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Controlling Polymer Adhesion with “Pancakes” Alfred J. Crosby,* Mark Hageman, and Andrew Duncan Department of Polymer Science and Engineering, University of Massachusetts, Amherst, Massachusetts 01003 Received June 27, 2005. In Final Form: September 19, 2005 Topographical patterns are used to selectively tune the adhesion of polymers. Although nature has provided guidance, relatively little is known of how topographic patterns can be intelligently used not only to enhance adhesion but, more importantly, to tune adhesion. We demonstrate that properly designed, low-aspect-ratio posts can alter adhesion from 20% to 400% the value of conventional adhesion descriptors for nonpatterned interfaces. This control is not related to the magnitude of interfacial area but, rather, to altering the local separation processes at an interface by geometry. We establish general relationships that govern the interaction between material properties, pattern length scales, and the control of adhesion. These relationships provide insight into the mechanisms of adhesion for examples in nature, such as the gecko, while also providing concrete guidance for the future design of “smart” adhesives and coatings with nanoscale patterns.
Introduction Nature often uses topographic patterning to control interfacial interactions, such as adhesion and release. Examples range from the lotus leaf, to the gecko, to the jumping spider.1-5 Each example demonstrates that in addition to chemistry and material properties, geometric structure is also critical for optimizing interfacial design. Although nature has provided guidance, little is known of how topographic patterns can be intelligently used not only to enhance adhesion but, more importantly, to tune adhesion. To tune adhesion with patterns, we must understand how material properties and pattern structures interact. Can we discover general relationships that link interfacial properties, near-surface bulk properties, and pattern characteristics to guide the future design of tunable interfaces? This concept is timely as technology moves into realms where interfaces dominate function and performance, i.e., nanotechnology and biotechnology. In this paper, we describe how a square array of posts amplify or disrupt the adhesion mechanisms of an elastomer/glass interface. We link material structure with mesoscale patterns to regulate polymer adhesion and release. Our results demonstrate how geometrically simple, low-aspect ratio posts, or “pancakes”, decorating a polymer surface dictate the adhesion mechanisms of an interface. (Figure 1) These results focus on noncovalent adhesion mechanisms that control the contact and separation of soft elastomers with inorganic surfaces, such as glass. Understanding the role of topographic patterns for this material system will have a direct impact on the development of “smart” adhesives, antifouling coatings, and * Corresponding author: mail.pse.umass.edu.
Alfred
J.
Crosby,
crosby@
(1) Autumn, K.; Liang, Y. A.; Hsieh, S. T.; Zesch, W.; Chan, W. P.; Kenny, T. W.; Fearing, R.; Full, R. J. Nature 2000, 405 (6787), 681685. (2) Nun, E.; Oles, M.; Schleich, B. Macromol. Symp. 2002, 187, 677682. (3) Autumn, K.; Peattie, A. M. Integr. Comp. Biol. 2002, 42 (6), 10811090. (4) Autumn, K.; Sitti, M.; Liang, Y. C. A.; Peattie, A. M.; Hansen, W. R.; Sponberg, S.; Kenny, T. W.; Fearing, R.; Israelachvili, J. N.; Full, R. J. Proc. Natl. Acad. Sci. U.S.A. 2002, 99 (19), 12252-12256. (5) Kesel, A. B.; Martin, A.; Seidl, T. J. Exp. Biol. 2003, 206, 27332738.
Figure 1. (left) Schematic of spherical probe contacting polymer surface decorated with low-aspect ratio posts, or “pancakes”. (right) Contact area image of spherical probe and patterned surface. Dark areas are established regions of contact. Newton’s rings indicate deformation profile of surface.
imprint lithography technologies, while the fundamental concepts discussed in this paper reach beyond these limits. Experimental Methods Pattern Surface Fabrication. Patterned polymer surfaces for these experiments are fabricated using conventional methodologies for producing stamps in soft lithography.6 In brief, a template, or mold, is fabricated by exposing a photoresist-coated silicon wafer to an ultraviolet light source (OAI, Inc.) through a printed transparency mask (PageWorks, Cambridge, MA). After exposure, the unexposed regions are removed, as in conventional photolithography. For low-aspect-ratio features, further etching is not necessary, and the thickness of the photoresist defines the height of the pattern features. After the template is fabricated, uncured polymer is poured into the template, degassed, and cured. The cured elastomeric layer is peeled from the pattern template, thus producing an elastomeric layer with defined topographic patterns on its surface. This patterned material is immediately characterized using a contact adhesion test. Gradient patterned libraries are used to fabricate a single elastomer sample containing 100 patterns with unique combinations of feature radius and spacing. This approach ensures equal comparisons for different parameters without complications introduced by different processing, storage, or testing histories. Patterns discussed here are cylindrical posts arranged in square arrays. The radius of the post (rp) ranges from 25 to 250 µm, and the edge-to-edge spacing between posts (L) ranges from 50 to 500 µm. (6) Xia, Y. N.; Whitesides, G. M. Annu. Rev. Mater. Sci. 1998, 28, 153-184.
10.1021/la051721k CCC: $30.25 © 2005 American Chemical Society Published on Web 11/03/2005
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Figure 2. Resultant force (P) versus time for contact adhesion test on representative “pancake” patterned poly(dimethylsiloxane) surface. Contact area images are shown in time progression on top of plot. (inset) Resultant force versus applied displacement. Note hysteresis pattern dictated by surface pattern features. Controlled displacement rate ) 0.86 µm/s. rp ) 40 µm; L ) 110 µm. Contact Adhesion Tests. We use a spherical probe contact adhesion test to measure the adhesion of patterned elastomers.7-12 Since the pattern features are on the micrometer length scale, we quantitatively track the force-displacement history and monitor the interfacial area optically. The contact adhesion experiments are conducted on a customdesigned/fabricated instrument that includes a sub-nanometer precision inchworm actuator (Exfo Burleigh IW-812), a force transducer (Sensotec model 34, 50g), and a fully automated, inverted optical microscope (Zeiss Axiovert 200M). Every component is controlled through custom-written software within a National Instruments Labview environment. The test procedure involves attaching a hemispherical probe to the inchworm actuator and securing the patterned layer to the stage of the inverted optical microscope. The hemispherical probe contacts and subsequently separates from the patterned elastomer layer at a controlled displacement rate. During contact and separation, the applied displacement (δ), resulting force (P), and contact area (A ) πa2) are continuously monitored and recorded. All experiments are performed at room temperature, and the applied displacement rate is 0.86 µm/s. For each pattern, the maximum compression force is controlled at -10 mN and the radius of curvature (R) of the probe is 5 mm. Due to the different patterned surfaces, a constant maximum compression force does not ensure equal contact areas for each pattern. Consequently, the true contact area is measured and quantitatively accounted for in our analysis. Materials. The spherical probe is polished fused silica (R ) 5 mm), and the patterned elastomer surface is cross-linked poly(dimethylsiloxane) (PDMS) (Dow Corning, Sylgard 184). After the PDMS liquid is degassed, the PDMS layer is cured at 75 °C for 2 h. A 1:10 cross-linker to prepolymer mixture is used for the PDMS. This ratio produces a cross-linked PDMS layer with an elastic modulus, E, of 2.1 MPa as measured using microindentation. For nonpatterned PDMS/fused silica interfaces, the adhesion can be described by the critical energy release rate for interfacial separation, Gc. Nonpatterned contact adhesion tests conducted in our laboratory determined that an average, rateindependent value of Gc equal to 0.13 J/m2 adequately describes our materials. The pattern feature height (h) is 4 µm, and the total thickness of the PDMS layer is 1 mm. The experimental and measured parameters are presented in Table 1. (7) Barquins, M.; Maugis, D. J. Mec. Theor. Appl. 1982, 1 (2), 331357. (8) Ahn, D.; Shull, K. R. Macromolecules 1996, 29 (12), 4381-4390. (9) Chaudhury, M. K.; Whitesides, G. M. Langmuir 1991, 7 (5), 10131025. (10) Crosby, A. J.; Shull, K. R. J. Polym. Sci., Part B: Polym. Phys. 1999, 37, 3455-3472. (11) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301-313. (12) Maugis, D.; Barquins, M. J. Phys. D: Appl. Phys. 1978, 11, 19892023.
Table 1. Experimental and Measured Parameters for Samples and Contact Adhesion tests R (µm) h (µm)
5000 4
E (MPa) Gc (J/m2)
2.1 0.13
Results and Discussion Contact Adhesion Experiments with Patterns. To understand the complexity of results derived from the patterned surfaces, it is important to first understand a contact adhesion test on an individual pattern. A representative force-time curve for a contact adhesion experiment between a smooth, fused silica hemisphere and a patterned, PDMS surface is shown in Figure 2. Initial contact with the center post occurs at an approximate time of 5 s, and this is defined as displacement (δ) equal to zero. This initial contact causes a tensile force to be transferred across the new interface due to the presence of adhesion forces. As the single, center post is further compressed, the force-time relationship is linear as predicted for a constant displacement rate under low strains for cross-linked PDMS. The slope of the forcedisplacement curve here is given by the stiffness for an elastic flat punch contacting a flat elastic substrate: dP/dδ ) (2Erp)/(1 - ν2).11 As adjacent posts are contacted, the force-time and force-displacement curves show a nonlinear response. This nonlinear response is related to the change in stiffness as the contact area increases, similar to a conventional Hertzian response. At an approximate time of 18 s, a small jump in the force is observed as the interfacial area incorporates the PDMS surface between posts. This response is unique to lowaspect-ratio widely spaced posts. At an arbitrary force of -10 mN (constant for all of our experiments), the direction of the spherical probe is reversed and separation is induced. During separation, several instability transitions are observed in the force history where the force changes suddenly with little change in displacement. These sharp transitions are associated with the contact edge being pinned by the pattern features. As the applied energy overcomes the propagation barrier, the contact edge accelerates and changes contact area significantly in a very small time span. This pinning is associated with two features of our interfacial pattern: (1) the interface “jumps” from material between posts to top of the posts and (2) the individual posts completing their separation. The magnitude of the hysteresis created by these pinning
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events is attributed to both the pattern features and the natural hysteresis of PDMS/fused silica interfaces. To quantitatively compare the adhesion of different topographic patterns, we rely on two adhesion descriptors, Wadh and Pm. Wadh is the overall work of adhesion, or the energy per unit interfacial area that is dissipated between formation and failure of an interface10
Wadh )
IPdδ 2πIGc a da ) Amax Amax
(1)
where Amax is the interfacial area at maximum contact. This descriptor is useful for polymer interfaces whose adhesion is dictated by the history of the interface. For example, for nonpatterned interfaces, these changes may be associated with phenomena such as chemical complexes formed upon contact, rearrangement of surface molecules, or viscous losses in the near-interface molecules.13-20 On the basis of previous research on the adhesion of crosslinked PDMS,17-20 Wadh is an appropriate descriptor for this material. This descriptor is not equivalent to the thermodynamic work of adhesion. In fact, for a material interface whose adhesion is only described by reversible, surface thermodynamics, the value of Wadh will equal zero. Pm is the maximum applied tensile force during failure of an interface. Pm for a smooth interface of a spherical probe geometry (Pm,s) that can be adequately described with a constant, rate-independent value of Gc, is predicted by the following relation11
Pm,s )
3 πGcR 2
Figure 3. Contour map of normalized Wadh as function of pattern spacing (L) and pattern feature diameter (rp).
(2)
Interestingly, eq 2, as originally predicted by Johnson, Kendall, and Roberts (JKR),11 is independent of maximum interfacial area, hence, independent of applied pressure, and bulk elastic properties, e.g., E. Tuning Wadh and Pm with Patterns. For our material system, over the range of patterns within our library, we reproducibly tune Wadh from 100% to 400% of the value for a smooth interface. Figure 3 illustrates this effect as a function of pattern feature radius (rp) and feature spacing (L). In this plot, Wadh for the pattern is normalized by Wadh for a nonpatterned interface of PDMS and glass, as measured in our laboratory. The changes in color represent the changes in Wadh as a function of pattern. This contour map demonstrates the capability of pattern parameters for tuning critical adhesion descriptors. Similar to Wadh, Pm is tuned by the patterned surfaces from 20% to 200% the value of a smooth interface. Figure 4 illustrates this effect as a function of the pattern characteristics, rp and L. For both adhesion descriptors, this control is interesting since Wadh accounts for the interfacial area and Pm is independent of the maximum interfacial area for a conventional smooth interface. These results suggest that the specific shape, not the magnitude, of the interfacial (13) Creton, C.; Lakrout, H. J. Polym. Sci., Part B: Polym. Phys. 2000, 38, 965-979. (14) Creton, C.; Leibler, L. J. Polym. Sci., Part B: Polym. Phys. 1996, 34, 545-554. (15) Crosby, A. J.; Shull, K. R.; Lin, Y. Y.; Hui, C. Y. J. Rheol. 2002, 46 (1), 273-294. (16) Shull, K. R.; Ahn, D.; Chen, W.-L.; Flanigan, C. M.; Crosby, A. J. Macromol. Chem. Phys. 1998, 199, 489-511. (17) Chaudhury, M. J. Phys. Chem. B 1999, 103 (31), 6562-6566. (18) Chaudhury, M. K.; Owen, M. J. Langmuir 1993, 9 (1), 29-31. (19) Silberzan, P.; Perutz, S.; Kramer, E. J. Langmuir 1994, 10 (7), 2466-2470. (20) Perutz, S.; Kramer, E. J.; Baney, J.; Hui, C.-Y.; Cohen, C. J. Polym. Sci. B: Polym. Phys. 1998, 36, 2129-2139.
Figure 4. Contour map of normalized Pm as function of pattern spacing (L) and pattern feature diameter (rp).
area controls the level of adhesion developed between a soft elastomer and an inorganic surface. Therefore, tunable interfaces can be strategically designed by matching material properties to a unique pattern and shape. Although Figures 3 and 4 indicate that both pattern parameters, rp and L, are important, it is evident that the length scale of rp plays a prominent role relative to L in controlling adhesion in our experiments. In the following sections, we discuss the particular mechanisms through which rp dictates the adhesion descriptors for the interface. Pattern Control of Maximum Separation Force (Pm). In Figure 5, three regions (I, II, III) define the different interfacial separation processes. In region III, Pm becomes larger than the force to separate a nonpatterned interface as rp decreases. Hence, P h , or Pm for the pattern normalized by the Pm,s, is greater than unity. This deviation indicates that the post geometry affects the separation mechanism and invalidates eq 2. To understand this deviation, we consider our test geometry from two perspectives. First, at large radii our test geometry is simply a sphere separating from a flat plane (Figure 6a). In this geometry, the maximum separation force, Pm,s, for an elastic material is defined by eq 2. This force occurs when the contact radius equals a critical value, ac11
ac ) (9πR2Gc/8E*)1/3
(3)
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As rp becomes small, the separation mechanism changes from interfacial fracture which is described by eq 5 to a uniform separation dictated by the materials “optimal” strength, σth.22 As Gao and other researchers explain,22-25 the material becomes less sensitive to defects as the post diameter shrinks, and an “optimal” interfacial shape is more probable to form upon contact. The transition from separation controlled by eq 5 to that controlled by the optimal strength scales as a function of a material-defined length scale, d
8E*Gc
d)
(6)
πσth2
On the basis of the numerically calculated data presented in Gao and Yao,22 we define an effective energy release rate, Ge
Figure 5. Normalized maximum separation force (P) versus feature radius (rp). Large data points correspond to patterns with L ) 450 mm. Small data points correspond to patterns with all other values of L. Trend lines for regions I, II, and III are defined by eqs 10, 8, and 9, respectively. Inset shows region II trend plotted on log-log scale.
Ge ) Gc
( )[
( ( ) )]
rp rp 1 - exp d d
-1/2
2
(7)
and substitute this quantity into eq 5 to quantitatively describe the separation process for all rp < ac. Accordingly, the maximum separation force in region II is
P h)
(){}
Pm,f rp )2 Pm,s ac
3/2
Ge Gc
1/2
(8)
For rp larger than ac, the separation mechanism is influenced by the stress distribution prescribed by the spherical geometry of the probe. Here, we describe region III by an empirical relationship
P h )1+
Figure 6. (a) Schematic of contact between spherical probe and large diameter pancake. Similar to sphere on flat contact. (b) Schematic of contact between spherical probe and small diameter post. Similar to flat punch cylinder contact.
which is given by the JKR theory. E* is defined as 2
2
1 - υ1 1 - υ2 1 ) + E* E1 E2
(4)
where Ei and νi are the elastic modulus and Poisson’s ratio of the two respective contacting bodies. If rp is smaller than ac, then the maximum separation force is no longer described by eq 2 since ac cannot be achieved. In this regime, the separation mechanism is described by the geometry of Figure 6b. Here, the radius of curvature of the spherical probe is “infinite” relative to the post radius, and the post can be considered as a flat, cylindrical punch contacting a flat substrate (R ) ∞). The maximum separation force for a flat, cylindrical punch (Pm,f) separating from a flat substrate is related to the radius of the punch, rp, by21 3
Pm,f ) (8πE*rp Gc)
1/2
(5)
(21) Maugis, D.; Barquins, M. J. Phys. D: Appl. Phys. 1983, 16, 18431874.
( )( ( | ) ac rp
3
2
Ge Gc
1/2
rp)ac
-1
)
(9)
For both regions II and III, the trend lines given by eqs 8 and 9 are plotted in Figure 5 with d ) 32 µm. From independently measured values, ac ) 160 µm. If the cylindrical post has a high aspect ratio (h/rp . 1), then eqs 8 and 9 apply over all values of rp. For our experiments, h/rp , 1; therefore, for small values of rp, interfacial contact is established on both the top of the post and with the material surrounding the base of the post (Figure 7b). This behavior defines region I in Figure 5. Here, the pattern feature presents a compressive zone beneath the features. This compressive zone causes Pm for spherical contact (eq 2) to be offset by the force required to compress a post with dimensions (h/rp), and the normalized maximum separation force for region I is
P h )1-
( )( ) 3Rh rp 2ac2 ac
(10)
The trend line defined by eq 10 is plotted in Figure 5. As the post height increases, the intersection of eq 8 and eq 10 decreases to smaller values of rp, and Pm decreases to lower values as region II is extended. Note that this behavior is valid for elastic materials with compression forces less than the force required to buckle the posts. (22) Gao, H. J.; Yao, H. M. Proc. Natl. Acad. Sci. U.S.A. 2004, 101 (21), 7851-7856. (23) Ghatak, A.; Mahadevan, L.; Chung, J. Y.; Chaudhury, M. K.; Shenoy, V. Proc. R. Soc. London, Ser. A 2004, 460, 2049, 2725-2735. (24) Jagota, A.; Bennison, S. J. Am. Zool. 2001, 41 (6), 1483-1483. (25) Jagota, A.; Bennison, S. J. Integr. Comp. Biol. 2002, 42 (6), 11401145.
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Figure 7. (a) Contact between spherical probe and only top of posts. Spacing between posts is smal. (b) Contact between spherical probe and posts. A second plane of contact is established between posts.
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known and an average asperity size (R) is defined for a contacting surface, then the length scales defined by ac and d guide the design of topographic patterns. Pattern Control of Adhesion Hysteresis (Wadh). Wadh is the magnitude of adhesion hysteresis as defined in eq 1. In Figure 8, for large posts Wadh approaches the value for a smooth sphere separating from a smooth, planar substrate. Hence, the normalized value, W h adh, approaches unity. As the radius of the post decreases from infinity, the value of Wadh increases until a critical radius is reached. For rp g ac, the stress distribution imposed by the spherical geometry influences the separation mechanism as described earlier for the maximum separation force. In this regime, we define, similar to eq 9, the empirical relationship
W h adh ) 1 + 2
Figure 8. Normalized Wadh versus feature radius (rp). Large data points correspond to patterns with L ) 450 µm. Small data points correspond to patterns with all other values of L. Trend line defined by eq 11.
Equations 8, 9, and 10 define the normalized Pm for all regions in Figure 5. If material properties, Gc and E, are
( )[ ( | ) ] ac rp
3
2
Ge Gc
1/2
rp)ac
-1
(11)
The trend line defined by eq 11 is also plotted in Figure 8 with d ) 32 µm. Equation 11 defines the overall work of adhesion for our patterns when the posts have proper spacing and aspect ratio to ensure independent and nonbuckling responses. As rp < ac, a transition occurs where the response of multiple posts is coupled. This coupling prevents eq 11 from properly describing the data. Coupling between Adjacent Features. Feature radius plays a prominent role in controlling Wadh and Pm, but interfacial tuning is also realized in the coupling of neighboring features. For our experiments, this control is demonstrated by varying L, the nearest-neighbor spacing for a square array. For Pm, an optimal spacing is not observed, but the ability of L to control the specific mechanism of separation is seen. Specifically, due to the low aspect ratio of our posts when L is sufficiently large and rp is small, interfacial contact is established both on the top of the posts and in the material surrounding the base of the posts (region I). This “second plane of contact” causes the separation mechanism to return to a spherical/ flat geometry; thus eq 10 describes the maximum force of separation in these cases. If L is small, contact with neighboring posts prevents contact with the material
Figure 9. Normalized Wadh as a function of post radius, rp, for two differently spaced patterns: L ) 50 µm and L ) 500 µm. The patterns with L ) 50 mm have significantly lower values of Wadh. Maximum contact area images for L ) 500 mm (top) and L ) 50 µm (bottom) are on right.
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between posts. This behavior allows eq 8 to describe the maximum separation force. For Wadh, L plays a more significant role. An inherent assumption of the model that describes the increase in Wadh with decreasing diameter is the ability of individual features to act independently. If the features do not act independently, the separation of the individual features is coupled and the overall separation mechanism is the same as a sphere separating from a flat substrate. Since our features have a low aspect ratio and they are connected by cross-linked PDMS at the base of the posts, the deformation of one post is coupled to neighboring posts. Therefore, when neighboring posts or the material surrounding the base of a post is contacted, the separation mechanism is altered and eq 11 does not apply. This influence of neighboring features is measured directly in Figure 9. Here, when only the center feature is contacted, Wadh increases with decreasing diameter. If adjacent posts couple during separation, Wadh is not altered and behaves similar to a nonpatterned interface. Neighbor interactions also provide unique control of interface history, or the path of the interfacial edge during formation and separation. This history control is evident in Figure 2 where the interfacial growth is dictated by the pattern, compared to the conventional axisymmetric interfacial growth of a spherical contact geometry. As the interface is separated, the force travels through several maxima and minima before final separation is achieved. As described earlier, these maxima and minima are attributed to the pinning of the contact edge by the pattern features. Changes in the pattern parameters alter the location and magnitude of these maxima and minima. In general, this control of local maxima and minima leads to “pockets” of hysteresis (see Figure 2 inset) which can be used to dictate when energy dissipation will occur for a given interface. This control of history may have applications in the development of warning systems for interfacial failure. Additionally, the frequency of these maxima and minima introduced by the spatial frequency of the neighboring features is hypothesized to impact that amplitude of interfacial viscous dissipation. The adhesion of polymer interfaces has long been related to the viscoelastic dissipative mechanisms of the polymer components.8,10,12,15,16,26,27 Our results on elastic interfaces demonstrate that the stress transfer can be altered periodically by controlling the spatial density of pattern features. This periodic stress transfer should alter viscoelastic dissipation near the polymer interface, and we are currently investigating these effects. Summary These results strongly confirm the existence of key relationships that link pattern dimensions to material (26) Barquins, M.; Maugis, D. J. Adhes. 1981, 13, 53-65. (27) Gent, A. N.; Petrich, R. P. Proc. R. Soc. London, Ser. A 1969, 310, 433-448.
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properties. The general relationships demonstrate that two length scales, ac and d, play significant roles in defining the pattern’s control of adhesion. These length scales are material-defined, not geometrically prescribed, as indicated by the ability to use the same values of these parameters to describe all of our data. Through these relationships, we have demonstrated how controlled geometric structures can regulate the energy required for interfacial separation and the maximum force for separating an interface. For our materials, this control ranges from 20% to 400% the strength of nonpatterned interfaces. This control is important for the future development of “smart” adhesives, for advanced coatings such as antifouling coatings, and for the understanding of interfacial control for natural systems, such as the gecko. Glossary a A ac d δ E Gc Ge h L ν P Pm Pm,f Pm,s R rp σth Wadh
contact radius contact area critical radius at which separation mechanism changes length scale as predicted by Gao et al. for the increased probability of forming “optimal” interfacial geometry relative displacement between contacting spherical probe and patterned surface elastic modulus critical energy release rate or adhesion energy for interface effective energy release rate for post separation height of surface posts in pattern edge-to-edge spacing between posts on square array pattern Poisson’s ratio force during contact and separation for contact adhesion test maximum separation force for general contact adhesion test maximum separation force for smooth, flat probe geometry maximum separation force for smooth, spherical probe geometry radius of curvature of sherical probe in contact adhesion tests radius of cylindrical post in surface pattern optimal, or theoretical, strength of interface total hysteresis measured between contact and separation per total interfacial area
Acknowledgment. We gratefully acknowledge the financial support of NSF CAREER Award DMR-0349078, a 3M Non-tenured Faculty Research Award, a CUMIRP Exploratory Research Award, and a University of Massachusetts Faculty Research Grant. LA051721K
