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Mechanism of Sliding Friction on a Film-Terminated Fibrillar Interface Lulin Shen,† Anand Jagota,*,‡ and Chung-Yuen Hui† Department of Theoretical and Applied Mechanics, Cornell UniVersity, Ithaca, New York 14850, and Department of Chemical Engineering, Lehigh UniVersity, Bethlehem, PennsylVania 18015 ReceiVed October 13, 2008. ReVised Manuscript ReceiVed December 23, 2008 We study the mechanism of sliding friction on a film-terminated fibrillar interface. It has been shown that static friction increases significantly with increasing spacing between fibrils, and with increasing rate of loading. However, surprisingly, the sliding friction remains substantially unaffected both by geometry and by the rate of loading. The presence of the thin terminal film is a controlling factor in determining the sliding friction. Experimentally, and by a simple model in which the indenter is held up by the tension in the thin film, we show how the indenter maintains a nearly constant contact area that is independent of the fibril spacing, resulting in constant sliding friction. By this mechanism, using the film-terminated structure, one can enhance the static friction without affecting the sliding behavior.
1. Introduction
2. Experimental Methods
A biomimetic film-terminated fibrillar surface has recently been shown to result in significantly enhanced adhesion and static friction.1-4 Curiously, while its adhesion and static friction can be modulated by changing geometry and rate of loading, the sliding friction remains nearly unchanged.4,5 In the first5 of this two-part study, we examined the connection between adhesion and static friction, and found that both operate by a crack trapping mechanism. Consistent with a model2,3 for adhesion enhancement by crack trapping, we found that rate and structural contributions are independent and coupled multiplicatively. In this work we examine the sliding friction mechanism. Specifically, we aim to understand why the usual connection between adhesion and friction6 found for static friction no longer operates in the case of sliding friction. This work has been motivated by studies of the contacting structures in insects and lizards in which a thin layer of microscale fibrillar structures plays an essential role in their adhering ability. Previous works on natural systems and recent progress toward its biomickry have been reviewed briefly in the companion paper.5 We begin in section 2 with a brief recapitulation of experimental methods; these too are described in greater detail in the first part of this study. Results of sliding friction experiments using a spherical or a cylindrical indenter are presented and discussed in section 3. Results on samples with terminal films using the cylindrical indenter are analyzed in some detail. To establish the essential role of the terminal film, we compare these results with experiments on two types of controls: flat unstructured samples and microfiber arrays without terminal films. We show that a constant friction stress model in which the indenter is supported by film tension can explain all the essential features of our experiments on film-terminated structures.
2.1. Fabrication of Specimens. Figure 1 contains scanning electron micrographs of typical samples. The fibril array is part of a poly(dimethylsiloxane) (PDMS) block, which has a thickness of 650 µm. The fibrils have a square cross-section 10 µm in width, are about 30 µm in length, and are arranged in a square pattern. The thickness of the terminal film was about 4 µm. With all other dimensions fixed as described above, a series of such fibrillar samples, with varying nearest center-to-center distance between fibrils (35 µm e w e 80 µm), were used for this study. We also studied control samples that either lacked both the fibril layer and the terminal film (flat unstructured controls, Figure 1b) or had the fibrils but no terminal film (roofless samples, Figure 1c). The fibrils, terminal thin film, and backing are all of the same material, PDMS (Sylgard 184, Dow Corning). PDMS is a highly elastic incompressible solid, the variety used had a Young’s modulus of 3 MPa. Fabrication procedures are described in detail in refs 1-3. 2.2. Friction Measurements. The behavior of microfibril arrays under shear is studied using the same apparatus first described in ref 4 (see the companion paper5). Briefly, a sample is placed on an inverted optical microscope. The sample is brought into contact with a glass indenter (either a sphere or a cylinder) by applying a compressive normal load P that is fixed by attaching the indenter to a mechanical balance (Ohaus 310D). We used a spherical indenter with a radius of 4 mm to study the roofless samples. Sliding experiments were also performed using a cylindrical indenter with a circular cross-section (radius 1 mm). The length of the cylinder was longer than the sample width. Since high adhesion between indenter and sample damages our fibrillar samples, indenters were precoated with a self-assembled monolayer (SAM) of n-hexadecyltrichlorosilane to reduce the interfacial adhesion. Details of this surface treatment can be found in the work of Glassmaker et al.1 Samples were driven by a variable speed motor (Newport ESP MFACC) and motion controller (Newport ESP300) at a fixed velocity. The imposed sample velocity varied from 0.05 µm/s to 0.3 mm/s. The frictional force was measured by a load cell (Honeywell Precision Miniature Load Cell model 31-50) attached on the balance arm in a direction parallel to the sliding motion.
* Corresponding author. † Cornell University. ‡ Lehigh University.
3. Results
(1) Glassmaker, N. J.; Jagota, A.; Hui, C.-Y.; Noderer, W. L.; Chaudhury, M. K. Proc. Natl. Acad. Sci. USA 2007, 104, 26, 10786–10791. (2) Noderer, W. L.; Shen, L.; Vajpayee, S.; Glassmaker, N. J.; Jagota, A.; Hui, C.-Y. Proc. R. Soc. London, A 2007, 463, 2631–2654. (3) Shen L., Hui C.-Y., Jagota A., J. Appl. Phys. Submitted for publication, 2008.
3.1. Sliding on Flat Control Samples. Figure 2 shows three optical micrographs of the contact (darker region) between a spherical indenter and a flat control. Figure 2a shows the initial contact, Figure 2b shows the contact just prior to sliding, and
10.1021/la803390x CCC: $40.75 2009 American Chemical Society Published on Web 02/05/2009
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Figure 1. (a) Scanning electron micrograph of a typical film-terminated sample. (b) Scanning electron micrograph of a flat control. (c) Scanning electron micrograph of a sample with fibrils but lacking the terminal film.
Figure 2. Optical micrograph of control sample under shear: (a) initial contact area before shear, (b) contact area before sliding, and (c) contact area during sliding. The letters ‘TL’ and ‘LD’ denote the trailing and leading edge, respectively, of the contact. The thin colored crescent at the trailing edge corresponds to a small depression behind the contact zone.
Figure 2c shows the contact during steady sliding. The letters ‘TL’ and ‘LD’ indicate the trailing and leading edge of the contact, respectively. Prior to the onset of sliding, the contact decreases in size, which we have previously explained4 using the theory of Savkoor and Briggs.7 Observations of sliding in the flat control samples suggest that there is uniform slip between the indenter and the substrate except at the very leading edge. If the entire contact region underwent uniform slip, then material points ahead of the leading edge would enter the contact zone smoothly at a rate given by the relative speed between the indenter and sample. However, on inspection of video recordings of the experiment, we found that the contact grows by small discrete additions at the leading edge that zip around the contour of the contact line. At the trailing edge, the deformation is similar to what we reported previously,4 that is, the surface forms a small valley immediately behind the trailing edge (see Figure 2c). In our previous sliding experiments on flat control samples,4 the surface immediately behind the trailing edge would periodically come into partial contact with the indenter, trapping a “dislocation”. As sliding proceeded, the dislocation would move toward the leading edge, shedding the shear jump across it. The shape of this region at the trailing edge would fluctuate between these two states. However, in this set of experiments, we did not observe such contact fluctuations. Instead, the deformation at the trailing edge during sliding maintained the constant shape seen in Figure 2c. We attribute these differences to the fact that our previous experiments were conducted on an indenter with a smaller radius (2 mm). (4) Shen, L.; Glassmaker, N.; Jagota, A.; Hui, C.-Y. Soft Matter 2008, 4, 618–625. (5) Vajpayee S., Long R., Shen L., Jagota A., Hui C.-Y. Langmuir 2009, 25, 2765-2771. (6) Johnson, K. L. Proc. R. Soc. London, A 1997, 453, 163–179. (7) Savkoor, A. R.; Briggs, G. A. D. Proc. R. Soc. London, A 1977, 356, 1684, 103–114. (8) Carpick, R. W.; Agrait, N.; Ogletree, D. F.; Salmeron, M. Langmuir 1996, 12, 3334–3340.
Figure 3. Sliding friction of the flat control and two film-terminated samples as a function of displacement rate.
In the first part of this study,5 we showed that static friction in the control samples increases with rate of loading, and that this rate dependence is similar to that exhibited by film-terminated samples. Figure 3 plots the sliding friction as a function of loading rate, showing little variation over the experimental range measured. 3.2. Sliding on Film-Terminated Samples. We have shown in the companion paper that friction experiments have two distinct phases. In the first phase there is no overall sliding between the indenter and sample. The shear force resisting sliding rises up to a peak (the static friction force), and then drops abruptly. After the abrupt load-drop, in the second phase, there is a transition to steady sliding, during which the shear force remains nearly constant, the sliding friction. The static friction is significantly larger than that of the flat control, and increases with increasing rate of loading. Sliding friction is lower and is substantially independent of shear rate. Figure 3 shows sliding friction of fibrillar samples with w ) 35 and 50 µm as a function of displacement rate. Each value on the plot represents the mean of five trials under the same conditions; the error bars are the standard deviation of the trials. (We have previously shown that sliding friction in the fibrillar samples is independent of fibril
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Figure 4. Typical force-displacement plots of the sample under shear (cylindrical indenter).
spacing, w, whereas static friction increases significantly with increasing w.4) The results in Figure 3 are obtained using a spherical indenter. The contact line underneath a spherical indenter during sliding evolves in a complicated manner, which makes it more difficult to analyze systematically. To study more systematically the deformation mechanism during sliding, we used instead a long SAM-coated glass cylindrical indenter with a radius of 1 mm, with its center line aligned perpendicular to the direction of relative motion. In this geometry, the contact axis lies parallel to the center line of the indenter, and the contact line undulates periodically along it. It has the advantage that one is able to follow the history of a representative line of fibrils during sliding. Our experiments studied one flat control sample and three fibrillar samples with nearest center-to-center distances w of 50, 65, and 80 µm. These samples were pulled at a rate of 5 µm/s. Force versus displacement curves for these samples are plotted in Figure 4. As with a spherical indenter, static friction increases with spacing (up to w ) 65 µm). For samples with w ) 80 µm, the static friction force carried by each fibril was large enough to tear the fibril away from the backing layer, limiting and ultimately reducing the static friction. However, the sliding friction remained unaffected by this damage since sliding friction forces were considerably smaller and were measured over an undamaged region of the sample. Figure 4 also shows that the sliding friction is much lower than the static friction and is practically independent of spacing, as observed with a spherical indenter.4 Using a synchronized video, pictures of the contact zone of all fibrillar samples were taken with a frequency of 1.5 Hz. To illustrate how we measure deformation of fibrils in the contact zone, we present several snap shots of a sample with w ) 80 µm. Figure 5a is the first still frame of the contact region. The top ends of the fibrils appear as fuzzy gray circles since the joint between the fibrils and the film is rounded (see Figure 1a). The small darker squares indicate the joints between the bottom of fibrils and the thick backing layer. In Figure 5a, no shear has been applied, so the squares and circles overlap. When sheared, they are mutually offset (see Figure 5b,c). By measuring the distance between them, one can determine the relative deflection of each fibril. To capture the micromechanics of fibril deformation before sliding, we analyzed a sequence of snap shots for a microfibril array with w) 65 µm to determine the deflections of one particular
Shen et al.
row of fibrils as a function of time. Since the focus of this paper is on sliding friction, results of deformation before sliding initiates are shown in the Appendix 1. The salient new finding is that overall sliding is preceded by some local microslip near the trailing edge. Figure 5b,c is from two consecutive frames capturing a sudden partial recovery of shear in one fibril (highlighted by a circle) near the leading edge of the contact during sliding. Because the sample is moved at a constant rate, the deformation history of each fibril entering the contact zone should be approximately the same. To study this deformation history, we marked a single row of fibrils aligned with the direction of sliding (highlighted in Figure 5b,c) with symbols 3-8. The deformation histories of these fibrils during sliding are followed by measuring the fibril deflections on a sequence of consecutive still frames. Figure 5b,c shows that the microscope view range is larger than the contact width, and the number of fibrils in a row is limited to 6. This means that the history of some fibrils can be traced only partially. Figure 6 plots the time history of deflection for fibrils 1-10. The time origin has been set to the time when fibril 1 just exits the contact zone. Note that fibrils 1-5 are already out of the leading edge, so complete deformation histories are not recorded. As each fibril enters the contact zone, it first shears at a rate determined by indenter velocity, shown by the slanted lines (fibrils 7 - 10). At a critical shear, a fibril suddenly slips backward (see also Figure 5b,c) and releases some of its elastic energy. This causes the previously buckled film between this fibril and the fibril to its right to unbuckle partially. These slip-stick events occur repeatedly with decreasing magnitude and are shown in more detail for a particular fibril in Figure 7. Note that, at the end of each slip/stick event, the fibril reloads at the approximately the same displacement rate of the indenter (5 µm/s), as shown by the slanted lines in Figure 7. As mentioned earlier, we expect that all the fibrils have a similar deformation history. That this is indeed the case is shown in Figure 8, which is a superposition of all 10 curves in Figure 6, with the time axis shifted so they all appear to load at the same time. Note in Figure 8 that, despite the stick-slip behavior of individual fibers, there appears to be a region of constant shear under the contact (B), and decaying shear in the noncontacting region after the trailing edge. Figure 8 shows a very important feature of our film-terminated microfibril array that is absent in an array without a continuous terminal film. It is that the shear force carried by the film outside the contact zone (behind the trailing edge) is nonzero and can actually be greater than the force carried by the fibrils inside the contact zone. The continuous film allows the shear force to be transmitted to noncontacting fibrils, as shown by the shear displacements of fibrils outside the contact zone in Figure 5b,c. The width of the noncontact region where shear force is nonnegligible is defined as the shear lag width. We have found that, with increasing spacing between fibrils, the shear deflection on each fibril and the shear lag width also increase, counteracting the decrease in fibril density and maintaining a constant total shear force.10 Two questions remain. Why is the sliding friction of our fibrillar sample independent of spacing (geometry) and displacement rate? In addition, why is the sliding friction for fibrillar samples so close to that of the flat control sample? In a previous work4 we suggested a scenario to explain why the sliding friction is independent of fibril spacing. We observed (9) Liu, J.; Hui, C.-Y.; Shen, L.; Jagota, A. J. R. Soc. Interface 2008, 5, 1087– 1097. (10) Shen L., Ph.D. Thesis, Cornell University, 2009.
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Figure 5. Images for the fibrillar sample with w ) 80 µm sliding at 5 µm/s under a SAM-coated cylindrical indenter with a 1 mm radius. (a) Initial contact and (b,c) two consecutive still frames illustrate the sudden recovery of one fibril located near the leading front (see inset circle).
that the film entering the contact is buckled. We suggested that the indenter motion is accommodated as a synchronization of film peeling at the leading edge of the buckled region and readhering at its trailing edge, that is, by traversal of a Schallamach-like wave. This mechanism implies that the energy needed to move the contact region a unit length must be equal to the difference of work of adhesion required to open and close a unit length of interface crack, that is,
Fsdu ) n(W+ - W-)ldu
(1)
where Fs is the shear force required to maintain sliding, u is the sample displacement, W+ and W- denote the work of adhesion corresponding to opening and closing a unit length of crack. respectively, l is the width of the specimen in the out-of-plane direction, and n is the number of buckles in the contact region. The number of buckles, n, is approximately given by the number of fibrils per unit area times the nominal contact area. In our experiments (see Table A1) the nominal contact area is almost independent of spacing. This means that n is inversely proportional to the spacing w. For our terminated film samples, W-, W+, and W+ increases with spacing w (see the work of Noderer et al.2). These counteracting influences could result in a spacingindependent sliding friction. However, in part 1 of this study, we showed that interfacial opening has a significant rate-dependence,
which ought to be inherited by the sliding friction according to eq 1; but this contradicts the data shown in Figure 3. 3.3. Sliding under Constant Interface Shear Stress. Recently, experiments of Carpick et al.8 have shown that, for PDMS, the sliding friction Fs is related to the actual contact area A by Fs ) τfA, where τf is a characteristic stress, which is a material constant that is independent of contact area. From our experiments, we can estimate how A varies with fibril spacing. To do this, we return to a previous set of friction tests that used a small indenter (2 mm in radius).4 These samples were compressed with nominally zero preload and sheared at a rate of 30 µm/s. There is some uncertainty about the exact contact region, especially for samples with wide spacing. We estimated the contact area by treating the contact region as a circle. As shown in Table A1, for w g 30 µm, the contact diameters during steady sliding are approximately independent of spacing (see Appendix 2 for details). Since Fs is found to be approximately independent of spacing, our results are consistent with the simple model that there is a constant sliding friction stress, τf. This model also explains why film-terminated samples have the same sliding friction as the unstructured flat control. There are two difficulties with the constant interfacial shear stress model presented above. First, Fs ) τfA applies only if the surfaces in contact are undergoing uniform slip, whereas, in our
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Figure 8. Deflection of a fibril as a function of its position relative to the indenter center line. These traces are a superposition of the curves shown in Figure 6 with position referenced to the center line of the indenter. The shear stress in the contact zone fluctuates slightly about a mean uniform stress.
Figure 9. Free body diagram of the section of the film in contact with the indenter. We show only the forces on the film. We have not indicated moments in the free body diagram. Moment balance is assumed to be satisfied independently. Figure 6. Deflections of 10 fibrils as a function of time. Fibril numbers correspond to those marked in Figure 5. The solid line has a slope of 5.0 µm/s, the indenter speed.
Figure 7. Stick-slip behavior of one fibril on a sample with w ) 50 µm. The solid line has a slope of 5.0 µm/s, the indenter speed.
case, the fibrils in the contact zone are undergoing stick-slip. This would mean that, at least locally, the film is not in perfect contact. If our argument is valid, then this region of imperfect contact, which we associate with local buckling of the thin film, must be small in comparison with the nominal contact area, which is plausible. A more difficult issue is why the nominal contact area is approximately independent of fibril spacing. At least before the static friction peak, the nominal contact area is found to increase with fibril spacing [Noderer et al.,2 and results in this paper]. The natural supposition is that, with increasing spacing, the normal compliance increases and, for a fixed normal load, so should the area of contact.
The deflection of fibers in the contact zone is observed to be very large, on the order of the length of a fibril. Liu et al.9 have shown that the normal compliance of such fibrils is so large that they lose most of their normal load-carrying capacity. Therefore, it is quite possible that the normal load is borne in some other manner. Two possibilities are (1) fibrils collapse on the backing layer so the thin film is in good contact with the indenter except near the top of the fibrils, where a small buckle must form, and (2) the indenter is held up by the thin film which is in tension. Note that, in both cases, the contact area is approximately independent of the spacing, since the normal compliance of the fibrils is practically infinite. In the first case, the contact area is determined by the thickness of the backing layer, while, in the second case, the contact area is determined by the tension in the film. We rule out the first hypothesis for our terminated film samples since contact of the fibrils and the film with the backing surface was not observed in experiments with film-terminated samples. It is also inconsistent with the picture of stick-slip in the contact zone. As will be shown in section 3.4, fiber collapse is observed for our roofless samples. As a result, the contact areas of these samples are also approximately independent of spacing (see Table A1). For our film-terminated samples, we hypothesize that case 2 applies. To examine this hypothesis more closely, we have developed a model for the case of a cylindrical indenter. The free body diagram of the thin film is shown in Figure 9. The origin, x ) 0, corresponds to the leading edge of contact, and x ) L is the location of the trailing edge. Because the radius R . L, for the purpose of force balance in the horizontal direction, we can ignore the curvature of the film. The shear force acting on the thin film in the contact zone (0,L) consists of the friction
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between the indenter and film (top surface, τf) and the reaction force exerted by the fibrils on the film. For x > L, the only force acting on the film is the shear force exerted by the fibrils and the tension force T in the film. The force acting at the leading edge, x ) 0, is approximately zero since the film to its left is buckled. Let u denote the horizontal displacement of the film. The film tensile stress, σ, is related to u by * du
σ)E
dx
τ ) k(u)u
(3)
The shear stiffness k is, in general, a nonlinear function of u. It is related to the shear stiffness of a single fibril K(u), by
k ) FK
(4)
where F is the number of fibrils per unit area. Analytic expressions for the dependence of K on u for nonlinear deformation of a beam can be found in Liu et al.9 In the shear lag region, that is, for x > L, force balance implies that
dσ ⁄ dx ) τ ⁄ h
(5)
where h is the film thickness. Substituting eqs 2 and 3 into eq 5 gives
E*h
du ) k(u)u dx2
x>L
Equation 6 can be reduced to the first-order equation
(6)
2ko
u skˆ(s)ds, ∫ 0 Eh
k(s) ) kokˆ(s)
*
(7)
where ko ) k(s ) 0) is the stiffness for small shear displacements. Note that ko is proportional to 1/w2 through its dependence on F, whereas kˆ is independent of w. Define a new function g() such that 2
2
) ∫0u skˆ(s)ds ≡ u g(u 2
(2)
where E* is the plane-strain modulus of the film. To simplify the analysis, the mechanical interaction between the fibrils and the film is model by a continuous elastic foundation with shear stiffness k(u). The foundation model replaces the discrete shear forces acting on the bottom film by a continuous shear distribution
2
du ) dx
(8)
Since kˆ is independent of w and positive, g is a positive function of u that is independent of spacing. The force T acting on the trailing edge (see Figure 9) is
To ) E*h
du dx
|
x)L
) -uo√E*hkog(uo2)
(9)
where uo is the displacement of the film at x ) L. For the special case of constant stiffness k(s) ) ko, g ) 1, so
To ) -uo√E*hk0
(10)
The governing equation for the film displacement in the contact zone can be derived in a similar way, it is
E*h
d2u ) k(u)u + τf, dx2
x ∈ (0, L)
(11)
The tension T can be decomposed into two parts:
To ) T∞ + ∆T
(12)
where T∞ is the force for an infinitely stiff film. Since fibrils deflect much more readily than the film stretches, the deflection of the fibrils in the contact zone is approximately uniform, so it is reasonable to assume T∞ . ∆T. Balancing forces in Figure 9, this uniform film displacement can be found using the condition
Figure 10. Typical force displacement curve of fibril sample without roof under shear.
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[
1 + kokˆ(uo)
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]
uo To ) τf τf L
(13)
Substituting eq 9 for To on the right-hand side of eq 13, we find
uo )
[
-τf L ⁄ √E*hko kˆ(u )L √g(uo2) + * o √E h ⁄ ko
( )]
where we have used the condition
≈
-τf L
√E*hkog(uo2)
(14a)
kˆ(uo)L
√Eh ⁄ ko
,1
(14b)
which is approximately satisfied in our experiments. Substituting eq 14b into eq 9 gives
To ≈ τf L
(15)
Since the normal load carrying capacity of the sheared fibrils is negligible, the applied normal load, N, is balanced by the vertical component of the film tension,
Figure 11. (a) Shear force versus shear displacement for a roofless sample with w ) 50 µm. The spherical indenter is sheared at a rate of 5 µm/s. (b-f) Sequences of images of the same sample. Labels b-f correspond to points in Figure 11a.
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N ) Tol sin θ ≈ Tolθ ) Tol
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L 2R
(16)
where l is the length of cylinder. Combining eqs 15 and 16 gives
L2 )
2RN τfl
(17)
Equation 17 shows that the constant interfacial shear model is consistent with the fact that the contact width during sliding is approximately independent of fibril spacing. In particular, it shows that it is primarily the film tension that resists the normal force exerted by the indenter. 3.4. Sliding on Fibrillar Samples Without a Terminal Film. It is interesting to compare the shear response of microfibril samples with and without terminal films or roofs. Roofless samples were made of the same material. The geometries of these samples were identical to the film-terminated samples, except that a wider range of w (from 20 to 125 µm) was used. All shear experiments were performed with a displacement rate of 0.5 µm/s using the same spherical glass indenter of radius 4 mm and the same normal force (much smaller than the shear force). Shear force versus displacement responses for eight samples with different spacings are shown in Figure 10. There are two main differences between samples with and without roofs. First, the shear force in the roofless sample initially decreases with shear displacement, reaches a minimum, then increases. Second, there is no static friction peak. A sequence of images of one sample (w ) 50 µm) is shown in Figure 11b-f. The labels (b-f) correspond to the points on the force-displacement curve in Figure 11a. Figure 11b is the image of the sample before shear is applied. The contact line is indicated by the dotted circle. Fibrils in the center of contact were slightly buckled as a result of the applied compressive load (see inset in Figure 11b). After a small amount of shear is applied, these fibers recover from their buckled state, releasing elastic energy, which performs work on the loading machine. Eventually, fibers bend in the direction opposite to the shear direction, and a force resisting motion develops. This adjustment is responsible for the initial decrease in shear force from b to c. Note that this phenomenon depends on the direction of initial buckling, which varies somewhat from sample to sample and depends on how the vertical load is applied. From c to e, contact increases with shear. In Figure 11d, the four fibrils in the center of contact (see highlighted region) are contacting the indenter on their lateral sides. (Recall that the fibrils have a square cross-section.) The region with lateral contact increases with shear displacement until e. After e, the contact zone as well as the number of fibrils making lateral contact remains unchanged with shear displacement. Normalized sliding friction of the roofless samples Fs/Fc is plotted against fibril spacing w in Figure 12, where Fs is the actual sliding friction force and Fc is the sliding friction obtained using a flat control sample under the same conditions. As a comparison, we include our previous data for film-terminated samples (with a spherical indenter of 2 mm radius) in the same figure.4 As shown in Figure 12, the sliding friction of filmterminated samples is independent of fibril spacing. On the other hand, sliding friction of the roofless samples follows a power law in spacing, that is,
Fs ⁄ Fc ∼ w-3⁄4
(18)
It is noted that the roofless sample with w ) 20 µm lies below the power law fit. For this case, fibrils do not make lateral contact with the indenter.
Figure 12. Comparison of sliding friction of fibril samples with and without a roof normalized by the sliding friction of a control sample.
Figure 12 again illustrates that the load-bearing capacity of the terminal thin film results in constant sliding friction. Some details of how sliding friction in the “roofless” samples scales with fibril spacing can be found in the work of Shen.10
4. Discussion and Summary In this paper we have examined the mechanism of sliding friction on a film-terminated fibrillar interface. Film-terminated samples show a static friction peak that increases with fibril spacing and loading rate for both spherical and cylindrical indenters. However, sliding friction remains approximately independent of spacing and rate. We have observed that, during sliding, the film is in partial contact with the indenter and contact regions are separated by buckled film. A typical fibril in the contact zone undergoes slip-stick. However, the average shear force in the contact region is constant. By direct measurement of fibril deflections, we determined the average force on a fibril at different locations. We found that the continuous film behind the trailing edge carries a substantial portion of the friction force. During sliding, the contact area of our film-terminated samples is approximately independent of fibril spacing. Fibrils shear so much that their normal stiffness becomes very small. The normal load is now balanced by the tension in the film. We show that this fact, with the assumption that sliding friction itself is constant, explains why overall sliding force is constant. Roofless samples do not exhibit a static friction peak. Sliding friction of these samples decreases with spacing. Its decrease with spacing is governed primarily by lateral contact of fibrils with the indenter. Acknowledgment. This work has been supported by a grant from the Department of Energy (DE-FG02-07ER46463). In addition, this work was performed, in part, at the Cornell NanoScale Science & Technology Facility, a member of the National Nanotechnology Infrastructure Network, which is supported by the National Science Foundation (Grant ECS 0335765). We would like to thank Ms. S. Vajpayee for help with SEM micrographs.
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Appendix 1 To capture the micromechanics of fibril deformation before sliding, we analyzed a sequence of snap shots for a microfibril array with w ) 65 µm to determine the deflections of one particular row of fibrils as a function of time. Time-dependent deflections of fibrils are shown as contours in Figure A1. The horizontal axis locates the bottoms of the fibrils. A vertical track follows the time-history of a given fibril as it enters the contact region, experiences shear, and leaves the contact. The slanted line
Shen et al.
indicates the position of the center of the indenter relative to the sample. The inverse slope of this line is the speed of the indenter relative to the sample. Before the static friction peak (t e 40 sec), fibrils III, IV, V, and VI are in contact with the indenter. Fibril V lies slightly to the left of the center line of the indenter. This figure shows that the fibrils in the leading edge (V and VI) have larger deflection than those in the trailing edge (III and IV), a phenomenon that we did not report in our previous paper.4 A careful reexamination of our previous data, which was obtained using a spherical indenter, also confirms this observation for fibril arrays without damage. The fact that fibrils near the trailing edge of the contact zone have lower deflections means that there must be some microslip occurring in this region. We attribute this to the fact that the film in this region is under tension and is peeled at a low angle. Microslip of the film is unlikely inside the leading edge of contact since the film ahead of it is under compression and buckles readily.
Appendix 2 Table A1. Nominal Contact Area of Film-Terminated Samples with Different Spacing during Sliding
Figure A1. Deflection development as a function of time of a row of fibers in an array with w ) 65 µm. The slanted black line indicates the location of the indenter center with respect to the sample.
fibril spacing w (µm)
effective contact diameter (µm)
0 (control) 20 35 50 65 80 110 125
160 360 490 550 520 480 495 470
LA803390X
