Many-Body Electrostatic Forces between Colloidal Particles at Vanishing Ionic Strength
Many-Body Electrostatic Forces between Colloidal Particles at Vanishing Ionic Strength, J. Merrill, S. Sainis, E. Dufresne, PRL 103, 138301 (2009) 
Soft Matter Keywords
Optical tweezer, Screening Length, Electrostatic Interactions
The electrostatic forces between pairs and larger numbers of colloids is studied in varying concentrations of surfactants. Non-pairwise additive electrostatic forces are directly observed when particle separation is less than the screening length.
In this paper the electrostatic interactions between 600nm poly(methylmethacrylate) (PMMA) particle immersed in hexadecane with surfactant is studied. The colloids are first held in place in a pair configuration at a given seperation by optical tweezers. The optical tweezers are periodically turned on and off (this technique is called "blinking optical tweezers"). When the trap holding a given particle is turned off, the particle is free to move. Its displacement during its untrapped excursion gives a velocity which in turn is related to the force it is experiencing via Stokes drag. Thus, the force on both particles can be extracted and measured as a function of seperation between the two spheres when they are brought closer together in successive measurements.
Figure 1 shows the forces experienced by colloids in the pair configuration at high and low surfactant concentration (corresponding to short and long electrostatic screening length). As expected, the pair interaction appears to follow that predicted by pairwise interactions (essentially a confirmation that the experiment returns accurate measurements). Successive geometries involving 3 and then 6 colloids arranged in an equilateral triangle and hexagon are shown to the right. The difference between pairwise additive predictions and actual force measurements in the short screening length experiments only becomes noticeable in the hexagonal arrangement involving 7 particles. At the longer screening length, however, the deviation from pairwise additivity is much more pronounced.
The origin of this discrepancy is that the differential equation (Poisson-Boltzmann equation - PBE) that governs the electrostatic potential in an ionic solution is non-linear. Thus, solutions to the equation from isolated pairs of particles may not be superimposed which is required if the potential is to be pairwise additive. The authors then state that in fact, the linearized version of the PBE is sufficient to explain their measured deviations from non-pairwise interaction. The physical origin of this is that the charge on a conducting sphere must change when two spheres are brought together in order to maintain a constant potential. The effect then becomes even more pronounced as more bodies are introduced. The authors appear to have concluded that the non-pairwise additivity of the forces can be explained by a linear theory and need not have anything to do with the non-linear nature of the PBE.
It is quite interesting to note that because of the form of the PBE (see the paper), the scale of the problem is defined by <math>\kappa a</math> where <math>\kappa</math> is the inverse screening length and <math>a</math> is the size of the particle, the experiment for micro particles in oil should be investigating the same physics as that of particle one-thousandth the size in a environment with one-one-thousandth the screening length (ie. nanoparticles in water). The latter system is of much interest but cannot be studied directly like microparticles which can be manipulated with optical tweezers. This is a nice feature of the scaling of the system, analogous to investigating geometrically similar flows defined by identical reynolds numbers but having vastly different length scales.