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Deformation and Adhesion of Elastomer Microparticles Evaluated by AFM Ivan U. Vakarelski,† Akihiro Toritani,‡ Masaki Nakayama,‡ and Ko Higashitani*,† Department of Chemical Engineering, Kyoto University, Yoshida, Sakyo-ku, Kyoto 606-8501, Japan, and Resins & Plastics Development Center, Mitsubishi Rayon Company, Ltd., Otake, Hiroshima 739-0693, Japan Received November 14, 2000. In Final Form: May 14, 2001 An atomic force microscope was used to investigate the adhesion behavior of a viscoelastic particle of micron size on a mica plate in water, by attaching a particle of polydiethylhexyacrylate on the end of the cantilever. Load-penetration curves were obtained by pressing the particle to the mica surface and retracting from it afterward. Our attention was especially focused on the individual contribution of the loading and unloading rates, the maximum load, and the contact time of surfaces to the load-penetration curve. A significant hysteresis between the loading and unloading regimes was observed. Two energy dissipation processes were distinguished in this hysteresis; one was correlated with the viscoelastic loss during the detachment, and the other was correlated with the increase of the adhesive force with the contact time. To analyze the features observed, we employed the Johnson-Kendall-Roberts theory, which was modified to allow the increase of the interfacial energy with the contact time. The proposed model was found to give a very good qualitative description for the particle adhesion, which enabled us to estimate the time dependence of the specific work of adhesion as well as the elasticity modulus of particles.
Introduction The deformation of a spherical particle on a plate under an external force has been of long-standing interest. As for elastic spheres with smooth surfaces, Hertz1 developed a theory for the deformation of the surfaces around a contact point when an external force was applied between them. Later Johnson, Kendall, and Roberts2 proposed the JKR theory into which the contribution of the adhesive energy between deformed surfaces in contact was introduced, and the adequacy was justified by measuring the contact between smooth rubber and gelatin surfaces. Derjaguin, Muller, and Toporov3 proposed the DMT theory in which the long-range interaction force outside the contact area was taken into account. It was shown that the JKR and DMT theories correspond to two extreme cases of the more complete theory of Maugis and Barquins;4,5 the JKR theory is applicable to large particles with high surface energy, while the DMT theory is applicable to small hard particles. As for elastomer materials, Maugis and Barquins investigated the contact between viscoelastic bodies6-10 extensively, where spheres of a few centimeters or millimeters11 in diameter were employed to make the * To whom correspondence should be addressed. Tel: +81-75-753-5562. Fax: +81-75-753-5913. E-mail: higa@ cheme.kyoto-u.ac.jp. † Kyoto University. ‡ Mitsubishi Rayon Co., Ltd. (1) Hertz, H. In Miscellaneous papers; McMillan & Co.: London, 1986. (2) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301. (3) Derjaguin, B. V.; Muller, V. M.; Toporov, V. J. Colloid Interface Sci. 1975, 53, 314. (4) Tabor, D. J. Colloid Interface Sci. 1977, 58, 2. (5) Muller, V. M.; Yushchenko, V. S.; Derjaguin, B. V. J. Colloid Interface Sci. 1980, 77, 91. (6) Maugis, D.; Barquins, M. J. Phys. D 1978, 11, 1989. (7) Barquins, M.; Maugis, D. J. Adhes. 1981, 13, 53. (8) Barquins, M. J. Adhes. 1982, 13, 53. (9) Maugis, D. J. Mater. Sci. 1985, 20, 3041. (10) Maugis, D. J. Adhes. Sci. Technol. 1987, 2, 105.
contact area observable by an optical microscope. More microscopic measurements were carried out for the deformation of surfaces covered by a thin elastic layer, using the surface force apparatus (SFA).12,13 The SFA allows us to measure optically the contact radius with the resolution of about 1 µm and provides the profile of surface separation around the contact zone, but the dimension of cylindrical surfaces used here is still of the order of centimeters. The growing application of microscopic particles of low elastic modulus to various industrial processes asks us to measure the adhesion of particles of much smaller scale. Microscopic measurements were carried out by Rimai et al.14,15 using scanning electron microscopy, and the deformation of spheres induced by the surface energy was observed. In the recent decade, an atomic force microscope (AFM) has been extensively used to measure in situ the force between two surfaces in controlled gaseous or liquid environments. In AFM force measurements, solid surfaces are usually employed; either a cantilever tip or a particle glued on the cantilever end16 is chosen as a surface with a curvature, and the surface fixed on the piezo system is chosen as a flat surface. However, sometimes the AFM is employed to evaluate the interaction of a solid surface with a deformable body in water, such as a gas bubble17,18 or an oil droplet,19,20 and to evaluate the mechanical (11) Chaudhury, M. K.; Whitesides, G. M. Langmuir 1991, 7, 1013. (12) Horn, R. G.; Israelachvili, J. N.; Pribac, F. J. Colloid Interface Sci. 1987, 115, 481. (13) Luengo, G.; Pan, J.; Heuberger, M.; Israelachvili, J. N. Langmuir 1998, 14, 3873. (14) Rimai, D. S.; DeMejo, L. P.; Browen, R. C. J. Appl. Phys. 1989, 66, 3574. (15) Rimai, D. S.; DeMejo, L. P.; Browen, R. C. J. Appl. Phys. 1990, 68, 6234. (16) Ducker, W. A.; Senden, T. J.; Pasheley, R. M. Langmuir 1992, 8, 1831. (17) Ducker, W. A.; Xu, Z.; Israelachvili, J. N. Langmuir 1994, 10, 3279. (18) Butt, H. J. J. Colloid Interface Sci. 1994, 166, 109. (19) Mulvaney, P.; Perera, J. M.; Biggs, S.; Grieser, F.; Stevens, W. G. J. Colloid Interface Sci. 1996, 183, 614.
10.1021/la001588q CCC: $20.00 © 2001 American Chemical Society Published on Web 07/10/2001
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property of bulk materials or thin films by pressing with a cantilever tip.21,22 Biggs and Sprinks23 first measured the deformation of a polystyrene particle attached on the cantilever by pushing it to a mica surface. To obtain a measurable deformation, they applied the load up to the elastic limit of the polystyrene particle such that a significant plastic deformation likely occurred, which complicated the interpretation of their data. Nevertheless, the particle deformation observed there is of particular interest, because it gives an image for the surface energy and the mechanism of surface contact. In the present work, we extended their procedure to evaluate the deformation of an elastomer particle by measuring the interaction force with a solid mica surface in pure water by an AFM. First, we describe the experiment procedure and the data analysis to obtain the relationship between the load and the particle deformation. Then, we display how the contact time, the loading rate, and the maximum load affect the deformation and the adhesion of the particle on the plate. To interpret the data obtained, we used the general framework of the JRK theory, but the contributions of the contact time and separation rate were taken into account, following the thermodynamic approach developed by Maugis and Barquins.6-10 Experimental Section Elastic particles, which were supplied from Mitsubishi Rayon Co., were made of a composite polymer of 98 wt % poly(diethyl hexyacrylate) (2EHA) and 2 wt % poly(acryl methacrylate) (AMA), and their surfaces were coated with a layer of poly(methyl methacrylate) (MMA) of a few nanometers in thickness. Because 2EHA belongs to a cross-linked amorphous polymer and the glass transition temperature is -55 °C, 2EHA behaves as an elastomer of low elasticity modulus above the glass transition temperature. Hence, the particle is called an elastomer particle here. A muscovite mica plate, which was freshly cleaved prior to use, was used as a solid flat surface. Water used was purified with a Millipore apparatus. Force measurements were carried out with a Multimode AFM of Digital Instruments connected to a Nanoscope III controller. The rectangular cantilever was cut off, and a particle was glued there by high-temperature melting epoxy resin. The spring constant was evaluated by measuring the resonant frequency of loaded and unloaded cantilevers.24 The image of the colloid probe was observed with a scanning electron microscope, as shown in Figure 1a, to ensure that the contact area of the particle surface is not contaminated by excess epoxy resin as well as to determine the particle diameter. Several series of experiments were carried out using different particles of 2-3 µm in diameter, and similar features were observed for all the particles, but the data for a 2.0 µm particle were employed here for the sake of quantitative consistency. The procedure to obtain the AFM force curve is given elsewhere,16 but it is briefly described below. A mica plate was attached on the piezo system which moved vertically with a constant scan rate. The cantilever deflection was detected by the voltage change of the split photodiode, u, onto which the laser beam reflected at the rear of the cantilever was focused. The deflection was then converted into the interaction force F between surfaces, using the spring constant of the cantilever kc. The relationship among the plate displacement ∆, the cantilever displacement δc, the penetration δ, and the radius of contact (20) Hartley, P. H.; Griezer, F.; Mulvaney, P.; Stevens, G. W. Langmuir 1999, 15, 7282. (21) Aime, J. P.; Elkaakour, Z.; Odin, C.; Bouhacina, T.; Michel, D.; Curely, J.; Dautant, A. J. Appl. Phys. 1994, 76 (2), 754. (22) VanLandingham, M. R.; McKnight, S. H.; Palmese, G. R.; Elings, J. R.; Huang, X.; Bogetti, T. A.; Eduljee, R. F.; Gillespie, J. W., Jr. J. Adhes. 1997, 64, 31. (23) Biggs, S.; Sprinks, G. J. Adhes. Sci. Technol. 1998, 12, 461. (24) Cleveland, J. P.; Manne, S.; Bockec, D.; Hansma, P. K. Rev. Sci. Instrum. 1993, 64, 1.
Figure 1. Image of colloid probe. (a) An elastomer particle glued on a cantilever observed with a scanning electron microscope. (b) Schematic drawings for relative positions of the particle and the plate before and after a load is applied, where a is the radius of contact area and δ is the penetration.
Figure 2. Correlation between raw data and loadingunloading curves. (a) The piezo displacement vs the voltage change for the deflection of cantilevers with and without a particle. (b) Loading-unloading curves derived from the data in Figure 2a. area a is shown schematically in Figure 1b, where the load F is applied to the particle by raising the piezo system. Figure 2a shows a set of row data of ∆ versus u for cantilevers with and without an elastomer particle, where the same cantilever of the spring constant kc ) 2.0 N/m was employed. Since forces
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due to the van der Waals attraction and electrostatic repulsion were not detected in the approaching force curve, their contributions were regarded as negligibly small, compared with the strong force due to the particle deformation on which our attention was paid in this study. After the probe contacts the mica surface, there appears a linear region of ∆ versus u. It is clear that the reciprocal slope of the linear region for the elastomer particle bp () d∆/du) is higher than bc for the bare cantilever. This difference enables us to evaluate the effective elastic constant of particle, kp, by the following equation.
kp ) kcbc/(bp - bc)
(1)
For the particular case in Figure 2a, bc ) 61 nm/V and bp ) 114 nm/V. Then, the value of kp is calculated to be 2.3 N/m, which is of the same order as the kc value for the cantilever. This elastic constant is used to evaluate the penetration distance δ attributable to the particle deformation, and the relationship of u versus ∆ is converted to that of F versus δ as shown in Figure 2b, using the following relations.
F ) kc(u - u0)bc
(2)
δ ) (∆ - ∆0) - (u - u0)bc
(3)
where ∆0 is the cantilever position at the initial contact with the particle surface, as illustrated in Figure 1b, and u0 is u for the nondeflected cantilever. Figure 2b shows a typical relation of δ versus F for a constant scan rate V. As the piezo system advances from the point A, the mica surface approaches to the particle and contacts at B. Then, the mica penetrates, deforming the particle until the point C where the maximum load Fm is applied. In the receding process, a hysteresis appears because of the energy dissipation, and the particle is kept adhering to the plate even at the point of zero load, D. When a negative load is applied, the particle passes through the maximum adhesive interaction at E, where the adhesive force Fad is defined, and is finally detached at F. This adhesive force is called the tackiness force in the case of viscoelastic contact.4-10 The energy dissipation Φd during the loading-unloading cycle is then obtained by the area defined by the following equation.
Φd )
∫ F dδ - ∫ F dδ C
A
F
C
Figure 3. Loading-unloading curves of a 2.0 µm particle at Fm ) 190 nN. (a) Loading curves for various values of V, where the solid line indicates the fitting curve of the Hertz theory. (b) Unloading curves corresponding to the curves in Figure 3a.
(4)
Experimental Results First, we examine the effects of the scan rate on the penetration in loading and unloading processes, by changing the value of V for a given maximum load Fm. V is inversely proportional to the contact time tc, when the piezo moves with a constant scan rate without stopping. Here, the contact time tc is defined as the time while the mica surface contacts with the particle, that is, the time spent from the points B to F in Figure 2b. Figure 3a shows the relation of δ versus F in the loading process at three different scan rates, V ) 10, 100, and 1000 nm/s. All the data are expressed by a master curve, independently of the large difference of V in magnitude. On the other hand, the unloading curves shown in Figure 3b do depend on the scan rate. The slower the scan rate, that is, the longer the contact time, the higher the values of Fad and Φd. Since the master curve in the loading process indicates that values of δ and the contact area πa2 are independent of V, the unloading process starts from the same values of δm and πam2 at Fm. Figure 4 summarizes the increase of Fad with increasing tc, that is, decreasing V, at Fm ) 190 nN. It is plausible to expect that values of Fad and Φd increase with the detachment rate V,6,25 because of the viscoelastic loss.
Figure 4. Variation of the adhesive force with the contact time at Fm ) 190 nN.
However, the opposite trend is observed here. We consider this to be attributable to the fact that the adhesion between surfaces, that is, the specific work of adhesion w, is strengthened by the bond formation between surfaces during their contact. To know whether this hypothesis is adequate or not, the individual contributions of V and tc are examined, comparing the unloading force curves obtained (i) by changing the value of V at a given value of tc and (ii) by changing the value of tc at a given value of V. The results are shown below. Dependence of δ and Fad on Detachment Rate. A load is applied up to a given maximum load with the scan rate V ) 45 or 800 nm/s, and then the piezo is left standing for a sufficiently long period tcm ) 80 s at the position of (25) Greenwood, J. A.; Johnson, K. L. Philos. Mag. A 1981, 43, 697.
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Figure 5. Unloading curves and adhesion at tcm ) 80 s and Fm ) 230 nN. (a) Unloading curves for V ) 45 and 800 nm/s. (b) Dependence of Fad on V. “Present model” indicates the values predicted by the model given in the discussion section.
Figure 6. Unloading curves and adhesion at V ) 100 nm/s and Fm ) 140nN. (a) Unloading curves for tcm ) 20 and 80s. (b) Dependence of Fad on tc. “Present model” indicates the values predicted by the model given in the discussion section.
the maximum load, using the holding function of the piezo system. Since the loading and unloading times, tcl and tcul, are much smaller than tcm, the total contact time tc () tcl + tcul + tcm) is nearly equal to tcm, and the contact time for any point within the contact area can be then regarded as tcm in the unloading process. The unloading curves of V ) 45 and 800 nm/s for tcm ) 80 s and Fm ) 230 nN are shown in Figure 5a, and the dependence of Fad on V for tcm ) 80 s and Fm ) 225 nN is shown in Figure 5b. These figures indicate clearly that values of Fad and Φd increase with V at a fixed contact time (tc ∼ tcm ) 80 s), as expected. This dependence agrees with the data given by numerous previous reports on elastomer materials.6,26-29 The effective work of adhesion w depends on the displacement velocity of the contact line vp ) da/dt, which is the velocity of peeling off two surfaces.6,26-28
for tcm ) 20 and 80 s are shown in Figure 6a, and the corresponding dependence of Fad on tc is in Figure 6b. It is clear that the value of Fad increases with tc (∼ tcm). This increase of Fad with tc is attributable to the fact that the bond formation between surfaces in contact increases with the contact time, especially in the case of viscoelastic materials. Hence, we consider that w0 in eq 5 must be an increasing function of tc, that is, w0 ) w0(tc). Dependence of δ and Fad on the Maximum Load. In this series of experiments, values of tcm and V were kept constant at tcm ) 80 s and V ) 40 nm/s, but the maximum load Fm was changed. The unloading force curves and the dependence of Fad on Fm are shown in parts a and b of Figure 7, respectively. The value of Fad increases with Fm initially but reaches the maximum value Fadm at Fm > 250 mN/m.
w ) w0[1 + R(T)vpn]
(5)
where w0 is the work of adhesion at vp ) 0 and R(T) and n are constants characterizing the elastomer material. The second term in the bracket characterizes the dissipation energy (viscoelastic loss) at the tip of the crack during the detachment. Since vp increases with V, eq 5 is consistent qualitatively with the increases of Φd and Fad with V shown in Figure 5a,b. Dependence of δ and Fad on Contact Time. Effects of tc on the unloading curve and Fad at a fixed detachment rate V ) 100 nm/s are examined. The unloading curves (26) Gent, A. N.; Schultz, J. J. Adhes. 1972, 3, 281. (27) Andrews, E. H.; Kinloch, A. J. Proc. R. Soc. London, Ser. A 1973, 332, 385. (28) Kendall, K. J. Phys. D 1973, 6, 1782. (29) Fuller, K. G.; Roberts, A. D. J Phys. D 1980, 14, 221.
Theoretical Background Since elastomer particles are treated in the present study, we extend the JKR theory, using the Maugis and Barquins6-10 thermodynamic approach. In this approach, the effect of the contact time on the work of adhesion is taken into account, as demonstrated below. Suppose that an elastic sphere of radius R contacts over the surface area A with a hard plate under a load F. The total energy of the system, UT, may be expressed as follows.
dUT ) dUE + dUF + dUS
(6)
where UE is the stored elastic energy, UF is the potential energy of load F, and US is the stored energy at the interface. Since the stored energy at the interface depends on the bonding energy between surfaces, dUS is expressed as dUS ) -w dA. If the mechanical work is applied to
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condition of opening the crack in the unloading process is given by
G g w0(tc)[1 + aνpn]
(11)
On the other hand, the crack closes if G < w. In closing the crack, the stress relaxation is so fast that the viscoelastic loss is negligible,6,25 and the contact time may be regarded as zero, because the strain energy is consumed to close the new surfaces at the crack front. Hence, the condition of closing crack in the loading process is expressed by
G e w0(0)
(12)
Loading Process. If the loading process is very fast, G < w0 and G f 0, and eqs 9 and 10 reduce to the Hertz equations.
a2 R
(13)
Ka3 R
(14)
δ) F)
On the other hand, if the loading is very slow, the equilibrium is nearly achieved, that is, G ) w0(0). Then, the following equations are derived by replacing G by w0(0) in eq 10. Figure 7. Unloading curves and the dependence of adhesion on the maximum loading at V ) 40 nm/s and tcm ) 80 s. (a) Unloading curves for Fm ) 50, 200, and 470 nN. (b) Dependence of Fad on Fm.
change the contact area by dA, the work can be expressed using the strain energy release rate G, as follow.
( ) ( )
(7)
dUT ) (G - w) dA
(8)
G)
∂UE ∂A
+
F
∂UF ∂A
F
Then
For systems in equilibrium, dUT is zero, or G ) w which is known as the Griffith criterion. For a Hertzian contact, w ) 0 is assumed and the contact area at equilibrium is simply given by the condition G ) 0. In the JKR theory in which the work of adhesion is taken into account, the penetration δ and G () w) at equilibrium are given as follows.2,6
δ) G)
2F a2 + 3R 3aK
(
(9)
)
Rδ 3a3K 1- 2 2 8πR a
2
(10)
where K ) (4/3)(E/(1 - ν2)), and E and ν are the Young’s module and the Poisson’s ratio of the particle, respectively. Strictly speaking, the JKR theory as well as the Hertz theory is valid only if a , R, which is not the case of our experiments. Nevertheless, we employed the JKR theory to estimate the general characteristics of elastic deformation. When G > w, the crack starts to open, decreasing the contact area. Following eq 5 and the data in Figure 6, the
δ) F)
(
)
4πw0(0)a a2 R 3K
1/2
Ka3 - a3/2[3πKw0(0)]1/2 R
(15) (16)
However, because no difference between the fast and slow loadings was observed in Figure 3a, eqs 13 and 14 must coincide with eqs 15 and 16, respectively. Hence, we consider that the contribution of w0(0) is insignificant especially at large penetrations. Unloading Process without Cracking. Here, we consider the unloading process, starting from the maximum load and contact radius, Fm and am. The contact radius must be constant until eq 11 is satisfied and the crack starts to open. Before the crack opens, vp ) 0 in eq 11 and G ) w0(tc). Hence, the point of crack open must be on the unloading curve predicted by eqs 9 and 10 into which G ) w0(tc) and a ) am are substituted. The critical pressure Fcr, below which the crack opens, is then given by the following equation.
Fcr )
Kam3 - am3/2[3πKw0(tc)]1/2 R
(17)
Since a ) am at Fcr e F e Fm, eq 9 is rewritten as follows.
δ)
am2 2F + 3R 3amK
(18)
This indicates that δ is a linear function of F. Then, the critical penetration where the cracking starts, δcr, is given by substituting Fcr into eq 18. The loading and unloading curves of the Hertz and JKR theories and the predicted unloading curve are compared in Figure 8. Unloading Process with Crack Opening. When the piezo system is further retracted such that F < Fcr, the
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Figure 8. Unloading curve predicted by the present model and the comparison with the Hertz and JKR models.
crack starts to open with the initial conditions of F ) Fcr and a ) am. The piezo displacement ∆ is expressed as follows using eq 3.
∆ - ∆0 ) δ + δc ) δ + F/kc
(19)
where ∆ ) ∆0 at δ ) 0. The relation of δ versus F in this region is solved, following the iterational numerical procedure proposed by Barquins.7 The rate of change of the contact radius is given by eq 11, as follows.
(
1 G da ) dt Rw0(tc) R
)
1/n
(20)
When eqs 9, 10, 19, and 20 are solved simultaneously for given values V ) d∆/dt and w0(tc), using the initial conditions of Fcr and am, variations of a, δ, F, and G with time are obtained. According to Barquins and Maugis,6 the final separation of surfaces is given by the following equation.
∂G )( ) (∂G ∂a ) ∂a ∆
-
δ
1 ∂F 2 )0 2πa(kc + kp) ∂a δ
( )
(21)
where kc and kp () (∂F/∂δ) ) 3aK/2) are the spring constants of cantilever and particle, as defined before. These equations enable us to evaluate the unloading curve up to the final separation, as shown in Figure 8. It is clear that the load decreases initially in the unloading process but increases after passing through the minimum by which the adhesion force is defined. Finally, surfaces detach completely when (∂G/∂a)∆ ) 0. Discussion The Hertz theory upon which the JKR theory is built is applicable only under the condition of a , R. Hence, when the propagation of strain and stress in the body is significant, the theory may not be applicable.30,31 However, loading curves in Figure 3a are fitted well with the Hertzian relation δ ∝ F2/3 up to δ ∼ 75 nm, which corresponds to a ∼ 275 nm and a/R ∼ 0.3. This coincidence gives us the value of elastic modulus E ∼ 2.5 × 106 N/m2, assuming ν ) 0.5 as usually employed. This value agrees well with the elastic modulus for typical elastomer materials.32 (30) Dongchan, A.; Shull, K. R. Macromolecules 1996, 29, 4381. (31) Deruelle, M.; Hervet, H.; Jandeau, G.; Leger, L. J. Adhes. Sci. Technol. 1998, 12, 225. (32) Sperling, L. H. Introduction to Physical Polymer Science; WileyInterscience: New York, 1992; Chapter 8.
Figure 9. Apparent elastic modulus of the particle estimated from the unloading curves and eq 18 and the comparison with the value given by the loading curve and the Hertz model.
Since the maximum penetration is greater than 75 nm in the most experiments here, the deviation from the assumption of a small deformation might generate significant effects on the unloading behavior. Nevertheless, we applied our model to the unloading processes where the relation of tcm . tcl + tcul holds as given in Figures 5-7. In these cases, w0(tc) must be constant around the maximum loading and so δ is linearly proportional to F. The apparent values of E and a are determined by fitting the linear part of the unloading process with eq 18 and shown in Figure 9. The apparent value of E is constant and nearly equal to the value given by the loading process at am < 300 nm, that is, a/R < 0.3, but the value of E increases linearly with am at am > 300 nm, which implies that the apparent stiffening of the particle occurs because of the restriction of particle deformation. The same features were reported by Deruelle et al.,31 who investigated the finite size effects using small elastomer lenses. To obtain the unloading curve at F < Fcr, eqs 9, 10, 19, and 20 are solved numerically as explained above, using the apparent values of E and a. According to previous reports on the peeling off of elastomer materials,6-10,28 the value of n was found to be 0.6. Since tc ∼ 80 s, w0(tc) must be constant but the magnitude is unknown. Hence, two parameters, R and w0(tc), are left to be unknown in solving the equations. Using two series of data of V ) 45 and 800 nm/s in Figure 5a, eqs 9, 10, 19, and 20 are solved to obtain several sets of R and w0(tc), such that Fad agrees with the corresponding experimental value. Because values of R and w0(tc) must coincide between V ) 45 and 800 nm/s, they are determined to be R ) 4850 SI system units (or (s/m)n) and w0 ) 41 mJ/m2 from the intersection of two curves, as shown in Figure 10a. The comparison with literature values indicates that the evaluated values are reasonable.4-10,26-29 Using these parameters, the unloading curves for V ) 45 and 800 nm/s are simulated as shown in Figure 10b, which should be compared with the experimental data given in Figure 5a. The simulation predicts the features of unloading curves well and the magnitude of Fad quantitatively. Once the value of R is determined, the variation of w0(tc) is obtainable, as shown in Figure 6b, which shows a good correspondence with the adhesion. These data are expressed by a function w0 ∼ tc0.4, which is expressed by a thin curve in the figure. This increase of w0 with tc is consistent with the data reported elsewhere.8,33 The most probable explanation for this increase is that the stress stored between rough surfaces at the beginning of contact relaxes toward the real contact, followed by the interdiffusion of molecular chains. Recently, we found that (33) Koszterszitz, G. Colloid Polym. Sci. 1980, 258, 658.
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Figure 10. Evaluation of unloading curves by the present model. (a) Determination of w0 and R using the unloading curves for V ) 45 and 800 nm/s in Figure 6 and (b) the unloading curves estimated using the values of w0 and R determined above.
w0 ∼ tc for the adhesion between hard surfaces covered with the adsorbed layer of water molecules, ions, and hydrated ions in aqueous solutions,34 and this dependence was explained such that it takes some time for surfaces to contact directly by destroying the adsorbed layers. As for the contact between polyurethane and glass surfaces in air, the relation w0 ∼ tc0.1 was reported.8 These results suggest that the dependence of w0 on tc in the present system may be attributable to the combination of the elastic relaxation of surfaces and the breakage of molecular layers adsorbed on surfaces. To know the mechanism more precisely, further investigation must be carried out. Finally, it is examined whether the present model can predict the effect of Fm on the unloading curve and Fad shown in Figure 7. The unloading curves simulated by the present model are shown in Figure 11a, which shows a good agreement with Figure 7a, at least quantitatively. The simulated values of Fad, as well as the values of aad at F ) Fad, are plotted as a function of am in Figure 11b. It is clear that features of Fad are very similar between Figures 7b and 11b. Fad and aad exhibit the maximum constant values, Fadm and aadm, respectively, if Fm is large enough to be am > 350 nm. In this region, the adhesion is independent of the applied load, as illustrated by the unloading curves c and b in Figure 11a. On the other hand, if Fm is small such that am < aadm, Fad < Fadm, only the linear part appears in the unloading curve as illustrated by curve a, so that the contact radius is kept constant, that is, am ) aad. In this case, the adhesion 0.2
(34) Vakarelski, I. U.; Ishimura, K.; Higashitani, K. J. Colloid Interface Sci. 2000, 227, 111.
Langmuir, Vol. 17, No. 16, 2001 4745
Figure 11. Estimated unloading curves and adhesion. (a) Unloading curves simulated for am ) 235, 412, and 510 nm and (b) dependence of Fad and aad on am.
decreases with decreasing load applied, as shown in Figure 11b. Conclusions An AFM technique was applied to investigate characteristics of the adhesion of a microscopic viscoelastic particle on a mica plate in water. Measurements were performed to distinguish the individual contributions of the scan rate, the contact time between surfaces, and the maximum load on the loading-unloading process. To interpret the data obtained, a JKR model modified using the Maugis-Barquins approach is proposed, in which the dependence of the surface energy on the contact time is taken into account. Strictly speaking, the present model is valid only for a small deformation. However, the model is found to be applicable for a larger deformation, if apparent values of the elastic modulus and contact radius are employed. The observed phenomena are well described especially in terms of the following aspects: (i) the loading process is independent of the loading rate, (ii) there exists a linear region in the unloading curve around the maximum load, and (iii) the adhesion increases with the applied load and reaches the maximum constant value if the load becomes large enough. As for the work of adhesion, the increase with the contact time is expressed by a function w0 ∼ tc0.4, but this function cannot be explained by the mechanisms proposed by the previous studies, such as the elastic relaxation of surfaces and the breakage of the adsorbed layer. Hence, we consider that the work of adhesion may be attributable to a more complicated mechanism, and further investigations are needed to clarify the detailed mechanism. LA001588Q
