*4.2. Equilibrium Surface Potential*

In Figure 8 we plot the change in surface potential, Δϕ, between pure water interface (ϕwater ~ −0.2 V) and the final stationary value at the surfactant solution-air interface, ϕ. Note that the surface potential difference seems to go through a maximum at a surfactant concentration between 0.2 and 0.4 mM. Above this concentrations, Δϕ diminishes continuously indicating the condensation of counterions onto the interface. This is consistent with the distance separating surfactant heads at the interface estimated from Gibbs equation, ~7.25 Å, which is very close to the value of the Bjerrum length *<sup>l</sup>*<sup>B</sup> (*l*<sup>B</sup> <sup>=</sup> *<sup>e</sup>*<sup>2</sup> <sup>ε</sup>*k*B*<sup>T</sup>* <sup>≈</sup> 7Å, *<sup>e</sup>* is the electron charge and <sup>ε</sup> is the water dielectric constant). We recall that, when the distance between charges is less than *l*b, condensation of counter-ions is expected, such that the effective distance between charges become equal to the Bjerrum length [23,24].

**Figure 8.** Change in the surface potential between water and surfactant solutions, Δϕ = ϕ − ϕwater, as a function of surfactant concentration. This curve includes the values for *c*s = 0.01 and 0.06 mM which are not included in Figure 7.
