Charge Renormalization for Effective Interactions of Colloids at Water Interfaces

We analyze theoretically the electrostatic interaction of surface-charged colloids at water interfaces with special attention to the experimentally relevant case of large charge densities on the colloid–water interface. Whereas linear theory predicts an effective dipole potential, the strength of which is proportional to the square of the product of charge density and screening length, nonlinear charge renormalization effects change this dependence to a weakly logarithmic one. These results appear to be particularly relevant for structure formation at fluid interfaces with arbitrarily shaped colloids.

Figure 1

Side view of a single colloid (homogeneously charged on the water side) trapped at the interface. Most of the counterions are confined in a layer close to the colloid surface with a width of the order of the Gouy-Chapman length $l_G=2{ϵ}_2{ϵ}_0/(\beta e{\sigma }_c)$. In many colloidal experiments, $l_G(\approx 1\mathrm{nm})<{\kappa }^{-1}(\approx 1\dots 300\mathrm{nm})<R(\approx 1\mu \mathrm{m})$.

Figure 2

The charge renormalization function in the nonlinear regime. For a colloid of radius $R=1\mu \mathrm{m}$, the two dimensionless charge densities ${\sigma }_c^{*}=500$ and 8000 correspond to charge densities of 0.9 and $15\mu \mathrm{C}/{\mathrm{cm}}^2$ which approximately bracket the charge densities occurring on polymeric colloids.