Published: July 20, 2011

r 2011 American Chemical Society 11008 dx.doi.org/10.1021/la201498u | Langmuir 2011, 27, 11008–11016

ARTICLE

pubs.acs.org/Langmuir

Effect of Fiber Geometry on Macroscale Friction of OrderedLow-Density Polyethylene Nanofiber ArraysDae Ho Lee,† Yongkwan Kim,† Ronald S. Fearing,‡ and Roya Maboudian*,†

†Department of Chemical and Biomolecular Engineering, and ‡Department of Electrical Engineering and Computer Science,University of California, Berkeley, California 94720, United States

’ INTRODUCTION

Since the finding that gecko’s unique capability of climbing isdue to the mechanical design of their foot-hair,1�3 great interesthas been focused on fabricatingmicro- or nanofibrillar structures,so-called synthetic gecko foot-hair. With the development ofrecent nanofabrication techniques, a variety of gecko-like struc-tures have been reported, including multilevel hierarchical fibers,4�10

angled fibers,9�14 and smart tip structures.14�20 Although therehave been many reports on the fabrication of ever more com-plex structures,21 there are only a few experimental reports23,25,27

on the effect of basic geometrical factors such as fiber diameter,length, and density on the macroscale adhesion and frictionbehavior.

As reviewed recently,22 the basic issue of geometrical effectremains unclear, as there have been contradictory results regard-ing the effect of aspect ratio on the adhesion or friction of fibrillarsurfaces. A systematic study was reported by Greiner et al.,23

where increased adhesion for higher aspect ratio of cross-linkedpolydimethylsiloxane (PDMS) microfibers was attributed tolarger elastic dissipation during the pull-off process. However,Glassmaker et al.24 have shown that measured pull-off stresses ofpoly(vinyl-butyral) fibers were nearly independent of fiberlengths, although the energy dissipation increased linearly withincreasing fiber length as expected due to the higher stored elasticstrain energy in a single fiber. In some cases, increasing aspectratio was observed to decrease adhesion or friction. Burtonet al.25 measured the pull-off forces of poly(methylmethacrylate)and polyurethane acrylate nanofibers and reported lower adhe-sion for higher aspect ratio at various humidities. Zhao et al.26

have reported a decrease in adhesive strength with increasingheight of multiwalled carbon nanotube arrays, which was attri-buted to the formation of canopy-like entangled surface layers as

MWCNT fibers become longer. In contrast, Qu et al.27 havemore recently demonstrated a dramatic increase in adhesion andfriction with increasing MWCNT fiber length. They have shownshear-induced alignment of top-entangled MWCNT fibers,which became more significant with increasing fiber length.Shear-induced fiber alignment was also demonstrated for poly-propylene (PP) fibers both experimentally and theoretically byLee et al.28 and Majidi et al.,29 respectively. Increase in shearstrength during sliding was attributed to the occurrence ofmicrofibers’ side contact on the substrate. However, the studieswere limited to fibrillar structures with fixed diameter, length, anddensity.

The purpose of this study is to systematically investigate theeffects of geometrical factors in the nanofibrillar structures ontheir friction characteristics and to provide a useful reference foroptimum conditions for high performance from a commercialpolymer. Low-density polyethylene (LDPE) was selected in thisstudy, which is the first report on fabrication and analysis of asynthetic gecko adhesive from this specific material. With thisaim, a method for fabricating ordered polymer nanofiber arrayswith varied aspect ratios is presented. Colloidal lithography30

combined with metal-assisted electroless etching of silicon31 isused to create silicon templates, which are then used to fabricateordered arrays of LDPE nanofibers. It is then demonstrated thatmacroscale friction of these ordered nanofiber arrays is verysensitive to nanoscale changes in relation to their complianceand tip�contact area. A friction design map is presented bymodifying the adhesion map previously reported,36 and a good

Received: April 22, 2011Revised: July 8, 2011

ABSTRACT: Ordered low-density polyethylene (LDPE) nano-fiber arrays are fabricated from silicon nanowire (SiNW)templates synthesized by a simple wet-chemical process basedon metal-assisted electroless etching combined with colloidallithography. The geometrical effect of nanofibrillar structureson their macroscale friction is investigated over a wide range ofdiameters and lengths under the same fiber density. Theoptimum geometry for contacting a smooth glass surface ispresented with discussions on the compromise between fibertip�contact area and fiber compliance. A friction design map isdeveloped, which shows that the theoretical optimum design condition agrees well with the LDPE nanofiber geometries exhibitinghigh measured friction.

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agreement with experimental results is observed regarding theoptimal geometry of the LDPE nanofibers.

’EXPERIMENTAL SECTION

Fabrication. LDPE nanofiber arrays were replicated from siliconnanowire (SiNW) templates synthesized by metal-assisted electrolessetching on Au-coated surfaces patterned by polystyrene microspheres(Figure 1) as follows.(i) Polystyrene microsphere arrays � float/transfer technique:

Si(100) wafer chips (p-type, 1�30 Ω cm, Universitywafer) of∼1� 1 cm2 size were cleaned by successive sonication in acetoneand isopropanol for 10 min each, followed by UV ozone cleaningfor another 10 min. Cleaned Si chips were thoroughly washedwith deionized (DI) water (18 MΩ) and then dried by N2 gas.Polystyrene (PS) microsphere (1 μm diameter, Duke ScientificCo.) arrays were prepared by a float/transfer technique.A mixture of microspheres solution in ethanol (1:1 volume ratio)was carefully introduced to a NaCl solution (∼0.5 wt % indistilled water, ∼30 mL), which produced a disordered film offloating PS microspheres. A close-packed film of PS microspherearray was formed with the addition of a small droplet (a few μL)of a surfactant solution (sodium dodecylsulfate, 3 wt % in distilledwater), and then transferred onto the clean silicon surface.A micromanipulating setup based on a stepping motor was usedfor controlled motion of the substrate during immersion andwithdrawal (∼2 mm/s). Close-packed PS microsphere arrays onSi surfaces could be obtained over a large area (up to 4” wafer-scale). Microsphere arrays could be repeatedly obtained from thesame solution, which further facilitates the fabrication process andminimizes consumption of the microsphere solution.

(ii) SiNW templates � metal-assisted electroless etching: SiNWswere synthesized by incorporating colloidal lithography intoAu-assisted electroless etching. The close-packed microspheresobtained by the above method were reduced in diameter byexposure to oxygen plasma (Plasma-Therm PK-12 RIE) at 30 Wand 50 mTorr. A gold film (∼20 nm) was deposited on thesesurfaces by an e-beam evaporator (Thermoionics VE-100vacuum evaporator). The microspheres were then removed bysonication in DI water to generate a patterned Au-coatedsubstrate defined by areas not previously covered by the micro-spheres. Afterward, the substrate was immersed in an etchingsolution containing hydrofluoric acid (HF, 48 wt %), hydrogenperoxide (H2O2, ∼35 wt %), and DI water. Au-coated regionsbecome catalytic sites during metal-assisted electroless etching,resulting in well-defined vertical SiNW structures. Acetonitrile(AN) was used as a cosolvent to improve etching uniformity overa large area (HF/H2O2/H2O/AN= 2/1/5/2, volume ratio). For

all cases, the etched Si surfaces were immersed and repeatedlywashed with isopropanol before N2 drying.

(iii) Ordered LDPE nanofiber arrays � nanomolding: LDPE nano-fiber arrays were replicated using the above SiNW templatesby a conventional melt process. Polycarbonate (PC) film(McMaster-Carr, ∼130 μm thickness) was first molded fromSiNWs templates to generate intermediate nanohole templatesby melt-process in a vacuum oven (∼300 �C, 1.5 h). SiNWtemplates were dissolved in the etching solution composed ofhydrofluoric acid/nitric acid/acetic acid (4/3/3 volume ratio);then LDPE films (McMaster, ∼150 μm thickness) were repli-cated from the PC nanohole templates by melt-process(∼160 �C, 1 h under vacuum). By dissolving the PC nanoholetemplates in methylene chloride, the LDPE nanofibrillar struc-tures were finally obtained with various diameters and lengthsunder a fixed center-to-center distance of 1 μm, as set by thediameter of the starting polystyrene microspheres.

Structural Characterization. The SiNWs and LDPE nanofiberarrays fabricated according to the above processes were imaged in ascanning electron microscope (FE-SEM, JEOL JSM 6490LV) at 5�20 kV.The LDPE nanofiber arrays were coated by Au (∼5 nm) prior to SEMobservation to minimize charging.Friction Measurements. The friction just prior to sliding was

measured using a simple pulley setup with a glass slide as the counter-surface under ambient condition (∼25 �C, ∼40% RH). Test sampleswere carefully placed on the clean glass surface without an intentionalpreload, followed by placing a rubber pad and a small piece of metal(total of ∼10 g, corresponding to the normal load of ∼0.1 N) tominimize any possible variations in contact between LDPE nanofibrillarsurfaces and the glass substrate during the sample loading for each test.Kapton tape was wrapped around one end of the sample to attach athread, which is connected to the measuring cup through a pulley. TheLDPE film was located near the glass edge to prevent undesired contactof the Kapton tape to the glass surface. Static friction force under theslight normal load (∼0.1 N/cm2) was measured by weighing the appliedamount of water in the cup to initiate sliding. The glass substrate wascleaned with acetone before every measurement to remove any possibleresidues and contaminants. Apparent contact area was simultaneouslyobserved during frictionmeasurement by CCD camera through a 45� tiltmirror. Light from the illuminator was aligned parallel to the glass/LDPE interface, and the optical fiber tip was covered with a black sheetto minimize the scattering.28 This provided a clear contrast between thecontacting bright regions and the noncontacting dark regions, which wasanalyzed by an image software (ImageJ 1.42q).

’RESULTS AND DISCUSSION

Fabrication Process.The overall fabrication process of LDPEnanofiber arrays is schematically illustrated in Figure 1 withspecific details provided in the Experimental Section.Figure 2 shows representative images obtained at various

steps. A highly uniform array of microspheres is obtained bythe float/transfer method (Figure 2a,b). The spheres are reducedin diameter by exposure to O2 plasma. The substrate is thencoated with a Au film and sonicated to remove the microsphere,which generates patterned Au-coated surface defined by areas notpreviously covered by the microspheres (Figure 2c). These surfacesare then immersed in the electroless etching solution containing HFand H2O2. This etching is generally understood to be a localizedelectrochemical process,31 with themetal (Au in this study) acting asa local cathodic site and the underlying Si as a local anodic site. Thus,the silicon in contactwith theAu region is etchedmuchmore rapidly.This results in SiNWs structure with the diameter corresponding to

Figure 1. Schematic presentation of the process for fabricating LDPEnanofiber arrays from SiNW templates: (a) polystyrene microspherearray prepared by the float/transfer method, (b) microsphere sizecontrol by oxygen plasma etching, (c) Au coating by e-beam evapora-tion, (d) microsphere removal by sonication, (e) SiNW formation bymetal-assisted electroless etching, (f) first replication using PC film forgenerating intermediate nanohole template, and (g) the final LDPEnanofiber array after second replication using LDPE film.

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the plasma-etched microsphere size and the length determined bythe etching time in the solution (Figure 2d).

LDPE nanofiber arrays are replicated using the above SiNWtemplates by a conventional melt-process. Figure 3 showsrepresentative images of LDPE nanofibers with various dia-meters and lengths at a fixed center-to-center distance of 1 μm.With increasing fiber length, some of the nanofibers start bendingand contacting each other (referred to as partial clumpingregime, e.g., the third row for D ≈ 600 nm); all nanofibers thenclump (all images on the fourth row) into bundles with furtherincreasing length (referred to as clumping regime). This behavioris discussed in more detail in the next section.Fiber Clumping. As shown in Figure 3, LDPE nanofibers are

observed to clump above a certain fiber length for a givendiameter. As will be discussed later, measured friction forcesare limited by the fiber clumping, and thus this clumpingcondition should be considered carefully. For the fibers contact-ing at their tips, a simple equation can be derived from JKRtheory and elastic beam theory:32

Lcrit ¼ ΔRt3E

W

!1=3

ð1Þ

where Lcrit is the maximum length before clumping, 2Δ is thespacing between adjacent fibers, Rt is the radius of curvature ofthe fiber tip, E is the Young’s modulus, and W is the adhesionenergy of the fiber. As shown in Figure 4a, experimentallyobserved values in this study coincide well with the theoretical

Figure 3. Representative images of LDPE nanofiber arrays with increasing aspect ratio (from top to bottom) for each fiber diameter. Fiber clumping isobserved above a certain critical length, discussed in Figure 4. Images with black boxes correspond to nanofiber structures corresponding to maximumfriction in Figure 6a,b (scale bar = 1 μm).

Figure 2. SEM image of PS microsphere array (a) showing well-orderedstructure inmicroscale.Optical imageof a 4 in.wafer (b), showinguniformPSarray. Au-coated patterned Si substrate defined by areas not previouslycovered by the plasma-etchedmicrospheres (c). Slight disordering is inducedduring the plasma etching process. Au-coated region becomes a catalytic siteduringmetal-assisted electroless etching, which results in well-defined verticalSiNW structures (45� tilt view) (d). Scale bar = 2 μm in all SEM images.

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prediction with Rt = R, fiber radius. This is a reasonable approxi-mation supported by the experimental observations shown inFigure 4b and c, where the tip-curvature of nanofiber is measuredfrom the side view SEM images (Figure 4b).Macroscale Friction of LDPE Nanofiber Arrays.The friction

before sliding was measured using the simple pulley setup with adetailed procedure explained in the Experimental Section. It wasoften observed that friction force continuously increased duringthe repeated tests, then decreased (Figure 5a). This appears to befrom fiber alignment and slanting by repeated shear, which mightallow more efficient contact along the shear direction. Indeed,LDPE nanofibers were often observed to be aligned along theshear direction (i.e., pulling direction by measuring cup) as

shown in Figure 5c. After reaching a certain value, the frictiondecreases, which is attributed to significant plastic deformation ofnanofibers. It remains a challenge to improve the mechanicaldurability of the LDPE fibers.Figure 6 shows the macroscale static friction of LDPE

nanofiber arrays for various diameters and lengths with fixed1 μm fiber-to-fiber spacing. For each sample, the highest value

Figure 4. (a) Experimental observation of fiber clumping: O, un-clumped; gray b, partially clumped; and b, clumped LDPE nanofibers.The solid line represents a theoretical prediction (eq 1) for E = 200MPa,γs =W/2 = 0.03 J/m2, and center-to-center distance between fibers = 1μm. (b) Representative image for evaluating the tip-curvature of LDPEnanofibers. ∼5� tilt (from a vertical position) view after cryogenicfracture (scale bar = 0.5 μm). (c) Average values of tip-curvature (Rt) ofLDPE nanofibers with respect to fiber radius (R). All data points fall nearthe dashed y = x line, which indicates Rt ≈ R.

Figure 5. Measured static friction often continuously increases duringrepeated tests and then decreases as shown in (a). Apparent contact areaduring friction measurement is shown in (b) from initial (beforemeasurement, left) to final (prior to detachment, right) state. Apparentcontact area increases during increasing friction (i.e., increased loadinginto measuring cup) with concomitant increase in brightness of thecontacting region (e.g., from ∼4 to ∼10% relative to the total test area,guideline (red) for evaluating the contact area by an image software, forD≈ 800 nm, L≈ 3 μm). In some cases, usually when nanofibers exhibithigh friction, permanent alignment and slanting were observed as shownin the SEM image after friction measurement (45� tilt view, scale bar = 1μm) (c) (arrow: pulling direction). Nanoscale plastic yielding isobserved at fiber tips (dotted circle).

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from repeated measurements before permanent deterioration(e.g., the peak in Figure 5a) was taken as the friction force for thatsample. Each data point plotted in Figures 6a,b represents theaverage of five samples. At each diameter, a maximum frictionforce is observed. This maximum point increases with diameterup to D ≈ 800 nm, then decreases for D ≈ 900 nm.Apparent Contact Area.As shown in Figure 5b, friction force

is not uniformly distributed across the surface, but rather

concentrated in the specific area under normal load with irregularshape. The bright zone in Figure 5b indicates that the contactingregion increases from the initial to final state with a concomitantincrease in brightness during the frictionmeasurement. Althoughnoncontacting dark regions are clearly distinguished from con-tacting bright regions by this method, detailed microscaleinformation inside the contact region is unknown. For example,the actual number of LDPE nanofibers in contact with thecounter surface may be smaller than that estimated from thearea defined by the bright region. It is noted that any difference inbrightness may be influenced by the actual number of contactingnanofibers or by morphological change in contacting nanofibers,for example, from tip to side contact. Thus, bright region isreferred to as the apparent contact area. Despite the lack ofdetailed microscale information, it is useful for friction analysis inmacroscale to visualize how the nanofiber contact is confinedwithin this apparent contact area.Because the static friction force is determined by the state just

prior to sliding, the apparent contact area at this moment shouldbe considered. It was found that this apparent contact fraction(fca = apparent contact area/test area � 100) was not signifi-cantly different between samples, typically 10�15% for thesample size of ∼1 � 1 cm2 as shown in Figure 6c, which issimilar to the previous results for polypropylene fiber arrays(∼11%).28 Distributions of each data point in Figure 6a and bmay reflect the variation in contact area for each test. Despite thisdistribution, it is clear that the measured friction forces arestrongly correlated with nanofibers geometry.JKR Analysis. Contact area generated between the nanofibers

and the flat glass surface can be calculated using contactmechanical theories. The normal load per fiber, FNf, can beestimated to be ∼7 nN, assuming all nanofibers, with arealdensity, FN of ∼108 cm�2, contact within the apparent contactfraction of ∼15%, that is, assuming the measured apparentcontact fraction fca to be that actual contact fraction, fc, at themoment just prior to sliding. Using the JKR equation33 with a tipradius of curvature Rt (≈R, fiber radius), the contact radius (ac)of LDPE nanofibers with diameter ranging from∼400 to 900 nmis estimated to be∼60�110 nmunder the estimated normal loadper fiber of ∼7 nN. Shear force before sliding of a single fiber isgiven by FSf = τ(πac

2), and assuming shear strength τ ≈ 6 MPafor LDPE34 against glass, it is calculated to be ∼75�220 nN forthe aforementioned range of fiber diameters. This corresponds toshear force of 1.1�3.2 N for 1 cm2 area with fc ≈ 15%, whichbecomes smaller for a smaller contact fraction. In contrast, theCoulomb friction per fiber, μsFNf, is estimated to be ∼2 nN,assuming static friction coefficient, μs, of about 0.3 for LDPEagainst glass34 and the normal load FNf of ∼7 nN. This thencorresponds to the Coulomb friction of ∼0.03 N. Consideringthe relatively small Coulombic contribution, one may concludethat measured friction in this study is mostly affected by theadhesion between nanofibers and the glass surface. It should benoted that even though variations in material parameters, such asE, τ, and γs, may cause variations in the measured friction values,the experimental range of values in Figure 6 is reasonably wellapproximated by the above analysis when assuming the actualcontact fraction is similar to the observed apparent contactfraction in Figure 6c.Although the contact area analysis based on JKR theory gives a

reasonable approximation as discussed above, it does not explainthe complex behavior of friction forces depending on fiberdiameter and length. Specifically, any variation in friction forces

Figure 6. (a) Friction forces of LDPE nanofiber arrays with variousfiber diameters and lengths (1 μm spacing) measured by the pulley setup(under ∼0.1 N normal load) described in the Experimental Section(all sample areas ∼1 � 1 cm2). Friction forces for D ≈ 900 nm areshown separately in (b) for clearer demonstration of decrease in frictionwith further increase in fiber diameter. (Dotted lines in (a) and (b) areguides to the eyes.) Maximum friction forces are observed at the aspectratio of∼4 for all fiber diameters. The data points highlighted by dottedcircles indicate the onset of partially clumped fibers. (c) Representativeplot of apparent contact area % (apparent contact area/test area� 100)with respect to fiber length (forD≈ 800 nm).While∼6-fold increase offriction force is observed, apparent contact fraction is not significantlyaltered between samples, typically 10�15% for all cases.

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according to fiber length cannot be explained. Intuitively, lateralflexibility of the nanofibers may play significant role in friction,for example, through further increase in contact area or fiberalignment during friction measurement. As mentioned in theIntroduction, elastic dissipation and side-contact models havebeen typically considered for explaining changes in adhesion andfriction with respect to fiber length (or more generally, aspectratio). Elastic dissipation theory considers trapped and dissipatedenergy induced by fiber stretching in normal direction during thepull-off from the substrate.23,24 Thus, this theory explains onlythe effect of fiber geometry on normal adhesion and is notdirectly applicable to frictional properties. On the other hand,side-contact models28,29 describe bending of sufficiently longfibers, which brings them into a stable side-contact with theopposing surface when adhesion is high enough to exceed theelastic bending forces. However, as shown in Figure 5c, nanoscaleplastic yielding is observed at the fiber tip, which clearly indicatesfiber contact is restricted to the tip. An approach developed toestimate the side contact length28 predicts no side contact for allof the geometries of LDPE nanofibers tested in this study, whichfurther supports that there is no side-contact induced by largebending of LDPE nanofibers.Possible Mechanism. In addition to the above explanation,

small spacing between ordered nanofibers (2Δ = 1 μm � 2R)would not allow a large bending for side contact. We suggestnanofibers have slanted contacts from fiber bending and align-ment by shearing during friction measurement as illustratedschematically in Figure 7. As compared to short nanofibers (a),long nanofibers (b) are much more compliant, and can be bentduring the alignment by shear; thus they can more easily toleratethe nanoscale height distribution (distribution of nanofiberlengths) inside the apparent contact (bright) region, whichfurther increases friction via contact area enhancement at the

microscale. However, even long nanofibers will not be able toovercome the macroscale height variation of the backing film(as observed by bright and dark regions in Figure 5b), whichresults in observation of similar apparent contact area irrespectiveof fiber geometry (Figure 6c). Further increase in fiber length (c)results in fiber clumping as discussed with Figure 4. In this case,significant decrease in contact area is expected by the engage-ment of several nanofibers into bundles that are not as compliantas individual nanofibers. Thus, significant reduction in friction isexpected as observed in Figure 6a,b.The degree of height tolerance induced by lateral bending of

nanofiber may be approximated by the vertical displacement(δN) generated when fibers are under large deflection as shownin Figure 8a. With the assumption of negligible Coulombicfriction, δN can be obtained by numerically solving the followingequation derived from elastic beam theory:35

EId2ϕds2

þ FSf cos ϕ ¼ 0 ð2Þ

where ϕ(s) is the fiber position at length s, with boundaryconditions of ϕ(0) = 0,ϕ0(s = L) = 0, and FSf = τπac

2 is theadhesive friction force per single fiber. Figure 8b is a representa-tive result for D≈ 800 nm, which compares δN for short (1 μm)and long (3.3 μm) fibers when the applied force increases up tothe theoretical shear force defined by FSf.As seen in Figure 8b, long nanofibers can have a significant

normal displacement up to ∼50 nm. Structural hindranceimposed by the fiber gap may make this value smaller, forexample, ∼10 nm if an upper limit of δN is assumed to be givenat δL ≈ 2Δ (i.e., at δL = 200 nm for D = 800 nm). Meanwhile,short nanofibers have only negligible δN value up to ∼0.1 nm(corresponding δL≈ 15 nm). Thus, while a short nanofiber arraywill be detached from the substrate by further increasing thelateral load (i.e., exceeding the sum of the theoretical friction ofall fibers in contact), long nanofiber arrays are able to withstandhigher load by increased contact from neighboring nanofibers(which were not in contact previously) induced by this heighttolerance. Our previous discussion based on the schematics inFigure 7 is well supported by this explanation. This also suggeststhe increased brightness in the apparent contact area duringfriction measurement (Figure 6c) is due to the increased numberof contacting nanofibers.Optimum Fiber Length. As indicated in Figure 3 with black

boxes and Figure 6 with gray dotted circles, the fiber length atmaximum friction (Lmax) for a given fiber diameter is close to the

Figure 7. Schematic picture illustrating possible contact morphologiesof nanofibers in the bright and dark zones during friction measurementfor increased fiber lengths (a�c). Arrows indicate shear direction.

Figure 8. (a) Schematic of a cantilever beamwith a large deflection (δL)generating a vertical displacement (δN). (b) A representative resultshowing δN with respect to the applied load for long (3.3 μm, close tomaximum friction), and short (1 μm) fiber with D = 800 nm.

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fiber clumping condition (Lcrit). As can be envisioned fromFigure 7c, fiber clumping is detrimental to friction as was alsoexperimentally observed; maximum friction values for D ≈ 400and∼600 nm are obtained for unclumped structure just prior toclumping (Figure 6a). However, it is interesting to note that themaximum friction forces forD≈ 800 and∼900 nm are observedfor a partially clumped structure (Figure 6b). The morphologydifference between unclumped and partially clumped nanofibersbecomes less distinguishable as the diameter increases. Becauseof the narrow spacing between thick nanofibers, partiallyclumped nanofibers of large diameter are touching neighboringfibers with only a small deflection, and many of the nanofiber tipsare still individually exposed to contact the opposing surface(Figure 3, third row images for D ≈ 800, 900 nm), which is notthe case for smaller diameter fibers (Figure 3, third and fourthrow images for D ≈ 400, 600 nm). Because thick nanofibers arenot as readily bent as thin nanofibers (expected from eq 2),increasing the fiber length into a slightly clumped state maystill be helpful to obtain higher friction by achieving morecompliance for lateral bending to further tolerate the fiberheight distribution and maximize contact area. Further clump-ing into the bundled structure with increasing fiber lengthcauses a drastic decrease in the fiber compliance and contactarea, and friction decreases as observed. A more accuratemodel that can deal with the complicated structure of(partially) clumped fibers would be desirable for theoreticalconsideration of this condition.Optimum Fiber Diameter. As seen in Figure 6a and b, the

maximum friction forces increase with fiber diameter up to∼800 nm, but decrease for ∼900 nm. This is contradictory toJKR analysis, which predicts increased friction for larger dia-meter. This can be understood by extending our previousexplanation based on fiber bending and height tolerance:although larger fiber diameter increases the tip�contact area, itwill concomitantly decrease the gap between fibers, which resultsin geometrical hindrance for lateral fiber bending with enhancedtendency for clumping. Thus, decreased friction from D ≈800 nm to D ≈ 900 nm can be attributed to the reduced heighttolerance due to the restricted lateral bending by smaller gapbetween adjacent nanofibers.On the basis of our results in this study, it is found that the

optimum geometry of LDPE nanofiber arrays for macroscalefriction is D ≈ 800 nm and L ≈ 3 μm with slight clumping,representing a compromise between the tip contact area and fibercompliance. It should be noted that this structure is optimal forthe specific case of having a flat glass as the counter-surface. Theoptimum geometry would be different for rough surfaces,possibly with smaller diameter and larger spacing to toleratethe height distribution of the rough substrates.Comparison with Theoretical Friction Design Map. Spole-

nak et al.36 have proposed theoretical adhesion design maps byconsidering limiting conditions of fiber fracture, ideal contactstrength, fiber clumping, and surface adaptability. It is usefulto see where the nanofibers in this study are located in thedesign map. For this, a friction design map was developed bymodifying the equations used in ref 36 so that friction can betaken into account instead of adhesion. Because the side-contact is likely excluded as discussed previously, this rendersreasonable approximation of contact area simply by JKRanalysis defined by fiber tip. Thus, development of a frictiondesign map from the adhesion map becomes straightforwardas follows.

• Fiber fracture: Friction stress exerted on a single fiber (σf)can be expressed as:

σf ¼ FLfπR2

=τπa2oπR2

¼ τ

R2

9πR2Wð1� ν2Þ2E

( )2=3

e σfth ð3Þ

where friction force (FLf) is assumed to be from the adhesiveterm defined by contact area under zero normal load (πao

2),ν is the Poisson ratio, and σth

f is the theoretical fracturestrength of the fiber. If the bulk shear strength is approxi-mated to be similar to tensile strength (∼E/10), the abovecondition is rearranged as:

Rf racture g103=2τ3=2

E5=29πWð1� ν2Þ

2

!ð4Þ

• Ideal contact strength: Because stress concentrated on theactual contact area, Af, is considered for ideal contactstrength, the limiting condition simply reduces to themaximum allowed τ as

σc ¼ FLfAf

¼ τAf

Af¼ τ e σth ¼ W

b

� �

and thus:

τ eWb

ð5Þ

where b is the characteristic length of surface interaction.The above condition sets a maximum parameter value thatcan be used for τ, which is well above typical τ values,34 forexample, W/b ≈ 0.1 (N/m)/2 � 10�10 (m) = 500 MPa.

• Fiber clumping: Equation 1 is used for the clumpingcondition, which is rearranged into

Rclumping e

ffiffiffiffiffiffiffiffi1

4FN

s� 8Wλ3

Eð6Þ

where FN is the fiber number density, and λ is the aspectratio (=L/2R). It is noted that the number density is usedinstead of the area fraction employed in the previousstudy.36 Because the number of fibers per unit area is fixedwith the center-to-center distance of∼1 μm (area fraction ischanged with respect to fiber diameter), the number densityis used for comparing nanofibers geometry.

• Adaptability: Effective elastic modulus (Eeff)37 for vertical

fibers can be expressed as

Eef f ¼ CENR2

ARL

� �2

¼ CEFNR

2

4λ2ð7Þ

whereN is the number of fibers on the area of A, and C is thegeometrical factor (typically ∼10).37 The adaptability con-dition becomes:

Radaptability e

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEef fE

4λ2

CFN

sð8Þ

Eeff needs to be arbitrarily chosen to ensure the contact

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adaptability, which will depend on the surfaceroughness.

It is noted that, in this study, apparent contact strength (σapp)increases with increasing fiber diameter under fixed fiber numberdensity, different from the previous report36 where larger dia-meter fibers mean lower number density under fixed area

fraction. σapp can be written as

σapp ¼ FLfAapp

¼ FNσfπR2 ¼ πFNτ

9πR2Wð1� ν2Þ2E

!2=3

which gives R as the following form:

R ¼ 2E9πWð1� ν2Þ� �1=2 σapp

πτFN

� �3=4

ð9Þ

The friction design map can be obtained by drawing eqs 4, 6,and 8 with apparent contact strength given from eq 9. Figure 9shows the friction design maps for three different aspect ratios.For this design map, Eeff was set at 5 MPa based on the estimatedEeff values from eq 7, which ranges from 1 to 10 MPa formaximum friction condition (L = Lmax) corresponding todifferent diameters.In Figure 9a, for a low aspect ratio (λ = 1), it can be seen that

most of the short nanofibers are out of the optimum regionlimited by the surface adaptability (R should be less than∼100 nm to be in the optimum region). This indicates that lessstiff material than E ≈ 200 MPa is desired to ensure the contactfor low aspect ratio geometry. For higher aspect ratio at λ = 4 inFigure 9b, most diameters are within the optimum region. This isin good agreement with the observed maximum friction at thisaspect ratio in Figure 6a. Maximum friction stress values of up to∼200 kPa are achieved by increasing diameter to ∼700 nm,which is quite similar to the experimental values of∼300 kPa forthe actual contact fraction of 15% (∼5 N/0.15 cm2). Furtherincreasing the diameter along the E = 200 MPa line is limited bythe fiber clumping condition. This indicates that the design mapin Figure 9b also predicts the existence of optimal diameter at∼700 nm according to the fiber clumping condition. Furtherincreasing the aspect ratio (λ = 7) results in exclusion of allnanofiber geometry out of the optimum region by the clumpinglimit, which is reflected by experimental observation that frictiondecreases for clumped nanofiber bundles.As discussed above, experimental results show that a slightly

higher aspect ratio above the clumping condition, that is, partialclumping, is allowed to achieve higher friction for larger dia-meter. This may indicate the clumping limit needs to be lessstrict, allowing some range rather than a fixed value. For example,if the red line (clumping limit line) in Figure 9b is allowed toslightly shift upward, increasing fiber diameter (along the E =200 MPa line) to achieve higher friction is bound by the surfaceadaptability limit rather than the clumping limit, as experimen-tally shown by partially clumped structures exhibiting maximumfriction forces (D ≈ 800, 900 nm). Although the friction designmap in Figure 9 is based on very simplified models, it successfullydescribes the optimum geometry of LDPE nanofibers observed,and thus appears useful for selecting the design parameters closeto the optimum condition. Additional experiments involvingdifferent materials and substrate roughness are needed and areunderway to further verify the proposed friction map.

’CONCLUSION

In summary, ordered LDPE nanofibers were fabricated fromSiNW template synthesized by metal-assisted electroless etchingcombined with colloidal lithography using PSmicrosphere arraysprepared by float/transfer method. Macroscale friction measuredby a pulley setup was investigated for ordered LDPE nanofiber

Figure 9. Friction design maps for LDPE nanofibers for three differentaspect ratios (λ = 1, 4, 7). Eeff = 5 MPa was arbitrarily chosen for thesurface adaptability limit. Shaded regions enclosed by three lines (gray,fiber fracture limit; red, fiber clumping limit; blue, adaptability limit) ineach figure define the optimum design area to achieve at least 10 kPa offriction stress. Dashed line indicates E = 200 MPa. Dotted circlesindicate the experimental region in this study (fiber radius, R, rangesfrom 200 to 450 nm along the E = 200 MPa line).

11016 dx.doi.org/10.1021/la201498u |Langmuir 2011, 27, 11008–11016

Langmuir ARTICLE

arrays with various diameters and lengths with the same fiberdensity. It is shown that the optimum geometry for macroscalefriction against a flat glass substrate isD≈ 800 nm and L≈ 3 μmwith slight clumping, which maximizes the fiber contact witha compromise between the tip contact area and compliance.A friction design map was developed on the basis of the adhesionmap previously reported, and the predicted optimum regionmarked on this map was found to be in good agreement with theexperimental observation.

’AUTHOR INFORMATION

Corresponding Author*Tel.: (510) 643-7957. E-mail: maboudia@berkeley.edu.

’ACKNOWLEDGMENT

This work was supported by the Korea Research FoundationGrant funded by the Korean Government (KRF-2008-357-D00049)and National Science Foundation grants EEC-0832819 (throughthe Center of Integrated Nanomechanical Systems) and DMR-0804646.

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