Counterion density profile around a charged disk: From the weak to the strong association regime

We present a comprehensive study of the two-dimensional one-component plasma in the cell model with charged boundaries. Starting from weak couplings through a convenient approximation of the interacting potential we were able to obtain an analytic formulation to the problem deriving the partition function, density profile, contact densities, and integrated profiles that compared well with the numerical data from Monte Carlo simulations. Additionally, we derived the exact solution for the special cases of $\mathrm{\Xi }=1,2,3,\cdots$, finding a correspondence between those from weak couplings and the latter. Furthermore, we investigated the strong-coupling regime taking into consideration the Wigner formulation. Elaborating on this, we obtained the profile to leading order, computed the contact density values as compared to those derived in an earlier work on the contact theorem. We formulated adequately the strong-coupling regime for this system that differed from previous formulations. Ultimately, we calculated the first-order corrections and compared them against numerical results from our simulations with very good agreement; these results compared equally well in the planar limit, whose results are well known.

Figure 1

The 2D cylindrical cell model. The disk with charge $Q_1$ and radius $R$ is surrounded by counterions with charge $-q$ within $R$ and the external boundary at $D$ with charge $Q_2$. The interior of the disk and the exterior of the cell have the same dielectric material.

Figure 2

The value for $j^{☆}$ as a function of $f_{M}N/(1+{\xi }_B/\xi )$. Notice that the solution is the composition of the red (dark) and green (light gray) parts which tells us that $j^{☆}=⌊f_{M}N/(1+{\xi }_B/\xi )⌋+1$ and signals the discontinuities precisely at the points where $f_{M}N/(1+{\xi }_B/\xi )$ is a whole number. The dashed line represents $j^{☆}=f_{M}N/(1+{\xi }_B/\xi )$.

Figure 3

Representation of $a_j$ as a function of $j$ with $j^{☆}$ the location of the minimum of the $\left\{a_j\right\}$ for the nondegenerate case.

Figure 4

Representation of $a_j$ as a function of $j$ with $j^{☆}$ the location of the minimum of the $\left\{a_j\right\}$ for the degenerate case with $2(1+{\xi }_B)/\mathrm{\Xi }$ an even number.

Figure 5

Representation of $a_j$ as a function of $j$ with $j^{☆}$ the location of the minimum of the $\left\{a_j\right\}$ for the degenerate case with $2(1+{\xi }_B)/\mathrm{\Xi }$ an odd number.

Figure 6

The density profile $\tilde{\rho }$ near the charged disk for different values of the Manning parameter for $N=10$. The data display the results for different box sizes. The dotted and dashed curves represent the analytic prediction Eq. (3.34). The values for the Manning parameter are chosen for small $\mathrm{\Xi }$ and very close to unity where the theory is no longer valid.

Figure 7

The integrated charge $N-Q\left(r\right)$ as a function of the logarithmic distance for $\mathrm{\Delta }={10}^2$, ${\xi }_B=0$, and $N=10$ for various $\mathrm{\Xi }$. The plots read for the coupling parameter, from top to bottom, $\mathrm{\Xi }=\frac{2}{5},\frac{10}{21},\frac{1}{2},\frac{10}{19},\frac{2}{3},\frac{10}{11},1,\frac{10}{9}$, and 2.

Figure 8

The density $\tilde{\rho }$ at contact in $r=R\left(a\right)$ and $r=D\left(b\right)$ as a function of the coupling parameter $\mathrm{\Xi }$ with ${\xi }_B=0$; here $\mathrm{\Delta }$ varies and $N=10$. The range of $\mathrm{\Xi }$ extends below the onset for condensation up to high couplings. The dashed curves represent the exact contact density taken from evaluating the density from Eq. (3.26). We considered the analytic exact result since $\mathrm{\Delta }$ is small.

Figure 9

Same as in Fig. 8 but with logarithmic scaling in the $y$ axis.

Figure 10

The density $\tilde{\rho }$ at contact in $r=R$ (squares, left vertical axis scale) and $r=D$ (up triangles, right vertical axis scale) as a function of the Manning parameter; here $\mathrm{\Delta }=100$ and $N=10$. The range of $\xi$ extends below the onset for condensation up to high couplings. The dashed curves represent the $\mathrm{\Delta }\to \infty$ formulation from Eqs. (3.45) and (3.46). The arrows indicate the location of $\mathrm{\Xi }=1$, the borderline to strong coupling.

Figure 11

The energy as a function of the Manning parameter; here $N=10$ and $\mathrm{\Delta }=20,{10}^2$. The dashed curve corresponds to the $\mathrm{\Delta }\to \infty$ prediction from Eq. (3.47). The dotted curve corresponds to MF's prediction, or $N\to \infty$, likely for $\mathrm{\Delta }\to \infty$.

Figure 12

The density profile $\tilde{\rho }$ near the charged disk for different values of the Manning parameter for $N=\xi$; here $\mathrm{\Delta }=20$. The dashed curves represent the exact profile from Eq. (4.12). The axis are in logarithmic to show the trend at short and long distances.

Figure 13

The integrated charge $Q\left(r\right)/N$ as a function of the logarithmic distance for $\mathrm{\Delta }=20$ and $\xi =3,6,9$. The exact result from integrating the density superimposes with the Monte Carlo curve in all cases. Notice the linear trend of the integrated charge for $ln(r/R)$ above 3.

Figure 14

The density $\tilde{\rho }$ as a function of the radial distance $r/R$ for $\mathrm{\Xi }=2$, $\mathrm{\Delta }=100$, $N=13$, and different $\xi$ as indicated. The dashed curves represent the exact result from Eq. (5.9) as compared to numerical results from Monte Carlo simulations (symbols).

Figure 15

The integrated charge $Q\left(r\right)$ as a function of the radial distance $r/R$ for $\mathrm{\Xi }=2$, $\mathrm{\Delta }=100$, $N=13$, and different $\xi$ as indicated. The solid curves represent the exact result from Eq. (5.10) as compared to numerical results from Monte Carlo simulations (symbols).

Figure 16

The ground state for the cylindrical cell model in two dimensions. The counterions sit on a position with ${\theta }_k=2\pi (k-1)/N$.

Figure 17

The density profile ${\tilde{\rho }}_{\text{SC-1}}$ [Eq. (6.18)] as it differs with the leading order 2D–SC-0${\tilde{\rho }}_{\text{SC-0}}$ [Eq. 6.11)] as a function of the distance near the charged disk for $\mathrm{\Xi }=3/2$ (a), $\mathrm{\Xi }=3$ (b), and $\mathrm{\Xi }=9/2$ (c) for $N=5,10$; here $\mathrm{\Delta }=20$. The dashed line represents the corrected profile 2D–SC-1 from Eq. (6.18). The symbols are numerical results from Monte Carlo simulations.

Figure 18

The ratio of density profile ${\tilde{\rho }}_{{\text{SC-0}}^{☆}}$ [Eq. (6.23)] and the leading order 2D–SC-0 as a function of the distance near the charged disk for $\mathrm{\Xi }=3/2$ (a), $\mathrm{\Xi }=3$ (b), and $\mathrm{\Xi }=9/2$ (c) for $N=5,10$; here $\mathrm{\Delta }=20$. The dotted curves represent the corrected analytic term 2D–SC-1 from Eq. (6.18) while the dashed represent the heuristic alternative 2D–SC-$0^{☆}$ from Eq. (6.23).

Figure 19

The correlation function $g_{\theta }$ as a function of the angle $\theta$; here the number of particles for the upper plot is $N=5$ and $N=10$ for the other. The data were obtained by means of Monte Carlo simulations. The normalization is enforced with $g_{\theta }\to 1$ for weakly correlated as in the small-coupling cases.

Figure 20

The energy as a function of the Manning parameter; here $N=5,10$ and $\mathrm{\Delta }=20,{10}^2$. The dashed curve corresponds to the prediction from Eq. (6.26) and the symbols were obtained from Monte Carlo simulations.