Contact conundrum conquered?
Drs. Wilfred T. Tysoe & Nicholas D. Spencer | TLT Cutting Edge June 2009
A new set of simulations of the contact in an atomic force microscope challenge our thinking.
One of the central issues in tribology is correctly determining the true area of contact between two surfaces. In large-scale contacts, the real contact area is made up of a number of contacting asperities. One of the central ideas behind using atomic force microscopy (AFM) to study friction is that the sharp tips used in this technique model a single-asperity contact. Moreover, it was thought that the contact mechanics could be described using analytical theories developed by Hertz, which showed that the area of contact was proportional to (Load)2/3. This theory has been modified to include the effects of adhesion between the contacting surfaces but still yields analytical formulae with sublinear dependences of contact area on load.
In this case, if the lateral force is proportional to the real contact area, where the proportionality constant is defined as the interfacial shear strength, this will result in the lateral (frictional) force being a sublinear function of load, and this has indeed been observed in a number of AFM experiments. However, in some cases, deviations have been observed from such simple elastic theories, where the frictional force is found to be proportional to the load or even where the two quantities were almost independent of each other. This raises the question of whether such analytical continuum contact models can really describe behavior at the nanoscale.
This issue was addressed by the group of professor Izabela Szlufarska at the University of Wisconsin-Madison by using largescale molecular dynamics simulations to calculate the frictional force at the nanoscale. These simulations were performed for hydrogen-terminated, amorphous-carbon tips and a diamond substrate that allowed both the tip and substrate to deform. The interaction potential included both short-range chemical interactions and longer-range Van der Waals’ forces. A key issue was to determine the real contact area correctly, and this was done by calculating the number of chemically interacting atoms in the contact, multiplied by the average surface area occupied per atom.
They first performed simulations in which they turned off the long-range interactions. In this case, they found a sublinear dependence of asperity contact area on applied load, in accord with the predictions from contact mechanics, but a linear dependence of lateral force on load. Furthermore, the composite Young’s modulus calculated using the simulations was substantially less than that predicted by continuum mechanics.
The authors argue that this apparent breakdown of single asperity theories arises because the real area of contact is less than the asperity contact area since not all atoms in the asperity are really in contact and suggest that friction laws should be formulated in terms of the real contact area, even at the nanoscale.
To explore the effect of long-range interactions, simulations were also performed by including Van der Waals’ interactions. It was found that the frictional force was again proportional to the real contact area. Surprisingly, the friction force was found to be a sublinear function of load, which would have previously been taken as confirming that elastic contact theories were valid. Simulations showed that the variation in real contact area with applied load could be successfully fitted by continuum mechanics theories (in this case using the most general Maugis-Dugdale model).
However, the authors suggest that this model, which includes three fitting parameters, does not correctly describe the contact mechanics at the nanoscale. In this case, the sublinear dependence of friction force on load arises from the inclusion of longer-range interactions, which leads to adhesion forces that do not scale linearly with the real contact area. This results in a total load, the sum of the loads due to both short- and long-range interactions that is not proportional to the real contact area, leading to a friction force that is a sublinear function of load.
The results of these simulations suggest that the picture of the nature of the contact at the macroscopic scale, that the real contact area arises from the sum of the areas of the contacting asperities, also applies at the nanoscale.
FOR FURTHER READING:
Tysoe, W.T. and Spencer, N.D. (2005), “Searching for the Fractal Truth,” TLT, 61 (2), p. 56.
Tysoe, W.T. and Spencer, N.D. (2004), “Why Does Amontons’ Law Work So Well?” TLT, 64 (8), p. 64.
Mo, Y., Turner, K.T. and Szlufarska, I. (2009) “Friction Laws at the Nanoscale,” Nature, 457 (7233), pp. 1116-1119.
Eddy Tysoe is a Distinguished Professor of Physical Chemistry at the University of Wisconsin-Milwaukee. You can reach him at wtt@uwm.edu.
Nic Spencer is professor of surface science and technology at the ETH Zurich, Switzerland. Both serve as editors-in-chief of STLE-affiliated Tribology Letters journal. You can reach him at nspencer@ethz.ch.