6518 | Soft Matter, 2019, 15, 6518--6529 This journal is©The Royal Society of Chemistry 2019

Cite this: SoftMatter, 2019,

15, 6518

Micropattern-controlled wicking enhancement inhierarchical micro/nanostructures†

Arif Rokoni, Dong-Ook Kim and Ying Sun *

Wicking in hierarchical micro/nanostructured surfaces has attracted significant attention due to its potential

applications in thermal management, moisture capturing, drug delivery, and oil recovery. Although some

studies have shown that hierarchical structures enhance wicking over micro-structured surfaces, others have

found very limited wicking improvement. In this study, we demonstrate the importance of micropatterns in

wicking enhancement in hierarchical surfaces using ZnO nanorods grown on silicon micropillars of varying

spacings and heights. The wicking front over hierarchical surfaces is found to follow a two-stage motion,

where wicking is faster around micropillars, but slower in between adjacent pillar rows and the latter stage

dictates the wicking enhancement in hierarchical surfaces. The competition between the added capillary

action and friction due to nanostructures in these two different wicking stages results in a strong

dependence of wicking enhancement on the height and spacing of the micropillars. A scaling model for

the propagation coefficient is developed for wicking in hierarchical surfaces considering nanostructures

in both wicking stages and the model agrees well with the experiments. This microstructure-controlled

two-stage wicking characteristic sheds light on a more effective design of hierarchical micro/

nanostructured surfaces for wicking enhancement.

1. Introduction

Wicking on textured surfaces has attracted significant attentiondue to its numerous applications in nature and engineeredsystems such as thermal management,1–6 moisture capturing,7,8

drug delivery,9,10 biomedical devices11–14 and oil recovery.15 Neara century ago, Washburn predicted the dynamics of liquidimbibition into capillary tubes, driven by the surface tensionforce, where the imbibition length is proportional to the squareroot of time.16 Recent developments1,6,15,17–27 in fabricationprocesses of micro, nano and hierarchical surfaces have createdample opportunities to tune the topography of the structuredsurfaces to achieve different wicking conditions. Meanwhile,several models have been developed to extend the classic Washburnmodel for a variety of engineered surfaces.

Bico et al.28 extended the Washburn model for wickingdynamics in capillary tubes to surfaces with micropillars byintroducing an empirical parameter b to account for the addedviscous resistance due to micropillars. Ishino et al.29 distin-guished wicking on micropillar surfaces into short and long-pillar scenarios, where the viscous resistance from pillar side-walls is ignored for short pillars whereas the pillar base

resistance is neglected for long pillars. Using Surface Evolver(SE)30 to predict the 3D meniscus shape and the 1D Brinkman’sequation to approximate the viscous resistance, Xiao et al.31

proposed a semi-analytical model to predict the liquid propa-gation rate on surfaces with micropillar arrays. Mai et al.20

correlated the viscous resistance enhancement factor b with thepillar height-to-spacing ratio by simplifying the pillars as channelshaving the same porosity and validated the model by wickingexperiments on surfaces with nanopillar arrays. Interestingly,recent boiling experiments32 have observed delays in the undesir-able Leidenfrost effect on surfaces with sparse micropillars, wherethe liquid wicking rate is found to be overpredicted by using theabove mentioned models. A recent study by Kim et al.33 consid-ered cases with pillar spacing ranging from dense to sparse andidentified a two-stage wicking motion for sparse pillars wherewicking is faster around pillars but slower in between pillar rows.The slow wicking stage between pillar rows is mainly due to itssmaller pre-wet surface area per wicking length, resulting in aweaker driving force in the system free energy. A scaling modelrelating the propagation coefficient with fluid properties andmicropillar geometries has been proposed, where the propaga-tion coefficient for sparse pillars is smaller than that of densepillars while keeping other parameters the same.33

Recently, wicking enhancements have been reported by addingnanostructures to microstructures for creating hierarchical surfaceswhich are superhydrophilic in an effort to enhance the criticalheat flux of pool boiling and thin film evaporation.1,6,19,34–38

Department of Mechanical Engineering and Mechanics, Drexel University,

Philadelphia, PA 19104, USA. E-mail: ys347@drexel.edu; Fax: +1-215-895-1478;

Tel: +1-215-895-1373

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9sm01055f

Received 26th May 2019,Accepted 17th July 2019

DOI: 10.1039/c9sm01055f

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Soft Matter

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Others have found that the low permeability of the addednanostructures, compared to microstructures alone, limitstheir role in assisting bulk wicking, resulting in a negligiblewicking improvement.2 For example, the added viscous resis-tance becomes important when the height of the ZnO nanorodsapproaches half of the micropillar spacing for hierarchicalsurfaces consisting of ZnO nanorods on micropillars of varyingheights.21 Despite these studies, general guidelines for predictingthe wicking enhancement in hierarchical surfaces of two verydifferent length scales are yet to be reported.

In this study, the wicking dynamics of hierarchical micro/nanostructured surfaces is examined by conducting systematicwicking experiments using ZnO nanorods grown on siliconmicropillars of varying spacings and heights. A two-stage wick-ing motion is observed on hierarchical surfaces, where wickingis faster around micropillars but slower in between pillar rows.The role of nanostructures is different in these two stages, whereno obvious wicking enhancement is observed with nanostruc-tures in the first stage, but significant enhancement is observedduring the second stage where wicking is slower in-betweenmicropillar rows and this second stage dictates the bulk wickingdynamics. Wicking improvement in hierarchical structures overbare micropillar surfaces is found to depend on the spacing andheight of the micropillars. A new scaling model relating thepropagation coefficient to the fluid properties and geometricalparameters of the hierarchical surfaces is proposed, consideringnanostructures in both wicking stages. The model predictionsare compared with our wicking experiments.

2. Experimental methods

In this study, circular micropillars were fabricated by deepreactive ion etching (DRIE) of silicon (100) wafers where theetching time was controlled to create micropillars of differentheights. The resulting micropillar diameter is fixed at 10 mm,the center-to-center spacing varies from 30 mm to 60 mm, andthe height varies from 8 mm to 26 mm. To create a hierarchicalsurface, ZnO nanorods were grown on bare silicon micropillarswith 25 mm � 25 mm sample dimensions via a chemicalprocess,39 where the growth time and concentration of reac-tants determine the height and diameter of the ZnO nanorods.

The nanoscale roughness, rn, is determined using rn ¼ 1þ phn

l2,

where p is the total perimeter of all nanorods within a squarewindow of length l of the nanostructure base plane and hn is theeffective height of the nanorods. The same growth time of 4 hoursand concentrations of hexamethylenetetramine (HMT) and zincnitrate (1 : 1) at 0.04 M were used, and hence the identical rough-ness of the ZnO nanorods was maintained for the hierarchicalsurfaces. However, the growth time varied from 2 to 5 hours andthe concentrations of HMT and zinc nitrate (1 : 1) varied from0.01 M to 0.05 M for the nanorod-only cases, resulting in differentnanoscale roughness values shown in the ESI.†

Fig. 1 shows the scanning electron microscopy (SEM) imagesof the fabricated micropillar (Fig. 1a and b) and hierarchical(Fig. 1c and d) surfaces. The growth time of 4 hours yields a

nanorod height of E1 mm, but an effective height of E350 nmcontributes to wicking due to the closely packed nanorods near thebase plane (see the SEM image in Fig. 1d). For the ZnO nanoroddiameter of E30 nm and the effective height of E350 nm, thecorresponding nanoscale roughness of rn E 2.5 is assumed inthe model for hierarchical surfaces, determined based on thetop- and side-view SEM images of nanorods.

To minimize the effects of inertia during wicking, a verticalwicking mechanism31 is applied in this study. Fig. 2a shows aschematic of the wicking test setup. The substrate was placedvertically in a deionized (DI) water bath and the wicking dynamicswas captured using a high-speed camera (Phantom V711) at aframe rate of 6000 frames per second using a 20� objectivewith a spatial resolution of 1 micron per pixel.

A high-powered LED (Thorlabs, Solis-445C) and a beamsplitter were used for illumination. Substrate surfaces werecleaned with isopropyl alcohol (IPA) and acetone and rinsedwith DI water, followed by air plasma cleaning (18 W and250 mTorr, Harrick Plasma PDC-32G). The wettability of puresilicon and ZnO nanofilm coated silicon40 after 4 minutes ofplasma cleaning was experimentally determined by interfero-metry imaging of a drop contact line region. An intrinsic watercontact angle of y E 41 was obtained for pure silicon andy E 51 for ZnO nanofilm coated silicon. The intrinsic contactangle remained the same for about 1 hour and then graduallyincreased. Thus, all wicking tests were done within 5 min ofplasma cleaning to ensure a constant intrinsic contact angle.

Fig. 2b shows the wicking process, where the water level andthe wicking front are marked in arrows. The instantaneouswicking length, the distance of the wicking front from the waterbath, is obtained by averaging at three different positions,equally spaced within the field of view of E10 mm, perpendi-cular to the wicking direction. Each test is repeated five timesto account for the imperfect flatness of the wicking front.Assuming the wicking length, a, scales with time to the powern, i.e., a = Gtn, where n is the wicking index (n = 0.5 in theWashburn model) and G is the propagation coefficient. In this

Fig. 1 (a) Schematic of a micropillar surface, showing pillar diameter d,height h and center-to-center spacing s. SEM images of (b) siliconmicropillars, (c) a hierarchical micro/nanostructured surface, and (d) ZnOnanorods on the base plane of the micropillar array.

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study, the wicking indexes for both microstructures alone andhierarchical surfaces have been found to be close to 0.5, asdescribed in the ESI.†

3. Experimental results

To understand the wicking dynamics of hierarchical surfaces, acomparison of wicking in micropillars with and without nano-rods has been given. The wicking dynamics of micropillar-onlysurfaces of varying pillar heights and spacings compared withthe existing wicking models is summarized in the ESI.†

Fig. 3 shows the wicking front positions over time as waterwicks over micropillars of different diameter-to-spacing ratios

without (the left panel) and with (the right panel) nanorods,wherein Fig. 3a shows a diameter-to-spacing ratio, d/s, of 0.333and Fig. 3b shows d/s of 0.2, both at a micropillar height of13 mm. Here, t = 0 is when the structured surface touches DIwater and water starts to wick vertically from the initial wickingfront position marked in a red dashed line. No significantwicking improvement is observed due to the presence of nano-structures for micropillar d/s of 0.333 as shown in Fig. 3a.However, wicking is much faster for hierarchical surfaces com-pared to the bare micropillar case at d/s = 0.2 as shown in Fig. 3b.Note that the diameter-to-spacing ratio is varied by changing thespacing between micropillars while keeping the pillar diameterfixed at 10 mm such that a lower diameter-to-spacing ratiocorresponds to a larger pillar spacing and a sparser pillar array.

Fig. 2 (a) Schematic of the vertical wicking test, where a high-speed camera is used to record the instantaneous wicking front location on structuredsurfaces. (b) An image of wicking over structured surface showing the instantaneous wicking front and the water bath level, which is the position ofwicking front at t = 0 s.

Fig. 3 Wicking dynamics of DI water over micropillar and hierarchical micro/nanostructured surfaces for micropillar diameter-to-spacing ratiosof (a) d/s = 0.333 and (b) d/s = 0.2. No significant wicking improvement due to nanostructures is observed for the d/s = 0.333 case but observableenhancement is found for d/s = 0.2. The initial wicking front position is marked in a red dashed line. For all cases, the pillar height is kept at h = 13 mm.

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Fig. 4 shows a quantitative comparison of the wicking lengthover time for micropillars without (black squares) and with (redsquares) nanorods for the cases shown in Fig. 3. Each experi-ment was repeated 5 times and the solid lines in Fig. 4 denotepower-law fittings assuming the wicking index n = 0.5.

For d/s = 0.333, the propagation coefficients, G, from curvefitting are 8.198 and 8.576 mm s�1/2 without and with nanorods,respectively, with only 4.6% increase in G due to the addednanostructures, very close to the error bar of 4.1% for micropillars.On the other hand a more obvious wicking enhancement withoutand with nanostructures has been observed for the d/s = 0.2 case,as shown in Fig. 4b, where the propagation coefficients, G, are4.505 and 5.697 mm s�1/2, respectively, with 26% increase due tothe nanostructures. Note that the wicking length measurement islimited to 10 mm due to the camera field of view. It is alsoimportant to note that the nanostructure roughness of E2.5 iskept the same for both micropillar diameter-to-spacing ratioswhereas very different wicking enhancements are achieved withand without nanostructures at these two d/s values, indicating thatmicrostructure geometries play an important role in wickingenhancement in hierarchical surfaces.

To obtain an in-depth understanding of how wickingimprovement in hierarchical surfaces depends on micropillargeometries, the wicking dynamics was observed at the indivi-dual micropillar level using 20� zoom lens. Fig. 5c showsthe instantaneous wicking front positions over time for bothmicropillar and hierarchical surfaces for the micropillardiameter-to-spacing ratio, d/s, of 0.2. The wicking front positionis measured in the middle of two adjacent pillars perpendicularto the wicking direction and plotted over two consecutive pillarspacings along the wicking direction, and t = 0 is defined herewhen the wicking front touches the row of pillars located ata = 6 mm both with and without nanorods. A two-stage wickingdynamics is observed in this zoomed-in experiment, where thewicking speed, i.e., the slope of the wicking position over time,is faster while around pillars but slows down as the wickingfront moves in-between two adjacent pillar rows both with andwithout nanorods.

Fig. 5a shows the wicking front positions at three differenttimes for the micropillar-only case, where the wicking front isuneven around pillars. It takes E1 ms for the wicking front tomove around pillars of 10 mm in diameter, but E25 ms to movethe next 40 mm in-between two pillar rows until the fronttouches the next pillar row for the bare micropillar case. Thisslower wicking stage between two pillar rows is believed to bedue to its smaller pre-wet surface area per wicking lengthcompared to the first wicking stage, resulting in a weakerdriving force in the system free energy. As the wicking fronttouches the next row of pillars, the two-stage wicking cyclerepeats itself.

Fig. 5b shows the bulk wicking front, marked in a red arrow,and the nano wicking front inside the nanorod region, markedin a blue arrow, at three different times for the hierarchicalsurface. A side-view schematic on the top row of Fig. 5b depictsthe locations of the bulk and nano wicking fronts. It takesE1 ms for the bulk wicking front to move around pillars, butE17 ms to touch the next pillar row, 8 ms shorter than that ofthe bare micropillar case. In other words, the effect of nanorodson wicking improvement is not observable in the first (fast)wicking stage but is obvious in the second stage. The secondstage dictates the wicking enhancement in hierarchical surfaceswhere the microscale capillary action is slower than the nano-scale one, and thus the bulk wicking speed becomes the speed-limiting factor. In an effort to identify the underlying physics ofthis wicking enhancement due to the nanorods in-between pillarrows, a side-view schematic of the wicking front for both micro-pillar and hierarchical surfaces is shown in Fig. 5d.

It is important to note that the intrinsic contact angles ofwater on pure silicon and ZnO nanofilm coated silicon are closeto each other (y E 41 and y E 51, respectively) and assumedequal. For the hierarchical surface, the apparent contact angleof the nano-rough micropillar base plane is lower than theintrinsic contact angle of water on ZnO-coated silicon or puresilicon due to the nano wicking effect, following the relationof cos yapparent � cos y = (1 � fs,n)(1 � cos y) Z 041 and henceyapparent o y for any y 4 0, where fs,n is the solid fraction

Fig. 4 Experimental wicking length over time for bare micropillars (black squares) and hierarchical surfaces (red squares) for different micropillardiameter-to-spacing ratios: (a) d/s = 0.333 and (b) d/s = 0.2 both at pillar height h = 13 mm. Solid lines denote power-law fittings using data from all fiveexperiments assuming the wicking index n = 0.5. Shaded areas correspond to the 90% prediction band.

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of nanorods.41 The lower wetting angle therefore enhances wickingof the hierarchical surface compared to the micropillar-only case.This reduction in the contact angle due to the presence of nano-structures is widely observed in droplet spreading scenarios,42,43

where a nanoscale precursor film assists wetting.44 As shown inFig. 5b, a nano wicking front of E40 to 60 mm ahead of the bulkwicking is observed on the hierarchical surface.

To examine why only limited wicking improvement isobserved for hierarchical surfaces of d/s = 0.333 as shown inFig. 4a, a comparison of the wicking dynamics of hierarchicalsurfaces for d/s = 0.333 and 0.2 is given in Fig. 6. The bulk andnano wicking front positions are shown at three different timesduring one wicking cycle for both cases. The side-view sche-matics are shown on the top in red and blue arrows indicatingthe positions of the bulk and nano wicking fronts, respectively.

For d/s of 0.333, shown in Fig. 6a, the nano wicking front isahead of bulk wicking at t = 0 ms, but two wicking fronts areclose to each other at t = 0.5 ms. In the case of d/s = 0.2, shownin Fig. 6b, the nano wicking front is always ahead of bulkwicking. As identified earlier, the second (slow) wicking stage

dictates the wicking improvement in hierarchical surfaces andis hence the focus of the current analysis. This second stage isanalogous to the scenario of droplet spreading on nanostruc-tured surfaces where two spreading stages are identified:42,43

the synchronous spreading stage, where the droplet apparentcontact line and the nano wicking front move at the same speed,followed by a hemi-spreading stage where the apparent contactline stops but the nano wicking front continues to advance.

In the present study, the bulk wicking front advances fasterfor the d/s = 0.333 case, analogous to the synchronous spreadingstage with non-distinguishable bulk and nano wicking fronts;however, bulk wicking is much slower for d/s = 0.2 where hemi-spreading takes place with nano wicking ahead of the bulkwicking front. In other words, the distance between the nanoand bulk wicking fronts is found be to a function of the bulkwicking speed. When the bulk wicking is faster in-between pillarrows for d/s = 0.333, the wicking capability of the nanostructurecannot catch up with the bulk wicking speed, resulting in asmaller nano-wicking length ahead of bulk wicking compared tothe d/s = 0.2 case, as shown in the middle column of Fig. 6.

Fig. 5 (a) Images showing wicking front positions (the red arrow) at t = 0, 1 and 26 ms when the wicking front touches the first pillar row, just passes thefirst pillar row, and touches the next pillar row, respectively. (b) Images showing bulk (the red arrow) and nano (the blue arrow) wicking front positionsat t = 0, 1 and 18 ms for the hierarchical surface when the bulk wicking front touches the first pillar row, just passes the first row and touches the next pillarrow respectively. (c) Wicking front positions over time for micropillars (black squares) and hierarchical surfaces (red squares) for d/s = 0.2 and h = 13 mmfor two consecutive pillar spacings where the wicking front is 6 mm away from the liquid reservoir. A two-stage wicking motion is observed for bothmicropillar and hierarchical surfaces. (d) Schematic diagram of the wicking front between two pillar rows for micropillar-only (top) and hierarchical(bottom) surfaces.

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To further explore the dependence of wicking improvementon the micropillar geometry, wicking experiments were con-ducted on micropillars of varying heights (8–26 mm) and pillar-to-pillar spacings (30–60 mm) without and with nanorods. Fig. 7shows the two-stage wicking dynamics for two consecutive pillarspacings for micropillar diameter-to-spacing ratios, d/s, of (a)0.333 and (b) 0.2 both with a pillar height of 26 mm. Similar tothose with a pillar height of 13 mm (shown in Fig. 3–6), nosignificant wicking enhancement is observed without and withnanorods for d/s of 0.333, as shown in Fig. 7a, but obviousenhancement is observed for d/s = 0.2, as shown in Fig. 7b. Thebulk wicking front positions for micropillar and hierarchicalsurfaces are close to each other in the first wicking stage aroundmicropillars, but the time it takes to wick in-between pillars isshorter for hierarchical surfaces compared to bare micropillars,all consistent with the findings from the h = 13 mm cases.

It is important to note that the micropillar height and pillardensity are varied in this study while the nanostructures on thehierarchical surfaces are kept the same. At a relative high pillar

density, e.g., d/s = 0.333, the wicking speed of the pure micro-pillar and hierarchical surfaces is almost the same, while at alower pillar density, e.g., d/s = 0.2, the wicking speed of hierarch-ical surfaces is faster. The wicking speed of the speed-limitingstage when the wicking front is in between two pillar rowsis increased when the pillar density of the hierarchical surfaceis lower. This is because the wicking front in the nanostructure isalways faster than the bulk wicking front in the microstructures.The bulk wicking speed is increased in this stage by decreasingthe apparent contact angle. However, the speed enhancement isnegligible when the wicking front is around pillars.

4. The wicking model for hierarchicalsurfaces

In this section, a scaling model relating the propagationcoefficient with the fluid properties and geometrical para-meters of micropillars has been developed by balancing the

Fig. 6 Bulk versus nano wicking front of the hierarchical surfaces for (a) d/s = 0.333 showing two wicking fronts close to each other at t = 0.5 ms and (b)d/s = 0.2 showing the nano wicking front ahead of the bulk wicking front at all times.

Fig. 7 A comparison of wicking dynamics over two consecutive pillar spacings showing two-stage wicking dynamics on both micropillars andhierarchical surfaces for a micropillar height of 26 mm and a diameter-to-spacing ratio, d/s, of (a) 0.333 and (b) 0.2. Black and red squares denotemicropillars and hierarchical surfaces, respectively.

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surface energy change with the frictional work as liquid wicksthe micropillars. The model is then extended to hierarchicalsurfaces by incorporating the effects of nanostructures onincreasing both the capillary pressure and viscous resistance.

Fig. 8 depicts the systems considered in the wicking modelsof micropillar and hierarchical surfaces. A unit cell of micro-pillars consists of two distinguished regions: Region 1 is aroundmicropillars and Region 2 is in-between pillar rows. Here, themicropillar spacing, height and diameter are shown as s, h andd, respectively. Wicking is along the x axis, the pillar height isalong the z axis, and y axis is perpendicular to the wickingdirection. The cross-sectional view of the unit cell is shown inFig. 8b assuming the top surface of the wicked liquid is having aflat meniscus. In the model, the circular micropillars are sim-plified as rectangular micropillars with length d and width s-L,shown in Fig. 8c, where Region 1 is considered to be a micro-channel bounded by three wall surfaces: two pillar side walls and

a micropillar base plane. Here, L ¼ sd � pd2

4

� ��d is the

equivalent channel width of the rectangular pillars, consideringRegion 1 of the rectangular pillars shown in Fig. 8c holds thesame volume of liquid as in Region 1 of the circular pillarsshown in Fig. 8a. Note that the length of the rectangularmicropillar, along the wicking direction, is kept the same asthe diameter, d, of the circular micropillar.

The following assumptions are considered in the model:(i) The meniscus shape of the top surface is considered to be

flat during wicking and the meniscus height is the same as theheight of micropillars as shown in Fig. 8b.

(ii) Circular micropillars are simplified as rectangular pillarswith length d, the same as the pillar diameter, and width s-L, asshown in Fig. 8c. Wicking in Region 1 is considered to bechannel flow bounded by three surfaces and in Region 2 is overa flat surface.

(iii) Flow in both Regions 1 and 2 is considered to be fullydeveloped.

(iv) Surface contamination and defects are not considered inthe model.

Fig. 8 (a) A unit cell of the structured surface consisting of two distinguished wicking regions: the blue box is Region 1 (around micropillars) and the redbox is Region 2 (in-between pillar rows). (b) Cross-sectional view of the structure along the wicking direction x, showing the flat meniscus on the top asassumed in the current study. The black dashed box is one unit-cell, consisting of Regions 1 and 2. (c) Wicking in circular micropillars is simplified aswicking in rectangular micropillars with length d and width s-L, where Region 1 is bounded by three wall surfaces, holding the same volume of liquid asRegion 1 around circular micropillars shown in (a). Schematic of the velocity profiles in Regions 1 and 2 is shown where u1(y,z) is the x-velocity of thewicking liquid in Region 1 (bounded by three wall surfaces) and u2(z) is the x-velocity in Region 2 (over the micropillar base plane). Wicking in hierarchicalsurfaces is modeled assuming nanostructures in both Regions 1 and 2.

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(v) Gravitational effects are not considered since the Bondnumber, Bo = rh2g/g,33 is very small O (10�6–10�5).

The model is based on balancing the total work done toovercome the frictional resistances as liquid wicks from the waterbath level to the wicking front with a change in the surface energyof the system as liquid wicks over one-unit cell, expressed

asa

sWtotal ¼ DE, where a is the wicking length and Wtotal is the

work done to overcome frictional resistance for one unit-cell oflength, s. The change in the surface energy of the system as liquidwicks over one-unit cell of the micropillar surface consisting ofRegions 1 and 2, as shown in Fig. 8a, is given by

DE = g[(rm � fs,m) cos y � (1 � fs,m)]s2 (1)

where rm ¼ 1þ pdhs2

is the roughness measure of micropillars,

fs;m ¼pd2

4s2is the solid fraction of a unit cell, and y the intrinsic

contact angle.Frictional losses arise due to liquid wicking in Regions 1 and 2

of the unit cell. In Region 1, for bare micropillars, wicking flow canbe considered to be a channel flow, bounded by two pillar sidewalls and a pillar base plane, as shown in Fig. 8c. The x-velocityof the wicking flow in Region 1, u1, depends on both y and zcoordinates and can be derived assuming no-slip conditions on

three side walls, i.e., u1(y,z) = 0 at y ¼ �L2

, y ¼ L

2, and z = 0, and

a free surface on the top, i.e.,@

@zu1ðy; zÞð Þ ¼ 0 at z = h, following

u1ðy; zÞ ¼ U1;max 2z

h

� �� z

h

� �2� �1� 4

y

L

� �2� �(2)

Here, U1,max is the maximum velocity of Region 1 at z = h andy = 0. Shear forces exerted by the no-slip walls are calculated

from the velocity gradient as F1;s ¼ 2A1;sm@u1@y

y¼�L=2

¼

2mdÐ h0

@u1@y

y¼�L=2

dz for the side walls and F1;b ¼ A1;bm@u1@z

z¼0¼

mdÐ L=2�L=2

@u1@z

z¼0

dy for the bottom surface, where A1,s is the area

of the side wall and A1,b is the area of the base plane of Region1. The work done to overcome the frictional resistance can becalculated by multiplying the shear force with the distancetravelled by the wicking flow. For Region 1, the wicking fronttravels a distance d such that the friction work for the side walls

becomes W1;s ¼ F1;sd � d2mU1;maxh

Land for the bottom surface

it follows W1;b ¼ F1;bd � mU1;maxd2L

h.

Flow in Region 2 can be considered to be flow over a flatsurface with x-velocity u2, as shown in Fig. 8c, whose profiledepends on z only and can be derived assuming no-slip condi-tions on the base plane (u2(z) = 0 at z = 0) and a free surface on

the top@

@zu2ðzÞð Þ ¼ 0 at z ¼ h

� �, following

u2ðzÞ ¼ U2;max 2z

h

� �� z

h

� �2� �(3)

Here, U2,max is the maximum velocity in Region 2 at z = h. Theshear force exerted by the bottom surface is calculated from

the velocity gradient as F2;b ¼ A2;bm@u2@z

z¼0¼ sðs� dÞm@u2

@z

z¼0

,

where A2,b is the area of the base plane of Region 2 and mis the liquid viscosity. For Region 2, the wicking flow travelsa distance s � d, leading to frictional work, W2;b ¼

F2;bðs� dÞ � sðs�dÞ2mU2;max

h.

Considering the same amount of liquid is flowingfrom Regions 1 to Region 2, i.e.,

:Q1 =

:Q2, where

:Q is the

volumetric flow rate, it leads to U1,avgL = U2,avgs, assuming themeniscus maintains a constant height of h. Here, the averagevelocity considering fully developed flow in Region 1 is

U1;avg ¼1

L

1

h

Ð L=2�L=2

Ð h0u1ðy; zÞdydz ¼

4

9U1;max and in Region 2 is

U2;avg ¼1

h

Ð h0u2ðzÞdz ¼

2

3U2;max. This leads to the relation

between maximum velocities of Regions 1 and 2 as2

3U1;maxL ¼ U2;maxs. Thus, the total work done to overcome the

shear forces within one unit cell combining Regions 1 and 2can be expressed as Wtotal = W1,b + W1,s + W2,b following

Wtotal � mU2;maxsðs� dÞ2

hþ sd2h

L2þ d2s

h

� �(4)

Introducing the microstructure roughness rm ¼ 1þ pdhs2

, eqn (4)

becomes

Wtotal � mU2;maxs3

h1� d

s1� d

s

� �þ d

srm � 1ð Þhs

L2

� �� �(5)

Considering that L2 E s(s � d) for s 4 d and L E s andU2,max B U2,avg, it yields

Wtotal � mU2;avgs3

h1� d

s1� d

s

� �þ d

srm � 1ð Þ h

ðs� dÞ

� �� �(6)

As observed in the experiment, the slow wicking stage, i.e.,Region 2, dictates the wicking dynamics such that U2,avg can beapproximated as the velocity of the wicking front, following

U2;avg �da

dt. This leads to

Wtotal � mda

dt

s3

h1� d

s1� d

s

� �þ d

srm � 1ð Þ h

ðs� dÞ

� �� �(7)

Substituting eqn (1) and (7) ina

sWtotal ¼ DE and integrating, it

follows

1

sms3

h1� d

s1� d

s

� �þ d

sðrm � 1Þ h

s� dð Þ

� �� �

�ðada � g rm � fs;m

�cos y� 1� fs;m

�� s2ðdt

and utilizing the wicking length versus time relation a = Gw/ot1/2,it yields the following relation for the propagation coefficient

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without nanorods, Gw/o, as

Gw=o � Iw=o ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirm � fs;m

�cos y� 1� fs;m

�1� d

s1� d

s

� �þ d

srm � 1ð Þ h

ðs� dÞ

� �ghm

vuuut (8)

Eqn (8) describes how the propagation coefficient for micropil-lars without nanorods is related to fluid properties, such asviscosity m and surface tension g, as well as the topography ofmicropillars, such as roughness rm, solid fraction fs,m, diameterd, height h, spacing s and wettability y. Note that this scalingmodel for the propagation coefficient of wicking in micropillarsis inspired by Kim et al.,33 where the shear forces in two differentregions are added to calculate the total viscous resistances.However, in the current study, the friction work instead of thefriction forces is added to balance the change in the surface freeenergy since forces in different regions should not be added.

The model for the propagation coefficient in micropillars isthen extended for hierarchical surfaces, where nanostructure isconsidered in both Regions 1 and 2. For a hierarchical surface,the change in the system free energy as liquid wicks one unitcell consisting of Regions 1 and 2 follows

DE = g[(rn(rm � fs,m)) cos y � (1 � fs,m)]s2 (9)

Here, rn is the nanostructure roughness. The nanostructure isconsidered on the bottom surface and two side walls of Region1, as well as the bottom surface of Region 2, such thatit increases the frictional resistance compared to bare micro-pillars. Considering b0 as the resistance enhancement factor,the work done to overcome the frictional resistances of the

bottom surface of Region 1 follows W1;b0 � b0mU1;maxd

2L

hand

for the side walls, W1;s0 � b0d2mU1;max

h

L. Similarly, for Region 2,

frictional resistances of the bottom surface can be scaled as

W2;b0 � b0sðs� dÞ2mU2;max

h. This leads to the total work done to

overcome the shear force for a hierarchical surface as

Wtotal0 � mb0U2;max

sðs� dÞ2h

þ sd2h

L2þ d2s

h

� �(10)

Eqn (10) can be rewritten for hierarchical surfaces as

Wtotal0 � mb0U2;avg

s3

h1� d

s1� d

s

� �þ d

srm � 1ð Þ h

ðs� dÞ

� �� �(11)

where the constant b0 can be calculated by solving the Brink-man’s equation for flow through a porous medium, i.e.,

md2u

dz2� e

dP

dx� ma2eu ¼ 0, where

dP

dxis the pressure gradient

driving the flow, e ¼ 1� fs;n ¼ 1� pdn2

4sn2is the porosity and

1

a2¼

sn2lnfs;n

�1=2 � 0:738þ fs;n � 0:887fs;n2 þ 2:038fs;n

3 þ o fs;n4

�4p

is the permeability of the nanorods and is valid fordn

sno 0:5731

dn

sn� 0:2 in the present study

� �. Here, dn, sn and fs,n are the

diameter, spacing and solid fraction of nanorods, respectively. Allparameters in the Brinkman’s equation are determined based onthe nanostructure geometry obtained from the SEM images. Theaverage velocity obtained from the Brinkman’s equation is given by

�u ¼ dP

dx

1

a2m1� e�a

ffiffiep

hn

affiffiep

hn eaffiffiep

hn þ e�affiffiep

hn �

"

þ eaffiffiep

hn � 1

affiffiep

hn eaffiffiep

hn þ e�affiffiep

hn �� 1

# (12)

which leads to the following expression for the viscous resistance

enhancement factor, b0, assuming the average velocity �u ¼ hn2

3b0mdP

dx

taking the format of flow over a flat surface

b0 ¼ hn2

3a2,

1� e�affiffiep

hn

affiffiep

hn eaffiffiep

hn þ e�affiffiep

hn �

"

þ eaffiffiep

hn � 1

affiffiep

hn eaffiffiep

hn þ e�affiffiep

hn �� 1

# (13)

Eqn (13) indicates that the viscous resistance enhancement factor,b0, depends on the nanostructure topography, such as porosity e,

permeability1

a2and height hn. The propagation coefficient for

micropillars with nanorods, Gw/, can hence be expressed as

Gw= � Iw= ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirn rm � fs;m

�cos y� 1� fs;m

�1� d

s1� d

s

� �þ d

srm � 1ð Þ h

ðs� dÞ

� � ghb0m

vuuut (14)

Eqn (14) describes how the propagation coefficient for micropillarswith nanorods is related to fluid properties and micropillartopography, similar to eqn (8), as well as nanorod topography,such as roughness rn and b0. For the micropillar-only case, bysetting rn = 1 and b0 = 1, eqn (14) reduces to eqn (8).

5. A comparison between models andexperiments

In this section, the propagation coefficients predicted bythe models are compared against the experimental results ofwicking on both micropillar and hierarchical surfaces. Fig. 9ashows the experimental values of propagation coefficient,Gw/o,exp., with respect to the parameters of micropillar geometries(diameter, spacing and height), material (the intrinsic contactangle) and fluid properties (surface tension and viscosity),defined as Iw/o in eqn (8). The experimental values of thepropagation coefficient for micropillars, Gw/o,exp., scale well withIw/o predicted by eqn (8). The black dashed line in Fig. 9a isthe least-square fitting with R2 = 0.99 and its slope is theproportionality constant, which is 0.35 for the bare micropillarcases, accounting for the effects of surface defects and theassumptions made in the modeling section.

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Similarly, Fig. 9b shows the experimentally determinedpropagation coefficients for the hierarchical surfaces, Gw/,exp., withrespect to the parameters of micropillar geometries (diameter,spacing and height), material (the intrinsic contact angle), nanorodgeometries (diameter, spacing and height) and fluid properties(surface tension and viscosity), defined as Iw/ in eqn (14). Goodlinearity is achieved between Gw/,exp. and Iw/ for the hierarchicalsurface with the red dashed fitting line as shown in Fig. 9bcorresponding to R2 = 0.99 and its slope is the proportionalityconstant, which is 0.94 in the present study.

The height of the micropillars is also varied in the presentstudy and the results show that the wicking improvement ofhierarchical surfaces over micropillars depends on the micro-pillar height. Fig. 10 shows the hierarchical to micropillarpropagation coefficient ratio as a function of the micropillardiameter-to-spacing ratio for different micropillar heightswith symbols denoting the experimental results and lines formodel predictions determined using eqn (8) and (14) with theproportionality constants obtained from linear fitting. Wickingenhancement in hierarchical surfaces is only observed whenthe ratio is higher than unity. The experimental results showthat for a fixed diameter-to-spacing ratio, such as d/s = 0.2,

wicking enhancement increases with the decrease in the micro-pillar height, consistent with model predictions. Agreementbetween experiments and the model is also obtained for acertain micropillar height. For all the micropillar heights,wicking enhancement is found to be a strong function of themicropillar diameter-to-spacing ratio. The smaller the d/s ratioor the larger the pillar-to-pillar spacing, the stronger theenhancement. This is consistent with our two-stage wickingmodel: for a larger pillar-to-pillar spacing, the length of theslow wicking stage (Region 2) increases such that the effect ofnanostructure wicking is more significant. Fig. 10 also showsthat the higher the micropillar height or the smaller thediameter-to-spacing ratio the stronger the bulk wicking inhierarchical surfaces and smaller the wicking enhancementwhen compared with bare micropillars. Thus, the relativeposition of nano and bulk wicking fronts or the nanostructurewicking dynamics in the two wicking stages is crucial indetermining the wicking enhancement in hierarchical surfaces.It is important to note that for a large height, h, and a large d/sratio, the hierarchical surface may experience wicking suppres-sion compared to micropillars alone, due to the increasedfriction losses of nanorods as a result of a higher bulk wickingspeed for high, dense pillars. The results also show largediscrepancies between the model predictions and experimentsfor shorter pillars, mainly due to the actual meniscus shapeduring wicking deviating from the model assumption of a flatmeniscus having the same height as the pillars.

6. Conclusion

In this study, the role of micropatterns in the two-stage wickingdynamics of hierarchical surfaces has been studied by changingthe micropillar geometry (e.g., pillar spacing and height) whilekeeping the nanostructures the same. Wicking improvementin hierarchical surfaces over micropillars is found to highlydepend on the micropillar geometry. The added capillary actiondue to the nanostructures is found to be trivial when thewicking front moves around pillars where the microscopic

Fig. 9 Experimental values of propagation coefficients, Gexp., for wicking on (a) micropillar and (b) hierarchical surfaces are plotted against predictedmodel I obtained from eqn (8) and (14) respectively, with good linearity.

Fig. 10 A comparison of hierarchical to micropillar propagation coefficientratio between the scaling model (lines) and experiments (symbols) as afunction of the diameter-to-spacing ratio for different heights of micropillars.

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capillary driving force is strong, but significantly helps thewicking process when the wicking front moves in betweenpillar rows with a weaker microscopic capillary force. Wickingenhancement in hierarchical surfaces is related to the relativepositions of the bulk and nano wicking fronts, which determinethe apparent dynamic contact angle, and is found to be moreobvious for the cases of larger pillar-to-pillar spacing. Hence,adding nanostructures to the existing microstructures forwicking enhancement is expected to be more effective forsparse microstructures.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

Support for this work was provided by the US National ScienceFoundation under Grant No. CBET-1357918 and CBET-1705745.

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