Determining the vibrational properties
- i.e. the vibration frequencies and the associated displacement fields -
of solid bodies with various shapes is a well studied area of continuum
mechanics. The early works of Lamb or Rayleigh, who determined these
vibrations using classical elasticity theory of isotropic materials,
have found applications in fields as different as planetary science and
nuclear physics. In today's materials sciences, the increasing development
of materials containing nanometer size structures naturally leads one to
question the limits of applicability of the continuum elasticity
theory, which is in principle valid only on length scales much
larger than the interatomic distances.
Investigating the vibration modes of nanometric objects using atomic level simulations is a natural way of probing this applicability.[1,2] Such an investigation is particularly relevant from an experimental viewpoint, since these properties, inferred from spectroscopic measurements, are systematically interpreted within the framework of continuum elasticity.
The snapshot shown on the right displays a
small disk-shaped aggregate formed by about 700 Lennard-
The second figure shows a large sample (10000 Lennard-Jones particles) with periodic
boundary conditions (L=104) has been submitted to an asymptotically weak
elongational strain in horizontal direction. The resulting
non-affine displacement field
is computed after relaxing the atoms to their new equilibrium
positions.[2] The length of the arrows shown is
proportional to the non-affine bead displacement. This field is highly
spatially correlated over 30 bead diameters and (as can be shown) of strong solenoidal
character.[5,6]
We also determine the linear response of amorphous elastic bodies to an external delta force is determined in analogy with recent experiments on granular aggregates.[4] For the generated forces, stress and displacement fields, we find strong relative fluctuations of order one close to the source, which, however, average out readily to the classical predictions of isotropic continuum elasticity. The stress fluctuations decay (essentially) exponentially with distance from the source. Only beyond a surprisingly large distance of again 30 interatomic distances, self-averaging dominates, and the quenched disorder becomes irrelevant for the response of an individual configuration.
In such a computer simulated system, in which all particle coordinates and interparticle forces are exactly known, it is possible to calculate exactly the vibration frequencies around an equilibrium position.[1,2] This is achieved by exact diagonalization of the so called dynamical matrix, a (d N)x(d N) matix (where d is the number of spatial dimensions and N the number of particles) matrix expressible in terms of the first and second derivatives of the interparticle interaction potentials. The density of states (DOS) may alternatively be also computed by Fourier transformation of the velocity correlation function of solids at very low temperatures.[5,6] This method is less rigorous (numerical errors, finite temperature effects) but allows the calculation of the DOS for large systems where the diagonalization of the dynamical matrix becomes impossible.
Investigation of the low frequency end of the vibrational spectrum of such nanometric disordered systems reveals that the classical continuum elasticity, applied to these systems, actually breaks down below a length scale of typically 30 to 50 molecular sizes.
We argue that the self-averaging length sets the lower wave length bound for the applicability of classical eigenfrequency calculations which explains the peculiar vibrational properties of glassy systems around the so-called "Boson-peak".[6]
J.P. Wittmer (principal investigator in Strasbourg), F. Léonforte (PhD thesis Lyon), A. Tanguy (LPMCN, Lyon), J.-L. Barrat (LPMCN, Lyon).