^bDepartment of Mechanical Engineering, University of California at Berkeley, CA 94720-1740, USA

What is nanofluidics?

The transport efficiency and selectivity are two special features of carbon nanotubes (CNT) because of which they will be used as membranes in future separation technology. The continuum theory (Navier Stoke’s equation) in fluid mechanics is time and position dependent and is independent of molecular structure and configuration of atoms in the molecule. Nanoscience poses a question: Is continuum approach valid for nanoscales? It is found that to obtain a physical property with less than 1% statistical fluctuation we need 10,000 molecules in continuum approach. For water, if 10,000 water molecules are confined in a sphere, the diameter of the sphere will be ~27 molecules or ~6.5 nm. A consensus could not yet be developed to decide the scale at which continuum approach breaks down. For carbon nanotubes (CNTs), if the lateral dimension is less than 2 nm, the systems falls in the category of noncontinuum fluidics.

Basic structure and properties of CNTs

A CNT is a rolled graphene sheet in the form of a cylinder. The structure of CNT is described by its role up vector (n,m), called chirality or helicity. The inner diameter of carbon nanotube is given by

Where a is the lattice parameter of graphene which is 2.5 ? and r_c is the van der Waals radius of carbon atom which is 1.7 ?. Two important properties of CNTs are it high aspect ratio with small dimensions: it can reach several millimeters in length keeping diameter to few nanometers, its atomic scale smoothness for flow, and inertness of graphene walls. CNTs can be prepared in the laboratory but preparation of a bundle of CNTs all having a specified chirality has been unsuccessful.

Simulations of water and gas in CNTs

Water inside CNTs

Water is hydrophilic and CNT is hydrophobic but still water fills the CNT. Is there any change in structure of water molecule or its properties? Single file of water molecules is observed in MD simulation that could not be seen in bulk water. It was found that water filled CNT and it remained filled throughout the simulation time. A single file of water molecules was observed that can’t be seen in bulk water. It was also found that water molecules inside and outside the nanotube remain in thermodynamic equilibrium. It offers a counterintuitive phenomenon that water molecules confined in a nanoscale enclosure narrows down the interaction energy distribution which lowers the chemical potential. Thus confining a liquid in a nanotube lowers its free energy. The filling equilibrium is sensitive to water-nanotube interaction parameters. A 40% reduction in interaction potential empties the CNT. A 25% reduction results in fluctuation (filling and emptying of CNT). Dependence of CNT hydration on other properties of CNT, for example, chirality, wall flexibility, charge, and diameter have also been studied.

CNTs as biological channel analogs

Water molecules form a chain bonded by hydrogen bonds which is similar to formation of water chain in biological systems like in aquaporins. This is because of hydrophobic nature of inside walls of both CNT and aquaporins. A combination of hydrophobic nature of inside walls and smoothness causes frictionless water transport. The friction is too low that Hagen-Poiseuille flow does not hold and the high flow under osmotic pressure depends on the events on the entrance and exit of the nanotube. Flow rate of 5.8 water molecules per nanotube per nanosecond has been observed and is similar to the flow rate inside aquaporins. One dimensional water wires play an important role in proton transport in biological structures like cytochromes via Grutthus mechanism. Similarly proton transport is also possible in nanotube (along nanotube axis) and a mobility of portons 40 times that of bulk water has been observed. Water plays an important role in the stability of confined proteins. MD simulations have shown that the stability of confined proteins is low in narrowest nanotube channels and this stability increases with the increase of nanotube diameter. It contradicts the classic polymer theory which claim that peptide helix stability should decrease. The classic theory suggests that confinement changes the solvent entropy which changes the utilization of free energy of protein water hydrogen bond formation and the relative stability of peptide helix. It means confinement of solvent entropy affects the rules of protein stability.

MD simulations of gas transport in CNTs

Researchers have shown that transport of gas increases three orders of magnitude inside nanotube than in zeolites with equivalent pore size. The diffusivities of light gases like hydrogen and methane on molecular and bulk scale are nearly the same (10^-1 cm²/s). With the increase in pore size the diffusivity increase (10 cm²/s). Based on density profiles, trajectories, and specular collisions of gas molecules with CNT walls, the gas molecules transport is in billiard ball like manner inside a nanotube. Maxwell’s coefficient or tangential momentum accommodation coefficient are of the order of 10-3 suggesting that only 0.1% of the gas molecules are thermalized or change their velocities upon collision with nanotube walls. Some gas molecules are adsorbed on the wall which may give rise to an alternative transport mechanism.

Testing platforms: fabrication of CNT membranes

Simulation results of fast transport through CNTs required the fabrication of a test platform. Experimental studies focus on a geometry consisting of multiple nanotubes in aligned arrays. The challenge in the fabrication of a test platform is to keep a high aspect ratio of the gaps between nanotubes. This aspect ratio should be 1000 or large.

Polymeric/CNT membranes

Hinds’ group prepared high density multiple wall carbon nanotubes (MWNT) encapsulated by polystyrene with ~7nm pore size. Being a liquid phase process, the procedures that confirm that CNTs will not be bundled together are required.

 

Silicon nitride CNT membranes

Low pressure chemical vapor deposition (CVD) process has been used to fabricate vertically aligned CNTs encapsulated by Si₃N_x. Extra Si₃N_x is removed from the ends by etching and uncapped by oxygen plasma. Transmission electron microscopy of these double walled nanotubes (DWNT) showed the pore size less than 2 nm. This group also developed MWNT of pore size ~6.5 nm. These nanotubes are robust and can bear a pressure gradient exceeding 1 atm.

CNT polymer network fabrication

Amine-functionalized CNTs after dispersion in tetrahydrofuran are filtered through polytetrafluoroethylene produced vertically aligned nanotubes. Spin coating of a dilute polymer solution gave protruded structure which exhibited enhancement in gas transport rates and non-Knudsen selectivity for a mixture of gases.

Gas transport measurements

Bulk flow rate of air through Si₃N_x/DWNT with a pressure gradient of ~1 atm was measured. To compare with simulation results, flow rate per pore was computed. Since all the nanotube pores were of different pore sizes, a lower and an upper limit of flow per pore was computed and it was found that gas transport rates were 100 times higher than the conventional Knudsen model for gas transport. The experiments could not determine the cause of gas flow rate enhancement and specular collisions of gas with walls were suggested. Nonhydrocarbon gases exhibited a Knudsen like scaling while hydrocarbon gases deviated from Knudsen like scaling. The deviation may be due to adsorption and diffusion of gas molecules by CNT walls.

Water transport in DW/MWNT membranes

After computation of active pore density, transport of water through DW and MWNT was found 4 to 5 orders of magnitude higher compared with the Hagen-Poiseuille model. The lower estimates report 3 to 4 orders of magnitude high water flux than no slip. The slip length was almost three orders of magnitude higher than the pore size and almost of the order of nanotube length. A polycarbonate membrane with a pore size of ~15 nm had a slip length of about 5 nm. This suggests that slip flow formalism is not applicable to CNTs because of length confinement. The measured flux (10-40 molecules/(nm²)(ns)) agrees with the simulation results (12 molecules/(nm²)(ns)).

Nanofiltration properties of CNT membranes

Nanofiltration using CNT membranes have shown that 10 nm Au particles are completely excluded while dyes (0.5 ? 2 nm) are diffused with low hindrance by MWNT membrane with inner pore diameter of ~7 nm. The aligned DWNT in pressure driven filtration excluded 2?5 nm Au particles with fast water permeation. Ruthenium bipyridine ions, Ru(bpy)₃⁺², (1.1 nm) also permeate fast though CNT showing that CNT diameter is in the range of 1.3 ? 2.0 nm and is in agreement with TEM analysis of CNTs.

MD simulation of ion exclusion

With nanoscale pores and high permeability, CNTs can be used in desalination and demineralization. MD simulations have shown that the removal of small ions like Na⁺, K⁺, and Cl^- etc require ~0.4 nm pore size of an uncharged CNT. At this pore size the ions lose their hydration shell and gain high energy barrier to cross the membrane. If the pore size increases (>1 nm), this free energy barrier drops to zero (~5 kJ/mol) and allow free access to small ions. MD simulations have shown that 0.34 nm CNT with negative charge on wall conduct K⁺ ions and exclude Cl^- ions while 0.47 nm CNT with positive charge conduct Cl^- ions and exclude K⁺ ions. Experimental evidences have not yet been received as the CNTs with such small pore size have not been prepared to date.

Membrane functionalization and transport

The chemical modification of the nanotube preserving its smooth surface and hydrophobicity can serve for required permeation and selectivity. For this purpose only the entrance and exit of the nanotube are modified. During synthesis process, removal of nanotube cap by oxidation attaches carboxylic groups at the mouth of the nanotube which can serve for variety of modifications. Researchers have attached a variety of molecules at the mouth of CNTs. They found that oxidation step did not attach carboxylic groups at the end only, but attached the groups up to 700 nm from the tip of CNT and after 50 nm of separation the density of functional groups decreases significantly. Another group of researchers have also attached aliphatic chains, charged molecules of dyes, and polypeptides. Attachment of C₄₀ carbon chain reduced the flux of test charged species (organic cations) by six times. An increase in the flux of test charged ions was observed due to attachment of charged dye possibly because of the attraction of oppositely charged ions. Long chain organic molecules are hydrophobic and align themselves with the nanotube and therefore have no effect on the overall flux. This group also modified CNT by charged organic dye in the presence of voltage. An inert solution was kept in flow through CNT so that only the mouth is modified. It was found that the applied voltage drew the dye molecule from end to the inner core of CNT. This phenomena could not be explained well because of the presence of both the positive and negative voltage making the process complex.

Outlook: CNT membranes as a scientific and technological platform

The unique properties of CNTs, its nanoscale pore size and atomically smooth surface offer a unique platform for research of ions removal in confined areas. MD simulations have provided counterintuitive results and few of them have been verified experimentally. It is hoped that a real consensus will soon develop in simulation and experimental studies. An understanding of transport mechanism and structure of CNTs will open new doors for separation and purification. The small pore size and high permeation will reduce the energy consumption of desalination and gas separation processes.

Reviewer: Aamir Alaud Din

Email: aamiralauddin@gist.ac.kr