Anomalous Underscreening in the Restricted Primitive Model

Underscreening is a collective term for charge correlations in electrolytes decaying slower than the Debye length. Anomalous underscreening refers to phenomenology that cannot be attributed alone to steric interactions. Experiments with concentrated electrolytes and ionic fluids report anomalous underscreening, which so far has not been observed in simulation. We present Molecular Dynamics simulation results exhibiting anomalous underscreening that can be connected to cluster formation. A theory that accounts for ion pairing confirms the trend. Our results challenge the classic understanding of dense electrolytes impacting the design of technologies for energy storage and conversion.

Figure 1

Phase diagram of the RPM for (a) the concentration $c$ and Bjerrum length ${\lambda }_B$ and (b) for reduced temperature $T^{*}=\sigma /{\lambda }_B$ and total number density ${\rho }^{*}=2{\rho }_s{\sigma }^3$. Each symbol marks a parameter set for which we have run MD simulations. The data points for $c=0.1\mathrm{mol}/\mathrm{L}$ and ${\lambda }_B=0.1,\dots ,5.0$ nm are highlighted by an orange rectangle. Special symbols show the sets used in a previous theoretical study (DFT) [24] and in experiments [5] with NaCl in water (NaCl), [C4C1Pyrr][NTf2] in propylene carbonate (C4mix), and an ionic liquid (IL) (further details in the Supplemental Material [46]). Horizontal lines mark the region of liquid-gas phase coexistence [63]; see Ref. [64] for further phases.

Figure 2

Charge-correlation function $h_{\mathrm{cc}}(r)$ in a representation that shows the decay length as the slope of the graph. This function was sampled by a MD simulation with $c=0.1\mathrm{mol}/\mathrm{L}$, ${\lambda }_B=5\mathrm{nm}$, and $\sigma =0.3\mathrm{nm}$. The pole fits have the analytical form of Eq. (1), respectively (see Supplemental Material [46] for further details on the fits).

Figure 3

Decay lengths ${\lambda }_1$ and ${\lambda }_2$ obtained by fitting ${\sum }_{i=1}^n{H}_i(r)$, $n\in \{1,2,3\}$ to bulk charge-correlation functions sampled from our MD simulations of the RPM as exemplarily shown in Fig. 2. ${\lambda }_1$ represents decay lengths of the structural decay and ${\lambda }_2$ represents decay lengths of the long-ranged monotonic decay (compare poles 1 and 2 in Fig. 2). Note that in some cases we used a third pole for the fit [46]. We show the decay length $\lambda$ in relation to the Debye length ${\lambda }_D$ against $\sigma /{\lambda }_D$ (depending on ${\lambda }_B$ and $c$), as common in the literature on underscreening [11]. For each given concentration, we varied only the Bjerrum length. Large black circles represent data from experiments on an ionic liquid ($|$), NaCl in water ($+$), and [C4C1Pyrr][NTf2] in propylene carbonate ($⨯$) [5]. Dotted lines depict power laws as noted.

Figure 4

Decay lengths represented as in Fig. 3. (a) Symbols without lines show decay lengths of anomalous underscreening (${\lambda }_2$ in Fig. 3) as obtained by fitting the charge-correlation functions from our MD simulations of the RPM. Triplets of vertically arranged blue symbols show decay lengths ${\lambda }^{\mathrm{MD},f}$ induced by free ions for different connectivity lengths $d=1.5\sigma$, $d=2\sigma$, $d=3\sigma$ (from top to bottom) in the cluster search algorithm. Yellow lines show the prediction ${\lambda }^{\text{theory}}$ of our minimal theory. (b) Directly measured ${\lambda }_1$ and ${\lambda }_2$ obtained by fitting the charge-correlation functions sampled from our MD simulations for parameter pairs $(c,{\lambda }_B)$ as used in the experiments (EX) of [5]. The experimental data are described in Fig. 3 and also listed in the Supplemental Material [46]. Blue symbols show the resulting ${\lambda }^{\mathrm{MD},f}$ from our cluster analysis for the same parameters. Yellow symbols (line added for clarity) show the corresponding ${\lambda }^{\text{theory}}$ from our minimal theory.