"Traditional models of contact approximate solids as a 'continuous media' with no internal structure, but all solids are made up of discrete atoms," Mark Robbins of John Hopkins University told nanotechweb.org. "Ignoring atomic structure may be a good approximation when modelling parts of macroscopic machines like cars, but what happens when the size of the parts and their contacts shrinks down to a few atomic diameters? How small is too small for continuum theory to apply? The answers to these questions are crucial to the function of man-made machines and many biological processes."

Robbins and colleague Binquan Luan say they modelled the displacements of up to ~107 atoms on a computer as two solids were pushed together. They compared the mechanical deformation, adhesion and friction forces to continuum calculations for a curved surface - cylinder or sphere - pushed into a flat surface. The pair varied the radius of the curve from about 100 atoms (~30 nm) to 1000 atoms.

"Knowing the exact atomic structure and how each atom moved allowed us to test the two key assumptions of continuum theory," said Robbins. "Being able to vary the placement of atoms allowed us to quantify the influence of different geometries on deviations from continuum theory."

According to the researchers, the continuum description of deformations inside solids worked surprisingly well, but the assumption that surfaces can be represented by smooth curves did not.

"Any surface made of discrete atoms has bumps of atomic dimension, and surfaces with the same average shape, but different atomic-scale structure behave very differently," said Robbins. "Surfaces formed by bending a flat crystal into a curve behaved most like the continuum models. Surfaces formed by cutting a sphere from a crystal or random, glassy material behaved quite differently."

The researchers reckon their findings have implications for avoiding unwanted adhesion and excessive friction in nanoscale devices; for varying materials at the nanometre scale to create optimal macroscopic mechanical properties; for examining the contacts between macroscopic objects, since the fractal structure of surfaces means these contacts are often extremely small; and for interpreting contact measurements such as those taken using atomic-force microscopes.

"Mechanical properties of small components are often determined by fitting contact measurements to continuum theory," said Robbins. "Our results indicate when continuum theory can be used to model such situations. They show that some quantities, like the stiffness, may be determined accurately, that others - contact area, adhesion and the yield stress where permanent deformation sets in - may be off by a factor of two, and others, like friction, may be off by more than an order of magnitude. Hopefully this will help in the creation of new tools needed to guide design of nanotechnology."

The scientists reported their work in Nature.