Polyelectrolyte Threading through a Nanopore

Abstract

Threading charged polymers through a nanopore, driven by electric fields E, is investigated by means of Langevin dynamics simulations. The mean translocation time 〈 τ 〉 is shown to follow a scaling law N^α, and the exponent α increases monotonically from 1.16 (4) to 1.40 (3) with E. The result is double-checked by the calculation of mean square displacement of translocation coordinate, which asserts a scaling behavior t^β (for t near τ) with β complying with the relation αβ = 2. At a fixed chain length N, 〈τ〉 displayed a reciprocal scaling behavior E^-1 in the weak and also in the strong fields, connected by a transition E^-1.64(5) in the intermediate fields. The variations of the radius of gyration of chain and the positions of chain end are monitored during a translocation process; far-from-equilibrium behaviors are observed when the driving field is strong. A strong field can strip off the condensed ions on the chain when it passes the pore. The total charges of condensed ions are hence decreased. The studies for the probability and density distributions reveal that the monomers in the trans-region are gathered near the wall and form a pancake-like density profile with a hump cloud over it in the strong fields, due to fast translocation.

Keywords: conformation; density distribution; ion condensation; molecular simulations; polyelectrolyte; probability distribution; scaling behavior; translocation.

Figure 1

(a) Snapshot of a system of $N=128$, where N refers to the number of monomers; (b) the membrane wall viewed from the trans side of the system. A pore is punched through the wall at the center. The gray beads represent the wall. The yellow, white, and green beads represent the monomers, cations, and anions, respectively.

Figure 1

(a) Snapshot of a system of $N=128$, where N refers to the number of monomers; (b) the membrane wall viewed from the trans side of the system. A pore is punched through the wall at the center. The gray beads represent the wall. The yellow, white, and green beads represent the monomers, cations, and anions, respectively.

Figure 2

Snapshots of a typical translocation run for $N=128$ in $E=0.5$ at time $t=0.1\tau$, $0.4\tau$, $0.7\tau$, and $1.0\tau$. The color scheme is the same as described in Figure 1.

Figure 3

Mean translocation time $\langle \tau \rangle$ as a function of (a) chain length N at a given E, and (b) field strength E at a given N. The error bar is smaller than the data symbol.

Figure 4

$w/\langle \tau \rangle$ as a function of E for different N. (Inset) Probability distributions $P\left(\tau \right)$ of translocation time at $E=4.0$. The number N is indicated near the curve.

Figure 5

$\tilde{t}$-variation of the averaged $R_{\mathrm{g}}$ for the chain of $N=128$ in the cis-(I), the trans-(III), and the whole (tot) region at $E=0.2$, $2.0$, $4.0$, $16.0$, and $32.0$. The gray-colored region denotes the distribution range of a curve.

Figure 6

Averaged z-coordinates of chain end, $\langle z_1\rangle$ and $\langle z_N\rangle$, and the difference $\langle z_1-z_N\rangle$, as a function of $\tilde{t}$ at different field strengths for $N=128$. The gray region denotes the distribution range of a curve.

Figure 7

$\tilde{t}$-variation of the averaged $N_{\mathrm{m}}$ for $N=128$ in the cis-region (I); the pore-region (II); and the trans-region (III). The field strength E is indicated in the figure.

Figure 8

MSD $\langle {(N_{\mathrm{m},\mathrm{III}}\left(t\right)-N_{\mathrm{m},\mathrm{III}}\left(0\right))}^2\rangle$ at $E=0.5$, plotted with the normalized time $\tilde{t}$, for $N=128$, 256, and 384. (Inset) Exponent β vs. E for different N.

Figure 9

(Left) Variations of the average number of condensed counterions $N_{\mathrm{c}}^{(+1)}$ in the cis-region (I), the pore-region (II), and the trans-region (III) for $N=128$. (Right) Fraction of charges neutralized on the chain, $\langle |Q_{\mathrm{c}}/Ne|\rangle$, during a translocation process. The field strength E is indicated in the legend.

Figure 10

Probability distributions in the z-direction for (a) monomers; (b) counterions; and (c) coions; at $E=0.2$, $2.0$, and $16.0$ ($N=128$). The gray color denotes the z-location of the pore. The value given near a curve is the normalized time $\tilde{t}$ at which the probability density was calculated. For clarity, the curves have been shifted upward with a fixed step value, one curve after the others.

Figure 11

Density distributions of monomer, viewed along x-axis, at (Left) $E=0.2$, (Middle) $E=2.0$, and (Right) $E=16.0$. The chain length is $N=128$. The value of $\tilde{t}$ is indicated in the figure.