Scaling Analysis of the Screening Length in Concentrated Electrolytes

Abstract

The interaction between charged objects in an electrolyte solution is a fundamental question in soft matter physics. It is well-known that the electrostatic contribution to the interaction energy decays exponentially with object separation. Recent measurements reveal that, contrary to the conventional wisdom given by classic Poisson-Boltzmann theory, the decay length increases with ion concentration for concentrated electrolytes and can be an order of magnitude larger than the ion diameter in ionic liquids. We derive a simple scaling theory that explains this anomalous dependence of the decay length on ion concentration. Our theory successfully collapses the decay lengths of a wide class of salts onto a single curve. A novel prediction of our theory is that the decay length increases linearly with the Bjerrum length, which we experimentally verify by surface force measurements. Moreover, we quantitatively relate the measured decay length to classic measurements of the activity coefficient in concentrated electrolytes, thus showing that the measured decay length is indeed a bulk property of the concentrated electrolyte as well as contributing a mechanistic insight into empirical activity coefficients.The structure of electrolytes near a charged surface and the resulting force between charged surfaces in an electrolyte solution is a fundamental question in soft matter physics. This question also underpins a plethora of applications, from supercapacitors [1] to colloidal selfassembly [2]. The classic Debye-Hückel theory [3], valid only for dilute electrolytes, predicts that the interaction between two charged surfaces in an electrolyte decays exponentially with the surface separation [4] with a decay length, called the Debye length, given bywhere is the dielectric constant of the medium (which is ion concentration-dependent), k B the Boltzmann constant, T the temperature, q the ion charge, c ion the ion concentration, andis the Bjerrum length. The Bjerrum length is the distance at which the interaction energy between two ions in a dielectric medium with dielectric constant equals the thermal energy unit k B T . The Debye-Hückel theory is a mean-field theory for asymptotically dilute electrolytes, i.e. where l 3 B c ion 1, so that the ion-ion separation is far greater than the Bjerrum length and thus the Coulomb interactions can be treated as a perturbation to ideal gas behaviour.The physical picture is less clear for concentrated electrolytes: Recent surface force balance (SFB) studies show that the interaction force between charged surfaces in an ionic liquid (molten salt at room temperature) decays exponentially, but with a decay length that is orders of magnitude longer than the Debye length or the ion diameter [5][6][7][8]. The screening lengths in concentrated inorganic salts are also long and increase with electrolyte concentration [8], in direct opposition to the prediction of the Debye-Hückel theory. Therefore, the anomalously long screening length is not a curiosity associated with ionic liquid chemistry, ...

Anion Layering and Steric Hydration Repulsion on Positively Charged Surfaces in Aqueous Electrolytes

“…The exception is perchlorate. According a recent discovery for ionic liquids and highly concentrated electrolytes unexpected longer Debye lengths can also appear above about 0.8–1 m , which is not the case here. Still in 0.1 m solution perchlorate exhibits an outer layer steric hydration repulsion that is well above the Debye length and at ≈2 times its diameter.…”

“…Above 100 m m that is, at Debye lengths that are smaller than 0.96 nm, the EDL thickness reduced less than 2–3 nm. At these dimensions the continuum based description of EDL tends to break down and unexpected trends can be observed . For small dimensions, density functional theory and molecular dynamics (MD) simulations can provide molecular insight into the distributions of atoms and molecules, making it a common method to analyze the interfacial molecular structures.…”

From Solvent-Free to Dilute Electrolytes: Essential Components for a Continuum Theory

“…Accordingly, increase in the concentration leads to the phenomenon of under-screening. 15,36 As introduced by Lee et al, 55 in a regime of concentration beyond the Kirkwood point, we observe that the screening length roughly scales as c 3/2 with respect to the Debye screening length.…”

Comments on the linear modified Poisson-Boltzmann equation in electrolyte solution theory

“…Smith et al [12], Lee et al [13] have recently investigated the experimental screening length of concentrated electrolytes. A scaling analysis [13] suggests that the decay length increases linearly with the Bjerrum length at higher concentrations. When applied to the restricted primitive model (RPM), the MSA and LMPB theories both predict a decrease in the screening length above a critical y c (y = κa).…”

“…The filled squares and the filled circles are the experimental results for the unhydrated (a = 2.94 × 10 −10 m) and hydrated (a = 5.2 × 10 −10 m) NaCl, respectively. The experimental results are taken from the Supporting Information of Smith A.M., Lee A.A., Perkin S.; electronic link given in reference[13]. The upper curve for y < y c is that for α 1 which would be unity if it was the DH value.…”