Article Cite This: Langmuir XXXX, XXX, XXX−XXX
Indentation versus Rolling: Dependence of Adhesion on Contact Geometry for Biomimetic Structures Nichole Moyle,† Zhenping He,† Haibin Wu,§ Chung-Yuen Hui,§ and Anand Jagota*,†,‡ †
Department of Chemical and Biomolecular Engineering and ‡Department of Bioengineering, Lehigh University, Bethlehem, Pennsylvania 18015, United States § Department of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14850, United States S Supporting Information *
ABSTRACT: Numerous biomimetic structures made from elastomeric materials have been developed to produce enhancement in properties such as adhesion, static friction, and sliding friction. As a property, one expects adhesion to be represented by an energy per unit area that is usually sensitive to the combination of shear and normal stresses at the crack front but is otherwise dependent only on the two elastic materials that meet at the interface. More speciﬁcally, one would expect that adhesion measured by indentation (a popular and convenient technique) could be used to predict adhesion hysteresis in the more practically important rolling geometry. Previously, a structure with a ﬁlm-terminated ﬁbrillar geometry exhibited dramatic enhancement of adhesion by a crack-trapping mechanism during indentation with a rigid sphere. Roughly isotropic structures such as the ﬁbrillar geometry show a strong correlation between adhesion enhancement in indentation versus adhesion hysteresis in rolling. However, anisotropic structures, such as a ﬁlm-terminated ridge-channel geometry, surprisingly show a dramatic divergence between adhesion measured by indentation versus rolling. We study this experimentally and theoretically, ﬁrst comparing the adhesion of the anisotropic ridge-channel structure to the roughly isotropic ﬁbrillar structure during indentation with a rigid sphere, where only the isotropic structure shows adhesion enhancement. Second, we examine in more detail the anomalous anisotropic ﬁlmterminated ridge-channel structure during indentation with a rigid sphere versus rolling to show why these structures show a dramatic adhesion enhancement for the rolling case and no adhesion enhancement for indentation.
1. INTRODUCTION The study of natural contact surfaces has been an area of interest for quite some time. These surfaces often have structures that produce desirable contact properties such as adhesion, static friction, sliding friction, and compliance.1−6 Many variations of biomimetic structured surfaces have been developed using elastomeric materials in attempts to capture the properties of natural surfaces in the lab.6−10 Various applications of these structures have been investigated such as climbing robots,11,12 high strength tapes and adhesives,13,14 skin friction applications for disposable razors,15 and wheels for in vivo medical imaging robots.16,17 Recently, a ﬁlm-terminated ridge-channel structure was developed which, when tested with a rigid spherical indenter, showed minimal adhesion and static friction enhancement but signiﬁcant sliding friction enhancement.18 These contact properties could be desirable in tire applications where low adhesion corresponds to better gas mileage and high sliding friction corresponds to better braking performance.19,20 A common experiment for measuring the adhesion of compliant materials is indentation by a rigid sphere, the socalled JKR geometry.21,22 Since the contact line can be identiﬁed as the tip of an external crack, moving the indenter into the sample/retracting it results in healing/separation of an © XXXX American Chemical Society
exterior crack that lies outside the perimeter of the contact region. In the simplest case, indentation of a homogeneous elastic-half space by a smooth spherical indenter, the resulting force−indentation data give a measurement of adhesion.21−24 Work done by external loads for opening the contact is usually signiﬁcantly greater than work done by adhesive forces to close the contact.22,24 Indentation is convenient due to the simplicity of the experimental setup and the fact that the spherical shape of the indenter mitigates alignment issues. While much has been learned about biomimetic structures using indentation, it is possible that contact geometry plays an important role in adhesion behavior when structures are both anisotropic and inhomogeneous, such as the ﬁlm-terminated ridge-channel structure. This raises the question of whether the properties measured during indentation experiments can be used reliably to predict behavior in geometries relevant to potential applications of the materials, such as rolling wheels. The rolling of an elastic cylinder on a ﬂat surface results in crack healing on the advancing edge and crack separation on the receding edge.25,26 For an isotropic or roughly isotropic Received: January 9, 2018 Revised: March 2, 2018
DOI: 10.1021/acs.langmuir.8b00084 Langmuir XXXX, XXX, XXX−XXX
and w125. Both ﬁbrillar and ridge-channel samples were cast to a thickness of approximately 700 μm, resulting in a very thick backing relative to dimensions of the surface structures. To ﬁlm-terminate the samples, PDMS was spun for 10 min at 3400 rpm and precured at 80 °C for 1 min and 45 s. Samples were then placed structure side down on the ﬁlm and cured in place at 80 °C for 2 h. Films obtained at these conditions were approximately 10 μm thick. For all stages, PDMS was used in a 10:1 base to cross-linker ratio, resulting in a Young’s modulus of approximately 3 MPa. A ﬂat sample that was cast on an area of the wafer which was not etched and underwent the same procedure as the structured samples, including ﬁlm termination, was used as a control. 2.2. Indentation Experiment. Indentation experiments were carried out in a manner similar to procedures detailed in previous studies.9,27 Figure 2 shows a schematic of the indentation setup, in
elastic material the interfacial hysteresis of this rolling geometry, deﬁned as the area-normalized work of crack separation minus the work of crack healing, should be similar to that found for a cycle of indentation by a hard spherical indenter (tested at similar rates of interfacial opening).21,25,26 Because the quantity extracted, adhesion hysteresis, is a scalar, one might argue that its value even for anisotropic structures is independent of rolling versus indentation. However, here we show that this is not the case. In this work, indentation experiments with a rigid sphere were performed on a roughly isotropic (ﬁlm-terminated ﬁbrillar) structure,9 an anisotropic (ﬁlm-terminated ridgechannel) structure,18 and ﬂat unstructured control samples to investigate the eﬀect of anisotropy on the mechanisms for adhesion enhancement. Next, rolling experiments were performed on the same structures. For unstructured control and isotropic structured samples we observed strong correlation between adhesion measured in indentation and by rolling. In striking contrast, we found adhesion measured by the two methods to be very diﬀerent for an anisotropic (ﬁlm-terminated ridge-channel) structure. Theoretical models were developed to explain the adhesion behavior observed.
2. EXPERIMENTAL SECTION 2.1. Sample Fabrication. Structured samples were made from poly(dimethylsiloxane) (PDMS, Dow Sylgard 184, Dow Corning) using molds created by microfabrication methods and ﬁlm terminating procedures as detailed in previous work.9,18 Figure 1 shows SEM
Figure 2. Schematic of indentation experiment. Sample substrates rest on a rigid backing while a rigid spherical indenter attached to a load cell is brought into contact to a set depth and is then retracted. which a glass indenter roughly 4 mm in diameter and coated with a self-assembled monolayer (SAM) of organosilane27 was brought into contact with the ﬁlm-terminated side of the structures. The SAM coating lowers the intrinsic adhesion of the materials, preventing damage of the ﬁlms. A variable speed motor (Newport ESP MFA-CC) brought the indenter to a depth of 20 μm at a rate of 5 μm/s. After the maximum indentation depth was reached, the indenter was retracted at the same speed until it was no longer in contact with the ﬁlm. The glass indenter was attached to a load cell (Honeywell Precision Miniature Load Cell), and distance was measured using a capacitive displacement sensor (MTI Instruments Accumeasure 9000). Three cycles were completed for each sample. The velocity of contact line motion provides insight into the mechanism of adhesion modulation. For the ﬁlm-terminated ridge/channel samples, this velocity was measured in two directions: parallel and perpendicular to the ridges. For the ﬁlm-terminated ﬁbrillar samples, the velocity was measured in the direction in which the contact line crosses over a ﬁbril. Further details are provided in the Supporting Information. 2.3. Rolling Experiment. A rolling experiment was designed as shown schematically in Figure 3. The backing of the ﬁlm-terminated or ﬂat control samples was attached to an aluminum wheel of 1.27 cm (0.5 in.) radius. Samples were placed such that ridges were perpendicular to the rolling direction of the wheel, as depicted in Figure 3a. The wheel had a shaft with an outer diameter of 2.38 mm (3/32 in.) inserted through a hole of only slightly larger diameter located in the center of the wheel, allowing for free rotation about the center axis. The ﬁlm side of the samples was brought into contact with a glass slide treated with the same SAM coating discussed in section 2.2. A load of 59 mN was applied normal to the glass slide, and the rolling attachment was connected to a load cell (Honeywell Precision Miniature Load Cell) measuring rolling forces, as shown in Figure 3b. Using a variable speed motor (Newport ESP MFA-CC), the slide was moved perpendicular to the normal load (and ridges) a distance of 4 mm backward and forward at a rate of 50 μm/s, creating a closed cycle. Cycles showed startup eﬀects and were run until they converged to steady state (constant force resisting rolling). For each experiment two steady-state cycles were recorded, and each sample was tested in two separate wheel locations. All samples were cut to widths of 6 mm
Figure 1. (a) Scanning electron micrograph showing the cross section of a ﬁlm-terminated ridge-channel structure. (b) Optical micrograph showing top view of a ﬁlm-terminated ridge-channel structure. (c) Scanning electron micrograph showing cross section of a ﬁlmterminated ﬁbrillar structure. (d) Optical micrograph showing top view of a ﬁlm-terminated ﬁbrillar structure. (scanning electron microscopy) and optical micrographs of the ﬁlmterminated ﬁbrillar and ridge-channel structures. The parameters used to deﬁne the PDMS structures are h (ﬁbrillar/ridge heights), t (ﬁlm thickness), d (ﬁbrillar/ridge width), and w (ﬁbrillar/ridge spacing). Brieﬂy, PDMS was cast into a silicon master with channels created using photolithography and deep reactive ion etching (DRIE). For the ﬁbrillar samples, molds contained hexagonally spaced square holes 10 μm in width and 30 μm in height, spaced at intervals ranging from 30 to 90 μm. Height and spacing were used to name the samples; for example, sample fh30w50 denotes a sample with a ﬁbril height of 30 μm and a ﬁbril spacing of 50 μm. The ﬁnal set contained (h30) w30, w50, w70, and w90. For the ridge-channel samples, mold channels were 1 cm in length, 40 or 30 μm in height, and 10 μm in width and were spaced at constant intervals ranging from 20 to 125 μm for various samples. Casting formed long thin PDMS ridges, with the ﬁnal set consisting of (h40 or h30) w20, w35, w50, w65, w80, w95, w110, B
DOI: 10.1021/acs.langmuir.8b00084 Langmuir XXXX, XXX, XXX−XXX
correlates directly with enhanced static friction, a property related to adhesive forces.9,28,29 The ﬁlm-terminated ridgechannel samples have nearly the same pull-oﬀ force as the ﬂat control for all spacings, indicating no adhesion enhancement for any of the samples tested. In contrast, the pull-oﬀ forces of the ﬁlm-terminated ﬁbrillar samples increase with spacing (up to about 3 times that of the ﬂat control), indicating an adhesion enhancement consistent with previous work.9 Examination of the contact region change during motion (Video S1 and Video S2 in the Supporting Information) gives insight into how the contact mechanics of both structures contribute to their adhesion properties. Figure 5 shows three sequential micrographs taken from the video at equal time intervals for ﬁlm-terminated ﬁbrillar sample fh30w90 (Figure 5a−c, 0.35 s time interval) and ﬁlm-terminated ridge-channel sample rh40w125 (Figure 5d−f, 0.55 s time interval), with arrows added to indicate important portions of the indenter− sample contact line. In Figures 5a.i and 5d.i we reproduce Figures 5a and 5d with the addition of drawn contours to indicate the contact line. For the ﬁlm-terminated ﬁbrillar samples, it is known that a crack-trapping mechanism contributes to their adhesion enhancement.9,27 The cracktrapping mechanism relies on periodic variation of energy release rate that sets up unstable jumps between energy release rate minima, which dissipate energy, leading to an enhancement in adhesion. A characteristic feature of crack growth under these conditions is that crack velocity is slow when the crack lies in compliant regions (here between ﬁbrils or ridges) and rapid as the crack jumps from one stable state to the next (jumps across a ﬁbril, for example). Crack trapping is illustrated by the movement of the contact line in the videos as the indenter is retracted. For the ﬁlm-terminated ﬁbrillar structure, the contact line goes through a series of alternating slow and abrupt movements as it approaches and crosses ﬁbrils, as shown in Figure 5a−c. In Figure 5a,b the two arrows indicate where the contact line (or crack front) slowly approaches, and is pinned (or trapped) near the edge of two ﬁbrils. In Figure 5c, the contact line has abruptly jumped across both ﬁbrils, becoming pinned at the ﬁbril between them. These jumps are caused by mechanical instabilities which rapidly release stored elastic energy from the ﬁbrils. This energy is lost, and thus the system requires a larger total energy input to open the crack than what is needed for the ﬂat unstructured control. In contrast, for the ﬁlm-terminated ridge-channel structures, the video shows that as the indenter is retracted from the sample the contact line shrinks much more
Figure 3. Schematics for rolling experiment. (a) Samples are attached to a free rolling wheel on the substrate side with ridges running perpendicular to the rolling direction. The ﬁlm side of the sample is exposed for contact. (b) Sample on wheel is brought into contact with the rigid glass surface as a ﬁxed normal load P is applied. Glass surface is moved perpendicular to the normal load.
(for the structured samples the ridges and channels were 6 mm in length) so that crack opening and closing areas would be equal across all runs. Experiments were also run with no sample attached to the wheel to estimate the internal friction of the rolling, as is discussed in more detail in section 3.2.
3. RESULTS AND DISCUSSION 3.1. Crack Trapping in Film-Terminated Structures under Spherical Indentation: Fibrillar versus Ridge Channel. Figure 4a shows force vs displacement (indentation depth) plots for typical indentation experiments on ﬁbrillar and ridge-channel ﬁlm-terminated structures as well as a ﬂat control. Because of the increased compliance,9 the force required for structured samples (compressive force is taken as positive) is smaller than that for the ﬂat control, at the same indentation depth/displacement. The pull-oﬀ force, which is the peak tensile force when retracting the indenter oﬀ the sample surface, is calculated as the diﬀerence between the baseline where there is no interfacial contact and the smallest (most negative) force value in Figure 4a obtained during pull-oﬀ. Its normalized value (normalized by pull-oﬀ force of the ﬂat control) is plotted against structure spacing w in Figure 4b. For an isotropic homogeneous elastomer this value can be used to calculate the work of crack separation, which is the work of adhesion, based on the JKR model for elastic adhesive contact.21,22 For the ﬁlm-terminated ridge-channel and ﬁbrillar structures, JKR theory does not apply, and so the work of adhesion cannot be obtained from pull-oﬀ force. Pull-oﬀ force is instead simply used as an indicator of interfacial strength. Previous works have shown an enhancement in pull-oﬀ force
Figure 4. Indentation results for ﬂat control, ﬁbrillar, and ridge-channel ﬁlm-terminated structures. (a) Force vs displacement plots for select ﬁlmterminated ﬁbrillar and ridge-channel structures as well as the ﬂat control. (b) Normalized pull-oﬀ force vs structure spacing; pull oﬀ force is normalized by the value obtained for the ﬂat control. C
DOI: 10.1021/acs.langmuir.8b00084 Langmuir XXXX, XXX, XXX−XXX
Figure 5. Still images from video of contact line motion. (a−c) Three consecutive images from the video at equal time interval of 0.35 s for ﬁlmterminated ﬁbrillar sample fh30w90 and (d−f) three consecutive images from the video at equal time interval of 0.55 s for ﬁlm-terminated ridge/ channel sample rh40w125. In panels a.i and d.i we have reproduced (a) and (b), additionally indicating the contact line, which is otherwise diﬃcult to see. Red arrows draw attention to certain sections of the sample−indenter contact line (scale bar indicates length of 200 μm).
Figure 6. (a) Plot showing how the contact line moves for a ﬁlm-terminated ﬁbrillar sample (fh30w90) along the lines drawn on the inset micrograph (distinguished by color). The data clearly show intermittent slow and rapid motion as the crack front traverses the region between ﬁbrils and jumps across them, respectively. (b) Plot showing how the contact line moves for a ﬁlm-terminated ridge-channel sample (rh40w125) along the lines drawn on the inset micrograph (distinguished by color). Crack motion is signiﬁcantly more rapid along the ridges than orthogonal to them, and there is no evidence of the rapid transitions seen for the ﬁbrillar samples.
previously, this mismatch in velocities results in the ridge coming out of contact before the channel separating it from the contact edge, leaving only the channel in contact and both adjacent ridges out of contact. This analysis of contact line motion highlights the fact that for a ﬁlm-terminated ridge-channel structure the opening of the crack in the direction parallel to the ridges eﬀectively eliminated the crack-trapping phenomenon observed in the indentation experiment. Speciﬁcally, the circular contact geometry is one in which the exterior crack opens from all directions. As contact shrinkage is accomplished primarily by the crack front motion parallel to the ridges, it essentially obviates the need to jump across barriers (that is crack front movement perpendicular to the ridges) that trap the crack between ridges. We turn next to the rolling geometry to investigate the extent to which indentation measurements of adhesion relate to those under rolling for the ﬁlm-terminated ridge-channel structure. (More data on indentation and rolling experiments on the isotropic ﬁlm terminated ﬁbrillar structure can be found in the Supporting Information.) 3.2. Indentation versus Rolling of Film-Terminated Ridge-Channel Structure. In order to compare quantitatively the adhesive response under indentation and rolling, we use the measured force−displacement data in either case to estimate the energy loss (a measure of adhesion) as the interface goes
smoothly and is not pinned at the ridges. Figure 5d−f shows that while, overall, the shrinkage of contact region is more or less isotropic, locally this is accommodated by diﬀerent contact front behaviors at the ridges and the channels. In Figure 5d the red arrows indicate the contact line of the ridges, which is retracting in the direction of the arrows. The blue arrow indicates the farthest contact line orthogonal to the ridge, which is in the area over a channel. The ridge between that point and the one indicated by the red arrow is already out of contact. Figure 5e,f shows the ridge contact receding further and then breaking contact altogether, all while the ﬁlm over the channel remains in contact. The principal feature of interest is the relatively smooth motion of the contact line, which shows the absence of crack trapping in the ridge-channel structure. Figure 6 shows graphically the diﬀerence in crack opening behavior for ﬁlm-terminated ﬁbrillar (6a) and ridge-channel (6b) structures. Figure 6a shows the displacement of the contact line on various paths along the ﬁbrils. It is evident from Figure 6a that for the ﬁbrillar sample the crack-front velocity goes through transitions of slow movement as the contact line approaches a ﬁbril to fast movement as the contact line jumps over a ﬁbril. Figure 6b shows the displacement of the contact line for ridge-channel structures, where the velocity of the contact line orthogonal to the ridges is signiﬁcantly slower than the velocity of the contact line along the ridges. As stated D
DOI: 10.1021/acs.langmuir.8b00084 Langmuir XXXX, XXX, XXX−XXX
it is clear that more force was required to move the wheel with the rh40w65 sample attached versus the wheel with the ﬂat control. The area normalized work of the cycle, ΔWtot, for the rolling experiments can also be calculated using eq 2, where ΔA is now the total sample area brought into and out of contact during a full cycle, P is the horizontal force to move the wheel during rolling, and δ is the displacement. However, now, in addition to the work of opening and closing the crack interface during rolling, there is also the work needed to overcome the internal friction of the wheel and the shaft it rotates about. We denote this work by Wf, which is the work of friction normalized by the distance traveled by the wheel. To estimate Wf, an experiment was run with a bare aluminum wheel in contact with a glass slide, where the work of adhesion is assumed to be zero, resulting in
through a closure/opening cycle. A typical plot of the data obtained during the cyclic indentation experiment is shown in Figure 7a for a ﬂat control and a 65 μm spacing, 40 μm ridge
Wtot,aluminum wheel = Wf =
∮ P dδaluminum wheel lualuminum wheel
where u is the distance the wheel is rolled and l is the width of the sample orthogonal to the rolling direction. The interfacial hysteresis for samples from the rolling experiments can then be calculated by subtracting the calculated frictional work such that ΔW = Wtot,sample − Wf = Figure 7. (a) Plot of indentation force vs displacement for indentation of a ﬂat control and ﬁlm-terminated ridge-channel structure rh40w65. Image i imbedded in plot shows optical micrograph of maximum contact area between indenter and ﬂat control while image ii shows the maximum contact area between indenter and rh40w65. (b) Plot of horizontal force vs displacement for rolling experiment on a ﬂat control and ﬁlm-terminated ridge-channel structure rh40w65. Image iii shows optical micrograph of ﬂat control contact area while image iv shows rh40w65 contact area during rolling. Both images show the leading edge at the bottom of the contact and the trailing edge at the top.
where Wtot,sample is the work of the cycle normalized by the area traversed by the wheel. Using eq 4, the interfacial hysteresis for the ﬂat control in the rolling experiments was found to be 64 ± 9 mN/m, which when including error is within 11 mN/m of the interfacial hysteresis calculated for the ﬂat control in the indentation experiments. Thus, for an isotropic unstructured material the indentation and rolling experiments give comparable values for interfacial hysteresis, as would be expected. Figure 8 plots adhesion enhancement ratios, ΔW̅ , obtained by dividing the interfacial hysteresis measured for the ﬁlm-
height ﬁlm-terminated ridge-channel structure (rh40w65). Interfacial hysteresis, ΔW, is deﬁned as ΔW = W + − W −
∮ P dδsample
where W is the work per unit area of separating or opening the interface and W− is the work per unit area of closing or healing the interface.27 It is calculated from the indentation data using the equation
∮ P dδ ΔA
where ΔA is the maximum contact area, P is the load on the indenter, and δ is the indenter displacement.27 As stated previously, none of the ﬁlm-terminated ridge-channel samples had a signiﬁcantly diﬀerent pull-oﬀ force from the ﬂat control. Interfacial hysteresis as computed by eq 2 also gives the consistent result that its value for all structured ridge-channel samples is very similar to the ﬂat control’s value of 88 ± 4 mN/ m, indicating again insigniﬁcant adhesion enhancement (Table S1). Detailed data and calculations from the indentation experiment including the ridge-channel interfacial hysteresis values can be found in the Supporting Information. Figure 7b shows a plot of force (horizontal force to move the roller) vs displacement during the rolling experiments for the same samples shown in Figure 7a. Upon inspection of the plot
Figure 8. Plot of enhancement ratios for indenter and rolling experiments for ﬁlm-terminated ridge-channel (rh40) samples. Enhancement ratio is deﬁned as the sample value divided by the equivalent ﬂat control value.
terminated ridge-channel structures by its value for the ﬂat control ΔW̅ = E
ΔWstructured sample ΔWflat control
(5) DOI: 10.1021/acs.langmuir.8b00084 Langmuir XXXX, XXX, XXX−XXX
Figure 9. Optical micrographs of crack opening for ridge spacings of (a) 35, (b) 65, and (c) 95 μm. Arrows indicate direction of crack front motion (parallel and perpendicular to the ridges). (d.i)−(d.iii) Schematic showing a sequence of steps for the 35 μm case: a straight crack front approaches a ridge, void nucleation leads to a pair of kinks that travel along the ridges, quickly moving the crack forward.
spacing increases. Thus, further study of the crack opening mechanism in the rolling geometry through both observation and modeling is needed and will be presented in the next section. 3.3. Model for Adhesion Enhancement in FilmTerminated Ridge-Channel Structures during Rolling. The resistance to rolling is primarily due to work required to open the trailing edge of the contact region, i.e., the opening and propagation of that external crack. The observed mechanism of crack opening for ﬁlm-terminated ridge-channel structures varied with ridge spacing, becoming more complex as spacing increased. Figure 9 shows optical images of crack opening for ridge spacings of 35, 65, and 95 μm, where the crack faces and areas in contact have been outlined and labeled for clarity. Videos of the rolling crack opening mechanisms can be found in the Supporting Information (Videos S3−S6). In most cases, the crack front moves both parallel (blue arrows) and perpendicular (green arrows) to the ridges at some point during the crack opening process. (The overall crack motion, i.e., the direction of rolling, is orthogonal to the ridges.) Only the smallest ridge spacing of 20 μm displayed crack movement exclusively in the perpendicular direction (with respect to the ridges). Figure 9 also details schematically (d.i−d.iii) the crack opening mechanism for the 35 μm spacing. As shown in Figure 9d.i, the contact line separating adhered and opened regions of the ﬁlm, i.e., the crack front, moves perpendicular to the ridges. As it approaches the next ridge, at some distance before it, a void nucleates ahead of the crack front at a point on the ridge, as shown in Figure 9d.ii. This nucleation point spawns two kink-like crack fronts that run parallel to the ridges in opposite directions. Motion of the crack along a ridge is much more rapid than the more sedate pace at which the crack front moves orthogonal to the ridges between them (an indication of crack trapping). Thus, as the kink sweeps through, the crack front quickly moves forward some fraction of the periodic distance between ridges. As shown in Figure 9d.iii, the kinks travel either
Also shown is the prediction of a model that will be described in further detail in the next section. Inspection of this plot shows strikingly diﬀerent trends and values for adhesion of the ﬁlm-terminated ridge-channel structures when tested in rolling versus indentation. Both the pull-oﬀ force and the interfacial hysteresis from the indentation experiments show no signiﬁcant adhesion enhancement for all spacings. In striking contrast, the interfacial hysteresis for the roller showed adhesion enhancement over the ﬂat control by ratios which increased as ridge spacing increased, reaching a value of more than eight for the ridge spacing of 65 μm, sample rh40w65, after which there was a gradual decrease with increasing ridge spacing. Other than the smallest ridge spacing of 20 μm, all samples showed an adhesion enhancement of more than 3 times that of the ﬂat control. These results clearly show that the ﬁlm-terminated ridge-channel structures produced an adhesion enhancement when used in a rolling geometry but produce no enhancement in the indentation geometry. That is, the value of adhesion energy obtained than indentation is vastly lower from that measured by rolling. This reiterates that for materials such as these, which are anisotropic, the contact geometry can play a signiﬁcant role in available adhesion mechanisms to the structure and thus the corresponding adhesion behavior. It is worth noting that supplementary rolling experiments were performed with the rolling direction now parallel to the ridges. In these experiments the crack trapping enhancement disappeared similar to the behavior for spherical indentation, as is detailed in the Supporting Information. This indicates there is a dependence of the adhesion behavior on the direction of rolling relative to the ridges. From section 3.1 it is understood why the ﬁlm-terminated ridge-channel structure does not have adhesion enhancement from crack trapping when tested with a spherical indenter. It is not clear why the adhesion enhancement is present in the rolling experiments, if the enhancement comes from crack trapping, or why the enhancement peaks at an intermediate ridge spacing of 65 μm and then steadily decreases as the F
DOI: 10.1021/acs.langmuir.8b00084 Langmuir XXXX, XXX, XXX−XXX
Figure 10. Schematic and ﬁnite element analysis for the model. (a) Schematic of the interfacial plane showing crack opening in only the perpendicular direction; the narrow strips shaded gray are ridges. (b) Finite element model with remotely applied K-ﬁeld. (c) Finite element model near the crack tip. (d) Plot of normalized energy release rate versus normalized crack tip position from ﬁnite element model.
crack, representing the trailing edge of the contact region. Let the remotely applied energy release rate be denoted by Gap and the work of adhesion (in the limit of very slow crack velocity) of a ﬂat interface be Wad. The local periodic structure modulates the remote energy release rate Gap, making the local energy release rate G a periodic function of crack location, x (x = 0 lies on the middle point of a ridge; see Figure 10c). We assume no energy dissipation in the elastic bulk but allow for ratedependent work of adhesion. By energy conservation, the remotely applied energy release rate Gap should be equal to the average energy release rate over a single ridge-channel interval ⟨G⟩, i.e., Gap = ⟨G⟩. The normalized local energy release rate with respect to Wad is denoted by Ĝ (x), such that
to the edge of the sample or until annihilation upon meeting another kink moving in the opposite direction. As channel spacings increase, such as for Figure 9b (65 μm spacing), a similar crack opening mechanism occurs as was shown schematically for the 35 μm spacing. Motion of the crack front perpendicular to the ridge direction brings the crack front toward the next ridge. Void nucleation occurs, and kinks are formed and begin to travel, with the parallel opening of the ﬁlm taking on a diﬀerent shape than in the smaller spacing, with the contact line over the ridge now slightly leading the contact line of the adjacent channel. With further increase in spacing, Figure 9c (95 μm spacing), once void nucleation occurs the mismatch in the kink velocity between the parallel opening of the ridge and channel becomes more severe. As the kink travels the contact line over the ridge dramatically leads the contact line of the adjacent channel, leaving long stretches of ﬁlm still in contact over just the channel region. This results in perpendicular opening of the contact line over the channel portion to occur orthogonal to both adjacent ridges, as indicated by the pairs of green arrows in Figure 9c. It has been shown that for a ﬁlm-terminated ﬁbrillar interface under the crack-trapping mechanism there is an increase in adhesion enhancement with increasing spacing, until an optimal spacing is reached.30 Beyond this optimal spacing adhesion enhancement begins to reduce, which has been interpreted as a transition from crack-trapping-control to void nucleation. Adhesion enhancement increases with increase in interﬁbrillar spacing whereas void nucleation attenuates it with increasing spacing. As a result, we anticipate that adhesion enhancement will be maximized at some intermediate value of spacing, as is evident in Figure 8 at a spacing of 65 μm. In order to illustrate the adhesion enhancement by crack trapping, a simple model with only perpendicular crack opening was developed. Modeling the transition to the void-nucleation controlled crack front motion requires a signiﬁcantly more complex model, which will be presented elsewhere. 3.4. Adhesion Enhancement by Crack Trapping. Figure 10a,b shows a schematic description of our model, in which we calculate the energetics of crack propagation of a semi-inﬁnite
Gap G (x ) f (x); Ĝ (x) = = ̂ Wad Wad
where f(x̂) is a dimensionless function with average value equal to one, which depends additionally upon geometrical parameters such as inter-ridge spacing, w. The function f(x̂) is periodic in x with a period of w. We use ﬁnite element simulations to determine f. Figure 10b shows a plane-strain ﬁnite element model used to simulate a crack front between regions adhered to and separated from the structured sample. The material (PDMS) is modeled as linearly elastic, which is known to satisfactorily represent the material constitutive behavior for such contact problems.22−24,26,27 To simplify calculations, we drive the system by subjecting the model to a remotely applied K-ﬁeld with K = 8μGap , where μ is the shear modulus of PDMS and we have the incompressibility condition ν = 0.5. Figure 10c shows the geometry of the ridgechannel structure near the crack tip. The local energy release rate, normalized by Gap, is plotted in Figure 10d for a number of diﬀerent ridge spacings. Further information on the ﬁnite element analysis is provided in the Supporting Information. For the sake of simplicity, we represent the ﬁnite element results by a simple sinusoidal function with the requisite period: f (x)̂ = (1 + a cos(2πx ̂ − 0.45π )) G
(7) DOI: 10.1021/acs.langmuir.8b00084 Langmuir XXXX, XXX, XXX−XXX
Langmuir where a is the crack trapping coeﬃcient, a numerical factor that depends only on the geometry of the ridge-channel structure. The shift factor of 0.45 was included to capture the position dependent energy minimum found using ﬁnite element analysis. As will be shown below, a is related to the adhesion enhancement factor due to crack trapping. Equation 7 has the beneﬁt of having a simple analytical form and is applicable to ridge channel spacing in the range of 20−65 μm as long as an appropriate value of a is chosen. In the rolling geometry, the interface opens at the trailing edge and closes at the leading edge. The net work that the external force needs to provide equals the work of separating the interface/crack at the trailing edge minus the work of closing the interface/crack at the leading edge. Generally, the work of crack opening greatly exceeds the work of crack closing. At the trailing edge, for continuous crack growth, the remote loading must be suﬃcient such that the opening crack propagates at the smallest value of G in eq 6, denoted by Gmin. That is, if we deﬁne aT as the ratio of the minimum to the average energy release rate, Gmin/Gap, then this condition is ̂T = Gap
T Gap T Wad
1 1 − aT
where v0 is a characteristic velocity and n is a constant which were found to be 9.28 m/s and 2, respectively, and H[.] is the Heaviside function, introduced because crack velocity vanishes if normalized energy release rate is less than one. Because the local energy release rate varies periodically, so will the velocity for a given value of remote applied K-ﬁeld. While eq 10 shows a rate eﬀect at the contact interface, the bulk of the material remains elastic, supplying the additional energy for dissipation. Equation 10 can be used to predict crack growth behavior for a given Gap. It can be integrated numerically or analytically; the analytical result is provided in the Supporting Information and was used to test the numerical model. Figure 11 contains plots generated by numerical integration for the 40 μm ridge spacing case showing Ĝ (x) versus x̂, x̂ versus normalized time, and v/v0 versus normalized time, with Ĝ ap values of 0.33, 1, and 7.5. Plots were generated by setting the crack trapping coeﬃcient a in eq 7 to the aT value obtained for the 40 μm spacing, 0.84, to model the crack trapping occurring at the trailing edge. Based on the criteria for crack growth, Figure 11a can be used to predict if the crack will remain stationary, grow to some length less than the period of the ridges and then stop, or grow continuously, for a given Gap. For normalized local energy release rate curves which have no values greater than 1, such as for Ĝ ap= 0.33, there will be no crack growth. For curves that have maximum values larger than 1 and minimum values less than 1, such as for Ĝ ap = 1, the crack will grow for some distance shorter than the period of the ridges and then stop. For curves with all values greater than 1, such as for Ĝ ap = 7.5, the crack will grow continuously over time. This crack growth behavior for the three Ĝ ap cases is displayed in Figure 11b, and the crack velocity behavior is displayed in Figure 11c. It is worth noting that the crack growth behavior in Figure 11b for the Ĝ ap = 7.5 case looks very similar to the experimental results plotted in Figure 6athis is because both represent crack trapping behavior. The eﬀect of the crack trapping parameter a on crack growth behavior is shown in Figure 12, where the natural logarithm of the average normalized velocity, vave/v0, is plotted versus Ĝ ap for four values of w (0, 20, 40, and 60 μm) which have respective corresponding a values (0, 0.36, 0.84, and 0.93, all taken from the aT analysis). The curves in Figure 12 converge to zero average velocity at some minimum value of the normalized average energy release rate. As a increases, the minimum normalized average energy needed for continuous crack growth increases, ranging from 1 for w = 0, a = 0 to over 14 for w = 60, a = 0.93. This illustrates the eﬀect of crack trapping on pushing the minimum average energy required for continuous crack growth to larger values. The data in Table 1 can be used to ﬁnd the theoretically predicted adhesion enhancement ratio related to the rolling experiments for a pure crack trapping case, Ĝ ap,model. Equations correlating aL and aT to w were ﬁt to the FEM data as detailed in the Supporting Information. Taking Gap,min from eq 8 for energy contribution from the trailing edge and taking Gap,max from eq 9 for the energy contribution from the leading edge, Ĝ ap,model can be calculated such that
where Ĝ Tap is the normalized average energy release rate at the trailing edge. The special case of aT = 0 corresponds to no crack trapping; in this case the local energy release rate equals the applied energy release rate, and there is no adhesion enhancement. For any 0 < aT < 1 eq 8 states that Gap must equal or exceed Wad/(1 − aT) in order for the crack to grow; hence, 1/(1 − aT) is interpreted as the crack trapping adhesion enhancement factor. For any given value of spacing, the minimum of the corresponding plot in Figure 10d is the value of 1 − aT for that spacing. Similarly, at the leading edge, the crack heals during rolling and the energy release rate must be smaller than the maximum value of G in eq 6, denoted by Gmax. If the ratio of the maximum to the average, Gmax/Gap, is denoted by aL, then at the leading edge ̂L = Gap
L Gap L Wad
1 1 + aL
where Ĝ Lap is the normalized average energy release rate at the leading edge. The maximum from the plot in Figure 10d is the value of 1 + aL for that spacing. Values of aT and aL are given in Table 1. Table 1. Finite Element Results for Crack Trapping Factor a w Gmax/Gap aL Gmin/Gap aT
20 1.34 0.34 0.64 0.36
40 2.42 1.42 0.16 0.84
60 3.54 2.54 0.07 0.93
80 4.70 3.70 0.03 0.97
100 5.78 4.78 0.01 0.99
120 6.90 5.90 0.00 1.00
3.5. Crack Growth Model. When the normalized local energy release rate exceeds unity, the crack has a ﬁnite velocity. On the basis of previous work,31 we represent the crack velocity−energy release rate function by a power-law v 1 dx = = H[(Ĝ − 1)n ] v0 v0 dt
̂ = Gap,model
T L − Gap Gap T L − Wad Wad
T Wad 1 − aT T Wad
L Wad 1 + aL L Wad
The value for of 24 mJ/m was taken from a prior study which calcualted the work of adhesion for crack healing between PDMS and a silanized glass indenter.24 The value for
DOI: 10.1021/acs.langmuir.8b00084 Langmuir XXXX, XXX, XXX−XXX
Figure 11. Plots showing the (a) normalized energy release rate versus normalized crack position, (b) normalized crack position versus time, and (c) normalized crack velocity versus time, all for Ĝ ap values of 0.33, 1, and 7.5. Plots were generated using values for a ridge spacing of 40 μm, a = 0.84, w = 40 μm, v0 = 9.2843 m/s, n = 2, and Wad = 60 mJ/m2.
ﬁlm thickness, also strongly inﬂuence crack-trapping and related phenomena. The eﬀect of geometrical parameters such as ﬁlm thickness can be anticipated by previous work.32 Essentially, thinner ﬁlms result in larger values of aT and aL and thus greater adhesion enhancement. The rate of contact opening is known to aﬀect the measured works of adhesion.33 In this work we took care to conduct experiments at slow enough rates that the indentation and rolling experiments are comparable and rate dependence of work of adhesion is negligible.
4. SUMMARY AND CONCLUSIONS It is commonly the role of simple experiments, such as indentation, to generate values for properties such as adhesion that can then be employed in other geometries (such as rolling contact). For isotropic ﬂat control samples, and an approximately isotropic ﬁlm-terminated ﬁbrillar structure, we ﬁnd that adhesion measured by indentation correlates very well with that measured in rolling, in both cases enhanced compared to the ﬂat control via a crack-trapping mechanism. However, for certain anisotropic (ﬁlm-terminated ridge-channel) structures there is a remarkable divergence between adhesion measured by indentation and by rolling. A study of this divergence has been the main focus of the work presented in this paper. We found that the ﬁlm-terminated ridge-channel structure had no adhesion enhancement when tested with a spherical indenter due to obviation of the crack trapping mechanism. However, adhesion enhancement and the crack trapping mechanism were recovered when tested in a rolling wheel geometry. For indentation, we showed through an analysis of the crackopening velocities that for the ﬁlm-terminated ridge-channel structures facile crack opening parallel to the ridges eﬀectively killed the crack-trapping mechanism. When the contact geometry was changed from indentation to rolling, the cracktrapping mechanism returned, resulting in dramatic adhesion enhancement which peaked at a ridge spacing of 65 μm, and then decreased as inter-ridge spacing increased past this point.
Figure 12. Natural logarithm of the normalized average crack velocity versus the normalized average energy release rate. Plot was generated using values of v0 = 9.28 m/s, n = 2, and Wad = 60 mJ/m2; w values are in μm.
WTad of 88 mJ/m2 was taken as the experimental rolling result for the ﬂat control sample, 64 mJ/m2, added to the WLad. The ﬁnite element predicted adhesion enhancement ratio is plotted in Figure 8 along with the experimental rolling interfacial hysteresis results. Comparison of the model prediction in Figure 8 to experimental rolling results shows that for small spacing the model predicts the experiment reasonably well, with a slight overprediction. This deviation between model and experiment is due to the void nucleation and kink formation which occurs over the ridges and limits pure crack trapping. The nucleation begins to attenuate the enhancement achieved by increasing ridge spacing, until at some critical point beyond a spacing of 65 μm it begins to decrease the adhesion enhancement to an extent which is greater than the spacing increases it. To further understand the eﬀect of nucleation, and to determine fully what causes the optimal adhesion peak at intermediate spacings, further modeling and ﬁnite element work which accounts for nucleation due to ridge stress must be done. In this work, we have focused on varying ridge periodic spacing. Other geometrical parameters, particularly the terminal I
DOI: 10.1021/acs.langmuir.8b00084 Langmuir XXXX, XXX, XXX−XXX
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To explain this enhancement peak, videos of the contact region during rolling were analyzed, and it was found that void nucleation occurred in most ridge-channel structures, which led to a mitigation of the crack-trapping eﬀect. An illustrative model for pure crack trapping on the ridge-channel geometries was developed which utilized ﬁnite element results to explain the enhancement phenomena observed in the rolling geometry. The model showed an increase in adhesion enhancement with an increase in ridge spacing, in reasonable quantitative agreement with experiments prior to void nucleation. Thus, the model shows that during rolling the measured increase in the adhesion as spacing increases is due to crack trapping. However, the model does not capture or explain the peak at 65 μm and the decrease in enhancement as spacings increase beyond that point. These results enforce that for anisotropic structures it is crucial to test their contact properties in geometries relevant to their proposed application, not just those which are experimentally convenient.
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.8b00084. A document detailing the speed analysis, video ﬁle list, tabled data from indentation vs rolling experiments, description of analytic model, description of ﬁnite element model, equations relating crack trapping coeﬃcients to ridge spacing, and justiﬁcation of model rate assumptions (PDF) Video S1 (AVI) Video S2 (AVI) Video S3 (AVI) Video S4 (AVI) Video S5 (AVI) Video S6 (AVI)
*E-mail [email protected]
Nichole Moyle: 0000-0002-6666-4841 Zhenping He: 0000-0002-3265-7539 Chung-Yuen Hui: 0000-0001-7270-6000 Anand Jagota: 0000-0002-0779-0880 Notes
The authors declare no competing ﬁnancial interest.
ACKNOWLEDGMENTS This work was supported in part by the Michelin International Corporation and by the National Science Foundation Award CMMI-1538002.
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