Microscopic contributions to the deviation from Amontons friction law

Using machine-learning-based molecular dynamics simulations, this study reveals that the nanoscale friction of MX2 monolayers on metal substrates deviates from Amontons' law due to a non-monotonic load dependence arising from the interplay of multiple sliding modes, with specific substrate-monolayer combinations like Au/MoSe2 significantly reducing friction by suppressing lateral slip.

The Gist

Imagine you are trying to slide a heavy book across a table. In the old days, scientists thought friction worked like a simple math equation: Push harder, and it gets harder to slide. This rule is called "Amontons' Law." It's like saying, "If you press your hand down on a table, it takes twice as much effort to push it if you press twice as hard."

But this new paper says, "Wait a minute! At the tiny, microscopic world, things don't work that simply."

Here is the story of what the researchers found, explained with some everyday analogies.

1. The Setup: A Tiny Dance Floor

The scientists built a tiny, virtual dance floor in their computer.

• The Floor: A sheet of metal (Gold or Silver).
• The Dancer: A single layer of special material (like a sandwich of Molybdenum and Sulfur) sitting on the metal.
• The Partner: A tiny silicon tip (like the needle on a record player) that slides across the dancer.

They used a super-smart computer program (trained by Artificial Intelligence) to watch how this tiny tip moves. They wanted to see if the "Push harder = Slide harder" rule still held up.

2. The Surprise: The "Wobbly" Slide

When they pushed the tip harder (increasing the load), they expected the friction to go up in a straight line. Instead, the friction acted weirdly. Sometimes, pushing harder actually made it easier to slide, or the friction jumped up and down unpredictably.

The Analogy: Imagine walking on a frozen lake.

• The Old Rule: If you carry a heavy backpack, you sink deeper and slide slower.
• The New Reality: Sometimes, carrying a heavy backpack changes the way your feet hit the ice. You might start "skating" or "sliding sideways" in a way that actually reduces the resistance. The relationship isn't a straight line; it's a bumpy, unpredictable path.

3. The Secret Culprit: The "Zig-Zag" Dance

Why did this happen? The researchers looked closely at how the tip moved. They found that the tip wasn't just moving in a straight line (forward). It was doing a complex dance:

1. The Slide: Moving forward (the main goal).
2. The Lateral Slip: Suddenly jerking sideways.
3. The Zig-Zag: Wiggling back and forth like a snake.

The Analogy: Think of a person trying to walk through a crowded hallway.

• If they walk in a straight line, they bump into people (high friction).
• But if they start wiggling and slipping sideways to dodge the crowd, they might actually get through faster with less effort!

The researchers found that when the tip started doing this "Zig-Zag" dance, the friction dropped. The more the tip wiggled sideways, the less "grip" it had, and the easier it was to slide.

4. The Magic Carpet: Why Some Combinations Work Better

The scientists tested different materials. They found that one specific combination—Gold + Molybdenum Selenide + Silicon—was a superstar.

The Analogy: Imagine two different types of shoes on two different floors.

• Shoe A on Floor A: You get stuck. You have to drag your feet. High friction.
• Shoe B on Floor B: You are wearing ice skates. You glide effortlessly.

In the "Gold + Molybdenum Selenide" case, the tip stopped doing the Zig-Zag dance entirely. It just slid straight forward. Because it didn't wiggle, it didn't get "stuck" in the sideways motions that usually lower friction. Wait, that sounds bad for sliding, right? Actually, in this specific case, the absence of the weird sideways wiggles meant the system behaved more predictably, but the overall friction was surprisingly low because the surface was so smooth and the atoms were arranged perfectly to let the tip glide without getting caught.

(Correction based on the text: The text actually says Au/MoSe2/Si has reduced friction because the lateral slip is suppressed. This implies that in other systems, the lateral slip was causing more energy loss or instability, but in this specific "perfect" combo, the lack of chaotic sideways motion made the sliding incredibly smooth and efficient.)

5. The Big Takeaway

The main lesson of this paper is that friction at the nanoscale is chaotic.

• Old View: Friction is a simple number (Coefficient of Friction).
• New View: Friction is a complex dance involving sideways slips, wiggles, and atomic vibrations.

If you want to build tiny machines (like micro-robots or future computer chips), you can't just use the old rules. You have to understand the "dance moves" of the atoms. Sometimes, making a surface smoother isn't enough; you have to control how the atoms wiggle sideways.

In a nutshell: The paper tells us that at the microscopic level, friction isn't just about how hard you push. It's about how the tiny particles "dance" and wiggle. If you can control the dance, you can control the friction.

Technical Summary

1. Problem Statement

The study addresses the breakdown of Amontons' Law at the nanoscale. While Amontons' Law posits a linear relationship between friction force ($F_f$) and normal load ($F_N$) ($F_f = \mu F_N$), experimental and theoretical evidence suggests this linearity often fails at the atomic scale.

• The Gap: Existing theoretical models, such as the Prandtl-Tomlinson model, often fail to capture complex atomic-scale phenomena like lateral slips, dynamic deformations, and thermal fluctuations.
• The Challenge: Understanding the microscopic origins of non-monotonic friction-load relationships and the significant uncertainty in extracting the coefficient of friction ($\mu$) via linear fitting in Transition Metal Dichalcogenide (TMD) heterostructures.
• Context: TMDs (e.g., MoS$_2$, WS$_2$) are promising solid lubricants, but their dynamic behavior during sliding on metallic substrates (Au, Ag) under a scanning probe tip remains complex and not fully understood.

2. Methodology

The authors employed a multi-scale computational approach combining Density Functional Theory (DFT) and Machine Learning (ML) driven Molecular Dynamics (MD).

• System Configuration:
  • Materials: MX$_2$ monolayers (M = Mo, W; X = S, Se) deposited on Au(111) and Ag(111) substrates.
  • Tip: A Silicon (Si) tip modeled as a rigid body.
  • Simulation Cell: A $3 \times 3 \times 1$ supercell containing 360 atoms (144 substrate, 36 M, 72 X, 108 Si).
• Force Field Development:
  • DFT Training: Structural optimization was performed using VASP (PAW method, PBE-GGA, Grimme D3 correction for van der Waals forces).
  • Machine Learning Force Field (MLFF): A Deep Neural Network (NNP) potential (DeepPot-SE model via DeepMD-kit) was trained on DFT data.
  • Active Learning: The DP-GEN workflow was used to iteratively generate training data by perturbing atomic positions and cell sizes. Convergence was achieved when the standard deviation of forces predicted by four independent models fell within a specific range ($0.05 < \sigma_{max} < 0.15$).
  • Validation: The MLFF achieved low Root Mean Square Errors (RMSE) for energy (1 meV/atom) and forces (0.02 eV/Å), and reproduced DFT-optimized geometries with high fidelity (RDF scalar products > 0.87).
• MD Simulations:
  • Software: LAMMPS using the trained MLFF.
  • Conditions: NVT ensemble at 300 K.
  • Parameters: Sliding velocities of 2, 3, 4, and 5 m/s; Normal loads of 0.0, 0.4, 0.6, and 0.8 nN.
  • Analysis: Friction forces were analyzed via time-series averaging and Spatial Fourier Transforms (FT) of the lateral force signals to decompose motion modes.

3. Key Contributions

• High-Fidelity MLFF for TMDs: Successfully developed and validated machine learning force fields for complex TM/MX$_2$/Si heterostructures, enabling large-scale, long-duration MD simulations with near-DFT accuracy but at a fraction of the computational cost.
• Decomposition of Friction Mechanisms: Introduced a Fourier analysis method to quantitatively distinguish between three distinct sliding modes:
  1. Longitudinal Sliding: Motion along the primary sliding direction ($x$).
  2. Lateral Slip: Transverse displacement ($y$-direction).
  3. Zig-Zag Motion: Relaxation movements coupled with lateral slip.
• Mechanism of Amontons' Law Breakdown: Demonstrated that the non-monotonic friction-load relationship is not a random fluctuation but a direct result of the varying relative contributions of these sliding modes as the load changes.

4. Key Results

• Non-Monotonic Friction-Load Relationship:
  • Across almost all systems (Au/Ag substrates with Mo/W and S/Se combinations), the average friction force did not scale linearly with the normal load.
  • Linear fitting to extract $\mu$ resulted in large errors (e.g., $\mu$ values ranging from 0.0009 to 0.051 with significant standard deviations), confirming the invalidity of Amontons' Law in these regimes.
• Role of Lateral Motion:
  • Correlation: The intensity of peaks in the Fourier transform corresponding to lateral slip and zig-zag motions ($I_2, I_3$) was inversely correlated with the average friction force.
  • Mechanism: When lateral slip and zig-zag motions are active, the tip explores multiple local minima on the potential energy surface, effectively reducing the energy dissipation along the primary sliding direction.
  • Exception (Au/MoSe$_2$/Si): This specific system exhibited significantly reduced friction. The Fourier analysis revealed the absence of lateral slip and zig-zag peaks; the tip moved strictly along the $x$-direction. Paradoxically, the suppression of lateral motion in this specific case led to a unique friction profile, highlighting the sensitivity of the system to the specific monolayer-substrate combination.
• Substrate Dependence:
  • Qualitative Similarity: The nature of the friction behavior (non-monotonicity, presence of multiple sliding modes) was largely independent of the substrate (Au vs. Ag).
  • Quantitative Differences: The magnitude of friction and the specific balance between sliding modes depended sensitively on the substrate. For instance, Ag/MoS$_2$/Si showed higher friction than Au/MoS$_2$/Si, while Au/WSe$_2$/Si showed higher friction than Ag/WSe$_2$/Si.
• Velocity Independence: The non-monotonic features and the breakdown of Amontons' law were observed consistently across all tested velocities (2–5 m/s), indicating that the phenomenon is driven by structural and energetic factors rather than kinetic effects.

5. Significance

• Theoretical Advancement: The study moves beyond simple 1D friction models by proving that 2D lateral motions are critical contributors to nanoscale friction. It provides a quantitative framework (Fourier analysis of force signals) to characterize these complex trajectories.
• Predictive Capability: The developed MLFFs offer a robust tool for probing nanoscale friction in a wide range of heterostructures without the prohibitive cost of DFT.
• Material Design: The findings suggest that friction can be tuned by selecting specific monolayer-substrate combinations that either suppress or promote lateral slip. For example, the unique behavior of Au/MoSe$_2$/Si suggests a pathway for designing ultra-low friction interfaces by controlling the suppression of specific lateral modes.
• Resolution of Uncertainty: The paper explains why experimental extraction of the coefficient of friction in nanoscale systems often yields inconsistent results, attributing it to the non-linear, mode-dependent nature of atomic sliding rather than measurement error.

In conclusion, the paper establishes that the deviation from Amontons' law at the nanoscale is a fundamental consequence of the coupling between longitudinal sliding and lateral atomic motions, which vary non-linearly with applied load and are highly sensitive to the specific chemical nature of the interface.