Effect of Fiber Geometry on Macroscale Friction of Ordered Low

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Effect of Fiber Geometry on Macroscale Friction of Ordered Low-Density Polyethylene Nanofiber Arrays Dae 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 ABSTRACT: Ordered low-density polyethylene (LDPE) nanofiber arrays are fabricated from silicon nanowire (SiNW) templates synthesized by a simple wet-chemical process based on metal-assisted electroless etching combined with colloidal lithography. The geometrical effect of nanofibrillar structures on their macroscale friction is investigated over a wide range of diameters and lengths under the same fiber density. The optimum geometry for contacting a smooth glass surface is presented with discussions on the compromise between fiber tip
contact area and fiber compliance. A friction design map is developed, which shows that the theoretical optimum design condition agrees well with the LDPE nanofiber geometries exhibiting high measured friction.

’ INTRODUCTION Since the finding that gecko’s unique capability of climbing is due to the mechanical design of their foot-hair,1
3 great interest has been focused on fabricating micro- or nanofibrillar structures, so-called synthetic gecko foot-hair. With the development of recent nanofabrication techniques, a variety of gecko-like structures have been reported, including multilevel hierarchical fibers,4
10 angled fibers,9
14 and smart tip structures.14
20 Although there have been many reports on the fabrication of ever more complex 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 friction behavior. As reviewed recently,22 the basic issue of geometrical effect remains unclear, as there have been contradictory results regarding the effect of aspect ratio on the adhesion or friction of fibrillar surfaces. A systematic study was reported by Greiner et al.,23 where increased adhesion for higher aspect ratio of cross-linked polydimethylsiloxane (PDMS) microfibers was attributed to larger elastic dissipation during the pull-off process. However, Glassmaker et al.24 have shown that measured pull-off stresses of poly(vinyl-butyral) fibers were nearly independent of fiber lengths, although the energy dissipation increased linearly with increasing fiber length as expected due to the higher stored elastic strain energy in a single fiber. In some cases, increasing aspect ratio was observed to decrease adhesion or friction. Burton et al.25 measured the pull-off forces of poly(methylmethacrylate) and polyurethane acrylate nanofibers and reported lower adhesion for higher aspect ratio at various humidities. Zhao et al.26 have reported a decrease in adhesive strength with increasing height of multiwalled carbon nanotube arrays, which was attributed to the formation of canopy-like entangled surface layers as r 2011 American Chemical Society

MWCNT fibers become longer. In contrast, Qu et al.27 have more recently demonstrated a dramatic increase in adhesion and friction with increasing MWCNT fiber length. They have shown shear-induced alignment of top-entangled MWCNT fibers, which became more significant with increasing fiber length. Shear-induced fiber alignment was also demonstrated for polypropylene (PP) fibers both experimentally and theoretically by Lee et al.28 and Majidi et al.,29 respectively. Increase in shear strength during sliding was attributed to the occurrence of microfibers’ side contact on the substrate. However, the studies were limited to fibrillar structures with fixed diameter, length, and density. The purpose of this study is to systematically investigate the effects of geometrical factors in the nanofibrillar structures on their friction characteristics and to provide a useful reference for optimum conditions for high performance from a commercial polymer. Low-density polyethylene (LDPE) was selected in this study, which is the first report on fabrication and analysis of a synthetic gecko adhesive from this specific material. With this aim, a method for fabricating ordered polymer nanofiber arrays with varied aspect ratios is presented. Colloidal lithography30 combined with metal-assisted electroless etching of silicon31 is used to create silicon templates, which are then used to fabricate ordered arrays of LDPE nanofibers. It is then demonstrated that macroscale friction of these ordered nanofiber arrays is very sensitive to nanoscale changes in relation to their compliance and tip
contact area. A friction design map is presented by modifying the adhesion map previously reported,36 and a good Received: April 22, 2011 Revised: July 8, 2011 Published: July 20, 2011 11008

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

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

’ EXPERIMENTAL SECTION Fabrication. LDPE nanofiber arrays were replicated from silicon nanowire (SiNW) templates synthesized by metal-assisted electroless etching 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 acetone and isopropanol for 10 min each, followed by UV ozone cleaning for another 10 min. Cleaned Si chips were thoroughly washed with deionized (DI) water (18 MΩ) and then dried by N2 gas. Polystyrene (PS) microsphere (1 μm diameter, Duke Scientific Co.) 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 % in distilled water, ∼30 mL), which produced a disordered film of floating PS microspheres. A close-packed film of PS microsphere array was formed with the addition of a small droplet (a few μL) of a surfactant solution (sodium dodecylsulfate, 3 wt % in distilled water), and then transferred onto the clean silicon surface. A micromanipulating setup based on a stepping motor was used for controlled motion of the substrate during immersion and withdrawal (∼2 mm/s). Close-packed PS microsphere arrays on Si surfaces could be obtained over a large area (up to 4” waferscale). Microsphere arrays could be repeatedly obtained from the same solution, which further facilitates the fabrication process and minimizes consumption of the microsphere solution. (ii) SiNW templates
metal-assisted electroless etching: SiNWs were synthesized by incorporating colloidal lithography into Au-assisted electroless etching. The close-packed microspheres obtained by the above method were reduced in diameter by exposure to oxygen plasma (Plasma-Therm PK-12 RIE) at 30 W and 50 mTorr. A gold film (∼20 nm) was deposited on these surfaces by an e-beam evaporator (Thermoionics VE-100 vacuum evaporator). The microspheres were then removed by sonication in DI water to generate a patterned Au-coated substrate defined by areas not previously covered by the microspheres. Afterward, the substrate was immersed in an etching solution containing hydrofluoric acid (HF, 48 wt %), hydrogen peroxide (H2O2, ∼35 wt %), and DI water. Au-coated regions become 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 over a large area (HF/H2O2/H2O/AN = 2/1/5/2, volume ratio). For

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all cases, the etched Si surfaces were immersed and repeatedly washed with isopropanol before N2 drying. (iii) Ordered LDPE nanofiber arrays
nanomolding: LDPE nanofiber arrays were replicated using the above SiNW templates by a conventional melt process. Polycarbonate (PC) film (McMaster-Carr, ∼130 μm thickness) was first molded from SiNWs templates to generate intermediate nanohole templates by melt-process in a vacuum oven (∼300 °C, 1.5 h). SiNW templates were dissolved in the etching solution composed of hydrofluoric acid/nitric acid/acetic acid (4/3/3 volume ratio); then LDPE films (McMaster, ∼150 μm thickness) were replicated from the PC nanohole templates by melt-process (∼160 °C, 1 h under vacuum). By dissolving the PC nanohole templates in methylene chloride, the LDPE nanofibrillar structures were finally obtained with various diameters and lengths under a fixed center-to-center distance of 1 μm, as set by the diameter of the starting polystyrene microspheres. Structural Characterization. The SiNWs and LDPE nanofiber arrays fabricated according to the above processes were imaged in a scanning electron microscope (FE-SEM, JEOL JSM 6490LV) at 5
20 kV. The LDPE nanofiber arrays were coated by Au (∼5 nm) prior to SEM observation to minimize charging. Friction Measurements. The friction just prior to sliding was measured using a simple pulley setup with a glass slide as the countersurface under ambient condition (∼25 °C, ∼40% RH). Test samples were carefully placed on the clean glass surface without an intentional preload, 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) to minimize any possible variations in contact between LDPE nanofibrillar surfaces and the glass substrate during the sample loading for each test. Kapton tape was wrapped around one end of the sample to attach a thread, which is connected to the measuring cup through a pulley. The LDPE film was located near the glass edge to prevent undesired contact of the Kapton tape to the glass surface. Static friction force under the slight normal load (∼0.1 N/cm2) was measured by weighing the applied amount of water in the cup to initiate sliding. The glass substrate was cleaned with acetone before every measurement to remove any possible residues and contaminants. Apparent contact area was simultaneously observed during friction measurement by CCD camera through a 45° tilt mirror. Light from the illuminator was aligned parallel to the glass/ LDPE interface, and the optical fiber tip was covered with a black sheet to minimize the scattering.28 This provided a clear contrast between the contacting bright regions and the noncontacting dark regions, which was analyzed by an image software (ImageJ 1.42q).

’ RESULTS AND DISCUSSION Fabrication Process. The overall fabrication process of LDPE nanofiber arrays is schematically illustrated in Figure 1 with specific details provided in the Experimental Section. Figure 2 shows representative images obtained at various steps. A highly uniform array of microspheres is obtained by the float/transfer method (Figure 2a,b). The spheres are reduced in diameter by exposure to O2 plasma. The substrate is then coated with a Au film and sonicated to remove the microsphere, which generates patterned Au-coated surface defined by areas not previously covered by the microspheres (Figure 2c). These surfaces are then immersed in the electroless etching solution containing HF and H2O2. This etching is generally understood to be a localized electrochemical process,31 with the metal (Au in this study) acting as a local cathodic site and the underlying Si as a local anodic site. Thus, the silicon in contact with the Au region is etched much more rapidly. This results in SiNWs structure with the diameter corresponding to 11009

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LDPE nanofiber arrays are replicated using the above SiNW templates by a conventional melt-process. Figure 3 shows representative images of LDPE nanofibers with various diameters and lengths at a fixed center-to-center distance of 1 μm. With increasing fiber length, some of the nanofibers start bending and contacting each other (referred to as partial clumping regime, e.g., the third row for D ≈ 600 nm); all nanofibers then clump (all images on the fourth row) into bundles with further increasing length (referred to as clumping regime). This behavior is 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 given diameter. As will be discussed later, measured friction forces are limited by the fiber clumping, and thus this clumping condition should be considered carefully. For the fibers contacting at their tips, a simple equation can be derived from JKR theory and elastic beam theory:32 Lcrit ¼

Figure 2. SEM image of PS microsphere array (a) showing well-ordered structure in microscale. Optical image of a 4 in. wafer (b), showing uniform PS array. Au-coated patterned Si substrate defined by areas not previously covered by the plasma-etched microspheres (c). Slight disordering is induced during the plasma etching process. Au-coated region becomes a catalytic site during metal-assisted electroless etching, which results in well-defined vertical SiNW structures (45° tilt view) (d). Scale bar = 2 μm in all SEM images.

!1=3 ΔRt 3 E W

ð1Þ

where Lcrit is the maximum length before clumping, 2Δ is the spacing between adjacent fibers, Rt is the radius of curvature of the fiber tip, E is the Young’s modulus, and W is the adhesion energy of the fiber. As shown in Figure 4a, experimentally observed 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 is observed above a certain critical length, discussed in Figure 4. Images with black boxes correspond to nanofiber structures corresponding to maximum friction in Figure 6a,b (scale bar = 1 μm). 11010

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

prediction with Rt = R, fiber radius. This is a reasonable approximation supported by the experimental observations shown in Figure 4b and c, where the tip-curvature of nanofiber is measured from 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 a detailed procedure explained in the Experimental Section. It was often observed that friction force continuously increased during the repeated tests, then decreased (Figure 5a). This appears to be from fiber alignment and slanting by repeated shear, which might allow more efficient contact along the shear direction. Indeed, LDPE nanofibers were often observed to be aligned along the shear direction (i.e., pulling direction by measuring cup) as

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

shown in Figure 5c. After reaching a certain value, the friction decreases, which is attributed to significant plastic deformation of nanofibers. It remains a challenge to improve the mechanical durability of the LDPE fibers. Figure 6 shows the macroscale static friction of LDPE nanofiber arrays for various diameters and lengths with fixed 1 μm fiber-to-fiber spacing. For each sample, the highest value 11011

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Figure 6. (a) Friction forces of LDPE nanofiber arrays with various fiber 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 are shown separately in (b) for clearer demonstration of decrease in friction with further increase in fiber diameter. (Dotted lines in (a) and (b) are guides to the eyes.) Maximum friction forces are observed at the aspect ratio of ∼4 for all fiber diameters. The data points highlighted by dotted circles indicate the onset of partially clumped fibers. (c) Representative plot of apparent contact area % (apparent contact area/test area  100) with respect to fiber length (for D ≈ 800 nm). While ∼6-fold increase of friction force is observed, apparent contact fraction is not significantly altered between samples, typically 10
15% for all cases.

from repeated measurements before permanent deterioration (e.g., the peak in Figure 5a) was taken as the friction force for that sample. Each data point plotted in Figures 6a,b represents the average of five samples. At each diameter, a maximum friction force is observed. This maximum point increases with diameter up 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

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concentrated in the specific area under normal load with irregular shape. The bright zone in Figure 5b indicates that the contacting region increases from the initial to final state with a concomitant increase in brightness during the friction measurement. Although noncontacting dark regions are clearly distinguished from contacting bright regions by this method, detailed microscale information inside the contact region is unknown. For example, the actual number of LDPE nanofibers in contact with the counter surface may be smaller than that estimated from the area defined by the bright region. It is noted that any difference in brightness may be influenced by the actual number of contacting nanofibers or by morphological change in contacting nanofibers, for example, from tip to side contact. Thus, bright region is referred to as the apparent contact area. Despite the lack of detailed microscale information, it is useful for friction analysis in macroscale to visualize how the nanofiber contact is confined within 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 should be considered. It was found that this apparent contact fraction (fca = apparent contact area/test area  100) was not significantly different between samples, typically 10
15% for the sample size of ∼1  1 cm2 as shown in Figure 6c, which is similar to the previous results for polypropylene fiber arrays (∼11%).28 Distributions of each data point in Figure 6a and b may reflect the variation in contact area for each test. Despite this distribution, it is clear that the measured friction forces are strongly correlated with nanofibers geometry. JKR Analysis. Contact area generated between the nanofibers and the flat glass surface can be calculated using contact mechanical theories. The normal load per fiber, FNf, can be estimated to be ∼7 nN, assuming all nanofibers, with areal density, FN of ∼108 cm
2, contact within the apparent contact fraction of ∼15%, that is, assuming the measured apparent contact fraction fca to be that actual contact fraction, fc, at the moment just prior to sliding. Using the JKR equation33 with a tip radius of curvature Rt (≈R, fiber radius), the contact radius (ac) of LDPE nanofibers with diameter ranging from ∼400 to 900 nm is estimated to be ∼60
110 nm under the estimated normal load per fiber of ∼7 nN. Shear force before sliding of a single fiber is given by FSf = τ(πac2), and assuming shear strength τ ≈ 6 MPa for LDPE34 against glass, it is calculated to be ∼75
220 nN for the aforementioned range of fiber diameters. This corresponds to shear force of 1.1
3.2 N for 1 cm2 area with fc ≈ 15%, which becomes smaller for a smaller contact fraction. In contrast, the Coulomb friction per fiber, μsFNf, is estimated to be ∼2 nN, assuming static friction coefficient, μs, of about 0.3 for LDPE against glass34 and the normal load FNf of ∼7 nN. This then corresponds to the Coulomb friction of ∼0.03 N. Considering the relatively small Coulombic contribution, one may conclude that measured friction in this study is mostly affected by the adhesion between nanofibers and the glass surface. It should be noted that even though variations in material parameters, such as E, τ, and γs, may cause variations in the measured friction values, the experimental range of values in Figure 6 is reasonably well approximated by the above analysis when assuming the actual contact fraction is similar to the observed apparent contact fraction in Figure 6c. Although the contact area analysis based on JKR theory gives a reasonable approximation as discussed above, it does not explain the complex behavior of friction forces depending on fiber diameter and length. Specifically, any variation in friction forces 11012

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

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

according to fiber length cannot be explained. Intuitively, lateral flexibility of the nanofibers may play significant role in friction, for example, through further increase in contact area or fiber alignment during friction measurement. As mentioned in the Introduction, elastic dissipation and side-contact models have been typically considered for explaining changes in adhesion and friction with respect to fiber length (or more generally, aspect ratio). Elastic dissipation theory considers trapped and dissipated energy induced by fiber stretching in normal direction during the pull-off from the substrate.23,24 Thus, this theory explains only the effect of fiber geometry on normal adhesion and is not directly applicable to frictional properties. On the other hand, side-contact models28,29 describe bending of sufficiently long fibers, which brings them into a stable side-contact with the opposing surface when adhesion is high enough to exceed the elastic bending forces. However, as shown in Figure 5c, nanoscale plastic yielding is observed at the fiber tip, which clearly indicates fiber contact is restricted to the tip. An approach developed to estimate the side contact length28 predicts no side contact for all of the geometries of LDPE nanofibers tested in this study, which further supports that there is no side-contact induced by large bending 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 suggest nanofibers have slanted contacts from fiber bending and alignment by shearing during friction measurement as illustrated schematically in Figure 7. As compared to short nanofibers (a), long nanofibers (b) are much more compliant, and can be bent during the alignment by shear; thus they can more easily tolerate the nanoscale height distribution (distribution of nanofiber lengths) inside the apparent contact (bright) region, which further increases friction via contact area enhancement at the

microscale. However, even long nanofibers will not be able to overcome the macroscale height variation of the backing film (as observed by bright and dark regions in Figure 5b), which results in observation of similar apparent contact area irrespective of 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 engagement of several nanofibers into bundles that are not as compliant as individual nanofibers. Thus, significant reduction in friction is expected 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 shown in Figure 8a. With the assumption of negligible Coulombic friction, δN can be obtained by numerically solving the following equation derived from elastic beam theory:35 EI

d2 ϕ þ FSf cos ϕ ¼ 0 ds2

ð2Þ

where ϕ(s) is the fiber position at length s, with boundary conditions of ϕ(0) = 0,ϕ0 (s = L) = 0, and FSf = τπac2 is the adhesive friction force per single fiber. Figure 8b is a representative result for D ≈ 800 nm, which compares δN for short (1 μm) and long (3.3 μm) fibers when the applied force increases up to the theoretical shear force defined by FSf. As seen in Figure 8b, long nanofibers can have a significant normal displacement up to ∼50 nm. Structural hindrance imposed by the fiber gap may make this value smaller, for example, ∼10 nm if an upper limit of δN is assumed to be given at 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 array will be detached from the substrate by further increasing the lateral load (i.e., exceeding the sum of the theoretical friction of all fibers in contact), long nanofiber arrays are able to withstand higher load by increased contact from neighboring nanofibers (which were not in contact previously) induced by this height tolerance. Our previous discussion based on the schematics in Figure 7 is well supported by this explanation. This also suggests the increased brightness in the apparent contact area during friction measurement (Figure 6c) is due to the increased number of contacting nanofibers. Optimum Fiber Length. As indicated in Figure 3 with black boxes and Figure 6 with gray dotted circles, the fiber length at maximum friction (Lmax) for a given fiber diameter is close to the 11013

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

fiber clumping condition (Lcrit). As can be envisioned from Figure 7c, fiber clumping is detrimental to friction as was also experimentally observed; maximum friction values for D ≈ 400 and ∼600 nm are obtained for unclumped structure just prior to clumping (Figure 6a). However, it is interesting to note that the maximum friction forces for D ≈ 800 and ∼900 nm are observed for a partially clumped structure (Figure 6b). The morphology difference between unclumped and partially clumped nanofibers becomes less distinguishable as the diameter increases. Because of the narrow spacing between thick nanofibers, partially clumped nanofibers of large diameter are touching neighboring fibers with only a small deflection, and many of the nanofiber tips are still individually exposed to contact the opposing surface (Figure 3, third row images for D ≈ 800, 900 nm), which is not the case for smaller diameter fibers (Figure 3, third and fourth row images for D ≈ 400, 600 nm). Because thick nanofibers are not as readily bent as thin nanofibers (expected from eq 2), increasing the fiber length into a slightly clumped state may still be helpful to obtain higher friction by achieving more compliance for lateral bending to further tolerate the fiber height distribution and maximize contact area. Further clumping into the bundled structure with increasing fiber length causes a drastic decrease in the fiber compliance and contact area, and friction decreases as observed. A more accurate model that can deal with the complicated structure of (partially) clumped fibers would be desirable for theoretical consideration 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 to JKR analysis, which predicts increased friction for larger diameter. This can be understood by extending our previous explanation based on fiber bending and height tolerance: although larger fiber diameter increases the tip
contact area, it will concomitantly decrease the gap between fibers, which results in geometrical hindrance for lateral fiber bending with enhanced tendency for clumping. Thus, decreased friction from D ≈ 800 nm to D ≈ 900 nm can be attributed to the reduced height tolerance due to the restricted lateral bending by smaller gap between adjacent nanofibers. On the basis of our results in this study, it is found that the optimum geometry of LDPE nanofiber arrays for macroscale friction is D ≈ 800 nm and L ≈ 3 μm with slight clumping, representing a compromise between the tip contact area and fiber compliance. It should be noted that this structure is optimal for the specific case of having a flat glass as the counter-surface. The optimum geometry would be different for rough surfaces, possibly with smaller diameter and larger spacing to tolerate the height distribution of the rough substrates. Comparison with Theoretical Friction Design Map. Spolenak et al.36 have proposed theoretical adhesion design maps by considering limiting conditions of fiber fracture, ideal contact strength, fiber clumping, and surface adaptability. It is useful to see where the nanofibers in this study are located in the design map. For this, a friction design map was developed by modifying the equations used in ref 36 so that friction can be taken into account instead of adhesion. Because the sidecontact is likely excluded as discussed previously, this renders reasonable approximation of contact area simply by JKR analysis defined by fiber tip. Thus, development of a friction design map from the adhesion map becomes straightforward as follows.

FLf τπa2o = πR 2 πR 2 ( )2=3 τ 9πR 2 Wð1
ν2 Þ ¼ 2 e σ fth R 2E

σf ¼

ð3Þ

where friction force (FLf) is assumed to be from the adhesive term defined by contact area under zero normal load (πao2), ν is the Poisson ratio, and σfth is the theoretical fracture strength of the fiber. If the bulk shear strength is approximated to be similar to tensile strength (∼E/10), the above condition is rearranged as: ! 103=2 τ3=2 9πWð1
ν2 Þ Rf racture g ð4Þ 2 E5=2 • Ideal contact strength: Because stress concentrated on the actual contact area, Af, is considered for ideal contact strength, the limiting condition simply reduces to the maximum allowed τ as   FLf τAf W σc ¼ ¼ ¼ τ e σ th ¼ b Af Af and thus: τe

W b

ð5Þ

where b is the characteristic length of surface interaction. The above condition sets a maximum parameter value that can be used for τ, which is well above typical τ values,34 for example, W/b ≈ 0.1 (N/m)/2  10
10 (m) = 500 MPa. • Fiber clumping: Equation 1 is used for the clumping condition, which is rearranged into sffiffiffiffiffiffiffiffi 1 8Wλ3
ð6Þ Rclumping e 4FN E where FN is the fiber number density, and λ is the aspect ratio (=L/2R). It is noted that the number density is used instead of the area fraction employed in the previous study.36 Because the number of fibers per unit area is fixed with the center-to-center distance of ∼1 μm (area fraction is changed with respect to fiber diameter), the number density is used for comparing nanofibers geometry. • Adaptability: Effective elastic modulus (Eeff)37 for vertical fibers can be expressed as   NR 2 R 2 F R2 ¼ CE N 2 ð7Þ Eef f ¼ CE A L 4λ where N is the number of fibers on the area of A, and C is the geometrical factor (typically ∼10).37 The adaptability condition becomes: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eef f 4λ2 ð8Þ Radaptability e E CFN Eeff needs to be arbitrarily chosen to ensure the contact 11014

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fraction. σapp can be written as σ app

FLf 9πR 2 Wð1
ν2 Þ ¼ ¼ FN σ f πR 2 ¼ πFN τ 2E Aapp

which gives R as the following form:   1=2  σ app 3=4 2E R ¼ 9πWð1
ν2 Þ πτFN

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

adaptability, which will depend on the surface roughness. It is noted that, in this study, apparent contact strength (σapp) increases with increasing fiber diameter under fixed fiber number density, different from the previous report36 where larger diameter fibers mean lower number density under fixed area

!2=3

ð9Þ

The friction design map can be obtained by drawing eqs 4, 6, and 8 with apparent contact strength given from eq 9. Figure 9 shows the friction design maps for three different aspect ratios. For this design map, Eeff was set at 5 MPa based on the estimated Eeff values from eq 7, which ranges from 1 to 10 MPa for maximum friction condition (L = Lmax) corresponding to different 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 region limited by the surface adaptability (R should be less than ∼100 nm to be in the optimum region). This indicates that less stiff material than E ≈ 200 MPa is desired to ensure the contact for low aspect ratio geometry. For higher aspect ratio at λ = 4 in Figure 9b, most diameters are within the optimum region. This is in good agreement with the observed maximum friction at this aspect 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 for the actual contact fraction of 15% (∼5 N/0.15 cm2). Further increasing the diameter along the E = 200 MPa line is limited by the fiber clumping condition. This indicates that the design map in Figure 9b also predicts the existence of optimal diameter at ∼700 nm according to the fiber clumping condition. Further increasing the aspect ratio (λ = 7) results in exclusion of all nanofiber geometry out of the optimum region by the clumping limit, which is reflected by experimental observation that friction decreases for clumped nanofiber bundles. As discussed above, experimental results show that a slightly higher aspect ratio above the clumping condition, that is, partial clumping, is allowed to achieve higher friction for larger diameter. This may indicate the clumping limit needs to be less strict, allowing some range rather than a fixed value. For example, if the red line (clumping limit line) in Figure 9b is allowed to slightly shift upward, increasing fiber diameter (along the E = 200 MPa line) to achieve higher friction is bound by the surface adaptability limit rather than the clumping limit, as experimentally shown by partially clumped structures exhibiting maximum friction forces (D ≈ 800, 900 nm). Although the friction design map in Figure 9 is based on very simplified models, it successfully describes the optimum geometry of LDPE nanofibers observed, and thus appears useful for selecting the design parameters close to the optimum condition. Additional experiments involving different materials and substrate roughness are needed and are underway to further verify the proposed friction map.

’ CONCLUSION In summary, ordered LDPE nanofibers were fabricated from SiNW template synthesized by metal-assisted electroless etching combined with colloidal lithography using PS microsphere arrays prepared by float/transfer method. Macroscale friction measured by a pulley setup was investigated for ordered LDPE nanofiber 11015

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Langmuir arrays with various diameters and lengths with the same fiber density. It is shown that the optimum geometry for macroscale friction against a flat glass substrate is D ≈ 800 nm and L ≈ 3 μm with slight clumping, which maximizes the fiber contact with a compromise between the tip contact area and compliance. A friction design map was developed on the basis of the adhesion map previously reported, and the predicted optimum region marked on this map was found to be in good agreement with the experimental observation.

’ AUTHOR INFORMATION Corresponding Author

*Tel.: (510) 643-7957. E-mail: [email protected].

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