Shell formation in short like-charged polyelectrolytes in a harmonic trap

Inspired by recent experiments and simulations on pattern formation in biomolecules by optical tweezers, a theoretical description based on the reference interaction site model (RISM) is developed to calculate the equilibrium density profiles of small polyelectrolytes in an external potential. The formalism is applied to the specific case of a finite number of Gaussian and rodlike polyelectrolytes trapped in a harmonic potential. The density profiles of the polyelectrolytes are studied over a range of lengths and numbers of polyelectrolytes in the trap, and the strength of the trap potential. For smaller polymers we recover the results for point charges. In the mean field limit the longer polymers, unlike point charges, form a shell at the boundary layer. When the interpolymer correlations are included, the density profiles of the polymers show sharp shells even at weaker trap strengths. The implications of these results are discussed.

Figure 1

(a) Schematic diagram showing polyelectrolytes trapped in a harmonic trap. (b) Concentric ringlike structures form due to the competing effects of the trap force and the electrostatic forces in 2D. In 3D (not shown) concentric shells are formed.

Figure 2

Mean field. (a) Monomer densities calculated from Eq. (17) for different values of trap strengths $\mathrm{\Gamma }$ for $N=100$ Gaussian polymers each containing $L=8$ monomers and monomer length (diameter) $\sigma =0.5$. Stronger interactions lead to sharper shells. (b) Monomer densities for various lengths $L$ of 100 polymers with $\sigma =0.5$ and $\mathrm{\Gamma }=8$. Also shown is the point-particle result, p-p. Longer polymers have sharper outermost shells.

Figure 3

Mean field. (a) Monomer densities for different values of the number of polymers in the trap $N$ for $L=8$ and $\sigma =0.5$ at $\mathrm{\Gamma }=8$. Increasing $N$ does not add any new shell. Instead the outermost shell moves outward. (b) Dependence of the monomer density on the monomer length $\sigma$ of the polymers under the same conditions as in (a).

Figure 4

Interpolymer pair correlation function for Gaussian polymers of lengths 2 and 4, respectively, $\mathrm{\Gamma }=4$ and $\sigma =0.5$. The longer polymer shows peaks in $g\left(r\right)$ due to stronger correlations.

Figure 5

Correlated densities. (a) The dependence of the correlated density profile (solid, filled) for 100 polymers with $L=4$ on $\mathrm{\Gamma }$. Also shown are the mean field density profiles (dashed, unfilled). Strong correlations at larger $\mathrm{\Gamma }$ produce sharper shells. (b) Correlated (solid) and mean field (dashed) monomer densities for different lengths $L$ of the polymers at $\mathrm{\Gamma }=4$. Longer polymers are more strongly correlated and hence have sharper shells. All the polymers have the same $\sigma =0.5$.

Figure 6

Correlated densities. On increasing the number of polymers with $L=4$ at $\mathrm{\Gamma }=4$ in the trap, new shells are formed unlike in the mean field case. Polymers of monomer lengths $\sigma =0.5$ (solid, filled) and 1 (dashed, unfilled) are considered.

Figure 7

Mean field. The monomer densities of 100 rodlike (solid) and Gaussian polymers (dashed) for different polymer lengths at $\mathrm{\Gamma }=4$. All the polymers have a monomer length of 1. The outermost shell is sharper for the rods than the Gaussian polymers. The three sharp peaks at the boundary are not individual shells but part of the outermost shell.

Figure 8

Correlated densities. The density profile of 100 rodlike polymers (solid) and Gaussian polymers (dashed) for different values of (a) $\mathrm{\Gamma }$'s at $L=4$ and (b) $L$'s at $\mathrm{\Gamma }=4$. The shells move outward for rodlike polymers due to stronger electrostatic repulsions. All polymers have $\sigma =0.5$.

Figure 9

Correlated densities. The density profile for rodlike polymers of length 4 at $\mathrm{\Gamma }=4$ showing more shells formed in the case of $N=100$ and 200 polymers in the trap. Polymers of different monomer lengths $\sigma =0.5$ (filled) and 1 (unfilled) are also considered. The density depends strongly on the diameter of the rods, unlike the Gaussian polymers in Fig. 6.