http://www.bibsonomy.org/burst/user/statphys23/counterionBibSonomy BuRST Feed for /user/statphys23/counterion2008-10-08T12:58:36+02:00http://www.bibsonomy.org/bibtex/2f390b2941dc3dc16c361970c437de17d/statphys23statphys232007-06-20T10:16:09+02:00condensation interaction counterion simulations molecular dynamics polyelectrolyte self-condensation topic-7 statphys23 electrostatic <span style="color:#555555;">H. <a href="http://www.bibsonomy.org/author/Takano">Takano</a> und I. <a href="http://www.bibsonomy.org/author/Baba">Baba</a> und D. <a href="http://www.bibsonomy.org/author/Kubota">Kubota</a> und S. <a href="http://www.bibsonomy.org/author/Miyashita">Miyashita</a> </span><em>Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, </em><em>Genova, Italy, </em><em>9-13 July2007. </em>Wed Jun 20 10:16:09 CEST 2007Genova, ItalyAbstract Book of the XXIII IUPAP International Conference on Statistical Physics9-13 JulyCounterion Condensation and Self-Condensation of Single Polyelectrolytes2007condensation interaction counterion simulations molecular dynamics polyelectrolyte self-condensation topic-7 statphys23 electrostatic \par For a highly charged polyelectrolyte, counterions are known to condense onto the polyelectrolyte. Moreover, under certain conditions, the polyelectrolyte itself condenses. In this study, the relation between the the counterion condensation and the conditions for the self-condensation of single polyelectrolytes is studied by performing molecular dynamics simulations of a single flexible polyelectrolyte. We consider a system consisting of a polyelectrolyte with $N_{\rm m}$ monomers, which is represented by a bead-spring model, and $N_{\rm c}$ counterions. The charge of each monomer is $-{e}<0$ and that of each counterion is $z_{\rm c}{e}>0$. We choose $N_{\rm m}=z_{\rm c}N_{\rm c}$. The Manning ratio $\lambda_{\rm M}=e^2/(\varepsilon ak_{\rm B}T)$ describes the strength of the electrostatic interaction relative to the thermal energy. Here, $\varepsilon$, $a$, $k_{\rm B}$ and $T$ are the dielectric constant of the solvent, the distance between two consecutive monomers, the Boltzmann constant and the temperature of the system, respectively. Note that $\lambda_{\rm M}=0$ corresponds to the case of the neutral polymer. The simulations are performed for various values of $\varepsilon$ at a constant temperature $T$ by using the Langevin-type equations of motion, where $N_{\rm m}=24,48$ and $96$ and $z_{\rm c}=1,2,3$ and $4$. \par As $\lambda_{\rm M}$ is increased from zero, the mean square average $\langle R_{\rm e}^2 \rangle$ of the end-to-end distance of the polyelectrolyte first increases and then decreases after reaching a maximum value. See Fig.\ 1. The increase is due to the increase of the strength of the repulsive electrostatic interaction between the monomers. Screening of this repulsive interaction by the counterions condensed onto the polyelectrolyte causes the decrease in $\langle R_{\rm e}^2\rangle$. For large values of $\lambda_{\rm M}$, $\langle R_{\rm e}^2\rangle$ is smaller than that for $\lambda_{\rm M}=0$, which signals the self-condensation of the polyelectrolyte. Note that the self-condensation can occur independent of the counterion valence. The self-condensation implies the existence of an effective attractive interaction among the monomers. If the repulsive interaction between the monomers is counterbalanced with the effective attractive interaction, the polyelectrolyte is expected to behave as an ideal chain, where the ratio of $\langle R_{\rm e}^2\rangle$ to the the mean square average $\langle R_{\rm g}^2\rangle$ of the radius of gyration becomes 6. By determining the onset of the self-condensation by the condition $\langle R_{\rm e}^2\rangle /\langle R_{\rm g}^2\rangle =6$, it is found that the self-condensation occurs when about 90\% of the charge of the polyelectrolyte is neutralized by the condensed counterions, which agrees with the experimental fact. It is also found that $\langle R_{\rm e}^2\rangle$ takes its maximum value when about 13\% of the charge of the polyelectrolyte is neutralized.