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Articles Surface Friction of Hydrogels with Well-Defined Polyelectrolyte Brushes Yutaka Ohsedo,† Rikiya Takashina,† Jian Ping Gong,*,†,‡ and Yoshihito Osada† Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan, and Presto, JST, Sapporo 060-0810, Japan Received November 25, 2003. In Final Form: March 17, 2004
Hydrogels of poly(2-hydroxyethyl methacrylate) (PHEMA) with well-defined polyelectrolyte brushes of poly(sodium 4-styrenesulfonate) (PNaSS) of various molecular weights were synthesized, keeping the distance between the polymer brushes constant at ca. 20 nm. The effect of polyelectrolyte brush length on the sliding friction against a glass plate, an electrorepulsive solid substrate, was investigated in water in a velocity range of 7.5 × 10-5 to 7.5 × 10-2 m/s. It is found that the presence of polymer brush can dramatically reduce the friction when the polymer brushes are short. With an increase in the length of the polymer brush, this drag reduction effect only works at a low sliding velocity, and the gel with long polymer brushes even shows a higher friction than that of a normal network gel at a high sliding velocity. The strong polymer length and sliding velocity dependence indicate a dynamic mechanism of the polymer brush effect.
Introduction Great efforts to pursue suitable polymer systems have been made on the design and production of artificial organs with a sufficient lubrication such as contact lens, catheter, artificial articular cartilage, artificial esophagus, etc.1,2 Studies on the surface sliding friction of water-swollen hydrogels on solid surfaces as well as between gels reveal the richness and complexity of gel friction.3-10 For example, hydrogels exhibit a wide range of frictional coefficients from an order of 10-3 to 100 in magnitude, depending on the interfacial interaction between the polymer network and the opposing substrate.3,5,6 It is considered that when the interfacial interaction is attractive, the force to detach the adsorbing chain from the substrate appears as friction to give a high frictional coefficient, such as the friction of rubber.11-14 On the other hand, when the interfacial † ‡
Hokkaido University. Presto.
(1) Freeman, M. E.; Furey, M. J.; Love, B. J.; Hampton, J. M. Wear 2000, 241, 129. (2) Wang, H.; Ateshian, G. A. J. Biomech. 1997, 30, 771. (3) Gong, J. P.; Higa, M.; Iwasaki, Y.; Katsuyama, Y.; Osada, Y. J. Phys. Chem. B 1997, 101, 5487. (4) Gong, J. P.; Osada, Y. J. Chem. Phys. 1998, 109, 8062. (5) Gong, J. P.; Iwasaki, Y.; Osada, Y.; Kurihara, K.; Hamai, Y. J. Phys. Chem. B 1999, 103, 6001. (6) Gong, J. P.; Kagata, G.; Osada, Y. J. Phys. Chem. B 1999, 103, 6007. (7) Gong, J. P.; Iwasaki, Y.; Osada, Y. J. Phys. Chem. B 2000, 104, 3423. (8) Kagata, G.; Gong, J. P.; Osada, Y. J. Phys. Chem. B 2002, 106, 4596. (9) Baumberger, T.; Caroli, C.; Ronsin, O. Phys. Rev. Lett. 2002, 88, 75509. (10) Baumberger, T.; Caroli, C.; Ronsin, O. Eur. Phys. J. E 2003, 11, 85. (11) Schallamach, A. Proc. Phys. Soc., London, Sect. B 1952, 65, 657661. (12) Chernyak, Y. B.; Leonov, A. I. Wear 1986, 108, 105-138. (13) Grosch, K. A. Proc. R. Soc. London, Ser. A 1963, 274, 21-39. (14) Vorvolakos, K.; Chaudhury, M. K. Langmuir 2003, 19, 6778.
interaction is repulsive, a solvent (water) layer, which serves as a lubricant, is retained at the interface even under a normal load, to give a very low frictional coefficient.3,4 Poly(2-acrylamido-2-methyl-propanesulfonic acid) (PAMPS) gels, for example, exhibit frictional coefficients in an order of 10-3 against glass in salt-free water due to the electrostatic repulsion between the polyanionic gel and the glass plate.5 It is found that the interfacial properties of hydrogels synthesized by radical polymerization of water-soluble vinyl monomers also depend on whether the substrate used for the preparation of the gel is hydrophobic (polytetrafluoroethylene, polystyrene, etc.) or hydrophilic (glass, mica, etc.). For example, PAMPS gel synthesized on the hydrophobic substrates exhibits a surface frictional coefficient of 10-4 against glass, which is 2 orders lower in magnitude than those of gels prepared on a glass plate under the same conditions.15 The decreased frictional coefficient of the gel prepared on the hydrophobic substrate was associated with the formation of loosely cross-linked network structures with a lot of dangling chains on the surface of the gel. The effect of polymer brush on the reduction of sliding friction was also observed between solid friction by surface force apparatus (SFA) measurements.16,17 For example, Klein et al. recently reported a massive lubrication between mica surfaces modified by repulsive polyelectrolyte brushes in water.18 These results show that polymer brushes on solid or gel surfaces can dramatically reduce the surface friction if (15) Gong, J. P.; Kurokawa, T.; Narita, T.; Kagata, G.; Osada, Y.; Nishimura, G.; Kinjo, M. J. Am. Chem. Soc. 2001, 123, 5582. (16) Klein, J.; Kumacheva, E.; Mahalu, D.; Perahia, D.; Fetters, L. Nature 1994, 370, 634. (17) Grest, G. S. Adv. Polym. Sci. 1999, 138, 149. (18) Raviv, U.; Giasson, S.; Kampf, N.; Gohy, J. F.; Jerome, R.; Klein, J. Nature 2003, 425, 163.
10.1021/la036211+ CCC: $27.50 © 2004 American Chemical Society Published on Web 07/01/2004
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the polymer brush has a repulsive interaction with the sliding substrate. Despite the great importance of this issue, the mechanism of the brush effect on the surface friction is not well understood. Quantitative relationships between the friction coefficient and the polymer brush properties, such as polymer length, density, and structure, have not been systematically studied, mainly due to the difficulty in obtaining a gel with well-defined polymer brushes. Several requirements are needed to quantitatively elucidate the friction reduction effect by polymer brushes. One is to control the length of polymer brushes precisely while keeping the polydispersity index (Mw/Mn) of the polymer brushes as low as possible. This is because if Mw/Mn of the polymer brushes is high, the quantitative relationship between µ and molecular weight may not be accurately evaluated. Another is to keep the bulk properties of gel unchanged before and after the reaction of making polymer brushes. This is because the frictional force strongly depends on the bulk viscoelasticity of the gel.7 In this study, poly(2-hydroxyethyl methacrylate) (PHEMA) gel with well-defined polymer brushes consisting of poly(sodium 4-styrenesulfonate) (PNaSS) is synthesized using the living radical polymerization in the presence of TEMPO (2,2,6,6-tetramethyl-1-piperidinyloxy) in aqueous media.19-21 Frictional behaviors of the gels with various PNaSS brush lengths are studied against a glass in pure water, keeping the polymer brush density constant. As glass has an isotropic point around pH 3-4 and is negatively charged in pure water,22 PNaSS brushes on the PHEMA network are electrostatically repulsive to the glass plate in water. Experimental Section Materials. Ammonia persulfate ((NH4)2S2O8) (Tokyo Kasei Co., Inc.), dimethyl sulfoxide, ethanol, ethylene glycol (EG), methanol (Junsei Chemical Co., Ltd.), sodium metabisulfite (Na2S2O5) (Kanto Chemical Co., Inc.), and 2,2,6,6-tetramethyl1-piperidinyloxy (TEMPO) (Aldrich) were used as received. N,N′Methylenebisacrylamide (MBAA) and potassium persulfate (K2S2O8) (Wako Pure Chemical Industries, Ltd.) were purified by recrystallization with ethanol and water. 2-Hydroxyethyl methacrylate (HEMA) (Aldrich) and methacryloyl chloride (Tokyo Kasei Co., Inc.) were distilled in a vacuum. Sodium 4-styrenesulfonate (NaSS) (Tokyo Kasei Co., Inc.) was purified by precipitation from acetone/water. Water was filtrated by a Millipore filter (Elix, Nihon Millipore Co., Ltd). All reagents were immediately used after purification. Preparation of PHEMA Gel. The procedures of preparing PHEMA gel with polymer brushes are summarized in Scheme 1. The polyHEMA (PHEMA) gel was prepared by radical polymerization of a 4 mol dm-3 aqueous solution of HEMA monomer in the presence of 1 mol % MBAA and 0.5 mol % Na2S2O5. The solution was bubbled with nitrogen gas for 30 min. Next, 0.1 mol % (NH4)2S2O8 as an initiator was added to the solution. The polymerization of HEMA monomer was carried out at room temperature for 3 h between two parallel glass substrates separated with a spacer 2 mm in thickness. The obtained PHEMA gel was washed in ethanol first and then in water seven times, respectively, for 24 h to remove unreacted reagents. Introduction of the Methacryloyl Group into PHEMA Gel. PHEMA gel was dried under vacuum at 60 °C for 12 h. The (19) Hawker, C. J.; Bosman, A. W.; Harth, E. Chem. Rev. 2001, 101, 3661. (20) Keoshkerian, B.; Georges, M. K.; Boils-Boissier, D. Macromolecules 1995, 28, 6381. (21) Bouix, M.; Gouzi, J.; Charleux, B.; Vairon, J. P.; Guinot, P. Macromol. Rapid Commun. 1998, 19, 209. (22) Shah, G.; Dubin, P. L.; Kaplan, J. I.; Newkome, G. R.; Moorefield, C. N.; Baker, G. R. J. Colloid Interface Sci. 1996, 183, 397.
Ohsedo et al. dried PHEMA gel (1.10 × 10-1 mol) was put into an excess amount of methacryloyl chloride at 0 °C for 5 min. The modified PHEMA gel was washed in ethanol first and then in water seven times, respectively, for 24 h to remove unreacted reagents. Addition of TEMPO into the Methacryloyl Group of PHEMA Gel. Na2S2O5 (4.00 × 10-3 mol) aqueous solution (20 mL), K2S2O8 (5.00 × 10-3 mol) aqueous solution (30 mL), and EG solution (150 mL) containing TEMPO (1.10 × 10-2 mol) were bubbled with nitrogen gas for 30 min. After nitrogen bubbling, the three solutions were mixed together. PHEMA gel with methacryloyl groups was immersed in this solution and then kept at 60 °C for 120 min. The modified PHEMA gel was washed in ethanol first and then in water seven times, respectively, for 24 h to remove unreacted reagents. Living Radical Polymerization of PNaSS Brush on PHEMA Gel. The methacryloyl group-modified PHEMA gel was immersed in an EG/water mixture (200 mL, 3:1 v/v). NaSS monomer (6.30 × 10-2 mol) was then added to this solution. The mixture was bubbled with nitrogen gas for 30 min. The living radical polymerization of PNaSS was performed in a sealed threenecked flask filled with nitrogen atmosphere at 125 °C, according to the literature.19-21 The polymerization was terminated by cooling the solution to room temperature. The molecular weight of the polymer brushes was controlled by converting the reaction time of monomer because the molecular weight of polymer brushes increases in proportion to the reaction time. The obtained gels were washed in ethanol and water for 24 h to remove unreacted reagents. The growth of the polymer brushes was confirmed by the electronic spectra using a model U-3000 doublebeam spectrophotometer (Hitachi, Ltd.) in a quartz cell (sample thickness is 1 cm in DMSO solution). The polymer brush density was kept constant in this study by fixing the reaction time for introducing the methacryloyl group into PHEMA gel. Interpenetrated Networks Composed of PHEMA and PNaSS. As a reference in the friction measurement, the interpenetrated networks gel (IPN gel) composed of PHEMA and PNaSS having the same molar ratio of NaSS/HEMA and swelling degree as those of PHEMA gel with PNaSS polymer brush was prepared by a two-step polymerization. During the first step, PHEMA gel was prepared as described above. NaSS aqueous solution with a prescribed concentration (0.16, 0.32, and 0.48 M) containing 0.1 mol % K2S2O8 as an initiator and a prescribed amount of MBAA as the cross-linking agent (8-30 mol %) was then prepared. PHEMA gel was immersed into this solution and was kept at 4 °C for 7 h. After that, the solution was bubbled with nitrogen gas for 30 min, and then was heated to 60 °C for 7 h for the polymerization of NaSS. The obtained IPN gels were washed with ethanol and water for 24 h to remove unreacted reagents.
Measurements Mw and Mn of Polymer Brushes. To measure the molecular weight of the polymer brush, the linear PNaSS chains were cut off from the gel by hydrolytic cleavage of the ester bond. The gels were put into methanol hydrochloric acid (1.0%). The solution was then refluxed for 3 h. The molecular weights of the PNaSS cleaved from the gel were assessed using gel permeation chromatography (GPC) with a Shodex SB-805HQ column and a refractive index detector. A series of linear PNaSS having regulated molecular weights (Polysciences, Ltd.) were used as standard. A buffer solution (1:4 (v/v) ) CH3OH/H2O 0.6 M NaNO3 + 0.02 M NaH2PO4; pH ) 8.40) was used as an eluent (flow rate: 0.7 mL min-1 at 40 °C). Swelling Degree of Gels. The amount of water contained in the gel was characterized by the swelling degree, q, which is defined as the weight ratio of the waterswollen sample to the dry sample. Dry gels were obtained by drying in a vacuum until a constant weight was reached. Friction Measurement. A commercially available rheometer “ARES” (Advanced Rheometric Expansion
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Scheme 1. Schematic Representation of Synthesis Procedures of Polymer Brushesa
a (a) Esterification at 0 °C for 5 min; (b) addition of TEMPO into the methacryloyl group at 60 °C for 120 min; (c) living radical polymerization of NaSS at 125 °C for various times.
System, Rheometric Scientific, Inc.) was used for measuring the frictional force of gels. Gels equilibrately swollen in water were cut into a disk shape of R ) 7.5 mm in radius by using a cylindrical gel-cutter. The gel was then glued on the upper surface of a coaxial disc-shaped plate with cyanoacrylate instant adhesive agent (Toagosei Co., Ltd.). A glass plate was used as the opposing substrate. The glass plate was much larger than the sample gels. During the measurement, the gel-glass plate was immersed in pure water. Two surfaces were compressed with each other under the normal load of 0.4 N to give a pressure of 2.3 kPa. The glass plate rotates at an angular velocity
ω of 10-2-101 rad/s, corresponding to a maximum sliding velocity v of 7.5 × 10-5 to 7.5 × 10-2 m/s at the edge of the disc-shaped gel. The torque of the rheometer was measured, and the shear stress (frictional force) corresponding to the maximum sliding velocity is σ(ωR) ) [(3 + R)T(ω)]/ 2πR3, supposing that the torque has a power relation with ω as T(ω) ≈ ωR where R lies between 0 and 1.8 As a good approximation, we use σ(ωR) ) [2T(ω)]/πR3 to calculate the frictional force. The frictional coefficient µ was calculated by dividing the frictional force by the normal pressure. Details of the friction measurement were described in previous papers.6,8
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Figure 1. UV spectra of PHEMA gel with polymer brushes at various reaction times of esterification.
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Figure 3. Polymerization time dependence of NaSS/HEMA molar ratio R in brush-gel.
Figure 2. Polymerization time dependence of number average molecular weight Mn (O), weight average molecular weight Mw (0), and the polydispersity Mw/Mn (4) of polymer brushes.
Results and Discussion Characterization of PHEMA Gel with Polymer Brushes. We attempted to confirm the presence of polymer brushes on gels by swelling the gel in DMSO, which is a good solvent both for PHEMA and for PNaSS, with the aid of spectroscopy. First, the reaction time of esterification of the alcohol group of PHEMA gels with methacryloyl chloride was varied for 5, 15, and 60 min, and then measured UV-vis spectra at the absorption peak around 290 nm originated from the benzene ring of PNaSS. Figure 1 shows UV-vis spectra of PHEMA gels after a living radical polymerization time of 50 h. The absorption intensity increased with an increase in the reaction time of esterification of PHEMA gel (Figure 1), indicating that the density of the PNaSS brush increases with time supposing the polymer brushes with the same chain length are obtained by living radical polymerization. To characterize the length of the polymer brush, the polymer brushes were removed from the gel by hydrolyzing the ester bonds of polymer brushes. Figure 2 shows Mn and Mw of the linear solubilized PNaSS separated from the gel. One can see both Mn and Mw increase with an increase in the reaction time of polymerization of polymer brushes. It is also seen that the polydispersity index (Mw/ Mn) is kept at a low value of 1.2 if the reaction time is less than 100 h. Figure 3 shows the molar ratio of NaSS to HEMA in gels obtained at various reaction times. The mass of PNaSS polymer brushes was obtained by the elementary analysis. As shown in the figure, the quantity of PNaSS in the gels increases in proportion to the reaction time of polymerization. PNaSS is a polyelectrolyte that has a strong ability for water solvation. Therefore, the grafted gel showed an increased swelling with an increase in the molecular weight of PNaSS because an increase in the molecular weight of PNaSS brush brings about an increase in the
Figure 4. Molecular weight Mn dependence of NaSS/HEMA molar ratio (0) and the degree of swelling of brush-gel (O) (a), and the relationship between NaSS/HEMA molar ratio R and the swelling degree q of the brush-gel (b).
molar ratio R of NaSS/HEMA. Figure 4a shows how the increase in the molecular weight of the polymer brushes increases both R and the swelling degree q of the gel. q increases almost in parallel with the increase in R as shown in Figure 4b. From the molecular weight Mn, molar ratio R, the swelling degree q of the brush-gel, and the density of the dried gel, we estimated the density of polymer brush in the gel, and the results are shown in Table 1. Even taking into consideration the swelling degree, the polymer brush density in gel is almost constant, around a value of chain/ (20 nm)3. That is, the average distance between the next neighboring brushes is about 20 nm. For the gels swollen in pure water, the polyelectrolyte brush takes an extended conformation, and it increases linearly with an increase of Mn. When Mn increases from the molecular weight of 6.9 × 103 to 1.4 × 104, the contour length of the polymer brush increases from 8.4 to 16.9 nm, which is close to the next neighboring distance between brushes (Table 1). A further increase in Mn to values of 2.9 × 104 and 5.8 × 104 leads to brush contour lengths of 35.1 and 70.2 nm, respectively (Table 1). Friction of PHEMA Gel Grafted with PNaSS Brushes. Figure 5 shows the relationship between the frictional coefficient, µ, and Mn of the polymer brushes
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Table 1. Characteristic Parameters of PNaSS Brushes Grafted to PHEMA Networka Mn
R (mol/mol)
q
brush density (10 nm)-3
ξ (nm)
vc (m/s)
0 1100 5500 6900 14 000 29 000 58 000
0 2.2 × 10-3 3.9 × 10-3 1.2 × 10-2 5.9 × 10-2 1.3 × 10-1 1.5 × 10-1
3.3 3.8 6.8 10.5 19.5 32.1 44.3
0 0.57 0.12 0.12 0.25 0.16 0.07
0 1.3 6.7 8.4 16.9 35.1 70.2
2.4 9.2 × 10-2 5.9 × 10-2 1.5 × 10-2 3.4 × 10-3 8.4 × 10-4
a R, NaSS/HEMA molar ratio; q, degree of swelling; ξ, brush length, estimated as the contour length of the polymer by using the monomeric molecular weight and length as 206.5 and 0.25 nm, respectively; vc, critical velocity, estimated from vc ) γ˘ c/ξ using ηs ) 10-3 Pa s, T ) 300 K.
Figure 6. Velocity dependence of the frictional coefficient for the gels sliding on a piece of glass surface in pure water at 25 °C under a normal pressure of 2.3 kPa. (O) Brush-gel; (0) IPNgel. Numbers in the figure are molecular weights of polymer brushes. R ) 1.2%. Brush-gel, q ) 10.5. IPN-gel, q ) 11. (b) q ) 19. Brush-gel, R ) 5.9%. IPN-gel, R ) 13%. (c) R ) 15%. Brush-gel, q ) 44. IPN-gel, q ) 52.
Figure 5. Relationship between Mn of polymer brushes and the frictional coefficient at various sliding velocity; (a) 7.5 × 10-5, (b) 7.5 × 10-4, (c) 7.5 × 10-3, (d) 7.5 × 10-2 m/s. Normal pressure: 2.3 kPa.
grafted onto PHEMA gel at various sliding velocities. As shown in Figure 5a, the pure PHEMA gel (Mn ) 0) shows a large friction coefficient, due to its adhesive nature to the glass plate. By incorporating the PNaSS brush to the PHEMA network, the friction coefficient decreases due to the repulsive nature of PNaSS to the glass in water. µ moderately decreases with an increase in Mn until Mn ) 1.1 × 103 and then decreases abruptly with an increase in Mn at a sliding velocity of 7.5 × 10-5 m/s (Figure 5a).
The Mn dependent on the frictional coefficient becomes slightly weaker when the velocity increases to 7.5 × 10-4 m/s (Figure 5b). At 7.5 × 10-3 m/s, an increase in Mn to a value larger than 1.4 × 104 even slightly increases the friction (Figure 5c). This velocity effect becomes clearer at 7.5 × 10-2 m/s where the friction increases with the brush length even at a smaller Mn (Figure 5d). Figure 6 shows the sliding velocity dependence of the friction coefficient of brush-gels with various brush molecular weight Mn. When Mn is 1.1 × 103, we observe a velocity weakening of the frictional coefficient, while the frictional coefficient increases with the sliding velocity when Mn is 6.9 × 103. The velocity dependence becomes more substantial with an increase in the molecular weight of the brush. The transition from velocity weakening (Figure 6a) to velocity strengthening (Figure 6b-d) of the frictional coefficient indicates a change in interfacial interaction from attraction to repulsion.8,14 That is, by an increase in the molecular length of PNaSS, the interaction between the gel and the glass changes from adhesion
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(between PHEMA network and the glass) to repulsion (between PNaSS to glass).8 To elucidate the effect of graft structure on friction, interpenetrating polymer network gels (IPN-gel) consisting of PHEMA as the first network and PNaSS as the second network were prepared, and their frictional behavior was studied. The composition and the swelling degree of these IPN gels were allowed to be almost the same as those of PHEMA gels grafted with PNaSS brush (brush-gel) by modulating the NaSS monomer concentration and the cross-linking density in polymerization. The frictional coefficient of IPN-gels having almost the same NaSS/HEMA molar ratio R as well as the degree of swelling q is shown in Figure 6. For the same R as well as q, µ of IPN-gels shows a velocity dependence similar to those of brush-gels. This indicates that similar to that of brush-gels, the interaction between the IPN-gel and the glass also changes from attraction in Figure 6a to repulsion in Figure 6b-d. However, µ values of the brushgels are much lower than those of the IPN-gel for short brushes (Figure 6a-c) or at a low velocity for long brushes (Figure 6d). The brush-gel with a large Mn ) 5.8 × 104 shows the lowest friction in the low velocity region, and the friction increases with the sliding velocity, reaching a value even higher than that of the IPN-gel at a sliding velocity of 7.5 × 10-2 m/s. In the literature,16-18 the lubrication effect of polymer brush modified on solid surfaces in a good solvent was attributed to the steric factor. That is, mutually compressed polymer brushes in a good solvent resist interpenetration, because of the excluded-volume effect arising from chain configurational entropy. For the polyelectrolyte brushes, this configurational excluded-volume effect is augmented by the large osmotic pressure exerted by mobile counterions within the brushes.18 Apparently, the massive reduction in friction of brushgels against the glass plate, a repulsive solid surface, observed in the present study is not due to the steric effect, because there is no interpenetration here. In addition, both the brush-gel and the IPN-gel should have the same osmotic repulsion exerted by mobile counterions at the gel-glass interface. Nevertheless, the former shows a much lower friction, especially in the low sliding velocity range (7.5 × 10-5 to 7.5 × 10-4 m/s). H. R. Brown has studied the friction behaviors of poly(dimethyl siloxane) (PDMS) rubber on a rigid surface coated with PDMS brush of various lengths. He found that the shear stress that appears on the brushes is strongly affected by the segment mobility of the materials on both sides, and when the materials on both sides of the interface are highly mobile considerably lower stress is observed.23,24 The strong velocity and molecular length dependences of the brush effect indicate that the brush effect observed in the present study is also due to an enhanced mobility of the brushes. When the negatively charged polyelectrolyte gel is slid against a glass substrate in pure water, an electric double layer (EDL) is formed at the interface even under a large normal load, and it serves as a lubricant layer to give a low friction.3-6 The thickness of the EDL due to the osmotic repulsion, Lrep, is determined by the balance of osmotic pressure exerted by the mobile counterion and the normal pressure applied to the gel. Lrep should be the same for both the brush-gel and the IPN-gel when they have the same counterion (Na+) concentration. During sliding, the frictional force (shear stress) induces polymer deformation that increases the effective thickness (23) Brown, H. R. Science 1994, 263, 1411. (24) Brown, H. R. Faraday Discuss. 1994, 98, 47.
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Figure 7. Sliding velocity dependence of the apparent hydrodynamic layer thickness for brush-gels and IPN-gels of similar degree of swelling q or NaSS/HEMA molar ratio R. Numbers in figure are contour lengths of PNaSS brushes. (O): q ) 19. Brush-gel, R ) 5.9%; IPN-gel, R ) 13%. (b): R ) 15%. Brush-gel, q ) 44; IPN-gel, q ) 52.
of the hydrodynamic layer to reduce the shear resistance (shear thinning effect). Therefore, the total lubricant layer thickness of the brush-gel becomes LT ) Lrep + Ldym upon sliding. The hydrodynamic Ldym should depend on the brush length or network mesh size as well as the sliding velocity. A polymer brush is highly mobile and more deformable than that of the cross-linked chain by the shear force. By the hydrodynamic lubrication mechanism, the apparent hydrodynamic layer thickness LT can be estimated from the frictional force (shear stress) σ by
σ ) ηsv/LT Here, ηs is the solvent viscosity and v is the sliding velocity. Figure 7 (n) shows that the hydrodynamic layer thickness LT thus obtained monotonically increases with the sliding velocity from ca. 1 nm to ca. 100 nm for a brush-gel with a length of 8.4 nm as well as for IPN-gels with a similar R and q, indicating the shear thinning effect. For the brushgel with a brush length of 70.2 nm, longer than the next neighboring distance, LT shows a maximum with the sliding velocity, while the IPN-gel with a similar R and q shows a monotonic increase in LT with v (Figure 7, b). Figure 7 also shows that for the same PNaSS content, the brush-gel shows a thicker LT than that of the IPN-gel when the brush is short (8.4 nm) or when the sliding velocity is not very high for a long brush (70.2 nm), indicating the drag reduction effect of the brush. What is interesting is that the change in LT with sliding velocity is about 100 nm, much larger than the brush length (Figure 7a) or the mesh size of the network (Figure 7b). This indicates that the shear thinning effect for a soft gel occurs not only on the top brush or mesh size level of the gel, but also in a deeper surface. The less prominent effect for a brush-gel with a long chain at a high sliding velocity should be attributed to the slow relaxation of the polymer brush. To be a mobile chain, the relaxation time of the brush, τ, should be much shorter than the inverse of the shear rate, (γ˘ )-1, of the sliding. That is, γ˘ τ , 1. In the present case, Zimm’s relaxation time is characteristic, and τ can be estimated from25
τ=
ηsξ3 kBT
(3)
where kB is the Boltzmann constant, T is absolute temperature, ηs is the viscosity of the solvent, and ξ is the brush radius, which is approximately the contour length (25) Doi, M.; Edwards, F. S. Oxford University Press: Oxford, 1986.
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for an extended polyelectrolyte. The shear rate corresponds to a sliding velocity v of γ˘ ) v/ξ. So,
γ˘ τ )
vηsξ2 kBT
(4)
From the condition γ˘ cτ ) 1, we estimate the critical sliding velocity vc for brushes of various lengths using ηs ) 10-3 Pa s, T ) 300 K, and the results are shown in Table 1. The polymer brush is effective in reducing the friction only when the sliding velocity is less than vc. The vc values in Table 1 are satisfactorily in agreement with the velocity dependence for brushes with various lengths in Figures 6 and 7. For example, a brush-gel with a brush length of 70.2 nm shows a maximum apparent hydrodynamic thickness (Figure 7a) around 7.5 × 10-4 m/s, which agrees well with the vc ) 8.4 × 10-4 m/s in Table 1. For a certain sliding velocity, we can also estimate the critical brush length. For example, for a velocity of 7.5 × 10-3 and 7.5 × 10-2 m/s, we obtain ξ , 23.4 nm and ξ , 7.4 nm, respectively, to satisfy the condition of γ˘ τ , 1. This estimation is in good agreement with the result in Figure 5c,d where the brush effect becomes less effective at a brush length of 35.1 and 6.7 nm for a velocity of 7.5 × 10-3 and 7.5 × 10-2 m/s, respectively.
Conclusions For a gel with polymer brushes of certain length, the drag reduction effect in comparison with the network structure gel is only observed at a sliding velocity lower than a critical value. The longer is the polymer brush, the lower is the critical value. A satisfactory correlation between the Zimm-like relaxation time τ of the polymer brush and the critical shear rate of the sliding γ˘ c is observed; that is, the brush effect is only effective when γ˘ τ , γ˘ cτ ) 1. These results indicate that the polymer brush effect observed in the gels here is due to the high mobility of the end-free brushes. This result may be informative in the understanding of the low friction phenomena in biological systems, such as the lubrication mechanism of proteoglycan in articular cartilage.26,27 Acknowledgment. This research was supported by Grant-in-Aid for the Fundamental Research A, and was also supported in part by a Grant-in-Aid for the Creative Scientific Research from the Ministry of Education, Science, Sports, and Culture of Japan. LA036211+ (26) Iozzo, R. V. Annu. Rev. Biochem. 1998, 67, 609. (27) Bernfield, M.; Go¨tte, M.; Park, P. W.; Reizes, O.; Fitzgerald, M. L.; Lincecum, J.; Zako, M. Annu. Rev. Biochem. 1999, 68, 729.
