Mechanism of Chain Collapse of Strongly Charged Polyelectrolytes

We perform extensive molecular dynamics simulations of a charged polymer in a good solvent in the regime where the chain is collapsed. We analyze the dependence of the gyration radius $R_g$ on the reduced Bjerrum length ${\ell }_B$ and find two different regimes. In the first one, called a weak electrostatic regime, $R_g\sim {\ell }_B^{-1/2}$, which is consistent only with the predictions of the counterion-fluctuation theory. In the second one, called a strong electrostatic regime, we find $R_g\sim {\ell }_B^{-1/5}$. To explain the novel regime we modify the counterion-fluctuation theory.

Figure 1

Variation of the gyration radius $R_g$ of a PE chain with the reduced Bjerrum length ${\ell }_B$ for different valencies of counterions. The chain length is $N=204$. The two power laws intersect at $R_g/a{N}^{1/3}\approx 0.63$ ($Z=3$), 0.66 ($Z=2$), and 0.69 ($Z=1$) with the corresponding crossover values ${\ell }_B^{*}(Z)\approx 3.71$ ($Z=3$), 5.58 ($Z=2$), and 13.70 ($Z=1$).

Figure 2

Snapshots of collapsed PE in (a) weak electrostatic regime, $Z=1$, ${\ell }_B=10.93$ (left) and (b) strong electrostatic regime, $Z=1$, ${\ell }_B=34.86$ (right). (c) The corresponding radial number density profile $\rho$ of the counterions, where $r$ is the distance of the counterion from the center of mass of the chain. $r^{\prime }$ is the distance at which the density is 50% of the density at $r=0$ and ${\rho }^{\prime }=N/V^{\prime }$, where $V^{\prime }=\frac{4}{3}\pi r^{\prime 3}$.

Figure 3

(a) Variation of the electrostatic energy $E_{\text{el}}$ with the radius of gyration $R_g$ for different valency. Inset: Variation with the chain length $N$. (b) Variation of the LJ energy of the system with the radius of gyration $R_g$ for different valency.