Intermittent Molecular Hopping at the Solid-Liquid Interface

The mobility of molecules on a solid surface plays a key role in diverse phenomena such as friction and self-assembly and in surface-based technologies like heterogeneous catalysis and molecular targeting. To understand and control these surface processes, a universally applicable model of surface transport at solid-liquid interfaces is needed. However, unlike diffusion at a solid-gas interface, little is known about the mechanisms of diffusion at a solid-liquid interface. Using single-molecule tracking at a solid-liquid interface, we found that a diverse set of molecules underwent intermittent random walks with non-Gaussian displacements. This contrasts with the normal random walk and Gaussian statistics that are commonly assumed for molecular surface diffusion. The molecules became temporarily immobilized for random waiting times between surface displacements produced by excursions through the bulk fluid. A common power-law distribution of waiting times indicated a spectrum of binding energies. We propose that intermittent hopping is universal to molecular surface diffusion at a solid-liquid interface.

Figure 1

Chemical structures of the four molecules studied.

Figure 2

(a) Schematic diagram of molecular surface diffusion that combines periods of immobilization with episodes of bulk diffusion above the surface. The black curve is the true 3D molecular trajectory and the magenta line is the effective 2D trajectory across the surface. (b) Example trajectories of PEG at the hydrophobic TMS-water interface. Each trajectory shown is greater than 4 s in length. A spectrum of intermittent behavior was observed, from completely mobile trajectories to trajectories that were immobile for their entire length.

Figure 3

Distributions of molecular surface displacements [Eq. (1)]. Displacement distributions at the annotated $\Delta t$ for (a) PEG, (b) BSA, (c) Atto6G, and (d) BODIPY. Symbols are experimental data measured using single-molecule tracking and the solid lines are simulated data using the model described in the text.

Figure 4

Mean square displacement as a function of time. Symbols are experimental data and solid lines are fits to the model $⟨\overline{r^2}⟩=4\Gamma (\Delta t{)}^{\alpha }$. The power-law exponents are $\alpha =0.79(2)$, 0.96(1), 0.93(1), and 0.88(2) for BODIPY, Atto6G, BSA, and PEG, respectively.

Figure 5

Distributions of the waiting time between surface displacements greater than $0.2\mu \mathrm{m}$. The data sets were translated vertically to allow easier interpretation (the PEG, Atto6G, and BODIPY data were shifted by a factor of ${10}^1$, ${10}^{-2}$, and ${10}^{-3}$, respectively). The dashed line illustrates $\psi (t)\sim t^{-2.5}$ behavior.