### Carbon Nanotube Capillarity

September 6, 2011

Capillary action, or capillarity, is a very useful physical phenomenon. I mentioned capillarity in the context of heat pipes in a previous article (Pumping Water, August 10, 2010).

A heat pipe is a passive device that uses the surface tension effect between liquids and solids, also known as capillarity, to transport heat. The liquid extracts heat by boiling, the vapors migrate to a part of the heat pipe that's below the boiling point, where they condense, and the liquid is drawn back to the hot zone via capillarity in a wicking material.

In a blatant use of the rhetorical ploy called ad auctoritatem, I'll mention that one of Einstein's first papers (Folgerungen aus den Capillaritätserscheinungen, 1901) was about capillarity.[1] As I wrote in a recent article (Trees, Wooden Variety, July 27, 2011), tall trees use capillarity to raise water to heights beyond the thirty foot limit of suction pumps.

Capillary action will lift water in a tube of inside radius r to an approximate height h given by
h = (1.4 x 10-5)/ r

where h and r are in the same units. A tube of one micrometer radius will raise water to a height of about 14 meters, and the equation asserts wonderful things for smaller radii. In reality, how small can we make a tube and still get water to suck up into it?

Scientists have taken the problem to an extreme by showing that water spontaneously flows into carbon nanotubes.[2-4] This is unexpected, since a cursory analysis of what should happen when water is confined at these near atomic scale dimensions indicates that entropy and bonding should both decrease; that is, water really shouldn't get sucked into nanotubes.

To discover why it happens, scientists at Caltech used molecular dynamics simulations of water confined in 0.8 to 2.7-nm diameter carbon nanotubes to calculate the entropy, enthalpy, and free energy. The simulations show that for all nanotube sizes, the water inside the nanotube is more stable than bulk water. However, the conformation of the water changes with nanotube size.

A cutaway simulation of a two nanometer-diameter carbon nanotube with confined water molecules.

(Caltech Image/Tod Pascal))

In small diameter nanotubes (0.8 - 1.0 nm), the water is vapor-like; for 1.1 - 1.2 nm, the water is ice-like; and for diameters larger than 1.4 nm, the water is liquid-like.[2] The parameter that's responsible for the stability of water inside carbon nanotubes is the entropy change.

Says William A. Goddard, Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics at Caltech, and director of its Materials and Process Simulation Center, "It's a pretty surprising result... People normally focus on energy in this problem, not entropy."[3]

The simplistic view is that water, which forms a network of hydrogen bonds, is very stable, and breaking these hydrogen bonds so that water can flow into the small volume of a nanotube requires energy. What happens is that the hydrogen bonds in bulk water restrict molecular motion, so the entropy of bulk water is low. The entropy that water gains by entering the nanotubes is enough to outweigh any energy losses of hydrogen bond breaking. It's energetically favored for water to spontaneously flow into the tubes.

Grave of Ludwig Boltzmann at the Zentralfriedhof (Central Cemetery) of Vienna, Austria

At the top is his famous entropy equation,
S = k log W.

The modern transcription uses Ω instead of W. Both of these letters are mapped to 'W' on the keyboard, so if you mistype, no one will notice.

(Photo by Daderot, via Wikimedia Commons))

The interesting ice-like structure that exists for water inside 1.1 - 1.2 nm diameter nanotubes is the result of a perfect match between the confined space, the arrangement of atoms in the water molecule, and the hydrogen bonding between molecules.[3]

The Caltech team devised a new simulation method, called the two-phase thermodynamic model, that helped in the entropy calculations. Said Goddard, "The old methods took eight years of computer processing time to arrive at the same entropies that we're now getting in 36 hours."[3] The Caltech team thinks that their research can be used to design supermolecules for water purification. A polymer with the same pore size as carbon nanotubes could suck water out of solution.[3]

### References:

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