Coulometry and Calorimetry of Electric Double Layer Formation in Porous Electrodes

Coulometric measurements on salt-water-immersed nanoporous carbon electrodes reveal, at a fixed voltage, a charge decrease with increasing temperature. During far-out-of-equilibrium charging of these electrodes, calorimetry indicates the production of both irreversible Joule heat and reversible heat, the latter being associated with entropy changes during electric double layer (EDL) formation in the nanopores. These measurements grant experimental access—for the first time—to the entropic contribution of the grand potential; for our electrodes, this amounts to roughly 25% of the total grand potential energy cost of EDL formation at large applied potentials, in contrast with point-charge model calculations that predict 100%. The coulometric and calorimetric experiments show a consistent picture of the role of heat and temperature in EDL formation and provide hitherto unused information to test against EDL models.

Figure 1

(a) The experimental setup to measure temperature and charge during (dis)charging of porous carbon electrodes (0.18 g each) in NaCl solution. (b) The dependence of the total charge $Q$ on $\mathrm{\Psi }$ is shown at ${\rho }_{\mathrm{s}}=1$, 10, 100, 1000 mM. Error bars are based on two or more duplicate measurements.

Figure 2

(a) In the coulometric experiment, the current (blue) initially exhibits a monotonic decay, onto which positive and negative peaks are superimposed when the electrolyte temperature (black) is decreased and increased. Plotted are data for $\mathrm{\Delta }{T}_{\text{step}}=2.5°\mathrm{C}$, ${\rho }_{\mathrm{s}}=1\mathrm{M}$, and $\mathrm{\Psi }=0.25\mathrm{V}$. (b) The temperature-scaled equilibrium charge difference $\mathrm{\Delta }Q/\mathrm{\Delta }{T}_{\text{step}}$ is shown at different potential and temperature steps.

Figure 3

In the calorimetric experiment, the cell voltage ${\mathrm{\Psi }}_{\text{cell}}$ (a) was switched to $\mathrm{\Psi }$ at $t_0$ and 0 V was applied at $t_1$. The current $I$ (b) and temperature difference $\mathrm{\Delta }T$ (c) were measured simultaneously. Plotted are data for $\mathrm{\Psi }=1\mathrm{V}$ and ${\rho }_{\mathrm{s}}=1\mathrm{M}$.

Figure 4

(a) The total heat production during charging ${\mathrm{\Pi }}_{\text{tot}}^{\text{ch}}$ (circles) and discharging ${\mathrm{\Pi }}_{\text{tot}}^{\text{dis}}$ (triangles) [see Eqs. (4) and (5)] versus the cell voltage $\mathrm{\Psi }$ at ${\rho }_{\mathrm{s}}=1$, 10, 100, 1000 mM. (b) The reversible heat ${\mathrm{\Pi }}_{\text{rev}}^{\text{ch}}$ [Eq. (6)] versus potential was measured at ${\rho }_{\mathrm{s}}=1\mathrm{M}$. Black lines in (a) and (b) indicate quadratic fits through the data as guides to the eye. Identifying ${\mathrm{\Pi }}_{\text{rev}}^{\text{ch}}={\mathrm{\Omega }}_{\text{ent}}$, the inset shows the ratio ${\mathrm{\Omega }}_{\text{ent}}/\mathrm{\Omega }$ versus potential.