Quenched charge disorder and Coulomb interactions

We develop a general formalism to investigate the effect of quenched fixed charge disorder on effective electrostatic interactions between charged surfaces in a one-component (counterion-only) Coulomb fluid. Analytical results are explicitly derived for two asymptotic and complementary cases: (i) mean-field or Poisson-Boltzmann limit (including Gaussian-fluctuations correction), which is valid for small electrostatic coupling, and (ii) strong-coupling limit, where electrostatic correlations mediated by counterions become significantly large as, for instance, realized in systems with high-valency counterions. In the particular case of two apposed and ideally polarizable planar surfaces with equal mean surface charge, we find that the effect of the disorder is nil on the mean-field level and thus the plates repel. In the strong-coupling limit, however, the effect of charge disorder turns out to be additive in the free energy and leads to an enhanced long-range attraction between the two surfaces. We show that the equilibrium interplate distance between the surfaces decreases for elevated disorder strength (i.e., for increasing mean-square deviation around the mean surface charge), and eventually tends to zero, suggesting a disorder-driven collapse transition.

Figure 1

Schematic representation of our model system: two ideally polarizable planar surfaces with quenched surface charge disorder at separation $D$ with mobile (pointlike) counterions of charge $Z{e}_0$ confined in the intersurface space. The surface charge density is characterized by a Gaussian distribution with fixed mean value of magnitude $\sigma$ and mean-square deviation $g$.

Figure 2

(Color online) Free energy per particle, Eq. (75), as a function of dimensionless interplate distance $D∕\mu$ for $\chi =0$, 1, and 2 (bottom line, middle line, and top line). $\mu =e_0∕\left(2\pi {\ell }_{\mathrm{B}}Z\sigma \right)$ is the Gouy-Chapman length for mean surface charge density. Inset: dimensionless equilibrium distance $D^{*}∕\mu$ as a function of the disorder coupling parameter $\chi$.