Theoretical interpretation of Warburg's impedance in electrolytic cells

We discuss the origin of Warburg's impedance in electrolytic cells containing only one group of positive and one group of negative ions. Our analysis is based on the Poisson-Nernst-Planck model, where the generation-recombination phenomenon is neglected. We show that to observe Warburg's like impedance the diffusion coefficient of the positive ions has to differen from that of the negative one, and furthermore that the electrodes have to be not blocking. We assume that the non-blocking properties of the electrodes can be described by means of an Ohmic model, where the charge exchange between the cell and the external circuit is described by means of an electrode conductivity. For simplicity we consider a symmetric cell. However, our analysis can be easily generalized to more complicated situations, where the cell is not symmetric and the charge exchange is described by Chang-Jaffe model, or by a linearized version of Butler-Volmer equation. Our analysis allows to justify the expression for Warburg's impedance proposed previously by several groups, based on wrong assumptions

Elastic continuum theory: Fully understanding of the twist-bend nematic phases

The twist-bend nematic phase, $N_{\rm TB}$, may be viewed as a heliconical molecular arrangement in which the director $\bf n$ precesses uniformly about an extra director field, $\bf t$. It corresponds to a nematic ground state exhibiting nanoscale periodic modulation. To demonstrate the stability of this phase from the elastic point of view, a natural extension of the Frank elastic energy density is proposed. The elastic energy density is built in terms of the elements of symmetry of the new phase in which intervene the components of these director fields together with the usual Cartesian tensors. It is shown that the ground state corresponds to a deformed state for which $K_{22} > K_{33}$. When the elastic free energy is interpreted in analogy with the Landau theory, it is demonstrated the existence of a second order phase transition between the usual and the twist-bend nematic phase, driven by a new elastic parameter playing a role similar to the one of the main dielectric anisotropy of classical nematics and being closely related to the bulk compression modulus representing the pseudo-layers of twist-bend nematic phases. A phase transition and the value of the nanoscale pitch are predicted in accordance to experimental results.Comment: 2 figure