Effective Debye length in closed nanoscopic systems: A competition between two length scales

Abstract

The Poisson-Boltzmann equation (PBE) is widely employed in fields where the thermal motion of free ions is relevant, in particular in situations involving electrolytes in the vicinity of charged surfaces. The applications of this non-linear differential equation usually concern open systems (in osmotic equilibrium with an electrolyte reservoir, a semi-grand canonical ensemble), while solutions for closed systems (where the number of ions is fixed, a canonical ensemble) are either not appropriately distinguished from the former or are dismissed as a numerical calculation exercise. We consider herein the PBE for a confined, symmetric, univalent electrolyte and quantify how, in addition to the Debye length, its solution also depends on a second length scale, which embodies the contribution of ions by the surface (which may be significant in high surface-to-volume ratio micro- or nanofluidic capillaries). We thus establish that there are four distinct regimes for such systems, corresponding to the limits of the two parameters. We also show how the PBE in this case can be formulated in a familiar way by simply replacing the traditional Debye length by an effective Debye length, the value of which is obtained numerically from conservation conditions. But we also show that a simple expression for the value of the effective Debye length, obtained within a crude approximation, remains accurate even as the system size is reduced to nanoscopic dimensions, and well beyond the validity range typically associated with the solution of the PBE.

Evaluation of Aromatic Boronic Acids as Ligands for Measuring Diabetes Markers on Carbon Nanotube Field-Effect Transistors

“…The Debye length is defined as the distance after which mobile charges, such as ions, will screen out the electric field strength. It is inversely proportional to the square root of the ionic concentration of the medium so at physiological ion concentrations (∼150 mM) the Debye length will be around 1 nanometer, which is close to the length of a 4-carbon spacer [28,29]. In some cases reducing the salt concentration can resolve the issue of distance [30,31].…”

Effects of molecular level surface roughness on electroosmotic flow

“…First, the scenario where the Debye length is comparable to surface roughness is seldom studied. Such a scenario usually arises when the electrolyte is at physiological concentrations (c % 0.15 M, Debye length k D = 8 Å ) or in closed nanofluidic system with moderate to high surface charge density where the effective Debye length can be small (Tessier and Slater 2006). Second, in most studies, the electrical double layer is modeled by the Poisson-Boltzmann equation, whose accuracy can be in question for thin EDL as the molecular nature of ion, water and surfaces and the discreteness of surface charges are not accounted in its formulation (Qiao and Aluru 2003).…”

Electrostatic free energy for a confined nanoscale object in a fluid

“…The first step in a free energy calculation for a system of charged entities involves obtaining the spatial electrostatic potential distribution by solving the nonlinear PB equation subject to appropriate boundary conditions. 23,24 Given the spatial distribution of the electrostatic potential, φ(x, y, z), all quantities that constitute the free energy can be directly obtained by integration, as shown in Eqs. (2)- (5).…”

“…This problem has been studied previously for the confined fluidic systems as well as in the context of charged lamellar membranes, which for the purpose of electrostatics is a good physical analog of the charged nanoslit. [22][23][24] In our calculation, we further modify the geometry of the bounding slit walls to reflect the geometric perturbation introduced in the experiment. We consider a parallel plate geometry with a gap of 2h = 215 nm where the top surface carries a disc-shaped indentation or "pocket" 200 nm in diameter and 100 nm in depth, as shown in Fig.…”

“…24 For the interior of the particle, we assume a uniform dielectric environment with dielectric constant given by ε p and zero space charge density, i.e., ∇ 2 ψ = 0. The equations were numerically solved using COMSOL Multiphysics for one half-space of the geometry, as depicted in Fig.…”