Surface stress, the lattice parameter of small crystallites and its response to pressure |
Joerg Weissmueller , Jürgen Markmann |
Forschungszentrum Karlsruhe, Institut für Nanotechnologie, Herrmann-von-Helmholtz-Platz 1, Karlsruhe 76344, Germany |
Abstract |
Nanometer-sized particles or polycrystalline solids with a nanometer grain size will usually exhibit a lattice parameter which differs from that of the same substance in coarse-grained form. This observation testifies to the fact that considerable forces, originating from the surfaces or grain boundaries, act on the crystal lattice. In fact, pressures of several GPa can be induced in this way, and experiments on nanoscale solids show millimeter-scale movements as a response to changes in the surface-induced pressure. Intriguingly, it is found that the pressure can be tensile as well as compressive, depending on the material and the state of its surface. One can derive rigorous and general results relating the surface-induced pressure to the surface forces. An important - and sometimes neglected - consequence of these considerations is that the cases of fluids and solids require separate treatment: in fuids, the Young-Laplace equation relates the jump in the pressure across a surface to the mean curvature and the surface tension, whereas in solids there are two equations, both involving the surface stress tensor: one for the local jump in the tangential component of the stress, and a second for the average of the full stress tensor over the microstructure. The second result also provides a relation between the pressure and a scalar surface stress, which is readily applied to the analysis of lattice parameter data. While surfaces are often considered as two-dimensional objects, recent experiments suggest - counterintuitively - that it is relevant to allow for a surface to undergo deformation in the third direction, that of its normal. This leads to a pressure-dependence of the surface stress, and to an apparent size-dependence of the compressibility of nanoparticles when subjected to an external pressure, which does not imply that the lattice becomes softer or more compliant at small size. |
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Presentation: invited oral at E-MRS Fall Meeting 2005, Symposium I, by Joerg Weissmueller Submitted: 2005-04-29 14:53 Revised: 2009-06-07 00:44 |