Scaling Analysis of the Screening Length in Concentrated Electrolytes

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 the classic Poisson-Boltzmann theory, the decay length increases with the 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 the 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.

Figure 1

Experimentally measured screening length ${\lambda }_S$ [8, 21, 22, 23, 24], normalized by ${\lambda }_D$, plotted against $a/{\lambda }_D$ for a range of pure ionic liquids (ILs), ionic liquid mixed with propylene carbonate (PC) molecular solvent, and various $1:1$ inorganic salts in water. One point for a pure protic ionic liquid, ethylammonium nitrate, arises from a new surface force balance measurement by us. See Supplemental Material [25] for a linear-linear plot.

Figure 2

The measured screening length ${\lambda }_S$ increases linearly with $l_B$: (a) Each data point corresponds to a 2M solution of $[{\mathrm{C}}_4{\mathrm{C}}_1\mathrm{Pyrr}][{\mathrm{NTf}}_2]$ in a different solvent and therefore different dielectric constant. Dielectric constants for the 2M solutions are calculated using the effective medium theory [19]. The vertical error bars arise from scatter between the experimental decay length measured in different experiments and different force profiles in the same experiment. The horizontal error bars correspond to scatter in the literature values of the dielectric constants. See Ref. [24] for an expanded discussion. (b) ${\lambda }_S\sim 1/T$ for pure ionic liquids $[{\mathrm{C}}_2\mathrm{mim}][{\mathrm{NTf}}_2]$ and $[{\mathrm{C}}_3\mathrm{mim}][{\mathrm{NTf}}_2]$. The data (open circles and squares) are taken from Ref. [7]. The dotted lines are the lines of best fit.

Figure 3

The excess chemical potential of aqueous sodium chloride solutions predicted using Eq. (6) and the experimentally measured screening length agrees with direct measurements [36].