Page 37 - Science at the nanoscale

RPS: PSP0007 - Science-at-Nanoscale
7:14
June 9, 2009
2.4. Motion at the Nanoscale
From the Gaussian relation in (2.15), we can define the charac-
teristic 2D diffusion length as:
1
/
x rms = (4Dt) 2
(2.17)
If we go back to the example in the previous section of the iron
nanoparticle of 1 nm radius falling through water with terminal
−1
velocity of 1 pms
, the corresponding Brownian diffusion length
1/2
x rms = 2D
is about 9 µm, which is the characteristic distance
displaced every second due to Brownian motion. This value is
much larger than 1 pm, and hence Brownian diffusive motion is
dominant for the 1 nm particle.
If however the iron particle radius was 1 µm, its diffusion length
is now 0.3 µm, which is almost comparable to its terminal velocity
−1
. Hence both diffusive motion and viscosity of the fluid
of 1 µms
need to be taken into account in describing the particle’s motion.
In general, Newton’s law of motion in such cases in the presence
of an external force F ext and taking into account the Brownian dif-
fusive force F(t) and viscosity η of the fluid can be written in 1D
as:
2
3

dx
d x
4πR ρ
(2.18)
=
F ext + F(t) − (6πηR)
2
dt
3
dt
This so-called Langevin equationLangevin equation is a stochastic
differential equation in which two force terms have been added
to Newton’s second law: One term represents a frictional force
due to viscosity, the other a random force F(t) associated with the
thermal motion of the fluid molecules. Since friction opposes mo-
tion, the first additional force is proportional to the particle’s ve- 27 ch02
locity (dx/dt) and is oppositely directed. This equation needs to be
solved to describe the complete motion of a nanosized-object in a
fluid.
2.4 MOTION AT THE NANOSCALE
It has been often hypothesised that in the not-too-distant-future,
micron-sized medical nanorobots will be able to navigate through
our bloodstream to destroy harmful viruses and cancerous cells
(see Figure 2.4). This is reminiscent of the 1966 science fic-
tion film Fantastic Voyage written by Harry Kleiner, which was