Page 34 - Science at the nanoscale

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RPS: PSP0007 - Science-at-Nanoscale
June 9, 2009
Classical Physics at the Nanoscale
24
−1
), and if we convert
typically use the unit of wavenumbers (cm
−1
) to frequency (Hz),
the wavenumber of the C-H stretch (2974 cm
13
we get about 9 × 10
frequencies are much higher than those of objects at larger scales.
2.2 VISCOSITY
The force F needed to move a sphere of mass m, density ρ, radius
R at a velocity v through a viscous medium of viscosity η (Stoke’s
Law) is given by:
(2.12)
F = 6πηRv
When the sphere reaches terminal velocity v t ,, the force on it
due to gravity (F = mg) is balanced by the retarding force due to
the viscosity of the medium:
4
3
2
πR ρg
mg
2ρgR
2
3
∝ R
(2.13)
=
=
v t =
6πηR
6πηR
9η
Since the terminal velocity is proportional to the radius squared,
it is clear that small particles fall very much more slowly. Note that
the above treatment is only valid under conditions of streamline
flow, for small particles and low velocities. This condition is met
when the Reynolds Number (Re) is less than about 2000, where Re
is a non-dimensional quantity that describes the type of flow in a
fluid defined by:
2Rρv
Inertial · f orces (ρv)
(2.14)
=
Re =

η
Viscous · f orces
η Hz. Hence, at the nanoscale, mechanical ch02
2R
As size decreases, the ratio of inertia forces to viscous forces
within the fluid decreases and viscosity dominates. Hence,
micro/nano-scale objects moving through fluids are dominated
by viscous forces, and their motion is characterised by a low
Reynolds number. This means that nanoparticles “feel” the vis-
cosity (or ‘gooeyness’) of the fluid much more than we do!
To give a quantitative example, consider an iron sphere of
radius 1 mm and density 7,000kgm −3 (i.e. a small ball bearing)
falling through water (η = 0.01 Pa.s, cf. Table 2.2). It has a ter-
minal velocity calculated from Eq. (2.13) of about 1 ms −1 . If
the sphere is now 1 µm in radius, its terminal velocity becomes