Weak- and strong-coupling electrostatic interactions between asymmetrically charged planar surfaces

We compare weak- and strong-coupling theory of counterion-mediated electrostatic interactions between two asymmetrically charged plates with extensive Monte Carlo simulations. Analytical results in both weak- and strong-coupling limits compare excellently with simulations in their respective regimes of validity. The system shows a surprisingly rich structure in terms of interactions between the surfaces as well as fundamental qualitative differences in behavior in the weak- and the strong-coupling limits.

Figure 1

(Color online) Geometry of the system comprising two asymmetrically charged planar surfaces located at $z=\pm a$ (at separation distance $D=2a$) with neutralizing pointlike counterions of valency $q$ distributed in between [33].

Figure 2

(Color online) Rescaled weak-coupling intersurface pressure, Eqs. (19, 29), as a function of the rescaled distance, $\tilde{D}$, between two charged plates for three different asymmetry parameters $\zeta =-0.5$, 0, and 0.5 as shown on the graph. The total pressure, $\tilde{p}={\tilde{p}}_0+{\tilde{p}}_2$ [dashed lines, plotted here for $\Xi =1$ using Eq. (43)], is always lowered from its mean-field PB value, $p_0$ (solid lines, obtained for $\Xi \to 0$), since quadratic fluctuations around mean field favor attraction.

Figure 3

(Color online) Rescaled strong-coupling intersurface pressure, Eq. (71), as a function of the rescaled distance, $\tilde{D}$, between two charged plates for different asymmetry parameters $\zeta =-0.5$, 0, 0.5, and 1 as shown on the graph. Long-distance attractive pressure is suppressed for the $\zeta =0$ case. For symmetrically charged surface $(\zeta =1)$, we recover the standard SC result $\tilde{p}\left(\tilde{D}\right)=-1+2∕\tilde{D}$ [9].

Figure 4

(Color online) Rescaled bound-state surface-surface separation ${\tilde{D}}^{*}=2{\tilde{a}}^{*}$ as a function of the asymmetry parameter $\zeta$ as predicted by the zeroth-order PB theory (solid line), Eq. (34), the second-order WC theory at $\Xi =1$ (dotted line), and the strong-coupling theory (dashed line), Eq. (74). Surfaces attract for $\tilde{D}>{\tilde{D}}^{*}$ and repel otherwise. Only in the SC limit can two surfaces of equal sign $(\zeta >0)$ attract. The second-order WC result is obtained by finding the zero-pressure point of the total WC pressure $\tilde{p}\left(\tilde{D}\right)={\tilde{p}}_0\left(\tilde{D}\right)+{\tilde{p}}_2\left(\tilde{D}\right)$, Eq. (43), for given $\Xi$ and $\zeta$.

Figure 5

(Color online) Rescaled counterion density profile $\tilde{n}\left(z\right)$ between two asymmetrically charged surfaces at half-separation $\tilde{a}=\tilde{D}∕2=1.34$ for three different asymmetry parameters $\zeta =0.5$, 0, and $-0.5$ (top to bottom). Solid lines represent the PB prediction, Eqs. (20, 30), and the dashed lines show the SC prediction, Eq. (76). Symbols correspond to MC simulations data at three different coupling parameters $\Xi =0.32$ (open squares), 8.6 (filled squares), and 86 (open circles).

Figure 6

(Color online) Rescaled interaction pressure, $\tilde{p}$, as a function of the rescaled intersurface distance, $\tilde{D}$, for three different values of the asymmetry parameter $\zeta =0.5$, 0 and $-0.5$ (top to bottom). Solid lines represent the PB prediction, Eqs. (19, 29), dotted lines show the second-order weak-coupling pressure (PB plus second-order corrections), Eq. (43), and the dashed lines are the SC prediction, Eq. (71). Symbols correspond to MC simulations data at three different coupling parameters $\Xi =0.32$ (open squares), 8.6 (filled squares), and 86 (open circles). Insets show details at small pressures along with the PB theory result as well as the second-order WC result for $\Xi =0.32$.

Figure 7

(Color online) Rescaled bound-state separation ${\tilde{D}}^{*}$ as a function of the coupling parameter $\Xi$ for different values of $\zeta$. Main set: Symbols are simulation data for $\zeta =0.5$ (filled squares) and $-0.5$ (filled circles). The SC results, Eq. (74), for $\zeta =-0.5$ and 0.5 are represented by dashed and dotted lines, respectively. The inset represents the detailed view for $\zeta =-0.5$, where we also show the zeroth-order PB result, Eq. (34) (solid line), and the result from the second-order WC approximation (dotted-dashed line) obtained numerically from Eq. (43).

Figure 8

(Color online) Rescaled bound-state separation ${\tilde{D}}^{*}$ as a function of $\zeta$. Symbols are simulation data for different values of the coupling parameter at $\zeta =0.5$ and $-0.5$. The region around $\zeta =-0.5$ is expanded in the inset showing the crossover from the PB prediction (solid line) to the SC prediction (dashed line) upon increasing the coupling parameter.