Sliding friction of a pillar array interface: part I.

Biology is replete with examples, at length scales ranging from the molecular (ligand-receptor binding) to the mesoscopic scale (wing arresting structures on dragonflies) where shape-complementary surfaces are used to control interfacial mechanical properties such as adhesion, friction, and contact compliance. Related bio-inspired and biomimetic structures have been used to achieve unique interfacial properties such as friction and adhesion enhancement, directional and switchable properties. The ability to tune friction by altering surface structures offers advantages in various fields, such as soft robotics and tire manufacturing. Here, we present a study of friction between polydimethylsiloxane (PDMS) samples with surfaces patterned with pillar-arrays. When brought in contact with each other the two samples spontaneously produce a Moiré pattern that can also be represented as an array of interfacial dislocations that depends on interfacial misorientation and lattice spacing. Misorientation alone produces an array of screw dislocations, while lattice mismatch alone produces an array of edge dislocations. Relative sliding motion is accompanied by interfacial glide of these patterns. The frictional force resisting dislocation glide arises from periodic single pillar-pillar contact and sliding. We study the behavior of pillar-pillar contact with larger (millimeter scale) pillar samples. Inter-pillar interaction measurements are combined with a geometric model for relative sliding to calculate frictional stress that is in good agreement with experiments.

Sliding friction of a pillar array interface: part II, contact mechanics of single pillar pairs.

Insects and small animals often utilize structured surfaces to create friction during their movements. These surfaces typically consist of pillar-like fibrils that interact with a counter surface. Understanding the mechanical interaction between such surfaces is crucial for designing structured surfaces for engineering applications. In the first part of our study, we examined friction between poly(dimethylsiloxane) (PDMS) samples with surfaces patterned with pillar-arrays. We observed that sliding between these surfaces occurs through the interfacial glide of dislocation structures. The frictional force that resists this dislocation glide is a result of periodic single pillar-pillar contact and sliding. Hence, comprehending the intricate interaction between individual pillar contacts is a fundamental prerequisite for accurately modeling the friction behavior of the pillar array. In this second part of the study, we thoroughly investigated the contact interaction between two pillars located on opposite sides of an interface, with different lateral and vertical offsets. We conducted experiments using PDMS pillars to measure both the reaction shear and normal forces. Contact interaction between pillars was then studied using finite element (FE) simulations with the Coulomb friction model, which yielded results that aligned well with the experimental data. Our result offers a fundamental solution for comprehending how fibrillar surfaces contact and interact during sliding, which has broad applications in both natural and artificial surfaces.

How chain dynamics affects crack initiation in double-network gels.

Double-network gels are a class of tough soft materials comprising two elastic networks with contrasting structures. The formation of a large internal damage zone ahead of the crack tip by the rupturing of the brittle network accounts for the large crack resistance of the materials. Understanding what determines the damage zone is the central question of the fracture mechanics of double-network gels. In this work, we found that at the onset of crack propagation, the size of necking zone, in which the brittle network breaks into fragments and the stretchable network is highly stretched, distinctly decreases with the increase of the solvent viscosity, resulting in a reduction in the fracture toughness of the material. This is in sharp contrast to the tensile behavior of the material that does not change with the solvent viscosity. This result suggests that the dynamics of stretchable network strands, triggered by the rupture of the brittle network, plays a role. To account for this solvent viscosity effect on the crack initiation, a delayed blunting mechanism regarding the polymer dynamics effect is proposed. The discovery on the role of the polymer dynamic adds an important missing piece to the fracture mechanism of this unique material.

How the geometry of patterned surfaces affects the thickness distribution of the oxidized silica layer on polydimethylsiloxane (PDMS) after ultraviolet/ozone treatment.

It is well established that a thin silica-like surface layer is formed when a cross-linked PDMS structure is subjected to ultraviolet/ozone treatment. Due to surface geometry, especially near the corners, this silica-like surface layer has non-uniform thickness, which can impact many mechanical properties, including adhesion and fracture strength. Here we use a simple analytic model based on diffusion of reactive species to predict the thickness of the oxidized surface layer near the corners. We demonstrate that these corner solutions can be patched together to determine the thickness of the oxidized layer in complex geometries.

Swelling induced debonding of thin hydrogel films grafted on silicon substrates.

We report on the delamination of thin (≈µm) hydrogel films grafted to silicon substrates under the action of swelling stresses. Poly(dimetylacrylamide) (PDMA) films are synthesized by simultaneously cross-linking and grafting preformed polymer chains onto the silicon substrate using a thiol-ene reaction. The grafting density at the film/substrate interface is tuned by varying the surface density of reactive thiol-silane groups on the silicon substrate. Delamination of the films from well controlled line defects with low adhesion is monitored under a humid water vapor flow ensuring full saturation of the polymer network. A propagating delamination of the film is observed under the action of differential swelling stresses at the debonding front. A threshold thickness for the onset of this delamination is evidenced which is increasing with grafting density while the debonding velocity is also observed to decrease with an increase in grafting density. These observations are discussed within the framework of a nonlinear fracture mechanics model which assumes that the driving force for crack propagation is the difference between the swelling state of the bonded and delaminated parts of the film. Using this model, the threshold energy for crack initiation was determined from the measured threshold thickness and discussed in relation to the surface density of reactive thiol groups on the substrate.

Slicing of a soft solid.

Cutting of soft materials is a complex problem, which is still not well understood at the fundamental level, especially for soft materials. The cutting process we consider is slicing, which starts with indentation, followed by sliding of a knife on the material to be cut. Here, we describe cutting experiments on PDMS elastomers with three different moduli. Our experiments reveal typical stages of this cutting process, starting with indentation and ending at steady state cutting. The process starts with a pre-cutting phase in which the blade does not slip grossly relative to the solid to be cut, and deformation is mostly elastic. Slip of the blade initiates suddenly and is often accompanied by initiation of cutting. Cutting is relatively smooth in the next stage, which requires a continuous increase in shear force. For soft PDMS, this smooth cutting stage is followed by one in which folds or creases form on the cutting surface. The corresponding shear force response is no longer smooth as "steady" sliding occurs in a stick-slip fashion with oscillatory forces. The average shear force reaches a plateau and no longer increases with shear displacement. Experimental observations of the various cutting stages are interpreted quantitatively.

Lubricated soft normal elastic contact of a sphere: a new numerical method and experiment.

An important problem in lubrication is the squeezing of a thin liquid film between a rigid sphere and an elastic substrate under normal contact. Numerical solution of this problem typically uses iteration techniques. A difficulty with iteration schemes is that convergence becomes increasingly difficult under increasingly heavy loads. Here we devise a numerical scheme that does not involve iteration. Instead, a linear problem is solved at every time step. The scheme is fully automatic, stable and efficient. We illustrate this technique by solving a relaxation test in which a rigid spherical indenter is brought rapidly into normal contact with a thick elastic substrate lubricated by a liquid film. The sphere is then fixed in position as the pressure relaxes. We also carried out relaxation experiments on a lubricated soft PDMS (polydimethysiloxane) substrate under different conditions. These experiments are in excellent agreement with the numerical solution.

Mechanical stress compromises multicomponent efflux complexes in bacteria.

Physical forces have a profound effect on growth, morphology, locomotion, and survival of organisms. At the level of individual cells, the role of mechanical forces is well recognized in eukaryotic physiology, but much less is known about prokaryotic organisms. Recent findings suggest an effect of physical forces on bacterial shape, cell division, motility, virulence, and biofilm initiation, but it remains unclear how mechanical forces applied to a bacterium are translated at the molecular level. In Gram-negative bacteria, multicomponent protein complexes can form rigid links across the cell envelope and are therefore subject to physical forces experienced by the cell. Here we manipulate tensile and shear mechanical stress in the bacterial cell envelope and use single-molecule tracking to show that octahedral shear (but not hydrostatic) stress within the cell envelope promotes disassembly of the tripartite efflux complex CusCBA, a system used by Escherichia coli to resist copper and silver toxicity. By promoting disassembly of this protein complex, mechanical forces within the cell envelope make the bacteria more susceptible to metal toxicity. These findings demonstrate that mechanical forces can inhibit the function of cell envelope protein assemblies in bacteria and suggest the possibility that other multicomponent, transenvelope efflux complexes may be sensitive to mechanical forces including complexes involved in antibiotic resistance, cell division, and translocation of outer membrane components. By modulating the function of proteins within the cell envelope, mechanical stress has the potential to regulate multiple processes required for bacterial survival and growth.

A surface flattening method for characterizing the surface stress, drained Poisson's ratio and diffusivity of poroelastic gels.

When a poroelastic gel is released from a patterned mold, surface stress drives deformation and solvent migration in the gel and flattens its surface profile in a time-dependent manner. Specifically, the gel behaves like an incompressible solid immediately after removal from the mold, and becomes compressible as the solvent is able to squeeze out of the polymer network. In this work, we use the finite element method (FEM) to simulate this transient surface flattening process. We assume that the surface stress is isotropic and constant, the polymer network is linearly elastic and isotropic, and that solvent flow obeys Darcy's law. The short-time and long-time surface profiles can be used to determine the surface stress and drained Poisson's ratio of the gel. Our analysis shows that the drained Poisson's ratio and the diffusivity of the gel can be obtained using interferometry and high-speed video microscopy, without mechanical measurement.

Constitutive modeling of strain-dependent bond breaking and healing kinetics of chemical polyampholyte (PA) gel.

A finite strain nonlinear viscoelastic constitutive model is used to study the uniaxial tension behaviour of chemical polyampholyte (PA) gel. This PA gel is cross-linked by chemical and physical bonds. Our constitutive model attempts to capture the time and strain dependent breaking and healing kinetics of physical bonds. We compare model prediction by uniaxial tension, cyclic and relaxation tests. Material parameters in our model are obtained by least squares optimization. These parameters gave fits that are in good agreement with the experiments.

Modeling of surface mechanical behaviors of soft elastic solids: theory and examples.

Surfaces of soft solids can have significant surface stress, extensional modulus and bending stiffness. Previous theoretical studies have usually examined cases in which both the surface stress and bending stiffness are constant, assuming small deformation. In this work we consider a general formulation in which the surface can support large deformation and carry both surface stresses and surface bending moments. We demonstrate that the large deformation theory can be reduced to the classical linear theory (Shuttleworth equation). We obtain exact solutions for problems of an inflated cylindrical shell and bending of a plate with a finite thickness. Our analysis illustrates the different manners in which surface stiffening and surface bending stabilize these structures. We discuss how the complex surface constitutive behaviors affect the stress field of the bulk. Our calculation provides insights into effects of strain-dependent surface stress and surface bending in the large deformation regime, and can be used as a model to implement surface finite elements to study large deformation of complex structures.

Lubricated steady sliding of a rigid sphere on a soft elastic substrate: hydrodynamic friction in the Hertz limit.

Lubricated sliding on soft elastic substrates occurs in a variety of natural and technological settings. It very often occurs in the iso-viscous elasto-hydrodynamic lubrication (EHL) regime (e.g., soft solid, low pressure). In this regime, for sliding of a smooth sphere on a soft solid, a "Hertz-like" effective contact region forms. Much of the fluid is squeezed out of the contact region although enough is retained to keep the solid surfaces fully separated. This is accompanied by complex deformation of the soft solid. The behavior of such soft lubricated contacts is controlled by a single dimensionless parameter 1/ß that can be interpreted as a normalized sliding velocity. Solving this fundamental soft-lubrication problem poses significant computational difficulty for large ß, which is the limit relevant for soft solids. As a consequence, little is known about the structure of the flow field under soft lubrication in the intake and outlet regions. Here we present a new solution of this soft lubrication problem focusing on the "Hertz" limit. We develop a formulation in polar coordinates that handles difficult computational issues much better than previous methods. We study how hydrodynamic pressure, film thickness and hydrodynamic friction vary with ß. Scaling laws for these relationships are given in closed form for a range of ß not previously accessible theoretically but that is typical in applications. The computational method presented here can be used to study other soft lubrication problems.

Time dependent fracture of soft materials: linear versus nonlinear viscoelasticity.

Toughness of soft materials such as elastomers and gels depends on their ability to dissipate energy and to reduce stress concentration at the crack tip. The primary energy dissipation mechanism is viscoelasticity. Most analyses and models of fracture are based on linear viscoelastic theory (LVT) where strains are assumed to be small and the relaxation mechanisms are independent of stress or strain history. A well-known paradox is that the size of the dissipative zone predicted by LVT is unrealistically small. Here we use a physically based nonlinear viscoelastic model to illustrate why the linear theory breaks down. Using this nonlinear model and analogs of crack problems, we give a plausible resolution to this paradox. In our model, viscoelasticity arises from the breaking and healing of physical cross-links in the polymer network. When the deformation is small, the kinetics of bond breaking and healing are independent of the strain/stress history and the model reduces to the standard linear theory. For large deformations, localized bond breaking damages the material near the crack tip, reducing stress concentration and dissipating energy at the same time. The damage zone size is a new length scale which depends on the strain required to accelerate bond breaking kinetics. These effects are illustrated by considering two cases with stress concentrations: the evolution of spherical damage in a viscoelastic body subjected to internal pressure, and a zero degree peel test.

Enhancement of elastohydrodynamic friction by elastic hysteresis in a periodic structure.

Lubricated contacts are present in many engineering and biological systems involving soft solids. Typical mechanisms considered for controlling the sliding friction in such lubricated conditions involve bulk material compliance, fluid viscosity, viscoelastic response of the material (hysteretic friction), and breaking of the fluid film where dry contact occurs (adhesive friction). In this work we show that a two-phase periodic structure (TPPS), with a varying modulus across the sliding surface, provides significant enhancement of lubricated sliding friction when the system is in the elastohydrodynamic lubrication (EHL) regime. We propose that the enhanced friction is due to extra energy loss during periodic transitions of the sliding indenter between the compliant and stiff regions during which excess energy is dissipated through the fluid layer. This is a form of elastic hysteresis that provides a novel mechanism for friction enhancement in soft solids under lubricated conditions.

A surface with stress, extensional elasticity, and bending stiffness.

We demonstrate that the surface of a commonly used polydimethylsiloxane formulation (PDMS, Sylgard 184) treated by ultraviolet ozonolysis (UVO) has significant surface stress, considerable extensional elasticity (the "Shuttleworth Effect"), and surface bending elasticity. For soft solids, phenomena such as wetting, contact, surface flattening, and stiffening by liquid inclusions are often governed by their surface, which is usually represented by a liquid-like constant surface stress. Whether the surfaces of soft solids can have more complex constitutive response is actively debated. We studied the deformation of three surface-patterned materials systems: untreated polydimethylsiloxane (PDMS), an organogel, and patterned PDMS with surface treatment by UVO. The last of these three, we found, has complex surface elasticity. This is analogous to the situation for liquids in which the presence of a second phase at the interface yields Gibbs elasticity. Our finding is of broad applicability because in soft solids the behavior of the surface can often dominate bulk deformation.

Effects of strain-dependent surface stress on the adhesive contact of a rigid sphere to a compliant substrate.

Recent experiments have reported that the surface stress of soft elastic solids can increase rapidly with surface strain. For example, when a small hard sphere in adhesive contact with a soft silicone gel is slowly retracted from its rest position, it was found that the retraction force versus displacement relation cannot be explained either by the Johnson-Kendall-Roberts (JKR) theory or a recent indentation theory based on an isotropic surface stress that is independent of surface strain. In this paper, we address this problem using a finite element method to simulate the retraction process. Our numerical model does not have the restrictions of the aforementioned theories; that is, it can handle large nonlinear elastic deformation as well as a surface-strain-dependent surface stress. Our simulation is in good agreement with experimental force versus displacement data with no fitting parameters. Therefore, our results lend further support to the claim that significant strain-dependent surface stresses can occur in simple soft elastic gels. However, significant challenges remain in the reconciliation of theory and experiments, particularly regarding the geometry of the contact and substrate deformation.

Fibrous nonlinear elasticity enables positive mechanical feedback between cells and ECMs.

In native states, animal cells of many types are supported by a fibrous network that forms the main structural component of the ECM. Mechanical interactions between cells and the 3D ECM critically regulate cell function, including growth and migration. However, the physical mechanism that governs the cell interaction with fibrous 3D ECM is still not known. In this article, we present single-cell traction force measurements using breast tumor cells embedded within 3D collagen matrices. We recreate the breast tumor mechanical environment by controlling the microstructure and density of type I collagen matrices. Our results reveal a positive mechanical feedback loop: cells pulling on collagen locally align and stiffen the matrix, and stiffer matrices, in return, promote greater cell force generation and a stiffer cell body. Furthermore, cell force transmission distance increases with the degree of strain-induced fiber alignment and stiffening of the collagen matrices. These findings highlight the importance of the nonlinear elasticity of fibrous matrices in regulating cell-ECM interactions within a 3D context, and the cell force regulation principle that we uncover may contribute to the rapid mechanical tissue stiffening occurring in many diseases, including cancer and fibrosis.

Indentation versus Rolling: Dependence of Adhesion on Contact Geometry for Biomimetic Structures.

Numerous biomimetic structures made from elastomeric materials have been developed to produce enhancement in properties such as adhesion, static friction, and sliding friction. As a property, one expects adhesion to be represented by an energy per unit area that is usually sensitive to the combination of shear and normal stresses at the crack front but is otherwise dependent only on the two elastic materials that meet at the interface. More specifically, one would expect that adhesion measured by indentation (a popular and convenient technique) could be used to predict adhesion hysteresis in the more practically important rolling geometry. Previously, a structure with a film-terminated fibrillar geometry exhibited dramatic enhancement of adhesion by a crack-trapping mechanism during indentation with a rigid sphere. Roughly isotropic structures such as the fibrillar geometry show a strong correlation between adhesion enhancement in indentation versus adhesion hysteresis in rolling. However, anisotropic structures, such as a film-terminated ridge-channel geometry, surprisingly show a dramatic divergence between adhesion measured by indentation versus rolling. We study this experimentally and theoretically, first comparing the adhesion of the anisotropic ridge-channel structure to the roughly isotropic fibrillar structure during indentation with a rigid sphere, where only the isotropic structure shows adhesion enhancement. Second, we examine in more detail the anomalous anisotropic film-terminated ridge-channel structure during indentation with a rigid sphere versus rolling to show why these structures show a dramatic adhesion enhancement for the rolling case and no adhesion enhancement for indentation.

Friction of Poroelastic Contacts with Thin Hydrogel Films.

We report on the frictional behavior of thin poly(dimethylacrylamide) hydrogel films grafted on glass substrates in sliding contact with a glass spherical probe. Friction experiments are carried out at various velocities and normal loads applied with the contact fully immersed in water. In addition to friction force measurements, a novel optical setup is designed to image the shape of the contact under steady-state sliding. The velocity dependence of both friction force Ft and contact shape is found to be controlled by a Péclet number, Pe, defined as the ratio of the time τ needed to drain the water out of the contact region to a contact time a/ v, where v is the sliding velocity and a is the contact radius. When Pe < 1, the equilibrium circular contact achieved under static normal indentation remains unchanged during sliding. Conversely, for Pe > 1, a decrease in the contact area is observed together with the development of a contact asymmetry when the sliding velocity is increased. A maximum in Ft is also observed at Pe ≈1. These experimental observations are discussed in the light of a poroelastic contact model based on a thin-film approximation. This model indicates that the observed changes in contact geometry are due to the development of a pore pressure imbalance when Pe > 1. An order-of-magnitude estimate of the friction force and its dependence on normal load and velocity are also provided under the assumption that most of the frictional energy is dissipated by poroelastic flow at the leading and trailing edges of the sliding contact.

Mechanics of an adhesive tape in a zero degree peel test: effect of large deformation and material nonlinearity.

The common pressure sensitive adhesive (PSA) tape is a composite consisting of a stiff backing layer and a soft adhesive layer. A simple and common way to test how adhesive tapes respond to large shear deformations is the zero degree peel test. Because the backing is very stiff compared to the adhesive layer, the region where the adhesive layer is subjected to large shear can be hundreds of times its thickness. We use a large deformation hyperelastic model to study the stress and deformation fields in the adhesive layer in this test. We present a closed-form solution for the stress field in the adhesive layer and use this solution to determine how load is transferred from the backing layer to the adhesive. Our analytical model is then compared with finite element results, and except for a small region near the peel front, the predicted stress and deformation agree well with the finite element model. Interestingly, we find very different results from the classical linear theory established by Kaelble. In particular for large deformations, our analysis shows that the lateral stresses (parallel to the rigid substrate) are much larger than the shear stress in the adhesive layer. The discrepancy in the stress state and the deformation state with the linear theory is particularly large near the peel front, which we study with a finite element model. These new results will be very useful to interpret experiments and in particular to identify the high stress regions where failure is likely to initiate in zero-degree peel tests also called shear resistance tests in the PSA industry.