Analysis of ionic conductance of carbon nanotubes

Abstract

We use space-charge (SC) theory (also called the capillary pore model) to describe the ionic conductance, G, of charged carbon nanotubes (CNTs). Based on the reversible adsorption of hydroxyl ions to CNT pore walls, we use a Langmuir isotherm for surface ionization and make calculations as a function of pore size, salt concentration c, and pH. Using realistic values for surface site density and pK, SC theory well describes published experimental data on the conductance of CNTs. At extremely low salt concentration, when the electric potential becomes uniform across the pore, and surface ionization is low, we derive the scaling G ∝ √ c, while for realistic salt concentrations, SC theory does not lead to a simple power law for G(c). DOI: 10.1103/PhysRevE.94.050601 The ionic conductance, G, of carbon nanotubes (CNTs) is of relevance for applications in membrane technology for water desalination, energy harvesting, and energy conversion [1][2][3][4][5][6]. Secchi et al. [7,8] recently reported the first experimental results for G of single carbon nanotubes of different radii and lengths, in a large salt concentration range (1-1000 mM) and at several values of pH. The observed dependence of G on pH, and the absence of a plateau in G at low salinity, were taken as evidence that CNTs acquire a surface charge by reversible adsorption of hydroxyl ions from water. A theoretical analysis led to a 1/3 power-law scaling of G with salt concentration, which is supported by the data.In the present work, to describe the same data of Secchi et al. [7,8], we use the general classical dilute solution theory for long and thin capillary pores, combining the extended Nernst-Planck equation with the Stokes equation for fluid flow and the Poisson-Boltzmann (PB) equation for the structure of the electrical double layer (EDL), evaluated in radial direction. This model was developed by Osterle and co-workers [9,10] and is known as the capillary pore model, or space charge (SC) theory. SC theory is based on ideal Boltzmann statistics of ions as point charges, and assumes validity of the equilibrium PB equation in the radial, r, direction [11][12][13][14][15][16][17]. SC theory also includes an axial salt concentration gradient, but this effect is neglected in the present analysis. Secchi et al. [7,8] use SC theory with several simplifications to arrive at an analytical expression for G versus pore size and salt concentration. For CNTs they introduce the key idea that the surface charge depends on pH (in the external bath) and surface potential, via a model for the reversible adsorption of hydroxyl ions.The structure of this report is as follows. We present the SC theory for the conductance G and show model simplifications when the Donnan approach, or uniform potential model [16][17][18][19] is used, valid for highly overlapped EDLs. We derive a scaling law of G with salt concentration in the low-salinity limit. We assess the assumptions made in the derivation of the analytical solution of Secchi et al. Finally we combine the full SC theor...

Role of charge regulation and flow slip in the ionic conductance of nanopores: An analytical approach

“…We thus formally expect at low enough intermediate c s a cross-over scaling law with an exponent of 1/3 (the low concentration scaling regime predicted in [5]). Our analysis, which differs from the thin double argument proposed in [25] to explain the origin of this scaling regime, shows that it can only be an intermediate one observable at sufficiently high values of maximum surface charge density σ max (pH) (for intermediate values of low c s ) because charge regulation will eventually drive the system into the low surface charge density (homogeneous) GCE regime with an exponent of 1/2. The first (dominant) term at sufficiently low intermediatec s is due to both electrical migration and the non-slip electro-osmotic contribution and the second (subdominant) one is due to the slip part of the electro-osmotic contribution.…”

“…Although the chemical nature of these various nanopores can differ a lot, some features, due to the nanoscale transport, are common and a simple theoretical model that rationalizes them is still missing. * Electronic address: manghi@irsamc.ups-tlse.fr To model these experimental results and therefore extract important nanopore characteristics such as the radius or surface charge density, either a simple interpolation formula is used [8,9] or the full space-charge model (Poisson-Nernst-Planck (PNP) and Stokes equations) [21][22][23][24] is solved numerically [1,[17][18][19][20]25]. Recently a formula has been proposed for the conductance of nanopores bearing a constant surface charge density with or without fluid slippage at the nanopore surface [11].…”

“…A similar observation has been made recently by Yazda et al [7], but with a different power law. The theoretical explanation proposed in the literature is charge regulation [5,25], i.e. charged groups appear at the nanopore surface when the pH is increased, which could be due various mechanisms, such as the adsorption of hydroxyde ions [5] or the dissociation of carboxylic groups COOH.…”

“…Since Secchi et al [5] and Biesheuvel and Bazant [25] each developed different approximate theoretical approaches that led to different ex-ponents (α = 1/3 and 1/2, respectively) at low salt concentration, the situation needs to be reexamined. Furthermore, in addition to charge regulation, the role of flow slip, neglected in [5,25], but already touched upon in [18], needs to be assessed. In this last reference the influence of slip on conductivity was investigated, but only for the single case of a relatively large nanopore radius (5 nm) and small slip length (1.25 nm).…”

Theory of Ion and Water Transport in Electron-Conducting Membrane Pores with p H-Dependent Chemical Charge

“…where we have assumed that the concentrations of H + and OHare much lower than the concentrations of cations and anions arising from salt dissociation, so that their presence is not taken into account in the electroneutrality condition (5) and neither in the total ionic flux and ionic current.…”