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Friction of Gels. 6. Effects of Sliding Velocity and Viscoelastic Responses of the Network Go Kagata,† Jian Ping Gong,†,‡ and Yoshihito Osada*,† Graduate School of Science, Hokkaido UniVersity, Sapporo 060-0810, Japan, and PRESTO, JST ReceiVed: June 21, 2001; In Final Form: February 10, 2002
The velocity dependence of gel friction was investigated in pure water and in salt solutions to elucidate the effect of interfacial interaction with substrates. When the gel and the substrate were repulsive, the frictional force depended strongly on the sliding velocity, whereupon the higher the normal compressive strain, the stronger the velocity dependence of the friction. The frictional force per unit area, f, was found to follow a power law as f ∝ Vβ, where the exponent, β, depends on the normal compressive strain. This result shows that the gel friction in the repulsive case cannot be explained in terms of the simple hydrodynamic mechanism, from which f ∝ V1.0 is predicted. On the contrary, in the attractive case, the frictional force showed a maximum value with increase in the sliding velocity, which qualitatively coincides with our repulsion-adsorption model proposed previously.
1. Introduction papers,1-5
In preceding we have reported that the frictional force of hydrogels, F, shows specific dependencies on the normal load, W, and the sliding velocity, V, when slid against themselves or against solid substrates. Most importantly, the frictional coefficient of gels, µ, which is defined as the ratio between the frictional force (shear stress) and the normal load (compressive stress), changes across a wide range and exhibits extremely low values (µ ∼ 0.001) under certain experimental conditions. At a constant sliding velocity, the frictional force per unit area, f, follows a power relation with the normal compressive stress (pressure), P, as f ∝ PR, where R is a constant between 0-1.0, depending on the chemical structure of the gel as well as the opposing substrate. Thus, it was found that Amontons’ law F ) µW for solid friction could be observed as a special case when R ) 1.0. The frictional behavior of a gel cannot be explained in terms of the hydrodynamic lubrication mechanism from a viewpoint of solid materials since the low friction is sustained even at a Sommerfield number as small as 10-11, where the hydrodynamic lubrication can usually never be realized for friction between two solids.6 To describe the frictional behavior of a gel sliding on a solid surface, we have proposed a thermodynamic model from the viewpoint of polymer-solid interface interaction.2 The main argument of the model is summarized as follows: In analogy to a polymer solution placed in contact with a solid wall, the polymer network on the surface of the gel will be repelled from the solid surface if the interface interaction is repulsive and will be adsorbed to the solid surface if it is attractive. In the former case, the viscous flow of the solvent layer between the solid surface and the polymer network will make a dominant contribution to the friction force. In the latter case, however, the adsorbing chain will be stretched when the solid surface is slid relative to the gel. The elastic force increases with the deformation and eventually the adsorbing polymer network detaches from the substrate, which in turn appears as the † ‡
Hokkaido University. PRESTO, JST.
frictional force. The theoretical analysis predicted a high frictional force with a weak dependence on the normal compressive stress for attraction and a low frictional force with a strong dependence on normal compressive stress for repulsion,2 and these predictions were in good agreement with the experimental observation.3 Furthermore, the frictional behavior between two chemically cross-linked polyelectrolyte gels carrying the same sign of charges has been investigated experimentally,4 and it was found that the friction is strongly dependent on the charge density of the gel surface, which further substantiated the polymer repulsion-adsorption model.4 In this paper, the velocity dependence of the frictional force produced by repulsive and attractive interactions have been studied experimentally and compared with the repulsionadsorption model. According to this model, the frictional force, f, for the repulsive case is ascribed to the viscous flow of solvent at the interface, which predicts that f should be proportional to the sliding velocity. For the attractive case, the predominant origin of the frictional force arises from the elastic stretching of the adsorbing chain on the substrate. Since the elastic force should be proportional to the deformation distance that is the product of the sliding velocity and the adsorbing lifetime, f is anticipated to show a maximum at a certain sliding velocity. 2. Experimental Section Materials. Poly(vinyl alcohol) (PVA: MW ) 90 000) was purchased from Wako Pure Chemical Industries, Ltd., and used without further purification. 2-Acrylamido-2-methylpropanesulfonic acid (AMPS) (Tokyo Kasei Co., Ltd.) was used as received and neutralized with sodium hydroxide (Junsei Chemical Co., Ltd.) in order to obtain the sodium salt of AMPS (NaAMPS). N,N′-Methylene bisacrylamide (MBAA) (Tokyo Kasei Co., Ltd.) used as a cross-linking agent was recrystallized from ethanol. Potassium persulfate (Tokyo Kasei Co., Ltd.), which was used as a radical initiator, was recrystallized from water. Gel Preparation. Physically cross-linked PVA gel was prepared by a repeated freezing (-20°C) and thawing (25°C) method from a prescribed PVA aqueous solution (11 wt %).
10.1021/jp012380w CCC: $22.00 © 2002 American Chemical Society Published on Web 04/17/2002
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Chemically cross-linked poly(2-acrylamido-2-methylpropanesulfonic acid sodium salt) (PNaAMPS) gel was prepared by a radical polymerization of a 1.0 M aqueous solution of NaAMPS monomer in the presence of a prescribed amount of MBAA and 0.001 M potassium persulfate. The polymerization was carried out at 60 °C for 12 h under a nitrogen atmosphere. The detailed procedure of the polymerization is described elsewhere.3,4 PVA gel was prepared between two parallel glass plates separated by a 3-mm-thick silicon spacer to give sheet-shaped gel. For PNaAMPS gel, a 1- (for swelling with pure water) or 2-mm-thick spacer (for swelling with aqueous NaCl) was used. After gelation, samples were immersed in a large amount of pure water for a week to equilibrate and wash away the residual chemicals. The amount of water contained in gels is characterized by the degree of swelling, q, which is defined as the weight ratio of swollen to dry sample. Dry gels were obtained by vacuum-drying until a constant weight was reached. Glass plates served as the opposing substrate and were polished carefully with a detergent, rinsed with distilled water, and dried in air before usage. Friction Measurement. A commercially available rheometer “ARES” (Advanced Rheometric Expansion System, Rheometric Scientific Inc.) was used for measuring the friction of gels. Samples (2-3 mm thick for PNaAMPS gel and 3 mm thick for PVA gel) were cut into a ring shape (inside radius ri, outside radius ro) and were glued on the upper surface of coaxial diskshaped platen with cyanoacrylate instant adhesive agent (Toagosei Co., Ltd.). It should be noted here that the attachment of hydrogels to the metal surface of platen with adhesive agent was not very strong, and it might be detached by a strong shear stress. Needless to say, the data is removed when this occurred in measurement. As the opposing substrate, a PNaAMPS gel or a glass plate was used in this study. The substrates were slightly larger than the upper ring-shaped gel and were glued on the lower platen. The gel-substrate interface was immersed in pure water or in aqueous NaCl solution and equilibrated, whereupon two surfaces were compressed with each other under the normal load. Here, we have to mention the methodological feature of a rheometer: the friction measurement was performed under a constant compressive strain mode, which led to a stressrelaxation in the normal stress (pressure). To account for this stress-relaxation, measurements were made after the normal strain was applied for at least 60 min at which the normal stress reached a quasi-equilibrium state. After achieving stressrelaxation equilibrium, an angular displacement with an angular velocity, ω, was applied to the lower platen to generate the frictional torque. The torque, M, as well as the normal stress were recorded during the rotation. The total frictional force, F, can be related to the torque by assuming that the unknown frictional shear at a radius r changes with the sliding velocity in a power law as f ∝ (ωr)R, where R is a constant. This assumption yields the relationship between F and M for the ringshaped gel as follows,
F)
(3 + R)(ro3 - ri3) (2 + R)(ro4 - ri4)
M
(1)
Since the value of R lies in a range of 0-1, which only makes an 8/9 times difference in the F value, we simply set R ) 1 in the calculation of F using the above equation. The average frictional force per unit area of the ring-shaped gel, f, is
Figure 1. Time profile of the frictional force for a ring-shaped PNaAMPS gel rotated against a piece of PNaAMPS gel under various angular velocities as measured by a rheometer in pure water at 25 °C. The numbers in the figure are the angular velocities in rad/s.
calculated as
f)
F π(ro - ri2) 2
(2)
A more detailed derivation of eq 1 is given in a previous paper.4 The angular velocity dependence at a certain experimental condition was studied by stepwise changes in the velocity from lower to higher values without separating the two rotating surfaces. Figure 1 shows a typical example of the time profile of the frictional force for a ring-shaped PNaAPMS gel rotated on a piece of PNaAPMS gel. As shown in Figure 1, the frictional force showed a remarkable decrease at the beginning (ca. 2030 min.) of the measurement. This relaxation-like decrease in friction force should be related to the generation of the high static friction, which will be reported in the next paper. The frictional force, however, became stable when the angular velocity was increased stepwise later in the experiment. The data were used only when a constant frictional force was obtained. Unless otherwise noted, the standard deviations of the data are less than 20%. The rheometer has several advantages over the tribometer (Heidon 14S/14DR, SHINTO Scientific Co., Ltd.) used previously for measuring the friction of gels by us.1,3,5 One is the possibility of measuring the gel friction with small gel samples under exactly controlled solvent temperatures using a Peltier device (stability: (0.1 °C). The other is its high sensitivity in detecting extremely small frictional torques. The disadvantage of the rheometer is the velocity distribution in the radial direction of the sample. The velocity dependence of the experimental data is therefore an average over the sample radius between ri and ro. With ri and ri values used in this experiment, the results obtained from the rheometer agree with results obtained from a sliding plate tribometer.1,3,5 The dynamic modulus of the PNaAMPS gel at various temperatures was measured by using the same rheometer. A cylindrical gel 15 mm in diameter and 4 mm in height was glued to the upper and lower parallel plates of the rheometer, respectively, and then, a normal strain of a few % was applied to the gel. After the normal stress reached an equilibrium value of 2.8 kPa, a shear strain of 0.5% was applied with a frequency of 1 Hz, and the shear force was measured to obtain the dynamic modulus.
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Figure 2. Angular velocity dependence of the frictional force for PNaAMPS gel rotated on a piece of glass surface in pure water at 25 °C. The numbers in the figure are the values of the normal compressive strains and normal stresses applied during the measurement. Sample thickness: 3 mm. Degree of swelling: 27.
3. Results and Discussion 3.1. Repulsive Case. According to the repulsion-adsorption model, a solvent layer is formed at the interface when the gelsubstrate interaction is repulsive, and the frictional force arises from the viscous flow of the solvent layer. Therefore, the frictional force (shear stress) should increase with the increase in the sliding velocity, that is, f ∝ V. As confirmed by atomic force microscopy (AFM) measurements,3 the anionic polymer gel placed in water has an electrostatic repulsion with the glass surface due to the presence of SiO- groups on the glass surface.7 Therefore, we investigated the velocity dependence of the friction for PNaAMPS gel sliding on a glass substrate. Figure 2 shows the relationship between the angular velocity, ω, and the frictional force for PNaAMPS gel rotated against a glass surface in water at 25 °C under various normal compressive strains. The average velocity of the ring-shaped gel is (ri + ro)ω/2, that is, 7.5 × 10-3 ω m s-1. As shown in Figure 2, the frictional force, f, increases with an increase in velocity, where the profiles depend on the applied normal compressive strain and therefore normal stress. At low normal strain (2.8%), f is almost constant for low velocities and then gradually increases at higher velocities. At higher normal strains, f increases moderately in the low velocity range but increases distinctly at higher velocities. In other words, the velocity dependence is less notable at low velocities and normal strains and becomes stronger at higher velocities and normal strains. We also measured friction between two pieces of PNaAMPS gels in water at 20 °C while changing the compressive strain (Figure 3b). At small compressive strain (1.5%), the frictional force shows no dependence on the velocity over a range of 3 orders in magnitude. At larger compressive strains (higher than 2.5%), the frictional force shows stronger dependence on the velocity. For reasons of simplicity, a first-order fit is used to approximate data,, f ∝ Vβ, where the exponent β increases with the increase in the compressive strain. β saturates to a value of about 0.55 at a high strain, as shown in Figure 4. Although the frictional force increases monotonically with an increase in angular velocity, the relationship between f and V is far more complicated than what can be expected from the simple hydrodynamic mechanism, which predicts f ∝ V1.0 (eq
Figure 3. Angular velocity dependence of the frictional force for a ring-shaped PNaAMPS gel rotated on a piece of PNaAMPS gel in pure water at 5 (a), 20 (b), and 50 °C (c). Sample thickness: 2.0-2.1 (a), 2.2-2.3 (b), and 2.3 ( 0.05 mm (c). The numbers in the figures are the values of the normal compressive strains and normal stresses.
25 in Gong and Osada2). In addition, f shows a complicated dependence on the normal strain, i.e., on the normal stress. These experimental results suggest that in addition to the hydrodynamic resistance of the liquid, the viscoelastic deformation of the polymer network might contribute to the gel friction. Since both the viscosity of the solvent (water) and the viscoelasticity of the polymer network change sensitively with temperature, we also investigated the velocity dependence of friction at 5 and 50 °C, and the results are shown in parts a and c, respectively, of Figure 3. The double-logarithmic plots of f against angular velocity show approximately linear relations for
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Figure 5. Dynamic modulus G′ as a function of temperature for cylindrical PNaAMPS gel (diameter, 15 mm; thickness, 4 mm) sheared dynamically in water. Frequency, 1 Hz; shear strain, 0.5%. The error ranges in the figure are standard deviations of the mean values over three samples.
Figure 4. Relationships between the normal compressive strain λ (a) or the normal compressive stress (b) and the scaling exponent β of f ∝ νβ for the ring-shaped PNaAMPS gel rotated against a piece of PNaAMPS gel at various temperatures in pure water. The figure is obtained from the results of Figure 3.
a first-order fit at each temperature, and the slopes of these lines, β, increase with increase in the compressive strain, similarly as in Figure 3b. The slopes β at 5, 20, and 50 °C are plotted as a function of compressive strain in Figure 4a. The slopes β, which are nearly zero at small compressive strains, rise abruptly over a narrow strain range, and then saturate to a value of 0.55 (Figure 4a). The critical value of compressive strain at which β rises abruptly decreases with increasing temperature. This means that the strong velocity dependence begins at a lower compressive strain when the temperature is higher. However, if the β is plotted against the normal stress (Figure 4b), we found that the β increases with the normal stress and is not sensitive to the temperature. This might be attributed to the increase in the elastic modulus of polyelectrolyte gels at a higher temperature since the counterion entropy of the network increases with increasing in temperature. To confirm this, the temperature dependence of the dynamic modulus, G′, of the gel was investigated. As shown in Figure 5, the elastic modulus increases modestly with the increase in the temperature. This result means that the polymer network in the gel becomes more rigid and requires more energy to deform at an increased temperature. Taking the above into consideration, it is understandable that the λ/β curve shifts to a lower strain at a higher temperature, as shown in Figure 4a. So far, we cannot explain why the frictional force does not depend on the sliding velocity (β ≈ 0) when the compressive strain (stress) is low. Two factors, (i) the viscoelastic nature of
a gel and (ii) the non-Newtonian behavior of water at the friction interface, might be important. (i) The highly hydrated polymer network of the gel might be extensively deformed under the shear stress (frictional force) at the interface due to its viscoelastic nature. Therefore, the larger the sliding velocity, the higher is the shear stress and the more the network is deformed. This, in turn, would result in an increased interface gap between two gel surfaces, which leads to a decrease in the frictional force. This explanation, however, is not sufficient if we consider the fact that the frictional force is constant over 3 orders in magnitude of rotational velocity. (ii) If water molecules form a layer at the friction interface and behave ideally as a Newtonian fluid, no static friction should be observed at the beginning of interface shearing. However, our recent studies, which will be reported in a separate work, show that two repulsive gel surfaces could not slip with each other until the shear stress acting on the interface exceeded over a critical value. This suggests that the water molecules, which are hydrated strongly to the polyelectrolyte chain and form a layer at the interface, possess extremely high viscosity or even behave like a non-Newtonian fluid under the constraint interface environment. The viscosity of the water at the interface might decrease with the increase in the shear rate, which results in a velocity-insensitive frictional resistance. However, we cannot explain the fact that the velocity dependence of the friction force becomes stronger with an increase in normal compressive strain (stress). The above results demonstrate that the mechanism of gel friction in the repulsive case cannot be explained simply in terms of hydrodynamic lubrication. Probably, two effects (the viscoelastic nature of polymer networks and the non-Newtonian behavior of water molecules) intertwined intricately with each other make a vitally important contribution to friction. The explanations given for the trends in the repulsive case are not conclusive and further studies are needed. 3.2. Attractive Case. As mentioned above, the predominant origin of the frictional force in this case arises from the elastic deformation of the polymer chains adsorbed to the substrate, which is expressed as the product of the deformation velocity and the lifetime of adsorption. Since the adsorption time decreases with an increase in deformation, the frictional force should show a maximum when the sliding velocity is increased.2 Our previous analysis showed that the frictional force f is associated with the normalized velocity parameter R, defined
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Figure 6. Angular velocity dependence of the frictional force for PVA gel rotated on a piece of glass surface in pure water at 25 °C. Sample thickness: 3 mm. Degree of swelling: 11. Normal strain: 30%. Normal stress: 8.5 kPa.
as
Figure 7. Frictional forces as a function of angular velocity for PVA gel rotated against a piece of glass surface in pure water at 5 (b) and 45 °C (9). Sample thickness: 3 mm. Degree of swelling: 11. The error ranges in the figure are standard deviations of the mean values over four samples.
where
N ) (RF/a)5/3 R ) ν/νf ) ντf/RF
(3)
where νf is the thermal fluctuation velocity, τf is the relaxation time, and RF is the radius of the “partial polymer chain” which is the term meaning the polymer segment between two adjacent cross-linking points.2 The frictional force exhibits a maximum at a certain R value that decreases with the increase in the attraction energy of one partial chain, Fads. For weak attraction (Fads/T , 1), the maximum of the frictional force occurs at R ≈ 2. By using the equation τf ≈ ηRF3/T,8
νmax ≈ 2RF/τf ) 2T/ηRF2
(4)
where T is the absolute temperature in energy unit (T ) Boltzmann constant × absolute temperature). PVA immersed in water has a weak attractive interaction with a glass surface as confirmed by AFM.3 Figure 6 shows the frictional forces when physically cross-linked PVA gel is slid against a glass surface. The measurements were performed in water at room temperature. A maximum is observed at an angular velocity of 5 × 100 rad s-1, which corresponds to an average sliding velocity of 3.8 × 10-2 m s-1. The appearance of a peak in the frictional force at a certain sliding velocity agrees with theoretical prediction.2 The theory also predicts the continuous increase in frictional force after a local maximum largely due to increased hydrodynamic friction at a higher velocity region, in agreement with experimental results (Figure 6). Thus, in the attractive case, the profiles of velocity dependence of the frictional force are quite different from those observed in the repulsive case. We estimated the radius of the partial polymer chain, RF, from the velocity at which the frictional force attains a maximum by using eq 4. Using the viscosity of water η as 0.90 × 10-3 Pa s at 25 °C9 and setting νmax ) 3.8 × 10-2 m s-1, we found RF ) 16 nm. From the C* gel theory,8 the degree of swelling of the gel can be related to the Flory radius of the polymer chain by
q ) RF3/Na3 ) (RF/a)4/3
(5)
(6)
is the number of Kuhn segments and a is the Kuhn length. For a flexible vinyl polymer like PVA, a is several Angstrom in water. The Flory radius of the polymer chain RF estimated from the swelling degree q ) 11 from eq 5 is about 3 nm supposing a ) 0.5 nm. Therefore, the RF estimated from the velocity of the maximum frictional force is about 5 times larger than that evaluated from the degree of swelling. This disagreement suggests that the degree of swelling close to the gel surface, where the friction occurs against glass surface, might be 5 times larger than that of the bulk (q ) 11). However, our estimates have been made using scaling relations, neglecting all numerical factors, and more systematic analysis is necessary in the future. As shown by eq 4, the thermal fluctuation velocity of the polymer chain should increase with an increase in temperature. This originates partly from the increased thermal energy and partly from the decreased viscosity of the solvent. Therefore, we can expect an increase in νmax by raising temperature or a decrease in νmax by lowering temperature. Figure 7 shows the frictional force measured at 5 and 45 °C plotted as a function of velocity. As shown in Figure 7, an increase in temperature results in a decreased frictional force. Furthermore, an increase in temperature leads to an increase in the velocity where friction force is maximal. When the temperature is raised from 5 to 45 °C, νmax increases 5 times from 0.5 to 2.5 rad s-1. Putting η (5 °C) ) 1.52 × 10-3 Pa s and η (45 °C) ) 0.60 × 10-3 Pa s,9 our theory ( eq 4) predicts about 3 times increase in νmax, which roughly agrees with the experimental results. As is well established, an addition of neutral salt into water decreases the electrostatic repulsion and favors a closer contact of the surface polymer networks between two gel surfaces carrying the same sign of charge. In this case, the short-range van der Waals attraction between two polymer networks comes into action, and the frictional behavior is changed from repulsive to attractive, as was reported in a preceding paper.4 Thus, as another attractive case, we investigated the velocity dependence of the frictional force for two pieces of PNaAMPS gels slide against each other at various ionic strengths. Figure 8 shows the velocity dependence of the frictional force for PNaAMPS gel slid against PNaAMPS gel immersed in 0.1
Friction of Gels. 6
Figure 8. Angular velocity dependences of the frictional forces for PNaAMPS gels rotated on a PNaAMPS gel in aqueous NaCl solutions at 25 °C. The numbers in the figure are the NaCl concentrations in M. Sample thickness: 2.4 (O), 2.9 (2), and 2.4 (0). Degree of swelling: 27 (O), 16 (2), and 7 (0).
and 1.0 M NaCl aqueous solution. As shown in the figure, the frictional forces for both cases are much larger than that in water. Furthermore, in strong contrast to the results in Figures 6 and 7, friction forces show a minimum as a function of angular velocity, and when the velocity increases further, they show a scaling exponent with velocity similar to that measured in pure water. This result indicates that the friction behavior between two PNaAMPS gels in the presence of salt gradually changes from the adsorption-dominated mechanism to the hydrodynamic friction-dominated mechanism with increasing velocity, which is in good agreement with our theoretical prediction.2 4. Conclusions The frictional force of a gel showed different dependences on the sliding velocity for different interfacial interactions between gel and the opposing surface. When the interfacial interaction is repulsive, the frictional force increases with an increase in the sliding velocity and the higher the normal compressive pressure (or strain), the stronger the dependence on the velocity. When two negatively charged polyelectrolyte
J. Phys. Chem. B, Vol. 106, No. 18, 2002 4601 gels were slid against each other, a simple scaling relation f ∝ νβ is observed, where the exponent β increases from near zero and saturates at a value of 0.55 when the normal compressive strain increases, but β never reaches 1.0, as predicted by the hydrodynamic friction mechanism. This result suggests that the simple hydrodynamic mechanism is not sufficient to explain the frictional behavior for the repulsive interaction. The viscoelastic feature of the gel and the non-Newtonian behavior of water at the interface must be considered in order to elucidate the friction mechanism. In the case of attractive interfacial interactions, a maximum in the frictional force was observed when the sliding velocity increased to a value comparable to the thermal fluctuation of the polymer chains. This behavior is in agreement with theoretical predictions and suggests that the frictional force is associated with the polymer chain deformation when the sliding velocity is not much larger than the velocity due to thermal motion of a polymer coil. When the sliding velocity is much larger than this characteristic velocity of the polymer coil, the hydrodynamic mechanism becomes dominant and the frictional force increases with the sliding velocity. Acknowledgment. This research was supported by Grantin-Aid for the Specially Promoted Research Project “Construction of Biomimetic Moving System Using Polymer Gels” from the Ministry of Education, Science, Sports and Culture of Japan. We thank Mr. Shinsuke Oogaki for his preparation and offer of samples for the dynamic modulus measurement. References and Notes (1) Gong, J. P.; Higa, M.; Iwasaki, Y.; Katsuyama, Y.; Osada, Y. J. Phys. Chem. B 1997, 101, 5487. (2) Gong, J. P.; Osada, Y. J. Chem. Phys. 1998, 109, 8062. (3) Gong, J. P.; Iwasaki, Y.; Osada, Y.; Kurihara, K.; Hamai, Y. J. Phys. Chem. B 1999, 103, 6001. (4) Gong, J. P.; Kagata, G.; Osada, Y. J. Phys. Chem. B 1999, 103, 6007. (5) Gong, J. P.; Iwasaki, Y.; Osada, Y. J. Phys. Chem. B 2000, 104, 3423. (6) Persson, B. N. J. Sliding Friction: Physical Principles and Applications; Springer: New York, 1998. (7) Shah, G.; Dubin, P. L.; Kaplan, J. I.; Newkomw, G. R.; Moorefield, C. N.; Baker, G. R. J. Colloid Interface Sci. 1996, 183, 397. (8) de Gennes, P. G. Scaling Concept in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (9) Chronological Scientific Tables; National Astronomical Observatory, Ed.; Japan, Maruzen Co., Ltd.: Tokyo, 1992.
