Non-Fickian single-file pore transport

Single-file diffusion exhibits anomalously slow collective transport when particles are able to immobilize by binding and unbinding to the one-dimensional channel within which the particles diffuse. We have explored this system for short porelike channels using a symmetric exclusion process with fully stochastic dynamics. We find that for shorter channels, a non-Fickian regime emerges for slow binding kinetics. In this regime the average flux $\langle \mathrm{\Phi }\rangle \sim 1/L^3$, where $L$ is the channel length in units of the particle size. We find that a two-state model describes this behavior well for sufficiently slow binding rates, where the binding rates determine the switching time between high-flux bursts of directed transport and low-flux leaky states. Each high-flux burst is Fickian with $\langle \mathrm{\Phi }\rangle \sim 1/L$. Longer systems are more often in a low-flux state, leading to the non-Fickian behavior.

Figure 1

Schematic of our model system, with a one-dimensional channel of length $L$ indexed by discrete positions $i\in \{0,L\}$. Positions are either occupied (solid shapes) or unoccupied (dashed circles) by particles. A concentration gradient across the system induces an average net flux, and is maintained by imposing $p_0=100\%$ occupancy of the first site and $p_L=0\%$ occupancy of the last site, as indicated. All particles can transition from mobile (green solid circles) to immobile (red solid octagons) states, or back, with transition rates $k_{\text{stop}}$ and $k_{\text{go}}$, respectively. The single-file constraint is imposed by allowing mobile particles to hop to an adjacent site at a rate $k_{\text{hop}}$ only if the destination site is not occupied.

Figure 2

The time-average flux $\mathrm{\Phi }$ divided by the simple-diffusion (SD) flux ${\mathrm{\Phi }}_0=D_0/L$ vs the system size $L$. (a) For $K_A=10$ and various dimensionless binding rates $R\equiv k_{\text{stop}}/k_{\text{hop}}$ as indicated in the legend. The horizontal black dotted line is the SD result $\mathrm{\Phi }/{\mathrm{\Phi }}_0=1$, while the other horizontal dotted lines are the expected $L\gg 1$ (Fickian) behavior from Eq. (1), with colors corresponding to the legend. (b) For $R_{\text{stop}}={10}^{-7}$, for a variety of association constants $K_A$ as indicated. The diagonal dashed black line indicates $\mathrm{\Phi }/{\mathrm{\Phi }}_0\sim 1/L^2$, i.e., $\mathrm{\Phi }\sim 1/L^3$. The curved colored lines are from the two-state model, Eq. (3). The open black circles are experimental data from Ref. [16]—see Discussion.

Figure 3

Two-state switching between fast and slow flux $\mathrm{\Phi }$ (black traces, with a logarithmic scale on the left-hand side) vs scaled time $t{R}_{\text{stop}}$. High flux coincides with no bound particles (red histograms, with a linear scale on the right-hand side) while low flux coincides with bound (immobile) particles in the channel. Three different representative time series are shown, with $R_{\text{stop}}={10}^{-6}, {10}^{-5}$, and ${10}^{-4}$ (top to bottom, as indicated) at fixed $K_A=10$ with $L=8$. The sawtooth decrease of bound particle numbers between high-flux events indicates the slow clearing of the plug in the slow state, followed by a burst of flux in the fast state. Time units are such that $k_{\text{hop}}=1$.

Figure 4

(a) Average duration $\langle \tau \rangle$ of states with any particles bound (orange squares, slow) and all particles free (blue circles, fast) vs system length $L$. For all plots, $K_A=10, L=8$, and $R_{\text{stop}}={10}^{-5}$, unless otherwise indicated; the two-state model calculations are indicated with colored dashed lines for $L\le L_X\equiv R_{\text{stop}}^{-1/3}$. (b) Average flux $\langle \mathrm{\Phi }\rangle$ in the slow (duration weighted) and fast states vs $L$. The dashed black lines indicate ${\mathrm{\Phi }}_{\text{Fick}}$ from Eq. (1). (c) Average duration $\langle \tau \rangle$ of slow and fast states vs $R_{\text{stop}}$. (d) Average flux $\langle \mathrm{\Phi }\rangle$ for slow and fast states vs $R_{\text{stop}}$. Time units are such that $k_{\text{hop}}=1$.