The atomic or molecular dynamics in condensed matter systems can
be probed by various ways, like e.g. by applying a deformation or sending
radiation. Under the applied field, every atom tends to be displaced from
its original position, which, in the case of solid states, represents an
equilibrium or quasi-equilibrium position. For small displacements in the
solid state, the harmonic approximation leads straightforward to standard
lattice dynamics and one recovers the vibrational spectrum and elastic
constants of perfect crystals. For disordered materials like glasses,
however, things are not so simple. When the atom reaches the position
prescribed by the applied field, the forces transmitted to it by its
nearest neighbours cancel mutually to zero in all crystals which possess
local center-inversion symmetry. In glasses and other non-centrosymmetric
and disordered systems, this is no longer true: a net force acts on every
atom in its displaced (affine) position because the nearest-neighbour
forces no longer balance. As a result, an additional displacement is
required to satisfy the mechanical equilibrium. In light of this simple
consideration, we have reformulated classical lattice dynamics to make it
applicable to glasses and disordered systems in general. The theoretical
framework is able to predict elastic constants and their softening due to
disorder, as well as rheological flow curves of different materials. For
example stress-strain overshoots in start-up shear experiments can be
reproduced as well as the shear modulus of polymers across the glass
transition. This conceptual framework can also be used to elucidate the
vibrational density of states of glasses and disordered crystals.
University Lecturer, University of Cambridge