# Understanding the Tomlinson Model of Sliding Friction

• Leo Liu
In summary, the Tomlinson model, also known as the Prandtl-Tomlinson model, explains friction at a nanoscale by using a spring dragged over a corrugated substrate. The potential energy curve of the substrate is periodic due to the corrugation. The potential energy between the nanotip and the substrate is due to intermolecular forces. The potential energy of the spring is quadratic due to the constantly changing length of the spring. The corrugation of the potential energy is proportional to the normal force. This is because the actual contact area on an atomic level is a small fraction of the total surface area, and the pressure, or force per unit area, determines the contact area. The model also includes
Leo Liu
I have just watched a video lecture about the Tomlinson model (aka Prandtl–Tomlinson Model) which explains the friction mechanism at a nanoscale by analogy with a tip dragged by a spring. The video has prompted me to ask several questions.

The spring is dragged over a corrugated substrate, as shown in the picture above. Because of the corrugation, the potential energy curve of the substrate is periodic.
However, I am not quite sure why there is some potential energy between the nanotip and the substrate-- my guess is that the potential energy is stored within the intermolecular forces between the tip and the molecules in the substrate. Could someone provide an explanation?

Then he mentions that the curve of the potential energy of the spring is quadratic. Because the length of the spring is constantly changing, I wonder why the EP curve of the spring does not show any periodicity.

Last, why is the corrugation of the potential energy proportional to the normal force?

I would also like to invite you to share your knowledge and insight into this model or the mechanism of friction on the atomic scale. Thank you.

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etotheipi
Hey Leo, it's quite an interesting topic! As far as I can tell, we take the extension of the spring to be approximately the horizontal displacement of the point of attachment (moving at ##v##) from the position of the particle, resulting in an elastic potential energy of ##\frac{1}{2}k(x-vt)^2##. We model the total potential energy as $$V = V_{\text{interaction}} + V_{\text{elastic}} = V_0\cos{\left(\frac{2\pi}{a}x \right)} + \frac{1}{2}k(x-vt)^2$$if ##x## is the position of the particle, ##a## the interatomic spacing of the substrate, and ##v## the speed of the tip. That is the squiggly potential he has drawn. Also, you must certainly be correct that the first term is due to electric interactions with the substrate. If we also add in a damping force ##-\eta \vec{v}## then the differential equation describing the motion is$$m\ddot{x} = k(vt-x) - N\sin{\left(\frac{2\pi}{a} x \right)} -\eta \dot{x}$$with ##N = -\frac{2\pi V_0}{a}##. From this you can derive a few special cases for the friction, as is nicely explained here, although they use ##F## for the external force on the particle from the spring, which I have called ##k(vt - x)## so that it is more general than just the case of uniform relative velocity.

However I do not know why the potential energy is proportional to the normal force. I too would be interested in a justification for this!

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Leo Liu
etotheipi said:
Hey Leo, it's quite an interesting topic! As far as I can tell...
Hey James. I am not sure what you meant by "damping force". For a simple harmonic oscillator in real life, the dumping force is basically the friction force. However, since we are talking about the essence of kinetic friction here, the macroscopic friction force should not exist at a nanoscale. I would be grateful if you can inform me what the dumping force is in the equation you wrote.

Redrafting after thinking about it a bit more...

The damping term is there to account for dissipation of heat. A bit more digging tells me that this is actually an example of a Langevin equation and is really within the realm of statistical mechanics.

I cannot add much more, since I too am struggling to make out what is happening. I am going through this reference: http://www.iop.kiev.ua/~obraun/book_md/book_md_4.pdf

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etotheipi said:
Redrafting after thinking about it a bit more...

The damping term is there to account for dissipation of heat. A bit more digging tells me that this is actually an example of a Langevin equation and is really within the realm of statistical mechanics.

I cannot add much more, since I too am struggling to make out what is happening. I am going through this reference: http://www.iop.kiev.ua/~obraun/book_md/book_md_4.pdf
Everything on a microscope level behaves differently than the macroscopic objects. Perhaps it is insanity to apply our intuition based on large objects to the behaviour of tiny particles .

etotheipi said:
However I do not know why the potential energy is proportional to the normal force. I too would be interested in a justification for this!
I think I have found an answer to this question in Kleppner's mechanics book.
An Introduction to Mechanics said:
"In many cases the maximum value of the friction between two surfaces is found to be essentially independent of the area of contact, which may seem strange. The reason is that the actual area of contact on an atomic scale is a minute fraction of the total surface area. This fraction is proportional to the pressure, that is, the force per unit area. If the areais doubled while the normal force is held constant, then the pressure is halved. Thus there is twice as much contact area, but only half as much microscopic force on each. The resultant total friction force, the product of contact area and microscopic force, is unchanged. This is an oversimplified explanation and non-rigid bodies like automobile tires are more complicated. A wide tire is generally better than a narrow one for good acceleration and braking."

etotheipi

## What is the Tomlinson Model of Sliding Friction?

The Tomlinson Model of Sliding Friction is a theoretical model that explains the phenomenon of sliding friction between two surfaces. It was developed by physicist Stephen Tomlinson in the 1920s and is based on the concept of atomic interactions between the surfaces.

## How does the Tomlinson Model explain sliding friction?

The Tomlinson Model proposes that when two surfaces are in contact, the atoms on the surface of one material interact with the atoms on the surface of the other material. This interaction creates a potential energy barrier that must be overcome in order for the surfaces to slide against each other. This barrier is what causes the resistance or friction between the surfaces.

## What factors affect sliding friction according to the Tomlinson Model?

The Tomlinson Model states that the magnitude of sliding friction is affected by the strength of the atomic interactions between the two surfaces, the roughness of the surfaces, and the applied load or pressure between the surfaces. These factors can all influence the height of the potential energy barrier and therefore the amount of friction between the surfaces.

## How is the Tomlinson Model different from other models of friction?

The Tomlinson Model differs from other models of friction in that it takes into account the atomic interactions between the surfaces, rather than just the macroscopic properties of the materials. This allows for a more accurate and detailed understanding of the factors that contribute to sliding friction.

## What are the practical applications of the Tomlinson Model?

The Tomlinson Model has been applied in various fields, including materials science, engineering, and nanotechnology. It has helped researchers to better understand and predict frictional behavior between different materials, leading to the development of more efficient and durable materials and coatings. It has also been used to study the effects of friction on atomic-scale processes, such as wear and lubrication.

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