*4.3. Dynamic Surface Tension*

Let us first discuss the short adsorption times. In the insets of Figures 4 and 5 the dynamics in this time range can be observed in an amplified scale. For *c*<sup>s</sup> = 0.1 mM and for adsorption times shorter than about 2 s, the dynamic surface tension data is beautifully fitted (see inset in Figure 4a) with the diffusional model [20,21], indicating that the adsorption dynamics is limited by diffusion (DLA),

$$\gamma(t) \cong \gamma\_0 + \left(\gamma\_0 - \gamma\_{\rm eq}\right) \left(1 - \sqrt{\frac{\tau\_{\rm D}}{t}}\right) \tag{3}$$

where τ<sup>D</sup> is the characteristic diffusional time. From the fitting with Equation (3), we obtain τ<sup>D</sup> = 0.25 ± 0.01 s. As the surfactant concentration increases, but always below the cmc, the experimental points become noisy and the fittings less reliable (see insets in Figure 4b–f). The corresponding times, τD, obtained from Equation (3) for each concentration are summarized in Table 1.


**Table 1.** Characteristic times obtained from fittings with Equation (3), and the diffusion coefficients, *D*, calculated with Equation (4). The surfactant concentrations are all below the cmc.

From the theory the characteristic time for a DLA process can be estimated [20,21],

$$
\tau\_{\rm D} = \frac{\phi\_0^4}{\phi\_{\rm b}^2} \frac{a^2}{\pi D} \tag{4}
$$

being ϕ<sup>b</sup> and *D* the bulk surfactant volume fraction and diffusion coefficient respectively. From the bulk concentration, the surfactant volume fraction can be estimated, <sup>φ</sup><sup>b</sup> = *<sup>a</sup>*3*c*s, for *<sup>c</sup>*<sup>s</sup> <sup>=</sup> 0.1 mM, <sup>ϕ</sup><sup>b</sup> <sup>=</sup> 3.1 <sup>×</sup> <sup>10</sup><sup>−</sup>5. Now, making use of the data on Figure 3, we estimate ϕ<sup>0</sup> ~ 0.83. With these values, from Equation (4) and <sup>τ</sup><sup>D</sup> <sup>=</sup> 0.25 s, we calculate *<sup>D</sup>* <sup>=</sup> 4.1 <sup>×</sup> <sup>10</sup>−<sup>10</sup> <sup>m</sup><sup>2</sup> <sup>s</sup><sup>−</sup>1. This value coincides perfectly with the literature value [25] of 4 <sup>×</sup> <sup>10</sup>−<sup>10</sup> m2 <sup>s</sup><sup>−</sup>1. The same calculation for larger surfactant concentrations deviates from this value but they fall in the correct order of magnitude except for *c*s = 0.5 and 0.6 mM. The results are summarized in Table 1.

For surfactant concentrations close and above the cmc, the adsorption dynamics is consistent with an exponential decay of surface tension even at very short adsorption times:

$$
\gamma(t) - \gamma\_{\text{eq}} = \left(\gamma\_0 - \gamma\_{\text{eq}}\right) \text{exp}\left(-\frac{t}{\tau}\right). \tag{5}
$$

This behavior can be seen in the insets of Figure 5, the characteristic times found by fitting with Equation (5) are labelled as τ<sup>1</sup> and shown in Table 2. In Equation (5), γeq and γ<sup>0</sup> are the equilibrium surface tension of the solutions, and pure solvent, respectively.

**Table 2.** Characteristic times, in seconds, found in dynamic surface tension for all surfactant concentrations studied.


Turning our attention to the last part of the adsorption process, close to equilibrium, the relaxation is consistent with a kinetically limited adsorption (KLA), following an exponential decay [10,20,21] as in Equation (5). We label the characteristic time of this final stage of the adsorption dynamics as τk.

In Figures 9 and 10 we plot, in a semi-log scale, Δγ = γ(*t*) − γeq as a function of time for surfactant concentrations below and close/above the cmc respectively. The results of the fittings with exponentials are shown as lines on the figures and labelled with the corresponding characteristic time. Note that there are several processes (and characteristic times) all consistent with an exponential decay. Those results are summarized in Table 2. Note that the number of processes increases with surfactant concentration until the cmc. The observation of several processes, each with its characteristic time, is consistent with what was found in surface rheology experiment on the same surfactant system [11].

**Figure 9.** Several relaxation processes following an exponential decay were observed in dynamic surface tension curves at concentrations below the cmc. (**a**) *c*s = 0.1 mM, (**b**) 0.2 mM, (**c**) 0.3 mM, (**d**) 0.4 mM, (**e**) 0.5 mM, (**f**) 0.6 mM.

The origin of the intermediate relaxations, between τ<sup>D</sup> and τk, seems to be related to charge redistribution as observed from surface potential results on Figures 6 and 7. This redistribution of charges may include the formation of aggregates at the interface, condensation of counterions onto those aggregates and further surfactant adsorption, as well as phase transitions [26].

Now, from the values of the characteristics times in the KLA regime, τK, we can estimate the average (ψ), the equilibrium surface (ψ0) and subsurface (ψa) electrostatic potentials, ψ = (ψ<sup>0</sup> + ψa)/2 [10,20,21]:

$$\frac{e\overline{\psi}}{k\_{\rm B}T} = \frac{\left(\alpha + \beta + \ln\left(\pi a^4 Dc^2 \tau\_{\rm k}\right)\right)}{2}.\tag{6}$$

From previous calculations we have *a* ~ 8 <sup>×</sup> 10−<sup>10</sup> m, <sup>α</sup> = 8.9 *k*B*T* and <sup>β</sup> = <sup>−</sup>3.24 *k*B*T*, using for the diffusion coefficient the value of *D* = 4 <sup>×</sup> 10−<sup>10</sup> m2 s−<sup>1</sup> and the <sup>τ</sup><sup>K</sup> values from the last column on Table 2, we calculate the average surface and subsurface electrostatic potential shown in Table 3. In that table ψ<sup>0</sup> was estimated from the Poisson-Boltzmann theory [21],

$$\frac{\epsilon\psi\_0}{k\_\text{B}T} \approx 2\ln(2b\phi\_0) \tag{7}$$

where *<sup>b</sup>* was defined above, *<sup>b</sup>* = [π*l*B/(2*a*φb)]1/2.

**Figure 10.** Several relaxation times following an exponential decay at concentrations above the cmc. (**a**) *c*s = 0.8 mM, (**b**) 1 mM, (**c**) 2 mM.



The obtained theoretical values for the surface electrostatic potential, ψ0, are larger than the average, ψ, obtained from dynamic surface tension experiments. Because the subsurface electrostatic potential, ψa, should be lower than the surface electrostatic potential, ψ0, the average should be ψ< ψ0, as it is in fact observed on Table 3. Note that the values of ψ<sup>0</sup> and ψ approach each other as the surfactant concentration increases, which could be explained by ions self-screening (being G12-2-12

a cationic surfactant, as the surfactant concentration increases, the ionic strength increases, and the Debye length diminishes reducing the distance of the electrostatic interaction, even in the absence of added salt). It is worth mentioning that the obtained values are similar to those found for the cationic surfactant DTAB [10].

We cannot use the previous analysis for surfactant concentrations above the cmc, in this case the adsorption-desorption and aggregation-disaggregation of micelles [27,28] could play a role in the observed dynamics both in γ(*t*) and ϕ(*t*). The fact that the dynamics of surfactant adsorption at short times follow an exponential relaxation (see insets on Figure 5) could indicate that we are not accessing times short enough to see the diffusion-limited adsorption step.
