Solitons Transport Water through Narrow Carbon Nanotubes

Transformative technologies for desalination and chemical separations call for understanding molecular transport through man-made and biological nanochannels. Using numerical simulation of single-file flow of water through carbon nanotubes, we find that flow is due to fast-moving density variations (solitons) that are additive so flow rate is proportional to number of solitons. Simulation results match predictions from a theoretical model for soliton propagation. From 1–300 K flow rates increase as temperature decreases. Our results build a fundamentally new understanding of nanochannel flows and suggest new principles for the design of nanoscale devices.

Figure 1

Defects in water density are formed when $S\ne 0$. (a) Single-file water (oxygen, blue; hydrogen, white) in a (5,5) CNT with $L=398$ carbon rings. A 39.3 Å length is shown. The CNT potential energy $U$ evaluated at the mean radial position of the water molecules shows two potential maxima (orange) and two minima (purple) for each carbon ring along the axial direction. (b)–(i) Each blue data point corresponds to a water molecule $i=1,\dots ,N$. Positions of waters $u_i$ are measured relative to potential minima shown in (a). (b) When $S=0$, water molecules are spaced one per carbon ring along the length of the CNT so particle density ${\rho }_i$ is constant for all $i=1,\dots ,N$ water molecules, and (c) all waters rest in minima of the CNT potential, $u_i=0$. (d) With one less water ($S=-1$), the reduced density is localized in two defects (centered on horizontal lines). Outside of the defects, the density tends to the same value as for $S=0$ shown in (b). The MD data are fit by the sech solution (red dashed line) from theory [11]. (e) From the left side of a defect to the right, $u_i$ increases by one-half of the length of a carbon ring as waters are displaced into neighboring potential minima. The MD data are fit by the atan solution (red dashed line) from theory [11]. (f) With one less water, but for an artificially smoothed channel, density is uniform with particles positioned evenly along the channel (g), and no defects are created. (h),(i) Compressive defects ($S=+1$). (b)–(i) Simulations performed at 1 K. At higher temperatures, thermally excited phonon modes are superposed on the defects obscuring their shape (Supplemental Material, Fig. 6 [11]).

Figure 2

Defects convect mass. (a) Displacement $u_i$ of water molecules from their positions at an earlier (red dashed) and 0.8 ps later (blue solid) time. (b) Velocity $v_i$ of water molecules within defects is positive, while outside of defects it is approximately zero, cf. Fig. 3. (c) Defect positions as a function of time. The slope of these lines is used to obtain the defect velocities in (d). (d) Water velocity (green line), obtained by averaging the velocity of all waters within the CNT matches well with results from defect tracking (black dashed line). Data from (4,4) CNT with $f=0.4\mathrm{pN}$ at 300 K. (e) For this panel only, the (4,4) diameter was reduced by 5% which reduces the width of a defect so that it and neighboring molecules can be displayed within a reasonably proportioned figure. [Defects in unmodified (4,4) CNTs have the same structure but longer length unsuitable for compact illustration.] Data from a sequence of MD snapshots at three times showing water molecules as (blue and yellow) spheres, carbon atoms as (black) spheres, and the location of the defect (red arrowheads). Top portion of CNT has been removed to reveal water molecule positions; only bottom hexagonal rings remain. Illumination from above. Shadows delineate the location and structure of the defect—light penetrates through the center of the rings, casting bright waxing crescents on the right of the defect; waning crescents are formed on the left. The switch from waxing to waning crescent illustrates the advance (by 1/2 ring length) of waters over the center of the ring. In the region of the defect, the water molecules’ positions near the centers of the rings occlude light, casting a full shadow. The defect is seen to span approximately five waters.

Figure 3

Contrasting views of single-file channel transport, shown at three times. (a) In the conventional view, all molecules move as one (aside from thermal motions). (b) Fluid flow by defect propagation. Most molecules are motionless, but near a region of low density, molecules quickly shift positions. The average velocity is the same in (a) and (b). Arrow length is proportional to velocity. Only a short portion of the channel is represented.

Figure 4

At all temperatures $T$, flow rate $v$ increases with amplitude of external forcing $f$ to a common maximum flow rate. Insets: Defect trajectories as a function of time at high temperature and high forcing, 300 K and 6 pN (red), are similar to those at relatively low temperature and low forcing, 5 K and 0.8 pN (blue). Data from (4,4) CNT with $S=-1$.

Figure 5

Flow rates near $S=0$ are proportional to the number of defects. When no defects are present ($S=0$) water molecules reside in potential minima and the flow rate is nearly zero. Flow rates due to expansion defects created by fewer water molecules $S<0$ lie close to the linear gray dashed line. Flow rates due to compression defects created by additional water molecules $S>0$ approach the linear black dotted line for small values of $S$. (5,5) CNT data at $f=0.02\mathrm{pN}$.