Several modern technologies for energy storage and conversion are based on the screening of electric charge on the surface of porous electrodes by ions in an adjacent electrolyte. This so-called electric double layer (EDL) exhibits an intricate interplay with the electrolyte's temperature that was the focus of several recent studies. In one of them, Janssen et al. [Phys. Rev. Lett. 119, 166002 (2017)] experimentally determined the ratio Qrev/W el of reversible heat flowing into a supercapacitor during an isothermal charging process and the electric work applied therein. To rationalize that data, here, we determine Qrev/W el within different models of the EDL using theoretical approaches like density functional theory (DFT) as well as molecular dynamics simulations. Applying mainly the restricted primitive model, we find quantitative support for a speculation of Janssen et al. that steric ion interactions are key to the ratio Qrev/W el . Here, we identified the entropic contribution of certain DFT functionals, which grants direct access to the reversible heat. We further demonstrate how Qrev/W el changes when calculated in different thermodynamic ensembles and processes. We show that the experiments of Janssen et al. are explained best by a charging process at fixed bulk density, or in a "semi-canonical" system. Finally, we find that Qrev/W el significantly depends on parameters as pore and ion size, salt concentration, and valencies of the cat-and anions of the electrolyte. Our findings can guide further heat production measurements and can be applied in studies on, for instance, nervous conduction, where reversible heat is a key element.

The underdamped, non-linear, generalized Langevin equation is widely used to model coarse-grained dynamics of soft and biological materials. By means of a projection operator formalism, we show under which approximations this equation can be obtained from the dynamics of the underlying microscopic system and in which cases it makes sense to introduce a potential of mean force. We discuss shortcomings of previous derivations presented in the literature and demonstrate the implications of our derivation for the structure of memory terms and for generalized fluctuation-dissipation relations. We show, in particular, that the widely used, simple structure which contains a potential of mean force, a memory term which is linear in the observable, and a fluctuating force which is related to the memory term by a fluctuation-dissipation relation, is neither exact nor can it, in general, be derived as a controlled approximation to the exact dynamics.

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Recently, the reversible heat production during the electric double layer (EDL) buildup in a sodium chloride solution was measured experimentally [Janssen et al., Phys. Rev. Lett. 119, 166002 (2017)] and matched with theoretical predictions from density functional theory and molecular dynamics simulations [Glatzel et al., J. Chem. Phys. 154, 064901 (2021)]. In the latter, it was found that steric interactions of ions with the electrode’s walls, which result in the so-called Stern layer, are sufficient to explain the experimental results. As only symmetric ion sizes in a restricted primitive model were examined, it is instructive to investigate systems of unequal ion sizes that lead to modified Stern layers. In this work, we explore the impact of ion asymmetry on the reversible heat production for each electrode separately. In this context, we further study an extension of the primitive model where hydration shells of ions can evade in the vicinity of electrode’s walls. We find a strong dependence on system parameters such as particle sizes and the total volume taken by particles. Here, we even found situations where one electrode was heated and the other electrode was cooled at the same time during charging, while, in sum, both electrodes together behaved very similarly to the already mentioned experimental results. Thus, heat production should also be measured in experiments for each electrode separately. By this, the importance of certain ingredients that we proposed to model electrolytes could be confirmed or ruled out experimentally, finally leading to a deeper understanding of the physics of EDLs.

We recently showed that the dynamics of coarse-grained observables in systems out of thermal equilibrium are governed by the non-stationary generalized Langevin equation [