Elastohydrodynamic Lift at a Soft Wall

Lift at low Reynolds number

Lift forces are widespread in hydrodynamics. These are typically observed for big and fast objects and are often associated with a combination of fluid inertia (i.e. large Reynolds numbers) and specific symmetry-breaking mechanisms. In contrast, the properties of viscosity-dominated (i.e. low Reynolds numbers) flows make it more difficult for such lift forces to emerge. However, the inclusion of boundary effects qualitatively changes this picture. Indeed, in the context of soft and biological matter, recent studies have revealed the emergence of novel lift forces generated by boundary softness, flow gradients and/or surface charges. The aim of the present review is to gather and analyse this corpus of literature, in order to identify and unify the questioning within the associated communities, and pave the way towards future research.

Effect of fluid viscoelasticity, shear stress, and interface tension on the lift force in lubricated contacts

We consider a cylinder immersed in viscous fluid moving near a flat substrate covered by an incompressible viscoelastic fluid layer, and study the effect of the fluid viscoelasticity on the lift force exerted on the cylinder. The lift force is zero when the viscoelastic layer is not deformed, but becomes non-zero when it is deformed. We calculate the lift force by considering both the tangential stress and the normal stress applied at the surface of the viscoelastic layer. Our analysis indicates that as the layer changes from the elastic limit to the viscous limit, the lift force decreases with the decrease of the Deborah number (De). For small De, the effect of the layer elasticity is taken over by the surface tension and the lift force can become negative. We also show that the tangential stress and the interface slip velocity (the surface velocity relative to the substrate), which have been ignored in the previous analysis, give important contributions to the lift force. Especially for thin elastic layers, they give dominant contributions to the lift force.

The Endothelial Glycocalyx as a Target of Excess Soluble Fms-like Tyrosine Kinase-1

Soluble fms-like tyrosine kinase-1 (sFlt-1) is a secreted protein that binds heparan sulfate expressed on the endothelial glycocalyx (eGC). In this paper we analyze how excess sFlt-1 causes conformational changes in the eGC, leading to monocyte adhesion, a key event triggering vascular dysfunction. In vitro exposure of primary human umbilical vein endothelial cells to excess sFlt-1 decreased eGC height and increased stiffness as determined by atomic force microscopy (AFM). Yet, structural loss of the eGC components was not observed, as indicated by Ulex europaeus agglutinin I and wheat germ agglutinin staining. Moreover, the conformation observed under excess sFlt-1, a collapsed eGC, is flat and stiff with unchanged coverage and sustained content. Functionally, this conformation increased the endothelial adhesiveness to THP-1 monocytes by about 35%. Heparin blocked all these effects, but the vascular endothelial growth factor did not. In vivo administration of sFlt-1 in mice also resulted in the collapse of the eGC in isolated aorta analyzed ex vivo by AFM. Our findings show that excess sFlt-1 causes the collapse of the eGC and favors leukocyte adhesion. This study provides an additional mechanism of action by which sFlt-1 may cause endothelial dysfunction and injury.

Mechanical response of a thick poroelastic gel in contactless colloidal-probe rheology

When a rigid object approaches a soft material in a viscous fluid, hydrodynamic stresses arise in the lubricated contact region and deform the soft material. The elastic deformation modifies in turn the flow, hence generating a soft-lubrication coupling. Moreover, soft elastomers and gels are often porous. These materials may be filled with solvent or uncrosslinked polymer chains, and might be permeable to the surrounding fluid, which further complexifies the description. Here, we derive the point-force response of a semi-infinite and permeable poroelastic substrate. Then, we use this fundamental solution in order to address the specific poroelastic lubrication coupling associated with contactless colloidal-probe methods. In particular, we derive the conservative and dissipative components of the force associated with the oscillating vertical motion of a sphere close to the poroelastic substrate. Our results may be relevant for dynamic surface force apparatus and contactless colloidal-probe atomic force microscopy experiments on soft, living and/or fragile materials, such as swollen hydrogels and biological membranes.

Elastohydrodynamic Relaxation of Soft and Deformable Microchannels

Hydrodynamic flows in compliant channels are of great interest in physiology and microfluidics. In these situations, elastohydrodynamic coupling leads to (i) a nonlinear pressure-vs-flow-rate relation, strongly affecting the hydraulic resistance; and (ii), because of the compliance-enabled volume storage, a finite relaxation time under a stepwise change in pressure. This latter effect remains relatively unexplored, even while the timescale can vary over a decade in typical situations. In this study we provide time-resolved measurements of the relaxation dynamics for thin and soft, rectangular microfluidic channels. We describe our data using a perturbative lubrication approximation of the Stokes equation coupled to linear elasticity, while taking into account the effect of compliance and resistance of the entrance. The modeling allows us to completely describe all of the experimental results. Our Letter is relevant for any microfluidic scenario wherein a time-dependent driving is applied and provides a first step in the dynamical description of compliant channel networks.

Rotation-translation coupling of soft objects in lubricated contact

We study the coupling between rotation and translation of a submerged cylinder in lubricated contact with a soft elastic substrate. Using numerical solutions and asymptotic theory, we analyze the elastohydrodynamic problem over the entire range of substrate deformations relative to the thickness of the intervening fluid film. We find a strong coupling between the rotation and translation of the cylinder when the surface deformation of the substrate is comparable to the thickness of the lubricating fluid layer. In the limit of large deformations, we show that the bodies are in near-Hertzian contact and cylinder rolls without slip, reminiscent of dry frictional contact. When the surface deformation is small relative to the separation between the surfaces, the coupling persists but is weaker, and the rotation rate scales with the translation speed to the one-third power. We then show how the external application of a torque modifies these behaviors by generating different combinations of rotational and translational motions, including back-spinning and top-spinning states. We demonstrate that these behaviors are robust regardless of whether the elastic substrate is thick or thin relative to the length scales of the flow.

A quartz crystal microbalance method to quantify the size of hyaluronan and other glycosaminoglycans on surfaces

Hyaluronan (HA) is a major component of peri- and extra-cellular matrices and plays important roles in many biological processes such as cell adhesion, proliferation and migration. The abundance, size distribution and presentation of HA dictate its biological effects and are also useful indicators of pathologies and disease progression. Methods to assess the molecular mass of free-floating HA and other glycosaminoglycans (GAGs) are well established. In many biological and technological settings, however, GAGs are displayed on surfaces, and methods to obtain the size of surface-attached GAGs are lacking. Here, we present a method to size HA that is end-attached to surfaces. The method is based on the quartz crystal microbalance with dissipation monitoring (QCM-D) and exploits that the softness and thickness of films of grafted HA increase with HA size. These two quantities are sensitively reflected by the ratio of the dissipation shift (ΔD) and the negative frequency shift (- Δf) measured by QCM-D upon the formation of HA films. Using a series of size-defined HA preparations, ranging in size from ~ 2 kDa tetrasaccharides to ~ 1 MDa polysaccharides, we establish a monotonic yet non-linear standard curve of the ΔD/ - Δf ratio as a function of HA size, which reflects the distinct conformations adopted by grafted HA chains depending on their size and surface coverage. We demonstrate that the standard curve can be used to determine the mean size of HA, as well as other GAGs, such as chondroitin sulfate and heparan sulfate, of preparations of previously unknown size in the range from 1 to 500 kDa, with a resolution of better than 10%. For polydisperse samples, our analysis shows that the process of surface-grafting preferentially selects smaller GAG chains, and thus reduces the average size of GAGs that are immobilised on surfaces comparative to the original solution sample. Our results establish a quantitative method to size HA and other GAGs grafted on surfaces, and also highlight the importance of sizing GAGs directly on surfaces. The method should be useful for the development and quality control of GAG-based surface coatings in a wide range of research areas, from molecular interaction analysis to biomaterials coatings.

Elastohydrodynamic friction of robotic and human fingers on soft micropatterned substrates

Frictional sliding between patterned surfaces is of fundamental and practical importance in the haptic engineering of soft materials. In emerging applications such as remote surgery and soft robotics, thin fluid films between solid surfaces lead to a multiphysics coupling between solid deformation and fluid dissipation. Here, we report a scaling law that governs the peak friction values of elastohydrodynamic lubrication on patterned surfaces. These peaks, absent in smooth tribopairs, arise due to a separation of length scales in the lubricant flow. The framework is generated by varying the geometry, elasticity and fluid properties of soft tribopairs and measuring the lubricated friction with a triborheometer. The model correctly predicts the elastohydrodynamic lubrication friction of a bioinspired robotic fingertip and human fingers. Its broad applicability can inform the future design of robotic hands or grippers in realistic conditions, and open up new ways of encoding friction into haptic signals.

Biomechanical properties of endothelial glycocalyx: An imperfect pendulum

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This review seeks to fuse the discoveries in cell biology with mechanical engineering to produce a comprehensive biophysical model of endothelial glycocalyx.

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The aperiodic oscillatory motions of glycocalyx and cortical actin web underlie our prediction of two functional pacemakers and their participation in the outside-in signaling, the basis for mechanotransduction, and the dampening action of the inside-out signaling.

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Advancing an idea that the glycocalyx, plasma membrane, and cortical actin web represent a structure-functional unit and proposing the concept of tensegrity model.

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Presentation of our recent data suggesting that erythrocytes are gliding or havering and rotating over the surface of intact glycocalyx, whereas the rotational and hovering components of their passage along the capillaries are lost when glycocalyx of either is degraded.

Endothelial glycocalyx plays a crucial role in hemodynamics in health and disease, yet studying it is met by multiple technical hindrances. We attempted to outline our views on some biomechanical properties of endothelial glycocalyx, which are potentially amenable to mathematical modeling. We start with the null-hypothesis ascribing to glycocalyx the properties of a pendulum and reject this hypothesis on the grounds of multiple obstacles for pendulum behavior, such as rich decoration with flexible negatively charged side-chains, variable length and density, fluid fixation to the plasma membrane. We next analyze the current views on membrane attachments to the cortical actin web, its pulsatile contraction-relaxation cycles which rebound to the changes in tension of the plasma membrane. Based on this, we consider the outside-in signaling, the basis for mechanotransduction, and the dampening action of the inside-out signaling. The aperiodic oscillatory motions of glycocalyx and cortical actin web underlie our prediction of two functional pacemakers. We next advance an idea that the glycocalyx, plasma membrane, and cortical actin web represent a structure-functional unit and propose the concept of tensegrity model. Finally, we present our recent data suggesting that erythrocytes are gliding or hovering and rotating over the surface of intact glycocalyx, whereas the rotational and hovering components of their passage along the capillaries are lost when glycocalyx of either is degraded. These insights into the mechanics of endothelial glycocalyx motions may be of value in crosspollination between biomechanics, physiology, and pathophysiology for deeper appreciation of its rich untapped resources in health and pharmacotherapy in disease.

Cross talk between endothelial and red blood cell glycocalyces via near-field flow

Vascular endothelial cells and circulating red blood cell (RBC) surfaces are both covered by a layer of bushy glycocalyx. The interplay between these glycocalyx layers is hardly measurable and insufficiently understood. This study aims to investigate and qualify the possible interactions between the glycocalyces of RBCs and endothelial cells using mathematical modeling and numerical simulation. Dissipative particle dynamics (DPD) simulations are conducted to investigate the response of the endothelial glycocalyx (EG) to varying ambient conditions. A two-compartment model including EG and flow and a three-compartment model comprising EG, RBC glycocalyx, and flow are established. The two-compartment analysis shows that a relatively fast flow is associated with a predominantly bending motion of the EG, whereas oscillatory motions are predominant in a relatively slow flow. Results show that circulating RBCs cause the contactless deformation of EG. Its deformation is dependent on the chain layout, chain length, bending stiffness, RBC-to-EG distance, and RBC velocities. Specifically, shorter EG chains or RBC-to-EG distance leads to greater relative deflections of EG. Deformation of EG is enhanced when the EG chains are rarefied or RBCs move faster. The bending stiffness maintains stretching conformation of EG. Moreover, a compact EG chain layout and shedding EG chains disturb the neighboring flow field, causing disordered flow velocity distributions. In contrast, the movement of EG chains on RBC surfaces exerts a marginal driving force on RBCs. The DPD method is used for the first time, to our knowledge, in the three-compartment system to explore the cross talk between EG and RBC glycocalyx. This study suggests that RBCs drive the EG deformation via the near-field flow, whereas marginal propulsion of RBCs by the EG is observed. These new, to our knowledge, findings provide a new angle to understand the roles of glycocalyx in mechanotransduction and microvascular permeability and their perturbations under idealized pathophysiologic conditions associated with EG degradation.

Regimes of Soft Lubrication

Elastohydrodynamic lubrication, or simply soft lubrication, refers to the motion of deformable objects near a boundary lubricated by a fluid, and is one of the key physical mechanisms to minimise friction and wear in natural and engineered systems. Hence it is of particular interest to relate the thickness of the lubricant layer to the entrainment (sliding/rolling) velocity, the mechanical loading exerted onto the contacting elements, and properties of the elastic boundary. In this work we provide an overview of the various regimes of soft lubrication for two-dimensional cylinders in lubricated contact with compliant walls. We discuss the limits of small and large entrainment velocity, which is equivalent to large and small elastic deformations, as the cylinder moves near thick or thin elastic layers. The analysis focusses on thin elastic coatings, both compressible and incompressible, for which analytical scaling laws are not yet available in the regime of large deformations. By analysing the elastohydrodynamic boundary layers that appear at the edge of the contact, we establish the missing scaling laws - including prefactors. As such, we offer a rather complete overview of physically relevant limits of soft lubrication.

Validity of Winkler's mattress model for thin elastomeric layers: beyond Poisson's ratio

Winkler's mattress model is often used as a simplified model to understand how a thin elastic layer, such as a coating, deforms when subject to a distributed normal load: the deformation of the layer is assumed proportional to the applied normal load. This simplicity means that the Winkler model has found a wide range of applications from soft matter to geophysics. However, in the limit of an incompressible elastic layer the model predicts infinite resistance to deformation, and hence breaks down. Since many of the thin layers used in applications are elastomeric, and hence close to incompressible, we consider the question of when the Winkler model is appropriate for such layers. We formally derive a model that interpolates between the Winkler and incompressible limits for thin elastic layers, and illustrate this model by detailed consideration of two example problems: the point-indentation of a coated elastomeric layer and self-sustained lift in soft elastohydrodynamic lubrication. We find that the applicability (or otherwise) of the Winkler model is not determined by the value of the Poisson ratio alone, but by a compressibility parameter that combines the Poisson ratio with a measure of the layer's slenderness, which itself depends on the problem under consideration.

Rotation of a submerged finite cylinder moving down a soft incline

A submerged finite cylinder moving under its own weight along a soft incline lifts off and slides at a steady velocity while also spinning. Here, we experimentally quantify the steady spinning of the cylinder and show theoretically that it is due to a combination of an elastohydrodynamic torque generated by flow in the variable gap, and the viscous friction on the edges of the finite-length cylinder. The relative influence of the latter depends on the aspect ratio of the cylinder, the angle of the incline, and the deformability of the substrate, which we express in terms of a single scaled compliance parameter. By independently varying these quantities, we show that our experimental results are consistent with a transition from an edge-effect dominated regime for short cylinders to a gap-dominated elastohydrodynamic regime when the cylinder is very long.

Direct Measurement of the Elastohydrodynamic Lift Force at the Nanoscale

We present the first direct measurement of the elastohydrodynamic lift force acting on a sphere moving within a viscous liquid, near and along a soft substrate under nanometric confinement. Using atomic force microscopy, the lift force is probed as a function of the gap size, for various driving velocities, viscosities, and stiffnesses. The force increases as the gap is reduced and shows a saturation at small gap. The results are in excellent agreement with scaling arguments and a quantitative model developed from the soft lubrication theory, in linear elasticity, and for small compliances. For larger compliances, or equivalently for smaller confinement length scales, an empirical scaling law for the observed saturation of the lift force is given and discussed.

Compliant Surfaces under Shear: Elastohydrodynamic Lift Force

In this work, we have investigated the behavior under shear and compression of mica surfaces coated with poly(N-isopropylacrylamide) cationic microgels. We have observed the emergence of velocity dependent, shear-induced normal forces, which can be large enough to entrain a fluid film that separates the surfaces out of contact, driving the dynamic system from conditions of boundary to hydrodynamic lubrication. By implementing a feedback-loop control on the surface separation, we were able to quantify the magnitude of the lift force as a function of the surface separation and driving speed. Our results illustrate how elastohydrodynamic effects can play an important role in the lubrication of compliant surfaces, providing pathways for control of friction and wear.

Migration velocity of red blood cells in microchannels

The lateral migration of red blood cells (RBCs) in confined channel flows is an important ingredient of microcirculatory hydrodynamics and is involved in the development of a cell free layer near vessel walls and influences the distribution of RBCs in networks. It is also relevant to a number of lab-on-chip applications. This migration is a consequence of their deformability and is due to the combined effects of hydrodynamic wall repulsion and the curvature of the fluid velocity profile. We performed microfluidic experiments with dilute suspensions of RBCs in which the trajectories and migration away from the channel wall are analyzed to extract the mean behavior, from which we propose a generic scaling law for the transverse migration velocity valid in a whole range of parameters relevant to microcirculatory and practical situations. Experiments with RBCs of different mechanical properties (separated by density gradient sedimentation or fixed with glutaraldehyde) show the influence of this parameter which can induce significant dispersion of the trajectories.