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# On the debris-level origins of adhesive wear

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved June 1, 2017 (received for review January 17, 2017)

## Significance

Wear causes a huge amount of material and energy losses annually, with serious environmental, economic, and industrial consequences. Despite considerable progress in the 19th century, the scientific understanding of wear remains mainly empirical. This study reveals the long-standing microscopic origins of material detachment from solids surface, at the most fundamental level, i.e., wear particles. It discloses that the detached particle volume can be estimated without any empirical factor, via the frictional work. This study unifies previously disconnected and not understood experimental observations. The results open the possibility for developing new wear models with drastically increased predictive ability, with applications to geophysics, physics, and engineering.

## Abstract

Every contacting surface inevitably experiences wear. Predicting the exact amount of material loss due to wear relies on empirical data and cannot be obtained from any physical model. Here, we analyze and quantify wear at the most fundamental level, i.e., wear debris particles. Our simulations show that the asperity junction size dictates the debris volume, revealing the origins of the long-standing hypothesized correlation between the wear volume and the real contact area. No correlation, however, is found between the debris volume and the normal applied force at the debris level. Alternatively, we show that the junction size controls the tangential force and sliding distance such that their product, i.e., the tangential work, is always proportional to the debris volume, with a proportionality constant of 1 over the junction shear strength. This study provides an estimation of the debris volume without any empirical factor, resulting in a wear coefficient of unity at the debris level. Discrepant microscopic and macroscopic wear observations and models are then contextualized on the basis of this understanding. This finding offers a way to characterize the wear volume in atomistic simulations and atomic force microscope wear experiments. It also provides a fundamental basis for predicting the wear coefficient for sliding rough contacts, given the statistics of junction clusters sizes.

The study of material loss at sliding surfaces, known as “wear,” has over two centuries of history (1). Substantial progress occurred in the mid-1900s with a systematic series of wear experiments that showed, within a certain range of applied load, (*i*) the wear volume (i.e., total volume of wear debris) is independent of apparent area of contact (2, 3), (*ii*) the wear rate (i.e., wear volume per sliding distance) is linearly proportional to the macroscopic load acting normal to the interface, i.e., Archard’s wear law (3, 4), and (*iii*) the wear volume is proportional to the frictional work (i.e., the product of frictional force and sliding distance), which was first hypothesized by Reye in 1860 (5) and intermittently discussed and observed experimentally (3, 5⇓⇓–8). The first observation is commonly rationalized by arguing that the wear process is a direct result of contact between elevated surface asperities and is consequently associated with the real area of contact (2, 3). The second observation can then be understood by noting that the real area of contact is observed to be proportional to the macroscopic normal load (2, 9). The third observation follows from the first two if one assumes a wear volume proportional to sliding distance and a tangential force proportional to normal force (i.e., Amontons’ first law of friction) (10).

Despite the passage of more than 50 years, these wear relations remain fully empirical, and their microscopic origins are still unclear (11). Single-asperity wear simulations (12⇓–14) and atomic force microscope (AFM) wear experiments (15⇓–17) challenge the origins of wear debris formation by reporting a gradual atom-by-atom asperity smoothing. This observation further challenges a long-standing question posed by Archard (4, 18): When does an asperity collision lead to the formation of a wear debris particle? This question was recently addressed (19) by the identification of a critical length scale that controls debris formation at asperity contacts. It was shown that debris particles form only at contact junctions with sizes above a critical junction size, which is a function of bulk and interfacial properties. This finding opens the possibility of quantifying the amount of detached materials in the form of debris particles and studying the origins of macroscopically observed wear relations.

Inspired by this finding (19), this report aims to address how much material is detached during sliding contact, by focusing on the quantification of wear and the above-mentioned wear relations at the most fundamental level, i.e., wear debris particles. It shows that the volume of individual wear debris can be evaluated, without any empirical factor. The contacting asperity junction size is found to be the governing factor, controlling the tangential force and sliding distance for debris creation such that the tangential work required to create a debris particle is always proportional to its volume. Discrepant microscopic and macroscopic wear observations and models are then contextualized on the basis of this understanding.

## Results

A large set of multimillion-atom simulations of sliding contact were performed, differing in system size, boundary conditions, asperity shape and size, and bulk and interfacial properties. The simulations spanned two distinct geometrical configurations, i.e., a single asperity in contact with an atomistically flat surface and two asperities on opposing surfaces. As shown in our recent study (19), there exists a critical junction size, with larger asperity junctions forming wear debris particles and smaller junctions smoothing plastically. Accordingly, the configuration of all simulations discussed here was chosen so that the junction size leads to wear debris formation. A set of 2D (19) and newly developed 3D model interatomic potentials and a recently developed diamond potential (20) are used. The formation process of 3D wear particles is simulated and analyzed. Details of the potentials and corresponding physical properties and critical junction size are presented in *Methods* (see also *SI Appendix*, Fig. S1 and Table S1). In all cases, the simulations represent the dry adhesive sliding of identical materials at a constant temperature to reduce the complexity of the model and subsequent analysis. Details of the simulations are given in *Methods* and *SI Appendix*, Fig. S2 and Table S2.

Universal features are apparent in all simulations despite the variety of parameters and configurations examined (Fig. 1). Initially, a strong adhesive bond (junction) forms between contacting asperities. Subsequent sliding leads to the buildup of tangential forces and stored elastic energy (Fig. 1 *E*–*H*). During this phase, the junction grows by localized inelastic deformation (*SI Appendix*, Fig. S3), until crack nucleation and growth ensues at the two corners of the junction loaded in tension. With subsequent sliding, the forces transmitted across the junction decrease as the cracks grow, ultimately creating a debris particle. *SI Appendix*, Fig. S4 depicts a similar observation in simulations with different asperity shapes. On the whole, the behavior observed in the simulations is consistent with the classical picture of adhesive wear hypothesized from experimental observations (21⇓⇓–24). Recent small-scale wear experiments on ceramics and rocks (25, 26) confirm the formation of cylindrical and spherical wear debris particles. We make the distinction that this work is focused on wear by fracture-induced debris formation, as opposed to surface folding and delamination (23) mechanisms that may occur at different scales and/or under different wear conditions, e.g., abrasive wear.

### Wear Is Predictive at Debris Level.

Inspired by Archard’s wear model (4) *A*), with *SI Appendix*, Fig. S5).

Inspired by (*i*) our recent study (19) that revealed the critical importance of the junction shear strength (*ii*) macroscopic laboratory observations of a linear correlation between the tangential work and wear volume (7, 25, 27, 28), we examined the relationship between tangential work and the volume of resultant debris particles. Remarkably, we found that, at the debris level, these two quantities are related with a proportionality constant of 1/*B*),*SI Appendix*, Figs. S2 and S4 and Tables S1 and S2), the ability of Eq. **1** to estimate the volume of wear debris particles without any empirical coefficients is the most remarkable result of this work. The emergence of Eq. **1** from an energy balance viewpoint (8, 19) is elaborated as a discussion in *SI Appendix*.

In contrast to the tangential work, the sliding distance required for debris particle creation and the force profile over that sliding distance are highly variable across the set of simulations that have been performed (Fig. 1). Thus, the force profile and sliding distance must be linked in such a way that the tangential work required to create a debris particle is proportional to the debris volume. To better understand these findings, we examine the kinematics of debris formation, i.e., the relation between debris particle volume and the asperity junction size.

### Junction Size Dictates Debris Volume.

As discussed in the Introduction, macroscopic wear experiments (2⇓–4) show that the wear volume is independent of the apparent area of contact. Based on this observation, Archard (4) hypothesized that the wear volume is directly correlated with the real area of contact. Following this hypothesis, we examined the relationship between the wear particle size and the junction size. Fig. 3*A* compares the diameter of the debris particle, computed from its volume considering an idealized particle shape (i.e., circular, cylindrical or spherical), and the maximum junction size during a debris creation event. See *SI Appendix*, Fig. S7 for further details on the junction size measurement. Remarkably, the diameter of the debris particle is found to be the same as the maximum junction size, independent of the parameters and configurations of a particular simulation. This one-to-one correspondence is attributed to the kinematics of debris particle formation being independent of material parameters and geometric configuration across the range of simulations that have been examined, i.e., the junction grows to a maximum size that is determined by the nucleation of a pair of cracks from its edges, which then grow to create a debris particle having a cross-section equal to the maximum junction size. In other words, we observe a direct correlation between the real contact area and the volume of the debris, which is a central tenet of Archard’s wear model (4).

Following this observation, we studied the relationship between the maximum junction size and maximum tangential force encountered during a debris creation event (Fig. 3*B*). We find a linear relation between these two quantities with a proportionality constant of *SI Appendix*, Fig. S3). Together with Fig. 3*A*, this result indicates that the maximum tangential force transmitted across a junction can provide an accurate indication of the debris volume that will result from an asperity contact event.

Considering Archard’s classical model, we also examine the correlation between the maximum junction size and the applied normal force, normalized by the hardness at the debris level (*SI Appendix*, Fig. S6). No relation is observed across the range of simulations performed. This finding can be understood, as the contact area of a single asperity is not proportional to the normal load and is largely affected by the asperity’s geometry (31). In addition to the geometrical effect, the real contact area also depends on interfacial adhesion and shear loading (32, 33) (see also *SI Appendix*, Figs. S9 and S10). This result is consistent with recent observations in single (29, 34, 35) and multiasperity (36, 37) contacts that confirm a nonlinear relation between the contact area and applied normal load, where the degree of nonlinearity is a function of adhesion, material properties, and roughness parameters.

## Discussion

Archard’s model (4), which has proven quite successful for predicting macroscopic wear rates, is built upon the assumption that a debris particle forms from an asperity junction of the same diameter. This assumption implies that the depth to which the material is worn is proportional to the junction size. Our simulations have confirmed this assumption across a wide range of conditions (Fig. 3), if one takes the maximum asperity junction size observed in the simulations to be the junction size to which Archard referred. However, our simulations show that the asperity-level junction size cannot be solely approximated by the applied normal load, due to the collision of opposing asperities, the presence of interfacial adhesion, and the role of plastic shearing. Instead, the tangential load is shown to be a good indicator of the asperity junction size. This finding is in agreement with observations in recent studies (29, 30, 35, 36, 38) that show that, although friction force is always linear with the real contact area at any length scale, the relation between the normal load and the real contact area is largely influenced by the roughness parameters and interfacial adhesion.

The above argument also explains the discrepancy between the asperity-level observation made here and the macroscopically observed linear correlation between wear volume and the applied normal force (3, 4). Although our simulations confirm a correlation between the wear volume and real contact area, this discrepancy can be attributed to the absence of a correlation between the real contact area and the normal load at single-asperity contacts. However, it was shown (39, 40) that a linear correlation between the normal load and the real contact area can be recovered via multiasperity contact models. Accordingly, the macroscopically observed linear relation between the wear volume and the normal force may be reconstructed via multiasperity contact models (39⇓–41). It has been shown (29, 35) that sliding between surfaces with low roughness and high adhesion (e.g., asperity-level contact) is an adhesion-controlled process, in which normal force is a sublinear function of real contact area and tangential force. On the other hand, the sliding between surfaces with large roughness and low interfacial adhesion [e.g., macroscopic multiasperity contacts (39⇓–41)] is a load-controlled process, in which both the normal and tangential forces are proportional to the contact area (i.e., Amontons’ friction law). In such conditions, the wear volume is expected to correlate with both the normal and tangential forces (see also *SI Appendix*, Fig. S5).

To arrive at the volume of material worn per distance slide, Archard assumed that the sliding distance required to create a debris particle from an asperity junction was equal to the diameter of the debris particle. Although our simulation data (Fig. 1) and the literature (8) make it clear that Archard’s assumption is an approximation of the real case, the data presented here show that this is a good approximation, provided one considers the maximum asperity junction size and the tangential force transmitted through the junction at its maximum size (*SI Appendix*, Fig. S8). More precisely, Fig. 2 shows that

An important component of Archard’s model is the proportionality factor, i.e., wear coefficient,

Considering that typical values of

## Conclusion

This study reveals the long-standing microscopic origins of experimentally observed macroscopic wear relations between the wear volume, the real contact area, and frictional work. Our simulations demonstrate that the asperity junction size directly controls the debris size, and, consequently, the wear debris volume can be quantified via the work done by the tangential component of the load carried by a junction. It is also shown that, at the asperity level and in the presence of high adhesion, the normal applied force does not linearly depend on the junction size and, as a result, cannot predict debris volume. This finding offers a pathway to quantify wear at the single debris level in atomistic simulations and AFM wear experiments. In addition, we confirm a wear coefficient of unity at the debris level, when asperity junction size is larger than the critical junction size (19). This finding provides a fundamental basis for predicting the adhesive wear coefficient for sliding rough contact, given the statistics of junction clusters sizes.

## Methods

### Interatomic Potentials and Physical Properties.

As extensively discussed in ref. 19 for 2D coarse-grained model potentials and now here extended in 3D, the wear debris formation can be studied within the atomistic framework. The unique feature of these potentials is that the corresponding critical junction size (19) is small enough, which allows the creation of fracture-induced debris particles within the atomistic framework. To keep elastic properties unchanged, the long-range character of the Morse potential (44) is modified without disturbing the short range interactions as follows:*SI Appendix*, Tables S1 and S2 present potentials parameters and corresponding coarse-grained mechanical properties. *SI Appendix*, Fig. S1 shows the corresponding indentation response for 2D and 3D model potentials (see also ref. 19). For diamond simulations using multimillion atoms, the Tersoff potential (45), enhanced with a screening function (20) for modeling brittle bond breaking, is used. Potential parameters and corresponding physical properties are taken from ref. 20. To control the interfacial adhesion, a different set of potentials with a modified well depth is used (see also ref. 19).

### Simulation Geometry, Loading, and Boundary Conditions.

All simulations were performed using the molecular dynamics software LAMMPS (46). For diamond simulations, the ATOMISTICA library (20) is used along with LAMMPS. A periodic boundary condition and a sliding velocity were applied in the *x* direction for all of the simulations. In the *y* direction, both constant pressure and fixed boundary conditions were examined (see *SI Appendix*, Fig. S2). Temperature was enforced along two thin layers just outside of this region using a Langevin thermostat. In the 2D simulations, both the collision between asperities and a welded junction (*SI Appendix*, Fig. S4*A*) are modeled. Noting the enormous computational cost of 3D simulations (e.g., *D*), we consider a welded junction for all of the 3D simulations.

We performed a large set of simulations, which confirmed that the results are independent of the simulation box size, boundary conditions (i.e., fixed displacement versus fixed applied pressure), initial asperity shapes (i.e., semicircular and partial circular segment, triangular, rectangular, and half sine; *SI Appendix*, Fig. S4), and geometrical configurations (i.e., single asperity versus interlocking asperities). *SI Appendix*, Table S2 shows the covered range in this study for the system dimension, sliding velocity, and applied load. For model potentials, a damping parameter of 0.05 and a Verlet algorithm with a time step of 0.0025 (both in reduced time unit) are used for numerical integration (see also ref. 19). A thickness of 5 (in reduced length unit) is considered for rigid boundary and thermostat layers. For diamond simulations, a damping parameter of 1 ps and a time step of 1 fs are considered. A thickness of 1 nm is considered for rigid boundary and thermostat layers. Tangential force profiles shown in Fig. 1 are averaged every 100 time steps. All simulations are visualized by Ovito software (47).

### Compilation of Simulations Data.

Surface atoms within the cutoff radius of the interfacial potential are considered to define the bonded area (48). Accordingly, the junction size is estimated as the number of bonded atoms at the junction. Inspired by the experimental approach for measuring contact area using thermal/electrical conductance, (49, 50) the minimum and maximum junction sizes are measured and reported in terms of horizontal error bars in Fig. 3 (see *SI Appendix*, Fig. S7). The volume of debris is measured at the steady-state regime (i.e., once the corresponding tangential force reaches zero) as the product of the number of atoms in debris and the atomic volume. For 2D simulations, a single atomic layer thickness is considered. Debris size is directly computed from the debris volume by considering an idealized particle shape (i.e., circular in 2D and cylindrical or spherical in 3D).

## Acknowledgments

J.-F.M. and R.A. acknowledge financial support from the Swiss National Science Foundation (Grant 162569, Contact mechanics of rough surfaces).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: jean-francois.molinari{at}epfl.ch.

Author contributions: R.A. designed research; R.A. carried out simulations; R.A. analyzed data; R.A., D.H.W., and J.-F.M. participated in the discussions; J.-F.M. supervised the research; and R.A., D.H.W., and J.-F.M. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1700904114/-/DCSupplemental.

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