Molecular dynamics simulation for flow characteristics in nanochannels and single walled carbon nanotubes

Flows in graphite-, diamond- and silicon-walled nanochannels are discussed by performing molecular dynamics simulations. Flows in carbon nanotubes (CNTs) and graphene- walled nanochannels are also investigated. It is found that the flow rate in the graphite-walled channel tends to be the largest because of its slippery wall structure by the short bond length and the high molecular density of the CNTs. The flow rate in the single walled CNT at a very narrow diameter tends to increase although such a tendency is not seen in the graphene-walled channel.


Introduction
Carbon nanotubes (CNTs) and nanochannels attract much attention because these nano-structures are expected for expanding the applications of the membrane engineering. Recently, it was reported that hydrocarbons or water molecules flowed through a CNT much faster than in the other materials [1]. After this sensational report, many research studies have been reported on flows in CNTs [2][3][4]. However, the mechanism of the fast transport in a CNT has not been clearly understood. Also, understanding the detailed flow physics in nano-structures is still underway. Therefore, in this study, molecular dynamics (MD) simulations of flows in nanochannels and CNTs are performed to understand the flow characteristics in nano-structures and the fast transport mechanism in a CNT.

Lennard-Jones fluid
The presently applied classical molecular dynamics simulation consists of the numerical solution of classical equations of motions of N molecules interacting via model potentials. The well-known model potential is the 6-12 Lennard-Jones (L-J) potential  which is defined as where r ij is the intermolecular distance between i and j molecules, r c is the cut-off radius, ε and σ are the well depth and the diameter of the molecules. From this potential, intermolecular force F is written as This model has been used in the classical MD simulation by argon (Ar) molecules. In the system which is modeled by the Lennard-Jones molecules, the classical equation of motion can be written as where m, are the mass, the nondimensional length, time and energy, respectively. However, to make a physical interpretation, it is sometimes expressed as dimensional values in terms of Ar. In addition, the nondimensional number density k is the Boltzmann constant. In actual calculations, to hold down the calculation effort of the interactions, the potential effects are truncated over cut off distance which is defined as c r . In the present study,   3 c r is employed.

Brenner and Tersoff potentials
To describe the covalent bond, the Brenner and Tersoff potentials [5,6] are applied to carbon and silicon walls. The covalent bond energy is written as For the Brenner potential, f R (r ij ) and f A (r ij ) are written as where D e , S and R e are determined by the physical properties of carbon. For the Tersoff potential, f R (r ij ) and f A (r ij ) are written as where A, B, λ 1 and λ 2 are physical properties of silicon. The cut off function c f is written as The bond order term is described as The function b ij is written as where ζ ij is and g c (θ) is described as where θ ijk is the angle between bonds ij and ik. The parameters of the Brenner and Tersoff potentials are tabulated in table 1.

Wall structures
In this study, five types of wall structures: diamond, silicon, graphite, graphene and CNT wall structures are discussed. These five structures can be categorized into two kinds of structures which are the diamond and six membered ring structures. Figure 1 shows the diamond structure. The diamond and silicon walls have this structure. The lattice constants of the diamond and silicon crystals are 0.356 and 0.543nm, respectively. The six membered ring structure for the graphene and graphite is shown in figure 2. Their bond length is 0.1421nm. The distance between the graphite layers is 0.3354 nm. The armchair CNT is considered in the present study.

Nanochannel flows of three types of wall structures
Firstly, flows in diamond-, silicon-and graphite-walled nanochannels are discussed. Argon molecules are applied for the fluid molecules. Figure 3 illustrates calculation domains for the nanochannels and CNTs. For the nanochannels, the calculation domain whose length L=3. 4 Figure 4 shows the number density profiles of the three wall cases. It is seen that the fluid molecules are densely distributed near the walls. This is because the channel walls strongly attract the fluid molecules. Figure 5 shows velocity profiles across the channels. In this figure, the flow in the graphite-walled channel has the largest velocity profile and the silicon-walled channel has the smallest one. One of the reasons why the velocity profiles have such tendencies is considered to be the roughness effect of the wall structure. In the three types of wall structures, the graphite wall has the largest atomic packing factor on the wall surface P f = 0.956 and the silicon wall has smallest atomic packing factor P f = 0.811 while the diamond wall has P f = 0.922. It is considered that the velocity profiles correspond to these atomic packing factors.        Figure 6 shows centerline and slippage velocities normalized by the bulk velocities. In all channel heights or diameters, centerline and slippage velocities are nearly the same corresponding to flat velocity profiles. Next, the tendency of flow rate depending on the tube diameter and the channel height is compared. The normalized flow rate is defined as

Flows in CNT and graphene-walled channel cases
Cs aH where ρ and a are the number density and the acceleration. The sound speed Cs of argon is defined as where γ = 1.67 and R = 208 m 2 /(s 2 ·K) are the specific heat ratio and the gas constant. Figure 7 shows the distribution of the normalized flow rates.   As shown in figure 7, the normalized flow rates Q of the two cases against H or D show a similar tendency where H or D are larger than 2.9 nm. However, the flow rate in the CNT at very narrow diameter (smaller than 2.9 nm) tends to be increase as the diameter decreases unlike in the graphene channel. This is because the attractive forces from the surrounding wall molecules are almost canceled each other. Moreover, the motions of fluid molecules in the narrow CNT are almost limited to the streamwise direction although those in the graphene-walled channel are allowed also in the spanwise direction. Thus the loss of the kinetic energy by collisions between fluid molecules in the CNT is less than that in the graphene-walled channel.

Conclusions
Nano-flows in several nano-structures are discussed by performing molecular dynamics simulations. Under a constant temperature and a pressure, three types of structures: diamond-, silicon-and graphite-walled nanochannel cases are calculated. The flow tendencies in the CNTs and the graphenewalled channels are also discussed. The number density profiles of the fluid molecules have peaks near the walls because the walls strongly attract the fluid molecules. Amongst the three types of wall structures, the flow in the graphite-walled channel has the largest velocity profile and that in the silicon-walled channel has the smallest velocity because of the roughness effect of the wall structure.
In the case of the flows in the CNTs and the graphene-walled channels, flat velocity profiles across the tubes or channels are seen regardless of their diameters or heights. The flow rates of the CNT have a different tendency when the tube diameter is less than 2.9 nm due to the effects of the surrounding wall molecules.