# Mechanical and Thermo-Mechanical properties of Materials

The theory of thermal expansion goes back over 80 years to the work of Born (1923) and Grüneisen (1924), and it requires a knowledge of the phonon frequencies and the way they change with changing unit cell size. This is a great computational challenge, even in materials with few atoms in the unit cell, and our success in explaining the thermal expansion of ß-eucryptite is quite remarkable. The high-temperature structure of ß-eucryptite is shown in the figure, and the low-temperature structure is even more complicated (84 atoms in the unit cell).

The main constituent of the CERAN cooking tops of the company Schott is the alumino-lithia-silica glass ceramic beta-eucryptite, which has a very small thermal expansion coefficient over a temperature range of around 1000 degrees. For the crystalline form of beta-eucryptite (see figure, oxygen atoms are green, lithium atoms are brown, silicon atoms blue, and aluminium atoms red), density functional calculations - which are free of adjustable parameters show that the thermal expansion coefficients parallel and at right angles to the lithium chains are almost constant. The coefficient parallel to the chains is negative and twice as large as the positive coefficient normal to the chains, and a polycrystalline sample should have a constant and very small thermal expansion. Since the atomic movements can also be followed in a precise way, it is possible to understand the reasons for this behaviour.

It is interesting to ask whether simpler and numerically less demanding theories can reproduce experimental measurements, and whether they can work in crystals that are not isotropic and/or close-packed. Several groups have been working on calculations of elastic constants, and it is generally accepted that they can be calculated with a quite reasonable precision. Many other challenges remain, including the calculation of the various entropic contributions to the free energy and the theoretical description of P-V-T diagrams. There is cause for optimism, although one must never forget that approximations are unavoidable in DF calculations.

( R. O. Jones, J. Harris)