Counterions at Charged Cylinders: Criticality and Universality beyond Mean-Field Theory

The counterion-condensation transition at charged cylinders is studied using Monte Carlo simulations. Employing logarithmically rescaled radial coordinates, large system sizes are tractable and the critical behavior is determined by a combined finite-size and finite-ion-number analysis. Critical counterion localization exponents are introduced and found to be in accord with mean-field theory both in two (2D) and three (3D) dimensions. In 3D, the heat capacity shows a universal jump at the transition, while in 2D, it consists of discrete peaks where single counterions successively condense.

Figure 1

Snapshot top views for (a) logarithmic box radius $\Delta =\ln (D/R)=100$ and Manning parameters $\xi =0.7$, 1.0, and 3.0, and (b) $\xi =1.0$ and $\Delta =10$ and 25 shown in logarithmic radial scale $y=\ln (r/R)$ (number of particles $N=100$ and the coupling parameter $\Xi =0.1$). (c) Radial counterion distribution function, $p(\tilde{r})$, for $\xi =1.0$, $\Xi =0.1$, $N=100$, and various system sizes. (d) Counterion distribution for $\xi =3.0$ and various $\Xi$ shows a crossover between MFT (solid curve) and SC (dashed curve) predictions, Eqs. (1, 2). (e) Order parameter $S_1=⟨1/\tilde{r}⟩$ as a function of Manning parameter, $\xi$, for different coupling $\Xi$ compared with MFT (solid curve) and SC (dashed curve) results, Eqs. (3, 4). In (d), (e) $\Delta =300$ and typically $N=200$.

Figure 2

(a) Rescaled internal energy, $\tilde{E}/\Delta$, and (b) rescaled (excess) heat capacity, $\tilde{C}/\Delta$, as a function of Manning parameter, $\xi$, for fixed $N=100$ compared with the exact MFT prediction for $\Delta \to \infty$ (solid line), $\Delta =300$ and 100 (broken lines, from top to bottom). (c) Location of the peak of the energy, ${\xi }_{*}^E$, as a function of $1/\Delta$ for $\Xi =0.1$ compared with MFT (solid line). Inset: peak location vs $\Xi$ for $N=200$ and $\Delta =300$. (d) Scale invariance of the order parameter, $S_1$, near ${\xi }_c$ with respect to the reduced Manning parameter, $\zeta =1-{\xi }_c/\xi$, and the counterion number, $N$, with exponents $a/c=2/3$ and $1/c=1/3$ (logarithmic size is $\Delta =300$). (e) Scale invariance of $S_1$ for various $\Delta$ with exponents $a/b=2$ and $1/b=1$ ($N=200$ is fixed).

Figure 3

2D results for $\Delta =300$: (a) order parameter $S_1=⟨1/\tilde{r}⟩$ as a function of Manning parameter, $\xi$, for different particle numbers, $N$, compared with MFT and SC predictions, Eqs. (3, 4). (b) Rescaled energy and (c) heat capacity (per particle) exhibit discrete singularities where single counterions condense; dashed lines denote analytical results.