Scaling and universality in the counterion-condensation transition at charged cylinders

Counterions at charged rodlike polymers exhibit a condensation transition at a critical temperature (or, equivalently, at a critical linear charge density for polymers), which dramatically influences various static and dynamic properties of charged polymer solutions. We address the critical and universal aspects of this transition for counterions at a single charged cylinder in two and three spatial dimensions using numerical and analytical methods. By introducing a Monte Carlo sampling method in logarithmic radial scale, we are able to numerically simulate the critical limit of infinite system size (corresponding to the infinite-dilution limit) within tractable equilibration times. The critical exponents are determined for the inverse moments of the counterionic density profile (which play the role of the order parameters and represent the mean inverse localization length of counterions) both within mean-field theory and within Monte Carlo simulations. In three dimensions (3D), we demonstrate that correlation effects (neglected within mean-field theory) lead to an excessive accumulation of counterions near the charged cylinder below the critical temperature (i.e., in the condensation phase), while surprisingly, the critical region exhibits universal critical exponents in accordance with mean-field theory. Also in contrast with the typical trend in bulk critical phenomena, where fluctuations become more enhanced in lower dimensions, we demonstrate, using both numerical and analytical approaches, that mean-field theory becomes exact for the two-dimensional (2D) counterion-cylinder system at all temperatures (Manning parameters), when the number of counterions tends to infinity. For a finite number of particles, however, the 2D problem displays a series of peculiar singular points (with diverging heat capacity), which reflect successive delocalization events of individual counterions from the central cylinder. In both 2D and 3D, the heat capacity shows a universal jump at the critical point and the internal energy develops a pronounced peak. The asymptotic behavior of the energy peak location is used to determine the critical temperature, which is also found to be in agreement with the mean-field prediction.

Figure 1

The three-dimensional model consists of a charged cylinder of infinite length $L$ and its neutralizing counterions confined in a cylindrical box (see the text for parameters).

Figure 2

Typical snapshots from the simulations on the counterion-cylinder system in 3D for (a) lateral extension parameter $\Delta =\ln (D∕R)=100$ and three different Manning parameters $\xi =0.7,1.0$, and 2.0 as indicated on the graph and (b) for Manning parameter $\xi =1.0$ and smaller lateral extension parameters $\Delta =10$ and 25. The snapshots show top views of the simulation box (see Sec. 5B and Fig. 1) with radial distances shown in logarithmic scale $y=\ln (\tilde{r}∕\tilde{R})$. Pointlike counterions are shown by black spheres and the charged cylinder by a circle in the middle. (c) gives the simulated radial density of counterions in rescaled units, $\tilde{\rho }\left(\tilde{r}\right)=\rho \left(r\right)∕\left(2\pi {\ell }_{\mathrm{B}}{\sigma }_{\mathrm{s}}^2\right)$, as a function of the (linear) distance from the cylinder axis. Main set shows the data for $\Delta =100$ and Manning parameters $\xi =2.0$ (open squares), 1.5 (open up triangles), 1.1 (open diamonds), 1.0 (solid squares), and 0.7 (solid circles) from top to bottom. The inset shows the same for $\xi =1.0$, but for various lateral extension parameters $\Delta =$10, 25, and 100 (top to bottom). Solid curves represent the mean-field PB prediction, Eq. (27). Number of counterions here is $N=100$ and the coupling parameter $\Xi =0.1$. Error bars are smaller than the size of symbols.

Figure 3

Cumulative density $n\left(\tilde{r}\right)$ per total number $N$ of counterions as a function of the logarithmic distance from the charged cylinder, $\ln (\tilde{r}∕\tilde{R})$. Dot-dashed curves are the simulation results for $\Xi =0.1$, $N=70$, and $\Delta =300$ and various Manning parameters as shown on the graph. These curves also closely represent the PB prediction (73), which is not explicitly shown. Solid curves show the simulation results for large coupling parameter $\Xi ={10}^2$ and for $\xi =3.0$ and $\xi =2.0$.

Figure 4

Main set: simulated condensed fraction of counterions, $\alpha$, as defined via the inflection-point criterion, as a function of the Manning parameter $\xi$ for $\Xi =0.1$ (diamonds). Solid curve displays the Manning or PB limiting value ${\alpha }_{\mathrm{M}}$, for $\Delta =\ln (D∕R)\to \infty$ [Eq. (72)]. Inset: condensed fraction as a function of the coupling parameter $\Xi$ for $\xi =3.0$. These data are obtained for $\Delta =300$ and $N=70$.

Figure 5

Simulation data for the order parameter $S_1=⟨1∕\tilde{r}⟩$ as a function of the Manning parameter $\xi$ for coupling parameters $\Xi =0.1$ up to ${10}^5$ as indicated on the graph. The mean-field (solid line) and the strong-coupling (dashed line) curves are calculated from Eqs. (40, 67), respectively. The lateral extension parameter is $\Delta =300$, and the number of counterions is $N=200$ for $\Xi =0.1$, $N=50$ for $\Xi ={10}^5$, and $N=70$ for other coupling parameters.

Figure 6

Rescaled radial density of counterions, $\tilde{\rho }\left(\tilde{r}\right)=\rho \left(r\right)∕\left(2\pi {\ell }_{\mathrm{B}}{\sigma }_{\mathrm{s}}^2\right)$, as a function of the (linear) distance from the cylinder axis for Manning parameter $\xi =3.0$ and various coupling parameters ($\Xi =0.1$ up to ${10}^5$) as shown on the graph. Here $\Delta =300$ and the number of counterions is $N=50$ for $\Xi ={10}^5$ and $N=70$ for other values of $\Xi$. The mean-field (solid line) and the strong-coupling (dashed line) curves are obtained from Eqs. (27, 63), which, for $\Delta =300$, roughly coincide with the asymptotic expressions (34, 65).

Figure 7

The one-dimensional pair distribution function of counterions at contact with the cylinder as defined in Eq. (76). Symbols show the simulation data for $\Xi ={10}^5$ and $\xi =4.0$ (solid diamonds), $\Xi ={10}^3$ and $\xi =2.0,3.0$, and 4.0 (open symbols from bottom to top) and for coupling parameters $\Xi =10$ and $\Xi =100$ with Manning parameter $\xi =3.0$ (cross symbols and solid up triangles respectively).

Figure 8

(a) The rescaled internal energy $\tilde{E}=E_N∕\left(N{k}_{B}T\right)$ and (b) the rescaled heat capacity $\tilde{C}=C_N∕\left(N{k}_B\right)$ of the counterion-cylinder system as a function of the Manning parameter $\xi$. Open symbols show the simulation data for small coupling parameter $\Xi =0.1$ and lateral extension parameters $\Delta =100,200$, and 300 as indicated on the graph. Solid symbols are the data for large coupling parameter $\Xi ={10}^2$ with $\Delta =200$. Number of counterions is $N=100$. Solid curves show the asymptotic PB prediction for $\Delta \to \infty$ [Eqs. (61, 62)]. Dashed curves are the full PB results for $\Delta =100$ and 300 (from bottom to top), which are calculated numerically using Eqs. (59, 81). The inset shows a closer view of the energy peak.

Figure 9

Main set: location of the peak of energy, ${\xi }_{*}^E$, as a function of the inverse lateral extension parameter, $1∕\Delta =\left[\ln \right(D∕R){]}^{-1}$. Open symbols are the simulation data for small coupling parameter $\Xi =0.1$ and different number of counterions (from bottom): $N=25$ (down triangles), 50 (squares), 100 (up triangles), 200 (circles), and 300 (diamonds). Solid curve shows the mean-field prediction for the peak location obtained by numerical evaluation of the full PB energy, Eq. (59). Inset: location of the simulated energy peak as a function of the coupling parameter $\Xi$ for $N=200$ and $\Delta =300$.

Figure 10

Main set: order parameter $S_1=⟨1∕\tilde{r}⟩$ as a function of the inverse lateral extension parameter $1∕\Delta$. Open symbols are the simulation data for $\Xi =0.1$ and Manning parameters (from top): $\xi =1.05$ (squares), 1.01 (circles), 1.0 (open diamonds), 0.99 (up triangles), and 0.97 (down triangles). Solid diamonds are the data for large coupling parameter $\Xi ={10}^2$ and $\xi =1.0$. Number of counterions, $N=100$, is fixed. Dot-dashed curves are the PB result (40) for the corresponding $\xi$. Inset: simulated $S_1$ as a function of the inverse number of counterions, $1∕N$, for $\xi =1.0$, $\Delta =300$, and coupling parameters $\Xi =0.1$ (open diamonds), ${10}^2$ (solid diamonds), and ${10}^3$ (open circles). Dashed line shows the power law $S_1\sim N^{-2∕3}$.

Figure 11

Rescaled order parameter $N^{a∕c}{S}_1$ as a function of the rescaled reduced Manning parameter $\zeta N^{1∕c}$ in the vicinity of the critical point ${\xi }_c=1.0$ for small and large coupling parameters $\Xi =0.1$ (main set) and $\Xi ={10}^3$ (inset). Symbols show the simulation data for various number of particles $N=50$ (down triangles), 70 (circles), 75 (squares), 100 (diamonds), 200 (cross symbols), and 300 (up triangles) at fixed $\Delta =300$. Here, the exponents are chosen as $a∕c=2∕3$ and $1∕c=1∕3$. Error bars are smaller than the symbol size.

Figure 12

Main set: simulated order parameter $S_1$ as a function of the lateral extension parameter $\Delta$ for increasing number of counterions from $N=70$ up to 300 as indicated on the graph (Manning parameter is $\xi =1.0$ and the coupling parameter $\Xi =0.1$). Dashed line shows the PB power law, Eq. (45). Inset: rescaled order parameter $S_n=⟨1∕{\tilde{r}}^n⟩$ as a function of the index $n$ for Manning parameters close to the critical value (from top): $\xi =1.03$ (solid circles) and $\xi =1.01$ (solid diamonds). Here, the coupling parameter is $\Xi =0.1$, number of counterions is $N=100$, and $\Delta =300$.

Figure 13

Rescaled order parameter ${\Delta }^{a∕b}{S}_1$ as a function of the rescaled reduced Manning parameter $\zeta {\Delta }^{1∕b}$ in the vicinity of the critical point ${\xi }_c=1.0$ for small and large coupling parameters $\Xi =0.1$ (main set) and $\Xi ={10}^5$ (inset). Symbols show the simulation data for various lateral extension parameters $\Delta =100$ (circles), 150 (cross symbols), 200 (diamonds), 250 (plus symbols), and 300 (up triangles). Number of counterions is fixed ($N=200$ for the coupling parameter $\Xi =0.1$ and $N=100$ for $\Xi ={10}^5$), and the exponents are chosen here as $a∕b=2.0$ and $1∕b=1.0$.

Figure 14

Order parameter $S_1=⟨1∕\tilde{r}⟩$ as a function of the Manning parameter $\xi$ for the 2D counterion-cylinder system. Symbols show the simulation data for different number of particles as indicated on the graph. The mean-field (solid) and the strong-coupling (dashed) curves are obtained from Eqs. (40, 67), respectively, which are valid in 2D as well. The lateral extension parameter here is $\Delta =300$. Thin dashed curves are guides to the eyes.

Figure 15

The rescaled internal energy of the 2D counterion-cylinder system, $\tilde{E}=E_N∕\left(N{k}_{B}T\right)$, as a function of the Manning parameter $\xi$ for $\Delta =300$ and different number of particles as indicated on the graph. Symbols show the simulation data, and dashed curves are the analytical results obtained from Eq. (125). Singular regions represent successive condensation (decondensation) of counterions.

Figure 16

The rescaled heat capacity of the 2D system, $\tilde{C}=C_N∕\left(N{k}_B\right)$, as a function of the Manning parameter $\xi$ for $\Delta =300$ and different number of particles, $N$, as indicated on the graph (for clarity, the data are multiplied by $N$). Symbols show the simulation data, and dashed curves are the analytical results obtained from Eq. (126). The peaks represent successive condensation (decondensation) of counterions.