Contact Dynamics in the Adhesion Process between Spherical

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Langmuir 2008, 24, 14059-14065

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Contact Dynamics in the Adhesion Process between Spherical Polydimethylsiloxane Rubber and Glass Substrate Yoshihiro Morishita,*,† Hiroshi Morita,‡ Daisaku Kaneko,§ and Masao Doi† Department of Applied Physics, The UniVersity of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan, Nanosimulation Research Group, Nanotechnology Research Institute, National Institute of AdVanced Industrial Science and Technology, 1-1-1 Umezono, Tsukuba, Igbaraki, 305-8568, Japan, and School of Materials Science, Japan AdVanced Institute of Science and Technology, Nomi, Ishikawa, 923-1292, Japan ReceiVed July 28, 2008. ReVised Manuscript ReceiVed October 10, 2008 The contact dynamics between a soft sphere and a rigid substrate on the micron scale was studied experimentally. The time evolution of the contact radius, contact angle, and the force acting on the sphere were measured simultaneously in the loading and the unloading cycle. There is little effect of repetition: the experimental results obtained in the second and third cycles agree completely with those of the first cycle. The contact angle changes dynamically in the loading and the unloading processes, and there are regions where the advancing angle and the receding angle remain constant. The experimental results were analyzed in terms of the extended Johnson-Kendall-Roberts theory, and it was found, to our surprise, that the theory works well: the theory predicts the force curve quite accurately if the apparent surface energy obtained from the contact radius is used. The apparent surface energy was experimentally obtained as a function of the contact line velocity, and it was found that (1) the curve agrees qualitatively with that predicted by Greenwood and Johnson, and (2) certain modification is needed when the velocity of the contact line changes the sign.

I. Introduction The contact dynamics between a soft material and a rigid substrate is an important problem in various industrial applications such as adhesion and copying, and has been studied in both academia and industry.1-4 There is a resurging interest for the problem in connection to recent technologies such as micromolding and nanoimprinting: in the nanoimprinting process, it is required to detach the mold from the resist completely without breaking the material. In the contact of the soft materials, it has been realized that the viscoelastic effect in the bulk and the surface effect (i.e., the effects localized at the surface) are both equally important, but the interplay of these two effects are not well understood.5 Various tests have been used in the study of the contact of soft adhesive materials. A conventional test is the peeling test, in which the force needed to peel off the adhesive tape from the substrate is measured with varying the peeling rates and temperature. It has been shown by Gent6-8 and Kinloch9 that the result has a strong correlation with the viscoelasticity of bulk materials, and theoretical expression for the fracture energy has been given by de Gennes.10 This is a convenient test for evaluating the performance of adhesive tapes in the steady peel-off process to study the “bulk” effect. * To whom correspondence should be addressed. E-mail: morige@ rheo.t.u-tokyo.ac.jp. † The University of Tokyo. ‡ National Institute of Advanced Industrial Science and Technology. § Japan Advanced Institute of Science and Technology.

(1) Haiat, G.; Huy, M. P.; Barthel, E. J. Mech. Phys. Solids 2003, 51, 69–99. (2) Liu, K. K. J. Phys. D: Appl. Phys. 2006, 39, 189–199. (3) Barthel, E. J. Phys. D: Appl. Phys. 2008, 41, 163001. (4) Shull, K. R. Mater. Sci. Eng. R 2002, 36, 1–45. (5) Luengo, G.; Pan, J.; Heuberger, M.; Israelachvili, J. N. Langmuir 1998, 14, 3873–3881. (6) Gent, A. N.; Kinloch, A. J. J. Polym. Sci., Part A-2 1971, 9, 659–668. (7) Gent, A. N. Langmuir 1996, 12, 4492–4496. (8) Gent, A. N.; Petrich, R. P. Proc. R. Soc. London, Sect. A 1969, 310, 433– 448. (9) Kinloch, A. J.; Layu, C. C.; Williams, J. G. Int. J. Fract. 1994, 66, 45–70. (10) de Gennes, P. G. Langmuir 1996, 12, 4497–4500.

Other typical experiment is to measure the force needed to press a microscale spherical sample against substrate or to pull it off from the substrate by micron-compression apparatus,11 or more recently by the surface force apparatus (SFA)5,12 or by the atomic force microscope (AFM).13-15 This type of experiment is suitable for studying the surface effect, and has the advantage that the geometry of the contact line remains simple, although the process is not in the steady state and includes the nonuniform deformation of sphere. Interpretation of the contacting sphere experiments has been done by using the Johnson-Kendall-Roberts (JKR) theory.16 The JKR theory is an extension of classical Hertz theory17 to take into account of the surface energy w between the sphere and the substrate. Although the original JKR theory is for the equilibrium contact between elastic spheres, the theory has been used for the nonequilibrium contact. Schapery18-21 and Maugis and Barquins22 proposed that the irreversible contact dynamics can be treated if the surface energy w is replaced by the apparent surface energy wapp, which is the work needed to separate a unit area of the adhering surfaces. For soft materials, wapp is not constant, and varies depending on the velocity of the contact line. Greenwood and Johnson23-26 recently gave an explicit form (11) Briscoe, B. J.; Liu, K. K.; Williams, D. R. J. Colloid Interface Sci. 1998, 200, 256–264. (12) Horn, R. G.; Israelechvili, J. N.; Pribac, F. J. Colloid Interface Sci. 1987, 115, 480–492. (13) Vakarelski, I. U.; Toritani, A.; Nakayama, M.; Higashitani, K. Langmuir 2001, 17, 4739–4745. (14) Wahl, K. J.; Asif, S.; Greenwood, J. A.; Johnson, K. L. J. Colloid Interface Sci. 2006, 296, 178–188. (15) Burnham, N. A.; Dominguez, D. D.; Mowery, R. L.; Colton, R. J. Phys. ReV. Lett. 1990, 64, 1931–1934. (16) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Sect. A 1971, 324, 301–313. (17) Hertz, H. J. Reine Angew. Math. 1882, 92, 156. (18) Schapery, R. A. Int. J. Fract. 1975, 11, 549–562. (19) Schapery, R. A. Int. J. Fract. 1975, 11, 369–388. (20) Schapery, R. A. Int. J. Fract. 1975, 11, 141–159. (21) Schapery, R. A. Int. J. Fract. 1989, 39, 163–189. (22) Maugis, D.; Barquins, M. J. Phys. D: Appl. Phys. 1978, 11, 1989–2023. (23) Greenwood, J. A.; Johnson, K. L. Philos. Mag. 1981, 43, 697–711. (24) Greenwood, J. A. J. Phys. D: Appl. Phys. 2004, 37, 2557–2569.

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for the velocity dependence of wapp. They compared the results with the experiments of the force-distance curve obtained by AFM,14 but the comparison was not complete since the data of the contact radius and the contact line velocity were not available for the SFA and the AFM experiments. In order to conduct a critical test for the contact dynamics, it is preferable to do the experiment for much softer and larger spheres, in which the contact area in the deformation process can be measured by an optical microscope directly. Such an experiment was first done by Briscoe et al.:11 they measured the force-displacement curve as well as the contact area by combining the mechanical tester and microscope. Their analysis, however, was limited to the force curve, and the data for the contact area has not been analyzed in detail. Extensive studies were done by Deruelle et al.27,28 and Vaenkatesan et al.29 They measured the contact radius and the displacement simultaneously as a function of the force, and studied the curves such as the force against the displacement and the force against the contact radius in terms of the JKR theory. We constructed our own experimental system which measures the force and the deformation of the adhesive sphere simultaneously in the loading and the unloading processes. Compared with the earlier works, we used a very soft (the Young’s modulus being 0.06 MPa) and small (radius being 0.4 mm) sphere. This is to see the effect of the adhesion more clearly. We took an image of the sphere from vertical and horizontal directions, and measured the time evolution of the contact area and the contact angle together with the force acting on the sphere when the soft sphere was pressed against and pulled off from the glass plate. From the measurement of the contact area and force, we obtained the apparent surface energy using the extended JKR theory. In this paper, we report our experimental results and discuss how the apparent surface energy depends on the dynamics of the contact line, i.e., how the adhesion force depends on energy dissipations at bulk and interface.

II. Experimental Section A. Samples. Spherical samples of polydimethylsiloxane (PDMS) rubber were synthesized by emulsion polymerization. The PDMS monomer (Silpot 184, Dow Corning Toray Co.) with 2.5 wt % of cross-linker (Catalyst Silpot 184, Dow Corning Toray Co.) were mixed completely. The mixture was deaired with a vacuum pump, and the resulting clear mixture of PDMS monomer and cross-linker was poured into water and stirred, first for 8 h at 40 °C to make spherical PDMS droplets, and then for 6 h at 60 °C to cross-link the PDMS. The sample was then dried in the oven (60 °C) for more than 12 h. The surface of the resulting sphere was very smooth. In this method, the radius of PDMS spheres can be controlled to be anywhere between 100 µm and 1 mm by the speed of stirring. Most experiments reported in this paper were done for the particle of radius 410 µm. The bulk elasticity of the sphere can be controlled by the mixing ratio of the cross-linker. The elastic modulus of the sample was estimated by a rheometer (Haake Co.) for a cylindrical sample having the same mixing ratio. The Young’s modulus of our sample was 6.0 × 10- 2 MPa, which is very small compared with that of typical PDMS rubber. The softness of the rubber made the particle adhesive. B. Experimental Apparatus. Figure 1 shows our experimental set up. Spherical PDMS rubber was glued on the aluminum plate spring by a hard bond composed of R-cyanoacrylate monomer (25) Greenwood, J. A.; Johnson, K. L. J. Colloid Interface Sci. 2006, 296, 284–291. (26) Greenwood, J. A. J. Phys. D: Appl. Phys. 2007, 40, 1769–1777. (27) Deruelle, M.; Leger, L.; Tirrell, M. Macromolecules 1995, 28, 7419– 7428. (28) Deruelle, M.; Hervet, H.; Jandeau, G.; Leger, L. J. Adhes. Sci. Technol. 1998, 12, 225–247. (29) Vaenkatesan, V.; Li, Z.; Vellinga, W.-P.; de Jeu, W. H. Polymer 2006, 47, 8317–8325.

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Figure 1. Experimental setup: (a) The sample (spherical PDMS rubber) is pressed against the glass plate by the vertical motion of the Z-stage. The bending of the plate spring is measured by the strain gauge. The displacement d of the PDMS rubber relative to the glass substrate is given by d ) ∆ - δ, where ∆ is the displacement of the Z-stage, and δ is the displacement of the plate spring. The loading force (F) is calculated from the bending of the plate spring. (b)The imaging system to take the top view and the side view of the deformed PDMS rubber.

(Toagosei Co.). The size of the plate spring was 4.98 mm × 11.6 mm with thickness of 0.19 mm. The plate spring was fixed to the Z-stage, whose vertical position could be controlled within an accuracy of 2 µm. The PDMS rubber was pressed onto the glass plate fixed above or pulled off from it by the vertical motion of the Z-stage. The force acting on the PDMS rubber was measured by the strain gauge (Kyowa Co.) attached on the plate spring. The plate spring was calibrated using the correction weight, and the accuracy of the plate spring was within 10- 4 N. As the Z-stage is displaced, the plate spring is bent. The net displacement d of the PDMS rubber relative to the glass plate is given by d ) ∆ - δ, where ∆ is the displacement of the Z stage, and δ is the displacement of the plate spring (see Figure 1a). (The reference position for ∆ will be explained later). In addition to the mechanical measurement, the deformation of the PDMS rubber was observed from the top and from the side (Figure 1b) by two digital microscopes. The observed video images were stored in a PC and were used for the structural analysis. C. Experimental Procedure. We conducted the cyclic loadingunloading experiments as it is shown in Figure 2a. We first set the sphere at a position separated from the glass substrate, and start to move up the Z-stage with a constant speed of 100 µm/s. At point A, the sphere starts to contact the glass plate. We define this position of the Z-stage as the reference position, and set d and ∆ equal to zero (notice δ is equal to zero at this state). We keep moving up the Z stage further until ∆ reaches a certain preset value ∆max (point B1). We then reverse the velocity of the Z-stage immediately and move down the Z-stage, until the Z-stage comes to the reference position (point C1). At point C1, we reverse the velocity of the Z-stage again, and go into the second cycle. During these loading-unloading cycles, the magnitude of the speed of the Z-stage is kept constant. In the unloading process of the third cycle, the Z-stage is kept moving down. The adhesive force becomes the largest at point D, and the PDMS rubber is detached from the glass substrate at point E. Figure 2b shows the side view of the sphere for states A-E.

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Figure 4. Top view of the contact area.

Figure 2. (a) Control of the Z-stage and the observed force F. The Z-stage is moved vertically upward and downward repeatedly three times, as indicated by the solid line. The measured force F is shown by the circles. (b) Side view of the sphere for states A-E indicated in panel a.

Figure 5. Definition of the contact radius (r) and the contact angle (θ).

Figure 3. Typical displacement-load curve. The maximum displacement is dmax ) 201 µm.

We repeated the experiments, changing dmax, the maximum value of the displacement. The adhesive force is characterized by the absolute value of the force at point D, which is denoted by Fad.

III. Experimental Results A. Typical Behavior in the Loading and the Unloading Cycle. Figure 3 shows an example of the plot of the force F against the displacement d for the loading-unloading cycle. It is seen that after the transient part of the first loading (the part near point A in Figure 3), the curves overlap with each other perfectly in the loading cycle. The force is zero at the reference position, where d ) 0, in the first cycle, while it is nonzero in the second and third cycles. The deviation, however, disappears

quickly as the loading proceeds, and the curve overlaps perfectly in the rest of the cycles. The perfect overlapping of the force curve indicates that the contact of the adhesive sphere does not alter the surface property of the substrate or the internal structure of the PDMS rubber. This is an attractive feature of our system. Although the adhesion may be a complex phenomena having hysteresis, the present system is clean and robust: the experimental results are quite reproducible. Figure 4 shows the top view of the deformed sphere. It is seen that, despite the large deformation, the shape of the contact area remains close to a circle. To characterize the deformation of the sphere, we measured the radius r of the contact area and the contact angle θ as a function of the displacement d. They are defined as it is explained in Figure 5. The definition of θ has some uncertainties, as it depends on the length scale of observation. To avoid the artifacts in the measuring process, the contact angle in our experiments were measured digitally in the region within 15 µm. In the side view image, the first line, which is the boundary between the glass and the sphere, was obtained as an almost horizontal line. In the same image, the surface points of the sphere were defined as the digitized points in every 2 µm within 15 µm from the first line. An averaged line passing through the obtained points was obtained as the second line. Our contact angle was defined as the angle between the two lines shown in Figure 5.

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Figure 6. The displacement-load curves for various dmax values are plotted in the same graph. Different symbols correspond to different values of dmax.

Figure 8. Change of the contact radius (a), and the contact angle (b) in the loading-unloading cycle. dmax ) 201 µm.

Figure 7. The adhesion force Fad (the maximum value of -F) and the maximum contact radius are plotted against maximum displacement dmax.

B. Effect of the Maximum Displacement. Figure 6 shows the plot of the load F against d for various dmax values ranging from 57.0 to 375.0 µm. For clarity of the plot, only the curves of the first loading process (the curve from point A to B1) and the curve of the last unloading process (the curve from points B3, D, and E) are shown. It is seen that the curves for different dmax values coincide with each other in the loading process. This again indicates that the contact of the adhesive sphere does not leave any irreversible change at the surface of the glass substrate. In the unloading process, the curves for different dmax values do not overlap completely. Deviation is seen in the region where the negative force -F becomes the largest. For larger dmax, however, the curves seem to converge to a common curve independent of dmax. Figure 7 shows the plot of the maximum adhesion force Fad, which is the largest value of -F, against the maximum displacement dmax. The figure also includes the maximum contact radius, the radius of the contact area at point B. It is seen that, as the maximum displacement increases, the maximum contact radius increases monotonically, while the maximum adhesion force approaches a constant value. The increase of the maximum adhesion force is considered to be due to the increase of the contact time rather than the increase of the maximum load: the force in the present experiment is too weak to cause any

irreversible change on the surface. Indeed the force in the second and the third cycles of the unloading follows the same curve as that in the first cycle (see Figure 3). It has been observed that the maximum adhesion force increases with the increase of the contact time, and the phenomena has been interpreted in terms of the molecular diffusion at the surface,5 or the viscoelastic effect in the region near the contact line.30 C. Contact Radius and Contact Angle. Figure 8 shows how the contact radius and contact angle change in the loading and the unloading processes. Points A-E in Figure 8 correspond to points A-E in Figure 3. When the loading starts (the plus symbol in the figure), the contact radius first increases quickly (arrow 0 in panel a), and then slowly (arrow 1 in panel a). The contact angle, on the other hand, first decreases a little (arrow 0 in panel b), and then stays almost constant (arrow 1 in panel b). When the first unloading starts at point B, the contact radius first decreases little (arrow 2 in panel a) and then starts to decrease drastically (arrow 3 in panel a). In contrast to this, the contact angle first decreases drastically (arrow 2 in panel b), taking the value of about 90°, and then remains almost constant (arrow 3 in panel b). (The contact angle remains about 90° also in the final cycle before the sample is detatched from the substrate; see the arrow connecting points C, D, and E in Figure 8). When the second loading cycle starts at point C, the contact radius does not change much (rather it decreases despite the fact that the PDMS sphere is now pressed against the glass plate; see arrow 4 in panel a). On the other hand, the contact angle increases (30) Barthel, E.; Haiat, G. Langmuir 2002, 18, 9362–9370.

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Figure 9. The contact angle is plotted against the contact line velocity.

significantly during this process (arrow 4 in panel b). After a while, the curves of the contact radius and that of the contact angle overlap with those of the first loading cycle. To summarize, the cycle of the loading and the unloading process can be classified into four regions, each indicated by arrows 1-4 in Figure 8. Immediately after the unloading starts (region 2), the contact line is almost pinned, while the contact angle changes greatly. In this region a large deformation of the sphere takes place near the contact line. After some time (region 3), the unloading process becomes quasi steady: the contact radius changes with constant rate, and the contact angle remains constant. Similar behavior is seen in the loading cycle. Immediately after the loading starts (region 4), the contact radius changes little, while the contact angle changes widely. After a while (region 1), the contact radius starts to change at a constant speed, while the contact angle remains almost constant. Such behavior can be qualitatively understood. Because of the adhesive force, the PDMS rubber tends to minimize the change of the contact area. Therefore, when the mode is switched from unloading to loading or vice versa, the PDMS rubber responds to the change of the external conditions by changing the contact angle (or by changing the shape near the contact line) without changing the contact radius. As the loading or unloading force increases, this becomes impossible, and the contact line starts to move. Once the contact line starts moving, the contact angle remains nearly constant. D. Advancing Angle and Receding Angle. Figure 9 shows the contact angle as a function of the velocity of the contact line. The contact line velocity is calculated from the time variation of the contact radius r(t) by V ) dr/dt. Although the experimental error of the contact angle is not small, the following characteristics is clearly seen: (a) When the contact area is increasing, the contact angle takes a steady-state value of around 160°. (b) The same trend is seen when the contact area is decreasing: the contact angle also takes a steady state value around 100°, although there is a considerable scatter in the experimental data. (c) On the other hand, the contact angle does not seem to have a fixed value when the contact velocity is nearly zero, or the contact angle increases sharply for small change of the contact velocity. These features are similar to the behavior of the contact angle in liquid. The steady state values of 160° and 100° correspond to the advancing angle and the receding angle, respectively. Our results suggest that the similar concept is useful for the dynamics of the adhesive contact of viscoelastic bodies.

Figure 10. Comparison between the extended JKR theory and the experimental results: experimental results are denoted by symbols, and the results of the JKR theory (eqs 1 and 2) are denoted by the dashed line and solid line.

IV. Discussion A. Comparison with the JKR Theory. We first analyze the experimental result using the JKR theory. The original JKR theory is an equilibrium theory for an elastic adhesive sphere having the contact energy w, and gives the following expression for the displacement d and the loading force F as a function of the contact radius r:

d) F)

8πwr r2 R 3K

(

1⁄2

)

Kr3 - r3⁄2(6πKw)1⁄2 R

(1) (2)

where K ) 4E/3(1 - ν2), and R, E, and ν are the radius of the sphere, the Young’s modulus, and the Poisson’s ratio, respectively. To compare the JKR theory with our experimental result, we use the K and w as the fitting parameter, and obtained the best fitting for the regions where the contact angle is nearly constant (regions 1 and 3). The result of the fitting is shown in Figure 10. In the loading process,we were able to get a nearly perfect fitting with the JKR theory, as seen in Figure 10. Notice that the maximum displacement here is 201.0 µm, which is about half of the sphere radius. Good fitting was obtained for even larger displacement such as dmax = 350 µm. It is remarkable that the JKR theory, which is based on the linear elastic model, works for such large deformation. This is perhaps due to the fact that our sample is very soft, and the elastic behavior is described by

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the Neo-Hookian model.31 The fitting gave E and w as E ) 6.0 × 10- 2 MPa and w ) 3.0 × 10- 3 J/m2 (ν was taken to be 0.5). The value of E is consistent with the bulk moduli obtained by the rheometrical measurement, the measured storage modulus (G′ ≈ 3 × 10-2 MPa) and loss modulus (G′′ ≈ 1 × 10-2 MPa), which gives E ) 3G ) 9 × 10- 2 MPa. Similar fitting was done for the unloading curve. In this case, we used the same value for the Young’s modulus E as that of the loading process, and regarded w as the only fitting parameter. The fitting was successful in the region where the contact angle is almost constant (the region indicated by arrows 3 and 5), but fails completely in the region where the contact angle changes widely (the region indicated by arrows 2 and 4). The apparent surface energy w obtained for the unloading process was 5.0 × 10- 1 J/m2, which is much larger than the value obtained for the loading process. This indicates that the adhesive force is not as important in the loading process, while it is very important in the unloading process. Although the experimental results can be fitted with the JKR theory reasonably well, the fitting is not perfect. In the next section, we shall re-examine the fitting assuming that w can vary with time. B. Apparent Surface Energy. In the previous fitting of the experimental results with the JKR theory, we used a constant value for the apparent surface energy wapp in the loading and the unloading processes, respectively. If wapp is regarded as a parameter determined by the JKR theory, it can be calculated by

wapp )

( )

3K r2 -d 8πr R

2

(3)

The JKR theory predicts that the load F is give by

F)

Kr3 - r 3 ⁄ 2(6πKwapp)1 ⁄ 2 R

(4)

Figure 11a shows the apparent surface energy wapp calculated by eq 3. It is seen that wapp is nearly 0 in the loading process, i.e., the effect of adhesion is negligible in the loading process. On the other hand, wapp takes a large value and changes widely in the unloading process. Figure 11b shows the comparison between the load calculated by eq 4 (filled symbols) and the experimental value (open symbols). In the calculation, we used E ) 6.0 × 10- 2 MPa and ν ) 0.5. The calculated value agrees quite well with the experimental result. Good agreement was also obtained for other values of dmax. Deviation is seen for the case of larger displacement of dmax > 375 µm, where the linear analysis, which is the base of the JKR theory, breaks down. Again it should noted that the linear analysis of the JKR theory works surprisingly well. The above results suggest that if the apparent surface energy wapp is used, the loading and the unloading process of the adhesive contact of viscoelastic spheres can be described by the JKR theory. As it has been discussed by Greenwood and Johnson, applying the JKR theory for the irreversible process using the apparent surface energy wapp is equivalent to assuming that the major part of the energy dissipation is caused near the contact line: wapp stands for the energy dissipated in the system when the contact area is increased or decreased by unit amount. Notice that the apparent surface energy wapp includes the contribution of the dissipation taking place in the bulk region near the contact line; in fact, this part is much larger than the normal surface energy, the interaction energy between the atoms in the small range (of nanometer order) near the interface. (31) Lin, Y. Y.; Chen, H. Y. J. Polym. Sci., Part B: Polym. Phys. 2006, 44, 2912–2922.

Figure 11. (a) The apparent surface energy calculated by eq 3 is plotted against the displacement. (b) The load calculated by eq 4 (filled symbols) is compared with the experimental results (open symbols).

Figure 12. The apparent surface energy wapp is plotted against the contact line velocity. In this graph the velocity of the contact line is expressed as an absolute value both in the loading and the unloading processes. dmax ) 201 µm.

In the conventional model, the apparent surface energy is assumed to be a function of the contact line velocity V. To check this assumption, we plotted wapp as a function of the contact line velocity V ) r˙. This is shown in Figure 12. In the loading process, wapp gradually decreases as the contact line velocity increases. On the other hand in the unloading process, wapp increases with the increase of the contact line velocity. These results agree qualitatively with the model of Greenwood and Johnson. However, deviation from the model is seen in the

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region where the velocity of the contact line is almost zero. This is the region where the displacement mode is changed from loading to unloading or vice versa. In such a region, the contact line is nearly pinned, and the contact angle changes from the previous steady-state value to the new value. Accordingly, a large energy dissipation is caused at bulk near the contact line, and this contributes to the apparent surface energy wapp. This situation may be taken into account by assuming that wapp depends on not only the contact line velocity V ) r˙, but also on the change of the contact angle θ˙ , i.e., wapp ) wapp(r˙,θ˙ ).

V. Conclusions In this paper, we have studied the contact dynamics between a soft sphere and flat substrate. We observed the dynamics of the contact line and the deformation directly by digital microscopes and analyzed the results using the extended JKR theory. We have found that the loading-unloading curve can be fitted quite well by the JKR theory if the apparent surface energy wapp is used in the JKR theory. The apparent surface energy is a function of the contact line velocity V in a quasi steady state, but deviation is seen in the region where the contact line is nearly pinned.

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The fact that the force curve can be reproduced quite well by the extended JKR theory supports the basic idea of the theory: the energy dissipation of our system takes place only in the small region near the contact line, and the remaining part of the bulk of the sphere can be regarded as an ideal elastic material. Our experiments indicate that further consideration is needed in the region where the contact line velocity is nearly zero. In such a region, although the contact line is pinned, the contact angle can change, and this causes a large energy dissipation. Therefore, the main contribution to wapp is from this large energy dissipation at bulk near the contact line rather than from the surface effects in this region. This effect can be taken into account by assuming that the apparent surface energy depends not only on the rate change of the contact radius, but also on the rate change of the contact angle. Acknowledgment. The authors thank the Ministry of Education, Science, Sports and Culture for support through Grantin-Aid No. 20244067. The authors are grateful to the Japan Science and Technology Agency for supporting this study. LA8024155