Enhanced diffusion on oscillating surfaces through synchronization

The diffusion of molecules and clusters under nanoscale confinement or absorbed on surfaces is the key controlling factor in dynamical processes such as transport, chemical reaction, or filtration. Enhancing diffusion could benefit these processes by increasing their transport efficiency. Using a nonlinear Langevin equation with an extensive number of simulations, we find a large enhancement in diffusion through surface oscillation. For helium confined in a narrow carbon nanotube, the diffusion enhancement is estimated to be over three orders of magnitude. A synchronization mechanism between the kinetics of the particles and the oscillating surface is revealed. Interestingly, a highly nonlinear negative correlation between diffusion coefficient and temperature is predicted based on this mechanism, and further validated by simulations. Our results provide a general and efficient method for enhancing diffusion, especially at low temperatures.

Figure 1

Schematic sketch of the simulated model. (a,b) The front and side views of the model, where molecules diffuse in a (8,0) carbon nanotube. (c) Potential energy profile for a molecule moving along axial coordinate $x$. The potential energy computed by the L-J potential (black) coincides well with the sinusoidal fitting (red).

Figure 2

Enhanced diffusion for a particle confined in a nanotube with translation oscillation along the axial direction. (a) Normalized diffusion coefficient $D/D_{\mathrm{free}}$ at different $A/b$ and $f/f_0$, where $A$ is the amplitude of oscillation, $b$ is the spatial period of the potential, $f$ is the frequency of oscillation, and $f_0$ is the intrinsic frequency for small oscillations at the minimums of the substrate potential. The values for other parameters are ${\eta }^{\prime }=3.2,T^{\prime }=0.27$. (b,c) Profiles of normalized diffusion coefficient at $A/b=0.4$ or $f/f_0=0.9$ which correspond to the dashed lines in (a).

Figure 3

Mechanisms of enhanced diffusion. (a) The splitting mechanism. (b) The synchronization mechanism. (c) Mean square displacement (MSD) for particles confined in oscillating nanotubes without thermal noise within a time interval of $100/f_0$. MSD is normalized to its maximum.

Figure 4

Mechanisms of enhanced diffusion in terms of mean free path. (a–c) Trajectories of particles at different temperature $T^{\prime }=k_{B}T/U_0$. (d) Probability density functions of displacement for particles at time $t{f}_0=3\times {10}^4$ with different temperatures.

Figure 5

Dependence of diffusion coefficient $D^{\prime }=D/b^2{f}_0$ on temperature $T^{\prime }=k_{B}T/U_0$ with $A/b=0.4$ and $f/f_0=0.93$ which correspond to the situation with maximum diffusion enhancement. The data points are calculated numerically according to Eq. (2). The dashed line is predicted by the Einstein relation $D_{\mathrm{Ein}}=m{f}_0{T}^{\prime }/\eta$. The position of the maximum enhancement in the $\left(A/b,f/f_0\right)$ plot does not change with temperature.

Figure 6

The estimated threshold temperature $T_{\mathrm{th}}$ of different fluids as a function of boiling points $T_{\mathrm{bp}}$. The enhancement coefficient $D/D_{\mathrm{free}}$ in the blue, green, yellow, and pink areas ranges from 4.7 to 10, 10 to 100, 100 to 1000, and $>1000$, respectively.