Extension of the Reynolds equation by a non-linear wall slip law

When an electric vehicle accelerates, the motor generates maximum forces and enormous pressure acts on the gears of the electric drive system. Surface meets surface, metal meets metal. If there was no lubricating film to allow the gears to slide more easily, they would not only get extremely hot, but also wear out quickly. “Without a lubricating film, many things in our everyday lives would be slower, squeakier and jerkier,” explains prof. Michael Moseler, Head of the Tribology Business Unit at the Fraunhofer IWM.

“The electric vehicle will certainly never reach such a high range,” adds dr. Kerstin Falk at, who heads the “Molecular Lubrication Design” team. Together they investigate the behavior of lubricating films in highly stressed tribological contacts to determine their suitability for low-friction operation.

Whether the material involved is metal, plastic or ceramic, ideal lubrication can save more than 20 percent of energy as machines operate with less resistance. It is also a promising field of research in terms of sustainability.

It is therefore no wonder that the partner companies of the MicroTribology Center µTC, a collaboration between the Fraunhofer IWM and the Karlsruhe Institute of Technology (KIT), are very interested in reducing friction in their systems as much as possible.

“Many tribological systems are now designed at their load limit, where lubricant film thicknesses are in the nanometer range and pressures in the gigapascal range. Our partners are wondering how to calculate the friction in a component with such highly loaded tribological contacts, as conventional .fluid dynamic computational approaches fail under these extreme conditions,” says Kerstin Falk, summing up the problem.

Together with their simulation team at the MicroTribology Center μTC, Falk and Moseler found an answer to this question. They have published their research Science advances.

Understand and optimize friction

How friction is calculated and therefore kept as low as possible depends on the lubrication regime a company is aiming for in its components. Usually, it wants to drive its tribosystems – where a force pushes the primary and counterbodies together – under elastohydrodynamic conditions.

A lubricating film, the thickness of which is much greater than the roughness of the two surfaces, is intended to reduce friction. In this case, the friction can be accurately predicted using a continuum mechanics approach. This involves solving the so-called Reynolds equation for the lubricant, which Osborne Reynolds derived in 1886.

In addition, the heat conduction equation for the overall system and the linear elastic equations for both surfaces are calculated. The only material data required are the elastic moduli and Poisson’s ratios of the friction partners, thermal conductivity and heat capacity of all materials involved, as well as accurate constitutive laws – for the density of the fluid and for its dynamic viscosity for a parameter field that exists ​​from pressure, temperature and local shear rate in the fluid. It is state of the art.

However, if the tribological system is operated in boundary lubrication, with a very thin lubricating film in which the asperity contact, ie the roughness peaks, is separated from the lubricant by only a few atomic layers, the friction coefficient is only roughly estimated. used in the calculations for the “dry” contact points.

“This is very unsatisfactory, because calculations with guessed material parameters are inaccurate, lead to suboptimal designs and end up costing companies a lot of money,” says Michael Moseler.

Kerstin Falk and Michael Moseler were not satisfied with this: Together with four partner companies of the MicroTribology Center µTC, they researched their own mathematical law for the behavior of extremely thin lubricating films in a three-year project and further developed the Reynolds equation, so to speak. “We wanted to understand how friction behaves in boundary lubrication,” explains Moseler.

The aim of the project is to explain below which lubricant film thickness the continuum mechanics fails and how the underlying equations can be extended so that a lubricant film thinner than the surface roughness can be calculated.

For this purpose, the molecular dynamics of a hydrocarbon lubricant was calculated in an asperity contact geometry, for example two diamond-like carbon (DLC) surfaces lubricated with a polyalphaolefin (PAO) base oil. The results of the molecular dynamics simulation were then compared with those of the Reynolds equation.

The resounding result: For pressures between the friction partners below 0.4 gigapascal and lubrication gap heights greater than 5 nanometers, the Reynolds description agrees well with the molecular dynamics reference calculations, provided an exact constitutive law for the viscosity of the lubricant is used.

In contrast, Kerstin Falk and Michael Moseler were able to show that under extreme boundary lubrication conditions, namely high pressures of approx. 1 gigapascal and small lubrication gap heights of approx. 1 nanometer, the adhesiveness of the lubricant to the surfaces is reduced, and therefore the slip between a friction partner and the lubricant must be included in the calculation to correctly predict the friction.

This requires a non-linear wall slip law. This relates the wall slip velocities (ie the difference in velocity between a friction partner and the adjacent lubricant) to the local shear stresses in the lubricant film.

Breakthrough in tribology: Making boundary friction predictable

With these research results, the researchers now present an innovative method to predict friction under boundary lubrication conditions. An additional piece of information needed for this non-empirical predictive continuum modeling of highly loaded tribological contacts is the atomic structure of the rubbing surfaces. It is determined using in-depth experimental analyzes and is a prerequisite for the wall slip law.

The new findings from the Fraunhofer IWM are now being used in follow-up projects to predict friction coefficients and friction behavior in specific applications – for example in gears and bearings – as well as to support the research partners in building up simulation expertise.

They can then perform test bench and component simulations, reduce uncertainties in the design of tribological systems and determine design parameters more precisely. It is an essential step towards knowledge-based lubricant, surface and component design and should be of great interest to lubricant and coating manufacturers as well as bearing and gear manufacturers.