Carbon nanotubes (CNTs) have been widely used as the components of nanoelectromechanical systems (NEMS)^{[1, 2, 3, 4, 5]} because of their superior mechanical properties,ideal geometry,and chemical inertness. Due to their quasi-one-dimensional nature,they are regarded as one of the promising and powerful tools for molecule transportation or mass delivery^[6]. In recent years,various designs have been proposed to achieve and control the transportation of water molecules across CNTs,such as the osmotic pressure gradient^[7],the thermal gradient^[8],and the external electric fields or charges^{[9, 10, 11, 12]} . Besides,manipulating the charged or uncharged macro-molecules confined inside nanochannels also attracts much attention. Yeh and Hummer^[13] observed that the charged ribonucleic acid (RNA) molecules could be driven through nanopores of CNT membranes by the electric field. Longhurst and Quirke^[14] used the strong capillary force and the temperature gradient to draw decane into a single-walled carbon nanotube (SWCNT). Xiu et al.^[15] achieved the controllable manipulation,both in space and time,of biomolecules with aqueous liquids inside an SWCNT by using an external charge or a group of external charges. Recently,Gong et al.^[16] presented a design for a controllable ion-selective nanopore based on an SWCNT with specially arranged carbonyl oxygen atoms modified inside the SWCNT.

In the present paper,we propose a new design to achieve the guided unidirectional motion of the SWCNT in water environments driven by an electric field. The systems consist of two coaxial SWCNTs,i.e.,a short SWCNT outside (System 1,denoted by S1 in Fig. 1(a)) a longer SWCNT and a short SWCNT inside (System 2,denoted by S2 in Fig. 1(b)) a longer SWCNT. After the electric field with a linear gradient is applied,the outer short SWCNT in S1 moves to the area with the lower field strength,while the inner capsule-like SWCNT in S2 travels to the area with the higher field strength.

Fig. 1. Snapshots of simulation system. (a) Sketches of physical models of S1 and S2. (b) Field intensity distribution of electric field with linear gradient by E = αz,where field direction is along x-coordinate,and two ends of SWCNTs are presented by dashed lines at z=1.925 nm and z=14.074 nm,respectively

Figure options

The sketches of the physical models considered in the present study are illustrated in Fig. 1. For S1,the inner SWCNT is 12.149 nm long with a chiral vector of (5,5) corresponding to a diameter of 0.678 nm,while the outer SWCNT is 1.858 nm long with a chiral vector of (10,10) corresponding to a diameter of 1.356 nm. S2 consists of an outer 12.149 nm long (10,10) SWCNT and an inner capsule-like SWCNT modeled as an open short 1.858 nm long (5,5) SWCNT with caps on two ends. The two systems are placed in a cubic box with the dimensions of L_x=6 nm, L_y = 6 nm,and L_z = 16 nm. The axes of the SWCNTs are along the z-direction. The water molecules with the density 998 kg/m³ are filled in the box but not inside the SWCNTs. Then, a non-uniform electric field with a linear gradient is applied to the system. The direction of the electric field is along the x-direction,and the strength is given by E = αz,where z is the z-coordinate,and α is the gradient coefficient. This kind of electric fields can be generated by a capacitor with two unparallel plates. If the two plates are close to each other at one open end while far away from each other at the other open end,the non-uniform electric field with a certain gradient will be produced in the interior region.

The molecular dynamics simulations are performed at a constant temperature 300 K with the large scale molecular dynamics package GROMACS 4.0.7^[17]. The extended simple point charge (SPC/E ) model is utilized for water molecules,which is reasonable when the external electric field strength is below 10 V/nm^[18] . In the simulations,the carbon atoms are modeled as the uncharged Lennard-Jones (LJ) particles with the parameters of σ_CC = 0.34 nm and ε_CC = 0.361 2 kJ/mol. The harmonic potentials are used for SWCNTs to maintain the bond length of 0.14 nm and the bond angle of 120 ◦ with the energy constants 393 960 kJ/mol and 527 kJ ·mol ⁻¹ ·rad ⁻²,respectively. Meanwhile,the bonds of the CNT are represented by weak proper dihedral angle potentials. The CNT-water interaction is considered by a carbon-oxygen LJ potential with the parameters of σ_CC = 0.33 nm and ε_CC = 0.48 kJ/mol. All these parameters employed in the present study have been widely used in the previous studies^[9] . The LJ interactions are treated with a cut-off distance of 1.2 nm,and the particle mesh Ewald (PME) method^[19] with a real-space cut-off of 1 nm is utilized to treat the long-range electrostatic interactions. The periodic boundary conditions are imposed on all directions. The time steps in all simulations are set to be 1 fs,and the data are collected every 1 ps. First,the short SWCNTs are constrained by position restraints,and the system is simulated for 1 ns in the absence of the electric field to reach equilibrium states. Afterwards,we remove the constraint of the short SWCNTs and conduct computation for additional 8 ns in the electric field with the linear strength gradients of α = 0.05,0.1,0.2,and 0.25 for data analysis. 3 Result s and discussion

For S1 and S2,the guided unidirectional motion of the short SWCNTs is observed,and the z-coordinates of the centers of mass (COM) of the short SWCNTs are measured during the simulations,which are shown in Figs. 2(a) and 2(b),respectively. For S1,the short (10,10) SWCNT moves to the area with a lower field strength and finally reaches the left open end of the long (5,5) SWCNT. The short SWCNT cannot escape from the long SWCNT because of the existence of the van der Waals potential energy barrier between the two SWCNTs^[20] . Because of the thermal motion induced by the interactions between the short SWCNT and the surrounding water molecules,apparent fluctuations can be observed on the curves. In addition, the movement of the short SWCNT depends on the gradient coefficient α. The short SWCNT moves faster as α increases. For α = 0.05,0.1,0.2,and 0.25,the values of the average velocity are 1.06,1.40,2.13,and 2.60 nm/ns,respectively. While for S2,the capsule-like (5,5) SWCNT moves to the area with a higher field strength in the direction opposite to that in S1. When the capsule arrives at the right open end of the long (10,10) SWCNT,it also keeps oscillating there and cannot escape. Since the capsule is confined in the interior of the long SWCNT, the thermal motion effects of the surrounding water molecules on translation are much weaker. Therefore,the curves of the COMs are much smoother than those in S1,and the motion is nearly steady statistically. The values of the average velocity of the capsule are 5.31,6.54,10.63,and 12.84 nm/ns for α = 0.05,0.1,0.2,and 0.25,respectively.

Fig. 2 z-coordinates of COMs of short SWCNTs in S1 and S2 for α = 0.05,0.1,0.2,and 0.25, respectively

Figure options

It might be interesting to seek the physical mechanism behind this kind of nanomotors. The potential of mean force (PMF),first introduced by Kirkwood in 1935^[21],provides a measure of the effective difference in the free energy between two states as a function of one or several “interesting” degrees of freedom. These degrees of freedom can be the real coordinates of the system or a combination of them. Usually,the distance between the two interacting groups is chosen as the degree of freedom,which is also called the reaction coordinate. A routinely used technique to compute the PMF along a given reaction coordinate ξ uses the umbrella sampling. In this method,one of the groups serves as a reference,while the other group is placed at a set of N_w different distances from the reference group with its position maintained by an umbrella potential. From each of the N_w umbrella simulations (sometimes referred to as “umbrella windows”),an umbrella histogram h_i(ξ) (i=1,2,· · · ,N_w) is recorded,representing the probability distribution of the reaction coordinate along the reaction coordinate ξ. Finally, the weighted histogram analysis method (WHAM)^[22] is used to assemble a PMF curve as a functio n o f the e ntir e r ea ctio n c o o r dina te ξ. In the present two systems,both short SWCNTs move along the z-coordinate. Thus,the z-coordinate is chosen as the reaction coordinate. The PMF value at z then reflects the work required to bring the short SWCNTs from the reference positions to the position of the coordinate z. The short SWCNT COM position of z=10.0 nm close to the right end of the long (5,5) SWCNT and the one of z = 4.0 nm close to the left end of the long (10,10) SWCNT are set as the reference positions for S1 and S2,respectively. In S1,the COM of the short SWCNT is pulled from z = 10.0 nm to z = 2.85 nm with a spring constant of 1000 kJ ·mol ⁻¹ ·nm ⁻² and a pull rate of 0.01 nm ·ps −1 when the values of the electric field gradient coefficient α are 0.0,0.05,0.1,0.2,and 0.25,respectively. From the trajectories, snapshots are taken to generate the starting configurations for the umbrella windows with the window spacing 0.2 nm COM separation,resulting in N_w = 35 windows for each value of α. In each window,molecular dynamics of 6 ns are performed for the total simulation time of 1080 ns utilized for the umbrella sampling. While in S2,the COM of the capsule is pulled from z = 4.0 nm to z = 12.75 nm with the same spring constant and pull rate. The window spacing is also 0.2 nm,and the N_w = 44 windows are presented for each value of α. Then,the total simulation time of 1320 ns is performed for the umbrella sampling. Finally,the WHAM is implemented by the tool g −wham^[23] to get the PMF curves.

From Fig. 3,when the electric field is absent,there is no obvious change of the PMF along the z-coordinate,indicating that the short SWCNT will not move along the long SWCNT unidirectionally. When the electric field with the linear gradient is applied,the values of the PMF decrease monotonously as the COMs of the short SWCNTs approach the end position. Because the system always tries to reach the position where the free energy is minimum,the descending profile of the free energy confirms the spontaneous unidirectional motion of the short SWCNTs. If one wants to make the short SWCNTs move in the opposite direction,the additional work must be done on the system. In addition,as the gradient coefficient α of the electric field increases,the free energy difference between the initial state and the final state increases. For α = 0.05,0.1,0.2,and 0.25,the values of the maximal free energy difference for S1 are 12.77,30.96,50.01,and 58.88 kcal/mol,respectively. While for S2,the similar situations are also found in Fig. 3(b). The increase of the value of α also produces the larger free energy difference between the initial state and the final state. The values of the maximal free energy difference for α = 0.05,0.1,0.2,and 0.25 are 55.33,89.63,112.08,and 115.22 kcal/mol, respectively. Moreover,the values of the free energy difference of S2 for different α are larger than those of S1. That is the reason that the motion of the short SWCNT in S2 is faster than that in S1.

Fig. 3 PMFs of short SWCNTs in S1 and S2 for α=0,0.05,0.1,0.2,and 0.25,respectively

Figure options

One issue which has to be addressed is that the curves of the motion of COMs of the short SWCNTs are made by the snapshots of each trajectory,but the curves of the PMF are obtained by the average one of the trajectories . Thus ,the correspondence is not exactly one to one . When α is larger than 0.25,if we repeat the simulation for the same parameters,the difference between trajectories becomes larger,and the correspondence with the PMF curves becomes not so direct. In that case,we may need to make average on the movements of the short SWCNTs. 4 Conclusions

In this paper,the molecular dynamics simulations have been performed to show that in aqueous environments,a short SWCNT guided by a long SWCNT,either inside or outside the longer tube,is capable of moving along the nanotube axis unidirectionally in an electric field with the linear gradient perpendicular to the CNT axis. The results show that the short SWCNTs move to the area with a lower or higher field strength,and the velocity increases as the gradient of the electric field becomes larger.

The PMF has been calculated to show the effective difference in the free energy between two states as a function of the z-coordinate of the COMs of the short SWCNTs. The system always tries to reach the configuration with a minimum free energy. The descending profiles of the obtained free energy confirm the spontaneous unidirectional motion of the short SWCNTs. The moving speeds of the short SWCNTs are governed by the free energy difference,and the larger the difference is,the faster the speed is. The design suggests a new way of molecule transportation or mass delivery.