Mark O. Robbins

The breakdown of continuum models for mechanical contacts

Forces acting within the area of atomic contact between surfaces play a central role in friction and adhesion. Such forces are traditionally calculated using continuum contact mechanics, which is known to break down as the contact radius approaches atomic dimensions. Yet contact mechanics is being applied at ever smaller lengths, driven by interest in shrinking devices to nanometre scales, creating nanostructured materials with optimized mechanical properties, and understanding the molecular origins of macroscopic friction and adhesion. Here we use molecular simulations to test the limits of contact mechanics under ideal conditions. Our findings indicate that atomic discreteness within the bulk of the solids does not have a significant effect, but that the atomic-scale surface roughness that is always produced by discrete atoms leads to dramatic deviations from continuum theory. Contact areas and stresses may be changed by a factor of two, whereas friction and lateral contact stiffness change by an order of magnitude. These variations are likely to affect continuum predictions for many macroscopic rough surfaces, where studies show that the total contact area is broken up into many separate regions with very small mean radius.

Origin of stick-slip motion in boundary lubrication

Molecular dynamics simulations of atomically thin, fluid films confined between two solid plates are described. For a broad range of parameters, a generic stick-slip motion is observed, consistent with the results of recent boundary lubrication experiments. Static plates induce crystalline order in the film. Stick-slip motion involves periodic shear-melting transitions and recrystllization of the film. Uniform motion occurs at high velocities where the film no longer has time to order. These results indicate that the origin of stick-slip motion is thermodynamic instability of the sliding state, rather than a dynamic instability as usually assumed.

A continuum and molecular dynamics hybrid method for micro- and nano-fluid flow

A hybrid multiscale method is developed for simulating micro- and nano-scale fluid flows. The continuum Navier–Stokes equation is used in one flow region and atomistic molecular dynamics in another. The spatial coupling between continuum equations and molecular dynamics is achieved through constrained dynamics in an overlap region. The proposed multiscale method is used to simulate sudden-start Couette flow and channel flow with nano-scale rough walls, showing quantitative agreement with results from analytical solutions and full molecular dynamics simulations for different parameter regimes. Potential applications of the proposed multiscale method are discussed.

Finite-element analysis of contact between elastic self-affine surfaces.

Finite-element methods are used to study nonadhesive, frictionless contact between elastic solids with self-affine surfaces. We find that the total contact area rises linearly with the load at small loads. The mean pressure in the contact regions is independent of load and proportional to the root-mean-square slope of the surface. The constant of proportionality is nearly independent of the Poisson ratio and roughness exponent and lies between previous analytic predictions. The contact morphology is also analyzed. Connected contact regions have a fractal area and perimeter. The probability of finding a cluster of area a(c) drops as a(-tau )(c ) where tau increases with a decrease in roughness exponent. The distribution of pressures shows an exponential tail that is also found in many jammed systems. These results are contrasted to simpler models and experiments.

Simple microscopic theory of Amontons's laws for static friction.

A microscopic theory for the ubiquitous phenomenon of static friction is presented. Interactions between two surfaces are modeled by an energy penalty that increases exponentially with the degree of surface overlap. The resulting static friction is proportional to load, in accordance with Amontonss laws. However, the friction coefficient between bare surfaces vanishes as the area of individual contacts grows, except in the rare case of commensurate surfaces. An area independent friction coefficient is obtained for any surface geometry when an adsorbed layer of mobile atoms is introduced between the surfaces. The predictions from our simple analytic model are confirmed by detailed molecular dynamics simulations.

Molecular origins of friction: the force on adsorbed layers.

Simulations and perturbation theory are used to study the molecular origins of friction in an ideal model system, a layer of adsorbed molecules sliding over a substrate. These calculations reproduce several surprising features of experimental results. In most cases, the frictional force on a solid monolayer has a different form from that observed between macroscopic solids. No threshold force or static friction is needed to initiate sliding; instead, the velocity is proportional to the force. As in experiments, incommensurate solid layers actually slide more readily than fluid layers. A comparison of experiment, simulation, and analytic results shows that dissipation arises from anharmonic coupling between phonon modes and substrate-induced deformations in the adsorbate.

Cracks and crazes: on calculating the macroscopic fracture energy of glassy polymers from molecular simulations.

We combine molecular dynamics simulations of deformation at the submicron scale with a simple continuum fracture mechanics model for the onset of crack propagation to calculate the macroscopic fracture energy of amorphous glassy polymers. Key ingredients in this multiscale approach are the elastic properties of polymer crazes and the stress at which craze fibrils fail through chain pullout or scission. Our results are in quantitative agreement with dimensionless ratios that describe experimental polymers and their variation with temperature, polymer length, and polymer rigidity.

Shear yielding of amorphous glassy solids: Effect of temperature and strain rate

We study shear yielding and steady state flow of glassy materials with molecular dynamics simulations of two standard models: amorphous polymers and bidisperse Lennard-Jones glasses. For a fixed strain rate, the maximum shear yield stress and the steady state flow stress in simple shear both drop linearly with increasing temperature. The dependence on strain rate can be described by either a logarithm or a power law added to a constant. In marked contrast to predictions of traditional thermal activation models, the rate dependence is nearly independent of temperature. The relation to more recent models of plastic deformation and glassy rheology is discussed, and the dynamics of particles and stress in small regions is examined in light of these findings.

Contact between rough surfaces and a criterion for macroscopic adhesion

Significance Macroscopic objects rarely stick together, yet the van der Waals interactions between surface atoms produce attractive pressures that are orders of magnitude larger than atmospheric pressure. This “adhesion paradox” has been linked to surface roughness, which reduces the area of intimate atomic contact to summits on the rough landscape. This paper presents a parameter-free theory that captures the interplay between elasticity, interatomic attraction, and surface roughness. It predicts how adhesion changes contact area and when surfaces are sticky. The results offer a simple explanation for why tape sticks to our desktops but a sheet of paper does not, and may aid in the design of adhesives and in engineering surface roughness to enhance or eliminate adhesion. At the molecular scale, there are strong attractive interactions between surfaces, yet few macroscopic surfaces are sticky. Extensive simulations of contact by adhesive surfaces with roughness on nanometer to micrometer scales are used to determine how roughness reduces the area where atoms contact and thus weakens adhesion. The material properties, adhesive strength, and roughness parameters are varied by orders of magnitude. In all cases, the area of atomic contact is initially proportional to the load. The prefactor rises linearly with adhesive strength for weak attractions. Above a threshold adhesive strength, the prefactor changes sign, the surfaces become sticky, and a finite force is required to separate them. A parameter-free analytic theory is presented that describes changes in these numerical results over up to five orders of magnitude in load. It relates the threshold adhesive strength to roughness and material properties, explaining why most macroscopic surfaces do not stick. The numerical results are qualitatively and quantitatively inconsistent with classical theories based on the Greenwood−Williamson approach that neglect the range of adhesion and do not include asperity interactions.At the molecular scale there are strong attractive interactions between surfaces, yet few macroscopic surfaces are sticky. Extensive simulations of contact by adhesive surfaces with roughness on nanometer to micrometer scales are used to determine how roughness reduces the area where atoms contact and thus weakens adhesion. The material properties, adhesive strength and roughness parameters are varied by orders of magnitude. In all cases the area of atomic contact rises linearly with load, and the prefactor rises linearly with adhesive strength for weak interactions. Above a threshold adhesive strength, the prefactor changes sign, the surfaces become sticky and a finite force is required to separate them. A parameterfree analytic theory is presented that describes changes in these numerical results over up to five orders of magnitude in load. It relates the threshold strength to roughness and material properties, explaining why most macroscopic surfaces do not stick. The numerical results are qualitatively and quantitatively inconsistent with classical theories based on the Greenwood-Williamson approach that neglect the range of adhesion and do not include asperity interactions. 1 ar X iv :1 31 1. 11 78 v1 [ co nd -m at .m tr lsc i] 5 N ov 2 01 3 Surfaces are adhesive or “sticky” if breaking contact requires a finite force. Few of the surfaces we encounter are sticky even though almost all are pulled together by van der Waals interactions at atomic scales.1 Gecko setae2,3 and engineered adhesives4 use this ubiquitous attraction to achieve pull off forces per unit area that are orders of magnitude larger than atmospheric pressure, and our world would come to a halt if these pressures operated on most macroscopic surfaces. The discrepancy between atomic and macroscopic forces has been dubbed the adhesion paradox.5 Experiments show that a key factor underlying this paradox is surface roughness, which reduces the fraction of surface atoms that are close enough to adhere.5–8 Quantitative calculations of this reduction are extremely challenging because of the complex topography of typical surfaces, which have bumps on top of bumps on a wide range of scales.9,10 In many cases they can be described as self-affine fractals from a lower wavelength λs of order nanometers to an upper wavelength λL in the micrometer to millimeter range.7,11 Here, we use an efficient Green’s function approach to calculate adhesive contact of surfaces with roughness from subnanometer to micrometer scales. Numerical results for a wide range of surfaces, adhesive interactions and material properties are presented and used to develop a simple, parameter-free equation that predicts the effect of adhesion on contact. The traditional Greenwood-Williamson (GW)12 approach for calculating contact of rough surfaces approximates their complex topography by a set of spherical asperities of radius R whose height distribution is determined from self-affine scaling. The long-range elastic interactions between different asperities are neglected. This approach is analytically tractable and provided a simple explanation for the observation that the area of contact between nonadhesive elastic surfaces is proportional to the normal force or load pushing them together. Later generalizations6,13 considered the effect of adhesion between surfaces and found that the key parameter was the ratio of the root mean squared (rms) height variation hrms to the normal displacement δc of a single asperity due to adhesion. If the work of adhesion gained per unit area of contact is w, then δ c = (3/4) 3R(πw/E∗)2 with contact modulus14 E∗ = E/(1 − ν) for an isotropic material with Young’s modulus E and Poisson ratio ν. GW based adhesion models6,13 predict that the force needed to separate surfaces drops rapidly as hrms/δc increases and is negligible for hrms/δc > 3. In the last decade, Persson has developed a scaling theory that includes an approximate treatment of asperity interactions.15,16 At the same time, large scale numerical calculations of contact between rough surfaces have become feasible.17–20 Both approaches reveal limitations in the GW treatment of nonadhesive surfaces. For example, the definition of R is ambiguous,22 the predicted

Yield conditions for deformation of amorphous polymer glasses

Shear yielding of glassy polymers is usually described in terms of the pressure-dependent Tresca or von Mises yield criteria. We test these criteria against molecular dynamics simulations of deformation in amorphous polymer glasses under triaxial loading conditions that are difficult to realize in experiments. Difficulties and ambiguities in extending several standard definitions of the yield point to triaxial loads are described. Two definitions, the maximum and offset octahedral stresses, are then used to evaluate the yield stress for a wide range of model parameters. In all cases, the onset of shear is consistent with the pressure-modified von Mises criterion, and the pressure coefficient is nearly independent of many parameters. Under triaxial tensile loading, the mode of failure changes to cavitation, and the von Mises criterion no longer applies.