Page 75 - Principles and Applications of NanoMEMS Physics

62 Chapter 2

interactions must be included to improve modeling results. Nevertheless,
applied to a SWNT beam of diameter r and suspended by a gap R, they
obtained the van der Waals energy per unit length of the CNT as,

2
2
U C σ 2 π 2 r (r + R )(3r + ( 2 R + r ) )
vdW = 6 , (49)
L ( ( 2 R + ) r 2 − r 2 ) / 7 2
−
where σ ≅ 38nm is the atomic surface density, L is the CNT length. The
2
corresponding van der Waals force is given by,
§ U ·
d¨ vdW ¸
F = © L ¹
vdW
dR . (50)
− (C πσ 2 2 r R (R+ r 2 )) ( R8⋅ 4 + 32 R 3 r+ 72 R 2 r + 80 Rr + 35 4 ) r
2
3
= 6
2 R 5 (R+ ) r 2 5
As mentioned previously, the van der Waals force is one contributor to
the phenomenon of stiction. Thus, its prominence must be accounted for in
the design of advanced structures, e.g., nanoelectromechanical frequency
tuning systems [54] based on quantum gears [81], as estimates of its
magnitude are useful in designing against it [18, 82].

2.3.2 Casimir Force

The Casimir force arises from the polarization of adjacent material
bodies, separated by distances of less than a few microns, as a result of
quantum-mechanical fluctuations in the electromagnetic field permeating the
free space between them [74-77]. It may also arise if vacuum fluctuations are
a classical real electromagnetic field [83]. The force may be computed as
retarded van der Waals forces or as due to changes in the boundary
conditions of vacuum fluctuations; these are equivalent viewpoints as far as
it is known [80].
When the material bodies are parallel conducting plates, separated by
free space, the Casimir force is attractive [74], however, in general whether
the force is attractive or repulsive [82], [84] depends on bot h the boundary
conditions, including specific geometrical features, imposed on the field as
well as the relationship among material properties of the plates and
the intervening space. For example, repulsive forces are predicted by