Nowadays, the Prandtl-Tomilson models (PT-models) in one- and two-dimensions are very popular for their explanation of nanotribological experiments performed by using a friction or atomic force microscope (AFM). In a PT-model, one can clearly distinguish between frictional regimes by considering as parameter the ratio between the average interaction and elastic energies. In this research, the (a)periodic corrugation of interest is separately derived and directly introduced into the corresponding PT-model resulting in a set of Newtonian equations of motion, which are then solved numerically by applying an adequate forth-order Runge-Kutta method. These Newtonian equations of motion form a set of stochastic differential equations involving a thermally induced random force mimicking the impact of temperature on sliding. Due to this randomness, both frictional regimes, either stick-slip or structural superlubric, must be statistically evaluated and interpreted, for example, in terms of distributions.

Blue bar chart of a Gaussian bell

© Maja Srbulovic

Histogram of the friction force at room temperature (blue) compared with the corresponding Gaussian distribution (red)

Having developed a statistically proper numeric scheme, in the immediate next steps, frictional performances calculated for various (a)periodic atomic structures will be compared, by also elucidating the role of irrational numbers in nature, e.g., that of golden ratio, which well describes the frequently observed arrangements of leaves and seeds in various plants.