Stick slip contact mechanics between dissimilar materials: effect of charging and large friction

Measurements of the contact radius as a function of applied force between a mica surface and a silica surface (mica/silica) in air are reported. The load/unload results show that the contact radius generally increases with applied force. Because of the presence of charging due to contact electrification, both a short-range van der Waals adhesion force and longer-range electrostatic adhesive interaction contribute to the measured force. The results indicate that approximately 20% of the pull-off force is due to van der Waals forces. The contact radius versus applied force results can be fit to Johnson-Kendall-Roberts (JKR) theory by considering that only the short-range van der Waals forces contribute to the work of adhesion and subtracting a constant longer-range electrostatic force. Also, an additional and unexpected step function is superimposed on the contact radius versus applied force curve. Thus, the contact diameter increases in a stepped dependence with increasing force. The stepped contact behavior is seen only for increasing force and is not observed when symmetric mica/mica or silica/silica contacts are measured. In humid conditions, the contact diameter of the mica/silica contact increases monotonically with applied force. Friction forces between the surfaces are also measured and the shear stress of a mica/silica interface is 100 times greater than the shear stress of a mica/mica interface. This large shear stress retards the increase in contact area as the force is increased and leads to the observed stepped contact mechanics behavior.     

General Equations Describing Elastic Indentation Depth and Normal Contact Stiffness versus Load

Continuum mechanics models describing the contact between two adhesive elastic spheres, such as the JKR and DMT models, provide a relationship between the elastic indentation depth and the normal load, but the general intermediate case between these two limiting cases requires a more complex analysis. The Maugis-Dugdale theory gives analytical solutions, but they are difficult to use when comparing to experimental data such as those obtained by scanning force microscopy. In this paper we propose a generalized equation between elastic indentation depth and load that approximates Maugis' solution very closely. If the normal contact stiffness can be described as the force gradient, that is the case of the force modulation microcopy, then a generalized equation between normal contact stiffness and load can be deduced. Both general equations can be easily fit to experimental data, and then interfacial energy and elastic modulus of the contact can be determined if the radius of the indenting sphere is known. Copyright 2000 Academic Press.     

Effect of contact geometry on the pull-off force evaluated under high-vacuum and humid atmospheric conditions

The effect of condensed water on pull-off forces under high vacuum (HV) and 0 to 83% relative humidity (RH) N2 atmospheric conditions was evaluated for different contact geometries using atomic force microscopy (AFM). The pull-off force was measured using two types of contact geometry: contact between hemispherical asperities and a flat silicon probe on an AFM cantilever (called a spherical-flat contact) and between a flat silicon substrate and a flat nickel probe on an AFM cantilever (called a flat-flat contact). The hemispherical asperities were fabricated using a focused ion beam (FIB) system, and each peak had a radius of curvature of between 70 and 610 nm. The flat nickel probe was fabricated by friction-induced wear. Measurement results showed that for the spherical-flat contact the pull-off force was proportional to the radius of curvature of the asperity peak and was slightly lower in HV than in humid 14% RH N2. For the flat-flat contact in HV, with increasing contact time, the pull-off force increased in HV but decreased in humid 62 and 83% RH N2. The pull-off force in HV was lower than that in humid N2 when the contact time was less than 10 s but was higher when the contact time was longer than 30 s. The estimated adhesion force based on the Laplace pressure from the contact geometry agreed reasonably well with the measured pull-off force.     

Using the Dugdale approximation to match a specific interaction in the adhesive contact of elastic objects

In the Maugis-Dugdale model of the adhesive contact of elastic spheres, the step cohesive stress sigma(0) is arbitrarily chosen to be the theoretical stress sigma(th) to match that of the Lennard-Jones potential. An alternative and more reasonable model is proposed in this paper. The Maugis model is first extended to that of arbitrary axisymmetric elastic objects with an arbitrary surface adhesive interaction and then applied to the case of a power-law shape function and a step cohesive stress. A continuous transition is found in the extended Maugis-Dugdale model for an arbitrary shape index n. A three-dimensional Johnson-Greenwood adhesion map is constructed. A relation of the identical pull-off force at the rigid limit is required for the approximate and exact models. With this requirement, the stress sigma(0) is found to be k(n)Deltagamma/z(0), where k(n) is a coefficient, Deltagamma the work of adhesion, and z(0) the equilibrium separation. Hence we have sigma(0) = 0.588Deltagamma/z(0), especially for n=2. The prediction of the pull-off forces using this new value shows surprisingly better agreement with the Muller-Yushchenko-Derjaguin transition than that using sigma(th) = 1.026Deltagamma/z(0), and this is true for other values of shape index n.     

Adhesion and debonding of soft elastic films: crack patterns, metastable pathways, and forces

We study the phenomenon of debonding in a thin soft elastic film sandwiched between two rigid plates as one of the plates is brought into intimate contact and then pulled away from contact proximity by application of a normal force. Nonlinear simulations based on minimization of total energy (composed of stabilizing elastic strain energy and destabilizing adhesive interaction energy) are employed to address the problems of contact hysteresis, cavitation, crack morphology, variation of contact area, snap-off distance, pull-off force, work done, and energy loss. Below a critical distance (d(c)) upon approach, simulations show the formation of columnar structures and nonrandom, regularly arranged nanocavities at the soft interface at a length scale of approximately 3h (h being the thickness of the film). The persistence of such instability upon withdrawal (distance >d(c)) indicates a contact hysteresis, which is caused by an energy barrier that separates the metastable states of the patterned configuration and the global minimum state of the flat film. The energy and the pull-off force are found to be nonequilibrium and nonunique properties depending on the initial contact, defects, noise, etc. Three broad pathways of debonding leading to adhesive failure of the interface, depending on the stiffness of the film, step size of withdrawal, and the imposed noise, are identified: a catastrophic column collapse mode, a peeling mode involving a continuous decrease in the contact area, and a column splitting mode. The first two modes are caused by a very high stress concentration near the cavity edges. These metastable patterned configurations engender pull-off forces that are orders of magnitude smaller than that required to separate two flat surfaces from contact.     

General solution to two-dimensional nonslipping JKR model with a pulling force in an arbitrary direction

In this paper, a generalized JKR model is investigated, in which an elastic cylinder adhesively contacts with an elastic half space and the contact region is assumed to be perfect bonding. An external pulling force is acted on the cylinder in an arbitrary direction. The contact area changes during the pull-off process, which can be predicted using the dynamic Griffith energy balance criterion as the contact edge shifts. Full coupled solution with an oscillatory singularity is obtained and analyzed by numerical calculations. The effect of Dundurs' parameter on the pull-off process is analyzed, which shows that a nonoscillatory solution can approximate the general one under some conditions, i.e., larger pulling angle (pi/2 is the maximum value), smaller a/R or larger nondimensional parameter value of Deltagamma/E*R. Relations among the contact half width, the external pulling force and the pulling angle are used to determine the pull-off force and pull-off contact half width explicitly. All the results in the present paper as basic solutions are helpful and applicable for experimenters and engineers.     

Volume of a nanoscale water bridge

Water bridges formed through capillary condensation at nanoscale contacts first stretch and then break during contact rupture. Atomic force microscopy (AFM) pull-off experiments performed in air with hydrophilic tips and samples show that stretched nanoscopic water bridges are in mechanical equilibrium with the external pull-off force acting at the contact but not in thermodynamic equilibrium with the water vapor in air. The experimental findings are explained by a theoretical model that considers constant water volume and decrease of water meniscus curvature during meniscus stretching. The model predicts that the water bridge breakup distance will be roughly equal to the cubic root of the water bridge volume. A thermodynamic instability was noticed for large water bridges formed at the contact of a blunt AFM tip (curvature radius of 400 nm) with a flat sample. In this case, experiments showed rise and stabilization of the volume of the water at the contact in about 1 s. For sharp AFM tips (curvature radius below 50 nm), the experiments indicated that formation of stable water bridges occurs in a much shorter time (below 5 ms).     

Origin of adhesion in humid air

The origin of adhesion in humid air is investigated by pull-off force measurements between nanoscale contacts using atomic force microscopes in controlled environments from ultrahigh vacuum through various humidity conditions to water. An equivalent work of adhesion (WOA) model with a simplified interface stress distribution is developed, combining the effects of screened van der Waals and meniscus forces, which describes adhesion in humid air and which self-consistently treats the contact stress and deformation. Although the pull-off force is found to vary significantly with humidity, the equivalent WOA is found to be invariant. Increasing humidity alters the nature of the surface adhesion from a compliant contact with a localized, intense meniscus force to a stiff contact with an extended, weak meniscus force.     

Adhesive Contact Deformation of a Single Microelastomeric Sphere

This paper reports on an experimental study of the adhesive contact of a single microscopic (about 300 μm) elastomer sphere compressed between two smooth parallel glass platens at small imposed deformations. An experimental arrangement that allows the simultaneous measurement of the compressive displacements and the reaction forces is described. A number of interesting phenomena, including the pull-off separation and the “jump” contact phenomena of the microsphere and the moving platen supported by a cantilever, are shown in the experimental force-displacement curve of a loading and unloading cycle. The pull-off forces are demonstrated to not depend upon the applied dimensionless approach (compressive displacement/initial particle diameter), while they increase with the increasing rate at which the interfaces are separated. The predictions of an established contact mechanical adhesive theory, Johnson–Kendall–Roberts (JKR) theory, in which the influence of the surface energy on the contact has been taken into account, are in good agreement with these experimental results. An application of the JKR analysis to the pull-off force provides a reasonable estimate of the interfacial free energy of the contact.     

Determination of solid surface tension from particle-substrate pull-off forces measured with the atomic force microscope

Atomic force microscopy (AFM) is capable of solid surface characterization at the microscopic and submicroscopic scales. It can also be used for the determination of surface tension of solids (gamma) from pull-off force (F) measurements, followed by analysis of the measured F values using contact mechanics theoretical models. Although a majority of the literature gamma results was obtained using either Johnson-Kendall-Roberts (JKR) or Derjaguin-Muller-Toporov (DMT) models, re-analysis of the published experimental data presented in this paper indicates that these models are regularly misused. Additional complication in determination of gamma values using the AFM technique is that the measured pull-off forces have poor reproducibility. Reproducible and meaningful F values can be obtained with strict control over AFM experimental conditions during the pull-off force measurements (low humidity level, controlled and known loads) for high quality substrates and probes (surfaces should be free of heterogeneity, roughness, and contamination). Any probe or substrate imperfections complicate the interpretation of experimental results and often reduce the quality of the generated data. In this review, surface imperfection in terms of roughness and heterogeneity that influence the pull-off force are analyzed based upon the contact mechanics models. Simple correlations are proposed that could guide in selection and preparation of AFM probes and substrates for gamma determination and selection of loading conditions during the pull-off force measurements. Finally, the possibility of AFM measurements of solid surface tension using materials with rough surfaces is discussed.     

Simulations of nanotribology with realistic probe tip models

We present the results of massively parallel molecular dynamics simulations aimed at understanding the nanotribological properties of alkylsilane self-assembled monolayers (SAMs) on amorphous silica. In contrast to studies with opposing flat plates, as found in the bulk of the simulation literature, we use a model system with a realistic AFM tip (radius of curvature ranging from 3 to 30 nm) in contact with a SAM-coated silica substrate. We compare the differences in response between systems in which chains are fully physisorbed, fully chemisorbed, and systems with a mixture of the two. Our results demonstrate that the ubiquitous JKR and DMT models do not accurately describe the contact mechanics of these systems. In shear simulations, we find that the chain length has minimal effects on both the friction force and coefficient. The tip radius affects the friction force only (i.e., the coefficient is unchanged) by a constant shift in magnitude due to the increase in pull-off force with increasing radius. We also find that at extremely low loads, on the order of 10 nN, shearing from the tip causes damage to the physisorbed monolayers by removal of molecules.     

Effects of contact geometry on pull-off force measurements with a colloidal probe

This paper examines the effects of contact geometry on the pull-off (adhesion) force between a glass sphere (colloidal probe) and a silicon wafer in an environment with controlled relative humidity. An atomic force microscope is used to measure the pull-off force between the colloidal probe and the sample mounted at different tilt angles. The results show that the measured pull-off force is very sensitive to the tilt angle. Through the use of a newly developed direct scanning method, the exact contact geometry is determined for the zero-tilt angle case. The obtained digital image is then rotated to determine the contact geometry for the cases with other tilt angles. A detailed examination of the contact geometry, along with a magnitude analysis of the capillary force, suggests that the adhesion is most likely dominated by the capillary force from the meniscus formed between the probe and the sample. The strong dependence of the adhesion on the tilt angle may result from the change of meniscus dimensions associated with the probe-sample separation, which in turn is controlled by the highest peak on the probe sphere. Our observation emphasizes the combined role of microsurface shape near the contact and nanoroughness within the contact in determining the colloidal probe pull-off force and also microadhesion force in general.     

The wall and multivalent counterion effects on the electrostatic force between like-charged spherical particles confined in a charged pore

The effect of wall confinement (wall charge and wall-sphere separation distance) on the electrostatic force between two charged spheres confined in a long charged pore in symmetric and asymmetric electrolytes have been quantified by solving the nonlinear Poisson-Boltzmann equation (PBE), using adaptive finite elements combined with error minimization techniques. The computed force indicated the strong effect of the wall potential on the reduction of the repulsive force for all type of electrolytes. The influence of the wall effect was reduced when the valence of the electrolyte was increased. A significant reduction in the repulsive force between the two spheres was also observed when the distance between the pore wall and the sphere surface was reduced. A smaller long-range repulsive interaction was observed between spheres when the solutions contained multivalent counterions as compared with a monovalent solution. However, at short ranges of separation distances multivalent counterions increase the electrostatic repulsive force between the spheres. The effect of the dimensionless radius of the spheres on the electrostatic force between them has been determined and a significant reduction observed as the dimensionless radius was reduced.     

An improved meniscus surface model for contacting rough surfaces

Based on the Extended-Maugis-Dugdale (EMD) elastic theory, a single asperity capillary meniscus model considering asperity deformation due to both contact and adhesive forces was developed. Specifically included in the single asperity meniscus model was the solid surface interaction inside the contact area. Subsequently, the single asperity model was coupled with a statistical roughness surface model to develop an improved meniscus surface model applicable to a wide-range of humidity levels and adhesion parameter values. Simulations were performed using typical surfaces found in microelectromechanical systems (MEMS) and magnetic storage hard disk drives to examine the effects of surface roughness and relative humidity. It was found that smoother surfaces give rise to higher adhesive and pull-off forces, and at higher relative humidity levels, the capillary force governs the adhesive behavior. As the humidity decreases, the solid surface interaction increases and needs to be included in the total meniscus adhesion. By integrating the adhesive force-displacement curves, the adhesion energy per unit area was calculated for MEMS surfaces and favorably compared with reported experimental data.     

Mechanics of contact and adhesion between viscoelastic spheres: An analysis of hysteresis during loading and unloading

Detailed finite element simulations are carried out to study the adhesive contact of viscoelastic spheres. The spheres are brought into contact by a compressive force that increases at a constant rate. Upon reaching a maximum load, the spheres are unloaded until they separate. We studied in detail the effect of loading and unloading rates on hysteresis and on the pull‐off force for a standard viscoelastic solid. The surface interaction is modeled by the Dugdale–Barenblatt model. Numerical results are compared with analytical models for bonding and debonding, including a recent theory proposed by Johnson. There is excellent agreement between analytical and finite element results for the bonding phase. However, for the debonding phase, current analytical models break down unless the loading and unloading rates are slow in comparison with the material relaxation time. Based on the finite element results, a simple approximate analytical model is proposed to quantify adhesive contact in the debonding phase. We also examine the dependence of hysteresis on interfacial parameters such as the cohesive strength and the intrinsic work of adhesion. Our results show that viscoelastic adhesive contact depends on the details of the surface interaction and cannot be determined solely by the work of adhesion. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 772–793, 2002     

Oscillatory loading of a viscoelastic adhesive contact

The cycle of loading and unloading of a spherically-tipped probe against an adhesive, viscoelastic plane specimen is studied by numerical integration of the relations between crack speed and apparent surface energy previously found for a linear 3-element viscoelastic solid with a Maugis-Dugdale law of force across the crack. It is found that even when the rate of loading is so slow that the loading and unloading curves almost coincide, suggesting purely elastic behaviour, the pull-off force can be appreciably greater than the elastic (JKR) value. When the normal force is modulated with a small amplitude sinusoidal variation during unloading--in order to find the contact stiffness--the contact radius barely changes, and the stiffness is close to that for a rigid flat punch instead of having the expected JKR value.     

Forces between polyethylene surfaces in oxyethylene dodecyl ether solutions as influenced by the number of oxyethylene groups

The atomic force microscopy (AFM) colloidal probe technique was used to study the effect of oxyethylene dodecyl ethers, C12En (n = 1-7), on interactions between hydrophobic polyethylene (PE) surfaces in aqueous solutions. Long-range (colloidal) and contact (pull-off) forces were measured between 10 to 20 microm PE spheres and a flat PE surface at concentrations of surfactant of 1 x 10(-6) and 1 x 10(-4) M. The surface tension of the surfactant solutions and contact angles at PE surfaces were also studied. The influence of the number of oxyethylene groups in the surfactant molecule was examined. Initially, long-range attractive (hydrophobic) forces between the PE surfaces were observed that decreased in range and magnitude with an increase in the number of oxyethylene groups in 1 x 10(-4) M solutions. Above four oxyethylene groups per molecule, repulsive forces were observed. The measured pull-off force between PE surfaces decreased monotonically from approximately 500 mJ/m2 for C12E1 to 150 mJ/m2 for C12E7. The interfacial energy was calculated on the basis of the JKR model, taking into account long-range forces operating outside the contact area. The interfacial energies decreased from 43-47 mJ/m2 for PE-water and PE-C12E1 (1 x 10(-4) M) interfaces to approximately 18 mJ/m2 for PE-C12E7 (1 x 10(-4) M). The interfacial energy was also calculated from measured contact angles and surface tensions using Neumann's equation of state and Young's equation. A similar relationship between interfacial energy and the number of oxyethylene groups was observed on the basis of contact and surface tension measurements. However, interfacial energy values were smaller, within 15-20 mJ/m2, than those calculated from AFM pull-off force measurements.     

Adhesive contact based on the Lennard-Jones potential: a correction to the value of the equilibrium distance as used in the potential

A Lennard-Jones type surface law is commonly used in adhesive contact modeling; however, one of its parameters, namely the equilibrium distance z0, is not well defined. In this paper, a self-consistent method is used to derive the Lennard-Jones surface law from the interatomic Lennard-Jones potential. The parameters of the surface law are directly related to the material lattice parameter and surface energy, and the equilibrium distance z0 values are obtained for various materials. The effect of using the z0 proposed in the present work is demonstrated via the study of adhesive contact behavior for a single sphere and a flat surface, as well as the contact between planar rough surfaces. For pull-off force prediction of the contact between a single sphere and a flat surface, the error of using the z0 suggested in previous studies could be as large as 10% at intermediate ranges of a dimensionless adhesion parameter. For the contact between planar rough surfaces, the error of using the previously proposed z0 is larger for smoother cases, and the prediction of pull-off force could be different by as much as a factor of 5.