Dichotomous collective proton dynamics in ice

Abstract

The collective proton dynamics in ice is studied on the basis of the two-dimensional (2D) nonlinear lattice model which takes the dichotomous branching of proton transfers in hydrogen-bonded networks into account. The essential point of this model is that the network topology of the square proton lattice satisfies the Bernal-Fowler ice rules of the 3D ice crystal structure. The model is considered as a straightforward extension of the standard 1D coupled double-well oscillator model described by the discrete nonlinear Klein-Gordon equation to two dimensions while imposing the ice rules. This generalization under the ice constraints has been shown to be unique. A relation between the boundary conditions and topological charge (like Gauss' law) is established. For any domain of the square ice lattice with nonzero topological charge and an ideal ice configuration chosen randomly on its boundary, the extended positive $({\mathrm{H}}_{3}{\mathrm{O}}^{+})$ and negative $({\mathrm{OH}}^{\mathrm{\ensuremath{-}}})$ ionic defects are described in terms of 2D vector topological solitons. The definition of the 2D kinks and antikinks is given by using Gauss' law. An anisotropic generalization of the 2D ice model and an appropriate numerical scheme allows us to study the dynamical properties of the 2D solitons in comparison with the corresponding 1D solutions. Particularly, contrary to the 1D case, the existence of a nonzero Peierls-Nabarro relief has been proved to exist in all cases, even if intersite proton-proton interactions are infinitely strong, so that the free 2D soliton dynamics is impossible in the ice crystal. On the other hand, our studies of the thermalization of the 2D ice lattice clearly demonstrate the crucial role of the cooperativity of hydrogen bonding in the nucleation and dynamics of the defect pairs ${\mathrm{H}}_{3}{\mathrm{O}}^{+}$ and ${\mathrm{OH}}^{\mathrm{\ensuremath{-}}},$ explaining their very low density known from experimental data in ice physics.