Sedimentation Equilibrium of Mixtures of Charged Colloids

Monte Carlo simulations of figure 3 of Ref.[15]: σ = 150nm, H = 1000σ,λ B = σ/128, L 1 = 3L 2 2 = 10σ, ¯η 1 = ¯η 2 = (π/6) × 10 −4 , Z 1 = 15, and Z 2varies between 15 and 45. The simulated system does not contain addedsalt but only counterions, which we can represent within our theory by theextremely low reservoir salt concentration ρ s = 1nM, which we checked tobe low enough to be in the zero-added-salt limit as regards the colloidalprofiles. In Fig.1(a) we show, for Z 2 = 45, the density profiles of the twocolloidal species, as well as that of the counterions (and the coions) in theinset. We observe profound colloidal segregation into two layers, with theheavy species floating on top of the lighter ones. The counterions are seen tobe distributed throughout the whole volume, i.e. much more homogeneouslythan when all colloids would have been barometrically distributed in a thinlayer of thickness L i just above the bottom (since in that case the net ioncharge would have been located in that same thin layer). The resulting gainof ion entropy is the driving force for the formation of the electric field thatpushes the heavy (highly charged) colloids to high altitudes against gravity[8, 14]. In Fig.1(b) we show the mean height h i of species i, defined ash i =∫ H0 dx x ρ i(x)∫ H0 dx ρ i(x) , (A.5)as a function of Z 2 . We replot the Monte Carlo simulation results of figure3 of Ref.[15] (symbols), together with the predictions for h i that follow fromthe present theory (continuous curves). Given that there is not a single fitparameter involved, the agreement is remarkable, certainly when comparedwith the theoretical analysis on the basis exponentially decaying density profilesas in Ref.[15]. Fig.1 shows that the heavy particles are on top of thelighter ones, h 2 > h 1 , provided Z 2 /Z 1 ≃ 1.6. For barometric profiles onewould find that h i = L i , but we see in all cases that h i ≫ L i due to the lifteffect of the induced electric field. The good agreement between theory andsimulation in Fig.1(b) also indicates that hard-core effects (which are takeninto account in the simulations but not in the theory) are not so relevant, atleast not in the parameter regime studied here.The next system we study is a ternary system (n = 3) in a solvent characterisedby λ B = 2.3nm (ethanol at room temperature) with a reservoirsalt concentration ρ s = 10µM and meniscus height H = 20cm. The colloidalcharges and gravitational lengths are Z i = (1000, 250, 125) and L i =(1, 2, 1)mm, respectively, and the system is equimolar with ¯η i = 0.005 for allthree species. Fig. 2 shows the density profiles as predicted by the presenttheory in (a), as well as the electric field in (b). We find almost perfect segregationinto three layers, such that the mean heights satisfy h 3 < h 2 < h 1 .57