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. 2020 Jun 30;5(27):16406–16412. doi: 10.1021/acsomega.9b04184

Friction Dynamics of Hydrogel Substrates with a Fractal Surface: Effects of Thickness

Koki Shinomiya †, Hina Okawara †, Kei Kikuchi †, Hiroyuki Mayama ‡, Yoshimune Nonomura †,*
PMCID: PMC7364433  PMID: 32685803

Abstract

graphic file with name ao9b04184_0007.jpg

Interfacial phenomena on soft and wet materials, such as hydrogels, are important for modeling physical phenomena, such as friction, wetting, and adhesion on hydrophilic biosurfaces. Interfacial phenomena on soft material surfaces are not only affected by the properties of the surface but also by the geometry of the substrate. However, there are few reports on the influence of geometry and deformability on friction behavior at gel interfaces. In this study, we evaluate the effects of the thickness (H) of the upper agar gel layer on the friction force between gels under a sinusoidal movement. Although H does not significantly affect the friction force or pattern, the normalized delay time (δ), which is the normalized time lag in the friction force response to the contact probe’s movement, increases with H. A regression analysis between δ and H shows that δ increased linearly with H. We present a simple model incorporating a shear modulus to qualitatively explain the experimental results. The analysis and our model indicate that one must not only consider surface properties, such as adhesion, but also thickness and rigidity when studying friction behavior at the gel–surface interface. These findings will be useful for understanding friction phenomena on soft biological systems, such as the tongue, throat, esophagus, and gut surfaces.

Introduction

Gels exhibit complex and specific mechanical properties. For instance, they show extremely low friction resistance and rheological behavior with both viscosity and elasticity. The friction coefficient on a gel surface ranges from 10–1 to 10–3, which is lower than that on common solid surfaces.1,2 These low friction forces are caused by both surface lubricity and the liquid-induced load support.3−9 A model incorporating both repulsion and adsorption between polymer–solid surfaces is generally accepted to explain the friction behavior on gel surfaces.3,4 If a thick water layer is formed at the interface because of osmotic repulsion between electrical double layers, friction is dominated by hydration lubrication. Additionally, a biphasic theory is generally accepted to explain the tribology of articular cartilage and hydrogels in boundary lubrication.6−9 Because the liquid phase supports the applied load, the friction resistance generated between the solid–solid phase is low. We discovered an anomalous friction phenomenon between flat agar gels under an accelerated motion.10 In our previous study, both asymmetric friction behavior between the outward and inward directions and friction phenomenon with extremely low friction coefficients were observed. These characteristic friction properties may be caused by gel deformations during the reciprocating motion. Additionally, the mechanical properties of materials are reflected in the normalized delay time (δ), which is the normalized time lag in the friction force response to the contact probe’s movement. The parameter δ is an important parameter for understanding the friction of viscoelastic materials. The parameter δ of the gel and elastic sponge exceeded that of the polymer film, which was hard to deform.10−14

Surface geometry is a factor that affects the friction properties of gels. The effect of surface roughness on friction resistance has been studied by others.15−19 For example, the surface roughness of whey protein gels was found to decrease when xanthan was added.16 Gels with the roughest surfaces showed the largest friction and load-dependent friction changes. The friction force of a hydrogel with a rough surface decreases as the sliding velocity increases.17 We evaluated and modeled interfacial phenomena, such as friction, wetting, and adhesion, on a fractal agar gel, which has a rough hierarchical surface structure.14,18−23 We evaluated the friction force between fractal agar gel surfaces under accelerated motion. An asymmetric profile which was observed in the friction evaluation on flat agar gel was not obtained.14 It is because the rough structure of the gel surface changes the contact state of the gel. Several hydrophilic fractal surfaces coated with mucus contribute to either effective nutrient absorption or sensitization of the senses. Sapid substances are perceived by taste buds via small projections (papillae) on the tongue.24 Villous surfaces on the nasal and esophageal mucosa efficiently trap odorous substances and exclude foreign matter.25 In the circular folds of the small intestine walls, hierarchical rough surfaces contain villi and microvilli that improve nutrient absorption efficiency.26

With most soft materials, the effect of their physical properties and geometry on the friction phenomena must be considered. The energy dissipation of soft materials may be mainly dominated by viscoelastic deformation.27−29 In addition, the balk deformability can affect tactile and food texture.30 Nakatani et al. reported that many women feel more sensitive than men when they touch objects with their fingers. This result can be related to their soft and deformable skin. Additionally, deformability strongly affects the wetting and dewetting processes on soft surfaces.31−33 Bulk deformability may also influence lubrication at the interface. For friction between poly(dimethylsiloxane) (PDMS) rubber and a glass substrate, the friction coefficient ranges from about 10–2 to 100 according to the thickness of the rubber.34 When a shear force is applied to a gelatin hydrogel, both a decrease in the stress and an increase in the energy release rate was observed with increasing hydrogel thickness.35 For a thick temporomandibular joint disk, the rate of increase for friction force was low.36

However, there are few reports on the effect of thickness on friction behavior at gel–gel interfaces. In the human oral cavity, the thicknesses of the maxillary mucosa and tongue are several mm and several tens of mm, respectively.37,38 Additionally, the thicknesses of the esophagus, stomach, small intestine, and large intestine walls are several mm.39,40 Thus, the geometry and deformability of entire soft organ systems may affect friction and interfacial phenomena in daily life, not just simple surface geometry. During the swallowing and digestive processes, food boluses of various geometries move over the esophagus or gut surfaces. To explain these interactions and nutrient transfer phenomena, it is important to consider friction under asymmetric conditions with various body geometries. When a rigid-glass prism is sheared off a thin film of silicone elastomer and bound to a glass plate, the critical shear stress of fracture decreases with the thickness following a square root relationship, as is the case with the removal of rigid punches from thin elastomeric films by normal pull-off forces.41

In this study, we assess the friction between two fractal agar gels in water under a sinusoidal motion to model friction phenomena on hydrophilic biological surfaces. The influence of the specimen thickness on the friction dynamics was evaluated for agar gel thicknesses of 1, 3, 5, 10, and 15 mm. The friction was systematically evaluated under various velocities and normal force conditions. Sinusoidal motion of our friction evaluation system was achieved by the Scotch-yoke mechanism to mimic the friction conditions in the body; this enabled a reciprocating sliding motion of the contact probe accompanied by acceleration.10−14

Experimental Section

Materials

Agar powder and tristearin [(C17H35COO)3C3H5] were obtained from Wako Pure Chemical Industries, Ltd. (Osaka, Japan). Plaster was purchased from Sanjo Co., Ltd. (Hokkaido, Japan). Water was purified with a Demi-Ace Model DX-15 demineralizer (Kurita Water Industries Ltd., Tokyo, Japan). The fractal agar gel was produced as follows:18 First, a surface structure of solidified tristearin with a hierarchical rough structure was transferred to the surface of a fractal plaster replica. Agar powder (9 g) was mixed with deionized water (216 g) and heated at 90 °C for 80 min under agitation at 500 rpm. If the heating temperature is lower or the heating time is shorter, the agar gel becomes brittle. The aqueous agar solution (225 g, agar concentration = 4 wt %) was poured into a glass Petri dish (11 cm in diameter) containing the fractal plaster replica in the center of the dish. The solution was left at room temperature for 90 min before cooling at 7 °C for 90 min. The agar gel substrate for the lower and upper sides was molded to dimensions of 80 × 30 × 10 mm3. The substrate for the upper side was molded to a size of 20 × 20 mm2, whereas the thickness H was adjusted to 1, 3, 5, 10, and 15 mm. Prior to friction evaluation, the gel substrates were immersed in deionized water at 7 °C for 2 days. The gel surface geometries were examined with a LEXT 3D measuring laser microscope OLS4000 (Olympus Co., Tokyo, Japan). Figure S1 presents the optical microscopy and scanning electron microscopy (SEM) images of the gel surface. The roughness parameters, Ra and Rz, were 4.07 ± 1.11 and 23.14 ± 8.39 μm, respectively, at a magnification ratio of 50. According to the analysis based on the box-counting method, the fractal dimension (D) of the gel surface was 2.23 ± 0.02 within a box size range r = 5–414 μm (Figure S2).18 As in our previous study, we obtained cross-sectional images by SEM (SU8000, Hitachi High-Technologies Co., Tokyo, Japan). Using the actual cross sections, D was determined using the box-counting method.

We determined these conditions based on the results of preliminary tests. On the thickness of agar gel, the suitable thickness was chosen to provide a homogeneous gel for all evaluations. If the substrate is too thick or too thin, it is difficult to obtain uniform substrates. In previous studies, we evaluated the effect of agar concentration. In this study, 4% gel was chosen because the elastic modulus is almost similar to that of the human tongue. We chose agar because it is the most common and readily available gelling agent.

To consider the effect of surface geometry on the friction profile and resistance, an agar gel with a flat surface (flat agar gel) was also prepared. Based on our previously reported method for creating the gel substrates,10 flat agar gels with different thicknesses were prepared at an agar concentration of 4 wt %. Samples for the lower and top sides were molded into the same shapes as the fractal agar gels; the thicknesses of the upper samples were H = 1, 3, 5, 10, and 15 mm. We evaluated the surface geometry of 4 wt % flat agar gel using a VK-X100 laser microscope (Keyence Co., Tokyo, Japan).22 The roughness parameters Ra and Rz were 26 and 316 nm, respectively.

Measurements

We determined the friction force between the two fractal agar gel substrates in deionized water using a sinusoidal motion friction evaluation system (Figure S3). Sinusoidal motion was achieved via the Scotch-yoke mechanism where the rotational movement of an eccentric disk was converted to the sinusoidal reciprocating movement of a contact probe.10−13 A direct-current servomotor was used with a rotation rate of 0.001–2.1 rad s–1 and a sliding distance of ±2.5–20 mm. The moving distance of the contact probe was measured with a displacement sensor (CD22100VM122, OPTEX FA Co., Ltd., Tokyo, Japan) attached to the side of the driving motor. The light source was a red laser diode (wavelength: 655 nm), the displacement accuracy was 20 μm, and the measurement range was ±50 mm. The friction and normal forces were measured with two load cells. The measurement ranges of the forces were: Fx = 0.03–9.7 N and Fz = 0.03–9.7 N. The minimum detection limits of Fx and Fz were both 0.01 N. Figure 1 shows a schematic of the evaluation system used in this study. The agar gel substrate (80 × 30 × 10 mm3) was fixed to an acrylic resin holder on the lower side with an adhesive (Aron Alpha, Toagosei Co., Ltd., Tokyo, Japan) and double-sided tape (Nichiban Co., Ltd., Tokyo, Japan). The lower substrate was covered with a 2 mm-thick water film. A contact probe covered with agar gel was prepared as follows. An acrylic plate (20 × 20 × 5 mm3) was fixed to the probe with double-sided tape. The agar gel substrate (20 × 20 × H mm3) was fixed to the acrylic plate with an adhesive and double-sided tape. The sliding velocity V under the sinusoidal movement was calculated from the stroke length (A), the angle rate (ω), and time T according to eq 1

graphic file with name ao9b04184_m001.jpg 1

Here, the friction conditions were A = ±14.5 mm; ω = 0.1, 1.0, and 2.1 rad s–1; sampling interval = 1 ms (2.1 rad s–1), 2 ms (1.0 rad s–1), and 20 ms (0.1 rad s–1); and the normal force (W) was 0.29, 0.39, 0.69, and 0.98 N. The maximum velocities (Vmax) for the angular velocities were 1.5 mm s–1 (0.1 rad s–1), 15 mm s–1 (1.0 rad s–1), and 30 mm s–1 (2.1 rad s–1). Additionally, the cycle times (T0) were 60 s (0.1 rad s–1), 6 s (1.0 rad s–1), and 3 s (2.1 rad s–1). In considering the influence of the surface geometry on the friction profile and resistance, the friction on the fractal agar gel surfaces was compared with that of flat agar gel surfaces at ω = 0.01 rad s–1 and W = 0.98 N. At ω = 0.01 rad s–1, the sampling interval, Vmax, and T0 were 200 ms, 0.15 mm s–1, and 600 s, respectively. Three different samples were evaluated in triplicate to verify the repeatability of the friction data. Nine evaluations at 25 ± 1 °C and 50 ± 5% relative humidity were conducted in total.

Figure 1.

Figure 1

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Schematic representation of the evaluation system and fractal agar gels in this study.

Results and Discussion

Friction Profile

Figure 2 shows typical profiles for the friction force, normal force, and velocity between the two fractal agar gel substrates under sinusoidal motion with gel thickness H = 1 mm, angular velocity ω = 0.1 rad s–1, and normal force W = 0.98 N. In regions where the sliding velocity (V) is positive or negative, the contact probe moves in an outward or inward direction, respectively. The friction force changed dynamically with time. Although W was almost constant during the friction process, V changed with time. For the outward direction, the static friction force (Fs) was 0.44 N at V = 0.39 mm s–1 and decreased with the sliding velocity. A similar friction profile was observed for the inward direction. The absolute friction force decreased with the sliding velocity after Fs occurred. Similar symmetric friction profiles were obtained for all friction conditions.

Figure 2.

Figure 2

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Temporal profile of the friction force, normal force, and velocity on fractal agar gel surfaces (H = 1 mm, ω = 0.1 rad s–1, and W = 0.98 N).

Additionally, we observed a time lag (Δt) in the friction force response to the contact probe movement. To catch the essence of the friction, we defined the normalized delay time δ as described by eq 2

graphic file with name ao9b04184_m002.jpg 2

where Δt is the observed time lag and T0 is time for a cycle. δ was 0.021 ± 0.003 (H = 1 mm, ω = 0.1 rad s–1, and W = 0.98 N). As shown in Table S1, the magnitude of δ was dependent on the gel thickness, angular velocity, and normal force.

Effects of Normal Force, Angular Velocity, and Upper Fractal Gel Thickness on the Friction Force

The friction force (F) was analyzed using the traditional power law of friction on a soft material surface

graphic file with name ao9b04184_m003.jpg 3

where a and n are constants, and W is the normal force.42,43 The proportion of variance (R2) was determined by regression of eq 3.44Figure S4 and Table S2 show a relationship between the friction and normal forces and a, n, and R2 values obtained under each condition. Additionally, Figure 3a shows the relationship between n and H. The vertical force was set in a narrow range because wear was observed on the gel surface at loads greater than 0.98 N. This narrowness in dynamic range led to a large variation in n and a. Therefore, we describe here a rough discussion of the effect of gel thickness. The n was 0.9–1.2 regardless of H and ω. These data predict that the Coulomb–Amonton law, F = μW [F: friction force, W: normal force, and μ: friction coefficient] is approximately true for friction between fractal agar gel surfaces (n ≈ 1.0) under most friction conditions. Thus, a can be regarded as a friction coefficient. As shown in Figure 3b, a did not change significantly with H. At ω = 0.1 rad s–1, the magnitudes of a for H = 1, 3, 5, 10, and 15 mm were 0.326, 0.310, 0.364, 0.391, and 0.298, respectively. Similarly, a did not change with H at ω = 1.0 and 2.1 rad s–1. These results indicate that the thickness of fractal agar gels does not significantly affect the friction force and load dependence. Although a decreased as the angular velocity increased, its order of magnitude did not change significantly.

Figure 3.

Figure 3

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Relationships between n and H (a) and between a and H of fractal agar gels (b). ω = 0.1 rad s–1 (circle), 1.0 rad s–1 (square), and 2.1 rad s–1 (triangle). Friction data (H = 5 mm, W = 0.29, 0.39, and 0.98 N) were obtained from ref (14).

Effect of Upper Fractal Gel Thickness on the Normalized Delay Time

To understand the characteristics of soft hydrogels, we evaluated the friction force response to changes in the sliding velocity. On viscoelastic soft material surfaces, such as elastomers, the occurrence of friction forces is delayed relative to the start of sliding as some force will be absorbed by material deformation.43 In this study, a and n did not change significantly with the gel thickness (H). On the other hand, the δ tended to increase with thickness. Figure 4 shows the typical relationship between δ and H (ω = 0.1 rad s–1 and W = 0.98 N). Under these conditions, the magnitudes of δ were 0.021 ± 0.003, 0.021 ± 0.002, 0.027 ± 0.005, 0.033 ± 0.007, and 0.029 ± 0.005 for H = 1, 3, 5, 10, and 15 mm, respectively. Even under other conditions, a similar tendency was observed. Figure 5 shows the dependence of W, ω, and H on δ. Here, we observed large deviations in the parameter δ as shown in Figure 4. We believe that this large deviation is an inherent property of soft materials. Figure 5 shows the overall trend of this parameter. Detailed data are shown in Figure S6. Similarly, δ increased with H. The number of conditions where δ exceeded 0.020 was one, three, four, nine, and ten at H = 1, 3, 5, 10, and 15 mm, respectively. Additionally, the number of conditions where δ exceeded 0.025 was zero, zero, two, four, and five for H = 1, 3, 5, 10, and 15 mm, respectively. Moreover, δ increased with a higher normal force and lower angular velocity.

Figure 4.

Figure 4

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Relationships between δ and H (circle) and between PbH and H (square) of fractal agar gels with ω = 0.1 rad s–1 and W = 0.98 N. Friction data (H = 5 mm) were obtained from ref (14).

Figure 5.

Figure 5

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Dependence of W, ω, and H on δ of fractal agar gels. Friction data (H = 5 mm, W = 0.29, 0.39, and 0.98 N) were obtained from ref (14).

Linear Regression Analysis between δ and H

To evaluate the mechanism causing the increase in δ with H, we approximated the dependence of δ on H by a linear function. Figure 4 shows the typical relationship between δ and H (ω = 0.1 rad s–1 and W = 0.98 N). The linear function is given by eq 4

graphic file with name ao9b04184_m004.jpg 4

where b and δ0 are constants. The R2 was determined by regression of eq 4.43Figure S6 and Table S1 show the b, δ0, and R2 values obtained for each condition. Here δ0 is regardless of the bulk deformation effects as δ0 is not influenced by the gel thickness. However, we believe that b affects δ based on the bulk geometry and its deformability. As shown in Figure 4, for ω = 0.1 rad s–1 and W = 0.98 N, b and δ0 were 0.0007 mm–1 and 0.021, respectively. Additionally, a linear relationship was observed between δ and H (R2 = 0.724). In this study, b and δ0 had magnitudes of 0.0003–0.0009 mm–1 and 0.013–0.021, respectively. For ω = 2.1 rad s–1 and W = 0.29 N, the linearity of the regression line was poor (R2 = 0.166). However, the linearity was good under other conditions (R2 = 0.719–0.995). These results indicate that the normalized delay time δ increases linearly with the gel thickness H.

We established two factors that determine δ: bH and δ0. We believe that bH represents the geometry and deformability of the body, whereas δ0 is regardless of the bulk deformability. Table S1 shows the contribution rate (PbH) which is defined by in eq 5.

graphic file with name ao9b04184_m005.jpg 5

As shown in Figure 4, PbH increased considerably with gel thickness. For H = 1, 3, 5, 10, and 15 mm, PbH was 4, 10, 13, 22, and 36%, respectively (ω = 0.1 rad s–1, W = 0.98 N). This tendency was significant even under other conditions. Thus, if the gel is thick, the thickness H affects δ strongly with H. The parameter δ should be dependent on the dissipative properties of the system, including stress-diffusion coupling, viscoelastic relaxation, and frictional dissipation because hydrogels have multiple relaxation processes (viscoelasticity and water diffusion).46 In the right side of eq 4, the first term, bH, is determined by the dissipative properties of the system, whereas the frictional dissipation affects the second term, δ0.

Effect of Surface Geometry on the Friction Dynamics of a Gel

To evaluate the effect of surface structure, the friction force was determined between flat agar gel surfaces under acceleration.10 The profile on this surface is symmetrical in the outward and inward directions when sliding velocity is low. Under these conditions, we can accurately evaluate and compare the friction parameters. Thus, we compared the friction on flat agar gel surfaces with that on fractal agar gel surfaces at H = 1, 3, 5, 10, and 15 mm (ω = 0.1 rad s–1, and W = 0.98 N, Section S5 in the Supporting Information, Figure S5, and Table S3). Although the static friction was slightly smaller in one direction for the flat agar gel, the friction profile did not change significantly with H for both flat and fractal agar gels. The kinetic friction coefficient also did not change significantly with H, regardless of the gel surface geometry. The magnitude of the kinetic friction coefficient was approximately 0.30 under all conditions. However, the δ increased linearly with H, and the PbH also increased with H, regardless of the gel surface geometry. Thus, if there are no anomalous phenomena, such as large deformations or the formation of a thick liquid film at the interface, a linear increase in δ with H can occur regardless of the gel surface geometry.

Why Does δ Increase Linearly with H?

In this study, we found that δ increased linearly with H. Kaneko et al. introduced an interfacial free energy term and an elastic energy in bulk deformation term to their friction model for studying friction resistance of PDMS rubber on glass.34 This model is consistent with their experimental results. However, it does not explain the delay in the mechanical response increase with thickness.

To understand how the geometry and deformability of the hydrogel body affected δ, we present a macro model based on body shear. Figure 6 shows a conceptual model diagram. The shear modulus (G) is defined as the ratio of the shear stress (σ) to the shear strain (γ). G is given by eq 6

graphic file with name ao9b04184_m006.jpg 6

where Q is the shear force, S is the cross-sectional area of the object, and Lc is the shear deformation distance just before the shear force is generated. The time tc at which the deformation reaches Lc is introduced to obtain an approximate eq 7 if we assume that the velocity is constant at the instant sliding commences (see Section S6 in the Supporting Information).

graphic file with name ao9b04184_m007.jpg 7

Figure 6.

Figure 6

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Macro-scale friction model based on the shear modulus (G).

If we assume that the shear force (Q) is the friction force (F), δ is both proportional to the friction force and gel thickness and inversely proportional to the rigidity and cross-sectional area of the gel. As the rigidity and cross-sectional area are constant in this study, δ is proportional to the friction force and gel thickness.

Additionally, our model is consistent with the relationship between δ and other factors. For instance, in this study, δ increased proportionally with W (Figure S4). Thus, the Amontons–Coulomb law (F = μW) could be applied and the friction force increased with normal force. According to our model and experimental results, the increase in δ with the friction force can be explained by the increase in the friction force with the normal force. In our previous work, we found that the δ of soft agar gels tended to become larger than that of hard agar gels when the gel thickness and cross-sectional area were constant.10 Thus, our model qualitatively agrees with this experimental result as δ was inversely proportional to the rigidity. Unfortunately, the inverse relationship between δ and S is difficult to confirm experimentally in this study. We expect that δ decreases when S increases, as this would be regarded as a thin object.

Very recently, it has been understood that it is important to observe the relaxation process in order to show the mechanism of the dependence on the geometry from the viewpoint of microstructure of gel materials.45 So far, significant correlations have been identified between the compressive relaxation properties of the gel and the friction properties.46 Yamaguchi et al. have discussed the stress–relaxation behavior of agarose hydrogels on the basis of the results of the unconfined compression tests and found that hydrogels have multiple relaxation processes because of viscoelasticity and water diffusion.47 The relaxation time τ1, which is caused by the viscoelasticity mode, was constant regardless of the size of gel and was about 5 s. On the other hand, the time τ2, which is caused by the water diffusion mode, increased from about 50 to 100 s with the size. In our previous study, it was shown that the relaxation time of the gel affects the friction profile. When the gel substrate is rubbed for a shorter time than the relaxation time, the morphology of the gel surface becomes unstable, and the characteristic friction profiles were observed. Here, we tried to analyze the relaxation process from static friction to dynamic friction. However, significant trend was not observed between the relaxation times and the thickness. At present, it is not possible to determine whether a biphasic model is suitable to understand the frictional data. In the near future, we will try to measure the relaxation process in friction more accurately and propose an appropriate model.

Conclusions

In this study, we systematically investigated the influence of gel thickness (H) on the friction force between fractal agar gels and its profile under a sinusoidal motion. A regression analysis between the normalized delay time (δ) and H indicated that δ increased linearly with H. We proposed a simple model based on the shear modulus and explained the experimental results qualitatively. The normalized delay time (δ) is dominated by the surface properties, but the bulk properties of the substrate, such as the thickness and rigidity, must also be fully considered to understand the friction behavior. The friction parameters are closely related to energy dissipation in the gel substrate on the level of macroscopic and microscopic scales. The measurement of viscoelastic properties and the theoretical discussion should be included in our future plan. The findings of this study will provide clues to understanding the friction phenomena on wet and soft material surfaces.

Acknowledgments

This study was supported by a Grant-in-Aid for Scientific Research (C) (grants nos. 26390001 and 18K03554) and Scientific Research (B) (grants no. 18H01402) from the Ministry of Education, Culture, Sports, Science and Technology, Japan (MEXT).

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.9b04184.

  • Details of sinusoidal motion friction evaluation system; parameters of friction resistance and response; effects of normal force, angular velocity, and thickness of upper gel on friction resistance; effect of surface geometry on the friction dynamics of gel; and derivation of eq 6 from eq 5 in this manuscript (PDF)

The authors declare no competing financial interest.

Supplementary Material

ao9b04184_si_001.pdf (486.3KB, pdf)

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