The Prandtl–Tomlinson Model's Resurgence in Atomic-Scale Friction Research

Despite being largely overlooked for decades, the Prandtl–Tomlinson model is now recognised as a fundamental framework for understanding atomic-scale friction.

The Prandtl–Tomlinson model, introduced in 1928, laid the groundwork for our understanding of single-atom contact friction. Initially described as a point mass being dragged over a sinusoidal potential by a spring, this model remained relatively obscure for many years. 

However, recent experimental validations have highlighted its importance for contacts involving tens to hundreds of atoms, bringing it back into the limelight. 

Today, the Prandtl–Tomlinson model is celebrated as a highly insightful mechanical analogue, offering deep insights into atomic-scale phenomena at sliding interfaces.

Nanotribology, the study of friction at the nanoscale, emerged as a significant field in the late 1980s. This marked a pivotal moment in friction research, driven by the development of tools like the atomic force microscope (AFM) that allowed scientists to study nanoscale contacts. 

One of the most notable breakthroughs was the observation of individual atomic “hopping events” over a corrugated interface potential with atomic periodicity. These hopping events revealed a “stick-slip” motion, sparking renewed interest in the Prandtl–Tomlinson (PT) model, an older conceptual framework.

Historically, the PT model is often attributed to G. A. Tomlinson’s 1929 paper, hence its alternate name, the “Tomlinson model.” However, the foundational theory was first presented in Ludwig Prandtl's 1928 paper, which was initially inaccessible to much of the scientific community due to its publication in German. 

This oversight was corrected in 2012, when an English translation of Prandtl's seminal work was published, finally clarifying its foundational role in modern nanotribology.

The PT model has since become a cornerstone of our understanding of atomic-scale friction. According to this model, a point mass moves over a periodic potential, providing a comprehensive explanation for a wide range of experimental observations. 

The basics of the PT model show a point mass connected to a supporting body via a spring with an effective spring constant. The mass interacts with a periodic potential, and during sliding, the supporting body moves with a certain velocity. 

If the spring is sufficiently soft, the resulting motion exhibits characteristic “stick-slip” behaviour, where the point mass remains in a potential minimum until the spring tension reaches a critical value, causing it to jump to the next minimum. Temperature effects can also be considered by introducing thermal oscillations of the mass, influencing the overall frictional behaviour.

At zero temperature, the lateral force displays a sawtooth-like function, with the mass jumping only when the critical force for that potential and spring constant is reached. Higher temperatures allow the mass to jump at lower forces due to thermal activation, leading to a decrease in the overall frictional force and the visibility of thermal noise in the force pattern.

Prandtl's macroscopic mechanical model demonstrated similar stick-slip behaviour, illustrating the applicability of the PT model. The PT model has successfully explained friction phenomena across various systems, including flat surfaces, different atom types, and ions in a trap. Despite its origins as a single-atom model, it provides intuitive explanations for many fundamental properties of dry friction.

However, the PT model does have its limitations. Extrapolating its findings to larger, macroscopic contacts remains a challenge, as scaling up from nanometre-sized contacts to macroscopic rough interfaces has not yet been fully resolved. 

The model primarily addresses idealised, single-atom contacts, whereas real-world surfaces often feature defects, impurities, and varying chemical activities that influence friction and are not adequately captured by the model.

Factors like “structural lubricity” or “super-lubricity” at atomic and small-scale contacts also pose challenges for the PT model, as it does not fully account for friction behaviour when the contact area is large. 

The scaling of friction with contact area remains a significant challenge, and the model's predictions for real, rough contacts are still incomplete.

 

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