Screening and Separation of Charges in Microscale Devices: Complete Planar Solution of the Poisson−Boltzmann Equation

Abstract

The Poisson-Boltzmann (PB) equation is widely used to calculate the interaction between electric potential and the distribution of charged species. In the case of a symmetrical electrolyte in planar geometry, the Gouy-Chapman (GC) solution is generally presented as the analytical solution of the PB equation. However, we demonstrate here that this GC solution assumes the presence of a bulk region with zero electric field, which is not justified in microdevices. In order to extend the range of validity, we obtain here the complete numerical solution of the planar PB equation, supported with analytical approximations. For low applied voltages, it agrees with the GC solution. Here, the electric double layers fully absorb the applied voltage such that a region appears where the electric field is screened. For higher voltages (of order 1 V in microdevices), the solution of the PB equation shows a dramatically different behavior, in that the double layers can no longer absorb the complete applied voltage. Instead, a finite field remains throughout the device that leads to complete separation of the charged species. In this higher voltage regime, the double layer characteristics are no longer described by the usual Debye parameter kappa, and the ion concentration at the electrodes is intrinsically bound (even without assuming steric interactions). In addition, we have performed measurements of the electrode polarization current on a nonaqueous model electrolyte inside a microdevice. The experimental results are fully consistent with our calculations, for the complete concentration and voltage range of interest.

Characterizing Generated Charged Inverse Micelles with Transient Current Measurements

“…n ̅ /ε 0 ε r kT, but in all cases where φ 1 ≫ 1 the initially present charged inverse micelles eventually get separated 27,28,30 corresponding to a total transported charge Q = zen ̅ dS. During this transient the concentration of newly generated inverse micelles gradually increases according to eq 3 until quasi steady-state is reached.…”

“…Since the formation of a charged inverse micelle comes at a large free energy cost, we can restrict to the case that β ≪ α and therefore that n ̅ ≪ n ̅ 0 . 10 When a voltage difference is applied across the electrodes, the transient current I (t) can be modeled with 27,30 …”

“…Then the equilibrium concentration of charges in the bulk is much lower than the equilibrium concentration. 27,31,34 In these surfactant systems, at applied voltages above 1 V, the initially present charged inverse micelles can be separated, leaving the bulk almost depleted of charge. 16, 27−30 The charged inverse micelles which are attracted to the electrodes do not adsorb at the interfaces but instead form a diffuse double layer.…”

Charging Dynamics of Aerosol OT Inverse Micelles

“…41,42 E is the position-and time-dependent electrical field (V/m), D is the diffusion coefficient (m 2 /s) which is related to the mobility of charged inverse micelles μ by D = μk B T/ze. The electric field is determined by the externally applied potential difference V A and by space charge present in the bulk through Gauss's law ϵ r ϵ 0 (∂E/∂x) = ρ where ρ is the space charge density conserving overall charge neutrality ∫ −d/2 d/2 ρ dx = 0.…”

Charge transport and current in non-polar liquids

“…The comparison between simulated and measured transient currents indicates that the cited mechanisms describe the behavior quite well, even if the concentration, cell thickness or applied voltage vary over several orders of magnitude [1][2][3][4].…”