Figure 9.1b gives an example of a complicated structure. Ice cream can
contain seven or more different structural elements, making up six phases:
the continuous phase (an aqueous solution), air, ice, oil, crystalline fat,
lactose. It may be noted that the system depicted is bicontinuous in that the
clumped fat globules also form a continuous network. During long storage
at low temperature, part of the air phase can become continuous. At a lower
temperature, a greater part of the water freezes and a continuous ice phase
can be formed; moreover freezing causes less room being available between
the various particles.
This brings us to the subject ofpacking: what is the maximum volume
fractionðjmaxÞof particles that a dispersion can contain? For hard, smooth,
monodisperse spheres, it is in theory for a cubic arrangement 0.52, for a
hexagonal arrangement, the closest packing possible, the value is 0.74. In
practice, one often observes for solid spheres that jmax& 0 :6, because
friction between spheres hinders attaining their closest packing. For
emulsions, where the drops are relatively smooth, a common value is 0.71.
If the particles are
More polydisperse,jmaxis larger, because small particles can fill the
holes between large ones.
More anisometric or have a rougher surface,jmaxis smaller, mainly
because these factors cause more friction between particles, making
it difficult for them to rearrange into a denser packing.
Aggregated,jmaxis smaller, often very much so. In most aggregates,
much of the continuous phase is entrapped between the primary
particles: see Figure 9.2, G.
More deformable,jmaxis larger, and values up to 0.99 have been
observed in foams, where the large bubbles can readily be deformed.
How much material would be needed to make acontinuous network,
i.e., a structure that encloses all of the continuous phase? This can readily be
calculated for a regular geometry. Assume anisometric particles, more in
particular cylinders of lengthLand diameterd. When making a cubical
configuration of these particles, i.e., each edge of a cube is one particle, we
obtain
L/d¼ 5102050
jneeded& 0.094 0.024 0.006 0.001It is seen that very little material can suffice, provided, of course, that
attractive interaction forces keep the network together. In practice the
situation is more complicated; some systems are discussed in Chapter 17.