Hydrodynamics of confined colloidal fluids in two dimensions

We apply a hybrid molecular dynamics and mesoscopic simulation technique to study the dynamics of two-dimensional colloidal disks in confined geometries. We calculate the velocity autocorrelation functions and observe the predicted $t^{-1}$ long-time hydrodynamic tail that characterizes unconfined fluids, as well as more complex oscillating behavior and negative tails for strongly confined geometries. Because the $t^{-1}$ tail of the velocity autocorrelation function is cut off for longer times in finite systems, the related diffusion coefficient does not diverge but instead depends logarithmically on the overall size of the system. The Langevin equation gives a poor approximation to the velocity autocorrelation function at both short and long times.

Figure 1

Top: deviation of the simulated diffusion coefficient $D_s$, from the random collision approximation $D_o$ predicted by Eq. (21), as a function of the heavy particle mass. We simulated fluid particles in 2D for a square geometry with walls separated by a distance $L=32a_0$. Bottom: deviation of the simulated diffusion coefficient $D_s$, from the random collision approximation $D_o$, as a function of the particle mean-free path $\lambda$. For larger values of the mean-free path, the diffusion coefficient reduces to the random collision approximation.

Figure 2

The top two plots show the temporal evolution of the self-diffusion coefficient of a fluid particle in a large box of size $256a_0\times 256a_0$ with periodic boundary conditions. The right plot shows that rather than saturate, the diffusion coefficient grows as $D\sim \text{ln}t$, as expected from theory. The bottom plot shows the hydrodynamic corrections to the diffusion coefficient $D$ compared to the random collision approximation expression $D_0$ given by Eq. (21) for different box sizes $L$. As expected, these corrections show a logarithmic growth with $L/a_o$.

Figure 3

Scaling of the velocity autocorrelation function: when VACF is normalized and plotted in terms of the reduced time $t/t_{\nu }$, all the data collapse to the same curve. The VACF was originally measured for varying solvent densities $\gamma$. The system size is $L=32a_0$, which implies that ${\chi }_s=L/2{\sigma }_{cs}=8$.

Figure 4

Normalized colloid VACFs simulated for two pipe widths $L=28,96a_0$, corresponding to ${\chi }_s=\frac{L}{2{\sigma }_{cs}}=7,24$. The symbols denote simulation results. The $x$ axis was scaled with the Enskog time $t_E=1.0888t_0$ calculated from Eq. (29). The dashed line represents the short-time decay from Eq. (30), the dashed-dotted line represents the decay from the long-time tail calculated from Eq. (27) whilst the dotted line and the solid line denote the Langevin decay with the friction $\xi$ for the two respective box sizes. Although the two simulated VACFs are very close to each other for smaller $t/t_E$, the Langevin approximation gives two significantly different results.

Figure 5

Log-log plot of the VACF for a colloidal particle in boxes of size $L=32,48,500a_0$ $\left({\chi }_s=8,12,125\right)$. The dashed line is from Eq. (27) and shows the expected $t^{-1}$ tail. The measured VACFs deviate from this simple algebraic decay for larger times; the larger the box, the later the deviation sets in.

Figure 6

Normalized velocity autocorrelation function of a colloidal particle confined between two walls for increasing values of ${\chi }_s=L/2{\sigma }_{cs}$. Simulations here were performed for pipe widths $L=8,10,12,24a_0$. The different plots denote the components of the normalized VACF parallel $C_x\left(t\right)$ (top) and perpendicular $C_y\left(t\right)$ (bottom) to the walls. Note that all plots are scaled with the kinematic time $t_{\nu }={\sigma }_{cs}^2/\nu$ and the time scale for momentum to diffuse the distance between the walls $t_w=L^2/4\nu$. The minimum of the negative tails observed for $C_x\left(t\right)$ scale on top of each other when time is scaled with the sonic time $t/t_{cs}$ (inset in the upper left panel). Similarly, the oscillating tails for $C_y\left(t\right)$ show the same period when scaled with $t/t_W$ (bottom right panel).

Figure 7

Log-log plot of the VACF of a confined colloidal particle for boxes of size $L=8,10,12a_0$ (${\chi }_s=2,2.5,3$, respectively). The dashed line and the straight line have slopes $t^{-1}$ and $t^{-3/2}$, respectively. Only the positive values of $-C\left(t\right)$ are plotted here.

Figure 8

Time evolution of the time-dependent diffusion coefficient of a buoyant colloid in two dimensions for SRD particle densities $\gamma =5,10,50$. The bottom two graphs show the diffusion coefficient normalized by $C_x\left(0\right)=k_{B}T/M$, with time scaled by $t/t_{\nu }$. We expect the graphs to scale on to each other for longer times where the long-time tails dominate (see also Fig. 3). The bottom right graph has a logarithmic scale and the dashed line has the slope $\frac{M}{8\pi \eta }$ and serves as a guide for the eyes.

Figure 9

Top: the effect of confinement on diffusion. The two curves represent the temporal evolution of the self-diffusion coefficient for colloids in varying confinement. The dashed line denotes colloids diffusing between plates a distance $L=32a_0$ apart $\left({\chi }_s=8\right)$, whilst in the case of the solid line the plate separation is $64a_0$ $\left({\chi }_s=16\right)$. For the smaller pipe, the kinematic wall time is $t_W\approx 64t_{\nu }$. The diffusion coefficient begins to plateau much earlier than that because the effect of the wall on the VACF kicks in earlier. Note that only the $x$ component of the diffusion is plotted here. Bottom: simulations were performed for 2D pipes of respective widths $L=8,12,16,20,24,28,32,64,96a_0$. The straight line is the fit from Eq. (32) and the circles are the simulation points.