The weak electrolyte model

The SM fails in particular to describe the Hopf bifurcation leading to travelling rolls, which is observed quite frequently. For a Hopf bifurcation two processes that compete on a comparable time scale are necessary. In our case, however, the director relaxation is much slower than the other processes and thus determines the dynamics. Charge relaxation can usually also be treated adiabatically. In addition, director relaxation and charge relaxation do not compete but rather support each other usually, which also excludes a Hopf bifurcation. 

Thus another slow process appears to be operating. A recently proposed model assumes that this process is a relaxation of the mobile ion densities n and n on the time scale Tree, which may result from a dissociation-recombination reaction. One then gets if single-charged ions are assumed 

Here p , Pf are the ionic mobilities perpendicular and parallel to the director, respectively. For simplicity the anisotropies were assumed to be the same for both types of ions so that Tal = So now n is an additional variable. From the balance equations 

in this model ion accumulation effects are included wdiereas ionic diffusion is neglected as in the SM. The charge accumulation counteracts the standard (Helfrich) mechanism of generation of space charges. If r c is sufficiently slow one can find an oscillatory behaviour of the system at threshold, i.e. a Hopf bifurcation (Section 13.5). 

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