When one adds salt to an aqueous suspension of charged colloidal particles, this suspension becomes unstable, and the particles aggregate. Initially they form particles dimers, and subsequently the aggregates grow larger. The kinetics of the particle dimers formation follows a second order rate law. The rate coefficient k of this reaction and the particle concentration set the time scale of the aggregation process. This rate coefficient has a characteristic dependence on the salt concentration. At low salt concentration, this coefficient is small, and strongly increases with the salt concentration. One refers to the slow regime, and here this coefficient is sensitive to details (i.e., solution composition, types of particles). At higher salt concentrations, typically above 0.1 mol/L, this coefficient attains a constant value. Here, one refers to the fast regime, since the aggregation is controlled by the diffusional approach of the particles to contact.
| Rate coefficients (×10–18 m3/s) | Amidine | Carboxyl |
| Experimental | 3.8 | 2.0 |
| Smoluchowski | 12 | 12 |
| Literature Hamaker constant | 8.8 | 8.8 |
| Measured Hamaker constant | 6.5 | 6.8 |
The classical approach to quantify such diffusion controlled aggregation is Smoluchovski's theory [4]. This theory assumes no interaction between particles, and the predicted values only depend on the solvent viscosity and the temperature. Smoluchovski's predictions are also given in the above table, and they are clearly too big.
More sophisticated treatment includes conservative and hydrodynamic forces between the particles. At high salt concentrations, the conservative forces are dominated by van der Waals attraction. The hydrodynamic forces are repulsive, and lead to a decrease of the pair diffusion coefficient at smaller distances. The rate coefficient can be also calculated including thee two effects. Smoluchowski's theory can be modified by including these additional effects [5]. The literature value of the Hamaker constant for polystyrene is around 1×10–20J, which allows one to estimate the strength of the van der Waals force. The resulting rate coefficients are given in the table. While this modification goes into the right direction, the calculated rate constant remains too large.
For these particular latex particles, we were recently able to measure the van der Waals force with the atomic force microscope [2,3]. To our surprise, we found substantially smaller values of the Hamaker constants than the established values, typically values around 3×10–21J. This smaller value can be in fact understood by considering the surface roughness of the particles. When these measured Hamaker constants are included in the calculation, the table shows that this step goes into the right direction. Still, however, an important difference remains.
We have no sensible explanation for the remaining discrepancy so far.
Michal Borkovec, March 6, 2015.
[1] Trefalt G., Szilagyi I., Borkovec M. (2013) Measuring particle aggregation rates by light scattering, www.colloid.ch/aggregation.
[2] P. Sinha, I. Szilagyi, F. J. M. Ruiz-Cabello, P. Maroni, M. Borkovec (2013) Attractive forces between charged colloidal particles induced by multivalent ions revealed by confronting aggregation and direct force measurements, J. Phys. Chem. Lett. 4, 648, 10.1021/jz4000609.
[3] F. J. Montes Ruiz-Cabello, G. Trefalt, Z. Csendes, P. Sinha, T. Oncsik, I. Szilagyi, P. Maroni, M. Borkovec (2013) Predicting aggregation rates of colloidal particles from direct force measurements, J. Phys. Chem. B, 117, 11853-11862, 10.1021/jp406061f
[4] Russel, W. B., Saville, D. A., and Schowalter, W. R. (1989) Colloidal Dispersions. Cambridge University Press Cambridge.
[5] Trefalt G., Borkovec M. (2014) Overview of DLVO Theory, www.colloid.ch/dlvo.