Molecular mechanisms of self-mated hydrogel friction Jan Mees1 2Rok Simi c3Thomas C. OConnor4 Nicholas D. Spencer3and Lars Pastewka1 2

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Molecular mechanisms of self-mated hydrogel friction

Jan Mees,1, 2 Rok Simiˇc,3Thomas C. O’Connor,4

Nicholas D. Spencer,3and Lars Pastewka1, 2, ∗

1Department of Microsystems Engineering, University of Freiburg,

Georges-K¨ohler-Allee 103, 79110 Freiburg, Germany

2Cluster of Excellence livMatS, Freiburg Center for Interactive

Materials and Bioinspired Technologies, University of Freiburg,

Georges-K¨ohler-Allee 105, 79110 Freiburg, Germany

3Laboratory for Surface Science and Technology,

Department of Materials, ETH Z¨urich,

Vladimir-Prelog-Weg 1-5/10, 8093 Z¨urich, Switzerland

4Department of Materials Science and Engineering,

Carnegie Mellon University, 4309 Wean Hall,

Pittsburgh, Pennsylvania 15213, USA

Abstract

Self-mated hydrogel contacts show extremely small friction coeﬃcients at low loads but a distinct

velocity dependence. Here we combine mesoscopic simulations and experiments to test the polymer-

relaxation hypothesis for this velocity dependence, where a velocity-dependent regime emerges

when the perturbation of interfacial polymer chains occurs faster than their relaxation at high

velocity. Our simulations reproduce the experimental ﬁndings, with speed-independent friction at

low velocity, followed by a friction coeﬃcient that rises with velocity to some power of order unity.

We show that the velocity-dependent regime is characterized by reorientation and stretching of

polymer chains in the direction of shear, leading to an entropic stress that can be quantitatively

related to the shear response. The detailed exponent of the power law in the velocity dependent

regime depends on how chains interact: We observe a power close to 1/2 for chains that can

stretch, while pure reorientation leads to a power of unity. Our simulations quantitatively match

experiments and show that the velocity dependence of hydrogel friction at low loads can be ﬁrmly

traced back to the morphology of near-surface chains.

∗lars.pastewka@imtek.uni-freiburg.de

arXiv:2210.13921v1 [cond-mat.soft] 25 Oct 2022

INTRODUCTION

Hydrogels are chemically crosslinked, hydrophilic polymer networks that swell in water

or aqueous solvents. They are responsible for the low friction coeﬃcients found in some

biological systems, such as cartilage or the cornea [1]. Due to their tribological properties,

hydrogels have been applied in soft contact lenses [2–4] and as synthetic articular cartilage

materials [5–7].

Self-mated hydrogel-on-hydrogel frictional contacts, also known as Gemini hydrogels,

have low friction coeﬃcients at low sliding velocities [8]. It has been shown that the poly-

merization conditions deﬁne a hydrogel’s degree of crosslinking close to the surface [9, 10].

Hydrogels whose polymerization close to the surface is inhibited by the presence of oxygen,

exhibit pronounced crosslinker gradients and sparse surfaces. Conversely, the lack of oxy-

gen during synthesis leads to a homogeneously crosslinked hydrogel [9, 10]. Nonetheless,

no polymer network is defect free and this will introduce dangling chains and loops on the

surface [11, 12].

In weakly crosslinked hydrogel surfaces, hydrodynamics plays a signiﬁcant role in the

tribological response of the system [13]. For highly crosslinked surfaces, which are the focus

of this work, the friction force is velocity independent or only weakly velocity dependent up

to a threshold velocity [8, 14]. Past this threshold, experiments have shown that the friction

coeﬃcient increases approximately with the square root of the velocity for highly crosslinked

surfaces [9, 13–15].

Competing models exist as explanations for this behavior: Pitenis et al. [15, 16] argue that

for hydrogels with mesh size ξ, solvent viscosity ηand temperature T, friction is determined

by a competition of ﬂuctuation-induced polymer relaxation with time scale τr=ξ3η/kBTand

the time scale introduced by the shear rate, τs=ξ/v. This gives rise to velocity-independent

(Coulomb) friction at small velocities v, where τrτs. Friction measurements at high

velocities collapse onto a single curve when plotted against the Weissenberg number [17],

Wi = τr/τs. This velocity-dependent friction regime can thus be attributed to the non-

equilibrium behavior of strained polymer chains and possibly adhesive interactions between

the surfaces [14]. Simple hydrodynamic-ﬂuid-ﬁlm friction models predict friction coeﬃcients

that scale with the square root of the velocity [18], but those models assume that surfaces are

rigid and do not deform during sliding, which might not be representative of hydrogel-like

biological surfaces. Furthermore, they require converging surfaces to generate hydrodynamic

lift, a condition not met by parallel surfaces [19]. In contrast, soft elasto-hydrodynamic

lubrication theory [18] allows the hydrogels to deform during sliding, but it also predicts

friction coeﬃcients that are lower than those measured experimentally and does not predict

the scaling of friction with velocity observed in experiments [20]. These hydrodynamic

lubrication theories also do not capture the transition between the velocity-independent and

velocity-dependent friction regimes that appear to depend on the mesh size ξ[14, 15].

We present evidence from molecular simulations in favor of the polymer-relaxation hy-

pothesis in highly crosslinked hydrogels. By choosing a model with an implicit solvent,

i.e. without long-range hydrodynamic momentum transport and no possibility of interfacial

ﬂuid-ﬁlm formation, we disentangle the role played by the solvent from that of the polymer

network. We ﬁnd that hydrogel friction is a purely interfacial phenomena at low contact

pressures. Furthermore, the shear response of interfacial polymers can be split into three

speed-dependent regimes. For small velocities, the friction coeﬃcient is speed independent

due to the eﬀect of thermal ﬂuctuations overwhelming chain alignment. This is followed by

a regime in which the friction coeﬃcient increases linearly with velocity. We trace this obser-

vation back to the reorientation of interfacial polymer chains in the direction of shear. Once

forces are strong enough, these interfacial chains start stretching, which leads to a friction

coeﬃcient that increases with the square root of velocity. We complement our simulations

with experiments that conﬁrm that the friction coeﬃcient increases with increasing mesh

size for highly crosslinked systems.

RESULTS & DISCUSSION

Our molecular dynamics simulations [21] resolve polymer-chain dynamics on length scales

of the order of a mesh size (5-20 nm). We use a purely repulsive, ﬂexible bead-spring

model [22], where a polymer chain is divided into uncorrelated segments (beads), each con-

taining one or more monomers and of a size corresponding roughly to the Kuhn length [23].

The polymer chains are thermalized by a dissipative particle dynamics thermostat, which

acts as an implicit solvent, but mediates no long-range hydrodynamic interactions [24, 25].

In order to construct our mesoscopic model, we mimic hydrogels used in friction exper-

iments. These are disordered networks with four-fold coordination, physical crosslinkers,

FIG. 1. (a) Slice of the hydrogel network during shear. The network has been colored, in order

to help distinguish the interface. The black line depicts the borders of the periodic simulation cell.

The inset shows a detailed view of the interface before relaxation. (b) Coeﬃcient of friction (CoF)

as a function of sliding velocity vand chain lengths for our computational model. (c) Sketch of

the interface for networks with and without surface variations. Monomers are colored in red and

crosslinkers in green. (d) Schematic of the friction experiments using a ﬂat, hydrogel pin on a ﬂat,

highly crosslinked gel. (e) Experimentally determined coeﬃcients of friction for polyacrylamide

hydrogels as a function of monomer concentration. The inset shows estimated mesh sizes ξas a

function of monomer weight percent.

entanglements and potentially dangling chains. Unfortunately, the detailed statistical prop-

erties of the connectivity of hydrogel networks are often unknown [11, 26] and inﬂuence

frictional behavior [9, 13]. We therefore simplify our computational network and build the

simplest regular structure with four-fold connectivity: a diamond lattice with dangling sur-

face chains (see inset to Fig. 1a). Such diamond-like structures have been used in the past

to test mechanical properties of polymer networks [27]. Four-fold coordinated crosslinkers

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Molecularmechanismsofself-matedhydrogelfrictionJanMees,1,2RokSimic,3ThomasC.O'Connor,4NicholasD.Spencer,3andLarsPastewka1,2,1DepartmentofMicrosystemsEngineering,UniversityofFreiburg,Georges-Kohler-Allee103,79110Freiburg,Germany2ClusterofExcellencelivMatS,FreiburgCenterforInteractiveMaterialsandBi...

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Molecular mechanisms of self-mated hydrogel friction Jan Mees1 2Rok Simi c3Thomas C. OConnor4 Nicholas D. Spencer3and Lars Pastewka1 2

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