**3. Adsorption on surfaces and interfaces**

## **3.1 Adsorption**

It is generally agreed that adsorption of lignosulfonates follows the Langmuir isotherm [70–73]. The according framework was first published by Irving Langmuir in 1916, which described the adsorption of gasses on solids [74]. A common expression of the Langmuir equation is given in Eq. (1), were Γ corresponds to the amount of adsorbed species, Γ max is the maximum amount adsorbed, *c* is the concentration of adsorbing species in bulk, and *K* is the Langmuir equilibrium constant.

$$
\Gamma = \frac{\Gamma\_{\text{max}} Kc}{1 + Kc} \tag{1}
$$

The parameters Γ max and *K* can be determined via mass-balancing, i.e., measuring the adsorbed amount in dependence of concentration. This approach is simple for adsorption on solid surfaces, as the amount of non-adsorbing surfactant is equivalent to the remaining bulk concentration. An example for the adsorption of lignosulfonates on carbon black is given in **Figure 6**. The characteristic behavior accompanying Langmuir adsorption isotherms is a sharp increase of adsorbed amount at low concentrations, whereas higher concentrations yield a plateau.

Determination of the Langmuir equation parameters to liquid–liquid systems is less straight forward than for liquid–solid dispersions. Emulsification is accompanied by an increase in specific surface area, which can be difficult to quantify. The surface or interfacial tension can be used instead of mass-balancing. Langmuir type adsorption implies the applicability of the Szyszkowski equation, as given in Eq. (2). Here, the surface pressure Π is calculated as the difference of surface or interfacial tension without surfactant (*γ*0) and at concentration *c* (*γ*ð Þ*c* ). The equation further involves the temperature *T*, the ideal gas constant *R*, and the constant *n*.

$$
\Pi = \chi\_0 - \chi(\mathcal{c}) = n \text{RT} \Gamma\_m \ln \left( \mathbf{1} + \mathbf{K} \mathcal{c} \right) \tag{2}
$$

The surface excess Γ*<sup>m</sup>* can further be determined using the Gibbs adsorption isotherm. As stated in Eq. (3), this framework relies on the linear-logarithmic progression of surface tension *γ* with concentration *c*. It is important to note that the linear-logarithmic regime does not span over the entire concentration range. At low surfactant concentrations, the surface or interfacial tension approaches the surfactantfree case *γ*<sup>0</sup> asymptotically. At high lignosulfonate concentrations, agglomeration and

#### **Figure 6.**

*Adsorption isotherm of lignosulfonate on 1 wt% carbon black along with Langmuir isotherm fitting. Image taken from [46].*

other effects will decrease the observed slope [38, 75]. A more detailed discussion of the surface and interfacial tension of lignosulfonates is given in Section 3.2.

$$
\Gamma\_m = -\frac{1}{n\text{RT}} \left( \frac{\partial \chi}{\partial \ln c} \right) \tag{3}
$$

Assumptions can furthermore be made to determine the constant *n*. In case of strong surfactant electrolytes, such as *R* � *O* � *SO*� <sup>3</sup> *Na*þ, both the surfactant ion and the counterion must absorb, yielding a value of *n* ¼ 2 [76]. A value of *n* ¼ 1 corresponds to non-ionic surfactants or ionic surfactants in presence of high concentrations of an indifferent electrolyte, such as NaCl. Both assumptions have been applied to lignosulfonates and were found to yield conclusive results [13, 66].

By applying the above framework, other information can also be extracted from surface and interfacial tension measurements. For example, the surface excess Γ*<sup>m</sup>* can be used to calculate the area per molecule, as stated in Eq. (4) [76]. Here, *N*av denotes the Avogadro's constant 6*:*<sup>022</sup> � 1023 mol�<sup>1</sup> .

$$\mathbf{A\_m} = \frac{1}{\Gamma\_m N\_{\text{av}}} \tag{4}$$

Several authors have studied the adsorption kinetics of lignosulfonates on solid surfaces [71, 72, 77–80]. The reports generally agree that pseudo second-order kinetics provided the best fit. Such kinetic can be expressed as written in Eq. (5), where *qt* is the adsorbed amount at time *t*, *qe* is the amount adsorbed at equilibrium, and *k* is the rate constant.

$$\frac{\mathbf{t}}{\mathbf{q}\_{\text{t}}} = \frac{1}{kq\_{\text{e}}^2} + \frac{t}{q\_{\text{e}}} \tag{5}$$

Eq. (5) can be fitted to experimental data by plotting *<sup>t</sup> qt* against *t*, yielding <sup>1</sup> *qe* as the slope and <sup>1</sup> *kq*<sup>2</sup> *e* as the intersection. This model is, however, limited to adsorption from liquid onto solid phase. Kinetic modeling of lignosulfonate adsorption at liquid–liquid interfaces has also been attempted; yet, the results indicated that water–oil adsorption was not diffusion-controlled [66]. It usually takes hours or days to attain an equilibrium state at the liquid-air or liquid–liquid interface [66, 81, 82]. This time span is greater than for simple monodisperse surfactants. Different explanations have been given for this behavior. The lignosulfonate can undergo diffusion exchange at the interface, replacing molecules with a lower diffusion coefficient but higher effect on interfacial tension. In addition, the individual macromolecules may be subject to rearrangement, both with respect to the interface and to each other [2]. Such realignment of conformation has been described, e.g., for petroleum asphaltenes [83, 84]. Both lignosulfonates and asphaltenes share common characteristics, e.g., both are polybranched, exhibit a tendency for selfassociation, and require overnight storage, as the emulsions would be less stable if processed immediately after emulsification [13, 85]. Analogies have therefore been drawn between these two species in terms of their interfacial behavior [2, 66].

### **3.2 Effect on surface and interfacial tension**

Measuring surface tension is a useful tool, e.g., for assessing the behavior of lignosulfonates in solution or their interactions with other components [38, 86]. In addition, the interfacial tension provides a measure for the ease of emulsification [13]. The effect on surface or interfacial tension is related to the amphiphilic property of lignosulfonates, containing both hydrophilic and hydrophobic moieties. From a molecular point of view, the lignosulfonate macromolecule can attain a lower state of energy by extending its hydrophilic moieties into the water, while facing the hydrophobic parts away from it. The decrease in surface tension at increasing surfactant concentration can be explained by different mechanisms. Firstly, adsorption and desorption are occurring simultaneously, where an equilibrium is attained if the adsorption rate is equal and opposite to the desorption rate. By increasing the surfactant concentration, it is statistically more likely for a surfactant molecule to be in the right position and conformation to undergo adsorption. This would hence drive the adsorption–desorption equilibrium towards a higher surface coverage. Secondly, lignosulfonates are polyelectrolytes. By increasing the lignosulfonate concentration, the electrolyte concentration is also increased. As has been shown by small-angle X-ray scattering (SAXS), the effective surface charge of lignosulfonates decreases at increasing concentration [87]. Higher concentrations of a common ion indeed enhance effects such as counterion condensation, charge screening, and the dissociation equilibrium [88]. Water-solubility is ensured by the ionic groups of lignosulfonates, so a lower effective charge would have destabilizing effect. At a lower water-solubility, the equilibrium would hence be shifted to increased surface or interface adsorption. This effect concurs with the observations that lignosulfonates can be precipitated by salt addition (salting-out) [89] and that emulsion stabilization is highest, if the stabilizers are on the verge of precipitation [90].

Lignosulfonate adsorption at aqueous surfaces is evidenced, among others, by a decrease in surface tension [13, 38, 91]. Within a certain range, this decrease follows a linear-logarithmic progression with concentration [13, 38]. Below this range, the surface tension of the pure liquid is approached asymptotically. At high lignosulfonate concentrations, the slope of the surface tension decreases. Some authors have explained this behavior with the onset of lignosulfonate aggregation [75, 92], but other effects cannot be ruled out. Measurements of the surface tension with respect to concentration are shown in **Figure 7**.

In comparison with commercial surfactants, the effect on surface or interfacial tension is often less at the same mass concentration [75, 82, 92]. However, in comparison with other polyelectrolyte polymers, the effect on surface tension can be considerably higher [38]. This circumstance is likely related to the chemical composition and structure, which define macroscopic properties such as solubility, hydrophilic–lipophilic balance (HLB), or surface coverage.

#### **Figure 7.**

*Equilibrium surface tension in dependence of surfactant concentration. Comparison of lignosulfonates (LS) with polyelectrolyte polymers (left) [38] or with the surfactants dodecyl benzenesulfonate (DBS), nonylphenyl polyoxyethylene glycol (PONP) and polysorbate 20 (PS20) (right) [92].*
