Biphasic Lubrication Theory of Hydrogels: Transient Response and Sample Size Effects

doi:10.21203/rs.3.rs-5303101/v1

Download PDF

Research Article

Biphasic Lubrication Theory of Hydrogels: Transient Response and Sample Size Effects

https://doi.org/10.21203/rs.3.rs-5303101/v1

This work is licensed under a CC BY 4.0 License

You are reading this latest preprint version

Hydrogels are soft and wet materials with remarkable properties. Among such properties, low friction has attracted much attention from scientists and engineers. While many studies investigated the correlation between material properties and frictional behavior, little attention has been paid to other effects, such as sample sizes. In this study, we developed a theoretical model based on the biphasic lubrication theory to describe the frictional behavior of hydrogels against a glass substrate in the rotational shear test. Consequently, we obtained analytical solutions for the time evolutions of friction coefficient, hydrostatic pressure, and elastic stress. To validate our model, we prepared cylindrical PVA (Polyvinyl alcohol) hydrogels with different radii and thicknesses and conducted friction experiments. We confirmed reasonable agreements with theoretical predictions for the transient responses and their size dependences. Interestingly, we found that the friction coefficient at the initial phase decreased drastically with decreasing thickness. Our results indicate the fundamental importance of bulk transport in surface friction of hydrogels, and the controllability of friction by varying the geometry of hydrogels.

Hydrogel

Friction

Size dependence

Theory

Biphasic lubrication

Hydrogels are soft and wet materials composed of sparsely crosslinked polymers and a large amount of water [1,2]. Their unique structures impart remarkable properties, including flexibility, optical transparency, water retention (swelling) ability, and stimuli (pH, temperature, electric field) responses [2,3]. In addition, due to their strong affinity to biological tissues, hydrogels are also considered bio-compatible materials, making them suitable for replacing damaged human tissues, such as artificial articular skins, corneas, and cartilages [2,4–8].

Among such striking and intriguing properties of hydrogels, their frictional behavior during sliding has attracted many scientists and engineers for decades [9–14]. When a hydrogel slides against a rigid and smooth substrate, it often exhibits extremely low friction [15–23]. Different mechanisms have been proposed to explain this low friction: (1) the formation of the static lubrication layer due to the osmotic repulsion by poly-electrolyte networks [9,10,15,16,24], (2) the formation of the protective layers or boundary films through protein/lipid adsorption [8,9,13,25,26], and (3) the reduction in effective normal stress of solid phase or the increase in fluid load support, due to the biphasic lubrication [26]-[33]. Of these three mechanisms, biphasic lubrication is important for neutral hydrogels in a pure water environment, as it operates without electric effects or specific additive molecules. The biphasic lubrication theory is the continuum mechanics-based theory that describes the elastic deformation, interstitial fluid flow, and frictional behavior of articular cartilages, which was introduced by Mow and then extended by other researchers [34]-[38]. Later, it was applied to artificial articular cartilages, that is, hydrogels. There have been several studies on the frictional behavior of natural or artificial articular cartilages in various operating conditions and with different sample geometries [11,14,18,23,24,28,39]. However, there have been very few systematic studies on the effects of sample geometry: how the sample sizes affect frictional behavior has not yet been fully understood. In addition, few theoretical studies have been reported; no formula for designing or optimizing the performance of hydrogels is available at this moment.

In this paper, we developed a theoretical model that describes the frictional behavior of hydrogels within the framework of biphasic lubrication theory. As discussed later, we successfully obtained analytical solutions for the time evolutions and their sample size dependences in the rotational shear test. In addition, we experimentally investigated the frictional properties of PVA (Polyvinyl alcohol) hydrogels swollen and lubricated with pure water to validate the model. As discussed later, the friction coefficient changed significantly with the geometry of hydrogels, including radius and thickness, and reasonable agreements with the theoretical results were confirmed.

2.1 Sample

PVA hydrogels were prepared from a 20wt% aqueous solution of PVA (Kishida Chemical Co. Ltd.). The PVA we utilized has a polymerization degree of 2,000 and an average saponification degree of 98.4–99.8 mol%. The solution was sandwiched between two glass plates separated by a spacer, and gelation was done by the repeated freezing-thawing method (-20℃ for 10h and 4℃ for 20h, 5 cycles)[30]. After gelation, the hydrogel samples were immersed in a large amount of distilled water for over a week to achieve swelling equilibrium. Finally, the samples were cut into cylindrical-shaped pieces using biopsy punches.

2.2 Rotational shear test using cylindrical hydrogels

To facilitate quantitative analysis, we conducted friction experiments with a simple setup. Figure 1 shows the schematic of our experimental apparatus. The hydrogel samples were fixed onto the parallel plate of a Rheometer (Physica MCR 301, Anton Paar) using a small amount of cyanoacrylate adhesive. The sample was immersed in distilled water and slid against the bottom glass plate by rotating the top plate at a constant angular speed ($\:\omega\:=0.6$ rpm) under a given normal pressure ($\:P=0.1$MPa). The frictional torque at the interface between the gel and the glass plate was measured. Specifically, with this setup and cylindrical geometry, the nominal contact area remained constant over time, in contrast to the Hertzian geometry. In addition, the rotational symmetry is preserved, eliminating front-to-rear variations that could lead to stress heterogeneity. This configuration helps reduce experimental complexity and allows us to describe the frictional behavior with a simple theoretical model, as discussed later.

The time-dependent friction coefficient was calculated using the following formula:

$$\:\mu\:\left(t\right)=\frac{3M\left(t\right)}{2{r}_{0}{F}_{N}}\:,\:\:\:\:\:\:\:\:\left(1\right)$$

where $\:M$ is the frictional torque, $\:{r}_{0}$ is the radius of the hydrogel, and $\:{F}_{N}=\pi\:{r}_{0}^{2}P$ is the applied normal load. Note that the numerical coefficient ($\:=3/2$) in Eq. (1) is valid only for uniform normal and frictional stresses, which is not always the case for friction of hydrogels. Nevertheless, we applied Eq. (1) to estimate the friction coefficients in experiments and theoretical analyses.

For neutral (non-charged) hydrogels that are swollen and lubricated with pure water, the biphasic lubrication mechanisms are crucial for understanding friction. In this study, we developed a theoretical model based on the biphasic lubrication theory (or equivalently, the stress-diffusion coupling model or the diffusio-mechanical coupling model for gel dynamics [38–40]).

The model geometry is depicted in Fig. 2. The top surface of the cylindrical hydrogel is fixed onto a rigid disk, and the bottom surface is slid against a glass plate. The disk (and the hydrogel) is rotated with a constant normal force $\:{F}_{N}$ [17].

3.1 Governing equations

We begin by applying the theoretical description of the polymer network deformation and the interstitial fluid flow [38,39]. Darcy’s law describes the relationship between the relative solvent velocity and the pressure gradient in a porous polymer network as,

$$\:{\overrightarrow{v}}_{s}(\overrightarrow{r},t)-\frac{\partial\:\overrightarrow{u}(\overrightarrow{r},t)}{\partial\:t}=-\frac{1-\varphi\:}{\zeta\:}\nabla\:p(\overrightarrow{r},t)\:,\:\:\:\:\:\:\:\:\:\left(2\right)$$

where $\:{\overrightarrow{v}}_{s}(\overrightarrow{r},t)$ is the velocity vector for the solvent, $\:\overrightarrow{u}(\overrightarrow{r},t)$ is the displacement vector for the polymer, $\:\varphi\:$ is the volume fraction of the polymer, and $\:\zeta\:$ is the internal friction constant associated with the polymer relative to the solvent. Here we assume that $\:\varphi\:$ is uniform and does not change over time. Next, we consider the force balance for the polymer,

$$\:\nabla\:\bullet\:\stackrel{⃡}{\sigma\:}(\overrightarrow{r},t)=\nabla\:\text{p}(\overrightarrow{r},t)\:,\:\:\:\:\:\:\:\:\:\left(3\right)$$

where $\:\stackrel{⃡}{\sigma\:}(\overrightarrow{r},t)$ is the stress tensor acting on the polymer network. Additionally, the incompressibility of the hydrogel is expressed as:

$$\:\varphi\:\nabla\:\bullet\:\frac{\partial\:\overrightarrow{u}\left(\overrightarrow{r},t\right)}{\partial\:t}+\left(1-\varphi\:\right)\nabla\:\bullet\:{\overrightarrow{v}}_{s}\left(\overrightarrow{r},t\right)=0\:.\:\:\:\:\:\:\:\:\:\left(4\right)$$

The relationship between the stress and deformation of the polymer network is described by the following constitutive equation,

$$\:{\sigma\:}_{ij}=K\sum\:_{k=1}^{3}\frac{\partial\:{u}_{k}}{\partial\:{x}_{k}}{\delta\:}_{ij}+G\left(\frac{\partial\:{u}_{i}}{\partial\:{x}_{j}}+\frac{\partial\:{u}_{j}}{\partial\:{x}_{i}}-\frac{2}{3}\sum\:_{k=1}^{3}\frac{\partial\:{u}_{k}}{\partial\:{x}_{k}}{\delta\:}_{ij}\right)\:,\:\:\:\:\:\:\:\:\:\left(5\right)$$

where $\:K$ and $\:G$ are the bulk osmotic modulus and the shear modulus of the hydrogel respectively, and $\:{\delta\:}_{ij}$ is the Kronecker delta. Combining Eqs. (3) and (5) gives:

$$\:\left(K+\frac{G}{3}\right)\nabla\:\left(\nabla\:\bullet\:\overrightarrow{u}\right)+G{\nabla\:}^{2}\overrightarrow{u}=\nabla\:p\:.\:\:\:\:\:\:\:\:\:\left(6\right)$$

By combining Eqs. (2) and (4), we obtain the Poisson equation for the pressure.

$$\:\nabla\:\bullet\:\frac{\partial\:\overrightarrow{u}(\overrightarrow{r},t)}{\partial\:t}=\kappa\:{\nabla\:}^{2}\text{p}\:,\:\:\:\:\:\:\:\:\:\left(7\right)$$

where $\:\kappa\:={\left(1-\varphi\:\right)}^{2}/\zeta\:$ is the permeability of the fluid. Substituting Eq. (6) into Eq. (7), we obtain the diffusion equation,

$$\:\frac{\partial\:\omega\:(\overrightarrow{r},t)}{\partial\:t}={D}_{c}{\nabla\:}^{2}{\omega\:}\:,\:\:\:\:\:\:\:\:\:\left(8\right)$$

where $\:\omega\:=\nabla\:\bullet\:\overrightarrow{u}$ is the volumetric strain and $\:{D}_{c}=\kappa\:(K+4G/3)$ is the collective diffusion constant [41]. Eqs. (6), (7), and (8) are the basic equations to determine $\:\overrightarrow{u}(\overrightarrow{r},t)$ and $\:p(\overrightarrow{r},t)$.

3.2 Deformation of thick cylindrical hydrogels during sliding

First, we consider the friction of cylindrical hydrogels with a thickness not much smaller than the radius, as schematically shown in Fig. 2a. For simplicity, we assume that the deformation is symmetrical about the rotational axis. In addition, we assume that there is no pressure gradient or heterogeneous deformation in the vertical direction, meaning that the sample thickness $\:{h}_{0}$ does not appear in this theoretical analysis. In the cylindrical coordinate, this gives $\:\overrightarrow{u}(\overrightarrow{r},t)=\left({u}_{r},{u}_{\theta\:},{u}_{z}\right)=\left({u}_{r}(r,t),{u}_{\theta\:}(r,z,t),{u}_{z}(z,t)\right)$ and $\:p=p\left(r,t\right)$. By setting $\:{u}_{z}\left(z,t\right)={\epsilon\:}_{z}\left(t\right)z$ (uniform compressive strain), the volumetric strain $\:{\omega\:}$ is given by,

$$\:{\omega\:}\left(r,t\right)=\frac{1}{r}\frac{\partial\:}{\partial\:r}\left(r\:{u}_{r}(r,t)\right)+{\epsilon\:}_{z}\left(t\right)\:.\:\:\:\:\:\:\:\:\:\:\left(9\right)$$

Thus, the diffusion equation (Eq. (8)) can be expressed as follows:

$$\:\frac{1}{r}\frac{\partial\:}{\partial\:r}\left(r\frac{\partial\:{u}_{r}(r,t)}{\partial\:t}\right)+{\dot{\epsilon\:}}_{z}\left(t\right)={D}_{c}\frac{1}{r}\frac{\partial\:}{\partial\:r}\left[r\frac{\partial\:}{\partial\:r}\left\{\frac{1}{r}\frac{\partial\:}{\partial\:r}\left(r\:{u}_{r}(r,t)\right)\right\}\right]\:.\:\:\:\:\:\:\:\:\:\:\left(10\right)$$

The special solution $\:{u}_{r,s}$ for Eq. (10) can be easily found:

$$\:{u}_{r,\:\:s}\left(r,t\right)=\left\{-\frac{1}{2}{\epsilon\:}_{z}\left(t\right)+\alpha\:\right\}r\:.\:\:\:\:\:\:\:\:\:\:\left(11\right)$$

Next, we solve the homogeneous equation, which can be obtained by applying the variable separation method.

$$\:{u}_{r,h}\left(r,t\right)={\sum\:}_{i}\left\{{A}_{i}\:{J}_{1}\left(\sqrt{\frac{{\lambda\:}_{i}}{{D}_{c}}}r\right)+\:{B}_{i}\:{Y}_{1}\left(\sqrt{\frac{{\lambda\:}_{i}}{{D}_{c}}}r\right)\right\}\:\text{e}\text{x}\text{p}(-{\lambda\:}_{i}t)\:,\:\:\:\:\:\:\:\:\:\:\left(12\right)$$

where $\:{J}_{1}$ and $\:{Y}_{1}$ are the Bessel functions of the first and second kind, respectively. Considering the boundary condition $\:{u}_{r,h}\left(0,t\right)=0$ and taking only the slowest relaxation mode for simplicity, the general solution $\:{u}_{r}$ can be expressed by the following equation.

$$\:{u}_{r}\left(r,t\right)=A\:{J}_{1}\left(\sqrt{\frac{\lambda\:}{{D}_{c}}}r\right)\text{e}\text{x}\text{p}\left(-\lambda\:t\right)+\left\{-\frac{1}{2}{\epsilon\:}_{z}\left(t\right)+\alpha\:\right\}r\:,\:\:\:\:\:\:\:\:\:\:\left(13\right)$$

where $\:{\epsilon\:}_{z}\left(t\right)$ and three constants $\:A$, $\:\alpha\:$, and $\:\lambda\:$ are determined by the force balance along the z-direction; $\:{F}_{N}=2\pi\:{\int\:}_{0}^{{r}_{0}}r\:dr\:\left(p-{\sigma\:}_{zz}\right)$, the initial incompressibility condition; $\:\stackrel{-}{\omega\:}\left(0\right)=0$ (the definition will be given in Eq. (21)), and the two boundary conditions at the sample edge; $\:p\left({r}_{0},t\right)={\sigma\:}_{rr}\left({r}_{0},t\right)=0$.

Applying Eqs. (6), (13) with $\:p\left({r}_{0},t\right)=0$, we obtain:

$$\:p\left(r,t\right)=A\left(K+\frac{4}{3}G\right)\sqrt{\frac{\lambda\:}{{D}_{c}}}\:\left\{{J}_{0}\left(\sqrt{\frac{\lambda\:}{{D}_{c}}}r\right)-{J}_{0}\left(\sqrt{\frac{\lambda\:}{{D}_{c}}}{r}_{0}\right)\right\}\text{e}\text{x}\text{p}(-\lambda\:t)\:.\:\:\:\:\:\:\:\:\:\:\left(14\right)$$

From the constitutive law (Eq. (5)), the radial stress can be given as,

$$\:{\sigma\:}_{rr}\left(r,t\right)=\left(K+\frac{4}{3}G\right)\frac{\partial\:{u}_{r}}{\partial\:r}\left(r,t\right)+\left(K-\frac{2}{3}G\right)\left(\frac{{u}_{r}\left(r,t\right)}{r}+{\epsilon\:}_{z}\left(t\right)\right)\:.\:\:\:\:\:\:\:\:\:\:\left(15\right)$$

Applying $\:{\sigma\:}_{rr}\left({r}_{0},t\right)=0$ and substituting Eq. (13) into Eq. (15), we find:

$$\:{\epsilon\:}_{z}\left(t\right)=A\left\{\frac{K+\frac{4}{3}G}{G}\sqrt{\frac{\lambda\:}{{D}_{c}}}\:{J}_{0}\left(\sqrt{\frac{\lambda\:}{{D}_{c}}}{r}_{0}\right)-\:\frac{2}{{r}_{0}}{J}_{1}\left(\sqrt{\frac{\lambda\:}{{D}_{c}}}{r}_{0}\right)\right\}\text{e}\text{x}\text{p}(-\lambda\:t)+2\frac{K+\frac{1}{3}G}{G}\alpha\:\:.\:\:\:\:\:\:\:\:\:\left(16\right)$$

From Eq. (5), the vertical stress $\:{\sigma\:}_{zz}$ is obtained as,

$$\:{\sigma\:}_{zz}\left(r,t\right)=\left(K+\frac{4}{3}G\right){\epsilon\:}_{z}\left(t\right)+\left(K-\frac{2}{3}G\right)\left(\frac{{u}_{r}\left(r,t\right)}{r}+\frac{\partial\:{u}_{r}}{\partial\:r}\left(r,t\right)\right)\:.\:\:\:\:\:\:\:\:\:\:\left(17\right)$$

Using the vertical force balance and Eq. (17), we obtain,

$$\:\frac{{F}_{N}}{\pi\:{r}_{0}^{2}}=-6K\alpha\:+A\left\{8G\frac{{J}_{1}\left(\sqrt{\frac{\lambda\:}{{D}_{c}}}{r}_{0}\right)}{{r}_{0}}-3\left(K+\frac{4}{3}G\right)\sqrt{\frac{\lambda\:}{{D}_{c}}}\:{J}_{0}\left(\sqrt{\frac{\lambda\:}{{D}_{c}}}{r}_{0}\right)\right\}\text{e}\text{x}\text{p}(-\lambda\:t)\:.\:\:\:\:\:\:\:\:\:\:\left(18\right)$$

Since the term in the parenthesis “{}” in Eq. (18) must equal to zero, we have:

$$\:\frac{3}{8}\frac{K+\frac{4}{3}G}{G}=\frac{{J}_{1}\left({x}_{0}\right)}{{x}_{0}{J}_{0}\left({x}_{0}\right)}\:,\:\:\:\:\:\:\:\:\:\:\left(19\right)$$

where $\:{x}_{0}=\sqrt{\frac{\lambda\:}{{D}_{c}}}{r}_{0}$. This gives the relaxation time $\:{\tau\:}_{thick}=\frac{1}{\lambda\:}=\frac{{r}_{0}^{2}}{{x}_{0}^{2}\:{D}_{c}}$. Figure 3 shows $\:{x}_{0}$ as a function of $\:\frac{K}{G}$ (the solution $\:{x}_{1}$ for the stress relaxation test is also plotted, see Appendix). We also obtain,

$$\:\alpha\:=-\frac{{F}_{N}}{6\pi\:{r}_{0}^{2}K}\:.\:\:\:\:\:\:\:\:\:\:\left(20\right)$$

Finally, we determine the amplitude $\:A$ by considering the initial condition where the solvent cannot evacuate from the gel and that the volume is conserved. From Eq. (9), we have:

$$\:\stackrel{-}{{\omega\:}}\left(0\right)=\frac{1}{\pi\:{r}_{0}^{2}}{\int\:}_{0}^{{r}_{0}}dr2\pi\:\:r\:\omega\:(r,0)=2A\:\frac{{J}_{1}\left({x}_{0}\right)}{{r}_{0}}-\frac{{F}_{N}}{3\pi\:{r}_{0}^{2}K}=0\:,\:\:\:\:\:\:\:\:\:\:\left(21\right)$$

and we obtain:

$$\:A=\frac{{F}_{N}}{6\pi\:{r}_{0}K{J}_{1}\left({x}_{0}\right)}\:.\:\:\:\:\:\:\:\:\:\:\left(22\right)$$

With these calculations, we derive the following formulas.

$$\:{\epsilon\:}_{z}\left(t\right)=-\frac{{F}_{N}}{\pi\:{r}_{0}^{2}}\frac{K+\frac{G}{3}}{3G\:K}\left\{1-\frac{1}{3}\frac{G}{K+\frac{G}{3}}\:\text{e}\text{x}\text{p}(-\frac{t}{{\tau\:}_{thick}})\right\}\:,\:\:\:\:\:\:\:\:\:\:\left(23\right)$$

$$\:p\left(r,t\right)=\frac{{F}_{N}}{6\pi\:{r}_{0}}\frac{K+\frac{4}{3}G}{K}\sqrt{\frac{\lambda\:}{{D}_{c}}}\frac{{J}_{0}\left({x}_{0}\frac{r}{{r}_{0}}\right)-{J}_{0}\left({x}_{0}\right)}{{J}_{1}\left({x}_{0}\right)}\:\text{e}\text{x}\text{p}(-\frac{t}{{\tau\:}_{thick}})\:,\:\:\:\:\:\:\:\:\:\:\left(24\right)$$

$$\:{\sigma\:}_{zz}\left(r,t\right)=-\frac{{F}_{N}}{\pi\:{r}_{0}^{2}}\left[1-\left\{\frac{2}{9}\frac{G}{K}+\frac{1}{6}\frac{K-\frac{2}{3}G}{K}\:\frac{{x}_{0}\:{J}_{0}\left({x}_{0}\frac{r}{{r}_{0}}\right)}{{J}_{1}\left({x}_{0}\right)}\right\}\text{e}\text{x}\text{p}(-\frac{t}{{\tau\:}_{thick}})\right]\:\:.\:\:\:\:\:\:\:\:\:\:\left(25\right)$$

3.3 Friction coefficient for thick cylindrical hydrogels

Within the framework of the biphasic lubrication theory, the local frictional stress is given by the following equation [36]:

$$\:{\sigma\:}_{z\theta\:}\left(r,t\right)={\mu\:}_{\infty\:}\left\{\varphi\:p\left(r,t\right)-{\sigma\:}_{zz}\left(t\right)\right\}\:.\:\:\:\:\:\:\:\:\:\left(26\right)$$

where $\:{\mu\:}_{\infty\:}$ is the friction coefficient at $\:t\to\:\infty\:$, indicating that the hydrostatic pressure inside the hydrogel is completely relaxed and all the load is sorely supported by the elastic stress of the polymer network. In the rotational shear geometry, the frictional torque is calculated as

$$\:=\frac{2}{3}{\mu\:}_{\infty\:}{r}_{0}{F}_{N}\left[1-\left\{\frac{2}{9}\left(\frac{G}{K}\right)+\frac{2}{15}\left(\frac{K-\frac{2}{3}G}{K}\right)\left(\frac{20-3{x}_{0}^{2}}{8-{x}_{0}^{2}}\right)-\frac{4}{15}\varphi\:\left(\frac{K+\frac{4}{3}G}{K}\right)\left(\frac{{x}_{0}^{2}}{8-{x}_{0}^{2}}\right)\right\}\:\text{e}\text{x}\text{p}(-\frac{\:t\:}{\:{\tau\:}_{thick}\:})\right]\:,\:\:\:\:\:\:\:\:\:\:\left(27\right)$$

where we approximate $\:{J}_{0}\left(x\right)$ and $\:{J}_{1}\left(x\right)$ as $\:{J}_{0}\left(x\right)\cong\:1-\frac{{x}^{2}}{4}$ and $\:{J}_{1}\left(x\right)\cong\:\frac{x}{2}-\frac{{x}^{3}}{16}$. Finally, we obtain the solution for the friction coefficient, by applying Eqs. (1) and (27):

$$\:\mu\:\left(t\right)=\frac{3M\left(t\right)}{2{r}_{0}{F}_{N}}={\mu\:}_{\infty\:}\left[1-\left\{\frac{2}{9}\left(\frac{G}{K}\right)+\frac{2}{15}\left(\frac{K-\frac{2}{3}G}{K}\right)\left(\frac{20-3{x}_{0}^{2}}{8-{x}_{0}^{2}}\right)-\frac{4}{15}\varphi\:\left(\frac{K+\frac{4}{3}G}{K}\right)\left(\frac{{x}_{0}^{2}}{8-{x}_{0}^{2}}\right)\right\}\text{e}\text{x}\text{p}(-\frac{\:t\:}{\:{\tau\:}_{thick}\:})\right]\:.\:\:\:\:\:\:\:\:\left(28\right)$$

Eq. (28) can be simplified to:

$$\:\mu\:\left(t\right)={\mu\:}_{\infty\:}-\left({\mu\:}_{\infty\:}-{\mu\:}_{0}\right)\:\text{e}\text{x}\text{p}\left(-\frac{\:t\:}{\:{\tau\:}_{thick}\:}\right)\:.\:\:\:\:\:\:\:\:\left(29\right)$$

where $\:{\mu\:}_{0}$ is the friction coefficient at $\:t=0$, expressed as,

$$\:{\mu\:}_{0}=\:{\mu\:}_{\infty\:}\:\left\{1-\frac{2}{9}\left(\frac{G}{K}\right)-\frac{2}{15}\left(\frac{K-\frac{2}{3}G}{K}\right)\left(\frac{20-3{x}_{0}^{2}}{8-{x}_{0}^{2}}\right)+\frac{4}{15}\varphi\:\left(\frac{K+\frac{4}{3}G}{K}\right)\left(\frac{{x}_{0}^{2}}{8-{x}_{0}^{2}}\right)\right\}.\:\:\:\:\:\:\:\:\:\:\left(30\right)$$

Once the parameters $\:{r}_{0}$, $\:G$, $\:K$, $\:\kappa\:$, and $\:{\mu\:}_{\infty\:}$ are given, the time evolution of the friction coefficient can be predicted using Eq. (28). The unconfined compression test can evaluate the two elastic moduli and the permeability. In the Appendix, we present a framework to determine the parameters for thick hydrogels.

3.4 Thin film approximation

Next, we consider the thin film case (see Fig. 2b). In this theoretical analysis, besides the assumptions of rotational symmetry and negligible twisting deformation, we assume that the thickness $\:{h}_{0}$ is much smaller than the radius $\:{r}_{0}$ of the hydrogel. Since the displacements $\:{u}_{r}$ and $\:{u}_{z}$ are of the same order of the thickness of the hydrogel $\:{h}_{0}$, the terms $\:\partial\:{u}_{i}/\partial\:r$ ($\:i=r,z$) are much smaller than $\:\partial\:{u}_{i}/\partial\:z$ [40]. Under the thin film approximation, the analytical solutions can be obtained and thus the thickness dependences can be discussed. First, Eq. (6) can be reduced to the following equations:

$$\:\frac{\partial\:p}{\partial\:r}\cong\:G\frac{{\partial\:}^{2}{u}_{r}}{\partial\:{z}^{2}}\:,\:\:\:\:\:\:\:\:\:\left(31\right)$$

$$\:\frac{\partial\:p}{\partial\:z}\cong\:\left(K+\frac{4}{3}G\right)\frac{{\partial\:}^{2}{u}_{z}}{\partial\:{z}^{2}}=0\:.\:\:\:\:\:\:\:\:\left(32\right)$$

Here we also assume that the pressure variation along $\:z$ direction is negligibly small, and thus $\:p=p\left(r,t\right)$. With these approximations, the displacements are described as,

$$\:{u}_{r}\left(r,z,t\right)=\frac{1}{2G}\frac{\partial\:p(r,t)}{\partial\:r}\left({z}^{2}-{h}_{0}^{2}\right)\:,\:\:\:\:\:\:\:\:\:\left(33\right)$$

$$\:{u}_{z}\left(z,t\right)={\epsilon\:}_{z}\left(\text{t}\right)\:\text{z}\:,\:\:\:\:\:\:\:\:\:\left(34\right)$$

The Poisson equation (Eq. (7)) can be also approximated as:

$$\:\frac{1}{r}\frac{\partial\:}{\partial\:r}\left(r\frac{\partial\:{u}_{r}}{\partial\:t}\right)+\frac{\partial\:}{\partial\:z}\left(\frac{\partial\:{u}_{z}}{\partial\:t}\right)=\kappa\:\frac{1}{r}\frac{\partial\:}{\partial\:r}\left(r\frac{\partial\:p}{\partial\:r}\right)\:.\:\:\:\:\:\:\:\:\:\left(35\right)$$

Substituting Eqs. (33) and (34) into Eq. (35), we obtain the following equation.

$$\:\frac{{z}^{2}-{h}_{0}^{2}}{2G}\frac{1}{r}\frac{\partial\:}{\partial\:r}\left(r\frac{\partial\:}{\partial\:t}\left(\frac{\partial\:p}{\partial\:r}\right)\right)+{\dot{\epsilon\:}}_{z}=\kappa\:\frac{1}{r}\frac{\partial\:}{\partial\:r}\left(r\frac{\partial\:p}{\partial\:r}\right)\:.\:\:\:\:\:\:\:\:\:\left(36\right)$$

Since both $\:p(r,t)$ and $\:{\dot{\epsilon\:}}_{z}\left(t\right)$ are independent of $\:z$, we need to choose a specific value for $\:z$ to proceed with our calculations. Here, we set $\:z=0$ to represent the behavior at the frictional interface. Then we obtain:

$$\:-\frac{{h}_{0}^{2}}{2G}\frac{1}{r}\frac{\partial\:}{\partial\:r}\left(r\frac{\partial\:}{\partial\:t}\left(\frac{\partial\:p}{\partial\:r}\right)\right)+{\dot{\epsilon\:}}_{z}=\kappa\:\frac{1}{r}\frac{\partial\:}{\partial\:r}\left(r\frac{\partial\:p}{\partial\:r}\right)\:.\:\:\:\:\:\:\:\:\:\left(37\right)$$

Eq. (37) can be solved by applying the boundary condition $\:p\left({r}_{0},t\right)=0$.

$$\:p\left(r,t\right)=p\left(r,0\right)\text{exp}\left(-\frac{2G\kappa\:}{{h}_{0}^{2}}t\right)+\frac{G}{2}\frac{{r}^{2}-{r}_{0}^{2}}{{h}_{0}^{2}}exp\left(-\frac{2G\kappa\:}{{h}_{0}^{2}}t\right)\:{\int\:}_{0}^{t}ds{\dot{\epsilon\:}}_{z}\left(s\right)\text{e}\text{x}\text{p}\left(\frac{2G\kappa\:}{{h}_{0}^{2}}s\right).\:\:\:\:\:\:\:\:\:\left(38\right)$$

$\:p\left(r,0\right)$ can determined using the incompressibility condition ($\:\nabla\:\bullet\:\overrightarrow{u}=0$) at $\:t=0$ and $\:z=0$:

$$\:-\frac{{h}_{0}^{2}}{2G}\frac{1}{r}\frac{\partial\:}{\partial\:r}\left(r\frac{\partial\:p}{\partial\:r}(r,0)\right)+{\epsilon\:}_{z}\left(0\right)=0\:.\:\:\:\:\:\:\:\:\:\left(39\right)$$

With $\:p\left({r}_{0},0\right)=0$, Eq. (39) becomes:

$$\:p\left(r,0\right)=\frac{G}{2}\frac{{r}^{2}-{r}_{0}^{2}}{{h}_{0}^{2}}{\epsilon\:}_{z}\left(0\right)\:.\:\:\:\:\:\:\:\:\:\left(40\right)$$

Considering the vertical force balance and substituting Eqs. (38), (40), and $\:{\sigma\:}_{zz}\left(t\right)=\left(K+\frac{4}{3}G\right){\epsilon\:}_{z}\left(t\right)$, we obtain:

$$\:{F}_{N}=-\:{\pi\:}{r}_{0}^{2}\left\{K+\frac{4}{3}G+\frac{G}{4}{\left(\frac{{r}_{0}}{{h}_{0}}\right)}^{2}\right\}{\epsilon\:}_{z}\left(t\right)+\frac{\pi\:}{2}{G}^{2}\kappa\:{\left(\frac{{r}_{0}}{{h}_{0}}\right)}^{4}exp\left(-\frac{2G\kappa\:}{{h}_{0}^{2}}t\right){\int\:}_{0}^{t}ds{\epsilon\:}_{z}\left(s\right)\text{e}\text{x}\text{p}\left(\frac{2G\kappa\:}{{h}_{0}^{2}}s\right)\:.\:\:\:\:\:\:\:\:\:\left(41\right)$$

To solve this integral equation, we set $\:f\left(t\right)={\epsilon\:}_{z}\left(t\right)exp\left(\frac{2G\kappa\:}{{h}_{0}^{2}}t\right)$ and substitute it into Eq. (41). We obtain:

$$\:{\epsilon\:}_{z}\left(t\right)=-\frac{{F}_{N}}{\pi\:{r}_{0}^{2}\left(K+\frac{4}{3}G\right)}\:\left\{1-\frac{\frac{G}{4}{\left(\frac{{r}_{0}}{{h}_{0}}\right)}^{2}}{K+\frac{4}{3}G+\frac{G}{4}{\left(\frac{{r}_{0}}{{h}_{0}}\right)}^{2}}exp\left(-\frac{t}{{\tau\:}_{thin}}\:\right)\right\},\:\:\:\:\:\:\:\:\:\left(42\right)$$

where $\:{\tau\:}_{thin}={h}_{0}^{2}\:\left\{K+\frac{4}{3}G+\frac{G}{4}{\left(\frac{{r}_{0}}{{h}_{0}}\right)}^{2}\right\}/\left\{2G\kappa\:(K+\frac{4}{3}G)\right\}$ is the relaxation time for thin samples. Substituting Eqs. (40) and (42) into Eq. (38), the pressure and the vertical stress can also be obtained.

$$\:p\left(r,t\right)=\frac{{F}_{N}}{2\pi\:{r}_{0}^{2}}\frac{{{r}_{0}^{2}-r}^{2}}{{h}_{0}^{2}}\frac{G}{K+\frac{4}{3}G+\frac{G}{4}{\left(\frac{{r}_{0}}{{h}_{0}}\right)}^{2}}exp\left(-\frac{t}{{\tau\:}_{thin}}\right),\:\:\:\:\:\:\:\:\:\left(43\right)$$

$$\:{\sigma\:}_{zz}\left(t\right)=-\frac{{F}_{N}}{\pi\:{r}_{0}^{2}}\:\left\{1-\frac{\frac{G}{4}{\left(\frac{{r}_{0}}{{h}_{0}}\right)}^{2}}{K+\frac{4}{3}G+\frac{G}{4}{\left(\frac{{r}_{0}}{{h}_{0}}\right)}^{2}}exp\left(-\frac{t}{{\tau\:}_{thin}}\:\right)\right\}.\:\:\:\:\:\:\:\:\:\left(44\right)$$

3.5 Calculation of friction coefficient for thin hydrogels

Similarly to the thick film case, the frictional torque is calculated as:

$$\:M\left(t\right)={\int\:}_{0}^{{r}_{0}}2\pi\:rdr{\sigma\:}_{z\theta\:}\left(r,t\right)r=2\pi\:{\mu\:}_{\infty\:}{\int\:}_{0}^{{r}_{0}}dr{r}^{2}\left\{\varphi\:p\left(r,t\right)-{\sigma\:}_{zz}\left(t\right)\right\}=\frac{2}{3}{\mu\:}_{\infty\:}{r}_{0}{F}_{N}\left\{1-\frac{5-4\varphi\:}{20}\frac{\:G{\left(\frac{{r}_{0}}{{h}_{0}}\right)}^{2}}{K+\frac{4}{3}G+\frac{G}{4}{\left(\frac{{r}_{0}}{{h}_{0}}\right)}^{2}}exp\left(-\frac{t}{{\tau\:}_{thin}}\:\right)\right\}.\:\:\:\:\:\:\:\:\:\left(45\right)$$

Finally, we obtain the solution for the friction coefficient by applying Eq. (1):

$$\:\mu\:\left(t\right)=\frac{3M\left(t\right)}{2{r}_{0}{F}_{N}}={\mu\:}_{\infty\:}\left\{1-\frac{5-4\varphi\:}{20}\frac{\:G{\left(\frac{{r}_{0}}{{h}_{0}}\right)}^{2}}{K+\frac{4}{3}G+\frac{G}{4}{\left(\frac{{r}_{0}}{{h}_{0}}\right)}^{2}}exp\left(-\frac{t}{{\tau\:}_{thin}}\:\right)\right\}\:.\:\:\:\:\:\:\:\:\left(46\right)$$

Eq. (46) can also be expressed in a simple form (Eq. (29)), with the initial friction coefficient:

$$\:{\mu\:}_{0}=\:{\mu\:}_{\infty\:}\frac{K+\frac{4}{3}G+\frac{G\varphi\:}{5}{\left(\frac{{r}_{0}}{{h}_{0}}\right)}^{2}}{K+\frac{4}{3}G+\frac{G}{4}{\left(\frac{{r}_{0}}{{h}_{0}}\right)}^{2}}.\:\:\:\:\:\:\:\:\left(47\right)$$

Eq. (47) can be approximated up to the second order in $\:{h}_{0}/{r}_{0}\:(\ll\:1)$:

$$\:{\mu\:}_{0}\cong\:\:\frac{4}{5}{\mu\:}_{\infty\:}\left\{\varphi\:+\left(5-4\varphi\:\right)\frac{K+\frac{4}{3}G}{G}{\left(\frac{{h}_{0}}{{r}_{0}}\right)}^{2}\right\}.\:\:\:\:\:\:\:\:\left(48\right)$$

This result indicates that the initial friction coefficient is determined by the dry friction of the polymer network, its volume fraction, and the thickness of the gel. Particularly, in the thin film limit, the initial friction is given by $\:{\mu\:}_{0}=\:\frac{4}{5}{\mu\:}_{\infty\:}\varphi\:$.

4.1 Effects of sample radius

As a typical example of our experimental results, Fig. 4a shows time evolutions of the friction coefficient for the hydrogel samples with thickness $\:{h}_{0}=$ 2 mm and different radii ($\:{r}_{0}=$ 2 and 4 mm).

It is seen that the friction coefficients for both samples follow the same trends: they start from an initial value

($\:\approx\:0.08$), increase over time, and then reach a steady value ($\:\approx\:0.12$) while the time constants are different between the samples. Then we fit the experimental curves using the theoretical formula (Eq. (29)): $\:\mu\:\left(t\right)={\mu\:}_{\infty\:}-\left({\mu\:}_{\infty\:}-{\mu\:}_{0}\right)\text{e}\text{x}\text{p}\left(-t/\tau\:\right)$. As a result, both curves fit well. For $\:{r}_{0}=$ 2 and 4 mm, the estimated values are $\:\left({\mu\:}_{0},\:{\mu\:}_{\infty\:},\:\tau\:\right)=$ (0.077, 0.13, 833 s) and (0.088, 0.12, 2292 s), respectively. The frictional behavior observed here is different from that in polyacrylamide hydrogels [39], particularly in the initial phase. This might be due to the differences in the preparation processes and adhesion properties of hydrogel samples. On the other hand, the mechanism for the difference in relaxation time can be intuitively understood as follows: when the friction experiment begins, the normal load is applied, the hydrogel generates the hydrostatic pressure inside and the pressure supports a part of the normal load. Due to the pressure gradient between the center and the edge, water flows from the inside to the outside of the sample through the polymer network, causing a gradual decrease in hydrostatic pressure. In such a situation, a sample with a larger radius takes more time squeezing water from the inside to the outside, resulting in a longer relaxation time.

Such behavior can also be explained by the theoretical model for thick gels (Eq. (28)). We choose $\:K=$ 37 kPa and $\:G=$ 55 kPa, $\:\kappa\:=2\times\:{10}^{-13}{m}^{4}/Ns$, $\:\varphi\:=$ 0.2 from the previous unconfined compression tests using the samples prepared in the same manner [30], giving $\:{x}_{0}=1.55$ (see Fig. 3). In addition, we estimated $\:{\mu\:}_{\infty\:}=$ 0.12 from the present experiment. Figure 4b shows theoretical curves for different sample radii. As is seen, the friction coefficient starts at about 0.08 and converges to 0.12. Furthermore, the relaxation time becomes larger for the samples with a larger radius. Therefore, these features are in reasonable agreement with our experiments. However, the relaxation time shows a large deviation between the experiment and the theory. To compare the results quantitatively, we plotted the radius dependence in Fig. 4c. The relaxation time predicted by the theory is about 10 times smaller than that measured in experiments. This suggests that the permeability of the samples for friction experiments is about 10 times smaller ($\:\kappa\:\approx\:2\times\:{10}^{-14}{m}^{4}/Ns$) than that obtained by the unconfined compression test. It might be because some of the pores in a hydrogel are filled with the cyanoacrylate adhesive used to fix the sample. Further study is required to evaluate permeability and relaxation time for such samples.

In contrast, as shown in Fig. 4d, the sample radius correlated poorly with the initial friction coefficient. According to our theoretical formula (Eq. (30)), $\:{\mu\:}_{0}$ is independent of the sample radius, and the comparison in Fig. 4d shows reasonable agreement with experiments. It is important to note that fitting with thin film approximation (Eq. (47)) did not work well: since the sample thickness is explicitly included in the formula and gives a non-negligible contribution to the initial friction coefficient.

4.2 Effects of sample thickness

Next, we investigated the effects of sample thickness. Time evolutions of the friction coefficient for the samples with the same radius ($\:{r}_{0}=$ 4 mm) but different thicknesses ($\:{h}_{0}=$ 5, 2, 0.4 mm) are shown in Fig. 5a. In contrast to the frictional behavior in Fig. 4 for the samples having different radii, the friction coefficient significantly decreases with decreasing gel thickness. We fitted the curves to characterize the behavior using Eq. (29) and obtained $\:\left({\mu\:}_{0},\:{\mu\:}_{\infty\:},\:\tau\:\right)=$ (0.087, 0.11, 793 s), (0.088, 0.12, 2292 s), and (0.0074, 0.094, 72598 s) for $\:{h}_{0}=$ 5, 2, and 0.4 mm respectively.

Surprisingly, the significant difference in friction coefficients is obtained from the same material property, in other words, the surface property, because all samples were prepared in the same manner, as described in the experimental section. The mechanisms for the decrease in friction coefficient can be explained by the confinement effect [41,43,44]: since the upper surface of the thin plate hydrogel is firmly adhered to the rigid disk, the hydrogel cannot freely expand in the radial direction under compressive loading. This induces excess hydrostatic pressure, enhancing fluid load support and reducing friction. This mechanism is greatly influenced by sample geometry, that is, the aspect ratio of the gel sample: thinner hydrogel samples generate larger hydrostatic pressure to support the normal load, resulting in a smaller friction coefficient.

To evaluate the effect of sample thickness, frictional evolutions within the thin film approximation are plotted in Fig. 5b. The same values for the elastic moduli and a 10 times smaller permeability than that of a thick gel are used. It is seen that the theoretical predictions capture observed features in experiments: the friction coefficient increases initially with time and reaches a steady value, and the initial friction coefficient becomes much smaller for the thinnest hydrogel sample ($\:{h}_{0}=$ 0.4 mm). On the other hand, there is a large difference in the relaxation behavior for the thinnest sample. As discussed previously, the sample’s pore is considered to be filled partially with an adhesive agent (used for fixing the sample onto the plate), leading to a decrease in the permeability $\:\kappa\:$ and slowing down of the water transport. This tendency is more striking for thinner samples. To test this, we plotted the frictional curve by applying a smaller permeability $\:\kappa\:/800$ for the thinnest sample ($\:{h}_{0}=$ 0.4 mm), which gives a better fitting with our experimental result. It would be interesting to see if the same behavior is observed when the gel is fixed with a different adhesive agent or by changing the fixing method to avoid the change in permeability. It would also be worthwhile to test if low friction is maintained using thin hydrogels with smaller permeability. Finally, to check the thickness effects, we plotted the relationship between the initial friction coefficient predicted from theory (Eq. (47)) and that measured in experiments in Fig. 5c. An abrupt change in $\:{\mu\:}_{0}$ observed at $\:{h}_{0}\sim\:1\:mm$ is well captured by the theoretical curve, indicating the crossover from the thin film limit ($\:{\mu\:}_{0}=\:\frac{4}{5}{\mu\:}_{\infty\:}\varphi\:$) to the thick film limit ($\:{\mu\:}_{0}=\:{\mu\:}_{\infty\:}$).

To examine the sliding frictional behavior of hydrogels, we developed a theoretical model within the framework of biphasic lubrication theory. The model predicts a significant dependence of the friction coefficients on sample geometry, where the sliding friction decreases drastically with reduced sample thickness. To validate our theory, we conducted friction experiments between a cylindrical PVA hydrogel and a glass plate. We found that friction coefficient changes not only with gel radius but also with gel thickness. These results suggest that the surface friction of hydrogels is strongly affected by bulk transport, i.e., water flow inside the polymer network. It would be interesting to study further reductions in friction by combining and optimizing different strategies, for example by introducing polyelectrolyte surfaces and thin plate geometry. More systematic experiments are needed to check the accuracy of the model. This will be reported elsewhere shortly.

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Open Access

Data will be made available on request.

Author Contribution

T. Y. designed the study, conducted experiments, data analysis, and theoretical analysis, and wrote the manuscript. Y. S. did project administration, funding acquisition, and supervised the study.

Acknowledgement

T. Y. acknowledges N. Sakai for fruitful discussions and C. Chang for reading the manuscript.

Data Availability

Data will be made available on request.

Buwalda, S.J., Boere, K.W.M., Dijkstra, P.J., Feijen, J., Vermonden, T., Hennink, W.E.: Hydrogels in a historical perspective: From simple networks to smart materials. JCR. 190, 254–273 (2014)
DeRossi, J.D., Kajiwara, K., Osada, Y., Yamauchi, A.: Polymer gels -Fundamentals and Biomedical Applications. Plenum, New York (1991)
Osada, Y., Gong, J.P.: Soft and wet materials: polymer gels. Adv. Mater. 10(11), 827–837 (1998)
Caló, E., Khutoryanskiy, V.V.: Biomedical applications of hydrogels: A review of patents and commercial products. Eur. Polym. J. 65, 252–267 (2015)
Freeman, M.E., Furey, M.J., Love, B.J., Hampton, J.M.: Friction, wear, and lubrication of hydrogels as synthetic articular cartilage. Wear. 241, 129–135 (2000)
Beddoes, C.M., Whitehouse, M.R., Briscoe, W.H., Su, B.: Hydrogels as a Replacement Material for Damaged Articular Hyaline Cartilage. Materials. 9, 443 (2016)
Blum, M.M., Ovaert, T.C.: Low friction hydrogel for articular cartilage repair: evaluation of mechanical and tribological properties in comparison with natural cartilage tissue. Mater. Sci. Eng. C. 33(7), 4377–4383 (2013)
Milner, P.E., Parkes, M., Puetzer, J.L., Chapman, R., Stevens, M.M., Cann, P., Jeffers, J.R.: A low friction, biphasic, and boundary lubricating hydrogel for cartilage replacement. Acta Biomater. 65, 102–111 (2018)
Yamaguchi, T.: Reduction and Control of Friction in Hydrogels. Surfactants in Tribology 6, CRC Press (2019)
Gong, J.P.: Friction and lubrication of hydrogels - its richness and complexity. Soft Matter. 2, 544–552 (2006)
Dunn, A.C., Uruena, J.M., Huo, Y., Perry, S.S., Angelini, T.E., Sawyer, W.G.: Lubricity of surface hydrogel layers. Tribol Lett. 49, 371–378 (2013)
Takata, M., Yamaguchi, T., Gong, J.P., Doi, M.: Electric Field Effect on the Sliding Friction of a Charged Gel. J. Phys. Soc. Jpn. 78, 084602 (2009)
Takata, M., Yamaguchi, T., Doi, M.: Friction control of a gel by electric field in ionic surfactant solution. J. Phys. Soc. Jpn. 79, 063602 (2010)
Itagaki, N., Kawaguchi, D., Oda, Y., Nemoto, F., Yamada, N.L., Yamaguchi, T., Tanaka, K.: Surface Effect on Frictional Properties for Thin Hydrogel Films of Poly (vinyl ether). Macromolecules. 52, 9632–9638 (2019)
Gong, J.P., Kurokawa, T., Narita, T., Kagata, G., Osada, Y., Nishimura, G., Kinjo, M.: Synthesis of Hydrogels with Extremely Low Surface Friction. J. Am. Chem. Soc. 123, 5582–5583 (2001)
Kaneko, D., Tada, T., Kurokawa, T., Gong, J.P., Osada, Y.: Mechanically Strong Hydrogels with Ultra-Low Frictional Coefficients. Adv. Mater. 17, 535–538 (2005)
Suzuki, R., Yamaguchi, T., Doi, M.: Frictional Property of Hydrogels Prepared under Electric Fields. J. Phys. Soc. Jpn. 82, 124803 (2013)
Chen, Q., Zhang, X., Chen, K., Feng, C., Wang, D., Qi, J., Li, X., Zhao, X., Chai, Z., Zhang, D.: Bilayer hydrogels with low friction and high load-bearing capacity by mimicking the oriented hierarchical structure of cartilage. ACS Appl. Mater. Interfaces. 14(46), 52347–52358 (2022)
Luo, C., Guo, A., Zhao, Y., Sun, X.: A high strength, low friction, and biocompatible hydrogel from PVA, chitosan and sodium alginate for articular cartilage. Carbohydr. Polym. 286, 119268 (2022)
Ishikawa, Y., Hiratsuka, K.I., Sasada, T.: Role of water in the lubrication of hydrogel. Wear. 261(5–6), 500–504 (2006)
Urueña, J.M., McGhee, E.O., Angelini, T.E., Dowson, D., Sawyer, W.G., Pitenis, A.A.: Normal load scaling of friction in gemini hydrogels. Biotribology. 13, 30–35 (2018)
Gombert, Y., Simič, R., Roncoroni, F., Dübner, M., Geue, T., Spencer, N.D.: Structuring hydrogel surfaces for tribology. Adv. Mater. Interfaces. 6(22), 1901320 (2019)
Meier, Y.A., Zhang, K., Spencer, N.D., Simic, R.: Linking friction and surface properties of hydrogels molded against materials of different surface energies. Langmuir. 35(48), 15805–15812 (2019)
Ohsedo, Y., Takashina, R., Gong, J.P., Osada, Y.: Surface friction of hydrogels with well-defined polyelectrolyte brushes. Langmuir. 20(16), 6549–6555 (2004)
Yarimitsu, S., Nakashima, K., Sawae, Y., Murakami, T.: Influences of lubricant composition on forming boundary film composed of synovia constituents. Tribol Int. 42, 1615–1623 (2009)
Murakami, T., Yarimitsu, S., Sakai, N., Nakashima, K., Yamaguchi, T., Sawae, Y.: Importance of adaptive multimode lubrication mechanism in natural synovial joints. Tribol Int. 113, 306–315 (2017)
Blum, M.M., Ovaert, T.C.: Experimental and numerical tribological studies of a boundary lubricant functionalized poro-viscoelastic PVA hydrogel in normal contact and sliding. J. Mech. Behav. Biomed. Mater. 14, 248–258 (2012)
Delavoipiere, J., Tran, Y., Verneuil, E., Heurtefeu, B., Hui, C.Y., Chateauminois, A.: Friction of Poroelastic Contacts with Thin Hydrogel Films. Langmuir. 34, 9617–9626 (2018)
Sakai, N., Hagihara, Y., Furusawa, T., Hosoda, N., Sawae, Y., Murakami, T.: Analysis of biphasic lubrication of articular cartilage loaded by cylindrical indenter. Tribol Int. 46, 225–236 (2012)
Murakami, T., Sakai, N., Yamaguchi, T., Yarimitsu, S., Nakashima, K., Sawae, Y., Suzuki, A.: Evaluation of a superior lubrication mechanism with biphasic hydrogels for artificial cartilage. Tribol Int. 89, 19–26 (2015)
Murakami, T., Sakai, N., Yarimitsu, S., Nakashima, K., Yamaguchi, T., Sawae, Y., Suzuki, A.: Evaluation of influence of changes in permeability with aging on friction and biphasic behaviors of artificial hydrogel cartilage. Biotribology. 26, 100178 (2021)
Reale, E.R., Dunn, A.C.: Poroelasticity-driven lubrication in hydrogel interfaces. Soft Matter. 13(2), 428–435 (2017)
Chen, K., Zhang, D., Yang, X., Cui, X., Zhang, X., Wang, Q.: Research on torsional friction behavior and fluid load support of PVA/HA composite hydrogel. J. Mech. Behav. Biomed. Mater. 62, 182–194 (2016)
Mow, V.C., Ratcliffe, A., Woo, S.L.-Y. (eds.): Biomechanics of Diarthrodial Joints 1. Springer- (1990)
Ateshian, G.A., Lai, W.M., Zhu, W.B., Mow, V.C.: An asymptotic solution for the contact of two biphasic cartilage layers. J. Biomech. 27(11), 1347–1360 (1994)
Ateshian, G.A.: The role of interstitial fluid pressurization in articular cartilage lubrication. J. Biomech. 42, 1163–1176 (2009)
Accardi, M.A., Dini, D., Cann, P.M.: Experimental and numerical investigation of the behaviour of articular cartilage under shear loading—interstitial fluid pressurisation and lubrication mechanisms. Tribol Int. 44(5), 565–578 (2011)
Moore, A.C., Burris, D.L.: An analytical model to predict interstitial lubrication of cartilage in migrating contact areas. J. Biomech. 47(1), 148–153 (2014)
Qiu, X., Yan, Y., Zhang, G., Huang, J., Zhao, Y., Xia, X., Cui, X., Zhang, X.: Investigation of the time-dependent friction behavior of polyacrylamide hydrogels. Coll. Surf. A. 659, 130753 (2023)
Yamaue, T., Doi, M.: Swelling Dynamics of Constrained Thin-Plate Gels under an External Force. Phys. Rev. E. 70, 011401 (2004)
Yamaue, T.: The stress diffusion coupling in the swelling dynamics of cylindrical gels. J. Chem. Phys. 122, 084703 (2005)
Doi, M.: Gel Dynamics. J. Phys. Soc. Jpn. 78, 052001 (2009)
Doi, M., Yamaguchi, T.: Analytical solution for the deformation of pressure sensitive adhesives confined between two rigid plates. J. Non-Newtonian Fluid Mech. 145, 52–56 (2007)
Yamaguchi, T., Muroo, H., Sumino, Y., Doi, M.: Asymmetry-symmetry transition of double-sided adhesive tapes. Phys. Rev. E. 85, 061802 (2012)

No competing interests reported.

GA.png
Graphical abstract

Download PDF

Reviewers agreed at journal
08 Nov, 2024
Reviews received at journal
04 Nov, 2024
Reviewers agreed at journal
24 Oct, 2024
Reviewers invited by journal
23 Oct, 2024
Editor assigned by journal
21 Oct, 2024
Submission checks completed at journal
21 Oct, 2024
First submitted to journal
21 Oct, 2024

You are reading this latest preprint version

Biphasic Lubrication Theory of Hydrogels: Transient Response and Sample Size Effects

Status:

Version 1

Abstract

Figures

1 Introduction

2 Experiment

2.1 Sample

2.2 Rotational shear test using cylindrical hydrogels

3 Theory

3.1 Governing equations

3.2 Deformation of thick cylindrical hydrogels during sliding

3.3 Friction coefficient for thick cylindrical hydrogels

3.4 Thin film approximation

3.5 Calculation of friction coefficient for thin hydrogels

4 Results and discussion

4.1 Effects of sample radius

4.2 Effects of sample thickness

5 Conclusion

Declarations

Open Access

Author Contribution

Acknowledgement

Data Availability

References

Additional Declarations

Supplementary Files

Status:

Version 1