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arxiv: 2604.06890 · v1 · submitted 2026-04-08 · ❄️ cond-mat.mtrl-sci

Recognition: unknown

Microscopic contributions to the deviation from Amontons friction law

Suresh Ravisankar , Ravikant Kumar , Antonio Cammarata , Thilo Glatzel , Tomas Polcar
Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:29 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords nanoscale frictionAmontons lawMX2 monolayersmolecular dynamicssliding modes2D heterostructuresmachine learning force fields
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The pith

Nanoscale friction in MX2 monolayers deviates from Amontons law through competing atomic sliding modes

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models a silicon tip sliding across MX2 monolayers supported on gold or silver using molecular dynamics with machine-learned force fields. It finds that friction does not rise steadily with normal load as classical Amontons law expects; instead the force shows a clear non-monotonic dependence. This arises because the tip follows one of several atomic-scale paths—straight longitudinal sliding, sideways lateral slips, or zig-zag motions—and the share of each path shifts as load increases. Different monolayer-substrate pairs change which path dominates, sometimes suppressing the high-friction mode and lowering overall resistance. The result matters for predicting and tuning friction in atomically thin contacts and coatings.

Core claim

We observe a pronounced non-monotonic dependence of the friction force on the applied normal load, indicating a breakdown of Amontons's law at the nanoscale. Analysis of lateral force signals and their spatial Fourier transforms reveals the coexistence of multiple sliding modes, including longitudinal sliding, lateral slip, and zig-zag motions. We show that the overall friction response is governed by the relative contributions of these motions. While the qualitative features of friction are largely substrate-independent, both the magnitude of friction and the balance between sliding modes depend sensitively on the substrate-monolayer combination. In particular, Au/MoSe2/Si exhibits a strong

What carries the argument

The relative weights of longitudinal sliding, lateral slip, and zig-zag motions whose changing balance with load produces the observed non-monotonic friction

If this is right

  • Au/MoSe2/Si exhibits significantly reduced friction because lateral slip motion is suppressed.
  • The breakdown of Amontons law occurs when the dominant sliding mode shifts with increasing normal load.
  • Qualitative friction features remain similar across substrates while quantitative values and mode balance vary with the specific monolayer-substrate pair.
  • The simulation method extends to probing friction in other 2D heterostructures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Selecting substrate-monolayer pairs that favor low-friction sliding modes could guide design of low-resistance nanoscale interfaces.
  • The same mode-competition mechanism may appear in other van der Waals layered systems where multiple registry states exist.
  • Atomic-force-microscopy experiments that track lateral-force periodicity while varying load could directly observe the predicted mode transitions.

Load-bearing premise

The machine-learning force fields accurately reproduce tip-monolayer and monolayer-substrate interactions at all loads and directions without introducing artificial non-monotonicity.

What would settle it

A measured friction-versus-load curve for the Au/MoSe2/Si system that rises monotonically instead of showing the predicted non-monotonic shape would falsify the claimed breakdown of Amontons law.

Figures

Figures reproduced from arXiv: 2604.06890 by Antonio Cammarata, Ravikant Kumar, Suresh Ravisankar, Thilo Glatzel, Tomas Polcar.

Figure 1
Figure 1. Figure 1: Initial setup to generate strucuture model of TM/MX [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Energy (a–d) and force (e–h) comparisons between predicted and DFT results for [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Energy (a–d) and force (e–h) comparisons between predicted and DFT results for [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: MD simulations setup. Results and Discussion Friction Force Analysis We begin our study of the friction response by considering the friction force profile using 1 ns tra￾jectories under different applied normal loads, as shown in [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Friction force profile of Au/MoS2/Si with tip moving at 2 m/s for under loads of a) 0 nN, b) 0.4 nN, c) 0.6 nN and d) 0.8 nN. where 𝑁𝑠 is the number of time steps, 𝑁𝑡𝑖 𝑝 is the number of atoms in the tip, 𝐹𝑥 𝑗 is the force acting on the 𝑗-th atom of the tip at the 𝑡𝑖 time step along the sliding 𝒙-direction, and the negative sign accounts for the fact that friction opposes the direction of motion. The coeff… view at source ↗
Figure 6
Figure 6. Figure 6: Average friction force as a function of applied load for all systems at tip velocities of (a) [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: COF obtained from a linear fit of the average friction force as a function of loads for all [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Friction force profile associated with the motion of the tip. [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: a) Spatial Fourier transform of Au/MoS2/Si with varying loads at velocity 2 m/s, b) Peak intensity ratio and their average friction force with varying loads for all the systems except Au/MoSe2/Si. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of friction force profile between Au/MoS [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Average friction force as a function of applied loads for all the systems with tip veloci [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Fourier transform of the force on a tip varying with loads and different substrate for ve [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Ratio of the peak intensity I1 and I2 varying with loads and different substrate for veloc￾ity 2 m/s. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Average friction force, varying with loads and different substrate for velocity 2 m/s. [PITH_FULL_IMAGE:figures/full_fig_p023_14.png] view at source ↗
read the original abstract

We investigate the nanoscale friction behaviour of MX2 monolayers (M = Mo, W; X = S, Se) on Au(111) and Ag(111) substrates with a silicon tip using classical molecular dynamics simulations with machine-learning-based force fields. This approach enables an accurate description of tip-surface interactions and friction mechanisms at the atomic scale. We observe a pronounced non-monotonic dependence of the friction force on the applied normal load, indicating a breakdown of Amontons's law at the nanoscale. Analysis of lateral force' signals and their spatial Fourier transforms reveals the coexistence of multiple sliding modes, including longitudinal sliding, lateral slip, and zig-zag motions. We show that the overall friction response is governed by the relative contributions of these motions. While the qualitative features of friction are largely substrate-independent, both the magnitude of friction and the balance between sliding modes depend sensitively on the substrate-monolayer combination. In particular, Au/MoSe2/Si exhibits significantly reduced friction due to suppression of lateral slip motion. Our results indicate that the method is broadly applicable for probing nanoscale friction in related heterostructures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript reports classical molecular dynamics simulations of friction between MX2 monolayers (M = Mo, W; X = S, Se) on Au(111) and Ag(111) substrates using a silicon tip and machine-learning force fields trained on DFT data. The central claim is a pronounced non-monotonic dependence of friction force on normal load, indicating breakdown of Amontons' law at the nanoscale. This is linked to the relative weights of coexisting sliding modes (longitudinal sliding, lateral slip, zig-zag motions) extracted from lateral force signals and their spatial Fourier transforms. Substrate-specific quantitative differences are reported, with Au/MoSe2/Si showing notably lower friction due to suppressed lateral slip.

Significance. If the non-monotonic behavior and mode analysis hold, the work supplies concrete microscopic insight into how sliding-mode balance produces deviations from Amontons' law in 2D heterostructures. The ML-force-field approach enables atomic-scale resolution of tip-monolayer and monolayer-substrate interactions that standard empirical potentials cannot capture. The finding that qualitative friction features are largely substrate-independent while magnitudes and mode weights are sensitive to the specific combination is potentially useful for interface engineering in nanoelectronics and coatings.

major comments (2)
  1. [Methods] Methods section (MLFF training and validation): The machine-learning force fields are stated to be trained on limited DFT data, yet no validation is reported that the models reproduce known load-dependent quantities (e.g., interlayer shear modulus or tip-substrate adhesion energy) for normal loads and sliding directions outside the training distribution. Because the headline non-monotonic friction-versus-load curve and the mode-weight analysis are obtained exclusively from these MD trajectories, the absence of such checks leaves open the possibility that the observed non-monotonicity is an interpolation artifact rather than a physical effect.
  2. [Results] Results section (friction curves and mode analysis): The reported friction-force versus normal-load curves lack error bars derived from independent runs, and no convergence tests with respect to system size or sliding velocity are presented. These omissions make it difficult to assess whether the location of the friction maximum and the extracted relative contributions of longitudinal, lateral-slip, and zig-zag modes are robust or sensitive to simulation parameters.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'lateral force' signals' contains a stray apostrophe that should be removed.
  2. [Results] The spatial Fourier-transform analysis of lateral forces would benefit from an explicit definition of the wave-vector range used to quantify each sliding mode.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We address the major comments point by point below and have made revisions to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Methods] Methods section (MLFF training and validation): The machine-learning force fields are stated to be trained on limited DFT data, yet no validation is reported that the models reproduce known load-dependent quantities (e.g., interlayer shear modulus or tip-substrate adhesion energy) for normal loads and sliding directions outside the training distribution. Because the headline non-monotonic friction-versus-load curve and the mode-weight analysis are obtained exclusively from these MD trajectories, the absence of such checks leaves open the possibility that the observed non-monotonicity is an interpolation artifact rather than a physical effect.

    Authors: We agree that additional validation would enhance confidence in the MLFF predictions. Although our training dataset incorporated DFT data across a range of relevant configurations including varying loads, we did not explicitly report out-of-distribution tests for shear modulus and adhesion energies. In the revised manuscript, we include new validation results demonstrating that the MLFF accurately reproduces these quantities at loads and directions not directly in the training set, confirming the physical origin of the non-monotonic friction behavior. revision: yes

  2. Referee: [Results] Results section (friction curves and mode analysis): The reported friction-force versus normal-load curves lack error bars derived from independent runs, and no convergence tests with respect to system size or sliding velocity are presented. These omissions make it difficult to assess whether the location of the friction maximum and the extracted relative contributions of longitudinal, lateral-slip, and zig-zag modes are robust or sensitive to simulation parameters.

    Authors: We acknowledge these omissions in the original manuscript. We have now conducted additional simulations to test convergence with respect to system size and sliding velocity, and have included error bars calculated from multiple independent runs in the updated figures. The position of the friction maximum and the relative mode contributions remain consistent across these tests, supporting the robustness of our findings. revision: yes

Circularity Check

0 steps flagged

No circularity; results are direct outputs of MD trajectories with ML force fields

full rationale

The paper presents no analytic derivation, fitted functional form, or parameter prediction. All reported friction curves, non-monotonic load dependence, and sliding-mode analysis are direct numerical outputs from classical molecular-dynamics trajectories. No self-citation chain, ansatz, or uniqueness theorem is invoked to justify the central observations; the ML force fields are used as a computational tool whose training details are external to the reported results. This is the most common honest non-finding for simulation-only studies.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract; the central claim rests on the assumption that the chosen ML force fields and classical MD setup capture the relevant physics without major artifacts.

axioms (1)
  • domain assumption Classical molecular dynamics with machine-learned force fields sufficiently reproduces the atomic-scale tip-surface and interlayer interactions under the simulated loads and velocities.
    Invoked by the choice of simulation method to generate the friction curves and mode analysis.

pith-pipeline@v0.9.0 · 5509 in / 1258 out tokens · 75991 ms · 2026-05-10T17:29:27.402144+00:00 · methodology

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Reference graph

Works this paper leans on

47 extracted references · 41 canonical work pages

  1. [1]

    doi:https://doi

    Sarkar, M.; Mandal, N.Materials Today: Proceedings2022,66, 3762–3768. doi:https://doi. org/10.1016/j.matpr.2022.06.030

    work page doi:10.1016/j.matpr.2022.06.030 2022

  2. [2]

    Vanossi, A.; Manini, N.; Urbakh, M.; Zapperi, S.; Tosatti, E.Rev. Mod. Phys.2013,85, 529–552. doi:10.1103/RevModPhys.85.529

    work page doi:10.1103/revmodphys.85.529 2013

  3. [3]

    doi:https://doi.org/ 10.1016/j.mattod.2018.12.002

    Zhang, S.; Ma, T.; Erdemir, A.; Li, Q.Materials Today2019,26, 67–86. doi:https://doi.org/ 10.1016/j.mattod.2018.12.002

    work page doi:10.1016/j.mattod.2018.12.002 2018

  4. [4]

    H.; Li, Q.; Dong, M.Materials Today Physics2022,27, 100771

    Zhang, D.; Li, Z.; Klausen, L. H.; Li, Q.; Dong, M.Materials Today Physics2022,27, 100771. doi:https://doi.org/10.1016/j.mtphys.2022.100771. 24

    work page doi:10.1016/j.mtphys.2022.100771 2022

  5. [5]

    L.; Rosenkranz, A.Advanced Materials Inter- faces2022,9(3), 2101622

    Marian, M.; Berman, D.; Rota, A.; Jackson, R. L.; Rosenkranz, A.Advanced Materials Inter- faces2022,9(3), 2101622. doi:https://doi.org/10.1002/admi.202101622

    work page doi:10.1002/admi.202101622

  6. [6]

    Wang, J.; Khosravi, A.; Vanossi, A.; Tosatti, E.Rev. Mod. Phys.2024,96, 011002. doi:10. 1103/RevModPhys.96.011002

    2024

  7. [7]

    doi: https://doi.org/10.1016/j.progsurf.2024.100753

    Cammarata, A.; Perviz, E.; Polcar, T.Progress in Surface Science2024,99(3), 100753. doi: https://doi.org/10.1016/j.progsurf.2024.100753

    work page doi:10.1016/j.progsurf.2024.100753 2024

  8. [8]

    M.; Donnet, C.; Le Mogne, T.; Epicier, T.Phys

    Martin, J. M.; Donnet, C.; Le Mogne, T.; Epicier, T.Phys. Rev. B1993,48, 10583–10586. doi:10.1103/PhysRevB.48.10583

    work page doi:10.1103/physrevb.48.10583

  9. [9]

    Nicolás, The bar derived category of a curved dg algebra, Journal of Pure and Applied Algebra 212 (2008) 2633–2659

    Mukhtar, S. H.; Gulzar, A.; Saleem, S.; Wani, M.; Sehgal, R.; Yakovenko, A.; Goryacheva, I.; Sharma, M. D.Tribology International2024,192, 109194. doi:https://doi.org/10.1016/j. triboint.2023.109194

    work page doi:10.1016/j 2023

  10. [10]

    doi:https://doi.org/10.1002/zamm

    Prandtl, L.ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Ange- wandte Mathematik und Mechanik1928,8(2), 85–106. doi:https://doi.org/10.1002/zamm. 19280080202

    work page doi:10.1002/zamm

  11. [11]

    doi:10.1080/14786440608564819

    Tomlinson, G.The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science1929,7(46), 905–939. doi:10.1080/14786440608564819

    work page doi:10.1080/14786440608564819

  12. [12]

    Cammarata, A.; Nicolini, P.; Simonovic, K.; Ukraintsev, E.; Polcar, T.Phys. Rev. B2019,99, 094309. doi:10.1103/PhysRevB.99.094309

    work page doi:10.1103/physrevb.99.094309

  13. [13]

    doi:10.1021/acsnano.3c02915

    Bai, H.; Zou, G.; Bao, H.; Li, S.; Ma, F.; Gao, H.ACS Nano2023,17(13), 12594–12602. doi:10.1021/acsnano.3c02915

    work page doi:10.1021/acsnano.3c02915

  14. [14]

    W.; Hone, J.Science2010,328 (5974), 76–80

    Lee, C.; Li, Q.; Kalb, W.; Liu, X.-Z.; Berger, H.; Carpick, R. W.; Hone, J.Science2010,328 (5974), 76–80. doi:10.1126/science.1184167

    work page doi:10.1126/science.1184167

  15. [15]

    doi: 10.1021/acs.nanolett.3c03642

    Li, Y.; Wu, B.; Ouyang, W.; Liu, Z.; Wang, W.Nano Letters2024,24(4), 1130–1136. doi: 10.1021/acs.nanolett.3c03642. 25

    work page doi:10.1021/acs.nanolett.3c03642

  16. [16]

    doi: 10.1088/1361-6528/aa712b

    Fang, L.; Liu, D.-M.; Guo, Y.; Liao, Z.-M.; Luo, J.-B.; Wen, S.-Z.2017,28(24), 245703. doi: 10.1088/1361-6528/aa712b

    work page doi:10.1088/1361-6528/aa712b 2017

  17. [17]

    R.; Hasz, K.; Zhao, M.-Q.; Johnson, A

    Vazirisereshk, M. R.; Hasz, K.; Zhao, M.-Q.; Johnson, A. T. C.; Carpick, R. W.; Martini, A. ACS Nano2020,14(11), 16013–16021. doi:10.1021/acsnano.0c07558

    work page doi:10.1021/acsnano.0c07558

  18. [18]

    URLhttps://advanced.onlinelibrary.wiley.com/doi/abs/10.1002/advs

    Yao, Y.; Song, Y.; Wu, B.; Scherb, S.; Huang, S.; Hinaut, A.; Glatzel, T.; Meyer, E.; Liu, Z.; Ouyang, W.Advanced Science2025,12(21), 2415884. doi:https://doi.org/10.1002/advs. 202415884

    work page doi:10.1002/advs

  19. [19]

    Gao, X.; Urbakh, M.; Hod, O.Phys. Rev. Lett.2022,129, 276101. doi:10.1103/PhysRevLett. 129.276101

    work page doi:10.1103/physrevlett 2022

  20. [20]

    doi:https://doi.org/10.1016/j.ceramint.2018.10.139

    Shi, Y.; Cai, Z.; Pu, J.; Wang, L.; Xue, Q.Ceramics International2019,45(2, Part A), 2258–2265. doi:https://doi.org/10.1016/j.ceramint.2018.10.139

    work page doi:10.1016/j.ceramint.2018.10.139 2018

  21. [21]

    H.; van Duin, A

    Ostadhossein, A.; Rahnamoun, A.; Wang, Y.; Zhao, P.; Zhang, S.; Crespi, V. H.; van Duin, A. C. T.The Journal of Physical Chemistry Letters2017,8(3), 631–640. doi:10.1021/acs.jpclett. 6b02902

    work page doi:10.1021/acs.jpclett

  22. [22]

    W.; Shenderova, O

    Brenner, D. W.; Shenderova, O. A.; Harrison, J. A.; Stuart, S. J.; Ni, B.; Sinnott, S. B.Journal of Physics: Condensed Matter2002,14(4), 783. doi:10.1088/0953-8984/14/4/312

    work page doi:10.1088/0953-8984/14/4/312

  23. [23]

    doi:https: //doi.org/10.1002/smll.202510186

    Koczanowski, P.; Nicolini, P.; Khaksar, H.; Gnecco, E.Small2025,21(50), e10186. doi:https: //doi.org/10.1002/smll.202510186

    work page doi:10.1002/smll.202510186

  24. [24]

    & Choudhary, A

    Agrawal, A.; Choudhary, A.APL Materials2016,4(5), 053208. doi:10.1063/1.4946894

    work page doi:10.1063/1.4946894

  25. [25]

    Gaussian Approximation Potentials:

    Bartók, A. P.; Csányi, G.International Journal of Quantum Chemistry2015,115(16), 1051–1057. doi:https://doi.org/10.1002/qua.24927

    work page doi:10.1002/qua.24927

  26. [26]

    E.; Toth, D

    Yao, K.; Herr, J. E.; Toth, D. W.; Mckintyre, R.; Parkhill, J.Chem. Sci.2018,9, 2261–2269. doi:10.1039/C7SC04934J. 26

    work page doi:10.1039/c7sc04934j 2018

  27. [27]

    Rizzo, T.Phys. Rev. B2021,104, 094203. doi:10.1103/PhysRevB.104.094203

    work page doi:10.1103/physrevb.104.094203

  28. [28]

    Behler, J.; Parrinello, M.Phys. Rev. Lett.2007,98, 146401. doi:10.1103/PhysRevLett.98. 146401

    work page doi:10.1103/physrevlett.98 2007

  29. [29]

    npj Computational Materials11(1), 273 (2025) https://doi.org/10.1038/s41524-025-01761-9

    Shaidu, Y.; Naik, M. H.; Louie, S. G.; Neaton, J. B.npj Computational Materials2025,11 (1), 273. doi:https://doi.org/10.1038/s41524-025-01761-9

    work page doi:10.1038/s41524-025-01761-9

  30. [30]

    doi:https://doi.org/10.1016/j.cpc.2020.107624

    Lu, D.; Wang, H.; Chen, M.; Lin, L.; Car, R.; E, W.; Jia, W.; Zhang, L.Computer Physics Communications2021,259, 107624. doi:https://doi.org/10.1016/j.cpc.2020.107624

    work page doi:10.1016/j.cpc.2020.107624 2020

  31. [31]

    F.; Selloni, A.Phys

    Calegari Andrade, M. F.; Selloni, A.Phys. Rev. Mater.2020,4, 113803. doi:10.1103/ PhysRevMaterials.4.113803

    2020

  32. [32]

    doi:10.1063/5.0025051

    Li, R.; Liu, Z.; Rohskopf, A.; Gordiz, K.; Henry, A.; Lee, E.; Luo, T.Applied Physics Letters 2020,117(15), 152102. doi:10.1063/5.0025051

    work page doi:10.1063/5.0025051 2020

  33. [33]

    doi:10.1063/5.0201527

    Liu, X.; Wang, B.; Jia, K.; Wang, Q.; Wang, D.; Xiong, Y.Journal of Applied Physics2024, 135(20), 205107. doi:10.1063/5.0201527

    work page doi:10.1063/5.0201527

  34. [34]

    doi:10.1021/acs.jpcc.4c08188

    Zhou, J.; Fu, Y.; Liu, L.; Liu, C.The Journal of Physical Chemistry C2025,129(13), 6414–6422. doi:10.1021/acs.jpcc.4c08188

    work page doi:10.1021/acs.jpcc.4c08188

  35. [35]

    Kresse, G.; Furthmüller, J.Phys. Rev. B1996,54, 11169–11186. doi:10.1103/PhysRevB.54. 11169

    work page doi:10.1103/physrevb.54

  36. [36]

    Blöchl, P. E.Phys. Rev. B1994,50, 17953–17979. doi:10.1103/PhysRevB.50.17953

    work page doi:10.1103/physrevb.50.17953

  37. [37]

    P.; Burke, K.; Ernzerhof, M.Phys

    Perdew, J. P.; Burke, K.; Ernzerhof, M.Phys. Rev. Lett.1996,77, 3865–3868. doi:10.1103/ PhysRevLett.77.3865

    1996

  38. [38]

    J.; Pack, J

    Monkhorst, H. J.; Pack, J. D.Phys. Rev. B1976,13, 5188–5192. doi:10.1103/PhysRevB.13. 5188. 27

    work page doi:10.1103/physrevb.13

  39. [39]

    The Journal of Chemical Physics , volume =

    Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H.The Journal of Chemical Physics2010,132 (15), 154104. doi:10.1063/1.3382344

    work page doi:10.1063/1.3382344

  40. [40]

    Zhang, L.; Han, J.; Wang, H.; Car, R.; E, W.Phys. Rev. Lett.2018,120, 143001. doi:10.1103/ PhysRevLett.120.143001

    2018

  41. [41]

    Fusion2022,62, 126013

    Wang, X.; Wang, Y.; Zhang, L.; Dai, F.; Wang, H.Nucl. Fusion2022,62, 126013. doi:10. 1088/1741-4326/ac888b

  42. [42]

    doi:https://doi.org/10.1016/j.cpc.2020.107206

    Zhang, Y.; Wang, H.; Chen, W.; Zeng, J.; Zhang, L.; Wang, H.; E, W.Computer Physics Com- munications2020,253, 107206. doi:https://doi.org/10.1016/j.cpc.2020.107206

    work page doi:10.1016/j.cpc.2020.107206 2020

  43. [43]

    and Aktulga, H

    Thompson, A. P.; Aktulga, H. M.; Berger, R.; Bolintineanu, D. S.; Brown, W. M.; Crozier, P. S.; in ’t Veld, P. J.; Kohlmeyer, A.; Moore, S. G.; Nguyen, T. D.; Shan, R.; Stevens, M. J.; Tranchida, J.; Trott, C.; Plimpton, S. J.Computer Physics Communications 2022,271, 108171. doi:https://doi.org/10.1016/j.cpc.2021.108171

    work page doi:10.1016/j.cpc.2021.108171 2022

  44. [44]

    doi: https://doi.org/10.1007/BF01174717

    Suh, I.-K.; Ohta, H.; Waseda, Y.Journal of Materials Science1988,23(2), 757–760. doi: https://doi.org/10.1007/BF01174717

    work page doi:10.1007/bf01174717

  45. [45]

    D.; Kharabadze, S.; Kolmogorov, A

    Hajinazar, S.; Thorn, A.; Sandoval, E. D.; Kharabadze, S.; Kolmogorov, A. N.Computer Physics Communications2021,259, 107679. doi:https://doi.org/10.1016/j.cpc.2020.107679

    work page doi:10.1016/j.cpc.2020.107679 2020

  46. [46]

    Song, Y.; Wang, J.; Hinaut, A.; Scherb, S.; Huang, S.; Glatzel, T.; Tosatti, E.; Meyer, E.Phys. Rev. Lett.2024,133, 136201. doi:10.1103/PhysRevLett.133.136201

    work page doi:10.1103/physrevlett.133.136201 2024

  47. [47]

    Momma, K.; Izumi, F.Journal of Applied Crystallography2008,41(3), 653–658. doi:10. 1107/S0021889808012016. 28 Supporting Information for Microscopic contributions to the deviation from Amontons friction law Suresh Ravisankar∗1, Ravikant Kumar∗1, Antonio Cammarata1, Thilo Glatzel2 and Tomas Polcar1 Address: 1Department of Control Engineering, Faculty of El...