Page 74 - Principles and Applications of NanoMEMS Physics

2. NANOMEMS PHYSICS: Quantum Wave-Particle Phenomena 61

materials [78] are separated by distances R>>r, where r is the atomic radius,
the wave functions decay exponentially and no bonding forces are operative.
At these distances, each molecule (atom) may be characterized as a dipole
antenna emitting a fluctuating field with a frequency distribution
characterized by an average frequency ϖ . For distances, R, smaller than the
Rϖ
average emitted wavelength, i.e., R < λ or << 1, the emitted fields are
c G
reactive in nature, i.e., they vary with distance as E ∝ 1 R . Therefore,
3
with reference to two emitting molecules (atoms), separated a distance R and
endowed with dipole operators d >=< ˆ ω αE , the van der Waals interaction
ω
energy between them derives from the self-consistent field induction at each
others’ site. In particular, atom 1 induces a field at the site of atom 2 given
3
by, E ˆ 1 ind () 2 ≈ d ˆ 1 R , which, in turn, induces a dipole at the site of atom 2
ω
3
given by, d ˆ ind =α () d ⋅ω ˆ R , where α () is the polarizability at the
2 2 1 2
site of atom 2. Similarly, the induced dipole at atom 2 induces a field at the
site of atom 1 given by, E ˆ ind () 1 ≈ d ˆ 2 ≈α ()⋅ω d ˆ 1 . Thus, the average
2 3 2 6
R R
ground state dipole energy of atom 1 is given by [78],
α
* ˆ ˆ ind 2 ˆ * ˆ
U ω () =< dR 1 ⋅ E 1 >= < d 1 ⋅ d 1 > and is a function of its average
R 6
dipole fluctuation. The signature of van der Waals forces is the
7
F = dU dR ∝ 1 R distance dependence.
vdW vdW
For calculations, Desquesnes, Rotkin, and Aluru [79] have modeled the
van der Waals energy by the expression,

R
U () = ³³ n 1 n 2 C 6 dV dV , (48)
vdW 6 ) 1 2
1 V V 2 R ( ,VV
1 2
where V and V embody two domains of integration of the adjacent
1 2
materials, n and n are the densities of atoms pertaining to the domains
1 2
V and V , R ( ,VV ) is the distance between any point in V and V ,
1 2 1 2 1 2
and C , with units [ ÅeV 6 ], is a constant characterizing the interaction
6
between atoms in materials 1 and 2. While a good first step for modeling
purposes, the exclusively pair wise nature of the contributions embodied by
(46) may not be accurate enough for tube geometry since it is known [80]
that, in exact calculations, one needs to consider three-particle, four-particle,
etc interactions, or equivalently multi-pole interactions. These multiple