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Deformation and Adhesion of Elastomer Poly(dimethylsiloxane) Colloidal AFM Probes Renato Buzio,*,† Alessandro Bosca,† Silke Krol,‡ Diego Marchetto,§ Sergio Valeri,§ and Ugo Valbusa† Nanomed Labs, ABC - AdVanced Biotechnology Center, Largo R. Benzi 10, 16136 GenoVa, Physics Department, UniVersity of GenoVa, Via Dodecaneso 33, 16146 GenoVa, Italy, CBM - Centre for Molecular Biomedicine, Area Science Park, BasoVizza SS 14, 34012 Trieste, Italy, and CNR-INFM Research Center S3 and Physics Department, UniVersity of Modena and Reggio Emilia, Via Campi 213/A, 41100 Modena, Italy ReceiVed April 2, 2007. In Final Form: June 10, 2007 We report on the synthesis and characterization of elastomer colloidal AFM probes. Poly(dimethylsiloxane) microparticles, obtained by water emulsification and cross-linking of viscous prepolymers, are glued to AFM cantilevers and used for contact mechanics investigations on smooth substrates: in detail cyclic loading-unloading experiments are carried on ion-sputtered mica, the deformation rate and dwell time being separately controlled. We analyze load-penetration curves and pull-off forces with models due respectively to Zener; Maugis and Barquins; and Greenwood and Johnson and account for bulk creep, interfacial viscoelasticity, and structural rearrangements at the polymersubstrate interface. A good agreement is found between experiments and theory, with a straightforward estimation of colloidal probes’ material parameters. We suggest the use of such probes for novel contact mechanics experiments involving fully reversible deformations at the submicrometer scale.
Introduction Experimental investigations on mesoscale contact junctions, formed by mechanical interaction of elastically soft solids, are expected to significantly contribute to our understanding of several physical phenomena and promote major advances into traditional and novel applied fields. Elastic junctions, from few tens of nanometers up to several microns in size, are primarily involved in rubber friction and adhesion: they in fact control the magnitude of contact area, shear forces, and viscoelastic losses at the contact interface and ultimately impart functional properties and technological interest to rubber interfaces, especially for sealings, car tires, and pressure-sensitive adhesive applications, and for engineering of active components (valves and pumps) in elastomer-based bioMEMS.1-3 Mesoscale junctions play a pivotal role for a number of emerging lithographies (e.g., transfer printing, soft lithography, and scanning probe contact printing4-6) that exploit elastomer viscoelasticity and adhesion to manipulate objects, pattern surfaces, and transfer molecules with nanoscale accuracy. Moreover, biological studies on cell adhesion and mechanical interaction with physical scaffolds recognize that focal adhesion contact spots (i.e., the key modulators in cell spreading, migration, proliferation, and differentiation) originate from cell active sensing of scaffold stiffness, morphology, and biochemical heterogeneity at the micrometric and submicrometric length scale.7 * To whom correspondence should be addressed. E-mail: buzio@ fisica.unige.it. Tel./Fax: +39 010 5737 382. † Advanced Biotechnology Center and University of Genova. ‡ Center for Molecular Biomedicine, Trieste. § University of Modena and Reggio Emilia. (1) Persson, B. N. J.; Albohr, O.; Tartaglino, U.; Volokitin, A. I.; Tosatti, E. J. Phys. Condens. Matter 2005, 17, R1. (2) Bhushan, B; Sayer, R. A. Microsyst. Technol. 2007, 13, 71. (3) Bhushan, B. Ed. Springer Handbook of Nanotechnology; SpringerVerlag: Berlin, 2004; Chapter 9. (4) Meitel, M. A.; Zhu, Z. T.; Kumar, V.; Lee, K. J.; Feng, X.; Huang, Y. Y.; Adesida, I.; Nuzzo, R. G.; Rogers, J. A. Nat. Mat. 2006, 5, 33. (5) Xia, Y.; Whitesides, G. M. Angew. Chem., Int. Ed. 1998, 37, 550. (6) Wang, X.; Liu, C. Nano Lett. 2005, 5, 1867.
Rationalization of junctions’ response mostly proceeds through generalized theories involving continuum and linearly viscoelastic solids. Such models allow us to appreciate a rich phenomenology of contact responses,1,8 emerging as well by a proper inclusion of morphological roughness into the contact process.9 They also display a remarkable predictive power when applied to complex biological systems, as filamentary attachment pads,10 and might therefore drive the development of biomimetic adhesives.11-12 So far, contact mechanics experiments addressing the interplay of viscoelasticity and morphological roughness at the mesoscale are missing. The few existing studies explore tack forces for relatively smooth interfaces, formed by well-characterized polymers: they are conducted with sophisticated, customdesigned instruments (e.g., surface force apparatus) that often suffer from reduced flexibility in terms of testing materials and contact geometries.13-15 Atomic force microscopy (AFM) offers the possibility of exploring mesoscale contact mechanics by performing indentation experiments with colloidal probes. This approach, originally introduced by Ducker et al.,16 is implemented by gluing a microsphere at the end of an AFM cantilever and then recording the interaction force between the sphere and a target surface. Colloidal probes with elastically stiff beads (silica, TiO2, ZnS, gold, and tungsten) have been successfully used to measure Van der Waals adhesion for a variety of working conditions and (7) Sniadecki, N. J.; Desai, R. A.; Ruiz, S. A.; Chen, C. S. Ann. Biomed. Eng. 2006, 34, 59. (8) Persson, B. N. J. Phys. ReV. Lett. 2001, 87, 11. (9) Persson, B. N. J. Phys. ReV. Lett. 2002, 89, 24. (10) Persson, B. N. J. J. Chem. Phys. 2003, 118, 7614. (11) Geim, A. K.; Dubonos, S. V.; Grigorieva, I. V.; Novoselov, K. S.; Zhukov, A. A.; Shapoval, S. Y. Nat. Mater. 2003, 2, 461. (12) Majidi, C.; Groff, R. E.; Maeno, Y.; Schubert, B.; Baek, S.; Bush, B.; Maboudian, R.; Gravish, N.; Wilkinson, M.; Autumn, K.; Fearing, R. S. Phys. ReV. Lett. 2006, 97, 76103. (13) Falsafi, A.; Deprez, P.; Bates, F. S.; Tirrell, M. J. Rheol. 1997, 41, 1349. (14) Ruths, M.; Granick, S. Langmuir 1998, 14, 1804. (15) Benz, M.; Rosenberg, K. J.; Kramer, E. J.; Israelachvili, J. N. J. Phys. Chem. B 2006, 110, 11884. (16) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991, 353, 239.
10.1021/la700941c CCC: $37.00 © 2007 American Chemical Society Published on Web 07/27/2007
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systems,17 albeit with negligible deformation of contact interface. On the contrary, reports concerning softer poly(styrene) and poly(ethylene) probes attested interfacial plastic failure to occur at nanoscale asperities protruding from their surface.18,19 Our experimental strategy for studying deformation and adhesion under fully reversible deformation conditions focused on the development of poly(dimethylsiloxane) (PDMS) colloidal probes, a solution explored up to now only by a limited number of research groups.20,21 PDMS is a soft and durable elastomer, with a surface that is low in free energy (ca. 22 mJ/m2) and chemically inert.5 It received considerable attention in macroscale investigations focusing on adhesion hysteresis, a phenomenon causing the energy released in forming an adhesive bond to be smaller than that necessary to break it.22-26 Soft lithography applications and nanoindentation studies on bulk PDMS samples,27-29 thin films, grafted monomer chains, and liquidlike droplets30,32 attested the occurrence of reversible deformations down to the nanoscale. There is no doubt that the deformation and adhesion of this model elastomer are by themselves interesting for fundamental questions on polymers tackiness, crucial for engineering of integrated silicon-PDMS MEMS,33,34 and instrumental, in the present case, for setting up experimental tools in mesoscale contact mechanics. We prepared PDMS microparticles exploiting a newly developed method, namely water emulsification of a viscous prepolymer, which is effective, quick, low-cost, and reliable; the resulting beads were attached to AFM cantilevers under optical microscopy using a micromanipulation stage. We tested particles response for “sphere-on-flat” boundary conditions, acquiring loaddisplacement curves on smooth mica samples. Data analysis was carried out with models by Zener,29 Maugis and Barquins,35,36 and Greenwood and Johnson,37 and finite-size effects38 were taken into account: this approach allowed us to quantify the elastic modulus and interface energy of PDMS beads and led to estimates of the characteristic relaxation times probed by the experiment. Forthcoming investigations will explore the deformation and adhesion of PDMS colloidal probes on a rigid countersurface with controlled roughness. (17) Butt, H. J.; Cappella, B.; Kappl, M. Surf. Sci. Rep. 2005, 59, 1. (18) Reitsma, M.; Craig, V.; Biggs, S. Int. J. Adhes. Adhes. 2000, 20, 445. (19) Tormoen, G. W.; Drelich, J. J. Adhes. Sci. Technol. 2005, 19, 181. (20) Zou, J.; Wang, X.; Bullen, D.; Ryu, K.; Liu, C.; Mirkin, C. A. J. Micromech. Microeng. 2004, 14, 204. (21) Cho, J. H.; Lee, D. H.; Shin, H. S.; Pattanayek, S. K.; Ryu, C. Y.; Cho, K. Langmuir 2004, 20, 11499. (22) Chaundhury, M. K.; Whitesides, G. M. Langmuir 1991, 7, 1013. (23) Chaundhury, M. K.; Owen, M. J. Langmuir 1993, 9, 29. (24) Perutz, S.; Kramer, E. J.; Baney, J.; Hui, C. Y.; Cohen, C. J. Polym. Sci., Part B: Polym. Phys. 1998, 36, 2129. (25) Pickering, J. P.; Van Der Meer, D. W.; Vancso, G. J. J. Adhes. Sci. Technol. 2001, 15, 1429. (26) Galliano, A.; Bistac, S.; Schultz, J. J. Adhes. 2003, 79, 973. (27) Bar, G.; Delineau, L.; Brandsch, R.; Bruch, M.; Whangbo, M. H. Appl. Phys. Lett. 1999, 75, 4198. (28) Ebenstein, D. M.; Wahl, K. J. J. Colloid Interface Sci. 2006, 289, 652. (29) Sun, Y.; Walker, G. C. Langmuir 2005, 21, 8694. (30) Gillies, G.; Prestidge, C. A.; Attard, P. Langmuir 2001, 17, 7955. (31) Al-Maawali, S.; Bemis, J. E.; Akhremitchev, B. B.; Leecharoen, R.; Janesko, B. G.; Walker, G. C. J. Phys. Chem. B 2001, 105, 3965. (32) Rau, K.; Singh, R.; Goldberg, E. Mat. Res. InnoVat. 2002, 5, 151. (33) Andersson, H.; van den Berg, A. Sens. Actuators B 2003, 92, 315. (34) Klemic, K. B.; Klemic, J. F.; Reed, M. A.; Sigworth. F. J. Biosens. Bioelectron. 2002, 17, 597. (35) Vakarelski, I. U.; Toritani, A.; Nakayama, M.; Higashitani, K. Langmuir 2001, 17, 4739. (36) Maugis, D.; Barquins, M. J. Phys. D 1978, 11, 1989. (37) Greenwood, J. A.; Johnson, K. L. J. Colloid Interface Sci. 2006, 296, 284. (38) Shull, K. L. Mater. Sci. Eng. 2002, R36, 1.
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Materials and Methods Preparation of PDMS Beads and Colloidal Probes. Common methods for synthesizing PDMS microparticles rely on preparing an emulsion of PDMS droplets in water through the basecatalyzed polymerization of dimethyl-diethoxy-silane and trimethyl-ethoxy-silane:39-42 an aqueous solution containing 9-10% (by volume) ammonia and 2-6% monomers mixture is vigorously shaken for several seconds, left to react for 5-15 h, dialyzed against Milli-Q water, and then filtered. The PDMS droplets are highly monodisperse, with average size (in the range 500 nm to 10 µm) and rheology (from liquid-like to solid-like) depending on the procedure details. A similar strategy was adopted in the present investigation, PDMS microparticles being synthesized by emulsification in water of a commercial viscous prepolymer, Sylgard 186 by Dow Corning: this is supplied as a two-component kit consisting of a viscous base and a liquid curing agent. In our procedure, a 10:1 (by mass) mixture of base and cross-linking agent was poured on the rotating disk (1 cm diameter) of a DC motor (EG-530AD-6F, Mabuchi Motor), immersed in a 500 mL glass beaker filled with Milli-Q water, and rotated at a speed of about 2000 rpm for few seconds. The beaker was then placed inside an oven and heated at 60 °C for 24 h, in order to cure the emulsified droplets to a flexible elastomer. The suspension was filtered by a 1 µm pore filter, on top of which PDMS micrometric spheres could be easily observed by optical microscopy; these were finally swollen in toluene to remove the fraction of unbounded monomers and decrease bulk viscoelasticity43 (Figure 1). Particles 20-30 µm in diameter were manipulated using a thin tungsten wire and attached to the apex of silicon cantilevers with molten Shell Epikote resin.16 Their surface morphology was inspected by the Dimension 3100 AFM (Digital Instruments, Veeco), operated in tapping mode: colloidal probes were placed on a silicon wafer, located by optical microscopy, and scanned by sharp tips. Surface roughness, evaluated on areas of 10 × 10 µm2, was typically ∼2 nm. We intentionally discarded colloidal probes whose AFM topographies displayed isolated asperities of few tens nanometers in size. Preparation of Mica Substrates. The substrates used for indentations experiments consisted of muscovite mica that was ionsputtered in ultrahigh-vacuum by a low-energy defocused Ar+ beam (at normal incidence, with energy Eions ) 1040 eV, flux Jion ) 400 µA/cm2, sputtering time tsputt ) 600 s). The surface roughness of irradiated specimens remains below 2 nm (i.e., slightly rougher than crystalline mica), but variations of surface stoichiometry take place after irradiation, leading to a significant decrease of hydrophilicity: in detail static contact angle measurements based on the sessile-drop method indicate a variation of the contact angle from 5° on freshly cleaved mica to about 50° on ion-sputtered mica. We used the latter to reduce contributions of water capillaries to the measured adhesive forces. Further details on irradiated mica samples are reported elsewhere.44 Contact Mechanics AFM Experiments. Contact mechanics experiments were performed at T ) 22 °C in a glovebox chamber with low-humidity atmosphere (relative humidity RH e 5%), by means of the Explorer AFM (ThermoMicroscopes, Veeco), equipped with the TrueMetrix closed-loop scanner linearizer (nonlinearity e 1%) and operated in contact-mode: to this purpose, we glued PDMS beads on relatively stiff cantilevers (MikroMasch NSC35AlBS, nominal stiffness 14N/m). Standard deflection-displacement curves were obtained by recording cantilever deflection u (in nA) while ramping scanner displacement z (in nm). A schematic of force plots (39) Obey, T. M.; Vincent, B. J. Colloid Interface Sci. 1994, 163, 454. (40) Goller, M. I.; Obey, T. M.; Teare, D. O. H.; Vincent, B.; Wegener, M. R. Colloids Surf. A 1997, 123, 183. (41) Gillies, G.; Prestidge, C. A.; Attard, P. Langmuir 2002, 18, 1674. (42) Gillies, G.; Prestidge, C. A. AdV. Colloid Interface Sci. 2004, 108, 197. (43) Ng Lee, J.; Park, C.; Whitesides, G. M. Anal. Chem. 2003, 75, 6544. (44) Buzio, R.; Toma, A.; Chincarini, A.; Buatier de Mongeot, F.; Boragno, C.; Valbusa, U. Surf. Sci. 2007, 601, 2735.
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Figure 1. Synthesis of PDMS colloidal AFM probes starting from a commercial viscous prepolymer. (a) Emulsification in water environment followed by curing and filtering processes. (b) Optical micrographs of few cross-linked PDMS beads on a filter paper; a densely populated region is reported in the inset. White bars are 50 µm. (c) Optical micrograph of a 30 µm PDMS particle glued to a tipless silicon cantilever. The inset shows AFM topography of a 5 × 5 µm2 region on top of the same bead: surface roughness is 2 nm. is shown for clarity in Figure 2a, where cantilever deflection u was converted into normal force F by the Hookean relationship F ) kCSphu
(1)
with Sph photodiode sensitivity (in nm/nA) and kC the measured elastic stiffness of the colloidal probe (kC is usually larger than the nominal cantilever stiffness, as discussed below). Careful inspection of Figure 2a allows us to appreciate the following passages: as the scanner elongates, the bead approaches the surface and finally contacts it with a “jump-to-contact” region. The particle is then deformed
Figure 2. (a) Schematics of force-displacement curve acquired by AFM; the relevant contact processes are the approach (1), the adhesive jump-to-contact (2), the elastic deformation at maximum force (3), and the adhesive detachment event (4). (b) (Top) Experimental procedure used to estimate the photodiode sensitivity Sph: fine positioning of the probe under optical microscopy allowed to indent the Si substrate fracture edge with the very end of the cantilever, thus neglecting the compliant response of the elastic bead. (Bottom) Magnification of the micrometric bead-lever contact region formed when gluing particles of 20-30 µm diameter. until the maximum load Fmax is applied. In the receding process, a hysteresis appears because of energy dissipation, and the particle is kept adhering to the plate even at the point of zero load; when a tensile force is applied, the particle passes through the maximum adhesive interaction, where the adhesive force (or tackiness) Fadh is defined, and finally, detachment occurs. Relevant controllable parameters of the cycle are the approaching/retracting speed V, the maximum compressive load Fmax, and the contact time tc, separating the “jump-to-contact” and detachment events. Force-displacement curves can be used to study the dependence of adhesion force Fadh on V, Fmax and tc, provided that Sph and kC are known with sufficient accuracy. In order to estimate Sph, we used colloidal probes to indent a silicon wafer close to a fracture edge; to ensure that only the very end of the cantilever was touching the surface, thus neglecting any compliant response coming from the
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elastic bead, fine positioning of the probe on the edge was achieved under the optical microscope of the AFM by using the sample holder translator stage and the scanner tube (Figure 2b). To estimate kC, we first calibrated silicon cantilevers using the analysis of Sader et al.,45 based on measurements of lever geometry, fundamental resonance frequency, and quality factor. Recalling that for a rectangular-shaped cantilever kC ∝ Lc-3, where Lc is the cantilever length, we estimated the effective stiffness of a colloidal probe by multiplying the elastic constant of the bare cantilever by the factor (Lc/Lb)3, where Lb is the distance of the glued PDMS bead with respect to the base of the lever. To this purpose, we note that for particles of 20-30 µm diameter a micrometric bead-lever contact region is formed, over which the stiffness of the lever might be certainly enhanced. To account for this effect, we chose an effective Lb value corresponding to the distance between the lever base and the point located in the middle of the gap separating the center of mass of the PDMS bead and the edge of the contact (Figure 2b). Experimental errors on length measurements done by optical microscopy (2 µm) resulted in a 35% uncertainty on kC. Other calibration methods46 might be also implemented. Gaining a deeper insight on dynamic adhesion processes and hysteresis demands that force-displacement curves be converted into force-deformation curves. The deformation δ is usually calculated as the scanner displacement minus the corresponding cantilever deflection, that is δ ) (z - z0) - Sphu
(2)
where z0 is the scanner displacement at zero tip-sample distance. We assigned z0 at the minimum detectable attractive force of the loading curve (i.e., immediately before the “jump-to-contact” region): with this choice, already suggested in similar studies,29 we have δ(z0) ) -Sphu(z0) ≈ 0. Thus, we are assuming that the particle is not deformed at incipient contact. We recognize that this is a questionable choice in any indentation experiment involving deformable surfaces, and there is some uncertainty in the actual degree of strain of PDMS particles at z0, since deformations might occur prior to contact due to long-range van der Waals forces. Since we are not dealing with charged deformable particles, sophisticated theories for defining the zero of separation are not useful,30 and we have to ignore such effect, assuming that longrange deformations are considerably smaller than those induced by compressive and tensile forces during the experiment. By using such criteria for z0 and adopting eqs 1 and 2, we routinely converted raw deflection-displacement curves into force-deformation curves. An external data acquisition board controlled by a PC via LabVIEW software (by National Instruments) allowed us to monitor cantilever deflection and scanner elongation during tack experiments and save experimental data for successive analysis. The deformation rates and dwell time were set using the AFM control software: in fact this allowed us to separately control the loading and unloading rate in a force-displacement cycle as well as to establish feedback at the maximum applied force Fmax for a given time: the latter possibility was used to keep PDMS particles in contact with mica from few seconds up to few minutes. Control experiments, based on contact-mode AFM imaging with sharp tips, were done to verify the smoothness and cleanliness of the contact region and detect the occurrence of substrate or probe damage during repeated pull-off tests: no appreciable wear or debris could be observed with these measurements.
Experimental Results It has been extensively demonstrated that bulk and interfacial viscoelasticity as well as interface rearrangements affect PDMS response during force-displacement curves. These conclusions (45) Sader, J. E.; Chon, J. W. N.; Mulvaney, P. ReV. Sci. Instrum. 1999, 70, 3967. (46) Cleveland, J. P.; Manne, S.; Bocek, D.; Hansma, P. K. ReV. Sci. Instrum. 1993, 64, 403.
Figure 3. Experimental deformation and adhesion data for a PDMS colloidal probe of radius R ) 13 µm interacting with ion-sputtered mica. (a) Typical AFM force-displacement curve. (b) Dependence of the adhesive force Fadh on the unloading velocity V at fixed dwell time tC ≈ 30 s and maximum applied load Fmax ) 6.7 µN. (c) Dependence of the adhesive force Fadh on the dwell time tC at fixed unloading velocity V ) 1 µm/s and maximum applied load Fmax ) 6.7 µN. Error bars correspond to statistical fluctuations of force values.
emerge through macroscale tack testing studies24-26 and refined nanoindentations experiments, conducted with micrometric or nanometric tips,28,29 and suggest that we experimentally probe the contact process by separately controlling the deformation rate and dwell time. To this purpose, we systematically examined the dependence of loading-unloading cycles F(δ) and pull-off force Fadh on V and tC, for a fixed value of Fmax. In Figure 3a, we report a representative force-displacement curve acquired on ion-sputtered mica by means of a colloidal PDMS probe of bead diameter D ) 26 ( 3 µm and elastic constant kC ) 51 ( 18 N/m (estimated from a bare cantilever stiffness of 14.5 ( 2.0 N/m and Lb ) 59 ( 3 µm). Experimental data closely resemble the schematic diagram of Figure 2a: on approaching the substrate, a relatively sharp jump-to-contact transition takes place (roughly within 50 nm of bead deformation) followed by a nonlinear increase of normal force F with penetration depth δ. The unloading process consists, on the contrary, of a linear decrease of F with δ, due to stress release within the contact junction at fixed contact area, followed by a nonlinear decrease of force associated with crack opening and detachment event.35-38
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As expected, we noted a remarkable dependence of F(δ) curves on V and tC. Several experiments were carried out using different PDMS particles, and qualitative similar features were observed for all of them; therefore, we will focus in the following, for the sake of quantitative consistency, on data obtained for the 26 µm particle of Figure 3a. Figure 3b,c summarizes the variation of the pull-off force Fadh with unloading velocity V and dwell time tC, respectively: for simplicity, we chose Fmax ) 6.7 µN and tC ) 30 s, in the former case, and Fmax ) 6.7 µN, V ) 1.0 µm/s, for the latter. We note that Fadh separately increases with V and tC, with an overall variation of about 30%. This response is ascribed to two concomitant effects, namely the dynamic Young modulus E increasing with V and the appearance of an effective work of adhesion w given by the following phenomenological relationship:
w ) w0(tC)[1 + f(T, VC)]
(3)
where w0 is the Dupre´ work of adhesion, VC is the crack speed, and f f 0 as VC f 0. The w0(tC) function attests the oftenobserved monotonic increase of w0 with tC due to the activation of the available reorganization processes of interface upon contact35 (see also the Discussion section); the second term in the brackets indicates, on the contrary, a dependence of w on VC arising from energy dissipation at contact edges, which has been proved experimentally by macroscale experiments36,38 and derived by general theoretical arguments on crack propagation in linear viscoelastic solids.47 Therefore, the physical significance of eq 3 is that w significantly overcomes w0 at finite values of crack velocity and dwell time. In order to estimate the most relevant parameters of our system (i.e., w0(tC) and f(T, VC); at fixed T ) 22 °C), we interpolated experimental curves with theoretical predictions accounting for interface rearrangements and crack tip losses; bulk viscoelasticity was also quantified through our analysis.
Theoretical Background The kinetics of adherence of viscoelastic bodies was originally described by Maugis and Barquins36 and Greenwood and Johnson48 for contacts dominated by interfacial viscoelastic losses; refined theories including bulk creep effects later appeared,49-50 and considerable theoretical efforts are in progress.37,47 Comparison of experiments with such sophisticated models usually demands the specific designing of measurements and complimentary knowledge on bulk rheological properties.51 Here on the contrary, we apply an approximate theoretical description providing, nevertheless, physical insight on results of Figure 3 and robust estimates of mechanical properties and interface energy w0(tC). In detail unloading curves are compared to the MaugisBarquins theory,36 extended to account for interfacial rearrangements35 and to a new model proposed by Greenwood and Johnson.37 They differ for f(T, VC) expression and provide complimentary knowledge for the studied system. The reader is referred to the original papers36,48-50 for a description of fracture mechanics concepts in contact mechanics: we simply recall that the kinetics of adherence can be described in terms of the strain energy release rate G; that is, crack opening occurs for G > w, whereas crack closing corresponds to G < w; at equilibrium, G ) w (Griffith criterion). (47) Carbone, G.; Persson, B. N. J. Eur. Phys. J. E 2005, 17, 261. (48) Greenwood, A. G.; Johnson, K. L. Phil. Mag. A 1981, 43, 697. (49) Hui, C. Y.; Baney, J. M.; Kramer, E. J. Langmuir 1998, 14, 6570. (50) Barthel, E. J. Colloid Interface Sci. 1998, 200, 7. (51) Giri, M.; Bousfield, D.; Unertl, W. N. Tribol. Lett. 2000, 9, 33.
For a sphere of radius R subject to the external load F, the dependence of load and indentation on the tip-sample contact radius a is solved by Johnson-Kendall-Roberts theory,52 showing that
δ) F)
( )
4πw0 a2 R 3K
1/2
(4)
Ka3 - (6πKw0a3)1/2 R
(5)
where K ) (4/3)[E/(1 - ν2)], E and ν are the Young modulus and Poisson ratio of the deformable junction, and w0(tC) ) w0. Moreover G is given by
G)
(
)
Rδ 3a3K 1- 2 8πR2 a
2
(6)
Loading Process. When G < w0, the crack starts to close: viscoelastic losses at the interface crack tip are negligible,36,48 whereas viscoelastic relaxations inside the bulk of the polymer provide the main source of energy dissipation. Bulk viscoelasticity leads to the appearance of a dynamic elastic modulus under external perturbations; that is, the polymer is hard at the start of the external perturbation (with modulus E0 at t ) 0) and softens over time (with relaxed modulus E∞ at infinite time). According to the approach suggested by Sun and Walker,29 expressions for the modulus relaxation can be extracted from the adhesion-induced indentation curves and compared to the theoretical predictions coming from constitutive models for linear viscoelasticity, as the extended Zener model. The latter assumes that a cross-linked polymer can be modeled as a set of springs and dashpots arranged in parallel and predicts a creep compliance E-1 given by
1 E(t)
)
1 E∞
Ei
N
-
∑ i)1 (E + E )E i
∞
∞
[
exp -
EiE∞ ηi(Ei + E∞)
t
]
(7)
Comparison of experimental and theoretical data, according to the guidelines reported in the Discussion section below, allows us to estimate Ei, ηi, and E∞ for the tested polymers and speculate the main relaxation pathways of polymer chains involved in viscoelastic losses. Unloading Process. The unloading process after a contact time tC is analyzed below into two regions, the first one occurring without cracking and the second with crack35,38 opening. The unloading process starts from the maximum force and contact radius, Fmax and amax, and proceeds with unloading rate V until G > w0(tC) and cracks start to open. Before crack propagation VC ) 0 and G ) w0(tC): from eq 5, we deduce that Griffith’s criterion is violated when F < Fcr
amax3K - (6π amax3Kw0(tC))1/2 Fcr ≡ R
(8)
For Fcr e F e Fmax, eqs 4 and 5 provide
amax3K 3amax K δF) 2 2R
(9)
indicating that the force plot is described in this regime by a linear relationship; this is valid for δ in the range δcr e δ e δmax, (52) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. A 1971, 324, 301.
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with
VjC ≡ 2
δcr ≡
amax 2Fcr + 3amaxK 3R
(10)
The quantity k0 ≡ 3amax K/2 represents the contact stiffness, and the whole system can be modeled by a series of two springs, the effective spring k0 and the effective colloidal probe spring kC, respectively. When the scanner is further retracted, F < Fcr, and the crack starts to propagate. Both the Maugis-Barquins and the Greenwood-Johnson36,37 models describe crack opening; a comprehensive discussion on their physical basis and qualitative differences is reported in ref 38, whereas technical details are summarized below. The Maugis-Barquins model assumes f(T, VC) ) R(T)VCn: the parameter R(T) is related to the Williams-Landel-Ferry shift factor aT of the elastomer by R(T) ) aTn, where n ≈ 0.6 based on macroscale peeling experiments of rubber-like materials.35 The crack velocity VC is thus given by
VC )
||
da ) dt
(
) [
G - w0(tC) Rw0(tC)
| |
daj ) dth
ln
0.24β 1 - E∞/E0
(
1 - β-1
)
(
)
]
1 3a3K Rδ 2 1- 2 -1 1/n 2 R 8πR w0(tC) a
1/n
(11)
In the case of an indentation device with infinite stiffness (i.e., kC . k0), the time evolution of indentation is simply
δ(t) ) δcr - Vt
(12)
(16)
δ h ) a2 - (2/3)x2βaj
(17)
F h ) aj3 - x2βaj3
(18)
Equations 15 and 16 provide the rate dependence of the apparent surface energy at crack opening, in analogy with eq 11 for the Maugis-Barquins theory: in detail, eq 16 is inadequate for very low speeds (β f 1); hence, eq 15 is proposed as a good empirical fit for this range. Equations 17 and 18 correspond to eqs 4 and 5 expressed in nondimensional variables and with w0 replaced by G. Equation 17 can be inverted to give β ) β(δ h ) and then integrated step-by-step with simultaneous eqs 15, 16, and 18 to obtain unloading curves. For an indentation device with infinite stiffness, we impose
δ h (th) ) δ h cr -
1/n
)
β > 3.7
( ) Rtr ar2
V ht
(19)
whereas for a finite stiffness device eq 13, expressed in nondimensional form, is integrated simultaneously with eqs 15, 16, and 18. It appears that the Greenwood-Johnson model differs from the Maugis-Barquins analysis for the use of a creep compliance function φ(t), and f(T, VC) ) φ-1(t*) - 1 with t* ≡ t*(E∞, w0, µ, V).37
Discussion which can be inserted into eq 11. The latter can be solved numerically with standard techniques, since the involved parameters and initial conditions are known. On the contrary for a finite stiffness indentation device (i.e., kC ≈ k0), the deformation and unloading rates are related by (see eqs 1, 2, and 12)
dδ 1 dF )V) dt kC dt
[
(
) ]
1 3K a2 da Vδ(13) 2kC R dt 3 aK 1+ 2 kC
Discussion on the Loading Process. Loading curves were analyzed according to the procedure reported by Sun and Walker,29 and the Young modulus E was estimated at the point of purely adhesion-induced indentation. At such point δ ) δ0, F ) 0 (see Figures 3a and 4a), and from eqs 4 and 5 we have
a0 ) x3Rδ0
(20) 2
6πR w0(0) 3 E ) (1 - ν2) 4 a3
(21)
0
Equations 11 and 13 are coupled and can be solved iteratively as shown by Barquins and Maugis:53 this predicts the unloading curve from crack opening up to the final separation. Greenwood and Johnson have recently proposed a model for a standard linear “three-element” solid described by a creep compliance function37
[ ( )]
φ(t) ) 1 - 1 -
E∞ E0
exp(-t/τ)
(14)
where τ is a creep time constant. Introducing the reference quantities Fr ) 3πRw0, ar ) (9πR2w0/4E∞)1/3, δr ) ar2/R, and tr ) (9π/4)1/3µ2τ (where µ ) (Rw02/E∞2h03) is the Tabor parameter and h0 is the equilibrium atomic surface separation) and the nondimensional variables F h ) F/Fr, aj ) a/ar, δ h ) δ/δr, ht ) t/tr, and β ≡ G/w0, authors demonstrate that
VjC ≡
| |
daj ) [0.2112(β - 1) + 0.3939](β - 1)1.1403 dth 1 < β < 3.7 (15)
(53) Barquins, M.; Maugis, D. J. Adhes. 1981, 13, 53.
where a0 is the purely adhesion-induced contact radius. Provided that w0(0) is known, both a0 and E can be estimated for different loading velocities V. In Figure 4a, we show the evolution of representative loading curves for increasing values of V, and in Figure 4b, we report the corresponding time dependences of indentation curves δ(t), plotted up to their adhesion-induced indentation points δ0 (where t ≡ z(t)/V). As mentioned above, viscoelastic creep dominates deformation process thus indentation depth δ(t) separately depends on indentation time and loading velocity V. From Figure 4b, and using eq 20, we estimated δ0 and a0 values (Figure 4b inset), confirming that a micrometric junction is formed between the PDMS bead and the surface, with contact radius a0 ) 3.23.8 µm according to the imposed velocity V. The time dependence of the reciprocal of the Young modulus E-1(t), averaged over several curves, is shown in Figure 4c: here we assumed w0(0) ) 53 mJ/m2, checked to be a posteriori in agreement with the analysis of unloading curves, and we used eq 21 to evaluate E (the Poisson ratio was V ) 0.5). The quantity E-1 increases in 2s from 0.37 MPa-1 to an asymptotic value around 0.57 MPa-1, corresponding to a decrease of the Young modulus from ∼2.7
PDMS Colloidal AFM Probes
Figure 4. Data analysis of loading curves for the PDMS colloidal probe of Figure 3. (a) Dependence of force curves on the loading rate V after the jump-to-contact; black dots are used to mark the points of purely adhesion induced indentation δ0. (b) Time evolution of curves δ(t) plotted up to their adhesion-induced indentation point δ0 (symbols are the same used in panel a); specifically the dash curve indicates the time-dependence of δ0 with loading time. In the inset we show the corresponding values of contact radius a0 and indentation δ0 versus loading velocity. (c) Time evolution of the reciprocal of the relaxed elastic modulus: continuous line provides the best fit to experimental data by means of a single exponentialdecay function.
to ∼1.7 MPa. The ∼10% uncertainty affecting E-1 (due to statistical errors on δ0 and measurements error on R in eqs 20 and 21) is virtually independent from the colloidal probe spring value kC since at δ ) δ0 we have F ) 0. As originally suggested by Sun and Walker,29 the plot in Figure 4c can be interpreted as the adhesion-induced viscoelastic creep relaxation of the compliant PDMS particle sitting on the hard substrate immediately after jump-to-contact. Interpolation of experimental data with the creep compliance expression of the Zener model (eq 7) provides an efficient way to estimate elastic and viscous components as well as relaxation times for PDMS beads. A single exponential-decay E-1(t) ≈ A1 - A2 exp[-t/τ1] was found to be most appropriate to interpolate our data, with parameters values A1 ) 0.561 ( 0.009 MPa-1, A2 ) 0.27 ( 0.03 MPa-1 and τ1 ) 0.42 ( 0.08 s (errors from the best fit procedure). Since A1 ) E∞-1, A2 ) E1/[E∞(E1 + E∞)], and τ1 ) η1(E1 + E∞)/E1E∞, we estimated E∞ ≈ 1.8 MPa, E1 ≈ 1.7 MPa, and η1 ≈ 0.37 MPa s.
Langmuir, Vol. 23, No. 18, 2007 9299
The asymptotic elastic modulus E∞ is in the range of values declared by manufacturer for bulk Sylgard 186 and in agreement with results reported for PDMS probed by macroscopic quasistatic contact mechanics experiments.22,23,25 The number and value of appreciable time constants τi are related to segmental cooperativity manifested during polymer relaxation, which in turn depends on cross-link density. In fact the density of cross-linking points determines the relaxation pathways of polymer chains: denser cross-linked networks usually display a single relaxation time (≈102 ms), due to the coupling of segmental motions around the cross-linkers and away from the cross-linking points; shorter, multiple relaxation times (≈1 ms) appear for low cross-linked networks, reflecting intramolecular cooperativity and physical entanglements of relatively mobile polymers segments. Accordingly, the observation of a single-exponential decay τ1 ≈ 420 ms would suggest that PDMS beads behave as dense cross-linked networks, a statement corroborated by direct comparison with ref 29, where a single relaxation time of comparable magnitude (τ1 ≈ 120150 ms) is found for densely cross-linked PDMS bulk samples with degree of polymerization 37 and 60 (corresponding to the molecular weights of oligomeric precursors 2.74 and 4.44 kg/ mol). Indeed, the observation of a single-exponential decay is in the present case significantly affected by the limited resolution of our loading experiment, particularly by the absence of experimental points for E-1 in the time window below 0.5 s (Figure 4c), where shorter times constants τi should play a role. We anticipate that an analysis of unloading curves based on the Greenwood-Johnson model indicates that a shorter relaxation time τ ≈ 10-5 s is involved in interfacial viscoelasticity effects. Discussion on Unloading Process without Cracking. Experimental unloading curves were processed by custom software. We first estimated the Young modulus E and the maximum contact radius amax for different values of unloading velocity V and dwell time tC by interpolating the linear part of the unloading process by means of eq 9, with E and amax as fitting parameters. In Figure 5a, we show the dependence of experimental unloading curves on V for a fixed dwell time tC ) 30 s: it clearly appears that adhesion force Fadh monotonically increases with the unloading velocity V (see also Figure 3b), due to an increase of energy dissipation at crack opening.36 The elastic modulus E and maximum contact radius amax were extracted by fitting the (linear) upper region of the unloading curve and plotted as a function of V. A similar graph, averaged over several realizations, is reported in Figure 5b: amax ≈ 4.0 µm does not depend on V, since this quantity is defined by loading conditions (tC ) 30 s, Fmax ) 6.7 µN, and a loading velocity of 1 µm/s). We also note that amax slightly exceeds the estimated adhesion-induced contact radius a0 ≈ 3.2-3.8 µm, that means that the PDMS-substrate junction is substantially formed at the “jump-to-contact” event, increasing less than 20% on changing the normal load from 0 nN to Fmax ) 6.7 µN. On the contrary, Figure 5b demonstrates that the Young modulus E logarithmically increases with V from ∼2.0 to ∼2.3 MPa, suggesting bulk strengthening of PDMS beads at higher unloading rates. In Figures 5c, we report a similar data analysis on unloading curves acquired for tC varying in the range of 2-100 s and V ) 1 µm/s. It appears that the maximum contact radius slightly increases from 3.9 to about 4.1 µm with tC varying on two decades, which indicates a small creep of contact area with time. On the contrary, the Young modulus of the probe stays around 2.0 MPa, since the unloading velocity is fixed at V ) 1 µm/s for all measurements.
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Figure 6. Comparison of representative data with theoretical predictions based on the Maugis-Barquins model. (a) Arrows show the point where linear fitting, providing estimates for E and amax, is replaced by numerical integration of eqs 11 and 13. Solid and dash curves were generated for V ) 1 µm/s, tC ) 30 s, with the following combinations of parameters: (1) w0(30) ) 72 mJ/m2, n ) 0.4, R ) 210 (s/m)0.4; (2) w0(30) ) 68 mJ/m2, n ) 0.4, R ) 210 (s/m)0.4; (3) w0(30) ) 72 mJ/m2, n ) 0.42, R ) 210 (s/m)0.4. (b) Logarithmic increase of interface energy w0 with dwell time tC, found after interpolating data with theoretical curves generated for V ) 1 µm/s, n ) 0.4, and R ) 210 (s/m)0.4. Figure 5. Data analysis of unloading curves for the PDMS colloidal probe of Figure 3. (a) Dependence of unloading curves on deformation rate V: adhesion force increases due to viscoelastic losses; (b) Contact radius amax and Young modulus E obtained by linearly fitting unloading curves acquired at different deformation rates V: the elastic modulus E increases due to bulk viscoelasticity. (c) Contact radius amax and Young modulus E obtained by linearly fitting unloading curves acquired at different values of dwell time tC: the contact radius amax slightly increases due to creep relaxation of contact junction.
Experimental uncertainty on amax values is ∼5%: in fact amax depends on the ratio of the two parameters of the linear fit; therefore, it is affected only by the error on R (fitting errors are negligible). On the contrary the experimental uncertainty on E rises up to 40% due to the combination of errors on amax and kC. Discussion on the Unloading Process with Cracking. Crack opening was separately described by means of the MaugisBarquins and Greenwood-Johnson models. In order to implement the Maugis-Barquins model for all values of unloading velocity V and dwell time tC, the Young modulus E and maximum contact radius amax were chosen according to Figure 5b,c (i.e., including their dependence on V and tC, respectively); this allowed us to effectively account for long-range creep phenomena not treated by the model. Having defined E and amax as described in the section above, the remaining three unknown parameters, w0(tC), n, and R, were arbitrarily chosen and used to calculate Fcr and δcr (from eqs 8 and 10) as well as to predict the whole force plot for F < Fcr (by numerically solving eqs 11 and 13 with a fourth order Runge-Kutta method). The generated curve was then compared with the experimental
one by visual inspection, and w0(tC), n, and R were adjusted to minimize the difference between experimental and predicted values. In Figure 6a, we compare an experimental unloading curve acquired at V ) 1 µm/s and tC ) 30 s with theoretical curves obtained by numerical integration of Maugis-Barquins theory: the two vertical arrows denote the starting point of numerical predictions, that is (δcr, Fcr) given by eqs 8 and 10. Agreement within 7% between theory and experiment was always found for V ) 0.1-1.84 µm/s by choosing the three parameters in the range w0(30) ) 70-80 mJ/m2, n ) 0.38-0.40, and R ) 210260 (s/m)0.4. This result, and more extensively the possibility to satisfyingly predict the whole unloading curve by numerically solving the Maugis-Barquins theory, represents a considerable breakthrough of the present investigation with respect to previous studies,35 which estimated structural and interface parameters from data analysis conducted at few force vs deformation points. The excellent agreement of theory and experiment is stressing the fact that we are capturing the physical origins of adhesion increase with V, with an ad hoc inclusion of long-range creep effects through a velocity dependent Young modulus. We finally recall that the experimental uncertainty on the absolute values of w0(tC) is strongly affected by the error on kC, whereas we empirically observed almost no role of kC on estimates for n and R. At the same time, we recognize that there is a disagreement between our estimation for the parameter n and the value obtained within macroscopic peeling experiments on several elastomer materials: we believe that this disagreement should be ascribed to experimental aspects that are not fully controlled in our
PDMS Colloidal AFM Probes
Langmuir, Vol. 23, No. 18, 2007 9301
Figure 8. Comparison of representative data with theoretical predictions based on the Greenwood-Johnson model: arrows show the point where linear fitting, providing estimates for E and amax, is replaced by numerical integration of eqs 15-18. Curves were generated for V ) 0.1 µm/s, tC ) 30 s, with the following combinations of parameters: (solid) w0(30) ) 76 mJ/m2, τ ) 15 µs, h0 ) 0.55 nm; (dash) w0(30) ) 72 mJ/m2, τ ) 15 µs, h0 ) 0.55 nm. Figure 7. Phase-imaging map acquired simultaneously to the tapping-mode AFM topography of Figure 1c (inset): white bar is 1 µm. Phase shift contrast reflects compositional heterogeneity of PDMS surface layer, with nanoscale variations of viscoelasticity and/or surface energy.
investigation, the most relevant being the parallelism between cantilever axis and mica substrate. Any misalignment should in fact cause the application of a component of normal force but also shear stresses to our contact interface, causing a considerable deformation of contact area and stress/strain fields with respect to Johnson-Kendall-Roberts predictions. Since we are assuming, within our model, a perfectly circular contact area, experimental deviations from this assumption are indeed accounted for by a variation of materials constant w0(tC), n, and R. For the case of unloading curves acquired at different dwell time tC, we obtained excellent agreement between theory and experiment choosing n ) 0.4, R ) 210 (s/m)0.4 and assuming w0(tC) to monotonically increase with tC according to the plot reported in Figure 6b. The dependence of interface energy w0 on dwell time is consistent with the data reported elsewhere25,35 and can be generally attributed to several concurrent factors, as stress relaxation due to surface roughness, diffusion, and interdigitation of polymer chains across the interface, chemical reactions, and breakage of adsorbed molecular layers. In particular Vakarelsky et al.35 found that w0(tC) ≈ tC0.2 for the adhesion between hard surfaces covered with adsorbed layers of water molecules, ions, and hydrated ions in aqueous solutions and w0(tC) ≈ tC0.4 for the adhesion of poly(acrylmethacrylate) microparticles to mica substrates in pure water. Whereas in the present case morphological roughness is negligible over micrometric contact areas, we cannot totally exclude that compositional heterogeneity of the PDMS surface layer might affect stress relaxation at the contact interface, as suggested by phaseimaging maps acquired simultaneously to tapping-mode AFM topographies on colloidal probes (Figure 7). These in fact display a clear contrast that should be ascribed to nanoscale variations of PDMS viscoelasticity and/or surface energy.54 A quantitative interpretation of phase shift signal is out of the scope of the present paper; nevertheless, we underline that the chemical roughness revealed by similar maps might have a role in defining the functional relationship w0(tC) ≈ log tC. Further investigations must be carried out to fully understand this aspect. (54) Garcia, R.; Gomez, C. J.; Martinez, N. F.; Patil, S.; Dietz, C.; Magerle, R. Phys. ReV. Lett. 2006, 97, 016103.
Interpolations by means of the Greenwood-Johnson model were restricted to experimental data acquired at slow loading rates (i.e., V ) 0.1 µm/s), showing the smaller bulk creep effects (see Figure 5b): again we fixed E() E∞) and amax has shown in the section above, and the remaining three unknown parameters, w0, τ, and h0, were arbitrarily chosen and used to calculate Fcr and δcr (from eqs 8 and 10) and to predict the whole force plot for F < Fcr (by numerically solving eqs 15-18 with a fourth order Runge-Kutta method). To account for the finite stiffness of the indentation device, we interpolated the actual deformation δ(t) of each unloading curve by a simple polynomial,55 which was used instead of eq 19 to integrate eqs 15-18. In Figure 8, we compare a representative unloading curve acquired at V ) 0.1 µm/s and tC ) 30 s with predictions of Greenwood-Johnson theory. A satisfying agreement was found by choosing fitting parameters in the range w0(30) ) 68-76 mJ/m2, τ ) 1.2 × 10-5-1.5 × 10-5 s and h0 ) 0.45-0.55 nm, noting however that the experimental portion of the curve immediately after the adhesion point showed a 20% discrepancy from theory. The w0 values agree with previous interpolations with the Maugis-Barquins model, and h0 corresponds to the effective range of interaction potential reported by Giri et al.55 for latex thin films; the τ values are not too far from the creep time constant ≈10-6 s found for unfilled PDMS by depth-sensing nanoindentation under oscillatory loading conditions.56 We underline that the calculated β values at unloading were always below 3.7; that is, eq 15 was principally used in numerical calculations: since it does not depend on E∞/E0, as originally stated by Greenwood and Johnson, theoretical curves did not allow us to evaluate E0. As noted for the Maugis-Barquins theory, our estimates for unknown parameters w0, h0, and τ are affected by the experimental uncertainty on kC. The previous discussion demonstrates that curve fitting techniques based on Maugis-Barquins or Greenwood-Johnson theory are necessary to provide reasonable estimates for interface energy and relaxed Young modulus: a simplified data analysis requiring knowledge of only two or three points from the loadingunloading cycles to estimate w0 and E∞, as recently suggested by Ebenstein and Wahl,28 is not applicable in the present case due to the large hysteresis affecting our loading cycles (e.g., Figure 3a). (55) Giri, M.; Bousfield, D. B.; Unertl, W. N. Langmuir 2001, 17, 2973. (56) Wahl, K. J.; Asif, S. A. S.; Greenwood, J. A.; Johnson, K. L. J. Colloid Interface Sci. 2006, 296, 178.
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We comment now on a few limitations of the described experimental method. The contact radius amax, as shown in Figure 5b,c, lies in the micrometer range, and hence, we are actually exploring microscale contact mechanics. At the same time, our system appears dominated by contact adhesion (since a0 ≈ amax), which results in a poor modulation of contact area through externally applied load F e Fmax. This certainly occurs for the integration of van der Waals adhesive force over the relatively large bead surface, thus screening of adhesive interaction, for instance by working under water environment, might decrease the importance of jump-to-contact event and improve our capability to tune junction size in the mescoscale range. Moreover, we assumed a priori the formation of a perfectly circular contact region between PDMS probe and substrate, neglecting any cantilever-substrate misalignment leading to uncontrolled off-normal loading, shear stresses, and rolling components. These aspects might be partially reduced, at least for contact areas above a few microns squared in size, by performing indentation experiments over an inverted optical microscope, equipped with a high-magnification objective, allowing us to monitor the contact area and reduce distortions of junction shape while ramping the load. A different approach would consist of studying contact mechanics with PDMS beads glued to the free end of a nanoindenter tip holder, to exploit the more controlled geometry and reduce some of the unavoidable uncertainties inherent in AFM. Discussion on Finite-Size Effects. The Hertz theory upon which the Johnson-Kendall-Roberts theory is built is applicable only under the condition of a , R: from Figure 5b,c, we have a/R ≈ 0.31; therefore, finite-size effects might play a role. Estimation of their impact on mechanical parameters demands that we introduce correction factors to the elasticity equations. For simplicity, we will perform such analysis for the MaugisBarquins theory, with correction factors proposed by Shull et al.38 Assuming h ) 2R to be the thickness of the tested compliant layer (i.e., the PDMS bead diameter) and a/h < 0.5 to be certainly satisfied, eqs 9, 11, and 13 are replaced by F) VC ) dδ ) dt
[
3aK a3K δ2(1 - a/h) 2R(1 - a/h)
(
)
1 2a Rδ 3a3K 1- - 2 1/n 3h a R 8πR2w0(tC)
2
(22)
1 -1 (1 - a/h)2
]
1/n
1 aK 3 1+ 2 kC(1 - a/h) 3K a2 da Vδ+ R dt 2kC(1 - a/h) 3aK da 1 a3K δ2 2(1 - a/h) dt 2R(1 a/h) hkC(1 - a/h)
[
(
(
)
)
(23)
]
(24)
We recognize that eqs 20 and 21, related to the adhesion-induced indentation point, are not affected by finite-size effects (see also
eq 22 for F ) 0), and data analysis for loading curves, together with Figure 4, remain unchanged. Application of eq 22 to the unloading process without cracking provides E and amax. The former, multiplied by (1 - amax/h) ≈ 0.85, gives E ≈ 1.7-1.9 MPa for V ) 0.1-1.84 µm/s (compare with Figure 5b): it is pleasant to see that through finite-size corrections we recover an excellent agreement of the corrected E values and E∞ given by the Zener model. The contact radius amax remains on the contrary unchanged, since related to the ratio of the two unknown parameters obtained by fitting experimental data with eq 22. Estimation of the three parameters w0(tC), n, and R, through visual comparison of unloading curves and the Maugis-Barquins theory based on eqs 22-24, shows negligible variations with respect to the ranges w0(30) ) 70-80 mJ/m2, n ) 0.38-0.40 and R ) 210-260 (s/m)0.4 established before.
Conclusion In summary, we synthesized PDMS microparticles by emulsification of a commercial viscous prepolymer in droplets, successively cured to a rigid elastomer. Deformation and adhesion of PDMS beads were investigated (after gluing them to silicon AFM cantilevers) by nanoindentations conducted on ion-sputtered mica in a low humidity air atmosphere. Loading-unloading cycles were acquired to distinguish adhesion contributions coming from bulk creep, interfacial viscoelasticity, and surface rearrangements. We used the Zener model to describe the loading process and compared the MaugisBarquins and Greenwood-Johnson theories for unloading ramps. To this purpose, we developed an interpolation protocol that allowed whole curves fitting and estimation of material parameters with satisfying accuracy. Observed phenomena are well described as follows: (i) loading curves depend on the loading rate, due to bulk creep; (ii) contact area achieves saturation immediately after the “jump-to-contact” point; (iii) adhesion force monotonically grows with unloading velocity for the increase of viscoelastic losses at contact edges; and (iv) interface energy logarithmically increases with dwell time. Qualitatively similar results were observed for nanoindentations on silicon wafers and crystalline mica (not shown), supporting the generality of the above conclusions. The present investigation provides the framework to carry out PDMS beads contact mechanics on rough interfaces. Specifically, our experimental approach is readily extendable to contact studies probing adhesion modulation by controlled surface roughness; from a broader perspective, it might be useful to validate theoretical models and test elastomer-based mechanical devices at the mesoscale. Acknowledgment. Renato Buzio acknowledges useful discussions with Giuseppe Carbone, Luca Repetto, Carlo Denurchis, Andrea Calvi, Giuseppe Firpo, Elisabetta Princi, and Silvia Vicini. The authors thank all of the members of the ESF program NATRIBO for comments. This work was supported by the Italian Ministry of University and Research MIUR within the projects PRIN “Nanotribology” and FIRB “NanoMed”. LA700941C
