Geometric dependence of the conductance drop in a nanopore due to a particle

We study the effect of a neutral particle on the ionic flow through a nanopore using a basic uniform field theory and the coupled Poisson-Nernst-Planck and Navier-Stokes (PNP-NS) equations. We consider hourglass and cylindrical pore profiles and examine how the difference in pore shape changes the position dependence of the current change due to the particle. Good quantitative agreement between both calculations is seen, though we find that the simple theory is unable to correctly capture the change in the access resistance of the pore if a particle is placed at the pore entrance. Finally, we examine the spatial variations in the solutions of the PNP-NS equations, finding that the electro-osmotic flow through the pore is completely disrupted for sufficiently large particles.

Figure 1

(a), (b) Diagram of hourglass and cylindrical nanopore with spherical particle. (c) Diagram of the coordinate system depicting axis symmetry. The $z$ axis is the symmetric axis and the $r$ axis is the radial direction. $r=f\left(z\right)$ defines pore radii at different $z$ positions reflecting the nanopore geometry. The particle geometry is given by the axis symmetric function $r=g\left(z\right)$. (d) Diagrams of different configurations with particles located at $z_0=L/2,L/4$, and 0.

Figure 2

Basic characterization of the ionic current for the particle-nanopore system. (a) Current drop with the spherical particle located at different positions along the transverse axis of hourglass and cylindrical pores. The particle radii are 1.75, 2.5, and 3.75 nm. (b) Current drop as a function of particle radius with the particle fixed in the center of the pore. Inset: Current drop dependence on $r_{\mathrm{p}}^3$. The linear fit is calculated for $r_{\mathrm{p}}\le$3.75 nm, indicated by the dashed vertical line.

Figure 3

Change in current with pore entrance radius for hourglass shaped pores. (a) Current drop for a spherical particle of radius $r_{\mathrm{p}}=$ 3.75 nm located at various positions along the pore axis. The minimum pore radius, $r_0=$ 5 nm, is the same for all pores; the maximum pore radius, $r_1$, is varied in the range 5 to 7 nm. (b) Current modulation with different particle radii in the case of different pore shapes. The pore geometries are the same as (a) and the particle locations are $z_0=$ 0 and 10 nm.

Figure 4

Comparison of the two calculation approaches. (a) Current drop in the presence of the spherical particle located at different positions within hourglass and cylindrical pores calculated using the two continuum approaches. The particle radii are 2.5 and 3.75 nm. (b) Current modulation of hourglass and cylindrical pores in the presence of spherical particles with different radii. Abbreviations: coupled Poisson-Nernst-Planck Navier-Stokes calculations (PNP-NS); simple calculation based on Eq. (4) (SC).

Figure 5

Contour maps of the electric potential for particles of different size at the center of cylindrical and hourglass shaped pores. (a)–(c) Cylindrical pores. (d)–(f) Hourglass pores. The particle radii are $r_{\mathrm{p}}=$ 0, 2, and 4 nm. Equipotential lines are indicated by dashed lines for negative potentials and solid lines for positive potentials.

Figure 6

Contour maps of the difference in electric potential between open and occupied pores for an hourglass shaped pore. (a)–(e) A particle of radius $r_p=$ 3.75 nm moved from one side of the pore to the other in steps of 5 nm. The comparison is made with the data of Fig. 5.

Figure 7

Contour maps and stream plots of the fluid velocity for particles of different size at the center of cylindrical and hourglass shaped pores: (a)–(c) Cylindrical pores. (d)–(f) Hourglass pores. The particle radii are $r_{\mathrm{p}}=$ 0, 2, and 4 nm.

Figure 8

Ion and velocity distribution around the particle. (a) Radial density profiles of the ionic charge distribution. Inset: the region over which averages are calculated is indicated. (b) The absolute value of the $z$ component of the fluid velocity. The dashed vertical lines indicate the outer edge of the particle.