the force (or pressure) required to dislodge the slug and flush it from a horizontal flow channel will depend on the effective diameter of the flow channel, the surface tension of slug liquid, as well as the advancing and receding contact angles between the liquid and channel walls: West, G. D. Proc. R. Soc. London 1912, 86A, 20-25. If the slug moves vertically, then the volume of the slug, its density, and its vertical displacement must also be considered. Also, the rate at which the slug exits the flow channels will be influenced by the viscosity of the liquid.
Figure 5. Normalized gas-liquid interfacial area, S/S0, of a small, sessile liquid drop versus contact angle, θ, eqs 13, 15, 17. For constants R and V, S0 is the gas-liquid interfacial area of the small spherical drop, θ ) 180°. For constant D, S0 is the gas-liquid interfacial area of a small hemispherical sessile drop, θ ) 90°.
surfaces exposed to the ambient environment.16 These highenergy surfaces quickly attract hydrocarbons and other lowenergy contaminants, and consequently their contact angles rise. Nevertheless, this analysis suggests that modest improvements in wettability can lead to substantial reductions in obstruction of small flow channels by condensed liquid. (16) Wu, S. Polymer Interface and Adhesion; Marcel Dekker: New York, 1982.
8434 Langmuir, Vol. 22, No. 20, 2006
Extrand
The same approach could be applied to other channel geometries. A more comprehensive analysis would also include the influence of gravity on the cross-sectional area of larger or inclined drops. Under the action of gravitation forces, the basic spherical geometry used in this study would not apply.
where
S0 ) 4πR02 For constant drop volume (V ) V0),
Concluding Remarks Increasing the wettability of hydrophobic surfaces reduces contact angles and can lead to a substantial decrease in crosssectional areas. For a small sessile drop of constant volume, moderate increases in wettability do little to reduce the crosssectional area. Here, the largest reductions are realized if the wettability of a hydrophobic surface is reduced such that θ < 45°. In contrast, the cross-sectional area of a small sessile drop of constant contact diameter decreases steadily with increased wettability, where even moderate reductions in θ lead to substantial decreases in drop profile. These reductions in crosssectional area can minimize the obstruction in small gas flow channels and therefore are believed to improve the performance of fuel cells. While the motivation for this study was fuel cells, the same method could be applied to any macro-, micro-, or nano-fluidic device where liquid and flowing gas coexist. Acknowledgment. I thank Entegris management for supporting this work and allowing publication. Also, I thank C. Metzger, L. Monson, and S. I. Moon for their suggestions on the technical content and text.
Appendix Gas-Liquid Interfacial Areas. Area ratios similar to those for cross-sectional area can be derived for gas-liquid interfacial area, S. For constant radius of curvature (R ) R0),6,8
S/S0 ) (1/2)(1 - cos θ)
(13)
(14)
S/S0 ) 2[4(1 - cos θ)(2 + cos θ)2]-1/3
(15)
S0 ) (36πV02)1/3
(16)
where
Finally, for constant contact diameter (D ) D0),8
S/S0 ) (1 - cos θ)/sin2 θ
(17)
S0 ) (π/2)D02
(18)
where
Figure 5 shows gas-liquid interfacial area ratios, S/S0, of a small, liquid drop versus contact angle, θ, for three cases: constant radius of curvature, constant volume, and constant contact diameter. As a small drop of constant volume begins to spread, its gas-liquid interfacial area initially decreases. As θ values fall further, S/S0 reaches a minimum value at θ ) 90° and then begins to climb. S/S0 reaches unity again at θ ) 42°. As θ tends toward 0°, S/S0 values diverge; at θ ) 10°, S/S0 ) 2.45. The variation in S/S0 assuming a constant contact diameter is also shown in Figure 5 for θ values between 0° and 90°. As the contact angle decreases from θ ) 90°, S/S0 declines more precipitously and then levels for small θ values, reaching S/S0 ) 0.5 at θ ) 0°. LA061325H
