Coulombic Surface-Ion Interactions Induce Nonlinear and Chemistry-Specific Charging Kinetics

While important for many industrial applications, chemical reactions responsible for the charging of solids in water are often poorly understood. We theoretically investigate the charging kinetics of solid-liquid interfaces and find that the time-dependent equilibration of surface charge contains key information not only on the reaction mechanism, but also on the valency of the reacting ions. We construct a nonlinear differential equation describing surface charging by combining chemical Langmuir kinetics and electrostatic Poisson-Boltzmann theory. Our results reveal a clear distinction between late-time (near-equilibrium) and short-time (far-from-equilibrium) relaxation rates, the ratio of which contains information on the charge valency and ad- or desorption mechanism of the charging process. Similarly, we find that single-ion reactions can be distinguished from two-ion reactions, as the latter show an inflection point during equilibration. Interestingly, such inflection points are characteristic of autocatalytic reactions, and we conclude that the Coulombic ion-surface interaction is an autocatalytic feedback mechanism.

Figure 1

Time-dependent relative deviations $s_{\pm }(t)-1$ from the equilibrium charge density as follows from the kinetic Langmuir-Gouy-Chapman equations (2a) and (3) in (a), and Eqs. (2b) and (3) in (b) and (c), for equilibrium zeta potentials $(k_{\mathrm{B}}T/e)|{\varphi }_{\mathrm{eq}}|$ equal to 50 mV (crosses) and 100 mV (circles) for valencies $z=0$, 1, 2, 3 (colors), in (a) $s_{-}(t)$ for desorptive reactions when ${\sigma }_{-,\mathrm{eq}}\ll \mathrm{\Gamma }$, in (b) $s_{+}(t)$ for adsorptive reactions when ${\sigma }_{+,\mathrm{eq}}\ll \mathrm{\Gamma }$, and in (c) $s_{+}(t)$ for adsorptive reactions when ${\sigma }_{+,\mathrm{eq}}\simeq \mathrm{\Gamma }$. Insets denote semilogarithmic representations of $|s_{\pm }(t)-1|$. The case ${\sigma }_{-,\mathrm{eq}}\simeq \mathrm{\Gamma }$ (not shown) is trivial, with single-exponential decay for all $z$.

Figure 2

Time dependence of the relative deviation from the equilibrium charge density $s(t)-1$ for the autocatalytic ion displacement reaction [Eqs. (7a) and (7b)] for ten different initial conditions $s(0)$ in the experimentally common regime ${\sigma }_{\mathrm{eq}}\ll \mathrm{\Gamma }$. Symbols are numerical solutions for the full Langmuir-Gouy-Chapman equation (Supplemental Material [48]) for ${\varphi }_{\mathrm{eq}}=2$ (open symbols) and ${\varphi }_{\mathrm{eq}}=4$ (crosses), for five initial conditions. The gray dashed line denotes the inflection point [top $s(t)-1\simeq 0.6$, bottom $s(t)-1\simeq -0.25$].