Correlation-function based elasticity theory for systems with anisotropic particles

We quantitatively work out the elastic equations of continuum mechanics and the elastic free energies based on our previous fundamental work developing a unified framework capturing the reversible couplings of Nambu-Goldstone modes. Starting point of our coarse-graining procedure is the microscopic level capturing the coupling of rotational and translational motion of particles with simple well-defined shapes of different symmetries. Working in reciprocal space enables us to include finite concentrations of local defects throughout. Density functional and correlation function theory can then be employed to study the thermodynamic and mechanical properties of complex defect-rich crystals.

2D solids formed by square-like particles: cubic crystal (left, upper panel) and tetratic phase (left, lower panel). Snapshots and dispersion relations are obtained from Monte Carlo simulations and are analysed according to the developed theory [P8]. Middle panel: The orientational fluctuations exhibit an acoustic dispersion relation in the tetratic phase, but an optic one, viz. with finite frequency for vanishing wavenumber, in the crystal solid. Right panel: The acoustic transverse and longitudinal phonon modes and the optic angular mode are shown in a crystal state. The lower panel shows two dislocations which do not prevent the analysis of the density fluctuations in the solid. The lateral length of the square sets the unit of length, and frequencies are rescaled to correspond to wavenumbers.