Mesoscale Modeling of Hydrogels Under Frictional Shear Stress

Abstract

Hydrogels are three-dimensional networks of hydrophilic polymers often used as a simplified model of hydrated biological materials, from cartilaginous joints to the ocular tear film. However, the lubrication mechanisms of hydrogels remain poorly understood, partly due to their complex polymeric structure, which creates blurred interfaces during sliding that are challenging to study experimentally. In this study, we employ dissipative particle dynamics (DPD) to investigate the frictional behavior of a polymeric hydrogel network sliding against a solid wall in an explicit viscous solvent. This computational approach enables us to model hydrodynamic interactions and mesoscale polymer dynamics, capturing key aspects of hydrogel friction. Our simulations reveal that hydrogel friction is governed by the interplay between polymer relaxation and viscous shear, characterized by the Weissenberg number (Wi). At low Wi, friction coefficient remain nearly constant, dominated by polymer relaxation. However, at higher Wi, friction is dominated by viscous drag within a near-wall solvent layer, leading to a linear increase in friction coefficient with Wi. Furthermore, our results demonstrate an inverse relationship between the friction coefficient and the applied normal load, consistent with experimental observations. This work provides new insights into the fundamental tribological properties of hydrogels, shedding light on the micromechanics of hydrogel friction. Improving our understanding of hydrogel structure and dynamics under friction advances our knowledge of the mechanisms regulating biological lubrication in health and disease.

Introduction

Hydrogels are three-dimensional networks of hydrophilic polymers that swell in water or aqueous solvents. − Biological hydrogel networks are responsible for providing exceptionally low friction across natural sliding interfaces such as in articular cartilage and ocular surface. In some cases, friction coefficients as low as 10^–4 have been reported. , The unique tribological properties make hydrogels promising materials for biomedical and engineering applications where minimal interfacial resistance is essential. − In artificial cartilages, ultralow friction is a critical parameter for reducing wear and maintaining smooth joint motion under load. − In soft contact lenses, enabling low friction across the cornea–lens interface is important for maintaining comfort and ocular health. In soft robotics, the use of low-friction hydrogels minimizes abrasion while conforming to delicate surfaces. Advancing our understanding of hydrogel friction is therefore highly important for both fundamental science and practical engineering applications. ,

Several models have been proposed to explain the mechanisms underlying hydrogel lubrication. Hydration lubrication and hydrodynamic lubrication are commonly accepted as dominant contributors to low-friction in hydrogels. The hydration lubrication model involves water molecules associated with hydrophilic polymer chains forming a lubricating layer that reduces friction between surfaces, resulting in fluid-like responses to shear forces. − Studies have demonstrated that this hydration layer plays a crucial role in minimizing frictional forces by acting as a boundary lubricant, preventing direct contact between sliding surfaces. , In contrast, the hydrodynamic lubrication model suggests that at low sliding velocities, friction is primarily controlled by fluid flow through the hydrogel mesh and a mesoscopic liquid film develops between the hydrogel and the solid surface, facilitating lubrication. ,

Experimental studies highlight the role of shear-induced polymer dynamics in controlling hydrogel friction. In experiments with like-charged hydrogels, Oogaki et al. reported three distinct lubrication regimes, including boundary, hydrated, and elastohydrodynamic regimes, pointing to the existence of multiple frictional mechanisms depending on sliding conditions. Cuccia et al. demonstrated that hydrogel friction on smooth surfaces exhibits a complex, velocity-dependent behavior comprising multiple regimes, with a notable linear increase in friction coefficient at low velocities attributable to Darcy-like flow through the polymer network and interfacial chain interactions. This regime transitions to a lubricated state at higher velocities, influenced by network mesh size and structural relaxation. Pitenis and Sawyer examined the lubricity of self-mated hydrogels, including polyacrylamide (PAAm), polyethylene glycol (PEG), and poly­(N-isopropylacrylamide) (PNIPAm), and reported a transition from velocity-independent to velocity-dependent friction. This crossover was attributed to a shift from thermally dominated behavior at low sliding velocities to shear-driven deformation and dissipation at higher velocities. Their findings pinpoint the critical role of the Weissenberg number Wi in governing the onset of nonlinear frictional behavior in hydrogels. Pitenis et al. reported that the balance between elastic and viscous responses under shear affects energy dissipation and, consequently, friction forces.

Rennie et al. investigated the frictional response of hydrogel-based contact lenses and identified three contributing mechanisms to the total friction force: viscoelastic deformation of the hydrogel network, interfacial shear stress due to polymer–surface interactions, and viscous shearing of a thin fluid layer. Their model predicted that the friction coefficient should decrease with load, with an approximate scaling of μ ∼ P ^–0.4 to μ ∼ P ^–0.5, depending on the dominant dissipation mode. Their work provided early experimental evidence for load-dependent friction scaling in soft, hydrated materials. Similarly, Urueña et al. investigated friction in Gemini hydrogel interfaces and reported a scaling of μ ∼ P ^–1/3 in the velocity-independent regime. They attributed this behavior to a combination of Hertzian contact mechanics, where real contact area increases with load, and constant interfacial shear stress. In their interpretation, friction arises primarily from shearing within a thin, solvent-rich interfacial layer rather than from the bulk gel, and the sublinear dependence reflects the mismatch between the growth of contact area and the frictional force. Shoaib et al. and Shoaib and Espinosa-Marzal used lateral force microscopy to investigate the frictional response of hydrogels under varying loads and sliding speeds including static friction. They identified friction regimes dominated by polymer adsorption–desorption, poroelastic relaxation, and network deformation, and reported a sublinear increase in friction force with load. Blum and Ovaert combined tribological testing with poro-viscoelastic finite element modeling to investigate the frictional behavior of PVA hydrogels functionalized with a DOPA-based boundary lubricant. They demonstrated that surface functionalization significantly reduces friction, and reported a decrease in friction with increasing normal load.

Experimental studies are limited in resolving the nanoscale interactions between the hydrogel network and solvent underpinning the fundamental mechanisms of frictional energy dissipation. Researchers have employed computational modeling to gain insight into hydrogel friction at molecular and mesoscopic scales. Müser et al. developed a computational model to simulate the frictional response of hydrogels with different stiffness, surface roughness, and confinement, reveling distinct lubrication regimes influenced by gel mechanics. Wu et al. used molecular dynamics to examine how water content affects the tribological properties of biologically relevant hyaluronic acid hydrogels. Their results show that increasing gel hydration reduces the friction by enhancing molecular mobility and interfacial lubrication. Mees et al. employed a mesoscale method with implicit solvent to reveal that friction at hydrogel–hydrogel interfaces arises from the entropic stress generated by the reorientation and stretching of surface polymer chains. In particular, they showed that chain reorientation dominates friction at intermediate Weissenberg numbers, yielding a velocity-independent regime for low Wi followed by a linearly velocity-dependent regime for the friction coefficient. Similar chain reorientation with Wi was found in grafted polymer chains in a shear flow, where chain align with the direction of shear and stretch, resulting in a linear increase of entropic shear stress. Zhu et al. used experiments, theoretical modeling, and coarse-grained molecular dynamics (CGMD) simulations with an explicit solvent to probe gel friction in an open-air environment. They showed that hydrogel friction is governed by hydrodynamic drag through the polymeric network and that the friction coefficient is influenced by the hydrodynamic layer thickness, mesh size, and effective viscosity, all of which depend on water transport and polymer–water interactions.

In this work, we develop a model of a hydrogel sliding along a flat solid wall in an explicit solvent to probe its friction. We focus on swollen, chemically cross-linked hydrogels that exhibit a near-surface polymer-depleted layer with a characteristic thickness of roughly one to a few mesh sizes arising from entropic (confinement) and excluded-volume effects. This gel structure is consistent with experiments that demonstrate sparser networks with dangling chains at the interface compared to the bulk. − Our model is based on dissipative particle dynamics (DPD), a mesoscale simulation technique that effectively captures both hydrodynamic interactions and polymer dynamics in hydrated polymer networks. , We have previously used DPD to model the mechanics of polymeric networks and microgels, − demonstrating its capability for probing solvent-mediated mechanics and deformation in hydrogels. Our computational model enables a direct examination of the micromechanics of hydrogel friction by capturing the interactions among a sliding polymeric network, viscous solvent, and a solid substrate. We quantify the effects of sliding velocity and normal load on the friction coefficient. By evaluating local density, deformation, and velocity profiles, we reveal the role of the gel structure and near-wall gel changes in mediating friction forces under load. Our results identify a crossover in frictional behavior governed by the Weissenberg number and demonstrate that viscous drag within the interfacial solvent layer is the dominant source of friction at the higher Weissenberg number regimes.

Methods

We use dissipative particle dynamics (DPD) to model friction between a solid wall and a polymeric network immersed in an explicit viscous solvent. DPD is a particle-based mesoscale method where particles, representing clusters of molecules, interact via soft potentials. This approach enables simulations to span longer time and larger length scales, making it computationally efficient for studying large dynamic systems over extended time periods. Additionally, DPD employs pairwise interactions that inherently conserve momentum, allowing it to properly capture hydrodynamic interactions critical for investigating hydrogel friction.

In DPD, the total force acting on a DPD particle is given by $F_i=\sum _{j\ne i}(F_{ij}^C+F_{ij}^D+F_{ij}^R)$ , where the force acting on a bead i arises from interactions with its neighboring beads j within a cutoff radius r _c. The conservative force $F_{ij}^C=a_{ij}w(r_{ij}){\mathrm{r̂}}_{ij}$ represents the repulsion between particles accounting for excluded volume. Here, a _ij is the repulsion parameter, $w(r_{ij})=1-\frac{r_{ij}}{r_c}$ is the weight function, and ${\mathrm{r̂}}_{ij}$ is the unit vector connecting the particles. The dissipative force $F_{ij}^D=-\delta w^2(r_{ij})({\mathrm{r̂}}_{ij}·v_{ij}){\mathrm{r̂}}_{ij}$ represents the effects of viscosity, with δ being the dissipative coefficient and v _ij the relative velocity between particles. The random force $F_{ij}^R=\sigma w(r_{ij}){\zeta }_{ij}{(\mathrm{\Delta }t)}^{-1/2}{\mathrm{r̂}}_{ij}$ accounts for thermal fluctuations, where σ is the noise amplitude, ζ _ij is a random variable with zero mean and unit variance, and Δt is the time step. The fluctuation–dissipation theorem relates σ and δ as σ² = 2δk _B T, where k _B is the Boltzmann constant and T is temperature.

The initial gel network is constructed by randomly distributing cross-link points within a simulation box. Then, neighboring cross-link points are connected by up to four polymer chains. , This statistically isotropic protocol yields a permanently cross-linked random polymer network with a thin, sparser outer gel layer on the order of the mesh size, in line with depletion expected for swollen hydrogels. − Neighboring polymeric chain beads are connected with a harmonic bond potential $U_{\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d}}=k_{\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d}}{(r-r_{\mathrm{e}\mathrm{q}})}^2$ , where k _bond is the bond stiffness, r _eq is the equilibrium bond length, and r is the distance between bonded beads. An angle potential $U_{\mathrm{b}\mathrm{e}\mathrm{n}\mathrm{d}}=k_{\mathrm{b}\mathrm{e}\mathrm{n}\mathrm{d}}(1+\cos \theta )$ is used to set polymer bending stiffness, where k _bend is the bending stiffness, and θ is the angle between three consecutive beads.

Figure a shows the schematic of the computational model. The bottom surface of the simulation box serves as the stationary solid wall against which the hydrogel slides. The top of the hydrogel network is firmly attached to a moving rigid wall that moves with a constant velocity V in the positive x-direction. This wall is permeable to the solvent but impermeable to gel. A constant compressive force F _n is applied to this top wall, which is allowed to move vertically. As a result, the gel network is compressed, and solvent is expelled through the permeable wall, maintaining a constant density in the gel–solvent system. Figure b shows a representative snapshot of the simulation.

1.

(a) Schematic of the computational setup. A polymeric gel network (yellow) slides along a rigid bottom wall (gray). The gel is attached at the top to a rigid moving wall (green) translating at velocity V along x-direction. The moving wall is subject to a normal compressive force F _n. The red dashed circle highlights the mesh size ξ of the network. (b) Snapshot of the simulation box showing the three-dimensional gel network (yellow) immersed in an explicit solvent (red), confined between the bottom stationary wall (gray) and the top moving wall (green); periodic boundary conditions are imposed laterally. (c) Probability density distribution of mesh size ξ in the equilibrated gel network. (d) Probability density distribution of contour length N (monomers per chain) in the gel network.

Figure c quantifies the internal structure of the equilibrated and unloaded gel by presenting the probability density distribution of the network mesh size ξ evaluated with the pore–size algorithm introduced by Gelb and Gubbins. In this method, the largest solvent-accessible sphere that can be inserted at each point in the polymer network is calculated, yielding a statistical measure of the local mesh size. Figure d shows the distribution of the number of monomers N in polymer chains connecting cross-linkers within the gel network.

Periodic boundary conditions are imposed on the lateral sides of the simulation box to ensure an infinite repeating hydrogel in the x- and y-directions. A harmonic repulsion is set at the top box boundary to prevent solvent from leaving the computational domain.

At the bottom boundary that represents the frictional rigid wall, a no-slip boundary condition is applied to the solvent using a bounce-back method. Specifically, after each integration step, any solvent bead whose center crosses the boundary plane is reflected with its velocity reversed, so both normal and tangential components invert at impact, yielding zero in-plane velocity at the boundary. This boundary condition is supplemented with a harmonic repulsion potential to prevent solvent particles from accumulating next to the bottom box surface. Harmonic potential at the bottom boundary is also used to prevent the gel crossing the rigid wall.

In our simulations, we set r _c = 1, γ = 4.5, k _B T = 1, Δt = 0.01, the number density of DPD beads ρ₀ = 3 and the repulsion parameter a _ij = 25. In the hydrogel network model, the mean number of monomers per chain is N ₀ = 6.2, the bond stiffness is k _bond = 16, and the bending stiffness is k _bend = 10 leading to the hydrogel’s effective elastic modulus E = 0.078, porosity ε = 0.89, solvent dynamic viscosity η = 0.84, and a mean mesh size ξ₀ = 3.2. The porosity ε denotes the solvent-filled volume fraction.

Harmonic potentials of 6.57 and 50 are applied at the bottom wall to the solvent and gel, respectively. At the top boundary, a potential of 50 is applied to the solvent. A domain size of 60 × 60 × 60 is used in the simulations. We verified that our results are independent of the domain size by conducting simulations with a larger domain size. Furthermore, we select the simulation parameters to ensure that the velocities are well below the DPD speed of sound, making compressibility effects negligible, and the maximum shear rate is below 0.1 so that the DPD thermostat keeps temperature variations below one percent.

The simulations were performed for 10⁷ time steps to eliminate the effects of initial transient. The gel data was collected over the last 2 × 10⁶ time steps. The above computational parameters are given in DPD units.

We express the problem in terms of dimensionless groups to facilitate the comparison of our simulation results with other models and experimental data. We introduce a dimensionless normal load P = F _n/(AE) where A is the nominal contact area. The friction coefficient is defined as μ = F _f/F _n, where the friction force F _f is evaluated as the horizontal force applied to move the hydrogel. To characterize hydrogel sliding, we use the Weissenberg number $Wi=V\eta {\xi }_0^2/(k_{\mathrm{B}}T)$ that represents the ratio of polymer relaxation time scale $\eta {\xi }_0^3/(k_{\mathrm{B}}T)$ to the time scale of the shear ξ₀/V. In our simulations, we change Wi by changing the sliding velocity V. Furthermore, we normalize all distances by the mean mesh size ξ₀.

We consider P in the range between 0.14 and 0.28 and Wi in the range between 10^–4 and 1. These ranges of dimensionless parameters are representative for normal stresses and shear rates reported for ocular and articular cartilage lubrication. , Furthermore, the parameters are relevant to cartilage-mimetic materials with E ∼ 0.1–1 MPa − and ξ ∼ 50–100 nm. ,

Results and Discussion

Figure a shows the variation of normalized local gel density ρ_g/ρ₀ as a function of distance from the rigid wall for different values of Weissenberg number Wi and normal load P. The distance from the wall is normalized by the characteristic network mesh size ξ₀. Away from the wall, the gel density remains nearly constant with slight fluctuations with a wavelength on the order of the mesh size, indicating a uniform network structure. We refer to this as the bulk region. Closer to the wall, within a distance of approximately 2ξ₀, the density sharply decreases, reflecting a structural transition at the gel boundary. This drop is associated with a lower cross-link density and the presence of dangling chains near the wall. We refer to this zone of rapid density change as the transition region. We further define interface as a narrow zone immediately adjacent to the wall at the lower part of the transition region, where the gel network is in direct contact with the rigid wall. The interface thickness is set to 0.3ξ₀. All three regions are illustrated in Figure a.

2.

Gel and solvent dynamics. (a) Normalized gel density profile ρ_g/ρ₀ versus normalized wall distance z/ξ₀ for different normal loads P and Weissenberg numbers Wi. The legend in panel (a) indicates the P values (by color) and the Wi values (by color intensity) and applies to both panels. (b) Normalized solvent velocity profiles v/V as a function of z/ξ₀ for different P and Wi. Inset: cropped snapshot of the hydrogel network near the rigid wall.

Changes in P have only a modest effect on the thickness of the transition region, resulting in a slight decrease in thickness with increasing P. Furthermore, the thickness of the transition region exhibits a slight increase with Wi, likely due to a stronger effect of near-wall hydrodynamic lift. Specifically, raising Wi from 10^–2 to 1 enlarges the transition region thickness by approximately 10%. These suggest that an increase in bulk density due to higher P and Wi leads to a sharper change in ρ_g, whereas the spatial extent of the structural rearrangement near the wall is relatively insensitive to these parameters.

Figure b shows the variation of normalized solvent velocity v/V with distance from the wall. The solvent velocity is zero at the wall due to the no-slip boundary condition and increases sharply within the transition region, approaching the bulk gel velocity around a distance of 2ξ₀. This suggests that the spatial extent of the transition region for solvent velocity is similar to that of gel density. Moreover, the normalized solvent velocity profiles for different P and Wi collapse onto a single curve, indicating that the distribution of solvent velocity near the wall is largely insensitive to P and Wi. The rapid increase in solvent velocity near the solid wall results from viscous drag between the moving gel filaments and the solvent. Since the gel and solvent move at the same speed in the bulk region, most viscous dissipation is localized in the transition region, playing a critical role in determining gel friction.

The inset in Figure b provides a visual representation of the hydrogel network near the rigid wall, highlighting the structural heterogeneity within the transition region. It reveals that the gel exhibits noticeably higher porosity near the wall, which is consistent with the reduced cross-link density and the presence of dangling polymer segments.

Figure a presents the normalized bulk gel density ρ _b /ρ₀ as a function of Wi for different values of P. For Wi < 0.1, corresponding to slower sliding velocities, the bulk gel density is nearly constant and independent of Wi. In this regime, the relaxation time of the polymer chains is shorter than the time scale associated with the applied shear deformation. As a result, the polymer chains have sufficient time to equilibrate following mechanical perturbations from sliding. This leads to minor structural changes in the gel network, maintaining a nearly constant bulk gel density at lower Wi. For Wi > 0.1, ρ_b increases with increasing Wi. In this regime, polymer network relaxation is slower than the shear time scale, resulting in significant shear deformation of the gel network. This, in turn, leads to a denser packing of the polymer network under compressive load.

3.

Bulk gel density ρ_b/ρ₀ as a function of (a) Weissenberg number Wi and (b) normal load P. The numbered data points in panel (a) correspond to snapshots in Figure . Error bars represent standard deviation.

Figure b shows the variation of bulk gel density ρ_b/ρ₀ with P. We find that ρ_b increases nearly linearly with P. The slope of this increase is roughly 0.23 and is insensitive to Wi. This result suggests that gel compressibility is unaffected by shear-induced structural changes of the polymer network.

Figure a,b show the normalized gel density at the interface with the rigid wall ρ_i/ρ₀ as a function of Wi and P, respectively. Figure a demonstrates, in contrast to ρ_b that increases with Wi, ρ _i slightly decreases with increasing Wi. This can be related to an increased contribution of the hydrodynamic lift force displacing the gel away from the rigid wall. This is consistent with Figure a, where the thickness of the near-wall transition layer slightly increases with increasing Wi, resulting in a broader polymer-depleted zone and reduced interfacial density ρ_i. Similarly to ρ_b, the interfacial gel density ρ_i increases linearly with P (Figure b). However, this slope is significantly lower than that for the bulk gel density with Weissenberg number, indicating nearly four-times lower compressibility of the interfacial gel.

4.

Interfacial gel density ρ_i/ρ₀ as a function of (a) Weissenberg number Wi and (b) normal load P. Error bars represent standard deviation.

Figure quantifies the mean shear strain of the hydrogel γ as a function of Wi. Shear strain γ is evaluated as the geometric distortion of the gel with respect to its unloaded configuration. As Wi increases, the gel undergoes progressively larger shear deformation, reflected in the increasing γ. For Wi < 0.01, γ is nearly zero, indicating that the gel is compressed vertically. For Wi > 0.01, γ gradually increases with Wi. Note that in this regime, larger P leads to a slightly higher γ. The increase in γ with Wi is correlated with increased bulk density ρ _b with Wi (Figure ). This suggests that gel shearing deformation leads to gel compression, which reduces network mesh size.

5.

Shear strain γ as a function of the Weissenberg number Wi for different normal loads P. Snapshots above the plot show the gel deformation for selected Wi values labeled 1–5 at P = 0.14. Error bars represent standard deviation.

The snapshots in Figure illustrate the steady-state hydrogel network deformation at various Wi and P = 0.14. The red strip indicates a portion of the gel oriented normally to the sliding surface at V = 0. At lower Wi, the hydrogel network appears isotropic, with no significant distortion as the red strip remains vertical. As Wi increases, the network experiences progressive shear deformation, as indicated by the tilt of the red stripe in the direction of gel motion. The deformation is uniform across the thickness of the gel layer.

Figure a presents the variation of normalized friction coefficient μ/μ₀ as a function of Wi. Depending on the magnitude of Wi, our simulations reveal two distinct regimes in the friction response of the hydrogel. At low Wi < 0.01, the friction coefficient remains nearly constant and independent of Wi. For each P, we define μ₀ as the average friction coefficient in the Wi-independent regime and use these values to normalize μ. Figure a shows that the normalized friction data for different P collapse onto a universal curve.

6.

(a) Normalized friction coefficient μ/μ₀ as a function of Weissenberg number Wi for normal loads P = 0.14, P = 0.21, and P = 0.28. (b) Friction coefficient μ as a function of P for different values of Wi. The dashed line indicates a power-law dependence of −0.77. Error bars represent standard deviation.

The constant gel friction at low Wi can be rationalized by the dominance of thermal fluctuations over polymer-chain deformation due to friction. In this regime, the applied shear stress is insufficient to overcome random motion of polymer chains, preventing significant deformation or reorganization of the hydrogel network. As a result, the friction coefficient is independent of Wi.

In the high-Wi regime, μ increases approximately linearly with Wi, suggesting that friction is dominated by solvent hydrodynamics. Recall that in our model the gel does not experience friction interactions with the solid wall. For a constant normal load F _n , μ is proportional to the friction force F _f, which in turn is proportional to wall shear stress τ_w = η dv/dz. We assume that the velocity gradient near the wall dv/dz scales as V/d, where d is the thickness of a thin solvent layer forming between the gel interface and the wall. The solvent-layer thickness d scales with the interfacial gel density ρ _i . Indeed, higher d requires lower ρ _i . Noting that ρ _i is nearly independent of Wi (Figure a) and, therefore, sliding velocity V, we conclude that μ scales linearly with V, which in turn implies μ ∼ Wi, consistent with our simulation.

A similar linear dependence of μ on Wi has been observed in coarse-grained MD simulations of self-mated hydrogels and in experiments with polyacrylamide and agarose hydrogels sliding under nonadhesive contact conditions. In the latter case, the dominant dissipation mechanism was attributed to fluid–network interactions within the hydrogel mesh. Our simulations show similar frictional scaling, reinforcing the idea that interfacial viscous drag, rather than adhesive or entropic effects, governs friction in this high-Wi regime. The difference in the magnitude of friction coefficients between simulation and experiment likely stems from differences in polymer structure, cross-linking density, and solvent–polymer interactions.

Figure b shows the variation of the friction coefficient μ as a function of normal load P. For all Wi, μ decreases monotonically with increasing P. The scaling of μ with P follows a power-law behavior with an exponent about −0.77, indicating that F _f ∼ P ^0.23. Increasing the normal load densifies the gel, particularly at the interface near the wall (Figure b), thereby decreasing the thickness of the sheared solvent layer d. This results in increasing F _f with P. However, the increase is relatively moderate as ρ _i increases with P significantly slower than the bulk density ρ _b .

Inverse dependence of μ on P is consistent with experiments. , Although the reported exponents range between −1/2 and −1/3, those differences stem from varying assumptions about contact area growth with load. In our simulations, the nominal contact area is constant, and the observed trend can be attributed to increasing gel density at the interface with the sliding wall. Our results are also consistent with explicit-solvent CGMD simulations, which likewise found that the friction coefficient decreases with increasing normal load.

Conclusions

We developed a mesoscale model to investigate the frictional behavior of hydrogels sliding along a rigid flat substrate in an explicit solvent environment. Our simulations capture the essential micromechanical interactions between the hydrogel polymeric network, viscous solvent, and solid substrate, enabling us to probe the interplay between structural deformation and viscous dissipation at mesoscopic scales. In our computational model, a no-slip boundary condition is imposed on the viscous fluid at the solid substrate, whereas the compressed by an external load hydrogel is allowed to slide freely along the substrate. We systematically vary hydrogel load and sliding velocity to reveal the micromechanics of gel friction.

We found that hydrogel friction exhibits two distinct regimes governed by the Weissenberg number Wi. At low Wi, the friction coefficient μ remains nearly constant, reflecting thermally dominated dynamics with minimal structural rearrangement. In contrast, at higher Wi, μ increases linearly with Wi, indicating a transition to shear-dominated behavior where viscous drag within a near-wall solvent layer becomes the primary source of friction. Notably, we observed that the thickness of this interfacial solvent layer near the wall, where solvent experiences the maximum velocity gradient, is largely independent of Wi, leading to a linear dependence of μ on Wi consistent with experimental data.

Our results also demonstrate that increasing normal load P leads to densification of the hydrogel, both in the bulk and near the wall, though the compressibility at the interface is significantly lower. This densification decreases the thickness of the sheared solvent layer and contributes to an inverse power-law dependence of μ on P with an exponent of approximately −0.77.

These findings highlight the critical role of interfacial solvent dynamics and network deformation in mediating hydrogel friction. By explicitly modeling both the polymer network and solvent, our approach bridges the gap between molecular–scale interactions and continuum-level frictional responses. This framework can be directly applied for designing soft materials with tailored tribological properties. Furthermore, the results facilitate our understanding of the role of gel mechanics in regulating biological lubrication.

The initial hydrogel geometry used in this study is available at https://github.com/gt-cfms/Mesoscale-Modeling-of-Hydrogels-under-Frictional-Shear-Stress. The repository includes files containing a bead–spring network that models a cross-linked gel structure, formatted for use with LAMMPS.

M.K., A.A.P., and A.A. conceptualized the research project. A.A.P. and A.A. secured the funding. M.K. developed the computational model, performed the simulations, analyzed the data and prepared the manuscript. A.P. contributed to model development and manuscript writing. All authors discussed the results. All authors reviewed and approved the final version of the manuscript.

The authors declare no competing financial interest.