Negative capacitance and related instabilities in theoretical models of   the electric double layer and membrane capacitors
  Various models leading to predictions of negative capacitance, C, are briefly reviewed. Their relation to the nature of electric control is discussed. We reconfirm that the calculated double layer capacitance can be negative under S-control - an artificial construct that requires uniform distribution of the electrode surface charge density, S. It is shown that the combined relaxation of the ionic and electronic contributions can result in C<0 even for the local statistical ionic models with strictly positive diffuse layer capacitance. In reality, however, only the total charge q (or the average surface charge density S) can be experimentally fixed in isolated cell studies (q-control). For those S where C becomes negative under S-control, the transition to q-control (i.e. relaxing the lateral charge density distribution, fixing its mean value to S) leads to instability of the uniform distribution and a transition to a non-uniform phase. As an illustration, a "membrane capacitor" model is discussed. This exactly solvable model, allowing for both uniform and inhomogeneous relaxation of the electrical double layer, helps to demonstrate both the onset and some important features of the instability. Perspectives for further development are discussed briefly. 
Michael B. Partenskii and Peter C. Jordan