Assignment

Recently carbon nanotubes have been the subject of intense research. Their special geometry and unique properties offer great potential in applications such as nanoelectronic devices, energy storage, field emission displays, etc.

The adsorption of gas molecules onto carbon nanotubes may disturb the electronic characteristics of nanotubes. In this problem we will investigate the possibility of defects in nanotube materials due to gas adsorption.

From this course, we have learned that the chemical potential of a dilute gas can be expressed as a function of pressure P and temperature T:

μ = kT In [(P/kT) (h²/2πmkT)^3/2]

where h is Planck's constant, m is the mass of a gas molecule and k is Boltzmann's constant. Assuming that carbon nanotube surfaces are composed of lattice sites and only one molecule can occupy a site at a time, the mean surface density (the fraction of sites occupied by molecules) ρ_s was derived as

ρ_s = 1 / e^{-(ε + µ) / kT}

where ε is the adsorption energy advantage. It is experimentally measured that Oxygen (O₂) has adsorption energy advantage 0.509eV per molecule on carbon nanotube surfaces (eV means electron volt).

1) Express 0.509eV in units of kT (set T=300K).

2) Find the pressure value P at which 10% of the surface sites are occupied by gas molecules. (Please be careful about the units of your calculation.)

Recent experiments (App. Phys. Lett. 75, 3017 (1999)) show that the adsorption of O₂ molecules is significant even at the pressure 10^-5Pa, which should be far below your answer above. Here is a suggestion to solve this discrepancy.

Suppose that more than one molecule can be adsorbed on a surface site. Assume that there is no limit to the number of allowed molecules on a surface site. Thus the state vector σ = {n_i}^A_i=1, where n_i is the number of adsorbed molecules at the ith site, and A is the total number of sites. Regardless of n_i, the energy gain for each molecule is ε.

3) Calculate the grand potential Ω(T,A,µ) for the nanotube surface. (Express e^-Ω/kT as a summation over all of the n_i's. Hence prove the relation Ω = AkT In (1 - e^{-(ε + µ) / kT}).)

4) Hence express the surface density of molecules ρ_s as a function of ε, µ and T.

5) Does this new approach explain the discrepancy?