Techniques for the characterization of physical gels - Physical Gels from Biological and Synthetic Polymers - page 51

h x 2 i¼ 2Dt :

In three dimensions this gives

r 2 h i¼ 6Dt : ð 2 : 30 Þ

Here D is the Fickian diffusion coef cient, and the Stokes - Einstein equation gives

k B T
6

D ¼

r p :

ð 2 : 31 Þ

π

Since k B is Boltzmann s constant and r p is the hydrodynamic or Stokes radius of the

probe particle, the Newtonian viscosity

'

is straightforwardly obtained.
Conversely, if the embedding system is purely elastic, then particles will move as a
result of thermal motion but never actually diffuse. The amount of movement depends
upon the elastic modulus of the medium; in this case the movement can be equated to that
of a spring with constant K e , so that, again in one dimension,
K e ¼h

η

x 2

i : ð 2 : 32 Þ

k B t

For viscoelastic materials, more sophisticated methods need to be employed. For example,
Mason and Weitz ( 1995 ) showed how the Stokes

- Einstein (SE) equation can be general-

ized to a new relation which can be

used for deriving the frequency-dependent viscosity, or as shown in the review by Waigh
( 2005 ), the creep compliance. A limitation of this simple approach is that it is less successful
if sets of single particles are tracked, since these may simply probe sample inhomogeneities.
In order to overcome this limitation, two-particle correlations can be followed.
The idea, as the description suggests, is that a set of the particles is tracked using, for
example, a laser and digital correlator set-up. A variety of experimental devices have
been reported, and among these are those which make use of dynamic light scattering.
The resultant intensity

- the generalized Stokes - Einstein relation (GSER) -

fluctuation correlation function can be inverted to give the mean
square displacement. However, there is no need to go to such trouble: results can be
obtained relatively directly, although smart software needs to be written and imple-
mented, using the technique of video particle tracking (VPT).
This, as the name suggests, images the probe particles using a video camera (Corrigan
and Donald, 2009a , 2009b ). Digital image sets (frames) of the sample are captured, and
the positions of embedded particles calculated frame by frame by the software, to form
particle trajectories and mean square displacements. This approach allows large numbers
of particles to be tracked simultaneously, while retaining information on their individual
positions. Consequently, such experiments are relatively fast to perform and the local
microstructure of the sample can be investigated.
It is appropriate to consider both advantages and disadvantages of the techniques.
Advantages are that the technique, as already pointed out, is potentially non-perturbing
and that the apparatus can be assembled from commercial scattering kit and so is
relatively cheap to construct. This is particularly true for the video particle tracking
method. Also, the sample volume can be extremely small, as little as 20 μ L, and for