Thisdepletion free energycan be calculated for a pair of spherical
particles at a distancehfrom each other by
Vdepl¼ 2 prpPpolð 2 dhÞ^204 h 42 d rp 4 dPpol¼RTc
MþBc^2 þð 12 : 12 Þwherecis concentration (mass per unit volume) andMmolar mass of the
polymer, andBis the second virial coefficient. Since it generally concerns
fairly dilute solutions, a third virial coefficient is not needed. [Note that
the form of the equation differs from Eq. (12.11). The relation between
the parameters of the second term is, if water is the solvent,
B¼ 28? 103 b=r^2 & 0 : 012 b, whereris the mass density of the polymer.]
As an example, consider spherical particles of radius 1mmanda
polymer concentration c of 1 kg?m^3 (i.e., 0.1%). The polymer may
be xanthan, radius of gyration 30 nm, M¼ 103 kg?mol^1 , and
B¼ 5? 10 ^4 mol?m^3 ?kg^2. It then follows that the depletion interaction
ath¼0 would be about 16 kBT. This is a fairly strong interaction, and it
will undoubtedly cause particle aggregation, unless a strong repulsive force
also exists, e.g., due to an electric charge. It is observed in practice that
depletion flocculation, as it is commonly called, can often be induced by
xanthan concentrations as small as 0.02%. For polysaccharides of smallerrg,
i.e., having a lower molar mass or chains that are less stiff than the xanthan
just mentioned, higher concentrations would be needed to induce aggrega-
tion.
Complications. In practice, precise calculation of the interaction
free energy is not always easy. Eq. (12.12) applies to a pair of identical hard
spheres, and even in that case the result may differ from the prediction. The
relationd¼rgis not exact, even if the polymer is monodisperse, because
some chains will protrude beyond rg. If the particles are somewhat
deformable, because they are very soft or because they have a deformable
adsorption layer, the depletion interaction forcing them together may cause
local flattening, by whichjVdepljbecomes even larger. Particle shape has a
large effect, and for platelets the depletion interaction is far stronger than
for spheres of equal volume (can you explain this?).
In practice, depletion flocculation in a dilute dispersion causes the
particles to sediment rapidly. Increasing the polymer concentration at first
leads to increasing sedimentation (rate), but at still higher concentrations,
sedimentation slows down or will be altogether absent. This occurs if the
polymer concentrationcis larger than the chain overlap concentrationc*;
see Section 6.4.2. At such conditions, the particles areimmobilizedin a