Limitations and strengths of uniformly charged double-layer theory: Physical significance of capacitance anomalies

Theoretical studies of electrical double layers typically consider the response of ionic conductors to the field of uniform charge-density distributions sigma (sigma-control). Many such analyses predict apparent anomalies of differential capacitance, C, including divergences and negative values. To clarify misconceptions regarding these predictions, we critically reexamine some theoretical approaches dealing with the admissible sign of C. We examine the anomalies' origin and stress its relation to the artificiality of sigma-control. We show that calculations based on sigma-control can illuminate the nature of instabilities and phase transitions under the physically attainable conditions of potential control, where applied voltage phi rather than sigma is fixed. For illustration, we discuss the physical nature of the ultimate anomaly, negative integral capacitance predicted by some recent analyses. We also show that sigma-control anomalies can explain some experimentally observed features of C(phi).