Multiscale Friction Simulation of Dry Polymer Contacts: Reaching Experimental Length Scales by Coupling Molecular Dynamics and Contact Mechanics

This work elucidates friction in Poly-Ether-Ether-Ketone (PEEK) sliding contacts through multiscale simulations. At the nanoscale, non-reactive classical molecular dynamics (MD) simulations of dry and water-lubricated amorphous PEEK–PEEK interfaces are performed. During a short running-in phase, we observe structural transformations at the sliding interface that result in flattening of the initial nanotopographies accompanied by strong polymer chain alignment in the shearing direction. The MD simulations also reveal a linear pressure – shear stress dependence and large adhesive friction in dry conditions. This dependence, summarized in a nanoscale friction law, is of central importance for our multiscale approach, since it forms a link between MD and elastoplastic contact mechanics calculations. An integration of the nanoscale friction law over the real area of contact yields a macroscopic friction coefficient that allows for a meaningful comparison with measurements from macroscopic tribometer experiments. Severe normal loading conditions result in significant wear and high experimental friction coefficients µ≈0.5–0.7, which are in good agreement with the calculated values from the multiscale approach in dry conditions. For milder experimental loads, our multiscale model suggests that lower friction states with µ≈0.2 originate in the presence of physisorbed molecules (e.g., water), which significantly reduce interfacial adhesion.


Introduction
Nowadays, polymers are increasingly used in tribological applications due to their advantageous properties of low friction, corrosion resistance, biocompatibility, and cost-effectiveness. In this framework, a quantitative understanding of friction between polymer interfaces is fundamental to optimize their application range and improve their lifetime.
However, a numerical modeling of polymer friction remains a challenging problem, since several contributions such as adhesion, viscoelastic, and thermal effects, as well as plastic processes and ploughing must be considered. Furthermore, changes in material properties, e.g., strain hardening and direction-dependent transformations, can occur under loading and sliding. Continuum theories and numerical calculations allow us to consider several aspects of polymer friction at the macroscopic scale [1]. Here, a key element is the real area of contact between rough surfaces [2], which can be extracted from either experiments or contact mechanics calculations [3,4]. In the contact spots between polymer and a countersurface, sliding is opposed by the shear strength of the adhesive junctions [5]. Accurate empirical estimates of this quantity require carefully chosen experimental conditions-even at model interfaces [3,6]. The underlying physical processes are also unclear, as shown by several competing models in the literature [5,7,8]. Furthermore, the shear strength at the contact spots strongly depends not only on the polymer and counter material but also on the presence of moisture, lubricants, or transfer layers [9][10][11] at the shearing interface. This makes a direct experimental determination of inaccessible for relevant technical surface pairings.
Atomistic simulations represent an alternative to gain insight into the tribological response of polymer interfaces. Classical molecular dynamics (MD) allows for a detailed understanding of structural transformations occurring under sliding and their relationship with friction in several classes of materials. Non-equilibrium simulations (NEMD) as initially developed by Ashurst and Hoover [12] and recently reviewed in tribology [13] are often employed for this purpose. For polymers, pioneering MD sliding simulations of PTFE and PE [14][15][16] analyzed in detail the possible wear mechanisms occurring at the molecular level. MD has also been applied to relate material properties to the bulk mechanical response [17,18] in order to investigate interaction of oligomer chains with fillers [19,20] and countersurfaces [21] under shear, or to simulate scratching behavior [22].
Despite the vast insights provided by MD approaches, these remain computationally expensive tools leading to a sizeable gap in length and time scales between simulations and experiments or typical applications. Efforts should therefore be made to correctly transfer the information gained from atomistic simulations into macroscopic systems and to validate corresponding theoretical predictions by experimental observations. First studies for liquid lubricants under severe conditions showed promising agreement between friction results from NEMD simulations and extrapolations from macroscale tribometer experiments at lower shear rates [23][24][25].
The scope of this work is to tackle the scale-bridging issue for friction processes in simple polymer interfaces through the development of a multiscale approach and its comparison with experimental results. Among several high-performance polymers, we consider Poly-Ether-Ether-Ketone (PEEK). This semi-crystalline thermoplastic presents outstanding mechanical properties, thermal stability, and chemical resistance, thus constituting an excellent material for bearing and sealing applications. For simplicity, a neat PEEK-PEEK interface is considered in the simulations, whereas additives and fillers usually found in polymers are not considered in this article. The same material pairing is chosen in tribometer experiments to allow for meaningful comparison. Here, the excellent mechanical properties of PEEK even without fillers help in probing highly loaded conditions.
Few atomistic studies exist on PEEK, which are mostly focused on its structural [26,27] and mechanical properties [28,29], but not on its tribological performance. In this open field, we will investigate what insights MD can provide into friction at the molecular scale, and especially on the shear strength of adhesive junctions. First a pristine PEEK-PEEK interface is modeled first, and interfacial water is added next to analyze the effect of physisorbed fluids. A parametric study on the loading conditions is also performed to assess the transferability of MD friction results to larger scales.
We will then bridge the gap between atomistics and continuum by integrating MD results into contact mechanics simulations to estimate the friction coefficient in macroscopic systems. Finally, tribometer experiments are performed and compared to the results of our multiscale approach.

Methods
Our multiscale methodology involves the study of two main scales. At the nanometer scale, atomistic simulations are employed to understand the details of tribological phenomena such as structural transformations in the polymers, and to formulate local friction laws for the relevant contact interfaces. At the macroscopic scale, the real area of contact between rough polymer surfaces is determined by using contact mechanics calculations. By applying the local friction laws to the real contact area, macroscopic friction coefficients can be estimated and compared to results from tribometer experiments.
Note that the polymer-polymer pairing in this work is less common in tribological applications than polymer-metal contacts. The simpler interface-especially with respect to atomistic simulations-allows for more straightforward testing of the proposed multiscale approach, which can then be applied to more complex and technically relevant systems.

Atomistic Modeling of Bulk PEEK
Non-reactive MD is employed to quantify nanoscale friction mechanisms at dry and moist PEEK-PEEK interfaces. This method involves integration of Newton's equations of motion for each atom in a system and provides its time evolution under tribological loading. Intra-and intermolecular interactions for the polymer are described by the OPLS-AA force field and in the BOSS/Tinker package [30,31], while the TIP3P potential [32] is employed for water molecules. All simulations are performed using the software package LAMMPS [33] with a time step of 1 femtosecond. OVITO [34] is used for the visualization of atomic trajectories.
As a first step, we benchmarked the OPLS-AA potential by modeling the structural and mechanical properties of bulk PEEK and by comparing the results to experimental data. For relaxed crystalline structures, the lattice constants, characteristic angles, and density were in good agreement with X-ray diffraction experiments [26], with a maximum difference of 5% (Table 1).
Several amorphous bulk PEEK structures with different oligomer lengths are then generated. We consider oligomers with molecular weights ranging from 1000 to 9000 g/mol, corresponding to chains lengths from 4 up to 36 repeating units. Starting from low density and high temperature conditions (0.1 g/cm 3 , 1000 K) to randomize the molecular configurations, each system is compressed to reach a realistic density and quenched down to 0 K at a rate of 0.25 K/ps. Finally, relaxation at ambient pressure is performed to obtain stress-free amorphous structures. Elastic constants at 0 K are then computed by applying small strains (up to 0.5%) to the relaxed structures and quantifying the resulting stresses [35]. The computed Young's modulus and Poisson's ratio agree well with experimental values [36], although the latter have been measured for semi-crystalline PEEK at ambient temperature (Table 1).
In summary, these findings suggest that the OPLS-AA force field is suitable for a PEEK/PEEK friction study since it reproduces realistic structural and mechanical properties of amorphous and crystalline bulk PEEK systems. We also find that properties of the amorphous systems do not vary significantly for chains longer than 12 repeating units. Therefore, we employed bulk systems consisting of hexadecamers for our subsequent simulations.

Atomistic Simulations of Amorphous PEEK-PEEK Shear Interfaces
Amorphous PEEK surfaces are created from amorphous bulk structures by splitting in half while ensuring that no polymer chains are cut in the process. Relaxation at ambient temperature without additional constraints leads to the formation of nanoscale roughness as dangling polymer chains fold onto the surface (Fig. 1a). Conversely, smooth surfaces or defined sinusoidal roughness (Fig. 1b-c) are obtained by reheating and quenching the surfaces under compression through artificial walls with the desired shape.
Each surface block has a volume of 12 × 8 × 18 nm 3 and contains 256 oligomers.
Since representative nanotopography data at scale lengths characteristic of our MD system are inaccessible through experiments, we use the three surfaces from Fig. 1a-c to probe the system response at different height/length aspect ratios of the nanoasperities. In particular, the smooth and sinusoidal surfaces should be, respectively, seen as the lower and upper limiting cases for the aspect ratio, while the nanorough surface constitutes an intermediate, more realistic scenario. We also limit the scope of the molecular dynamics analysis to amorphous PEEK surfaces, as typical spherulitic crystalline structures in this polymer [38] possess amorphous regions with bent and dangling chains at their surface.
Amorphous PEEK-PEEK tribological interfaces are simulated by pairing two surface blocks which are then subject to compression and shear. Periodic boundary conditions are applied along the system length and width. Each surface block is divided into three regions (Fig. 1d): a 4 nm-thick rigid layer where normal load and shear velocity are applied, a 1 nm-thick thermostated domain, and a free unconstrained region around the shearing interface.
The temperature T =300 K is set through a Nosé-Hoover thermostat with time constant of 0.1 ps acting in the y-direction only to avoid affecting system dynamics during compression and sliding. Normal loading is applied through the barostat described in [39] with a critical damping constant of 4⋅10 -10 kg/s. The considered external pressures vary in the range P = 1-100 MPa.
Shearing is then performed under constant normal load with a reference velocity of U = 100 m/s. This high value is due to the high computational cost of all-atom non-reactive MD simulations which limits realistic simulation times to a few nanoseconds. In our simulations, a sliding distance of 0.2 µm (simulated time of 2 ns) is typically required to reach steady state and to quantify the shear response of the interface. Nonetheless, lower velocities of 2 m/s and 10 m/s are also considered to assess the speed dependence of friction processes.
Finally, two types of interfaces are simulated. In the "dry" case, the contact occurs between pure PEEK surfaces. Conversely, water is added between rough PEEK surfaces in the "moist" case to investigate the influence of physisorbed liquids. This is motivated by the occurrence of water sorption into PEEK in moist air [40], indicating that water molecules are present on the surfaces to allow for the diffusion into the bulk polymer. Surface coverages up to 1500 H 2 O molecules between the two 12 × 8 nm 2 surfaces are considered as a first estimate. This corresponds to a maximum surface density of 0.5 ng/mm 2 , representing a single water monolayer physisorbed on each PEEK surface.

Contact Mechanics Simulations
Elastoplastic contact mechanics calculations are performed to obtain the real contact area and pressure distribution in the rough ball-on-plate system used in the experiments. The calculations employ the code pyCo by L. Pastewka [41], based on an FFT-based Boundary Element Method [42,43] together with a conjugate gradient optimization method [44] for fast convergence.
Representative surface topographies of the PEEK sphere and countersurface used in the calculations were obtained from AFM measurements of unworn experimental samples (Fig. 2). The power spectral densities of all samples show a similar trend at high wavenumbers with a Hurst exponent of approximately 0.8. It should be noted that the topography data of the spheres also include their macroscopic curvature, which explains the higher PSD at lower wavenumbers compared to the flat countersurfaces.
In the contact mechanics code, the topographies of the sphere and countersurface are combined and pressed against a half space with equivalent elastic modulus E * = Plasticity is included by limiting the contact pressure to the penetration hardness of the materials [3]. The material parameters used in the contact mechanics calculations were experimentally determined for PEEK samples used in this work and are summarized in Table 2.
Finally, the sliding motion of the surfaces is not simulated explicitly in this work. Instead, the ball is pressed against the countersurface at several locations along the sliding direction, so that the sliding is reduced to a series of quasi-static cases with pure normal loading.

Experimental Setup
Neat PEEK samples with two different degrees of crystallinity were prepared in-house at Freudenberg Sealing Technologies. Differential calorimetry was used to assess the degree of crystallinity within the bulk thermoplastic: hereafter samples containing a 20% and 40% crystalline phase will be referred as "amorphous" and "crystalline," respectively. AFM morphology measurements reveal the presence of flake-like structures ranging from 150 to 500 nm in length on the surfaces of both amorphous and crystalline samples. These domains are interpreted as crystalline regions on the surface, with a local surface coverage up to ~ 25%. However, due to the non-uniform distribution of the flake-like structures, it is difficult to extract the degree of crystallinity over the entire surfaces from the aforementioned measurements.
To further differentiate between amorphous and crystalline samples and extract their elastic modulus and hardness at the surface, nanoindentation are performed on a Hysitron TI950 device. A Berkovich tip is used on 10 × 10 indent grids, with two indentation depths of 0.3 µm and 1.5 µm. The mechanical properties of the amorphous sample decrease with lower indentation depth. This is confirmed by indentation on a sample cross-section performed at different distances from the surface. Here, the thickness of  the surface zone is found to be 0.4 µm. A change in the mechanical properties with indentation depth is not observed on the crystalline samples. Instead, increased variation in elastic modulus and hardness across the indentation grid is found, which hints at a larger heterogeneity of the crystalline sample. Averaged values for the mechanical properties are summarized in Table 2. Here, both the elastic modulus and hardness decrease significantly when going from the maximum crystalline content (40%) of the "crystalline" sample to 20% of the "amorphous" one.
A BASALT-MUST ball-on-plate reciprocating tribometer by TETRA GmbH is used for the experimental analysis of PEEK-PEEK friction. In the tribometer, a sphere with radius of R = 1.5 mm is placed in contact with a tensile rod serving as the plane counter surface. Both samples were produced using the same molding tool.
The operating conditions for the experiments are summarized in Table 3. The four loads correspond to Hertzian pressures ranging from 10 to 75 MPa (the upper limit being close to the maximum operating limit of PEEK). Thus, the maximum sliding velocity at the stroke center is set to a low value of 2 mm/s to avoid overloading and excessive heat generation at the contact interface. Breakaway forces at reversal points are excluded from the evaluation of the coefficient of friction.

Atomistic Simulations of Dry PEEK-PEEK Interfaces
As a first step in the multiscale approach, we aim at understanding the atomic-scale tribological processes occurring at dry PEEK-PEEK interfaces. Figure 3 shows the evolution of the shear stress with the sliding distance for the three different interface geometries from Fig. 1a-c. For all three simulations, the loading conditions P =100 MPa, U =100 m/s, T =300 K are chosen. At the beginning of shearing, increases linearly as the surfaces are elastically deformed over their entire height before the onset of localized sliding at the interface. For the nanorough surface, this occurs after a sliding distance of 5 nm corresponding to a stress peak of 120 MPa. Friction then decreases, and stabilizes to a value of 80 MPa as steady state is reached.
Interestingly, a friction response similar to the nanorough case is recovered for both the smooth-smooth and sinusoidal-smooth pairings shortly after the onset of shearing, despite the significant differences in the initial nanoscale topography. In order to explain these similarities in the friction response of the three interfaces, we look at transformations in terms of surface roughness and molecular configurations of the oligomer chains. The line root mean square height along the shearing direction is defined as: where the outer sum corresponds to averaging over the system width W , h(x, y) represents the envelope of the surface atoms with a sphere of radius 1.4 Å [45], and h(y) is the average line height in the x−direction. Figure 3b shows the evolution of the line RMS height with the sliding distance. After reaching steady state, the h line,RMS value of the nanorough surface is reduced by half relative to the starting configuration. The reduction in roughness under shearing is even more pronounced for the sinusoidal profile, which is completely flattened in less than 20 nm sliding distance. It should be noted that this phenomenon is favored at elevated pressures such as P=100 MPa, which is close to the penetration hardness of PEEK. Nonetheless, the same flattening process was also observed at lower pressures such as 10 MPa.
A peculiar behavior is observed for the smooth-smooth interface, whose friction evolution with the sliding distance follows the nanorough case very closely. While the initial surfaces are almost atomically flat, roughening at the interface occurs as soon as shearing is applied: h line,RMS is doubled, then decreases slightly as steady state is reached. This behavior can be explained through the change in orientation of the oligomer chains at the interface as seen in Fig. 3c-d. Initially, the ends of the oligomer chains in the interfacial region are oriented randomly but tend to align along the sliding direction as soon as shearing is applied. This reconfiguration process results in significant roughening of the initially atomically flat interface, leading to h line,RMS and shear stress similar to the nanorough case.
We quantify chain alignment by analyzing a simplified representation of the PEEK molecules (Fig. 4a). From the all-atom representation, segments linking the Ether-Oxygens and the Ketone-Carbon atoms are extracted. Each segment is characterized by its center point and an angle with respect to the shearing direction � ⃗ x . While the distribution of at the PEEK-PEEK interface prior to sliding is uniform as expected for an isotropic amorphous polymer material (blue curve in Fig. 4b), after steady-state sliding is reached a strong alignment along the shearing direction can be observed (red curve in Fig. 4b). Next, we analyze the angle distribution of PEEK chains in the vicinity of the sliding interface in more detail. To do so, we chose as criterion for aligned segments that their angle with the shearing direction must be lower than a certain cutoff value (here | | < ∕4 , see Fig. 4b-c). Figure 4d shows the percentage of aligned segments along the system height before shearing and once steady-state friction is reached. All initial configurations show approximately 29% of aligned segments over the whole system height, corresponding to chains without preferential orientation as explained in Fig. 4c. During sliding the oligomer chains within a ± 3 nm proximity of the sliding interface align along the shearing direction. This process occurs over a sliding distance of 0.1 µm, which is also required to reach steady-state friction. Afterwards, no growth of the structurally aligned region is detected, although this may be due to the sliding distance being limited to 0.2 µm in our molecular dynamics simulations. In summary, the previous MD results indicate that both the friction response and the structural transformations in dry interfaces are independent on the exact surface topography at the nanometer scale for the operating conditions chosen in this work.
Next, we quantify friction in the PEEK-PEEK systems as a function of the loading conditions. First, the external pressure is varied from P =1 to 100 MPa. For all considered interfaces, the shear stress in steady-state sliding is generally high and increases linearly with the applied pressure (Fig. 5). The slope of a linear fit to the (P) curve is interpreted as the atomic-scale friction coefficient MD . Interestingly, MD is approximately 0.2 irrespective of the initial roughness of the PEEK surfaces. Furthermore, strong adhesion between Fig. 4 a Representation of simplified PEEK chain used in the analysis. b Segment angle distribution at the sliding interface. The continuous red line is a Gaussian curve with standard deviation corresponding to | | = ∕4 . c Aligned segments are contained within two spherical sectors featuring an aperture angle of ∕4 with respect to the x-direction. In systems without preferential orientation, the percentage of the aligned segments is given by the surface of the two spherical sectors, divided by the sphere surface. This gives 1 − cos( ∕4) ≈ 0.29 . d Number of aligned segments over the system height, with the sliding interface being located at z =18 nm. Full markers correspond to the beginning of shearing, whereas empty markers indicate steady-state shear results

Fig. 5
Shear stress in dry amorphous PEEK-PEEK interfaces as a function of the applied pressure ( U =100 m/s, T = 300 K). The dotted lines represent linear fits of the data points. Nanorough 2 represents an additional realization of a nanorough dry interface, obtained by removing water from one of the systems in Sect. 3.2 the PEEK surfaces occurs due to their chemical affinity, the absence of interposed media, and full contact at the sliding interface, resulting in a large shear stress offset adh. . This adhesive contribution to friction is of the order of the bulk shear strength of PEEK, akin to what was found for polystyrene through experiments in [3].
Based on the similar behavior exhibited by all interfaces considered here, a single law can be employed to quantify the steady-state friction response as a function of the applied pressure: where − represents the average of the (P) curves for different interfaces. The linear pressure-shear stress dependence with non-zero intercept is fitted here through the usual Amontons-Coulomb friction law with Derjaguin's modification to account for the adhesive term adh. [46]. One can note that the form of the nanoscale friction law is similar to the typical friction force dependence on load in macroscopic contacts; however, a direct comparison with experimental results is not possible yet. In fact, in our multiscale view, Eq. 2 quantifies the local friction response of the contact spots between rough macroscopic surfaces, i.e., the shear strength of adhesive junctions, which is typically unknown in continuum simulations. Hence, the nanoscale friction law is of central importance for our multiscale approach since it forms a link between the molecular and the continuum scales.
To complete the MD study of the dry interface, the sliding velocity is reduced from U = 100 m/s to U=10 m/s and 2 m/s at P=1 MPa to investigate the polymer response under loading conditions usually found in technical systems. Figure 6a (2) MD (P) = MD P + adh. = 0.18P + 50.5MPa shows that for very high and low velocities, the shear stress only differs by 10% at the stress peak in the transient regime, and by only about 7% in the steady-state regime. Additionally, as shown in Fig. 6b, the structural transformations at the PEEK-PEEK interface depend only on the sliding distance. These results indicate that the shear stress response is almost independent on the shear velocity for the considered interfaces. Thus, the nanoscale friction law in Eq. 2 derived from simulations at U=100 m/s will be integrated in the calculation of macroscopic friction and the subsequent comparison with tribometer experiments, which were performed at velocities lower by five orders of magnitude.

Atomistic Simulations of Moist Contact Interfaces
One should keep in mind that the shearing of pure polymer-polymer interfaces in the previous section only provides a very simplified representation of reality. This special situation could be characteristic of wear processes that rub away the initial surfaces and expose pure PEEK material at the sliding interface. Still, unworn samples generally exhibit an injection skin with modified crystallinity and material composition at their surface. Contaminants, impurities, or fluid molecules can also be found on top of the surfaces. Our previous results indicate that shear processes are strongly linked to the structure and topography in a nanometer-thin zone at the sliding interface. Thus, adhesion and friction can be strongly modified even by a small amount of physisosorbed fluid. As a first attempt to investigate a more realistic sliding interface, H 2 O is inserted between the two amorphous surfaces with a surface density below one nanogram per mm 2 . This Fig. 6 a Shear stress as a function of the sliding distance at different sliding velocities and low load ( P =1 MPa, T = 300 K). b Chain alignment after a sliding distance of 0.1 µm for three shear velocities aims at modeling physisorption of water from air humidity [47], which is likely to occur onto the ketone and ether groups of the PEEK chains. Shearing simulations of water-lubricated PEEK-PEEK interfaces are performed with the same pressures and sliding speed as in Fig. 5. Under tribological loading, the water molecules help to separate the surfaces by forming nanochannels aligned along the sliding direction, thereby significantly reducing friction (Fig. 7). Under shearing, small amounts of water penetrate the surface-near region of PEEK, but the present simulations are too short to evaluate diffusion of H 2 O within the polymer bulk. It should also be noted that water squeeze-out is not possible here, due to the lateral periodic boundary conditions of the Molecular Dynamics model.
Water between the surfaces acts as a lubricant and reduces interfacial friction due to the partial separation of the PEEK surfaces. Results in Fig. 8 show a linear dependence between b Dependence of the adhesive stress adh. on the amount of water con-fined between PEEK surfaces. The dotted lines represent linear fits to the data points applied pressure and shear stress. The slope of the (P) line, i.e., the friction coefficient from MD, is MD = 0.18 as in the dry case. However, the offset adh. is significantly lower due to the smaller domains of direct polymer-polymer contact where intersurface adhesion takes place (Fig. 5b). This result is coherent with previous NEMD simulations for metal surfaces, where higher surface coverages of polar molecules lead to reduced adhesion [46]. The decrease in adhesive shear stress can be approximated by a linear fit: where H 2 O is the surface density of water molecules in [number/nm 2 ] units. It should be noted that, although Eq. 3 reasonably describes the adhesive stress in presence of single water monolayers on pristine PEEK, it is eventually expected to break down as the amount of water at the interface is increased. In this case, the surfaces become fully separated, and the rheology of the nanoconfined liquid and nanohydrodynamics becomes the determining factors for the shear stress under sliding.

Contact Mechanics Simulations and Coupling with Molecular Dynamics
In this section, the friction laws obtained from MD are coupled with continuum contact mechanics simulations to obtain a macroscopic friction coefficient. We assume that the MD (P) friction law from Eq. 2 applies to the contact domains between PEEK surfaces with microscopic roughness. The tangential force in the macroscopic contact is then obtained by integrating MD over the real area of contact A real : where P(x, y) is the local contact pressure. As shown in Sect. 3.1, the coefficients MD and adh. are constant for all considered dry interfaces, leading to: The normal force F N is given by: with A 0 being the apparent contact area and P ext. the average normal pressure applied to the system.
Finally, the ratio F T ∕F N gives the macroscopic friction coefficient: The term A real A 0 (P ext. ) is determined from our contact mechanics simulations of real rough contact geometries obtained from AFM scans with elastoplastic material properties from Table 2. The relative contact area is computed for multiple sample topographies, i.e., for one ball and two rods for both the crystalline and amorphous samples, for several relative positionings of the surfaces, and for external pressures P ext. ranging from 0.1 to 50 MPa, corresponding to loads up to 0.3 N. The resulting values of A real do not depend significantly on the exact surface topography. This can be traced back to the similarity among the power spectra for all considered topographies, especially at high wavenumbers (Fig. 2). Additionally, the contact pressures are limited by the indentation hardness of PEEK, which further reduces the contact area dependence on local topographical features. It should also be noted that accounting for plasticity significantly increases the contact area compared to purely elastic simulation, akin to [3]. Finally, the presence of plastic deformation indicates that in our system, in addition to the structural transformations at the nanoscale reported in Sect. 3.1, flattening of the asperities also occurs at the continuum scale [48]. Figure 9 shows the relative contact area at different external loads, where each point corresponds to the average value over all considered surface topographies and relative positions of the rod-sphere pairings. A linear dependence is found, in agreement with results from reference [3]. The The difference in values is mainly due to the significantly higher penetration hardness of crystalline PEEK, and, to a more limited extent, to its larger Young's modulus. Finally, Eq. 2 and the ratios from contact mechanics are used in Eq. 7 to estimate the macroscopic friction coefficient of dry PEEK-PEEK interfaces. We obtain macro = 0.68 and macro = 0.50 for the amorphous and crystalline pairings, respectively. The higher friction coefficient in dry conditions for the amorphous case originates in the softer material properties which lead to a larger contact area at all loads.
In presence of small amounts of water at the interface, friction is reduced due to the lower adhesion offset adh. between polymer surfaces (Fig. 8). For instance, with the maximum water quantity simulated in one of the previous sections, one finds macro = 0.28. The reduction in friction due to the presence of water agrees qualitatively with previous experimental results [9]. Note that our calculated macro reflects only changes in friction at the nanometer scale, while modifications of mechanical bulk properties due to absorbed fluids within the polymer [10] are not accounted for. Finally, if adhesion is fully suppressed with adh. = 0 ,  . 10 a-b Experimental friction coefficient as a function of the number of reciprocating cycles for the amorphous and crystalline pairings, respectively. c Confocal microscope image of a worn crystalline sphere and material transfer to the rod after a high friction run at 0.310 N load one obtains macro = MD = 0.18 for both the crystalline and amorphous pairings as the second term in Eq. 7 vanishes. We summarize the main results of the multiscale approach in Table 4. One can recognize the parameters MD and adh. from the nanoscale friction law (Eq. 2), as well as the results from macroscopic contact mechanics. Their combination into Eq. 7 gives a macroscopic friction coefficient, finally allowing a meaningful comparison with tribometer experiments. This will be done in the following section. At the lowest applied load of 6 mN low friction outcomes were predominant. For normal forces of 130 mN, 310 mN and 1.05 N, high friction was observed in approximately half of the performed experiments, as the applied load approached the mechanical limit of the thermoplastic. Additionally, in the high friction cases a value of exp =0.65 ± 0.06 was observed for the amorphous pairing (20% crystallinity), which is generally higher than exp ≈0.51 ± 0.11 found for PEEK with maximum crystalline content of 40%.

Experimental Results and Comparison with Simulation
Finally, the two friction outcomes of the experiments correlate with the occurrence of wear. Stable, low friction runs showed only a slight flattening of the sphere sample. Conversely, massive abrasive wear occurs at high exp , as material is transferred between the sphere and a wear track on the rod (Fig. 10c). The wear process is likely related to the large temporal fluctuations in the friction coefficient, as material detachment requires localized shearing of the bulk polymer as well as transient changes in surface topography. Interestingly, "crystalline" samples feature larger variations of exp with time compared to the "amorphous" ones, indicating that wear events could be linked to the removal of crystalline spherulites from the surfaces.
Next, we compare the results of the multiscale approach with our tribometer results. The calculated friction coefficients for dry PEEK-PEEK interfaces ( macro = 0.68 and 0.50 for the amorphous and crystalline pairings) are in good agreement with the high friction states observed in the experiments ( exp =0.65 and exp =0.51) -see Fig. 10a, b. As explained in the previous section, the differences in friction between different degrees of crystallinity can be traced back to the elastoplastic material properties of the PEEK samples: the softer the material, the higher the contact area and friction will be.
The good agreement between simulations and experiments in high friction runs hints that, under loading conditions close to the mechanical limit of the polymer, the simplifications performed in our multiscale approach are justified. For instance, recent work on semi-crystalline polymers shows that the roughness spectra at high wavenumbers remains largely unchanged after plastic deformation occurs [48], indicating that changes in surface topography during sliding may not need to be considered in detail for the calculation of the contact area and macroscopic friction. Thus, AFM topography data from unworn samples used in our contact mechanics calculation should suffice. Instead, accounting for mechanical properties, e.g., hardness of different samples is crucial to correctly quantify friction and to explain the lower values consistently observed for the crystalline pairing compared to the amorphous one at high loads. Strain hardening under shearing, which would lead to lower contact area and friction and may explain the slight decrease in friction at long sliding distances, is not accounted for in our simulations as it appears to have only a minor impact on .
Finally, the nanoscale friction law used in our multiscale approach was obtained for the sliding of pure PEEK-PEEK. Although this ideal interface appears to be a strong simplification compared to real systems, it is expected to occur as bulk polymer material is sheared and removed during wear processes characteristic of high friction states from tribometer tests. The results also suggest that our choice to limit the scope of the atomistic modeling to amorphous interfaces is justified, as hard crystalline spherulites are unlikely to be sheared while sliding is localized within the weaker amorphous phase.
It should be noted that our approach does not directly predict low frictions values occurring in approximately half of the experiments. The reason can be pinpointed to the fact that the simulated pure PEEK-PEEK interface is not fully representative of the initial molding skin, which is preserved under sliding in the absence of wear. For more accurate atomistic simulations of the initial interface, further knowledge on the chemical composition of the last nanometers of PEEK surfaces is required.
Another aspect which must be analyzed is the presence of physisorbed molecules from the environment at the tribointerface. In this work water was considered as a first example, showing that its presence in extremely small quantities can significantly reduce adhesion between pure PEEK surfaces. Interestingly, if the adhesion-related shear stress component is fully suppressed from the nanoscale friction law, one obtains a macroscopic friction coefficient macro = MD in very good agreement with the low friction experimental runs. Further work is therefore required to better understand and generalize this promising result, especially on the hydrodynamics and squeeze-out of fluids, such as water or other lubricants, both at the nanometer and macroscopic contact scales.

Conclusions
In this letter, we employ a multiscale simulation method to investigate friction of neat PEEK-PEEK polymer interfaces and compare its results to tribometer experiments. As a first step in the modeling approach, non-reactive Molecular Dynamics simulations are considered. Surfaces with different roughness are cut from nanometer-sized amorphous PEEK blocks with realistic material properties. The surfaces are then paired together and sheared under different normal loads. Tribological processes at the nanoscale involve flattening of nanoscale roughness occurring in a transient regime, and strong alignment of PEEK chains along the sliding direction in the surface-near region.
The steady-state shear stress of the dry PEEK-PEEK interface is high due to the strong adhesion between the chemically affine pure surfaces in dry conditions. Steadystate friction increases linearly with the applied pressure and does not depend significantly on the sliding speed nor the initial geometry of the amorphous surface.
As a first test of the impact of physisorbed fluids on friction, water monolayers are considered in-between the surfaces. Their presence reduces the adhesion component of the shear stress, as the surfaces were separated through water nanochannels aligned in the shearing direction. The linear dependence between shear stress and pressure with adhesion is summarized in the nanoscale friction law MD P, H  In the second step of our multiscale approach, the nanoscale friction law is coupled with elastoplastic contact mechanics simulations based on AFM surface topography and material properties from nanoindentation experiments. Here, the real area of contact is determined at several loads for samples with 40% and 20% crystallinity, the latter material showing larger contact due to its lower penetration hardness. Finally, the nanoscale friction laws are integrated over the calculated contact area, giving friction coefficients of macro ≈0.5 and macro ≈0.7 for the aforementioned PEEK samples in dry conditions. These results are compared with reciprocating tribometer experiments on a ball-on-plate geometry. Excellent agreement is found for high friction cases occurring at high loads close to the operating range of PEEK. Under these conditions intersurface adhesion is high, massive wear occurs and the shearing of bulk polymer is expected during the experiments.
Low friction cases with exp ≈0.2 and no signs of wear were also observed in the experiments, especially at low loads. Although these are not directly reproducible by the multiscale simulations (assuming completely dry contacts), the present results hint at a reduced intersurface adhesion, e.g., through physisorbed fluids. We show as an example that water monolayers can reduce the adhesion component of friction by half. If adhesion is completely suppressed in the calculation of macroscopic friction, macro ≈0.2 is predicted, which corresponds to the value measured during the low friction states. Thus, further work could aim at a more detailed analysis and modeling of surface chemistry, especially regarding the role of lubricants, contaminants, or the polymer molding skin.
Finally, the present multiscale method could be applied to technically relevant surface pairings, such as polymer-metal interfaces, in order to understand the mechanisms and structural transformations leading to transfer film formation and its consequences on friction.