Particle Tracking as a Probe of Microrheology in Food Colloids - Food Colloids: Self-Assembly and Material Science - page 306

the mean-square displacement (MSD) of particles as a function of lag time t . 5
For instance, the two-dimensional time-averaged MSD is defined by
h Dr 2 ( t ) i¼h [x(t + t ) x(t)] 2 +[y(t + t ) y(t)] 2 i , (1)

where x(t)andy(t) are the time-dependent coordinates of the centres of the
particles and the angular brackets indicate an average over many starting times
for the ensemble of particles in the field of view. For spherical monodisperse
particles of radius a in a Newtonian fluid, the viscosity Z of the surrounding
medium can be determined from the measured diffusion coefficient D using the
standard Stokes-Einstein relation,

D ¼ kT/6 pZ a, (2)

where k is Boltzmann's constant and T the absolute temperature. For particles
tracked in viscoelastic media, the MSD may be analysed using a generalized
Stokes-Einstein equation to give frequency-dependent viscous and elastic
moduli. 6-8 For a macroscopically uniform system, the analysis allows direct
comparison of the viscosity Z or the complex modulus G*( o ) with the equiv-
alent property determined by conventional bulk rheometry.
For particles undergoing free diffusion, the ensemble-averaged MSD is a
linear function of time, i.e., in two dimensions, we have
MSD ¼h Dr 2 ( t ) i¼ 4D t .

(3)

When there is a steady drift or flow in the system (e.g., due to convection), the
plot of MSD versus t is no longer linear, but has positive curvature. For a
constant drift velocity V, the MSD is given by 9
MSD ¼ 4D t +(V t ) 2 . (4)

Even if the drift is only slow, the contribution from the quadratic term (V t ) 2
becomes predominant at longer times. It is as if the particle diffusive motion
becomes more vigorous the further the particle moves. In contrast, for diffusion
in a finite region (e.g., a pore or cage), the plot of MSD versus t has negative
curvature. The observed behaviour is most simply interpreted in terms of
interactions between diffusing particles and other mobile or immobile structures. 9
Different scenarios for the way in which colloidal probe particles may be
embedded in a biopolymer network have been discussed recently by Weitz and
co-workers. 10 One important factor is the particle size in relation to the
characteristic structural length-scale(s) of the material. The ensemble-averaged
MSD can be reliably related to G*( o ) of the bulk biopolymer network for the
case of probe particles that are large compared to macromolecular network
length scales (Figure 1A). In contrast, the bulk viscoelastic response does not
influence so much the dynamics of probe particles that are smaller than the
structural length-scales (Figure 1B). In this case, individual particle movements
are determined by the solvent viscosity, the effects of macromolecular crowding
and interactions with the network (electrostatic, steric, hydrodynamic). For
probing microenvironments of a heterogeneous material with particles smaller