Negative thermal expansion (NTE) refers to the peculiar property exhibited by some materials of contracting rather than expanding when they are heated. This is driven by unusual entropy considerations.

The central question regarding the phenomenon of negative thermal expansion is: what is the relationship between structure at the atomic scale and the occurrence of negative thermal expansion? Of particular interest is the possible role of structural under-constraint.

The structure of real crystals exhibiting negative thermal expansion can be complex; however, one can get an understanding of the essence of the relationship between constraint and expansion from relatively simple models. If one counts each short-range bond as a constraint, then the structure to the right (figure 2) is fully constrained.

The structure shown to the left (figure 3), however, has infinitely more degrees of freedom than constraints. In this sense it is macroscopically under-constrained. This causes it to manifest an * infinite manifold of softening modes* which lead to negative thermal expansion[1].

Examples of these modes, which involve complex mixing of twist and translational motion, can be seen by clicking on particular points in figure 1a. We believe that the connection between negative thermal expansion and nano-scale structure resides in these modes. Modes of this type provide the entropy contribution to the Gibbs free-energy which drives negative thermal expansion.

1) Z. Schlesinger, J. A. Rosen, J. N. Hancock, A. P. Ramirez: **Soft Manifold Dynamics Behind Negative Thermal Expansion**, Phys. Rev. Lett. 101, 15501 (2008)

Abstract: Minimal models are developed to examine the origin of large negative thermal expansion (NTE) in under-constrained systems. The dynamics of these models reveals how underconstraint can organize a thermodynamically extensive manifold of low-energy modes which not only drives NTE but extends across the Brillioun zone. Mixing of twist and translation in the eigenvectors of these modes, for which in ZrW_{2}O_{8} there is evidence from infrared and neutron scattering measurements, emerges naturally in our model as a signature of the dynamics of underconstraint.