Capacitive charging rate dependence of heat from porous carbon in aqueous salt solution

Porous capacitive electrodes are applied in supercapacitors and capacitive deionization of aqueous salt solutions. In both cases, electric charge and ions are stored in the electrical double layer at the surface of the pores. Recently, we revealed that the equilibrium potential energy of the ions in the pores can be obtained from isothermal calorimetric measurements. On that basis, we now introduce a model for the time-dependent heat production at any charging or discharging rate. The model centers on a mathematical expression for the time-dependent internal energy of the double layer, which depends only on constant system parameters and the time-dependent electric potential drop across the double layer. Semiquantitative agreement is found with experiments on a porous carbon electrode in aqueous salt solution. The theory applies both in the case of abrupt application of a potential to the electrode, generating a maximum amount of Joule heat, and in cases where the potential is applied more slowly, even when charging becomes so slow that heat production is essentially reversible. These results not only provide fundamental insight into the electrical double layer of porous capacitive electrodes, but they are also a new way to describe and to predict the time-dependent generation of heat from such electrodes.