%0 Book Section %1 statphys23_0930 %A Takano, H. %A Baba, I. %A Kubota, D. %A Miyashita, S. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K condensation counterion dynamics electrostatic interaction molecular polyelectrolyte self-condensation simulations statphys23 topic-7 %T Counterion Condensation and Self-Condensation of Single Polyelectrolytes %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=930 %X \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 $N_m$ monomers, which is represented by a bead-spring model, and $N_c$ counterions. The charge of each monomer is $-e<0$ and that of each counterion is $z_ce>0$. We choose $N_m=z_cN_c$. The Manning ratio $łambda_M=e^2/(ak_BT)$ describes the strength of the electrostatic interaction relative to the thermal energy. Here, $\varepsilon$, $a$, $k_B$ 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 $łambda_M=0$ 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 $N_m=24,48$ and $96$ and $z_c=1,2,3$ and $4$. \par As $łambda_M$ is increased from zero, the mean square average $R_e^2 \rangle$ 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 $R_e^2\rangle$. For large values of $łambda_M$, $R_e^2\rangle$ is smaller than that for $łambda_M=0$, 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 $R_e^2\rangle$ to the the mean square average $R_g^2\rangle$ of the radius of gyration becomes 6. By determining the onset of the self-condensation by the condition $R_e^2/R_g^2=6$, 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 $R_e^2\rangle$ takes its maximum value when about 13\% of the charge of the polyelectrolyte is neutralized.

@incollection{statphys23_0930, abstract = {\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.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Takano, H. and Baba, I. and Kubota, D. and Miyashita, S.}, biburl = {https://www.bibsonomy.org/bibtex/2f390b2941dc3dc16c361970c437de17d/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {3f02a1ab63318efc2bb5d7efce9e1d18}, intrahash = {f390b2941dc3dc16c361970c437de17d}, keywords = {condensation counterion dynamics electrostatic interaction molecular polyelectrolyte self-condensation simulations statphys23 topic-7}, month = {9-13 July}, timestamp = {2007-06-20T10:16:34.000+0200}, title = {Counterion Condensation and Self-Condensation of Single Polyelectrolytes}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=930}, year = 2007 }