Locally optimal geometry for surface-enhanced diffusion

Molecular diffusion in bulk liquids proceeds according to Fick's law, which stipulates that the particle current is proportional to the conductive area. This constrains the efficiency of filtration systems in which both selectivity and permeability are valued. Previous studies have demonstrated that interactions between the diffusing species and solid boundaries can enhance or reduce particle transport relative to bulk conditions. However, only cases that preserve the monotonic relationship between particle current and conductive area are known. In this paper, we expose a system in which the diffusive current increases when the conductive area diminishes. These examples are based on the century-old theory of a charged particle interacting with an electrical double layer. This surprising discovery could improve the efficiency of filtration and may advance our understanding of biological pore structures.

Figure 1

Surface-enhanced diffusion. (a) A concentration difference $\mathrm{\Delta }c$ of tracer particles (green dots, diameter $d$) leads to a diffusive current $I$ across a slender channel of length $L$ and uniform cross-sectional area ${\sigma }_0\sim {\ell }^2$. (b) Tracer particles interact with the channel boundaries through an energy potential $U$. (c) Surface interactions impact the diffusive current $I$ and can either reduce or enhance transport. We demonstrate an optimal case in which the current grows as the geometric pore area diminishes (marked by the dashed blue line).

Figure 2

Surface-enhanced diffusion in a slit. (a) A positively charged tracer particle (green) diffuses along a pore with negatively charged boundaries. The presence of an electrolyte induces the energy potential $U\left(y\right)$; see Eq. (5). (b)–(d) Diffusion cross-section $\sigma /\left(w{\lambda }_D\right)$ plotted as function of the channel height $2h/{\lambda }_D$ for energies $U_0/\left(k_{B}T\right)=\{-1,-3.05,-3.8\}$ [Eq. (6)]. The thin solid line corresponds to the interaction-free $\sigma ={\sigma }_0$ case. At low energies (b), the diffusion cross-section exceeds the geometric area $\sigma >{\sigma }_0$ and increases monotonically with channel height $2h$. However, near the energy $U_0/\left(k_{B}T\right)=-3.05$, a saddle point (i) appears where the current $I\sim \sigma$ is approximately independent of channel height in the range $2<2h/{\lambda }_D<5$. (d) Remarkably, for stronger interactions [e.g., $U_0/\left(k_{B}T\right)=-3.8$], local extrema (ii and iii) in the diffusion cross-section appear. (e) Contour plot of the diffusion cross-section $\sigma /\left(w{\lambda }_D\right)$ as function of energy $U_0/\left(k_{B}T\right)$ and relative channel height $2h/{\lambda }_D$. Experimental data from Ref. [12] [points in panels (b) and (e)] are not inconsistent with theory.

Figure 3

Comparison of diffusion cross-sections. (a) Top-view of a concentric geometry with outer radius $a$, constant inner radius $b=10{\lambda }_D$ and geometric cross-sectional area ${\sigma }_0=\pi (a^2-b^2)$. (b) Top-view of a circular geometry with radius $a$ and ${\sigma }_0=\pi a^2$. All boundaries are charged with surface potential $\zeta$. (c) Diffusion cross-sections $\sigma /\left(w{\lambda }_D\right)$ plotted as functions of the geometric cross-sectional area ${\sigma }_0/{\lambda }_D^2$ for energy $U_0/\left(k_{B}T\right)=-4$. Blue (parallel plates), orange (concentric), and red (circular). The width of the parallel plates is chosen as $w=2\pi b$ to match the circumference of the inner concentric cylinder.

Figure 4

Surface-enhanced diffusion in a circular geometry. (a) Top-view of circular channel with radius $a$ and surface potential $\zeta$. (b)–(d) Diffusion cross-section normalized by the Debye length squared $\sigma /{\lambda }_D^2$ plotted as a function of the channel radius $a/{\lambda }_D$ for energies $U_0/\left(k_{B}T\right)=\{-1,-3,-4\}$ [Eq. (A1a)]. The thin solid line corresponds to the interaction-free case $\sigma ={\sigma }_0=\pi a^2$. The diffusion cross-section exceeds the geometric area $\sigma >{\sigma }_0$ but increases monotonically with channel radius $a$. (e) Contour plot of the diffusion cross-section $\sigma /{\lambda }_D^2$ as function of energy $U_0/\left(k_{B}T\right)$ and relative channel radius $a/{\lambda }_D$.

Figure 5

Surface-enhanced diffusion in a concentric geometry. (a) Top-view of concentric circular channel with outer radius $a$ and inner radius $b$. Transport occurs in the gap between the two cylinders. Both surfaces are charged with surface potential $\zeta$. (b)–(d) Diffusion cross-section normalized by the Debye length squared $\sigma /{\lambda }_D^2$ plotted as a function of the channel gap $(a-b)/{\lambda }_D$ for energies $U_0/\left(k_{B}T\right)=\{-1,-3,-4\}$ [Eq. (A2a)]. The inner radius is kept constant $b=10{\lambda }_D$. The thin solid line corresponds to the interaction-free case $\sigma ={\sigma }_0=\pi (a^2-b^2)$. The diffusion cross-section exceeds the geometric area $\sigma >{\sigma }_0$ but increases monotonically with channel gap $(a-b)/{\lambda }_D$ at low energies (b). (d) For stronger interactions (e.g, $U_0/\left(k_{B}T\right)=-4$), a local maximum in the diffusion cross-section appears. (e) Contour plot of the diffusion cross-section $\sigma /{\lambda }_D^2$ as function of energy $U_0/\left(k_{B}T\right)$ and relative channel opening $(a-b)/{\lambda }_D$.

Figure 6

Dipole interactions. (a) Diagram of an electric dipole (separation distance $s$ and charge $q$) near a charged wall in an electrolyte solution. The dipole is oriented with the positive charge closest to the negatively charged wall. (b) Enhanced diffusion cross-section for a dipole with $s=0.68{\lambda }_D$ and $U_0/\left(k_{B}T\right)=\{-1,-5,-8,-10\}$ (bottom to top). The thin black line is the geometric cross-sectional area ${\sigma }_0=2w(h-s/2)$ and the black squares are experimental data from Ref. [13] on aspirin.

Figure 7

Offset in the slit geometry. When the gap between the parallel plates is much larger than the Debye length, $2h\gg {\lambda }_D$, the diffusion cross-section increases linearly with channel height, $2h$, but with an offset $\mathrm{\Delta }h$ (thin dashed line). Here with energy scale $U_0/\left(k_{B}T\right)=-4$, tracer particle valence $Z=2$ and surface potential $\zeta =-52$ mV. The dashed blue line is the diffusion cross-section found using the electric potential from the nonlinear Poisson-Boltzmann equation. The thin solid line corresponds to the interaction-free case $\sigma ={\sigma }_0=2hw$.

Figure 8

Surface-enhanced diffusion in a slit. A local maximum in the diffusion cross-section $\sigma$ as a function of channel height $2h$ also exists when the electric potential in the slit is found by solving the nonlinear Poisson-Boltzmann equation numerically (dashed blue lines). Here shown for relative interaction energy $U_0/\left(k_{B}T\right)=-4$ and electrolyte valence $Z_{\text{el}}=1$. (a) Tracer particle valence $Z=2$ and surface potential $\zeta =-52$ mV. (b) Tracer particle valence $Z=3$ and surface potential $\zeta =-34$ mV. (c) Tracer particle val- ence $Z=4$ and surface potential $\zeta =-26$ mV. (d) Tracer particle valence $Z=6$ and surface potential $\zeta =-17$ mV. The solid blue line shows the corresponding diffusion cross-section when the electric potential in the channel is found using the linearized Poisson-Boltzmann equation for $U_0/\left(k_{B}T\right)=-4$. The thin solid line corresponds to the interaction-free case $\sigma ={\sigma }_0$. Note that when the electric potential, and hence interaction energy $U\left(y\right)$, is found by solving the nonlinear Poisson-Boltzmann equation numerically, the diffusion cross-section depends on the tracer particle valence, $Z$, and channel surface potential, $\zeta$, separately and not only on their product as seen when the Debye-Hückel approximation is used.