pubs.acs.org/Langmuir © 2009 American Chemical Society
Experimental and Theoretical Analysis of a Dynamic JKR Contact E. Charrault, C. Gauthier,* P. Marie, and R. Schirrer Institut Charles Sadron, CNRS UPR 22 23, rue du Loess, F-67034 Strasbourg Cedex, France Received October 16, 2008. Revised Manuscript Received February 23, 2009 Analysis of the adhesive contact between solids makes use of contact mechanics, surface physics, and fracture mechanics. The elastic adhesive contacts have been intensively studied, and now, interest still remains about how the viscoelasticity of the solids may be taken into account for the calculation of the work of adhesion, the major difficulty being to separate the surface and bulk energy dissipations. This paper describes a new and original experimental device for “dynamic JKR” tests, which allows us to study dynamic adhesive contacts under a cyclic normal load. PDMS contacts on PDMS were studied at different frequencies and temperatures, and it was possible using the JKR model to determine the hysteretic value of the work of adhesion. The results obtained follow the same evolution as the loss factor of the material.
Introduction One of the most popular adhesion tests involves contact between a sphere and a flat surface and is based upon the theory of Johnson, Kendall, and Roberts, which will hereafter be referred to as the JKR technique.1 Contrary to typical methods of measuring adhesion such as peel tests, which are dominated by the bulk viscoelastic energy losses arising from macroscopic deformation of the sample, the JKR technique may be used to probe the molecular causes of adhesion in a number of elastomeric systems by minimizing the sample volume to reduce the bulk viscoelastic losses. In JKR adhesion tests, an adherence energy G is measured, which may be viewed as the energy required to decrease the contact area A by unit area. G is similar to the adherence energy used in fracture theory. In the case studied here, it is equal to the Dupre thermodynamic work of adhesion W for an ideal reversible equilibrium test and exceeds it when dissipation (Wdiss) processes occur: G = W + Wdiss. This energy loss may be due to an interfacial process, associated with van der Waals forces or in the case of polymers with the rupture of interfacial bonds and chain pullout,2-4 or to a bulk process, associated with the viscoelasticity of the material. One of the major difficulties in contact science is distinguishing among the different contributions to energy dissipation during dynamic contact, namely, the surface and volume contributions. Several studies5-7 concerning the velocity dependence (v) of the adherence energy G are in agreement with the empirical formalism of Gent and Shultz, which separates the interfacial and bulk contributions to G by defining a viscoelastic loss function: G = G0(1 + Φ(v)), where Φ(v) is the dissipative function of the material and G0 is the interfacial strength of the surface, i.e., the value of the strain adherence energy at vanishing crack speed. It is easier to determine the dissipative function on reversible substrates, for which the value of G at equilibrium is the same for *
[email protected]. (1) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324(1558), 301–313. (2) Creton, C.; Brown, H. R.; Shull, K. R. Macromolecules 1994, 27(12), 3174–3183. (3) Raphael, E.; De Gennes, P. G. J. Phys. Chem. 1992, 96(10), 4002–4007. (4) Leger, L.; Raphael, E.; Hervet, H. Adv. Polym. Sci. 1999, 138, 185–225. (5) Maugis, D.; Barquins, M. J. Phys. D: Appl. Phys. 1978, 11(14), 1989–2023. (6) Ahn, D.; Shull, K. R. Macromolecules 1996, 29(12), 4381–4390. (7) Deruelle, M.; Hervet, H.; Jandeau, G.; Leger, L. J. Adhes. Sci. Technol. 1998, 12(2), 225–247.
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loading and unloading (G0 = W). Such classical JKR studies are performed at thermodynamic equilibrium, and the viscoelastic loss function determined is related to the creep of the materials. Some recent publications (experimental and theoretical) point to an increasing interest in including viscoelastic and dynamic effects in JKR experiments.8-15 Whatever the probe scale, viscoelastic behavior is often studied in indentation tests due to forcedisplacement or contact radius-force curves via the JKR theory to finally exhibit the link between the measured adherence energy and the crack speed. In their simulations of a contact between two viscoelastic solids, Lin and Hui pointed out that the unloading rate was more important in controlling the energy hysteresis than the loading rate.14 The bulk dissipation used to be studied by changing the volume characteristics of the materials. However, there is another way of proceeding, which is to use the time and temperature dependence of their viscoelastic properties. The purpose of this work was to use an original “dynamic JKR” apparatus to analyze the energy dissipated (surface and volume) during an adhesive dynamic JKR test, i.e., contact between a flat surface and a PDMS hemisphere submitted to an oscillating normal load. One particularity of our JKR apparatus16 is that it allows us to vary the viscoelastic properties of the materials by adjusting the temperature and frequency of the force waveform. It is possible to measure an adhesion energy hysteresis defined by ΔW = Wul W1 ≈ G - W by using the JKR model to adjust the loading (Wl) and unloading (Wul) curves of the dynamic test. Keeping in mind that a JKR test is basically a fracture test with a weak fracture energy, in simple JKR experiments the contact radius, load, and normal displacement are recorded. Unfortunately, the JKR theory requires a knowledge of the bulk modulus (8) Barthel, E.; Haiat, G. J. Adhes. 2004, 80(1-2), 1–19. (9) Ebenstein, D. M.; Wahl, K. J. J. Colloid Interface Sci. 2006, 298(2), 652–662. (10) Greenwood, J. A.; Johnson, K. L. J. Colloid Interface Sci. 2006, 296(1), 284–291. (11) Vaenkatesan, V.; Li, Z.; Vellinga, W. P.; de Jeu, W. H. Polymer 2006, 47 (25), 8317–8325. (12) Wahl, K. J.; Asif, S. A. S.; Greenwood, J. A.; Johnson, K. L. J. Colloid Interface Sci. 2006, 296(1), 178–188. (13) Basire, C.; Fretigny, C. Tribol. Lett. 2001, 10(3), 189–193. (14) Lin, Y. Y.; Hui, C. Y. J. Polym. Sci., Part B: Polym. Phys. 2002, 40(9), 772–793. (15) Morishita, Y.; Morita, H.; Kaneko, D.; Doi, M., Langmuir . (16) Charrault, E.; Gauthier, C.; Marie, P.; Schirrer, R. In 30th Annual Meeting Proceedings, 30th Annual Meeting of the Adhesion Society, Tampa, Florida, 18-21 Fevrier, 2007; Tampa, Florida, 2007; pp 204-206.
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derived from the normal displacement, which depends on the properties of the whole half-spherical sample and the stiffness of the apparatus and hence cannot be measured with precision. Moreover, there are high strains in the rubbery sphere at the interface, in the region where the energy is dissipated. The elastic modulus of the rubber is nonlinear at high strain, and therefore, the mean value of the modulus obtained by determining the simple normal displacement is not helpful. This poor precision is similar to that of an ordinary tensile test with “dog bone” necking of the sample and without measurement of the local strain in the neck or to that of a nonlinear fracture test in which the nonlinear local strain at the crack tip is unknown. Precise fracture mechanics require a knowledge of the local strain at the crack tip and thus give results that are independent of the sample geometry. Similarly, the JKR test with local measurements at the contact boundary should give precise results in a manner independent of the geometric properties of the sample. Therefore, another important particularity of our apparatus is to allow local interferometric displacement measurements close to the circle of the contact boundary. In addition, as in common nonlinear fracture tests performed under loading and unloading with crack closure, the use of cyclic loading with a large frequency range in a JKR test has a great advantage: some bulk mechanical properties close to the crack tip may be determined during the loading phase and then used in the model to estimate more precisely the fracture energy during unloading and crack propagation. Last but not least, the model of cyclic loading over a large frequency range in a JKR test is close to real situations such as the sensory properties of materials or the complex dynamic adhesion mechanisms encountered in the case of structured soft surfaces. In the first part of this paper, we give a description of the dynamic JKR apparatus. We then report the macroscopic analysis of the typical results of an experiment with classical radius of contact versus load profiles. After that, we discuss the evolution of the adhesion hysteresis as a function of the frequency. Finally, a local analysis involving interferometric fringes is described and not only emphasizes the macroscopic results, but also justifies the use of the elastic JKR model to study our contacts showing some viscoelastic behavior. Some details of the macroscopic analysis are given in the Appendix.
Materials and Experimental Techniques Experimental JKR Setup. The experimental device (Figure 1) brings an elastomeric hemisphere deposited on a base connected to a force sensor into contact with a horizontally fixed plane surface. Observations are made from above the plate, perpendicular to the plane of contact, using a microscope and a CCD camera. The contact zone is lit by a source of incoherent white light or monochromatic sodium or He-Ne laser light, which allows interferometric analyses. The atmosphere of the measuring cell is controlled and its temperature can be varied from -30 to 60 °C by cooling or heating with, respectively, liquid nitrogen or non-humid, oil-free compressed air. The objective is to carry out several cycles of loading-unloading through an elastomeric lens onto a rigid surface. The force applied is triangular, sinusoidal, or square with a frequency adjustable between 0.02 and 100 Hz. After application of an initial force and measurement of the initial contact radius ai, the CCD camera records the evolution of the contact radius a as a function of the oscillating applied load 4P. In high-frequency tests ( f > 1 Hz), the camera uses a stroboscopic acquisition procedure which permits the analysis. Whatever the frequency, the number of force cycles studied is brought back to two. The stability of the 5848
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Figure 1. Experimental “dynamic JKR” device. The applied load (P) and associated radius of contact (a) are measured.
Figure 2. In the case of contact between PDMS and glass at 10 Hz and 30 °C, a plot of the load versus the contact radius shows that the JKR (unloading) and Hertz (loading) best fits are independent of the amplitude of the cyclic loading and of the mean value of the static load superimposed on the dynamic load. Depending on the modulus of the elastomeric material, the confidence interval of the adhesion energy calculated using the JKR fit is about 0.001 J/m2 < δW < 0.03 J/m2.
cycles, i.e., the reproducibility of the a = f(P) profiles during a number of loading and unloading cycles, is checked. Experimental Measurements. In the present work, to study self-adhesion of PDMS, PDMS hemispheres of radius 12.5 mm were pressed against a plate, and for each hemisphere, a series of measurements was performed using a triangular force at frequencies ranging over 4 decades (from 0.02 to 10 Hz) and constant temperature (-30, 0, 30, or 60 °C). During the unloading process, the PDMS hemisphere remains stuck to the surface and the crack at the interface starts to grow only when a certain tensile load is reached. The amplitude of the cyclic loading must therefore be sufficiently large to ensure a steady-state mechanism during both the loading and unloading phases. The data of Figure 2 were obtained under three different triangular cyclic loadings large enough to achieve this steady-state situation and show that, for a high-frequency contact between PDMS and glass, the JKR and Hertz fits remain independent of the amplitude of the cyclic loading and of the mean static load superimposed on the cyclic load. Polymer, Sample Preparation, and Dynamic Bulk Mechanical Properties. The polymer used was R,ω-divinyl-terminated polydimethylsiloxane (PDMS, Rhodia, Rhodorsil RTV141 A+B). Cross-linked sampled were prepared from a silicone base and curing agents and is supplied in a two-part kit consisting of liquid components. The base and curing agents were mixed in a weight ratio of 10 parts base to 1 part curing agent with stirring for about 2 min. These proportions ensure an optimal elastic network. The mixture was then placed under vacuum for 30 min to remove air bubbles. A 25 mm diameter steel ball was indented into a mold plate made of PMMA, and the mixture was poured onto the plate and cured at 80 °C for 2 h in an oven. The resulting Langmuir 2009, 25(10), 5847–5854
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by about 2%. The measurements were performed at temperatures ranging from -30 to 60 °C and different frequencies (0.02, 0.2, and 2 Hz) and results are shown in Figure 3. According to the linear theory of elastomers, E is proportional to the temperature. Although the PDMS network is considered to be a good elastic material (which is true at high temperatures), it is obvious (Figure 3a) that the more closely we approach the glass transition temperature (Tg = -130 °C) the more the E value departs from the theory. This may be explained by the viscoelasticity of the material. The energy dissipation due to the viscoelastic properties of the polymer can be represented by the loss factor tan(δ), which is defined as the ratio of the loss modulus (E00 ) to the storage modulus (E0 ) and can also be determined by spectroscopy [tan(δ) = E00 /E0 ]. Figure 3b depicts the evolution of tan(δ) as a function of the frequency for a PDMS sample, and Figure 3c shows the evolution of E00 at 30 °C.
Results and Discussion An adhesive contact of radius a under a load P may be defined by the JKR theory if the value of the Tabor’s parameter μ defined by: RW 2 μ≈ 2 3 E z0
!13 ð1Þ
where 1/R is the relative curvature of the contacting solids [1/R = 1/R1 + 1/R2], W is the surface energy, E* is the reduced modulus [1/E* = (1 - ν21)/E1 + (1 - ν22)/E2] (Ei and νi being, respectively, the Young’s moduli and the Poisson’s ratios of the materials), and z0 is the equilibrium surface separation (less than 1 nm), is larger than 5 (for large, compliant, high surface energy solids). On account of the use of large (R = 12.5 mm) and compliant (C = δ/P) hemispheres, the JKR model is applicable in our experiments (μ ≈ 3000). The purpose of the present work was to see whether this model could be used to analyze dynamic adhesive contacts. In the self-adhesion of PDMS experiments described here, as the radius of curvature is measured, the JKR equation defined by: a3 ¼ Figure 3. (a) Tensile modulus as a function of the temperature at different frequencies, measured by dynamic mechanical thermal analysis (DMTA) of a cylindrical PDMS sample. (b) Loss factor tan(δ) of a PDMS sample as a function of the frequency at different temperatures. (c) Evolution of the loss modulus of PDMS at 30 °C.
PDMS samples were immersed in 0.01% dodecanethiol in heptane for 10 h to remove free polymer chains, which as shown by Creton2 can be involved in adhesion hysteresis. After this cleaning procedure, the hemispheres were stuck onto a flat surface to prevent any slippage during the tests and avoid any nonplanarity of the base. The radius of curvature and the mean roughness of the samples were measured by optical profilometry and were, respectively, 12.5 mm and less than 20 nm. The thickness of the hemisphere was high enough that the sample could be considered a semi-infinite elastic body and the finite size effect could be disregarded. Two kinds of plane PDMS surface were prepared: thin plates were obtained by cross-linking of the PDMS mixture spread onto a glass surface using a 15 μm film maker, while thick plates were obtained by molding. The Young’s modulus (E) of the elastomer was determined by compression spectroscopy in a tensile machine. Cylindrical samples of PDMS (radius 5 mm, height 25 mm) were compressed Langmuir 2009, 25(10), 5847–5854
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R P þ 3πWR þ 6πWRP þ ð3πWRÞ2 K
ð2Þ
exhibits only two unknown variables: the elasticity constant K, defined by K = 4E*/3, and the adhesion energy W. If two identical materials are in contact, the interfacial term γ12 in the YoungDupre equation of energy is zero: W = γ1 + γ2 - γ12 = 2γ1, so that W is known and equal to twice the surface energy of the material (γ1). The surface energy of PDMS at ambient temperature is about 21.7 mJ/m2. Some contact angle measurements were performed to check the quality of the surfaces, and the results were in good agreement with the classical value for silicones. We therefore carried out all tests by bringing two PDMS samples (a hemisphere and a planar substrate) into contact, in order to avoid any effect of the interfacial term of the work of adhesion. Macroscopic Analysis. Young’s Modulus of PDMS in a JKR Test. Two PDMS-PDMS contact experiments were carried out, one on a thick layer of PDMS (almost 10 mm) and the other on a thin layer (about 15 μm), both deposited on a rigid glass surface. The ratio of substrate thickness to mean contact radius was 8 in the first case and 10-2 in the second, and owing to the incompressibility of the materials (νPDMS = 0.5), the respective K values of the system were K = (16/18)E and K = (16/9)E. At 0.02 Hz (Figure 4), the loading and unloading paths were DOI: 10.1021/la803434b
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Figure 4. Contact radius versus applied load for a PDMS hemisphere pressed against a compliant PDMS surface (O) and against PDMS deposited on a rigid substrate (b) at T = 30 °C and f = 0.02 Hz. In both cases, E = 1.65 MPa and Wl = Wul = 44 mJ/m2 for R = 12.5 mm.
identical, which means that the system was in ideal quasi-static JKR equilibrium with no energy dissipation.17,18 One may note the difference with the uniaxial compression spectroscopy value of E (Figure 3a, E = 1.85 MPa), due to the different geometrical configurations of the two experiments. In the JKR test, the apparent local modulus near the crack tip is not identical to the small strain uniaxial bulk modulus. Temperature and Frequency Effects. Certain properties of our material change with time and temperature. Among these, two are relevant to our experimental situation, the compliance of the network, which is related to Young’s modulus and the surface energy. Because of their weaknesses,19 the surface energy values variations were disregarded in our experiments. However, this was not the case for the volume property E (see Figure 3a). Therefore, as the value of K has a strong influence on that of W, we had to determine the evolution of this parameter with temperature and frequency (the method is explained in the Appendix). Several experiments were performed at ambient temperature and different frequencies over four decades, and the results are shown in Figure 5. One observes a transition between a nondissipative JKR test (at 0.02 Hz) with similar loading and unloading paths and a dissipative JKR test (at 1 or 10 Hz) where the loading and unloading paths are different. As the frequency of the test increases, the hysteresis cycle becomes larger. This means that, unlike in the low frequency test, there is some energy dissipation, which must be due to the bulk viscoelasticity of the material (on account of its weak variation over the range of our experimental conditions, the time and temperature dependence of the surface tension of PDMS may be completely neglected). The evolution of the adhesion hysteresis ΔW estimated in similar PDMS-PDMS contact experiments is represented in a master curve as a function of frequency at a reference temperature of 30 °C in Figure 6. One can see that the ΔW values display opposite behavior, decreasing with temperature and increasing with frequency. Under certain conditions, the values drop to almost zero, which means that there is no more energy dissipation. The precision of the measurement of W is about 0.005 J/m2 at low frequencies and about 0.030 J/m2 at high frequencies. If one considers the DMTA results for a PDMS sample and, in particular, those for the loss factor in Figure 3b, it is easy to see the (17) Deruelle, M.; Leger, L.; Tirrell, M. Macromolecules 1995, 28(22), 7419–7428. (18) Chaudhury, M. K.; Whitesides, G. M. Langmuir 1991, 7(5), 1013–1025. (19) Van Krevelen, D. V. Khimiya, Moscow 1976, 416.
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Figure 5. Contact radius versus applied load for contact between a PDMS lens and PDMS deposited on a rigid substrate at 30 °C. At low frequency (9, 0.02 Hz), the loading and unloading paths are similar (ideal reversible JKR system). At high frequencies (O, 1 Hz; 2, 10 Hz), an adhesion hysteresis loop appears.
Figure 6. Evolution of the adhesion hysteresis for contact between a PDMS lens and PDMS deposited on a rigid substrate: master curve at a reference temperature of 30 °C, the shift coefficients are varying from 0.5 (at 60 °C) to 16 (at -30 °C).
correlations between these two evolutions. In experiments far above the Tg, the solid behaves as an almost perfectly elastic network at low frequencies (30 °C and 0.02 Hz). As a consequence, it cannot dissipate any energy in the volume and ΔW = 0. Since increasing the temperature makes the network even more elastic, even if the frequency increases the energy dissipation still remains very low (low frequencies). On the contrary, as the system approaches the Tg (high frequencies) the loss factor progressively increases, which means that the viscoelastic behavior of the material becomes more important. Consequently, the energy dissipation represented by ΔW becomes increasingly larger to reach 2 J/m2 at -30 °C and 10 Hz. This dynamic JKR energy dissipation definitely originates from the bulk volume, since there cannot be any surface dissipation for two reasons. First, as the two materials in contact are identical, the test involves no creation of new bonds between the surfaces. Second, at very low frequency there is little or no dissipation, and hence, there cannot be any interdigitation of residual pendant chains on the two surfaces. ΔWDynamicJKR ¼ ΔWbulk
ð3Þ
when ΔWsurface = 0. This bulk dissipation may be associated with the dissipation at the crack tip zone, which is referred to as smallscale viscoelasticity.20 This would mean that the viscoelastic (20) Shull, K. R. Mater. Sci. Eng., R 2002, 36(1), 1–45.
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This propagation equation is based on that of Gent and Schultz24 and is applicable whatever the geometry of the adhesive test. The power law followed by the evolution of the dissipative function seems to be related to the loss modulus E00 of the viscoelastic material and the n exponents indeed appear to be the same.23 This kind of relationship has been studied in classical static JKR tests by many authors5-7 but is not yet well-understood. The dissipative functions derived from our experiments do not describe the kinetics of crack propagation, since the speed of propagation is imposed by a continuous unloading procedure with a controlled unloading rate dP/dt. In this case, we should speak of the dynamics of crack propagation. The difficulty is to correctly define the force applied per unit length of crack (G - W ), which makes the crack evolve, by choosing W as either W0 or Wl. Considering the dissipative function as Figure 7. Evolution of the adherence energy as a function of the crack propagation speed and the frequency at 30 °C.
dissipation zone is much smaller than the elastically deformed region of the solid and would explain why the JKR model, which only involves an elastic analysis, is nevertheless useful. Loading Frequency and Crack Velocity. The frequency of the force waveform is related to the periodical applied load and as such is not a pertinent parameter for this type of study. On the contrary, the crack propagation speed, which can be defined as da/dt, is directly related to the energy dissipation process. Two speeds of equal absolute value may be defined during the steady-state situations of an experiment: the first is linked to the crack recession (loading) and the second to the crack progression (unloading). Since the surfaces remain in contact during the whole process, no variation of the crack speed due to the approach of the pulloff conditions is seen. The evolution of the adherence energy G (during either loading or unloading) is identical as a function of either the crack propagation speed or the frequency. On the scale of our measurements, a frequency-speed superposition is readily established (Figure 7). For G = Wul, the evolution as a function of the crack velocity follows a scale law with an exponent of n = 0.36 ( 0.04. For G = Wl, this evolution is approximately constant (G ≈ W0) at speeds of less than 1 mm/s, while at higher speeds, it drops to almost zero (Wl ≈ 0 mJ/m2). This drop and the Hertzian behavior involved may be explained in terms of “sudden loading”, as formulated by Maugis in his representation of the equilibrium between penetration and radius of contact on reduced coordinates with superposition of the curves at constant load.21 The adhesive effects vanish during loading at high crack velocity. The same results are observed qualitatively in Greenwood and Johnson model on viscocoelastic contacts.22 These results are representative for the unloading branch and might not be similar to the adherence energy derived from the pulloff force measurements on viscoelastic contacts.10 Dissipative Function. In a study of viscoelastic bodies, in which the dissipative zones are located near the crack tip (as in the present work), Maugis and Barquins23 proposed the following equation to describe the crack propagation kinetics: G -W0 ¼ W0 ΦðaT vÞ ¼ W0 ðv=vÞn , where W0 is the adhesion energy at vanishing crack speed, v is the crack speed, v* is a characteristic speed, and aT is the shift factor of the WLF equation. (21) Maugis, D. Contact, Adhesion, and Rupture of Elastic Solids. Springer: Berlin, 2000. (22) Greenwood, J. A. J. Phys. D: Appl. Phys. 2007, 40(6), 1769–1777. (23) Maugis, D. J. Mater. Sci. 1985, 20(9), 3041–3073. (24) Gent, A. N.; Schultz, J. J. Adhes. 1972, 3(4), 281–294.
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ðG -W0 Þ=W0 ¼ ðWul -W0 Þ=W0 power law exponent n ¼ 0:36 ( 0:02 ðG -W0 Þ=W0 ¼ ðWul -Wl Þ=Wl power law exponent n ¼ 0:52 ( 0:02 The evolution of E00 (Figure 3c) can also be represented by a power law (as a function of the frequency) with an exponent n = 0.36 ( 0.02. Thanks to the speed-frequency equivalence, it is thus possible to establish the connection between the frequencydependent parameter E00 (ω) and the speed-dependent parameter Φ(v) defined by. (Wul - W0)/W0. Local Confirmation: Crack Tip Interferometry. Using a source of He-Ne laser light (λ = 632.8 nm), it was possible to carry out some efficient interferometric analysis. Newton’s ring patterns, provided by the interferometric system, give information about the shape of the lens outside the contact zone. Indeed, fringes of equal intensity are located in zones where the air gap thickness (e) equals e = kλ/2, where k is a natural, could indicate an effective way to determine the appropriate model to use in these types of adhesive tests. Thus, one important difference between the common contact theories concerns the prediction of the deformation of a sphere while in contact with a surface. Whereas in the Hertzian and DMT theories, the surfaces are tangential to the plane of contact, there is a right angle contact in the JKR model. The theoretical Hertzian and JKR separations z(r) of the surfaces outside the contact zone are defined by the following equations:25 2 3 !12 ! !12 2 2 2 2 a 6 r r r 7 Hertz : zðrÞ ¼ 4 2 -1 þ 2 -2 arctan 2 -1 5 πR a a a ð4Þ 2
!12 a2 6 r2 JKR : zðrÞ ¼ 4 2 -1 þ πR a 0 3 0 132 1 !12 1 2 2 2 3 r 4 ð6πWR =KÞ r B 7 AC @ 2 -2 þ @ A arctan 2 -1 5 ð5Þ 3 a a a
(25) Horn, R. G.; Israelachvili, J. N.; Pribac, F. J. Colloid Interface Sci. 1987, 115(2), 480–492.
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Figure 8. (a) Laser interferometric fringes during loading at f = 0.02 Hz and 30 °C. (b) z(r) profiles determined from interferometric
fringes (0) and calculated for Wl = Wul = 44 mJ/m2 and Wl = 0. f = 0.02 Hz, 30 °C, E = 1.65 MPa. (c) Analogous z(r) profiles for f = 1 Hz and 30 °C. E = 1.68 MPa, Wul = 90 mJ/m2. (d) Laser interferometric fringes during loading at f = 10 Hz and 30 °C. (e) z(r) profiles determined from interferometric fringes during loading (0) and calculated for Wl = Wul = 44 mJ/m2 and Wl = 0. f = 10 Hz, 30 °C, Wl = 0.
Figure 8a-e summarizes the different situations observed in our PDMS-PDMS contact experiments at 30 °C. On each profile, the ideal JKR case is compared to the Hertzian and experimental JKR cases. In low-frequency tests, in the quasi-equilibrium state (Figure 8a,b), small, closely spaced fringes are visible and the interference pattern is identical during loading or unloading, as equilibrium is reached at all times. There is a weak adhesion effect and the lens is slightly deformed outside the contact zone. In this example of a very low frequency test, the situation corresponds to that of an ideal reversible JKR system and Wl = Wul = 44 mJ/m2. The local tensile modulus may be derived from this experiment and is 1.65 MPa, consistent with the macroscopic modulus derived from Figure 4. In high-frequency tests (Figure 8d,e), one observes large, wellspaced fringes during loading, which implies a very small angle between the two surfaces and no deformation of the lens. This situation corresponds to a Hertzian model with Wl ≈ 0.00 J/m2. During unloading, at the resolution of the He-Ne laser, the fringes are not visible, because they are too thin and too closely spaced. 5852
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In intermediate-frequency tests (Figure 8c), the interferometric pattern is not Hertzian during loading (0 < Wl < 44 mJ/m2), and since there is generally some adhesion hysteresis, the unloading fringes are even thinner than in the ideal JKR case (Wul > 44 mJ/m2). This means that the deformation of the lens is important and the contact angle between the two surfaces is large (almost 90°). Fitting the interferometric profiles represents a completely independent way of estimating both the adhesion energy and the constant elasticity locally. Figure 9 shows the accuracy of the match between the two different ways of determining the adhesion energy: from the fit of the a3 curves or the fit of the interferometric profiles. Location of the Energy Dissipation: The Dissipative Volume. Since the energy dissipation is thought to take place near the crack tip in the vicinity of the surface, the interferometric fringe profiles, which provide information about the deformation of the sample outside the contact area, may be used to estimate a dissipative volume. According to fracture mechanics, the crack propagation results in an ellipsoidal deformation of the sample (elliptical hole: R long;in our case, infinite axis-, β wide). The radius of Langmuir 2009, 25(10), 5847–5854
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Figure 9. Comparison of the ΔW values estimated using two different methods: the fit of the a3 = f(P) curves and the fit of the interferometric profiles.
curvature r at the crack tip during this deformation may be estimated geometrically as shown in the inset of Figure 10. The characteristic length of the dissipative volume in a dynamic experiment may be defined as the difference between the radii of curvature in the unloading (rul) and loading (rl) phases. As seen in Figures 2 and 5, for a test at one temperature and one frequency, one value of the adhesion energy Wul is estimated during unloading and is independent of the radius of contact. Consequently, the geometry of the problem is axisymmetric. Moreover, during unloading, the value of Wul is associated with a constant crack propagation speed da/dt (Figure 7). The theoretical JKR profiles are supposed to be representative of an adhesive contact of radius a under load P. In our experiments, during each steady situations (loading or unloading) one adherence energy value was associated with different (a, P) contacts. We checked whether one specific radius of curvature of the hemispherical deformation could be determined. We found that, either experimentally or theoretically, the profiles were slightly different in a range far from the contact edge (from a distance x where x > 1.05a). However, close to the contact edge (x ≈ a), the profiles were equivalent, indicating that the radii of curvature are the same. Hence, just like the adhesion energy and the crack velocity, the radius of curvature of the deformation is constant during the unloading. This means that the local deformation remains the same during the unloading step, which supports the idea that one crack propagation speed corresponds to one dissipative volume and consequently one adherence energy. Using the theoretical profiles, it is easier to determine the different values of the radius of curvature associated with the adherence energy than by geometrical estimation. The curves in Figure 10 were calculated for a radius of contact of 1 mm and three different adhesion energies Wul1 < Wul2 < Wul3. Figure 11 depicts the evolution of the adherence energy as a function of the radius of curvature at the crack tip. The linearity observed is a well-known characteristic of linear elastic fracture mechanics,26 where the adherence energy is proportional to the sharpness of the crack. This also justifies that, even if there is some viscoelastic dissipation located near the crack tip, as long as the whole solid mainly behaves in an elastic manner the contact may be accurately described by the elastic theories. (26) Anderson, T. L., Fracture Mechanics: Fundamentals and Applications; CRC Press: Boca Raton, 2005.
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Figure 10. Radius of curvature of the deformation for several values of the adhesion energy. The representation of the ellipsoidal crack with semi axes R and β showing the radius of curvature of the deformation rc at the crack tip is in the inset.
Figure 11. Evolution of G = Wul as a function of the radius of curvature at the crack tip at 30 °C.
Conclusion Using an original “dynamic JKR” apparatus and a powerful local interferometric analysis, we have shown by demonstrating agreement with some fundamental characteristics of linear elastic fracture mechanics that a JKR approach is useful to study the viscoelastic energy dissipation during dynamic normal contact between two 3D networks of PDMS (a flat surface and a hemispherical lens). The viscoelastic behavior of the material was checked by controlling the time and temperature dependence of the polymer properties. Values of the adhesion hysteresis, associated with given crack propagation speeds, were estimated and seemed to follow the same evolution as the PDMS loss factor tan(δ). An energy calculation and interferometric data suggest that the energy dissipation volume is located at the perimeter of the contact area and is related to the radius of curvature of the crack tip. This kind of study could also be useful to assess the adhesive effects during loading or unloading, i.e., adhesion or adherence. Once the energy dissipation in the bulk viscoelastic material is known, one may determine the surface dissipation by changing the surface properties of one of the solids (surface treatment of either the flat surface or the hemisphere): ΔWsurface ¼ ΔWtotal -ΔWbulk
ð6Þ
Acknowledgment. The authors thank D. Favier for improvements in the “dynamic JKR” apparatus. DOI: 10.1021/la803434b
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Appendix Estimating the Tensile Modulus in a JKR Test
determine the evolution of E with frequency at a given temperature by calculating the following ratio:
The difficulty in such tests is to assess the adhesion energy term while taking into account the changes in the Young’s modulus of the network induced by changes in the temperature and/or the frequency of the force waveform. One of the major differences between traditional JKR adhesion tests (at quasi-static equilibrium) and dynamic tests like those presented in this study lies in determining the elastic modulus E of the network. It is of utmost importance to define a correct value of E since small fluctuations can cause significant variations in W (experimentally, ΔE = (10% f ΔW = (50%). The local interferometry technique is very powerful because it gives access to information about W and E without having to take into account parameters such as the geometry of the sample. Within the context of classical fracture mechanics, for which deformations near the crack are not equivalent to those within the material, obtaining information about the local deformation of the sample may represent a good way to determine E. Thanks to interferometric analyses and calculation of the loading profiles of a hemispherical lens at high-frequency (10 Hz), we know that the Hertzian model is applicable whatever the temperature (Wl = 0 J/m2). Consequently, the highest value of E is known and is equal to the value required to fit the loading curve a3 = f(P) with a Hertzian model. At 30 °C, the estimated value of E at 10 Hz is 1.72 MPa and is obviously a little higher than at 0.02 Hz. The unloading adhesion energy Wul can now be estimated by fitting the unloading curve using this new value of the elastic modulus (Wul = 0.40 J/m2). In intermediate frequency tests, it is more difficult to estimate E due to the influence of W. Using DMTA results, it is possible to
Eintermediate E0:02 Hz
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Applying this DMTA ratio to the calibration value of E obtained at 0.02 Hz in a dynamic JKR test, the different intermediate values of E may be estimated:
Eintermediate E0:02 Hz
¼ DMTA
Eintermediate E0:02 Hz
¼ constant JKR
At 10 Hz and 30 °C, this procedure yields the same value of E (1.72 MPa) and an adhesion energy Wl of less than 0.001 J/m2 (Hertzian contact).
Nomenclature a E, υ G K P r R tan(δ) v W Wl, Wul ΔW γ Φ(v)
contact radius elastic properties of the material adherence energy elastic constant applied normal load radius of curvature of the deformed profile radius of the hemispherical solid loss factor of the material crack propagation speed adhesion energy loading and unloading adhesion energy adhesion hysteresis surface tension dissipative function
Langmuir 2009, 25(10), 5847–5854
