On the role of intermolecular vibrational motions for ice polymorphs. IV. Anisotropy in the thermal expansivity and the nonaffine deformation for ice IX and III

Abstract

We explore anisotropic properties in the thermal expansivities of hydrogen-ordered ice IX and its hydrogen-disordered counterpart, ice III. The free energies of these ice forms are calculated to obtain the lattice constants for the tetragonal unit cell and the thermal expansivities at various thermodynamic conditions in the framework of quasi-harmonic approximation, taking account of their anisotropic nature. The thermal expansivities are also examined by applying a thermodynamic relation that connects them with the Grüneisen parameters and the elastic compliances. Both calculations suggest that ice III and IX exhibit a negative thermal expansion along the a-axis but have a positive one along the c-axis at low temperatures. It is found that nonaffine deformation in the variation of the lattice constant beyond affine transformation (the Born approximation) is essential in the theoretical calculation of the thermal properties of ice III and IX. We also find that the nonaffine deformation is described by the shift of the minimum energy positions in the potential manifold of hydrogen-ordered ice along a limited number of the normal mode coordinates, which is irrelevant to the system size. These modes become unstable against an applied strain, so that the potential minimum moves along those normal coordinates away from that of the affine-transformed structure. The unstable modes are all symmetry-preserving modes, and the space-group symmetry is an invariant under displacement along either of those normal coordinates. The number of the unstable modes in ice IX is 8 while it is 1 in another hydrogen-ordered ice VIII.