Physical Chemistry of Foods

The clusters formed should not show significantsedimentationbefore

they reach the size Rg, since otherwise demixing would occur.

Significant sedimentation will occur if

R>Rsed¼

3 kBT

2 pa^3 
DgjDrj

 1 =ðDþ 1 Þ

ð 13 : 17 Þ

whereRsedis the critical cluster radius for sedimentation andDris

the density difference between particle and liquid.Rsedwill thus be

smaller for larger values ofaandjDrj.

If significantrestructuringof a cluster occurs, its fractal properties may

be lost. If this occurs before the clusters have grown out to the size

Rg, a gel may still be formed ifjis not very small.

Vessel size. Equation (13.15) implies that a gel will always form,

however smalljis, although the gel may be extremely weak at very

smallj. (In practice, presence of a yield stress cannot be perceived if

jis below about 0.001.) There is, however, also a geometrical lower

limit to the value ofj. If the smallest dimensionxof the vessel is

smaller than 2Rg, the vessel does not contain enough particles to

make a network spanning the length x. From Eq. (13.15) the

condition then becomes

x>dj^1 =ðD
^3 Þ ð 13 : 18 Þ

wheredis the particle diameter. Sincedis of micrometre scale, this
condition is nearly always met for macroscopic vessels. For
instance, for d¼ 1 mm, j¼0.005 and D¼2.30, it follows that
x>2 mm is needed. However, if the vessel is also of microscopic
size, say an emulsion droplet containing aggregating particles, a
high value ofjis needed to obtain a network throughout the whole
droplet.
Thefractal dimensionalitycan vary considerably with the conditions
that prevail during aggregation: values between 1.7 and 2.4 have been
obtained. This is of importance, since the magnitude ofDaffects several
parameters. For instance, the example given just below Eq. (13.16) yields for
D¼2 a value ofNp,gof 100. ForDvalues of 1.7 and 2.4, the result would be
20 and 10^4 particles per cluster, respectively.
Simulation studies fordiffusion-limited aggregation(i.e., forW¼1)
generally yieldDvalues of 1.7–1.75. Careful experiments lead to about the
same or a slightly higher value, but only if the particle volume fraction is
0.01 or smaller. For increasingj, the fractal dimensionality considerably
increases, thereby invalidating the basic Eq. (13.12); moreover, the