Verschueren Alwin R M, Notten Peter H L, Schlangen Luc J M, Strubbe Filip, Beunis Filip, Neyts Kristiaan
Philips Research Laboratories, High Tech Campus 34, Eindhoven, The Netherlands.
J Phys Chem B. 2008 Oct 16;112(41):13038-50. doi: 10.1021/jp800675w. Epub 2008 Sep 19.
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.
泊松 - 玻尔兹曼（PB）方程被广泛用于计算电势与带电粒子分布之间的相互作用。在平面几何结构中的对称电解质情况下， Gouy - Chapman（GC）解通常被视为PB方程的解析解。然而，我们在此证明，这种GC解假定存在一个电场为零的本体区域，而这在微器件中是不合理的。为了扩展有效范围，我们在此获得了平面PB方程的完整数值解，并辅以解析近似。对于低施加电压，它与GC解一致。在此情况下，电双层完全吸收施加的电压，从而出现一个电场被屏蔽的区域。对于更高的电压（在微器件中为1 V量级），PB方程的解表现出截然不同的行为，即双层不再能吸收全部施加电压。相反，整个器件中会保留一个有限的电场，导致带电粒子完全分离。在这个更高电压 regime 中，双层特性不再由通常的德拜参数κ描述，并且电极处的离子浓度本质上是受限的（即使不假设空间相互作用）。此外，我们对微器件内的非水模型电解质上的电极极化电流进行了测量。对于感兴趣的完整浓度和电压范围，实验结果与我们的计算完全一致。
