293 research outputs found
The activity coefficient of :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
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
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
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 Gâsqrt[c], while for realistic salt concentrations, SC theory does not lead to a simple power law for G(c)
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