Model-free test of local-density mean-field behavior in electric double layers

We derive a self-similarity criterion that must hold if a planar electric double layer (EDL) can be captured by a local-density approximation (LDA), without specifying any specific LDA. Our procedure generates a similarity coordinate from EDL profiles (measured or computed), and all LDA EDL profiles for a given electrolyte must collapse onto a master curve when plotted against this similarity coordinate. Noncollapsing profiles imply the inability of any LDA theory to capture EDLs in that electrolyte. We demonstrate our approach with molecular simulations, which reveal dilute electrolytes to collapse onto a single curve, and semidilute ions to collapse onto curves specific to each electrolyte, except where size-induced correlations arise.

Figure 1

Universality of local-density approximation electric double layers: Any ion located a distance $z+\Delta z$ from an interface of surface charge density $\Sigma$ (top right) would “feel” no different than were the interface located a distance $\Delta z$ closer (bottom right) with lower effective surface charge density $\Sigma -\Delta \Sigma$ given by Eq. (3). All LDA EDLs represent a portion of a single, universal charge density profile ${\rho }_{\mathrm{LDA}}$ (left).

Figure 2

MD simulations of dilute primitive model electrolytes (${\Phi }^{\mathrm{B}}\le 8\times {10}^{-4}$) show underlying LDA behavior. (a) For a wide range of conditions, free charge density profiles all collapse onto a universal curve, when plotted against a similarity coordinate $S_{\mathrm{PM}}$, which is derived from (6) using ${\rho }_{\mathrm{PM}}(0,\Sigma )$ obtained from simulation data. (b) The charge density $\rho (z,\Sigma )$ collapses when plotted against ${\Sigma }_{\mathrm{eff}}\left(z\right)$, which is itself indicative of LDA behavior. (c) The Gouy-Chapman similarity variable (8) also collapses simulation data well. (d) The similarity variables $S_{\mathrm{PM}}$ and $S_{\mathrm{GC}}$, independently derived, are practically indistinguishable for $S_{\mathrm{GC}}\lesssim 0.4$, and differ by less than 4% for $1.6<S_{\mathrm{GC}}\lesssim 3.2$.

Figure 3

MD simulations of PM electrolytes with large ($\sigma /{\lambda }_{\mathrm{D}}\simeq 3/15$), weakly charged (${\lambda }_{\mathrm{B}}/{\lambda }_{\mathrm{D}}\simeq 0.1/15$) ions at constant bulk volume fraction ${\overline{\Phi }}^{\mathrm{B}}=0.044\pm 0.001$ show LDA behavior beyond a surface monolayer (with packing fraction ${\Phi }^0$). (a) Charge density profiles for various surface charge densities $\tilde{\Sigma }$ collapse onto a single curve when plotted against $S_{\mathrm{PM}}^{\prime }$, obtained by evaluating (6) using measured data. (b) EDL density profiles collapse when plotted against ${\tilde{\Sigma }}_{\mathrm{eff}}$ that is reduced by the measured charge density of the monolayer $\delta \tilde{\Sigma }$. (c) The similarity coordinate $S_{\mathrm{CS}}$ generated using the LDA-Carnahan-Starling approach also collapses simulated density profiles beyond the correlated region. A one-parameter (${\Phi }^{\mathrm{B}}$) family of universal EDL curves is generated from (6) and (7) using the LDA-CS approach. (d) The theoretical $S_{\mathrm{CS}}$ matches the measured $S_{\mathrm{PM}}^{\prime }$ to within 10% for these simulations.