Article pubs.acs.org/Langmuir
Intrusion Pressure To Initiate Flow through Pores between Spheres C. W. Extrand* and Sung In Moon Entegris, Incorporated, 101 Peavey Road, Chaska, Minnesota 55318, United States ABSTRACT: In this work, clusters of three or four spheres were used to examine intrusion pressure. Polytetrafluoroethylene (PTFE) or polyamide 66 (PA66) spheres were arranged horizontally to create a single pore. Liquid drops of water or ethylene glycol were gently introduced from above. If the spheres were too large, drops flowed through as soon as they were deposited. If the spheres were too small, liquid was suspended in the neck of the pore and could not pass through; drops became unstable and fell to one side. Alternatively, if spheres of a certain size were chosen, then capillary forces initially prevented drops of lesser stature from breaking though. However, as these drops grew taller, they eventually reached a height where the gravitational force exceeded the capillary force and the liquid flowed through the pore. A simple model for intrusion pressure was derived. Estimates from the model agreed well with experimentally measured values.
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INTRODUCTION The first analysis of intrusion pressure is attributed to Lucas1 and Washburn.2 Their equation allows for estimation of the critical hydrostatic pressure (Δpc) that must be applied to force liquid into a lyophobic cylindrical pore of diameter D
spheres that are uniformly organized and defect-free. Moreover, edge effects can complicate interpretation of data. To avoid these impediments yet still try to capture the essence of multiple sphere systems, we studied the wetting interactions and intrusion pressures associated with three- and four-sphere clusters by means of experimental observation and firstprinciple calculations.
− 4γ cos θ (1) D where γ is the surface tension of the intruding liquid and θ is its contact angle. The Lucas−Washburn equation has been widely applied to media having other pore geometries; however, the agreement between the estimated and actual pore size is often poor.3,4 Attempts have been made to model impregnation or intrusion into noncylindrical geometries.3−10 For example, Mayer and Stowe3 extended the analysis of Lucas−Washburn to the pores between packed spheres using free-energy arguments. Their theoretical analysis gave “breakthrough” pressures in terms of the porosity of packed spheres. Morrow and Mason4 examined the position of liquid menisci between spheres as a function of the contact angle and inferred a correction factor for the cosine function of eq 1, replacing cos θ with (2/3)cos θ. Bán et al.6 developed an intrusion criterion based on the wettability of hexagonal arrays of packed spheres. For liquids that effectively have no hydrostatic pressure, they showed that intrusion is independent of the sphere size and can be estimated from the contact angle. More recently, McHale and colleagues have used hexagonal arrays of packed spheres as models for soils. They examined contact angles by free-energy methods7,8 and also determined a wettability criterion for liquid intrusion.9,10 While models have been constructed for the intrusion pressure of well-organized spherical systems, little has been done experimentally. Why? Very small spheres and/or many of them can hamper observation of liquids as they intrude. Also, it is difficult to create multiple two- or three-dimensional arrays of
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Δpc =
© 2012 American Chemical Society
THEORY In contrast to earlier models that analyze the interfacial areas and their associated energies, we focus on the capillary forces at the contact line. Thus, in our analysis, two competing forces determine how far a liquid can intrude into the pore of a sphere cluster and whether it can flow all the way through the pore: a gravitational force and a capillary force. The gravitational force is always directed downward. In contrast, the capillary force can act up or down. The direction and magnitude of the capillary force depends upon the inherent wettability of the spheres, as gauged by a contact angle and the position of the contact line. If the capillary force (fc) is greater than or equal to the gravitational force ( fg)
fc ≥ fg
(2)
then the liquid is trapped inside the pore. Otherwise, if fc < fg, liquid flows through the pore. Description of the Four-Sphere Cluster. Panels a and b of Figure 1 show a four-sphere cluster. The spheres have a diameter of 2R and lie on a regular array. The unit cell dimension is also 2R. The space between the four spheres creates a pore. If a liquid drop is placed in the top of the pore of the four-sphere cluster, it may immediately flow through the pore and wet the underlying surface. Alternatively, the drop may partially intrude Received: December 24, 2011 Revised: January 17, 2012 Published: January 19, 2012 3503
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(g = 9.81 m/s2).
Δp = ρgh
(4)
The hydrostatic pressure is distributed over the unwetted crosssectional area of the pore
A = A u − As
(5)
where Au is the area of the unit cell
A u = 4R2
(6)
and As is the wetted cross-section of each sphere.
A s = πR2 sin 2 ϕ
(7)
The vertical component of the capillary force (fc) that can oppose (or aid) the downward pull of gravity on a liquid can be calculated from the product of the length of the contact line on the spheres and the vertical component of the liquid surface tension.12−25
fc = −π(2R )γ sin ϕ sin(θ + ϕ) Figure 1. Depictions of sphere clusters. (a) Plan and (b) side views of a four-sphere cluster. (c) Plan and (d) side views of a three-sphere cluster.
Combining eqs 2, 3, and 5−8 yields an expression that relates the hydrostatic pressure and extent of intrusion.
Δp =
before being suspended. In Figure 2, the position of a suspended liquid drop is described by the angle ϕ, which
− 2γ sin ϕ sin(θ + ϕ) R(4/ π − sin 2 ϕ)
(9)
By equating eqs 4 and 9, the height of the liquid can be related to its properties and the size of the spheres.
h=−
γ 2 sin ϕ sin(θ + ϕ) ρgR 4/π − sin 2 ϕ
(10)
Limiting Values for Preventing Breakthrough or Flow. It will be shown later that the location of the zero and maximum values of the capillary force is useful in understanding the wetting behavior and critical intrusion pressures. The intrusion angle where the capillary force is equal to zero (ϕ0) can be found by setting eq 8 equal to zero
Figure 2. Side-view depictions of water in the pore of a four-sphere cluster. (a) Far-field view showing the intrusion angle (ϕ) and the height (h) of the liquid drop. (b) Close-up view of the advancing liquid front showing the local inherent contact angle (θ) and the orientation of the surface tension (γ) vector.
sin(θ + ϕ) = 0
(11)
and solving for ϕ in terms of θ. Equation 11 has solutions or critical points where (θ + ϕ) is equal to multiples of π (0, π, 2π, ...). The immersion angle where the capillary force is zero occurs at
ranges from 0 radians (0°) for no intrusion to π radians (180°) for complete intrusion. The liquid drop has a surface tension of γ, density of ρ, and height of h. We assume that the curvature at the top and bottom of the drop are equal, and thus, contributions from Laplace pressure can be ignored. The drop advances across the solid spheres with an inherent contact angle of θ.11 Any contortions of the contact line or the air/ liquid interface are also neglected. The ability of a sphere cluster to resist intrusion of liquid can be cast as a competition between the gravitational force acting on the mass of the drop and capillary forces acting at the contact line between the liquid drops and spheres within the pore. The gravitational force can be described as the product of the hydrostatic pressure (Δp) and the cross-sectional area of the pore (A).
fg = ΔpA
(8)
ϕ0 = π − θ
(12)
Maximum values of the immersion angle (ϕmax) can be determined by differentiating eq 8 with respect to ϕ
∂fc /∂ϕ = −π(2R )γ[sin ϕ cos(θ + ϕ) + cos ϕ sin(θ + ϕ)] simplifying with the appropriate angle-sum relation
∂fc /∂ϕ = −π(2R )γ sin(θ + 2ϕ)
(13) 26
(14)
and then setting eq 14 equal to zero.
sin(θ + 2ϕ) = 0
(3)
(15)
Equation 15 has solutions or critical points where (θ + 2ϕ) is equal to multiples of π (π, 2π, ...). The immersion angle where the capillary force exhibits a maximum vertical value occurs at13
The hydrostatic pressure (Δp) depends upon the density of the liquid drop, its height (h), and the acceleration due to gravity 3504
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ϕmax = π −
θ 2
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Incorporating eq 16 into eqs 9 and 10 and simplifying gives expressions for the critical liquid pressure and height for breakthrough in terms of sphere size and wettability
Table 1. Properties of the Liquids and Inherent Contact Angles of the Various Liquid−Solid Combinations
2 γ 1 2 R ρg (4/π)csc (θ/2) − 1
(18)
− 2γ sin ϕ sin(θ + ϕ) R(2 3 /π − sin 2 ϕ)
(19)
γ 2 sin ϕ sin(θ + ϕ) ρgR 2 3 /π − sin 2 ϕ
(20)
and
h=−
The corresponding critical liquid pressure and height for threesphere clusters are
2γ 1 R (2 3 /π)csc 2(θ/2) − 1
(21)
2 γ 1 R ρg (2 3 /π)csc 2(θ/2) − 1
(22)
Δpc = and
hc =
Comparison of Pores from Cylinders and Spheres. To adapt the Lucas−Washburn equation to noncylindrical geometries, it is customary to estimate an equivalent pore size from the solid surface area3,27 or pore volume.28,29 In contrast, the differences between the Lucas−Washburn equation and the expressions derived here can be compared in terms of the cross-sectional area of the pore (A) and recast as dimensionless intrusion pressures. The critical dimensionless intrusion pressure for the Lucas−Washburn equation is 1/2
Δpc A
1/2
/γ = − 2π
cos θ
(23)
(24)
and
Δpc A1/2 /γ = 2
[2 3 − π sin 2(θ/2)]1/2 (2 3 /π)csc 2(θ/2) − 1
solid
θ (deg)
water water ethylene glycol
72 72 47
998 998 1110
PTFE PA66 PTFE
108 70 89
RESULTS AND DISCUSSION Pore between the Spheres. Figure 3 shows the dimensionless cross-sectional area of the pore between fourand three-sphere clusters (A/πR2) plotted against the intrusion angle (ϕ), where the area of the pore (A) is divided by the equatorial cross-section of the spheres (πR2). While the crosssection area of a Lucas−Washburn cylindrical pore remains constant, the cross-sectional area of the pores between the spheres is convergent−divergent;4 the pores for both three- and four-sphere clusters are relatively large at the upper apexes of the spheres (ϕ = 0°), narrow to a minimum at ϕ = 90°, and then open again. For the pore between the four-sphere cluster, A/πR2 ranges between 1.27 and 0.27. For a given ϕ, the foursphere pore is bigger than the three-sphere pore. The absolute
[4 − π sin 2(θ/2)]1/2 (4/π)csc 2(θ/2) − 1
ρ (kg/m3)
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For the four- and three-sphere clusters
Δpc A1/2 /γ = 2
γ (mN/m)
All solid spheres were purchased from McMaster Carr. The spheres were composed of polytetrafluoroethylene (PTFE) or polyamide 66 (PA66 or nylon 66) and ranged in size from 1.6 mm (1/16 in.) to 6.4 mm (1/4 in.). The diameters (2R) of the spheres were measured by two methods. The diameter of the larger spheres was gauged with a micrometer (Mitutoyo IP65). Alternatively, the sphere images were captured using a drop shape analyzer (Krüss DSA10), and the sphere diameters (2R) were estimated from images using Image-Pro Plus software. Diameters from the micrometer and the image analysis method generally differed from the nominal values of the supplier by less than 2%. To measure contact angles, small liquid drops were gently extruded from a 1 mL, glass syringe (M-S, Tokyo, Japan) and deposited on the spheres. Liquid was added to advance the contact line, and then a drop shape analyzer (Krüss DSA10) was used to capture the images. Contact angles were calculated from a three-phase interface using the tangent line method, and the curvature of sphere was normalized using a previously reported method.11 The standard deviation of the measured θ values was ±2°. Advancing contact angles of the various liquid−solid combinations are listed in Table 1. PTFE is quite hydrophobic and yielded θ = 108°. The lower surface tension liquid, ethylene glycol, wet PTFE more strongly than water and produced a lower contact angle, θ = 89°. PA66 is relatively hydrophilic and, thus, produced the lowest intrinsic contact angle, θ = 70°. These values agreed reasonably well with published values.31−33 To prevent rolling during the intrusion experiments, spheres were immobilized with double-side adhesive tape on a glass slide. Three- or four-sphere clusters was prepared by packing them closely with tweezers. Initially, a liquid drop was placed on top of the pore using a plastic pipet (3 mL transfer pipet, Samco Scientific Corp.). The size of the initial drop was chosen to be larger than the open pore between the spheres. Additional liquid was gently added. Images were captured using a drop shape analyzer (Krüss DSA10). The intrusion angle and drop height were measured from the images using Image-Pro Plus software. For each liquid-sphere combination, at least three trials were made. Errors in the critical intrusion angle and critical height were estimated to be ±5° and ±0.2 mm. All measurements were performed at 25 ± 1 °C.
Three-Sphere Cluster. Clusters having three spheres were also analyzed here. As depicted in panels c and d of Figure 1, these spheres have a diameter of 2R and lie on a hexagonal array. Using the same approach described above, the general expressions for the intrusion pressure and liquid height of three-sphere clusters are
Δp =
liquid
(17)
and
hc =
EXPERIMENTAL DETAILS
The liquids used in the experiments were 18 MΩ cm deionized water and ethylene glycol (Sigma-Aldrich, anhydrous 99.8%). Density (ρ) and surface tension (γ) of the liquids are given in Table 1.28,30
(16)
2γ 1 Δpc = 2 R (4/π)csc (θ/2) − 1
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Figure 5. Minimum and maximum intrusion angles (ϕi = ϕ0 or ϕmax) as a function of the inherent contact angle of the sphere material (θ), where i = 0 for the minimum intrusion angle and i = max for the maximum.
Figure 3. Dimensionless cross-sectional area of the pore between fourand three-sphere clusters (A/πR2) plotted against the intrusion angle (ϕ). The area of the pore (A) is divided by the equatorial cross-section of the spheres (πR2).
a cluster had an inherent wettability of θ = 108°, then the potentially stable range would be between 72° and 126°. Comparison of Pores from Cylinders and Spheres. Figure 6 shows a comparison of normalized critical intrusion
difference between the two is 0.17. On a comparative basis, this difference between the two configurations equates to 15% at ϕ = 0° or 180° but increases to more than 2.5× in the narrowest portion of the pore, where ϕ = 90°. Estimates of Capillary Forces and Intrusion Angles. Because the gravitational force/hydrostatic pressure is always directed downward, the ability of sphere clusters to impede flow largely hinges upon the capillary force. Therefore, let us examine how the capillary force interacts with spheres. Figure 4
Figure 6. Dimensionless critical intrusion pressures (ΔpcA1/2/γ) as a function of the contact angle (θ) for a cylindrical pore versus the pores between spheres.
pressures (ΔpcA1/2/γ) for the cylindrical pore and the pore between spheres as a function of the contact angle (θ), according to eqs 23−25. According to the Lucas−Washburn equation, for a smooth cylindrical pore to hold back liquid intrusion, its θ must be >90°. Not true for the pores between spheres. While a higher θ improves resistance to intrusion, it is not a requirement. Even low θ values can stop flow. For an equivalent area, pores between spheres always produce greater Δpc values than cylindrical pores. Because we have normalized for pore area, there is not much difference between four- and three-sphere clusters. Ability To Intrude Sphere Clusters. Figures 7−9 show images for intrusion experiments for water drops on clusters of four PTFE spheres. Figure 7 shows a cluster of small PTFE spheres, 2R = 3.2 mm. A 20 μL sessile drop was deposited on top of the cluster, and water was sequentially added. The water drop was suspended in the pore. Here, the maximum capillary force was much greater than the gravitational force, fc,max ≫ fg. With the addition of more and more water, the drop incrementally grew to a height of h = 5.4 mm, became unstable, and fell to one side. Because its ultimate height was less than the critical value for breakthrough (hc = 9.8 mm), none of the water passed through the pore. The PTFE spheres in Figure 8 are relatively large with 2R = 6.4 mm. A 100 μL sessile drop was deposited on top of the cluster, and the water immediately flowed through the pore. In this case, the maximum capillary force was less than the gravitational forces, fc,max < fg. This is the anticipated result, because
Figure 4. Variation of the vertical component of the capillary force ( fc) with the intrusion angle (ϕ).
shows variation in the vertical component of the dimensionless capillary force ( fc/2Rγ) plotted against the intrusion angle (ϕ) for various values of the contact angle (θ). Prior to the addition of liquid, the intrusion angle is zero (ϕ = 0°). As liquid intrudes into the pore and ϕ increases, the magnitude of fc/2Rγ initially decreases from zero. Here, the capillary force is directed downward, acting in concert with gravity. The fc/2Rγ values pass through a minimum value (i.e., the largest downward capillary force) before rising to a maximum value (i.e., the largest upward capillary force) and then falling back to fc/2Rγ = 0 at ϕ = 180°. If the capillary force is directed downward in concert with gravity, then the intruding liquid moves deeper into the pore. Otherwise, if directed upward and of sufficient magnitude, the liquid may be suspended and trapped within the pore. This applies for both three- and four-sphere clusters. Figure 5 shows the zero-force and maximum-force intrusion angles (ϕ0 and ϕmax) as a function of the contact angle. The values of ϕ0 range between 0° and 180°, while ϕmax values span 90−180°. Our model assumes that a liquid could be suspended in the pore for ϕ values between ϕ0 and ϕmax. If ϕ < ϕ0, then the surface tension vector will be directed downward and the liquid will advance. If ϕ > ϕmax, then the capillary force decreases with further intrusion and, thus, cannot hold back an increasing gravitational force. For instance, if the spheres within 3506
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Figure 9 shows images of the same experiment repeated with a cluster of intermediate-sized PTFE spheres, 2R = 4.8 mm. Again, 100 μL water drop was deposited, but this time, the drop was suspended in the pore (Figure 9b). The height of the drop was h = 4.9 mm and had intruded into the pore such that ϕ = 102°, which was well within the predicted ϕ0−ϕmax range for water on PTFE spheres, 72−126°. With the incremental addition of more water, the contact line advanced in a stepwise fashion. Figure 9c shows intrusion of the drop where the gravitational force was nearly equal to the maximum capillary force. Here, the measured critical intrusion angle was ϕmax = 120°, and the height of the drop had grown to h = 6.9 mm. The next increment of water caused flow through the pore (Figure 9d). Other liquid/solid combinations behaved similarly. If the spheres were too large, drops ran through the pore as soon as they were deposited. If the spheres were too small, liquid could not pass though the pore. Alternatively, if spheres of a certain size were chosen, then capillary forces initially prevented drops of lesser stature from breaking though the pore. However, as these drops grew taller, they eventually reached a height where the gravitational force exceeded the maximum capillary force and the liquid flowed through the pore. Figure 10 shows the
Figure 7. Images of water drops on a four-sphere cluster of small PTFE spheres, 2R = 3.2 mm. (a) Side view of the four-sphere cluster before deposition. (b) Water drop is deposited and was suspended in the pore. (c) With added water, the drop grew incrementally. (d) Drop became unstable and fell to one side; none of the water passed through the pore.
Figure 8. Series of images showing water drops on a four-sphere cluster, where the large PTFE spheres have a diameter of 2R = 6.4 mm. (a) Side view of the four-sphere cluster before deposition. (b) Drop was deposited and immediately flowed through the pore.
Figure 10. Additional combinations of liquids and four-sphere clusters at the critical condition where fc,max ≈ fg. (a) Ethylene glycol on 2.4 mm PTFE spheres. (b) Water on 3.2 mm PA66 spheres.
critical condition where fc,max ≈ fg for two additional liquid/ sphere combinations. Experimentally measured values of critical intrusion angles and heights for breakthrough are summarized in Table 2 for the various liquid/sphere combinations. Table 2. Experimental and Estimated Values of Critical Intrusion Angle (ϕmax), Drop Height (hc), and Critical Intrusion Pressure (Δpc) for the Various Liquid/FourSphere Cluster Combinationsa solid spheres liquid water ethylene glycol water
measured values
estimated values
material
2R (mm)
ϕmax (deg)
hc (mm)
ϕmax (deg)
hc (mm)
Δpc (kPa)
PTFE PTFE
4.8 2.4
120 125
6.9 4.2
126 135
6.5 4.5
64 50
PA66
3.2
144
2.9
145
3.3
32
Estimates of ϕmax were made with eq 16. Estimates of hc and Δpc for the four-sphere clusters were made from eqs 17 and 18. a
Figure 9. Images of water drops on a four-sphere cluster of intermediate-sized PTFE spheres, 2R = 4.8 mm. (a) Side view of the four-sphere cluster before deposition. (b) Water drop was deposited and suspended in the pore. (c) Addition of water increases intrusion to the critical point. (d) Next increment of water triggers flow.
More lyophilic spheres allowed liquid to penetrate deeper into the pores. As previously noted by Morrow and Mason,4 the maximum capillary force generally does not occur where the pore is most narrow, i.e., ϕ = 90°. Measured values of ϕmax ranged from 120° for water on PTFE spheres to 144° for water on PA66 spheres. While the intrusion angle is solely dependent
from eq 18, the height of the drop (h = 5.6 mm) was larger than this pore should have supported, hc = 4.9 mm. 3507
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upon the wettability of the spheres, the critical height of liquid also depends upon the properties of the liquids. Water and ethylene glycol had similar ϕmax values on PTFE, but with a greater surface tension and lesser density (i.e., greater capillary length), water was able to support taller drops and greater hydrostatic pressures. The behavior of liquid in the pores between spheres is markedly different from what is expected from cylindrical pores. With θ = 70°, water would be expected to spontaneously flow through cylindrical PA66 pores. However, the hydrophilic PA66 spheres were able to impede the flow of water. Predicted value of ϕmax and hc generally agreed quite well with measured values (Table 2). Three- versus Four-Sphere Clusters. Values of ϕmax are expected to be the same for both three- and four-sphere clusters. However, with everything else being equal, our model suggests that the critical heights and intrusion pressures of the three-sphere clusters should be greater than those of the foursphere clusters. Indeed, they were. Figure 11 shows images of
Figure 12. Larger sessile water drop suspended on top of a single layer, nine-sphere cluster of 4.8 mm PTFE spheres arranged as a regular 3 × 3 array.
interfaces at both the top and bottom should be nearly flat and, therefore, have approximately the same infinite curvature.)
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CONCLUSION Small clusters allow for observation of wetting interactions, extent of intrusion, and critical intrusion pressures in the pores between packed spheres. As expected, the smaller the pore between spheres, the greater the intrusion pressure to initiate flow. Liquids with greater surface tension or lesser density produce greater intrusion pressures. Less intuitively, the greatest intrusion pressures did not occur in the narrowest portion of the pore. Also, the extent of intrusion and the magnitude of the intrusion pressure are strongly coupled to the wettability of the spheres. The behavior of liquids in the pore between spheres is far more complicated than that observed in cylindrical pores. While intrusion into cylindrical pores occurs spontaneously for θ < 90°, pores between spheres can prevent flow even with low θ values. For an equivalent area, pores between spheres require a greater pressure to initiate flow than cylindrical pores.
Figure 11. Images of 6.4 mm PTFE sphere clusters after deposition of a water drop. (a) Four-sphere cluster, where the drop flowed through the pore. (b) Three-sphere cluster, where capillary forces impeded flow.
three- and four-sphere clusters of 6.4 mm PTFE spheres. A 100 μL drop of water has been deposited on top of each cluster. The water drop immediately passed through the pore of the four-sphere cluster. On the other hand, the three-sphere cluster impeded flow. For water on PTFE, the critical intrusion pressure is expected to be almost 40% greater for the threesphere cluster than the four-sphere cluster (88 versus 64 kPa). Many Spheres in a Single Layer. In practical situations, liquids may interact with many spheres. We reckon that the critical intrusion pressure to instigate flow should be the same for one pore or for many pores. To test this supposition, larger, single-layer arrays were constructed and evaluated. Indeed, with everything else being equal, multiple-pore clusters exhibited the same critical intrusion angle as single-pore clusters. In contrast, the critical height of drops on many sphere clusters was generally more than the predicted value. Why? The model assumes that the curvature at the top and bottom of the liquid drop are equal and, thus, can be ignored. While this was generally true for the smaller drops placed on three- and foursphere clusters, larger drops that spread across many spheres had a much larger radius of curvature at their apex compared to their lower extremities inside the pores. This unbalanced Laplace pressure allowed for larger drops that covered many spheres to grow taller. For example, Figure 12 shows a nine-sphere cluster of 4.8 mm PTFE spheres arranged as a regular 3 × 3 array. The critical intrusion angle here was still about 120°, but the critical height was greater than for a four-sphere cluster, 7.8 mm for nine spheres, as compared to 6.9 mm for just four spheres. (As an aside, we think the equal curvature argument also should hold for very large, shallow puddles or pools of liquid that cover a single layer of very small spheres; here, the air−liquid
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AUTHOR INFORMATION
Corresponding Author
*Telephone: 952-556-8619. E-mail: chuck_extrand@entegris. com. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank Entegris management for supporting this work and allowing publication. Also, thanks to M. Amari, B. Arriola, T. Edlund, V. Goel, L. Monson, M. Ngo, J. Pillion, B. Powell, A. Rashed, R. Sheppard, S. Sirignano, and S. Tison for their suggestions on the technical content and text.
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