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NEGATIVE THERMAL EXPANSION ORIGINS: The importance of being under-constrainedgreyline

Zack Schlesinger

a) Brillioun-zone

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Figure 1a: Brillioun-zone map showing clickable points at which modes can be seen as gif movies.

b) Eigenmode Nature

Click on a point on the graph on the left to see the mode.

Figure 1b: Example of a member of the infinite manifold of softening modes which can lead to negative thermal expansion in under-constrained systems.

Figure 2: Fully constrained structure
fully constrained crystal structure

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.

Entropy is a pretty simple concept

When your stuff is all over the place -- that's high entropy; when it's neatly put away in a small pile that's low entropy.

It is the same with crystalline solids. When they are bigger their entropy is higher; when they're smaller their entropy is lower. That is why materials tend to expand when they are heated. (Temperature and entropy go hand in hand.)

In a quantum perspective one can recast this in terms of phonon frequencies. Higher phonon frequencies correspond to more compactness, hence lower entropy. Lower phonon frequencies imply higher entropy. In the unusual circumstance where phonon frequencies decrease when a material contracts, it will contract when it is heated to increase its entropy. (Temperature and entropy go hand in hand.) This is called negative thermal expansion.

While ice does this (becomes more compact) when it melts to make water, the continuous negative thermal expansion of ZrW2O8, is particularly surprising and unusual. ZrW2O8 contracts continuously from near absolute zero to very high temperature (1000 K) without relying on a phase transition. Here we show results from simple model calculations designed to be relevant to understanding negative thermal expansion in ZrW2O8.

Figure 3: Under-constrained structure (c.f., Figure 1b)
macroscopically under-constrained crystal structure

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 ZrW2O8 there is evidence from infrared and neutron scattering measurements, emerges naturally in our model as a signature of the dynamics of underconstraint.

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