Interaction modes between asymmetrically and oppositely charged rods

The interaction of oppositely and asymmetrically charged rods in salt—a simple model of (bio)macromolecular assembly—is observed via simulation to exhibit two free energy minima, separated by a repulsive barrier. In contrast to similar minima in the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, the governing mechanism includes electrostatic attraction at large separation, osmotic repulsion at close range, and depletion attraction near contact. A model accounting for ion condensation and excluded volume is shown to be superior to a mean-field treatment in predicting the effect of charge asymmetry on the free-energy profile.

Figure 1

(a) Total force per length between the rods and (b) osmotic contribution to the force from the cations in the interrod space, for varying rod charge ratio $|{\tau }_1/{\tau }_2|$ and $\kappa$=0.7 $l_B^{-1}$. A positive force corresponds to repulsion. At right, simulation snapshots of the system at two rod separations.

Figure 2

Interaction “phase diagram” for two oppositely charged rods in terms of rod charge asymmetry ($|{\tau }_1/{\tau }_2|$) and inverse Debye length $\kappa$. “Attraction” denotes attraction at all separations, “Attraction-Repulsion” long-range attraction with short-range repulsion, and “Attraction-Repulsion-Attraction” a repulsive barrier separating long- and short-range attraction. Linear interpolation between simulated data points (marked by x) is used to determine phase boundaries. Theoretical prediction for the “Attraction”–“Attraction-Repulsion-Attraction” boundary based on Eqs. (3, 4, 5) is presented as a dashed line.

Figure 3

The potentials of mean force (PMF) per length for rod charge ratios (a) $|{\tau }_1/{\tau }_2|=0.250$ and (b) $|{\tau }_1/{\tau }_2|=0.125$ at different $\kappa$.

Figure 4

The force per rod length as predicted by nonlinear Poisson-Boltzmann theory $\beta F_{\mathrm{PB}}/L$ (solid lines), condensed ion theory $\beta F_{\mathrm{IC}}/L$ (dashed lines), and simulation data (symbols), for increasing $|{\tau }_1/{\tau }_2|$ at $\kappa =0.7l_B^{-1}$. The effect of increasing salt for $|{\tau }_1/{\tau }_2|$=0.125 is presented in the insert.