A coarse-grained model for a nanometer-scale molecular pump

Model

We propose a simple coarse-grained 2D lattice gas model to describe the nanometer channel (for example, a nanotube) that is connected to a wall separating two reservoirs of fluid. A sketch of the model is shown in Fig. 1.

The dynamics of the fluid is described within the FHP-III model based on a triangular grid (13). In the FHP-III model, identical fluid particles travel with unit speed from one site to the other on the grid and collide when they meet on the same site, conserving both particle density and particle momentum. Only one particle is allowed to travel in each direction along the six possible links connecting adjacent sites. Each site can also include a rest particle that can take part in the collision process. The equations of motion can be expressed in terms of a vector, n(r, t) = {n ₀(r, t),..., n ₆(r, t)}, with binary variable components that are proportional to the density of particles moving from site r in one of the possible directions c _j: where c _j = (cos(2πj/6), sin(2πj/6)), j = 1,..., 6 forthesix possible direction, or c _j = (0, 0) for the rest particle. The collision operator, Δ_j(n(r, t)), includes 76 possible two-, three-, four-, and five-particle collisions. We also include straightforward higher order collisions. These collisions conserve mass Σ_jΔ_j(n(r, t)) = 0 and momentum Σ_j c _jΔ_j(n(r, t)) = 0 and are described elsewhere (14).

The channel is introduced to the model by labeling certain lattice sites and introducing different collision rules at these sites. The choice of the asymmetric structure of the channel studied in this work is shown in Fig. 1. The channel sites are labeled by black circles connected by a thick solid line. The diameter of the wide (D) and narrow (d) ends, and the length of the channel (L) are also shown in Fig. 1. We use no-slip conditions at the boundaries by forcing any particle colliding with the channel to return along the link on which it approached the channel (27). We apply similar collision rules between the fluid particles and the wall separating the two reservoirs. The walls are labeled by the thick solid line in Fig. 1.

We start from a random initial configuration of fluid on the lattice with uniform density (unless otherwise noted),† at each reservoir. The propagation in time involves the following three steps: (i) a collision step, (ii) a travel step, and (iii) a channel movement step arising from the external driving force, with a period ω. The details are as follows:

1. At each time step particles at all sites collide according to the rules specified above and in ref. 14. These include particle–particle, particle–channel, and particle–wall collisions.

2. After the collision step each particle travels in a straight line along one of the lattice links (according to its velocity vector), unless it is a rest particle, until it arrives to the next site.

3. The channel movement step arising from the applied external force is slightly more involved. First, we apply collision rules at the predicted new sites of the channel. Particles are not allowed to penetrate the channel through the walls or move along the boundaries of the wall. An example for a single collision of a particle moving along direction C_j before the collision, and along direction C′_j after the collision (C_j→C′_j), with the upper wall moving to the right is given by: {E→ E; SE→ E; SW→ NE; W→ E; NW→ NE; NE→ NE; stationary→E}, where N, S, E, and W stand for north, south, east, and west directions, respectively. If after the collision with the walls of the channel a link is occupied with more than a single particle, the excess particles are propagated along their new direction. This procedure is repeated until all links with excess particles are occupied by not more than a single particle. After this step we move each channel site along a straight line in the direction of the external force until it arrives to the next site, preserving the structure of the channel.

Results and Discussion

To study the fluid dynamics through the channel we have performed simulations based on the lattice gas approach described above for different model parameters. The effects of the dimensions of the channel (diameters d, D, and length L), the frequency of the external driving force (ω, defined as the reciprocal time for one full cycle), and the density of the fluid on the flow were studied in detail. We have used N_grid = 500,000 lattice sites, unless otherwise stated, and imposed periodic boundary condition at the grid edge perpendicular to the channel axis. No-slip (reflection) boundary conditions were imposed at the remaining edge, unless otherwise noted. Each propagation step included a full cycle of the three steps described in Model. The propagation was stopped after the system approached its steady-state solution.

In Fig. 2 we show a snapshot of the velocity field at each site averaged over the initial 1,000 time steps. The frequency of the external force is ω = 0.1 (in units of reciprocal propagation steps), the density of the fluid is ρ = 0.5, the diameters of the narrow and wide ends are d = 5 and D = 11 lattice sites, respectively, and the total length is L = 20 lattice sites. A net flow of fluid through the channel from the narrow end to the wide end, as indicated by the directions of the arrows, is clearly seen. The direction of the flow can be traced to the fact that any particle inside the channel placed near the boundary between the narrow and wide ends can only be pushed by the channel in the direction of the wide end. It is interesting to note that the flow is more directional and the current is larger at the narrow side of the channel. Similar behavior is expected for macroscopic channels, where a narrow end is used to increase the current for a given pressure gradient.

Typical kinetics of the time dependence of the average density in each reservoir (ρ_l and ρ_r for the left and right densities, respectively) is shown in Fig. 3 on a semilogarithmic scale. The density, frequency, and other parameters are identical to those of Fig. 2. The buildup of density gradient follows exponential kinetics, with a steady-state solution that is independent of the initial conditions. To illustrate this we have used three different initial conditions that result in the same steady-state solution. As can be seen in Fig. 3, the final fluid density in each reservoir is the same in all three cases. We have also studied the kinetics starting from the steady-state solution and switched off the external oscillatory force (data not shown). Typically, we find that the decay of the density to equilibrium is faster than the buildup of density gradient in the presence of the external driving force.

The fact that the density in each reservoir approaches a steady state is a consequence of the finite size of the system, namely the finite size of the two reservoirs. We find that steady-state densities are independent of the dimensions of the reservoirs; however, the rate of approach to steady state decreases as the dimensions of the system increase. Therefore, a more meaningful observable that is independent of the size of the reservoirs is the average fluid flow through the channel, which can be obtained from the short time dynamics of the density gradient. The magnitude of the flow obtained this way is identical to the flow (at steady state) for infinite reservoirs. (Note that for a finite system the average flow approaches zero at long times.)

A 2D plot of the flow as a function of the density of the fluid (ρ) and the frequency of the external oscillatory force (ω) is shown in Fig. 4. The diameters of the narrow and wide ends are d = 5 and D = 11 lattice sites, respectively, and the total length is L = 20 lattice sites. The flow, J, is calculated from , where N_occ = 7 for the FHP-III model, and Δρ = ρ_l – ρ_r is the average density difference between the two reservoirs (namely, J ∝ d(N_l – N_r)/dt). This definition of the flow assures that J is independent of grid size. For a given fluid density, a nonmonotonic behavior, with a well pronounced global maximum, is observed for the average flow as a function of ω. At lower values of ω the system has enough time to relax back to the equilibrium state with equal densities at both sides of the channel, so that the net flow is very small. At higher values of ω there is a separation of time scales between the motion of the fluid and the channel. In this limit using a mean field treatment, one finds that the flow through the channel is sensitive to the average position of the channel. This leads to a diminishing net current at higher values of ω. In between these limiting cases the flow is larger, giving rise to a maximum of flow versus ω. In addition, we find that the flow of the liquid through the channel increases with increasing density for all values of ω, as can clearly be seen in Fig. 4.

We have also studied the effects of the dimensions of the channel on the resulting flow and on the density of the fluid in the reservoirs at steady state. The results for a set of simulations of different channel dimensions are summarized in Tables 1 and 2. The particle density is ρ = 0.5, and the frequency of the external driving force is ω = 0.15 (in units of reciprocal propagation steps). We observe a maximum value of flow when we increase the width of the wide and narrow ends such that the difference D – d = 6 is constant. This is a result of two competing effects. On one hand, as the dimensions of the channel increase the pressure gradient required for fluid flow decreases (28). But at the same time, the efficiency of the pump reduces with increasing dimensions, giving rise to a maximum of flow at an intermediated channel width. Unlike the nonmonotonic behavior observed for the fluid flow, the difference in fluid densities (Δρ) at steady state decreases monotonically (within the statistical noise) with the diameters of the channel.

We have also studied the effects of increasing the length of the channel on the fluid current and on Δρ at steady state. We find that increasing the length of the channel always results in a decrease in the fluid flow (see Table 2). We have tripled the length of the channel from 20 to 60 lattice sites. Within this range fluid current decreases linearly with the length of the channel. On the other hand, increasing the size of the channel has a negligible effect on the steady-state fluid densities at each reservoir.

So far we have discussed the properties of our nanopump in terms of its reduced lattice-gas dimensions. To make connections to a realistic system we have calculated the Reynolds number for several simulation cases and used the definition of this number to provide values for the relevant control parameters. The Reynolds number, R _e, is given by where u is the average fluid velocity in the pump region and is proportional to u ∝ Aω, where as before ω is the frequency of the external force and A is its amplitude. In the above equation, d is the diameter of the pump and ν is the kinematic viscosity of the fluid. Thus, for a given fluid kinematic viscosity, and for a given channel dimension, we can estimate the frequency of the driving force, or vice versa.

For the FHP-III lattice gas model we have used the following definitions to calculate the Reynolds number. The time averaged velocity field is given by where N_i(r) is the mean occupation number calculated as a time average of the occupation number n_i(r, t) defined above. The (scaled) kinematic viscosity is given by (29) The results for the Reynolds numbers for different simulation conditions are summarized in Table 3. In these simulations we have used periodic boundary conditions in both the longitudinal and transverse directions. The Reynolds number increases monotonically with the dimensions of the channel. The values obtained for the Reynolds number fall in intermediate range (0.06–7.2) for the densities and channel dimensions studied. These values correspond to a frequency of the external driving force in the range of ω ≈ 0.5–50 GHz for liquid water (kinematic viscosity ν = 10^–6·m²·s^–1), for pump diameter of d = 10 nm and oscillation amplitude of A = 10 nm. Clearly, smaller pump frequencies are required for smaller Reynolds numbers or larger channel dimensions. We have also studied the effects of increasing the amplitude of oscillation, A. A sublinear increase of the Reynolds number with A was observed, indicating that larger amplitudes would also lead to lower frequencies of the driving force. It is interesting to note that gigahertz oscillators based on the motion of the inner shell of a multiwall carbon nanotube have been discussed recently (30). This system maybe useful as a building block for a nanometer-scale pump based on the concepts introduced above.

Conclusions

A simple asymmetric geometry of a nanometer-scale channel connecting two reservoirs of fluid, coupled to an external periodic or stochastic driving force, can be used to pump material on a nanometer-length scale. The rate of flow through the device depends on the fluid density, the frequency and amplitude of the external force, and the dimensions of the channel.

We have developed a simple coarse-grained lattice gas model with FHP fluid dynamics that satisfy the Navier–Stokes equation to study the dynamic flow of fluid through a nanometer channel. The only well supported assumption made is that the fluid can wet the inside of the channel. Using simulation techniques we have studied the flow through the channel for different model parameters.

We showed that the flow is a nonmonotonic function of the frequency of the external force (ω) and the diameters of the channel. At low and high values of ω the fluid current through the channel is very small, whereas in between these limiting cases the flow is higher, giving rise to a maximum of flow at finite ω. Because of the competition between the pump efficiency and the pressure gradient required to push the fluid through the channel, a maximum in the current was also observed when the dimensions of the channel were increased. On the other hand, a monotonic behavior was observed for the fluid current with respect to fluid density and length of the channel.