Rates of cavity filling by liquidsDongjin Seoa,b, Alex M. Schradera, Szu-Ying Chena, Yair Kaufmanc, Thomas R. Cristianid, Steven H. Pagee,Peter H. Koenige, Yonas Gizawf, Dong Woog Leeg,1, and Jacob N. Israelachvilia,d,1

aDepartment of Chemical Engineering, University of California, Santa Barbara, CA 93106; bDepartment of Chemical Engineering, Brigham YoungUniversity, Provo, UT 84606; cThe Zuckerberg Institute for Water Research, The Jacob Blaustein Institutes for Desert Research, Ben-Gurion University of theNegev, 84990 Midreshet Ben-Gurion, Israel; dMaterials Department, University of California, Santa Barbara, CA 93106; eModeling and Simulation/Computational Chemistry, The Procter & Gamble Co., West Chester, OH 45069; fWinton Hill Business Center, The Procter & Gamble Co., Cincinnati, OH45224; and gSchool of Energy and Chemical Engineering, Ulsan National Institute of Science and Technology, 689-798 Ulsan, Republic of Korea

Contributed by Jacob N. Israelachvili, June 13, 2018 (sent for review March 14, 2018; reviewed by Michael Nosonovsky and Rafael Tadmor)

Understanding the fundamental wetting behavior of liquids onsurfaces with pores or cavities provides insights into the wettingphenomena associated with rough or patterned surfaces, such asskin and fabrics, as well as the development of everyday productssuch as ointments and paints, and industrial applications such asenhanced oil recovery and pitting during chemical mechanicalpolishing. We have studied, both experimentally and theoretically,the dynamics of the transitions from the unfilled/partially filled(Cassie–Baxter) wetting state to the fully filled (Wenzel) wettingstate on intrinsically hydrophilic surfaces (intrinsic water contactangle <90°, where the Wenzel state is always the thermodynamicallyfavorable state, while a temporary metastable Cassie–Baxter state canalso exist) to determine the variables that control the rates of suchtransitions. We prepared silicon wafers with cylindrical cavities of dif-ferent geometries and immersed them in bulk water. With bright-fieldand confocal fluorescence microscopy, we observed the details of, andthe rates associated with, water penetration into the cavities from thebulk. We find that unconnected, reentrant cavities (i.e., cavities thatopen up below the surface) have the slowest cavity-filling rates, whileconnected or non-reentrant cavities undergo very rapid transitions.Using these unconnected, reentrant cavities, we identified the variablesthat affect cavity-filling rates: (i) the intrinsic contact angle, (ii) theconcentration of dissolved air in the bulk water phase (i.e., aeration),(iii) the liquid volatility that determines the rate of capillary condensa-tion inside the cavities, and (iv) the presence of surfactants.

wetting transition | wetting dynamics | Wenzel | Cassie–Baxter

Phenomena involving the spreading (or “wetting”) of liquids,especially water, on rough or textured surfaces arise in many

different areas of surface science such as water and oil “re-pellency” (1–4), personal care products, epidermal medicines,and industrial applications. For example, understanding pittingcorrosion, which requires liquid penetration into narrow pores, iscrucial for improving industrial processes such as reser-voir flooding during oil recovery and chemical mechanicalpolishing. In many practical situations, the surfaces on whichsome paint, pharmaceutical products, or other fluids are ap-plied are never smooth. Rather, they have undefined rough-ness or random structures on which the spreading liquids mayonly partially wet the surfaces, forming vapor cavities underthe menisci. This partially filled state, or the Cassie–Baxterstate (5), may or may not transit to a fully filled state, or theWenzel state (6), where the entire solid surface is in contactwith liquid.Understanding the variables that control the cavity-filling

process (the “wetting transition”) is important not only forpractical applications such as product design, but also for manydifferent areas of surface science. Many fundamental issues needfurther study, such as the wetting phenomena on complex sur-face structures, including superhydrophobic structures, con-nected or unconnected cavities or “pillars,” reentrant cavities(i.e., cavities that open up below the surface), and in particularnonequilibrium (dynamic, time-dependent) effects. The maindynamic metric of interest is the time required for liquids to

penetrate into the cavities, both under the spreading liquid andahead of it, which manifests itself in “contact-angle hysteresis”(7, 8) and nonuniform (e.g., stick–slip) motion of advancing orreceding three-phase contact boundaries (9, 10).As for superhydrophobic and/or self-cleaning surfaces, main-

taining the partially wetted Cassie–Baxter state keeps the macro-scopic contact angle high. Even when the surfaces have anintrinsic contact angle (which, in conventional definition, isthe contact angle measured on a molecularly smooth surface)<90°, temporary superhydrophobicity can be achieved whenmicro- and nanostructures are prepared, which can kineticallymaintain the Cassie–Baxter wetting state (11–14). However, forintrinsic contact angles <90°, cavities in the metastable Cassie-Baxter state will eventually undergo a transition to the Wenzelstate, which decreases the macroscopic contact angle, and resultsin the loss of the surfaces’ self-cleaning ability (15, 16).Some studies have found that decreasing the droplet size through

evaporation affects the wetting state (15–18) on superhydrophobicsurfaces, which were prepared by building hydrophobic micropillars.Jung and Bhushan (15) prepared micropillars coated with PF3, ahydrophobic material, and found that a small pitch-to-pitch distancebetween pillars enabled the Cassie–Baxter state to be maintained bykeeping the water menisci at the top of the pillars. However, smallerdroplets (<0.1 mL) tended to transit to the Wenzel state upon re-ducing their size through evaporation. Tsai et al. (16) attributed thistransition to the increased Laplace pressure from the reduced

Significance

In engineering and natural phenomena, various fluids contactrough/textured surfaces, e.g., wicking, facial creams, corrosion-preventive paints, and rain on plant leaves. Liquids on roughsurfaces, especially those with cavities, pits, or pores, may ormay not transit from the unfilled or partially filled (wetted)state to the fully filled (fully wetted) state. Either one of thesestates may be desired for a given application (compare super-hydrophobicity) or even survival (compare oil-soaked feathers).In this article, we present five variables that control the wet-ting behavior (cavity filling) of water on intrinsically hydrophilicsurfaces with micrometer-sized cavities. Our experimental re-sults and theoretical analysis provide criteria for maintainingeither the partially filled state, or quickly transiting to the fullyfilled state, and insights into other related wetting phenomena.

Author contributions: D.S., A.M.S., and D.W.L. designed research; D.S., A.M.S., S.-Y.C.,Y.K., and D.W.L. performed research; T.R.C. contributed new reagents/analytic tools;D.S., A.M.S., S.-Y.C., Y.K., D.W.L., and J.N.I. analyzed data; and D.S., A.M.S., S.-Y.C., Y.K.,T.R.C., S.H.P., P.H.K., Y.G., D.W.L., and J.N.I. wrote the paper.

Reviewers: M.N., University of Wisconsin–Milwaukee; and R.T., Ben Gurion University andLamar University.

The authors declare no conflict of interest.

Published under the PNAS license.1To whom correspondence may be addressed. Email: dongwoog.lee@unist.ac.kr orjacob@engineering.ucsb.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1804437115/-/DCSupplemental.

Published online July 19, 2018.

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radius of the liquid–vapor interface, which led to a larger drivingforce to push the menisci toward the bottom.While many reports studied the wetting transition of water

droplets on patterned hydrophobic surfaces, only a few (19–21)studied the wetting transition when patterned surfaces wereimmersed under bulk liquid. Additionally, most studies usedintrinsically hydrophobic materials, or applied mechanical forcesto promote liquid penetration into the cavities. Here, we presentvariables affecting the time-dependent wetting transition (rep-resented by cavity filling) by immersing a patterned surface ofmicrocavities made of an intrinsically hydrophilic material, sili-con dioxide (intrinsic contact angle, θ0 < 60°) in bulk quiescentliquids. We shed light on the mechanisms and parameters thatdetermine the duration over which omniphobicity can be main-tained when textured surfaces are immersed in liquids. This sit-uation more closely resembles actual applications than theprevious droplet experiments on hydrophobic microstructures,since the applied liquids generally cover a large area comparedwith the size of cavities. Note that, throughout this paper, “in-trinsic contact angle” (or the Young contact angle), θ0, denotesthe contact angle measured on smooth, flat parts of our samples∼1 min after droplet deposition, by which point the contact anglehad stabilized.For our study, we fabricated cylindrical cavities with different

geometries, namely (i) reentrant cavities, (ii) nonreentrant cav-ities, and (iii) connected reentrant cavities, by adapting theprocesses developed by Kaufman et al. (22) The reentrant cav-ities shown in Fig. 1 are 9-μm-deep cylindrical cavities with adiameter of 60 μm and a center-to-center distance of 62 μm.

These reentrant cavities were used for all of our experiments,except for the study on the effect of geometry (Variable 5: CavityGeometry). More details about the fabrication processes aregiven in SI Appendix, section 1 and Fig. S1. The cavity-fillingprocesses were observed with bright-field and confocal fluores-cence microscopy to count the number of filled cavities and toobserve the details of cavity filling, as explained in Methods andin SI Appendix, sections 2 and 3 and Figs. S2 and S3. All ex-periments were done at 21 ± 1 °C and atmospheric pressure,unless otherwise specified.Fig. 2 summarizes our observations on the process through

which the cavity filling generally progresses when bulk water isspread above a hydrophilic, reentrant cavity. As shown in A, rightafter water is placed on a cavity, the water forms a concavemeniscus. As water evaporates into the cavity, condensate formsaround the bottom corners, as in B. When the condensate growslarge enough to meet the bulk water above, the three-phase linedrops toward the bottom, reaching the situation depicted in C.The trapped air bubble rises toward the cavity entrance due tobuoyancy while θ0 is kept constant as shown in D. Finally, the airinside the cavity leaves the cavity by buoyancy or dissolution.Depending on the variables that would drive the overall process,cavity filling may proceed right into D or E. For example, weobserved almost immediate filling of all of the cavities withoutever being able to observe the situations in B–D when θ0 is verylow (<10°). Through these processes, the vapor volume isexpected to decrease due to (i) dissolution of air into the water(from Fig. 2A to Fig. 2B) and (ii) the transition of the vaporpocket into a spherical bubble (from Fig. 2B to Fig. 2E), whichminimizes the liquid–vapor surface area, but also decreases theradius of curvature of the liquid–vapor interface and increasesthe Laplace pressure of the vapor. Apart from θ0, we found othervariables that control the cavity-filling rate. In the next section,five different variables affecting the cavity-filling rate, includingthe effect of meniscus shape and the pressure inside the cavities,are discussed.

Results and DiscussionVariable 1: Intrinsic Contact Angle θ0. To investigate the effect ofθ0 on the cavity-filling rate, we varied θ0 by controllably con-taminating the sample surface by varying the oven incubationtime. We first cleaned the surface of reentrant cavities (“thesample”) using water, ethanol, and UV ozone, resulting in θ0 <10°. The sample was then incubated in an oven at 75 °C forvarying amounts of time. Incubating the sample for 0, 3, 60, 480,and 720 min resulted in θ0 values for water of 0, 24, 35, 40, and50°, respectively. After 10 mL of 50% aerated water was pouredinto the Petri dish with the sample, the cavities on the samplewere observed with a low-magnification microscope (2–5×).Note that we used the same sample and imaged approximatelythe same area for every experiment to isolate θ0 as a sole

silica

silicon

reentrant

silicon wafer

60 μm

0.1 μm2 μm

silica

30 μm

1 μm

1 μm8 μm oxidized

silicon layer

A CB

Fig. 1. Micropatterned surfaces fabricated for this study. (A) Schematic il-lustration showing a cross-section of a single reentrant cavity. (B) Tilted SEMimage of the reentrant cavities. (C) High-resolution image showing the areawithin the red rectangle in B, showing a reentrant feature.

h

α

A B C D E

Fig. 2. Cavity-filling stages, showing the multistep processes by which wa-ter fills a reentrant, hydrophilic cavity (θ0 < 90°) initially filled with air at 1atm and 21 °C. (A) After water (either aerated or deaerated) initially con-tacts the cavity opening, the water penetrates slightly and partially fills thecavity, trapping the air inside and causing the liquid–vapor meniscus tobulge upward as the cavity becomes saturated with water vapor (blue cir-cles). However, the rate of air (white circles) dissolution and diffusion intothe bulk water depends on the degree of aeration of the water. If the water(or other test liquid) is fully deaerated, the cavity-filling process can quickly(<10 s) proceed to D or E. (B) If the liquid is fully aerated, then capillarycondensate, which is visible with a microscope, begins to form around thecorners within ∼10 min. (C) As the condensate on the upper corner ofthe cavity grows, it eventually connects with the bulk water on the top ofthe cavity, resulting in the bulk water rapidly advancing to the cavity bottom(over the course of <5 s), leaving an air bubble trapped. (D) The trapped airbubble rises due to buoyancy, while maintaining a constant θ0, and (E)eventually, the trapped bubble either floats away or dissolves into the liquid.h is the distance measured from the meniscus to a location inside the cavity;α is the angle between the slope of the surface at which the three-phase linehangs and the flat surface at the top of the patterned sample.

0

250

500

750

1000

0 20 40 60

,seitivaClliFo t

emiT

e garevAt a

ves][

Intrinsic Contact Angle, θ0 [°]

θ0 [°]1020304050 0

0

2

4

6

8

0.6 0.7 0.8 0.9 1

ln (t

ave)

cos θ0

A B

Fig. 3. Cavity-filling times and rates. (A) Average time to fill initially emptycavities (∼5,000 cavities), tave, for different θ0, which was varied by in-cubating the patterned surface in an oven at 75 °C for different times up to720 min. (B) The same data shown in A replotted as ln(tave) vs. cosine θ0 tocompare the data with first-order kinetics (dotted lines), showing that tworegimes control the cavity-filling process.

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variable. Fig. 3 shows the average time to fill the cavities tave vs.θ0 within the view of the microscope (about 5,000 cavities). tavevalues were obtained by fitting the plots of the fraction of filledcavities φ vs. time after water spreading t to the cumulativedistribution function of the normal distribution (CDF) and cal-culating the average times from the fits (SI Appendix, Fig. S4).φ was calculated by counting the number of filled cavities anddividing it by the total number of cavities in the field of view(refer to Methods). Note that the data points at θ0 = 0° in Fig. 3Acorrespond to θ0 values that were less than the minimum re-solvable value (10°). In such cases, we could not measure tavebecause all of the cavities were all filled within 30 s, which wasthe time required to prepare for imaging after pouring the water.We set tave to 0 s in those cases. The trend in Fig. 3 clearly showsthat tave is controlled by θ0. However, the SD of the CDF, σ, canbe affected by other variables, which is briefly discussed in SIAppendix, section 4.The same data in Fig. 3A are replotted in Fig. 3B as ln(tave) vs.

cos θ0. We attempted to see the relation between tave and ad-hesion energy,Wadh, as described by the Young–Dupré equation:

Wadh = γwð1+ cos θ0Þ, [1]

where γw is the surface energy of the air–water interface. Since γwdid not change over this series of experiments, cos θ0 is relatedlinearly to Wadh. While we recognize that there are other meth-ods (23, 24) of measuring Wadh, the Young–Dupré equation pro-vides sufficiently accurate values to analyze the effect of contactangles on the rate of cavity filling. Considering Wadh to impactthe energy barrier to overcome for the meniscus to advance

toward the bottom, higher Wadh (or lower θ0) values are corre-lated with higher cavity-filling rates due to a lower energy barrieror activation energy for liquid penetration. Therefore, we hy-pothesized that ln(tave) would vary linearly with cos θ0, as de-scribed by the Arrhenius equation. Fig. 3B shows the linearrelation but with two distinct regimes. The low contact-angleregime (θ0 < 35° or cos θ0 > 0.82, dotted black line in Fig. 3B)is thought to be controlled by the local force at the three-phaseline. Kaufman et al. (22) developed a local force equation bycalculating the change in the total surface energy as the three-phase line penetrates into the cavity and showed that the changein the total surface energy per unit distance that the three-phaseline travels (units of force) increases as θ0 decreases when α =90°, and where α is the surface slope at the height of the menis-cus (Fig. 2B). However, in the high contact-angle regime (θ0 >35° or cos θ0 < 0.82, dotted green line in Fig. 3B), the trendshows a different slope. When θ0 is high, it takes longer to fillthe cavities due to a lessened downward force on the meniscus.In this case the cavity-filling process is controlled by a differentmechanism in addition to the surface energy balance. For thiscase, where only θ0 changes, condensation is the only controllingvariable. Higher values of θ0 provide more time for capillarycondensate to grow and make contact with the bulk liquid above,as will be discussed later in Variable 3: Growth of Condensate andVolatility of Liquids.

Variable 2: Concentration of Dissolved Air in the Bulk Water. To in-vestigate the effect of the concentration of dissolved air on thecavity-filling rate, we immersed a sample (θ0 = 40°) in water atdifferent dissolved air concentrations. Fig. 4 shows the numberfraction of filled cavities φ as a function of the time elapsed afterwater spreading. While it only took about 20 min for 0% aeratedwater to fill all of the cavities on the patterned surfaces, it tookmore than a week for 100% aerated water. The abrupt changeswhen using 100% aerated water were due to the daily atmo-spheric temperature cycles without properly functioning airconditioning: (i) when the temperature increases, the solubilityof air in water decreases, which accordingly expels dissolved airto the reservoir, and (ii) when the temperature decreases, thesolubility of air in water increases, partially taking air from insidethe cavity which induces a decrease in pressure within the cavi-ties and causes abrupt filling of a portion of the cavities. Whenthe bulk water was not saturated with air, the air inside thecavities rapidly dissolved into the water. This dissolution allevi-ates the pressure inside the cavities, making it easier for thewater–solid–air three-phase line to advance toward the cavitybottom or pockets of condensate within the cavity.Due to the complex mechanism of the cavity-filling process

shown in Fig. 2, we found it beyond the scope of this work tomodel φ vs. t with mass transfer models. However, the differencein concentration between air in the liquid at the liquid–vaporinterface (necessarily at saturation) and air in the bulk water isproportional to the mass flux of air out of the cavity, and thus,the pressure within the cavity. A lower air pressure within thecavity increases the cavity-filling rate by decreasing the resistanceof the three-phase contact line to advance into the cavity.

1

A Brightfield B Fluorescence

filled filled

partially filled

Condensed water, no fluorescein

water with fluorescein

vapor

condensed water

condensed waterpartially

filled

filled

partially filled

Fig. 5. Image of cavities that are fully filled and partially filled withfluorescein-dyed water. The same area of the patterned sample was simul-taneously imaged with (A) bright-field and (B) fluorescence microscopy. AnFITC filter was used with fluorescein-dyed water to obtain the fluorescenceimages. Note that in the bright-field image, partially filled cavities arebrighter than fully filled cavities, whereas in the fluorescence image, par-tially filled cavities are darker than fully filled cavities, as explained in SIAppendix, Fig. S3. Schematics of fully filled and partially filled cavities areshown in the middle.

Table 1. Properties of the three liquids tested for thetime-dependent cavity-filling rates due to volatility

Properties Unit C16 DMSO DMF

MW g/mol 226 78 73bp °C 281 189 153Psat kPa 3.1 × 10−4 9.8 × 10−2 7.2 × 10−1

at °C 28 28 30γ, 20 °C N/m 0.028 0.044 0.037Viscosity, 20 °C mPa s 3.4 2.0 0.92

0.0

0.2

0.4

0.6

0.8

1.0

1 10 100 1000 100000.4

0.6

0.8

1.0

1000 10000

Time a�er water spreading [t, min]

,seitivacdelliffo

noitcarfrebmu

Nφ

50 % aerated water

0 % aerated water

100 % aerated water

Temperature change

Temperature increase (28 °C)

Temperature decrease (22 °C)

A B

Fig. 4. Effect of dissolved air concentration in the water on the cavity-fillingrate. (A) Plot showing how the time evolution of φ differs with the dissolvedair concentration. “0% aerated water” means deaerated water. (B) Thedetails of A for t = 1,000–13,000 min (marked with the red box) showing thatthe sudden increase in φ was likely due to temperature changes.

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Variable 3: Growth of Condensate and Volatility of Liquids. Observ-ing cavities with higher-magnification microscopes (20× and50×) revealed a partially dark cavity with a lens-shaped feature,marked with “condensed water” in Fig. 5A. Considering thatfully filled cavities appeared darker than partially filled cavities,the dark lens-shaped part suggests that there was a pocket ofwater at the bottom. It was not immediately clear whether thepocket of water was condensate, or water that was splashed fromthe bulk water above. To determine the origin of this waterpocket and also to obtain 3D structures of the liquid–vapormenisci, we conducted confocal microscopy experiments usingwater mixed with fluorescein. SI Appendix, Fig. S3 and Fig. 5,Middle show the experimental setup and interpretation of themicroscopy images in Fig. 5. The image contrast in the fluores-cent image is the opposite of that in the bright-field image: Filledcavities in the fluorescent image were brighter due to the emittedlight from fluorescein, while the absence of fluorescein made par-tially filled cavities darker. If the lens-shaped features in the partiallyfilled cavities were condensate, they would not emit light, as fluo-rescein would not evaporate from the bulk water as depicted in Fig.5, Middle. The appearance of water pockets in the partially filledcavities of the bright-field image (Fig. 5A), and the relative lackthereof in Fig. 5B shows that the water pockets are indeed con-densate, and are not connected to the bulk liquid above the dropletthrough liquid bridges or channels. We hypothesize that the quickerthe condensate volume grows, the sooner the bulk water above thecavity makes contact with the pockets of condensate, which willaccelerate the step from B to C in Fig. 2.The observation of condensate led to another question: Would

using a liquid with a higher vapor pressure accelerate the cavity-filling rate by accelerating the growth of condensate? Althoughthe temperature could be changed to control the vapor pressure,Psat, of water, we decided instead to use different liquids whichhad significantly different vapor pressures, but similar θ0 values(after adjusting the surface chemistry), viscosities, and surfacetensions. Three liquids were tested: dimethyl sulfoxide (DMSO),dimethylformamide (DMF), and hexadecane (C16), and theirphysical properties are listed in Table 1. A series of cavity-fillingexperiments was carried out with these liquids on the samesample used for studying the two variables above, but the surfacewas functionalized with various ratios of (3-Glycidyloxypropyl)trimethoxysilane (GOPTS) to octadecyltrichlorosilane (OTS), sothat each liquid had a similar θ0 to the other two, as shown in thedroplet images in Fig. 6B. Experiments measuring φ over timewere repeated three times for each liquid. One representativedata set for each liquid is shown in Fig. 6A, including lines thatshow the fits to a first-order adsorption model. Here, we comparethe cavity-filling rate to the “mesoscopic” adsorption of bulkwater to the bottom of empty cavities, and the process of emptying

the cavities to the mesoscopic desorption of bulk water from themicroscopic cavities. Therefore, the governing equation for thisprocess can be written as a simple first-order kinetic model:

dφdt

= kadsð1−φÞ− kdesφ, [2]

where kads and kdes are the rate constants for mesoscopic adsorp-tion (or cavity filling) and desorption (or cavity emptying) ofwater, respectively. We assumed that adsorption is much moreprevalent than desorption, as judged by the lack of any cavityemptying after a long time (∼1 wk). With kads >> kdes, the solu-tion to Eq. 2 is given by

φðtÞ=h1−C  exp

�−tτ

�i,   τ=

1kads

. [3]

Here, 1 − C denotes the fraction of cavities that were filledbefore the first image was taken––usually 30 s after immersion.The inverse of kads is defined as τ, denoted as the characteristicfilling time for each liquid. In this case, we argue that Psat orliquid volatility explains the differences in τ. Thus, the charac-teristic filling time τ can be used to estimate how fast liquidvolatility accelerates the cavity-filling rate. Fig. 6A shows theaverage values of τ for each liquid, indicating that DMF fillsthe cavities the quickest, followed by DMSO, and then C16. Thistrend is also apparent in the time to fill all of the cavities by eachliquid as shown in Fig. 6B. When the times to fill all of thecavities were averaged over the three experiments performedfor each of the three liquids, the trend was the same as shownin Fig. 6B. Niknejad and Rose (25) reported, after examiningvarious theoretical approaches explaining mass flux during con-densation, that the net condensation mass flux is generally pro-portional to Psat. This implies that the time to fill all of the cavitiesis inversely proportional to Psat if Psat is the only major variableaffecting the cavity filling, as will be discussed in the next para-graph. This inversely proportional relation is confirmed with thered trend line in Fig. 6B.Next, we examined how the differences in properties between

the three liquids (other than Psat) may have contributed to ourresults. The force that pulls the meniscus toward the bottom, −F,describes the balance of surface energies at the three-phase line,i.e., γLV, γSV, γSL, where γ represents surface tension/energy withsubscripts L for liquid, V for vapor, and S for solid, assuming thatthe meniscus is a spherical cap (22). These F values are similarfor all three liquids (Fig. 6), which disregards the possibility that thisforce is responsible for quicker or slower cavity filling. Indeed,the −F value for DMF is lower than that for DMSO, meaning thatDMSO should have filled the cavities quicker should this force beresponsible for controlling the cavity-filling rate, yet the opposite isobserved. From Table 1, one might imagine that differences inviscosity might have caused the differences in cavity-filling time.Assuming that it takes 1 s for an advancing liquid front to penetratea 9-μm-deep cavity (the approximate time to reach Fig. 2C once thecondensate connects with the bulk liquid), the calculated capillary

Table 2. Properties of surfactants and time to fill all ofthe cavities

Surfactant Charge CMC, mMγLV at CMC,

mN/mθ0 on silicaat CMC, ° Time,* min

SDS Anionic 7 37 5 5C12(EO)4 Nonionic 0.03 29 31 6.5DTAB Cationic 11 41 33 8.5Water

only— — 35 20

*Time to fill all the cavities.

DMFθ0 = 49 °

-F = 39 μN

DMSOθ0 = 50 °-F = 45 μN

C16θ0 = 41 °-F = 32 μN

0

5

10

15

20

0.0 0.2 0.4 0.6 0.8

Tim

e to

Fill

All

the

Cavi

�es [

min

]Vapor Pressure [Psat, kPa]

F < 0, meniscus pulled downwardCapillary number < 10-5

B

0.80

0.85

0.90

0.95

1.00

0 5 10 15 20

φdelliF

noit carF ,)t(

Time [t, min]

C16: τ = 3.5 min

DMF: 2.8 min

DMSO: 3.3 min

A

Fig. 6. (A) Fraction of cavities that were fully filled with C16, DMSO, and DMFas a function of time. This graph shows a representative data set for each ofthe three liquids, with data points for experimental data and lines for the first-order adsorption model. (B) The time to fill all of the cavities within the field ofview of the camera (averaged from three experiments for each liquid). F is theforce exerted on the three-phase line around the entrance of the cavities.Negative F values mean that the force acts to pull the meniscus downward (seetext for details). The red line in B is the fit to an inversely proportional relation.

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numbers (Ca, see SI Appendix, section 5) for C16, DMSO, andDMF are 1.2 × 10−6, 4.1 × 10−7, and 2.2 × 10−7, respectively, whichare considered to be capillary-force dominant (Ca < 10−5) ratherthan viscous-flow dominant. Therefore, we concluded that thevapor-pressure difference of these three liquids controlled thecavity-filling rates, but only after ruling out the other prominentvariables that might have caused the difference.

Variable 4: Types of Surfactants. Generally, surfactants decrease thesurface tension/energy of interfaces between hydrophilic and hy-drophobic media (such as the air/water interface), thus helping thewetting transition as discussed in Variable 1: Intrinsic Contact Angleθ0 regarding θ0. Here, we studied how different kinds of surfactantsaffected the cavity-filling rate. We used three different deaeratedaqueous solutions of either sodiumdodecylsulfate (SDS, anionic),dodecyltrimethylammonium bromide (DTAB, cationic), or poly-tetraoxyethylene dodecyl ether [C12(EO)4, nonionic] at their criticalmicelle concentrations, which gave γLV values that were similar toeach other. Table 2 shows the properties of the surfactants, and Fig.7 shows where the surfactants would be present when the solidsurface (silica) is negatively charged. Nonionic and cationic surfac-tants should behave similarly; they adsorb to the LV and SL in-terfaces. Cationic surfactants should adsorb onto the negativelycharged surface more strongly than nonionic surfactants, hinderingthe advancing of water fronts. SDS, an anionic surfactant, would berepelled from the negatively charged surfaces, but still attracted tothe liquid–vapor interface, thus giving the smallest θ0 value. Thesephenomena explain the differences in θ0 between the three sur-factants, and thus, the cavity-filling time.

Variable 5: Cavity Geometry. Before the rates and dynamics of cavityfilling were studied, we studied the effect of cavity geometry inhopes of finding a geometry that maintains the metastable Cassie–Baxter state long enough for contact-angle measurements. Based onreports in the literature (16, 22, 26), it is well known that geometryhas a significant effect on the wetting transition rate, but controlledtests of the quantitative impact of reentrant features and cavityconnectivity on the cavity-filling time are lacking. Above, we ex-amined unconnected cavities with reentrant features, but here weexamine (i) connected cavities with reentrant features and (ii) un-connected cavities without reentrant features, whose geometries areshown in Fig. 8 A and B, respectively.In both cases, it is likely that the cavities are filled within 5 s of

water deposition when 100% aerated water is used. For instance,the macroscopic (apparent) contact angle decreases to 0° withinthat time with the connected cavities (Movie S1). As confirmationof cavity filling, fluorescent confocal images were taken, but imagescould only be acquired roughly 5 min after water deposition (Fig. 8).In the case of connected cavities, water can fully penetrate quicklybecause air in the cavities can escape laterally through the con-nected cavity network, as opposed to being trapped and pressurized,which prolongs the metastable Cassie–Baxter state. Air pressuriza-tion in unconnected cavities with reentrant features is evidenced bythe concave curvature of liquid–vapor interface in the cavitiesshown in Fig. 9. On the unconnected cavities without reentrant

features, water also penetrates quickly, but that is because there isno observable (or even theoretical) metastable Cassie–Baxter state.

Shape of Menisci and Laplace Pressure. To study the shape of theliquid–vapor meniscus, cavities were imaged with a confocal fluo-rescence microscope using fluorescein-dyed water to construct 3Dimages as seen in Fig. 9. The meniscus was bent toward the bulkwater, which became the basis for drawing the shape of menisci asbulging toward the bulk water throughout this paper. To furthermeasure the radius and cap height of the meniscus, the cross-sectionwas analyzed (Fig. 9B). With this image, the radius was calculated tobe about 120 μm, and the cap height 4 μm. When considering themeniscus to be a spherical cap, the Laplace pressure is around1,200 Pa (0.01 atm). Due to the low resolution at the three-phaseline, θ0 was not able to be measured from these images.

ConclusionsThis paper discusses the variables controlling the cavity-filling(wetting transition) rates when bulk water was brought into contactwith hydrophilic silicon oxide surfaces patterned with microcavities.We identified the different stages (both the metastable and finalequilibrium states) of cavity filling, and the parameters and variablesthat determine their filling rates. (i) We found that smaller θ0 valuesdecrease the cavity-filling time, thereby accelerating the wettingtransition time. We found an exponential relation, similar to theArrhenius equation, between the average time to fill a certain largenumber (∼5,000) of cavities, tave, and θ0, i.e., ln(tave) ∝ cos θ0, whichexhibits two different time regimes (Fig. 3). (ii) The amount ofdissolved air in the bulk water has a profound effect on the cavity-filling rates––undoubtedly due to the rapid mass transport of air outof the cavities (Fig. 4). (iii) The geometry-dependent ease of cap-illary condensation (e.g., in corners, cracks) and further growth ofthe condensates within the cavities accelerate the cavity-filling rateby bridging the condensate with the bulk water phase. In connectionwith the study of condensation, we further investigated the effect ofliquid volatility, hypothesizing that increased vaporization rates ofmolecules should promote more rapid formation of condensates.We confirmed that higher liquid volatility increased cavity-fillingrates, which followed a first-order adsorption kinetics model(Fig. 6). (iv) Surfactant solutions at the CMC generally increasedthe cavity-filling rate; the anionic surfactant increased the ratethe most, followed by the nonionic, and then the cationic sur-factant. This trend can be explained through the Coulombic in-teractions of the three surfactants with the silica surface, and thesubsequent effect on decreasing θ0 to varying degrees (Fig. 7). (v)Removing reentrant features from the cavity openings decreasedthe cavity filling time to less than 5 min, as there was no meta-stable Cassie–Baxter state. Connecting the reentrant cavitiescaused similarly rapid filling of the cavities, since as soon as thefirst cavity is filled, the rest get filled immediately after (Fig. 8).(vi) The liquid–vapor meniscus near the top of the reentrant

SDS DTABC12(EO)4

vapor

- - - --- --- - ---

- - - - - - - - - - - - - - ---surface charges

silica

surfactant solution

- - - - - - - - - - - - - -surface charges

silica

- - - - - - - - - - - - - -surface charges

silica

++

++ + + + + + + +

+ + + + +

increasing cavity filling rate

θ0 θ0 θ0

Fig. 7. Schematics of how the surfactants are partitioned on the liquid–vapor, liquid–solid, and solid–vapor interfaces, which also explains θ0 of thesurfactant solutions on silica, and, thus, the cavity-filling rates.

60 μm

focal plane

62 μm

3 μm

12 μm 60 μm

62 μm

focal plane

1 μm

3 μm

A B

5 minutes after pouring water

5 minutes after pouring water

air escaping through connections

Fig. 8. Schematics and fluorescent confocal microscopy images of fluorescein-dyed water on cavities that are (A) reentrant and connected, and (B) non-reentrant and unconnected/separate. In both cases, water had completelyfilled all of the cavities by the time the first image was taken––at ∼5 minafter the water was placed on the surface.

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cavities bulges toward the bulk water with a radius of ∼120 μm,corresponding to a compressive Laplace pressure on the vapor inthe cavity of ∼1.2 kPa (Fig. 9B). This knowledge of how cavitiesunder bulk water are filled and what variables control the rate offilling can provide insights into the engineering of temporarily orpermanently superhydrophobic surfaces, and the designing andmanufacturing of various products that are applied to rough,textured, or patterned surfaces. Many of the fundamental insightsgained can also be applied to other liquids (e.g., oils), contactangles, and cavities or pores of different dimensions or geometries.

MethodsObservation and Counting of Filled Cavities. To calculate φ, we imaged a largenumber of cavities (∼5,000, 5 mm × 3 mm) with a low-magnification mi-croscope (2–5×). For each image, we counted the number of fully filled andpartially filled cavities according to their brightness (Fig. 5). The same ex-periments were also done with a high-magnification microscope (50×) to

observe the details of the process. More details of the experimental setupand analyses are provided in SI Appendix, section 2.

Controlling the Dissolved Air Concentration in Water. We prepared deaeratedwater, or 0% aerated water, by pouring deionized (DI) water into a glass vialdirectly from a dispenser (Milli-Q; Millipore) and then stirring the water witha Teflon-coated magnetic stirrer while applying vacuum (∼0.01 atm) for 1 h.The aerated water, or 100% aerated water, was water from the DI waterdispenser that was in contact with air for more than 1 h thereafter. Toprepare partially aerated water, we mixed the aerated water and deaeratedwater at the desired ratios.

Confocal Fluorescence Microscopy.We used an inverted confocal fluorescencemicroscope (Olympus Fluoview 1000 Spectral Confocal) with fluorescein inthe bulk water to confirm the formation of condensate. To overcome theeffects of gravity, we utilized capillary force to prevent the fluorescein-mixedwater from falling. To construct the 3D shape of the liquid meniscus, a water-immersion objective lens was used. Details of the setups can be found in SIAppendix, section 3.

Controlling the Intrinsic Contact Angle of the Three Nonaqueous Liquids onSilica. To have the same θ0 of DMSO, DMF, and C16 on the silicon wafer,pieces of silicon wafer were coated with the different ratios of GOPTS andOTS. They were immersed under different GOPTS:OTS ratios in dry toluene,followed by incubation at room temperature for 8 h. After incubating,rinsing with clean toluene, and drying the samples, the contact angles of thethree liquids on the coated silicon wafer pieces were measured. We foundthat all three liquids had θ0 ∼ 40° when the ratio of GOPTS:OTS on the silicasurface was 80:20 for C16, 97.5:2.5 for DMF, and 100:0 for DMSO. After aseries of experiments for one liquid was done, the same sample wascleaned with piranha solution, and then refunctionalized with a differentGOPTS:OTS ratio for the next series of experiments with a different liquid.

ACKNOWLEDGMENTS. This work was supported by a grant from the Procter& Gamble Company. We acknowledge the use of the Neuroscience ResearchInstitute and the Department of Molecular, Cellular and Developmental Bi-ology Microscopy Facility and the Spectral Laser Scanning Confocal sup-ported by the Office of the Director, National Institutes of Health underAward S10OD010610. We thank the Saudi Arabian Oil Company (SaudiAramco) for funding the purchase of the contact-angle measurement instru-ment. D.W.L. was supported by grants from the National Research Founda-tion of Korea funded by the Korean Government (NRF-2016R1C1B2014294).Y.K. was supported by the Adelis Foundation.

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filled cavities

partially filled cavities

3D view from bottom

partially filled cavity

filled cavity

4 μm

R ≈ 120 μm P ≈ 1.2 kPa

A

B partially filled cavity

Fig. 9. Confocal fluorescence microscope images of partially and fully filledcavities. (A) Three-dimensional construction of the fluorescein-dyed water asif viewed from underneath the cavities. (B) A cross-section of A showing theconcave curvature of the liquid meniscus.

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