Contrasting Dielectric Properties of Electrolyte Solutions with Polar and Polarizable Solvents

We examine the static dielectric constant of electrolyte solutions with a polar and/or polarizable small-molecule solvent using a classical field-theoretic approach. We compute corrections to the dielectric constant and screening length due to intra- and intermolecular correlations via a renormalized one-loop approximation, accounting for the excluded volume of both solvent and electrolyte. In the salt-free case, we verify the one-loop theory by comparison with full numerical solutions of the field theory. The one-loop theory predicts either a nonlinear dielectric decrement or increment with increasing salt, depending on whether the fluid correlations are dominated by the dipolar or polarizable nature of the solvent. These contrasting regimes of nonlinear dielectric behavior are consistent with experimental trends in high- and low-dielectric constant electrolyte solutions.

Figure 1

Irreducible one-loop diagrams for the corrections to the electrostatic propagator $⟨{\widehat{\phi }}_{\mathbf{k}}{\widehat{\phi }}_{-\mathbf{k}}⟩$. Wavy lines correspond to $⟨\widehat{\phi }\widehat{\phi }{⟩}_0$, dashed lines to $⟨\widehat{w}\widehat{w}{⟩}_0$, and circles denote the appropriate vertex function. Arrows indicate momentum flows. Note that in the renormalized one-loop theory, the internal $⟨\widehat{\phi }\widehat{\phi }{⟩}_0$ propagators are replaced by $⟨\widehat{\phi }\widehat{\phi }{⟩}_1$ (we leave $⟨\widehat{w}\widehat{w}{⟩}_0$ uncorrected).

Figure 2

(a) Dependence of the dielectric constant on the repulsion parameter $u_0$ for examples of purely dipolar (blue, $\mu =3\mathrm{D}$) and polarizable (red, ${\alpha }_v=54{\AA }^3$) fluids. The mean-field dielectric constants are also shown (dashed line) for both cases. (b) Comparison of the dielectric function ${\widehat{\epsilon }}_r(k)$ in the renormalized one-loop theory (solid lines) and complex Langevin field-theoretic simulations (CL-FTS, points) of the same model for the purely dipolar and polarizable fluids, with excluded volume parameters $\beta u_0=0.04255{\mathrm{nm}}^3$ (light colors) and $\beta u_0=0.4255{\mathrm{nm}}^3$ (dark colors). The dashed lines are the corresponding Gaussian approximations. In (a),(b), we use $a_p=2\AA$. (c) Dielectric constant as a function of salt concentration, for various $a_s$, for electrolyte solutions of purely dipolar (blue, $\mu =3\mathrm{D}$) and purely polarizable (red, ${\alpha }_v=22.5{\AA }^3$) solvents with $a_p=1.1\AA$ and $\beta u_0=2.5{\mathrm{nm}}^3$. All of (a)–(c) use $T=300\mathrm{K}$ and $c_p={10}^{28}{\mathrm{m}}^{-3}$.

Figure 3

(a) Dielectric constant as a function of salt concentration according to the one-loop theory, for polarizable and polar (solid lines) and purely dipolar (dashed lines) solvents with varying dipole moment. Parameters are $T=300\mathrm{K}$, $c_p={10}^{28}{\mathrm{m}}^{-3}$, $\beta u_0=2.5{\mathrm{nm}}^3$, ${\alpha }_v=22.5{\AA }^3$, $a_p=1.1\AA$, $a_s=0.55\AA$. (b) Examples of experimental data for polar solvents exhibiting dielectric decrement and increment: NaI in propylene carbonate [3] (diamonds), NaI in dimethylacetamide [3] (squares), ${\mathrm{Bu}}_4{\mathrm{NClO}}_4$ in dichloromethane [29] (circles), ${\mathrm{LiAsF}}_6$ in dimethyl carbonate [4] (triangles).