Depletion Flocculation

Quick Start

From DLVO theory we know that adding polymers to a particulate system can stabilize it . Frustratingly, if we add a little too much polymer, we can de-stabilize the system via a non-intuitive effect called Depletion Flocculation. To get a feel for the effect, play with the radius of the particles, the radius ("radius of gyration") of the polymer, its concentration and MWt. The key is the attractive energy at short distances. If this is large (negative) then it is unlikely to overcome whatever steric or charge stabilization is available.

Still confused? Well, most of us are. But depletion flocculation is a very significant effect in real-world formulations, so it's worth trying to understand what is going on so you can avoid the problem.

Depletion Flocculation

Particle r nm
Polymer δ nm
Polymer c g/l
Polymer MWt
Π Pa
N.bonds
Bond length nm
Rg nm

An extra attractive force not included in DLVO arises when a polymer that has no attraction to the particles is added to the dispersion. At first it is hard to see why a polymer that does not interact with the particles can have any effect on them. If the polymer is attracted to the particle then it can cause steric stabilisation at high concentrations and bridging flocculation (it attracts two particles and therefore brings them together) at low concentrations.

Solubility Science Depletion Flocculation The explanation of depletion usually involved a diagram showing the polymer chains as little spheres that are unable to get close to the particles when, by chance, the two particles happen to be near each other. This means that there is a high concentration of polymer in the bulk and a low concentration between the particles. This is a classic case of osmotic pressure difference and solvent will go from the low concentration to the high concentration, i.e. the solvent will flow out of the gap between the particles, causing them to clump.

The approach used here is the popular Asakura and Oosawa formula based on:

  • Radius of the particle, r
  • Effective Radius of the polymer, δ
  • Polymer Concentration
  • Polymer MWt

The combination of the polymer concentration and MWt gives the "number density" ρ (in molecules/m³) of the polymer from which the osmotic pressure Π can be calculated via Π=ρkT. The value R in the formula is the centre-to-centre distance, but in the plot (as with DLVO) the distance is shown as the particle-to-particle gap, h. The formula is valid for when h<2δ - above this the polymer is able to enter the gap and there is no osmotic force. The graph automatically scales in the x-axis to just above 2δ.

G=-4/3π(r+δ)³(1-3R/4(r+δ)+R³/(16(r+δ)³))Π

The downside of A&O is that there is no term describing the stabilisation effect at high concentration. More sophisticated approaches can simulate this, but A&O is good enough for the typical effects seen at lower polymer concentrations.

First play with the inputs and see the effect, then read more details below.

The first obvious problem for the user is to know the effective radius of the polymer. Classically this is the radius of gyration Rg plus whatever extra shell is provided (e.g. a Debye layer as in DLVO). The usual approximation is that Rg²=Nb²/6 where N is the number of polymer bonds and b is a "bond length", though N and b are more to do with "statistical bond length" rather than actual bond length. Feel free to use or ignore the calculation in the app if you know your Rg via some other route.

The second problem is that Rg and the MWt used to calculate the osmotic pressure are inter-related. Again you have to sort this out for yourself.

The third problem is knowing whether your polymer is completely unable to interact with the particle (a key assumption behind the theory). If this is true (e.g. both polymer and particle have the same ionic charge) then the calculation is OK. But if the polymer has a modest affinity for the particle then any flocculation is likely to be of the bridging type - or if the polymer level is large and it is well-attached to the particle then you get steric stabilisation as in DLVO.

The fourth problem is alluded to above. You need a reasonably high concentration of polymer to get a significant osmotic pressure to drive the flocculation. But at even higher concentrations stabilisation can set in for complicated reasons.

Finally, don't blame me that there are so many uncertainties behind depletion flocculation!