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 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.

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.

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.

The transition from Elasto-Hydrodynamic to Mixed Regimes in Lubricated Friction of Soft Solid Surfaces.

Lubricated contacts in soft materials are common in various engineering and natural settings, such as tires, haptic applications, contact lenses, and the fabrication of soft electronic devices. Two major regimes are elasto-hydrodynamic lubrication (EHL), in which solid surfaces are fully separated by a fluid film, and mixed lubrication (ML), in which there is partial solid-to-solid contact. The transition between these regimes governs the minimum sliding friction achievable and is thus very important. Generally, the transition from EHL to ML regimes is believed to occur when the thickness of the lubricant layer is comparable with the amplitude of surface roughness. Here, it is reported that in lubricated sliding experiments on smooth, soft, poly(dimethylsiloxane) substrates, the transition can occur when the thickness of the liquid layer is much larger than the height of the asperities. Direct visualization of the "contact" region shows that the transition corresponds to the formation of wave-like surface wrinkles at the leading contact edge and associated instabilities at the trailing contact edge, which are believed to trigger the transition to the mixed regime. These results change the understanding of what governs the important EHL-ML transition in the lubricated sliding of soft solids.

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.

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.

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.

Meso-scale dislocations and friction of shape-complementary soft interfaces.

The interface between two surfaces patterned with complementary shapes such as arrays of ridge-channel structures or pillars accommodates relative misorientation and lattice mismatch by spontaneous production of dislocation arrays. Here, we show that the relative sliding of such an interface is accomplished by dislocation glide on the interfacial plane. An exception is the singular case where the lattices are perfectly matched across the sample dimension, in which case sliding is accompanied by motion of edge-nucleated defects. These are meso-scale analogues of molecular sliding friction mechanisms between crystalline interfaces. The dislocations, in addition to the long-range elastic energy associated with their Burgers vectors, also cause significant out-of-plane dilation, which props open the interface locally. For this reason, the sliding friction is strongly pressure dependent; it also depends on the relative orientation of the patterns. Sliding friction can be strongly enhanced compared with a control, showing that shape-complementary interfaces can be engineered for strongly enhanced pressure- and orientation-dependent frictional properties in soft solids.

How surface stress transforms surface profiles and adhesion of rough elastic bodies.

The surface of soft solids carries a surface stress that tends to flatten surface profiles. For example, surface features on a soft solid, fabricated by moulding against a stiff-patterned substrate, tend to flatten upon removal from the mould. In this work, we derive a transfer function in an explicit form that, given any initial surface profile, shows how to compute the shape of the corresponding flattened profile. We provide analytical results for several applications including flattening of one-dimensional and two-dimensional periodic structures, qualitative changes to the surface roughness spectrum, and how that strongly influences adhesion.

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.

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.

Fiber-Reinforced Viscoelastomers Show Extraordinary Crack Resistance That Exceeds Metals.

Soft fiber-reinforced polymers (FRPs), consisting of rubbery matrices and rigid fabrics, are widely utilized in industry because they possess high specific strength in tension while allowing flexural deformation under bending or twisting. Nevertheless, existing soft FRPs are relatively weak against crack propagation due to interfacial delamination, which substantially increases their risk of failure during use. In this work, a class of soft FRPs that possess high specific strength while simultaneously showing extraordinary crack resistance are developed. The strategy is to synthesize tough viscoelastic matrices from acrylate monomers in the presence of woven fabrics, which generates soft composites with a strong interface and interlocking structure. Such composites exhibit fracture energy, Γ, of up to 2500 kJ m-2 , exceeding the toughest existing materials. Experimental elucidation shows that the fracture energy obeys a simple relation, Γ = W · lT , where W is the volume-weighted average of work of extension at fracture of the two components and lT is the force transfer length that scales with the square root of fiber/matrix modulus ratio. Superior Γ is achieved through a combination of extraordinarily large lT (10-100 mm), resulting from the extremely high fiber/matrix modulus ratios (104 -105 ), and the maximized energy dissipation density, W. The elucidated quantitative relationship provides guidance toward the design of extremely tough soft composites.

The surface stress of biomedical silicones is a stimulant of cellular response.

Silicones are commonly used for lubrication of syringes, encapsulation of medical devices, and fabrication of surgical implants. While silicones are generally viewed as relatively inert to the cellular milieu, they can mediate a variety of inflammatory responses and other deleterious effects, but the mechanisms underlying the bioactivity of silicones remain unresolved. Here, we report that silicone liquids and gels have high surface stresses that can strongly resist deformation at cellular length scales. Biomedical silicones, including syringe lubricants and fillings from FDA-approved breast implants, readily adsorb matrix proteins and activate canonical rigidity sensing pathways through their surface stresses. In 3D culture models, liquid silicone droplets support robust cellular adhesion and the formation of multinucleated monocyte-derived cell masses that recapitulate phenotypic aspects of granuloma formation in the foreign body response. Together, our findings implicate surface stress as a cellular stimulant that should be considered in application of silicones for biomedical purposes.

Extreme cavity expansion in soft solids: Damage without fracture.

Cavitation is a common damage mechanism in soft solids. Here, we study this using a phase separation technique in stretched, elastic solids to controllably nucleate and grow small cavities by several orders of magnitude. The ability to make stable cavities of different sizes, as well as the huge range of accessible strains, allows us to systematically study the early stages of cavity expansion. Cavities grow in a scale-free manner, accompanied by irreversible bond breakage that is distributed around the growing cavity rather than being localized to a crack tip. Furthermore, cavities appear to grow at constant driving pressure. This has strong analogies with the plasticity that occurs surrounding a growing void in ductile metals. In particular, we find that, although elastomers are normally considered as brittle materials, small-scale cavity expansion is more like a ductile process. Our results have broad implications for understanding and controlling failure in soft solids.

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.

Coupled flow and deformation fields due to a line load on a poroelastic half space: effect of surface stress and surface bending.

In the past decade, many experiments have indicated that the surfaces of soft elastic solids can resist deformation by surface stresses. A common soft elastic solid is a hydrogel which consists of a polymer network swollen in water. Although experiments suggest that solvent flow in gels can be affected by surface stress, there is no theoretical analysis on this subject. Here we study the solvent flow near a line load acting on a linear poroelastic half space. The surface of this half space resists deformation by a constant, isotropic surface stress. It can also resist deformation by surface bending. The time-dependent displacement, stress and flow fields are determined using transform methods. Our solution indicates that the stress field underneath the line load is completely regularized by surface bending-it is bounded and continuous. For small surface bending stiffness, the line force is balanced by surface stresses; these forces form what is commonly known as 'Neumann's triangle'. We show that surface stress reduces local pore pressure and inhibits solvent flow. We use our line load solution to simulate the relaxation of the peak which is formed by applying and then removing a line force on the poroelastic half space.

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.