Classical density functional treatment of polydisperse polarisable clusters

TL;DR

The paper develops a generalized classical density functional framework for mixtures of two monomer types that form living, polydisperse ion clusters capable of polarising near charged surfaces. By treating clusters as polarisable polymers with reversible A–A, A–B, and B–B bonds, the authors derive a self-consistent cDFT formalism and explore two equilibrium paradigms: full electrochemical equilibrium and a semi-restricted case where clusters neutralise themselves between surfaces while dissociated ions neutralise surface charges. In full equilibrium, clustering exerts only modest influence on inter-surface forces due to the strong presence of dissociated ions; under semi-restricted equilibrium, even a small cluster fraction yields strong, long-range repulsion, offering a possible mechanism for anomalous underscreening observed in experiments. The results highlight how constraints on cluster correlations can qualitatively and quantitatively alter surface forces, suggesting that non-equilibrium or constrained clustering could play a role in the observed screening anomalies in concentrated electrolytes.

Abstract

Ion clustering has been proposed as a mechanism leading to the peculiar 'anomalous underscreening' phenomenon seen for electrostatic interactions between charge surfaces immersed in concentrated electrolytes. These interactions have been measured using the Surface Force Apparatus, according to which there are strong repulsive interactions between like-charged surfaces, with a range that increases upon further addition of salt, above some threshold concentration. A common suggestion is that ionic aggregates, if they form in sufficient numbers, will reduce the concentration of free ions and thereby increase the nominal Debye length. In previous work, we investigated a cluster model using classical Density Functional Theory (cDFT) and a polymer-like description of the ion clusters. These clusters were monodisperse and of either a linear or branched architecture, and a fixed charge sequence along the chains. In this work, we generalise the cDFT to treat 'living polymers' with variable chain lengths and charge arrangements along the chain. This approach allows clusters to become polarised by the presence of charged surfaces, manifested by like-charged bonding. We find that even with a small degree of like-charged bonding a full equilibrium treatment of our model predicts only weak repulsion between like-charged surfaces. When a global constraint is applied so that the charged surfaces are neutralised only by the dissociated ions, while the clusters contribute overall zero charge, even a very small fraction of clustering ions generate strong and long-ranged forces. Moreover, if the cluster fraction increase substantially upon the addition of further salt, then the strength of the surface forces will also increase, although the range remains roughly constant.

Figures (6)

• Figure 1: Visualisation of the $\langle N_{seg}\rangle, \langle n\rangle$ dependence on $\kappa_{++}$ and $\kappa_{+-}$. Regions with $\kappa_{++} > \kappa_{+-}$ depict cases for which the backbone charges primarily (but not completely) alternate (in the absence of external fields), while $\kappa_{+-} > \kappa_{++}$ promotes the formation of block charges. The blue line and shaded area indicate the unphysical region, where $1/K$ diverges, or becomes negative.
• Figure 2: a$A/B$ mixtures: net interaction free energies. b$A/B$ mixtures: density profiles. Solid and dashed lines display distributions of $A$ and $B$ monomers.
• Figure 3: A comparison between the long-ranged net pressure tails at approximately 1 M dissociated salt, in the absence and presence of 20 mM "monomer salt". The latter ion type can polymerise as well as polarise: $\langle r \rangle=25$ and $\langle n \rangle\approx 1.14$. While not shown here, the results are very similar with fully alternating chains, $\langle n \rangle=1.00$.
• Figure 4: The net pressures, at semi-restricted equilibrium, with $\langle r \rangle=25$ and various degrees of polarisability, $\langle n \rangle$. The bulk solution contains $980$ mM dissociated salt and $20+20$ mM monomers (cat- and anionic). The inset displays the long-range decay, as illustrated by the logarithm of the net pressure. The dotted line, for $\langle n \rangle=1.22$, demonstrates the equivalence between the analytic ($p_{net} = -\partial g_s/\partial h$) and discrete ($p_{net} = -\delta g_s/\delta h$) free energy derivatives. The corresponding net pressure at full equilibrium is shown for reference, by a green line with diamonds.
• Figure 5: a Illustrations of possible effects on surface interactions, obtained by adding salt, under the assumption of semi-equilibrium conditions and concentration-induced clustering. Bond parameters ($\kappa_{++}$ and $\kappa_{+-}$) are chosen such that $\langle r \rangle=25$ and $\langle n \rangle \approx 1.14$ in all cases. b Long-range $p_{net}$ decay. c Comparing density ($n(z)$) and d charge density ($\rho(z)$) profiles at a surface separation of $100$ Å. The dashed lines in graph c show anion profiles. The bulk concentration of monomers and dissociated salt is $50$ mM and $1950$ mM (commensurate with a $2$ M salt solution, $50$ mM of which form clusters). Results are shown from calculations at full and semi-restricted equilibrium. e Comparing the variation of restriction potentials $\psi^{Donnan}$ (full equilibrium) and $\psi_d^{D_d}$ (semi-restricted equilibrium) with separation. The bulk concentrations of monomers and dissociated salt are $50$ mM and $1950$ mM, respectively.