# Adhesion-Induced Instability in Asperities

## Abstract

Adhesive forces between two approaching asperities will deform the asperities, and under certain conditions this will result in a sudden runaway deformations leading to a jump-to-contact instability. We present finite element-based numerical studies on adhesion-induced deformation and instability in asperities. We consider the adhesive force acting on an asperity, when it is brought near a rigid half-space, due to van der Waals interaction between the asperity and the half-space. The adhesive force is considered to be distributed over the volume of the asperity (body force), thus resulting in more realistic simulations for the length scales considered. Iteration scheme based on a “residual stress update” algorithm is used to capture the effect of deformation on the adhesion force, and thereby the equilibrium configuration and the corresponding force. The numerical results are compared with the previous approximate analytical solutions for adhesion force, deformation of the asperity and adhesion-induced mechanical instability (jump-to-contact). It is observed that the instability can occur at separations much higher, and could possibly explain the higher value of instability separation observed in experiments. The stresses in asperities, particularly in case of small ones, are found to be high enough to cause yielding before jump-to-contact. The effect of roughness is considered by modeling a spherical protrusion on the hemispherical asperity. This small-scale roughness at the tip of the asperities is found to control the deformation behavior at small separations, and hence are important in determining the friction and wear due to the jump-to-contact instability.

### Keywords

Surface roughness Adhesion Nanotribology Jump-to-contact instability## 1 Introduction

Adhesion-induced mechanical instability or “jump-to-contact instability” [1, 2, 3, 4, 5] of the asperities plays an important role in determining the friction [6, 7] and wear characteristics of the sliding surfaces [8]. When two nominally contacting surfaces slide past each other, the asperities on a surface make and break contact intermittently with the asperities of the counter-surface. As asperities approach each other, they will encounter an attractive force resulting from van der Waals force, and hence undergo deformation. This results in a reduced separation between the interacting asperities, and leads to increased force, causing still higher deformations. At large enough separations, equilibrium would be attained when the adhesive force and the final configuration are consistent with each other. However, when the resistance to deformation, determined by the geometry and the elastic moduli of the asperity, is less than the adhesive force gradient with respect to the separation, there will be a mechanical instability resulting in a sudden jump-to-contact [1]. The energy loss associated with this process will contribute toward the macroscopic friction between the bodies [6, 7], and the sudden jump-to-contact resulting from this instability could cause localized wear [8]. In this paper, using a finite element (FE)-based numerical model, we study the deformation, stresses and the resulting instability in asperities due to this adhesive force, and the effect of roughness on the instability.

Bradley [9] developed an expression for adhesive force between two rigid spheres, and Derjaguin approximation [10] provides a nice way to apply this relation for a variety of geometries. Adhesive forces have been incorporated into contact mechanics by two well-known analytical models, for interaction between two elastic spheres, called JKR and DMT models. JKR model [11] represents the case of contact between large spheres made of soft materials with high surface energy, while DMT model [12] is a good approximation for small spheres of hard materials with low surface energy. For the transition regimes, which lie between these two extremes, Maugis [13] developed an approach similar to Dugdale model in fracture mechanics, using a simplified adhesive force distribution outside the contact zone. The regimes of suitability of these different models were summarized on an “adhesion map” by Johnson and Greenwood [14]. Many researchers have also solved the adhesion problem in the transition regime using self-consistent numerical calculations, e.g. [15, 16]. All these studies were mainly concerned with the deformed profile, force distribution and pull-out force, which are the significant parameters after the contact has been established.

Pethica and Sutton [1] explicitly looked for the adhesion-induced instability (jump-to-contact) by analyzing the interaction between a rigid indenter and elastic half-space, when they are brought close. Comparing the gradient of the adhesive force and surface stiffness, they observed that the instability separation between a rigid sphere of radius *R* and elastic half-space is proportional to \( (A\sqrt R /E^{*} )^{2/7} \), where *A* is the Hamaker constant, and *E** is the effective elastic constant [16]. Attard and Parker [3], using a different criterion for instability, arrived at a similar expression for instability separation, except for a difference in the proportionality constant. Vinogradova and Feuillebois [17] developed simplified analytical expressions for adhesive force and pressure distribution, based on a “reference deformation length” parameter \( (\Uplambda = (R)^{1/7} (A/6)^{2/7} /(E^{*} )^{2/7} ) \), which characterizes the relative significance of deformation as compared to the initial separation.

Recent advances in in situ experiments in transmission electron microscope [18, 19, 20] have enabled direct observation of the deformation of the asperities as they approach each other. Erts et al. [19] observed that the jump-to-contact of an AFM tip occurs when the gap between the tip and the sample is about three times that predicted by the rigid analytical solution. Anantheshwara and Bobji [21] observed that the normal jump-to-contact occurs even when the bodies have appreciable tangential velocity simulating the asperity contacts. To interpret these experiments, there is an increasing need to understand the stress state resulting from the adhesive interactions.

van der Waals forces that give rise to adhesion are due to the interactions of the dipoles of the atoms or molecules of the interacting bodies [22]. These forces are significant over a range of few nanometers, beyond which their effects are negligibly small. Therefore, the adhesion force is usually approximated as a surface force [23], by employing Derjaguin approximation [10, 22]. The surface force approximation is suitable when the objects are big compared to the range of van der Waals force, since only the atoms close to the surface contribute significantly to the adhesion in such cases. In the present work, we perform finite element analysis (FEA) of adhesive interactions between a single asperity and a rigid half-space, by including van der Waals force as a volume force (body force) as the size of objects we study is comparable to the range of van der Waals forces. The force experienced by a finite volume of the deformable body is obtained by summing up the interaction of the atoms in that volume with all the atoms of a rigid half-space [10]. We have used finite element method (FEM) for this purpose, which can handle any complex geometrical shapes by discretizing into simpler elemental volumes, as against the analytical integration adopted in a surface force approximation, which can be used only for simple shapes such as spheres. By FEM, we can analyze interacting bodies of more complicated geometries, which are better representations of asperities, and more accurate results can be obtained for nanoscale bodies. FEAs considering van der Waals force as body force have been explored by other researchers also, notably by Cho and Park [24] and Sauer and his co-workers [25, 26].

In this work, we assume the asperity to be a smooth elastic hemisphere, and the effect of the rigid half-space is incorporated in the finite element formulation by summing up the contribution due to the atoms of the half-space [10]. We have studied the effects of separation between the asperity and the surface, radius of asperity and Young’s modulus on deformation, adhesive force, mechanical instability (jump-to-contact) and sub-surface stresses in the asperity. We have also studied a more realistic model of an asperity, by modeling a smaller asperity riding on the back of a larger hemispherical asperity (sphere-on-sphere model or Archard model [27]), which gives interesting insights into the effects of smaller scale roughness on asperities that are often ignored.

## 2 Formulation and Finite Element Modeling

*d*from a half-space is given by [10]:

*A*is Hamaker constant. This adhesion force is incorporated into finite element frame-work as a body force (force distributed over the volume of the body). Equilibrium is reached when the elastic restoring force in the deformed configuration becomes equal to the force due to adhesion. However, if the gradient of the adhesion force with separation is greater than the stiffness, then adhesion force cannot be balanced by the elastic restoring force [1]. The resulting instability would mean that the tip deformation increases till contact is established with the surface of the half-space. To capture this adhesion-induced instability (jump-to-contact) through finite element simulations, we adopt an iteration technique based on “residual stress update algorithm”. The general expression for elemental force vector ({

*r*

_{e}}) at any iteration in the absence of initial strain and surface tractions is given by

*N*] is the shape function matrix, {

*F*} is the body force vector, [

*B*] is the strain–displacement matrix and {

*σ*

_{0}} is the residual stress. The residual stress is initialized to zero for the first iteration, i.e. {

*σ*

_{0}} = {0} for all points. In subsequent iterations, we analyze the deformed configuration (updated using the deformation up to the previous iteration) for the incremental change in deformation and adhesion force. For example, the incremental elemental load vector for the second iteration is obtained by subtracting nodal forces induced in the first iteration from the nodal forces evaluated at the deformed configuration after first iteration. Thus, elemental load vector for

*m*th iteration becomes

*m*− 1) indicates that the quantity in parenthesis is evaluated for (

*m*− 1)th step. Since we solve for incremental deformations, global equilibrium equation from second iteration step onwards takes the general form:

*m*indicates the iteration number. The stiffness matrix [

*K*] for each iteration is evaluated at the corresponding deformed configuration, {

*δ*}

^{m}is the incremental displacement vector and {Δ

*R*} is the incremental load vector. The total displacement at each node after any iteration step is the sum of the incremental displacements of that node up to that iteration. Reaction forces and stresses are also calculated similarly:

The results are said to be converged when displacement for any iteration falls below a tolerance limit (in present study, we have used 0.5% of the displacement in the first iteration as the tolerance limit). It is found that the analysis diverges when the adhesion-induced instability sets in. If the initial separation is larger than this instability separation, a converged solution is obtained after a few iterations.

*s*is set to zero. Figure 2a shows the mesh near the tip of the asperity, where refined mesh is used to capture adhesion forces properly. As indicated in Fig. 1, the top nodes of the model are constrained in the axial direction, and the nodes along the axis are constrained in radial direction. The body force at any point in the hemisphere is calculated based on the axial distance of that point from the half-space. The analyses are performed for hemispheres of various radii, and by varying the initial separation (

*h*) between the points in the asperity closest to the half-space (referred to as the “tip” of the asperity) and the half-space. Corresponding to any one separation, output quantities such as deformation (displacements), strains, stresses and reaction forces are obtained, after reaching the equilibrium separation. Starting at a large initial separation, a series of analyses are carried out for each hemisphere model by progressively reducing the separation. Finally, when the instability separation is reached, the analysis shows divergence.

The effect of smaller scale roughness on an asperity is captured by modeling a spherical bulge of smaller radius on the tip of a bigger hemisphere (sphere-on-sphere model), i.e. when height *s* in Fig. 1 is greater than zero. Keeping the radius (*r*) of the spherical bulge as 25 nm, and that of the hemisphere (*R*) as 100 nm, the height of protrusion of the smaller asperity (*s*) is varied. Figure 2b shows the highly refined mesh that is used near the tip of the model so as to obtain the results with minimal error.

## 3 Results and Discussions

### 3.1 Deformation, Adhesive Force and Instability of a Single Asperity

In the finite element simulations, a value of 5×10^{−19} J has been used for Hamaker constant which is a typical value for metals [10]. The other parameters used in the simulations were chosen to match the typical values for in situ transmission electron microscope experiments [21, 28, 29]. Young’s modulus was varied from 70 GPa (aluminum) to 400 GPa (tungsten), and Poisson’s ratio value used was 0.3. Asperity radii ranging from 10 to 200 nm are used in the model with the aim of capturing the behavior of the typical probes used in the experiments. The effect of roughness is studied with a spherical asperity of 25-nm riding on the back the larger 100-nm asperity.

*F*) is obtained by summing up the reaction forces acting at the axial support locations corresponding to each initial separation (

*h*). The force values obtained from finite element simulation is compared with the analytical expressions for rigid sphere [1, 9]:

This expression had been obtained by them as a first-order correction to Eq. 6 using the deformation resulting from the rigid sphere adhesion force [1]. All the curves match very well at higher separation (>0.5 nm) which validates the finite element mesh and numerical procedure used in the simulation.

The simulation has been carried out starting at a large initial separation, and progressively reducing it till the instability point *h*_{inst} is reached. As seen from the Fig. 3, the FEA gives much higher values for force in comparison with that given by Eqs. 6 and 7 for the separations *h* closer to the instability point *h*_{inst}. This is because in FEA we obtain the equilibrium adhesive force, corresponding to a final separation (*d*), which will be much lower than initial separation (*h*) at low separations, due to the deformation of the asperity. At large separations, the deformation is too small to make any significant change in separation, and hence the values given by analytical expressions and FEM are similar.

*w*) at different initial separations (

*h*). The maximum deformation occurs at the tip of the asperity (point in the asperity that is closest to the half-space), and hence called “tip deformation”. The comparison between the maximum deformation obtained from our analyses, and that given by analytical expression derived by Pethica and Sutton [1], is shown in the figure. Similar to the case of force, it can be seen that the results for analytical expression and that from FE analysis match very closely at large initial separation (>0.4 nm), since the deformations are very small at large separations. However, closer to the instability separation (0.31 nm for 25-nm radius hemisphere), the tip deformation from FE solution is found to be much higher than that from the analytical expression. As in case of force, this can be attributed to the significant reduction in separation due to deformation of the asperity, which is not considered in the analytical expression. The slightly lower value of tip deformation at large separations (>0.45 nm) in FE analysis as compared to the analytical expression could be due to the difference between the surface force approximation used in the analytical models, and the body force distribution used in our simulations.

*h*

_{inst}and

*R*in nanometers):

### 3.2 Stresses and Yielding in a Single Asperity

The stress induced in the asperities due to the attractive forces could be very high. Under certain conditions, it could reach even the breaking strength of the material [30]. For metals, the asperities would undergo plastic deformation much before the breaking stresses are reached. Once the material becomes plastic, then the local stiffness reduces drastically leading to higher instability separations. Assuming that there is no size effect or the presence of defects in asperities, and von Mises yielding criterion, we studied the initiation of plastic deformation in an asperity.

*E*= 70 GPa, yield strength

*σ*

_{ys}= 400 MPa; copper:

*E*= 128 GPa,

*σ*

_{ys}= 70 MPa, carbon steel AISI 1006:

*E*= 205 GPa,

*σ*

_{ys}= 285 MPa, tungsten:

*E*= 400 GPa,

*σ*

_{ys}= 550 MPa) are analyzed. Figure 7 shows the variation of maximum von Mises stress, with change in separation for these spheres (25-nm radius) with different Young’s moduli. Assuming the engineering value of yield strength, this figure shows that the material would start yielding at an initial separation (yield point separation) of 0.37 nm for the aluminum alloy, while the yield point separation for copper is 0.59 nm, for carbon steel is 0.38 nm, and for tungsten is 0.3 nm. All these values are higher than the corresponding instability separation for the materials (0.31 nm for aluminum, 0.27 nm for copper, 0.23 for steel and 0.18 for tungsten). In general, the difference between yield point separation and the instability separation will be higher for a material with higher Young’s modulus or lower yield strength, assuming the geometry to be the same.

The difference in stresses between similar spheres of different materials is due to the difference in deformation, and hence the maximum stresses are nearly the same at large separations (>0.5 nm), as the deformations are very small. However, at smaller separations, stresses in material with higher Young’s modulus are found to be less compared to the material with lower Young’s modulus (at same separation). This is because a material with higher value of Young’s modulus deforms less, and hence experiences a lesser adhesive force resulting in lower values of stress. This being an elastic analysis, the predicted values of stresses are not expected to be correct after yielding has initiated; however, it is interesting to note that the stresses near the instability separation for tungsten (or in general for a material with higher Young’s modulus) is much higher than that for aluminum. Hence, even with the high value of yield strength, tungsten spheres are very much prone to yielding before reaching the instability.

### 3.3 Sphere-On-Sphere Model

*s*(Fig. 1) of the smaller asperity. This figure is similar to Fig. 3 except for the logarithmic scales, and the equation for adhesion force (

*F*) for rigid spheres (Eq. 6) will give a family of parallel lines for spheres of different radii. At large separations, the forces for 25- and 100-nm hemispheres match very closely with these lines corresponding to rigid spheres as explained in Sect. 3.1 and deviate considerably at separations close to the instability separation. It can be seen from the figure that for a given height of the protrusion,

*s*, at larger separations, the force is similar to that of the larger asperity (

*R*= 100 nm), and tends toward smaller asperity (

*R*= 25 nm) values at the smaller separations.

For separations close to instability separation, even when the height of smaller asperity (*h*) is as small as 1 nm, the adhesion force is tending toward the value for the smaller hemisphere (*R* = 25 nm). Also, the instability separation is found to be matching very closely with that of 25-nm hemisphere, for all the values of *h* considered in this study. These observations suggest that the local radius of curvature decides the behavior of an asperity, when the separation between with the mating surfaces is very small. This has the important implication for the multi-scale rough surfaces that the instability point is determined by the smallest scale of the roughness present. Assuming that this kind of continuum analysis is valid, the smallest scale one can think of is about one inter-atomic distance (few tenths of nanometer). The energy dissipated during such a jump-to-contact could be obtained, and could be related to the mechanism of friction. It should be noted that this is an elastic analysis and the actual material would undergo inelastic deformations, and the energy dissipated per jump will be higher. When two rough surfaces are rubbing against each other, there will be many locations at which the asperities will be experiencing the jump-to-contact instability. The number of such locations and the magnitude of the energy dissipated would then depend on the nature of the roughness that determines the asperity geometry. Assuming the asperity distributions, an estimate of the contribution toward the macroscopic friction could be estimated.

## 4 Conclusions

We studied the deformation instability in asperities, when they are brought close to a rigid half-space, using FEM. The adhesive force (van der Waals force) was considered to be distributed over the whole volume of the asperity. It was observed that the instability due to adhesion occurs at larger separation than those predicted by earlier models that assume surface force approximation. Further, it was observed that the stresses in the asperity could be much higher than the yield strength of the common engineering materials, and hence there exists a possibility that the asperities would undergo plastic deformation during this jump-to-contact instability. The plastic deformations would become more severe when the size of the asperities is reduced. The observation that yielding begins before adhesive instability is very important as it increases the instability separation itself, and also leads to additional dissipation of energy due to plastic deformations. Both these effects increase the energy loss associated with making and breaking contact between the asperities.

We have also studied the effect of roughness with a smaller asperity riding on the back of a larger asperity. It was observed that the distance at which the instability occurs is determined by the smaller asperity, and is almost independent of the height of the smaller asperity above the larger one. This observation has significant implications for the friction and wear behavior of the contacting surfaces in that the energy dissipated during the jump-to-contact events is determined by the smallest scale of roughness present.

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