The time-dependent development of electric double-layers in saline solutions

Abstract

We have studied the time-dependent development of electric double-layers (ionic sheaths) in saline solutions by simultaneously solving the sodium and chlorine ion continuity equations coupled with Poisson's equation in one dimension. The study of the effects of time-varying electric fields in solution is relevant to the possible health effect of radio-frequency electric fields on cells in the human body and to assessing the potential of using external electric fields to orient proteins for attachment to surfaces for biosensing applications. Our calculations, for applied voltages of 10–175 mV between the electrode and the solution, predict time scales of ∼0.1–110 µs for the formation of double-layers in solutions of concentration between 0.001 and 1.0 M. We develop an empirical equation that can predict the double-layer formation time to within 10% over this wide parameter range. The method has been validated by comparing the solutions obtained, once the program has run to a steady state, with the standard non-linear Poisson–Boltzmann equations. Excellent agreement is found with the Gouy–Chapman solution of the non-linear Poisson–Boltzmann equation. Thus the method is not restricted in accuracy and applicability as is the case for the linear Poisson–Boltzmann equation. The method can also provide solutions for cases where there are orders of magnitude changes in the ion densities; this has not been the case for previous studies where small perturbation analysis has been employed. The method developed here can readily be extended to two and three dimensions using time-splitting methods.

The time-dependent development of electric double-layers in pure water at metal electrodes: the effect of an applied voltage on the local pH

“…Previously, a time-dependent formalism was developed to study the doublelayer formation in salt (NaCl) solutions and, when run to the steady state, the results were in excellent agreement with the Gouy-Chapman solution of the nonlinear Poisson-Boltzmann equation (Morrow et al 2006). However, H 3 O + and OH − ions arising from dissociation of water were ignored.…”

“…The higher the voltage, the more ions must move to create the ion doublelayers; thus, the decay time for the current increases rapidly with voltage. These results can be compared with previous work with NaCl solutions (Morrow et al 2006); an empirical equation for the relaxation time for that case was obtained as follows:…”

“…The concentration of ions and the distribution of potential were first solved as a steady-state problem by Gouy (1910) and Chapman (1913). The time-dependent solution for the formation of a double-layer in an aqueous electrolyte containing extrinsic ions on the other hand has only recently been solved (Morrow et al 2006). The pH of a solution is an important property, especially when the solution contains molecules for which the charge is pH-dependent, as it is for biological macromolecules.…”

Time-dependent electrical double layer with blocking electrode

“…7 Linearized Poisson-Boltzmann equation is no longer valid in predicting time-dependent EDL; instead, the continuum equations together with Poisson's equation are used in numerical modelings. 8 There are also some obstacles present in the experimental study, e.g., the thickness of EDL is usually very thin, making it difficult to use most electrical probes. At steady state, the thickness of diffuse layer is given by the Debye screening length 9…”

“…Nevertheless, a more rigorous solution to the changes in would be numerical modeling with proper boundary conditions. 8 The condition of large potential drop across the EDL is often preferred for controlling the movement of charged biomolecules because the electrostatic force will be large. Our analysis shows that in addition to the requirement of low ionic strength, which provides sufficiently thick EDL to immerse biomolecules, the frequency of alternating electric field has also to be considered.…”