The activity coefficient of zz:1 ionic solutions scales with the cube root of salt concentration

To describe the activity coefficient of ionic solutions, we develop a model where Coulomb's law is evaluated between nearest-neighbour ions that together form a neutral `soft ion ensemble'. For this ensemble we calculate the electrostatic energy and the contribution to the activity coefficient of an ion. The calculation analyses the probability distribution of all positions of the ions relative to one another within a space limited by the requirement that they are nearest neighbours and that they cannot overlap. For a symmetric salt, we only have to analyze one cation and one anion, i.e., a `soft ion pair', and we can derive an expression for the dilute limit that depends on the Bjerrum length and the cube root of salt concentration, identical to a result already put forward by Bjerrum. For a 1:1 salt, this cube root dependence describes data for the activity coefficient very well, both at low and intermediate salt concentrations, with also the prefactor correctly predicted. We also analyze an asymmetric 2:1 salt and 3:1 salt which requires numerical solution of all possible orbits of two or three monovalent anions around a central multivalent cation, and this calculation also leads to a very good prediction of the mean activity coefficient, and again the cube root law is followed up to moderate concentrations. For 2:2 and 3:3 salts, the full theory of soft ion pairs again fits data very well, but the cube root limiting law does not apply. The theory can be extended to include ion volume effects, for instance via the Carnahan-Starling equation of state

Tutorial on the chemical potential of ions in water and porous media: electrical double layer theory and the influence of ion volume and ion-ion electrostatic interactions

In this tutorial we address the chemical potential of ions in water (i.e., in a salt solution, in an electrolyte phase) and inside (charged) porous media such as nanoporous membranes. In water treatment, such membranes are often used to selectively remove ions from water by applying pressure (which pushes water through the membrane while most ions are retained) or by current (which transports ions from a feedwater stream through the membrane). Chemical equilibrium across a boundary (such as the solution-membrane boundary) is described by an isotherm for neutral molecules, and for ions is described by an electrical double layer (EDL) theory. EDL theory describes concentrations of ions inside a porous medium as function of the charge and structure of the medium. There are many contributions to the chemical potential of an ion, and we address several of these in this tutorial, including ion volume and the effect of ion-ion Coulombic interactions

The sediment of mixtures of charged colloids: segregation and inhomogeneous electric fields

We theoretically study sedimentation-diffusion equilibrium of dilute binary, ternary, and polydisperse mixtures of colloidal particles with different buoyant masses and/or charges. We focus on the low-salt regime, where the entropy of the screening ions drives spontaneous charge separation and the formation of an inhomogeneous macroscopic electric field. The resulting electric force lifts the colloids against gravity, yielding highly nonbarometric and even nonmonotonic colloidal density profiles. The most profound effect is the phenomenon of segregation into layers of colloids with equal mass-per-charge, including the possibility that heavy colloidal species float onto lighter ones

Attractive forces in microporous carbon electrodes for capacitive deionization

The recently developed modified Donnan (mD) model provides a simple and useful description of the electrical double layer in microporous carbon electrodes, suitable for incorporation in porous electrode theory. By postulating an attractive excess chemical potential for each ion in the micropores that is inversely proportional to the total ion concentration, we show that experimental data for capacitive deionization (CDI) can be accurately predicted over a wide range of applied voltages and salt concentrations. Since the ion spacing and Bjerrum length are each comparable to the micropore size (few nm), we postulate that the attraction results from fluctuating bare Coulomb interactions between individual ions and the metallic pore surfaces (image forces) that are not captured by meanfield theories, such as the Poisson-Boltzmann-Stern model or its mathematical limit for overlapping double layers, the Donnan model. Using reasonable estimates of the micropore permittivity and mean size (and no other fitting parameters), we propose a simple theory that predicts the attractive chemical potential inferred from experiments. As additional evidence for attractive forces, we present data for salt adsorption in uncharged microporous carbons, also predicted by the theory.Comment: 19 page

Coulometry and Calorimetry of Electric Double Layer Formation in Porous Electrodes

Coulometric measurements on salt-water-immersed nanoporous carbon electrodes reveal, at a fixed voltage, a charge decrease with increasing temperature. During far-out-of-equilibrium charging of these electrodes, calorimetry indicates the production of both irreversible Joule heat and reversible heat, the latter being associated with entropy changes during electric double layer (EDL) formation in the nanopores. These measurements grant experimental access --for the first time-- to the entropic contribution of the grand potential; for our electrodes, this amounts to roughly 25% of the total grand potential energy cost of EDL formation at large applied potentials, in contrast with point-charge model calculations that predict 100%. The coulometric and calorimetric experiments show a consistent picture of the role of heat and temperature in EDL formation and provide hitherto unused information to test against EDL models.Comment: 11 pages, 10 figure

Analysis of ionic conductance of carbon nanotubes

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∝sqrt[c], while for realistic salt concentrations, SC theory does not lead to a simple power law for G(c)