*3.1. Equilibrium Surface Tension Isotherm*

In Figure 3a we show the surface tension isotherm for aqueous solutions of the G12-2-12 surfactant. The critical micelle concentration, cmc, obtained from these measurements is 0.9 mM. This value is in good agreement with values previously reported [5,14,17] which are in the range of 0.7 to 1 mM.

**Figure 3.** (**a**) Surface tension isotherm for Gemini 12-2-12. The lines are two different fittings with Equation (2), see Discussion section for details. (**b**) Equilibrium surface concentration as a function of bulk concentration calculated from Equation (2) (**c**) Surface tension as a function of surfactant area fraction at equilibrium (see Equation (2)).

#### *3.2. Dynamic Surface Tension*

In Figure 4 we present the results of the dynamic surface tension for 6 surfactant concentrations below the cmc. Those concentrations are *c*<sup>s</sup> = 0.1; 0.2; 0.3; 0.4; 0.5, and 0.6 mM. All the curves are well-behaved (smooth) and in all cases, they reach the equilibrium values obtained from Wilhelmy plate technique, considering the errors (in the figures the errors are represented by the dashed regions). In the insets in said figures we show, amplified, the very short adsorption times region (≤2 s). The lines shown are fittings using Equation (3) (see discussion below).

**Figure 4.** Dynamic surface tension curves for 6 surfactant concentrations well below the cmc (**a**) *c*s = 0.1 mM; (**b**) *c*s = 0.2 mM; (**c**) *c*s = 0.3 mM; (**d**) *c*s = 0.4 mM; (**e**) *c*s = 0.5 mM; (**f**) *c*s = 0.6 mM. The insets are amplifications of the very short adsorption times regions, the curves in them are fittings with Equation (3). The shaded regions represent the equilibrium surface tension (with error) measured by the Wilhelmy plate technique.

In Figure 5 we show the results for the same kind of experiments but for solutions at concentrations close and above the cmc.

**Figure 5.** Dynamic surface tension for surfactant concentrations close and above the cmc. (**a**) 0.8 mM; (**b**) 1 mM; (**c**) 2 mM. The insets show the dynamics of adsorption at very short times (<0.5 s). The lines are fittings with Equation (5).

The behavior at short adsorption times (insets) is different from those observed for *c*s < cmc. The lines are fittings with exponentials (Equation (5), see discussion).

#### *3.3. Time-Resolved Surface Potential*

In order to shed light on the processes involved in the adsorption dynamics, and the role played by the charges at the interface, we performed time-resolved surface potential experiments, as explained in the methods section. In Figure 6 we present the results for *c*s = 1 mM, superimposed to the dynamic surface tension curve in a semi-log plot. We show simultaneously three independent measurements of the surface potential as a function of time. The three curves are different but all of them present general common features. Note that, at short times, there is a fast increase of surface potential followed by a decrease and several oscillations, those oscillations were not replicated on the surface tension dynamics. In the figure, we included the relaxation times obtained by fitting certain parts of the curves with exponentials (see discussion).

**Figure 6.** Surface potential curves (symbols in color, left axis) superimposed to the dynamic surface tension, γ(*t*) (circles, black, right axis) for a solution at a surfactant concentration of 1 mM. We show three independent surface potential experiments (Exp 1, 2 and 3) to illustrate the "reproducibility" of such measurements.

In Figure 7 we show the results corresponding to *c*s = 0.1 mM, *c*s = 0.2 mM and *c*s = 0.5 mM. Note that the surface potential curves have the same features mentioned above, a fast increase of the surface potential and oscillations.

For all surfactant concentrations the initial (water) surface potential was about −0.2 volts and reaches the stationary value of 0.3–0.5 volts, depending on surfactant concentration. The insets on those figures are included to show that ϕ tends to a stationary value, a fact that is difficult to see when time is in logarithmic scale.

**Figure 7.** Time dependence of the Surface potential compared with dynamic surface tension (Surface tension in black, right axis) for three surfactant concentrations. (**a**) *c*s = 0.1 mM, (**b**) *c*s = 0.2 mM, (**c**) *c*s = 0.5 mM. The insets show the surface potential but with the time axis in linear scale.

#### **4. Discussion**

## *4.1. Equilibrium Surface Tension*

The first step in the discussion of the results shown in previous sections is the equilibrium isotherm of Figure 3a. Our point of departure is the Gibbs adsorption equation [12]:

$$
\Gamma = -\frac{1}{nk\_\text{B}T} \frac{d\gamma}{d\ln(c\_\ast)}\tag{1}
$$

From the dependence of the surface tension, γ, on the surfactant bulk concentration, *c*s, the surface excess, Γ, (surface surfactant concentration) can be obtained by means of this equation. Here *k*<sup>B</sup> is the Boltzmann constant and *T* the absolute temperature. The constant *n* takes the value 3 for a dimeric surfactant made up of a divalent surfactant ion and two univalent counterions, such as G12-2-12, in the absence of added salt. However, by means of neutron reflectivity measurements [18,19], it was found that, for G12-2-12, *n* = 2. This means that one counterion is condensed onto the Gemini polar head at the interface and thus G12-2-12 behaves like a monovalent ionic surfactant (*n* = 2). In Figure 3b we show the surface excess obtained from the experimental data of Figure 3a and the Gibbs adsorption equation, Equation (1). Note that the surface saturates at about 0.2 mM with a surface concentration of <sup>Γ</sup> ~ 1.9 <sup>×</sup> <sup>10</sup><sup>18</sup> molec.m<sup>−</sup>2, given an area per surfactant head of 5.3 <sup>×</sup> <sup>10</sup>−<sup>19</sup> m2, which gives 7.25 Å as an estimation for the distance between surfactant heads at the interface. On the other hand, from a free energy approach [10,20,21], the adsorption isotherms and the equation of state for a 1:1 ionic surfactant (note that we consider that one of the surfactant charges is also condensed in bulk because the distance between charges in the surfactant head [22], ~4 Å, is less than the Bjerrum length) can be expressed as,

$$\begin{aligned} \phi\_0 &= \frac{\phi\_0}{\phi\_\oplus + \left[ b\phi\_\theta + \sqrt{\left( b\phi\_0 \right)^2 + 1} \right]^2 \exp(-a - \beta \phi\_0)} \\ \gamma &= \gamma\_W + \frac{k\_\mathbb{R} T}{a^2} \left[ \ln(1 - \phi\_\mathbb{O}) + \frac{\beta}{2} \phi\_0^2 - \frac{2}{b} \left( \sqrt{\left( b\phi\_\mathbb{O} \right)^2 + 1} - 1 \right) \right] \end{aligned} \tag{2}$$

*a* being the average size of a surfactant molecule, ϕ<sup>0</sup> is the surfactant area fraction at equilibrium (ϕ<sup>0</sup> = Γ*a*2), βthe Frumkin lateral interaction parameter, αthe Langmuir adsorption parameter and *b* a parameter characterizing the strength of electrostatic interactions [10], *<sup>b</sup>* = [π*l*B/(2*a*φb)]1/2 (ϕ<sup>b</sup> is the surfactant volume fraction, φ<sup>b</sup> = *a*3*c*s). In Figure 3c we show the fitting of the data, γ(ϕ0), with Equation (2). The lines in Figure 3a correspond to the same fitting for the curve γ(*c*s). We show two fittings, in one case keeping the solvent surface tension constant at 72 mNm−<sup>1</sup> and for the other leaving this parameter free to adjust the curve. From the former fitting we found: *a* = (0.89 ± 0.2) nm, β= (−3.24 ± 1) *k*B*T*; α = (8.9 ± 2) *k*B*T*; from the latter, *a* = (0.77 ± 0.2) nm, β = (−0.6 ± 0.2) *k*B*T*; α = (10 ± 2) *k*B*T* and γ<sup>0</sup> = 72.8 mNm<sup>−</sup>1. The molecular size, *a*, obtained from the fittings with this model is 7.7–8.9 Å, which is close but larger than that obtained for DTAB solutions [10], for which *a* = 7.2 Å was obtained. This result is somehow expected because G12-2-12 has two DTAB chains linked by the heads with an ethyl group. Note that the Frumkin interaction parameter β is negative, this indicates a repulsive interaction among surfactant molecules at the interface.
