Abstract
A field theoretic representation of the classical partition function is derived for a system composed of a mixture of anisotropic and isotropic mobile charges that interact via long range Coulomb and short range nematic interactions. The field theory is then solved on a saddle-point approximation level, leading to a coupled system of Poisson–Boltzmann and Maier–Saupe equations. Explicit solutions are finally obtained for a rod-like counterion-only system in proximity to a charged planar wall. The nematic order parameter profile, the counterion density profile and the electrostatic potential profile are interpreted within the framework of a nematic–isotropic wetting phase with a Donnan potential difference.
Highlights
IntroductionFor rod-like charged cylinders, a generalized Onsager theory could be used to describe the ordering transition with electrostatic interactions strongly modifying the hard core diameter of the rods as well as providing a mechanism for the twisting interaction as first described by Odijk [20,21,22]
By solving the mean-field level equations that emerge as a coupled system of the Maier–Saupe and the Poisson–Boltzmann equation, we are able to derive some salient properties of inhomogeneous nematic ordering induced by the charged interface, as well as the modifications in the electric double layer distribution wrought by the presence of nematic order
To describe the orientational relaxation effects one would need the elastic deformation energy, which would stem from the expansion of the nematic interaction potential w.r.t. the gradient of the tensorial order parameter as in the general inhomogeneous Landau-de Gennes Ansatz [48]
Summary
For rod-like charged cylinders, a generalized Onsager theory could be used to describe the ordering transition with electrostatic interactions strongly modifying the hard core diameter of the rods as well as providing a mechanism for the twisting interaction as first described by Odijk [20,21,22] This approach has seen many further developments with different level modifications and extensions [23,24,25,26,27]. As well as electrical double layers in ionic liquid crystals, have been analyzed in the work of Bier within the density functional approach [34,35] that was formulated for homogeneous, as well as inhomogeneous, systems with interfaces [36,37] It is the latter case that is interesting, as it should exhibit features of both, the nematic ordering as well as the Gouy–Chapman-type electrostatic double layers. By solving the mean-field level equations that emerge as a coupled system of the Maier–Saupe and the Poisson–Boltzmann equation, we are able to derive some salient properties of inhomogeneous nematic ordering induced by the charged interface, as well as the modifications in the electric double layer distribution wrought by the presence of nematic order
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