Biomedical Engineering Reference
In-Depth Information
1
1
2 2
0
m u
ω
=
k T
(8.9)
B
2
2
k
m
ω
=
m
is the mass associated with the oscillator,
u
its displacement,
0
, its reso-
nance frequency,
k
is the spring constant of the cantilever.
Therefore:
k T
k
=
(8.10)
2
u
Equation (8.10) is used practically to calibrate the spring constant of the can-
tilever: when
k
~50 mN/m, the thermal fluctuations are of the order of a few ang-
stroms, which is readily measurable by the detector. Practically, a power spectral
density is plotted and fitted with the theoretical lorentzian shape for a harmonic
oscillator. The area under this curve is then used to access the spring constant.
8.3.2 Physical Characterization: Light Scattering
Light scattering is routinely used to get molecular weight information out of poly-
mers solutions in its static version; it is also a powerful tool to directly measure
hydrodynamic radii when in the form of dynamic light scattering.
The increase in computing capabilities and the reduction in size and cost of
lasers have greatly popularized the use of these techniques.
8.3.2.1 Static Light Scattering (SLS) [34]
It is a common observation that, when light hits a suspension, some of it is scattered
along all directions. Rayleigh scattering describes quantitatively this scattering for
particles smaller than the wavelength of the light. Here, we do not take into account
the temporal fluctuations of the scattered light but average the signal over long
times. The dynamic aspect will be treated in the next part.
The theory proceeds by computing the interactions of the electric field associ-
ated with the incident light with the polarizability of the particle it interacts with.
When applied to a solution of polymers of molecular weight
M
, the intensity
I
(
θ
) at
an angle of incidence
θ
is then given by:
2
I c
α
0
2
(8.11)
I
( )
θ
»
(1 cos
+
θ
)
2 4
r M
λ
where
α
is the particle polarizability,
I
0
the incident beam intensity,
c
the concentra-
tion in particles,
r
is the distance to the detector, and
λ
the wavelength.
As the polarizability varies linearly with the molecular weight, at a given angle,
the intensity is thus also proportional to
M
:
(8.12)
I
( ) K( ) c M
θ θ
=
×
×
If we consider a mixture of polymers of different masses or a polydisperse sam-
ple, we have to sum over all the contributions:
