Investigation of Interfacial Free Energy of Three-Phase Contact on a Glass Sphere in Case of Cationic-Anionic Surfactant Aqueous Mixtures

: The wetting of adsorbed surfactants solids is important for various technological applications in particular for the process of foam ﬂotation. The present work aims at calculating the surface tensions of the three phase interfaces at di ﬀ erent surfactant concentrations using the Girifalco and Good method. For this purpose, the surface tension and contact angle vs. surfactant concentration of the test substances amines and sulfonates and their mixture were measured for liquid–air interface. Calculated surface tension of solid–air interface vs. concentration for C10 amine and mixed systems are close to those for the liquid–air surface, but are slightly lower. In the case of mixed systems, the graph has a speciﬁc structure similar to that of liquid–air surface dependence. In contrast to the solid–air interface results, the solid–liquid surface tension values are signiﬁcantly lower. In case of the mixed surfactant systems, C10amine / C10 sulfonate, a synergetic e ﬀ ect on the surface tension is observed. The speciﬁc behavior of the mixed systems is interpreted with the emergence of aggregates consisting of the anionic and cationic surfactants. It is shown that in the whole area of concentrations complete wetting does not occur. The weakening of the dependence increases in the order C8sulphonate, C10sulphonate, and C12sulphonate. The optimal concentrations for maximal level of hydrophobicity shifts to smaller concentrations in the same order of the added anionic surfactant—C8sulphonate, C10sulphonate, and C12sulphonate.

Refs. [11][12][13] study surfactant mixtures as potential collectors to enhance both flotation recovery and selectivity. Refs. [14][15][16] established specific molar ratios (1:3 and 3:1) between the quantities of the anionic surfactants with regard to the cationic surfactants. They explain the emergence of synergy with the formation of stable two-dimensional hexagonal net at the surfaces. Ref. [17] confirms this finding. These 2-D nets of tight packing are due to matching of chain lengths of the different surfactants, thus affecting the interfacial properties, thus resulting in increased surface viscosity, more stable micelles, better foamability, lower values of the surface tension, contact angles, and bubble radii [18]. The formation of three phase contact is the most important elemental act in a froth flotation process [19]. Investigations involving the stability, rupture, contact angle, and interface tension of three phase contact find a number of industrial applications, such as mining [19], oil [20], food [21], pharmaceutical, cosmetics, and other industries. An important element in the flotation of mineral particles is the rupture of the thin wetting film between the bubble and the mineral particle [19]. The flotation of mineral particles involves collision between the latter and the bubbles accompanied by attachment of the particles to the bubbles. For a particle to become attached to an air bubble, two successive processes must take place successfully: (1) Rupture of the thin film between the particle and the bubble surface to produce a contact point (formation of a three-phase contact (TPC)); and (2) expansion of the three-phase contact line (TPCL) to increase the contact angle and consequently to strengthen the attachment. The interfacial viscoelasticity plays an important role in this phenomenon [22]. The theoretical methods and experimental techniques for wetting contact angles shed light on the surface forces and the different types of interaction, properties of the interfaces, etc. [19,23]. In addition, the measurements of the wetting contact angle at the solid/liquid interface gives one the possibility to have a better understanding about the adsorption and the conformation of the surfactants, and to distinguish between hydrophilic and hydrophobic surfaces. The polymeric surfactant's behavior at the different surfaces (air/liquid, liquid/liquid, and solid/liquid), is decisive for the properties of the dispersed systems (foams, emulsions, suspensions).
A basic problem of surface science is to measure and calculate the properties of the interfaces and the related total interfacial energy, and the free energy. In case of three-phase contact, the surface tension, liquid/vapor, and contact angle can be measured. The surface tensions of the solid/liquid or solid/air interfaces cannot be measured. The problems with the measurement of the surface energy and free energy of solids remain unsolved.
In our calculations below, we use the suggested method from Girifalco and Good [24] that we think is as general as possible in approach to the problem. We discuss it in detail in the relevant section.

Materials
A pure cationic (n-decyl amine) and a series of mixtures of cationic and anionic surfactants at 1:1 molar ratio (n-decyl amine/1-octanesulfonic, n-decyl amine/decanesulfonic, n-decyl amine/1 dodecylsulfonic acid) were used. The n-decylamine with 99% purity was purchased from Sigma Chemical Co. (St. Louise, MO, USA). The sodium salts of 1-octanesulfonic and 1-decanesulfonic acids were purchased from Sigma Aldrich CHEME GmbH (Steinheim, Germany). 1-Dodecanesulfonic acid was purchased from Fluka Chemika (Vaduz, Switzerland). They were used without purification. The pH values of the solutions and the ionic strength were controlled with a Britton-Robinson universal buffer, containing 0.2 mol/L NaOH, 0.12 mol/L CH 3 COOH, 0.12 mol/L H 3 PO 4 , and 0.12 mol/L H 3 BO 4 . The ionic strength was kept constant at 0.2 mol/L. The pH values were monitored with a glass electrode. Deionized (DI) water produced by Mili-Q plus 185 system (Merck, Burlington, VT, USA) was used in the present study.

Surface Tension Measurements
The surface tension γ of the aqueous surfactants solutions was measured by the du Noüy ring method [25,26]. The method involves a platinum ring, and measures the force required to detach the ring from the liquid's surface related to the liquid's surface tension as follows: where l = 2r, r = (r 1 + r 2 )/2, r 1 , r 2 are the inner and outer radii of the platinum ring and F is the maximum force requisite to break the meniscus, as measured by the dynamometer. C f is a correction factor based on the shape of the air/liquid interface, and is normally close to unity 1.0. Two fluid standards of known surface tension values, such as deionized water and ethyl alcohol, are used for an accurate two point calibration of the instrument to determine this factor. The measurements were performed at 20 ± 0.5 • C in a closed chamber, thus preventing the effects of evaporation. Sets of measurements were

Contact Angle Measurements
The receding contact angle was determined from the attachment of silica particles onto a double concave drop in the film holder of Scheludko-Exerowa cell [27]. The radii of the particles R p and the TPC wetting perimeter R tpc were visualized via optical microscope and recorded by camera. The receding contact angle of the particles was calculated by the approximate formula sinθ = R tpc /R p [27] where the gravitational deformation of the meniscus is neglected. The accuracy of the contact angle measurements was ±0.3 • in the range of 1 • -5 • and ±1.0 • in the range of 5 • -40 • .

Calculating Interface Free Energies-Discussion
The Young Equation [28] is the following: where γ s is the surface tension of the solid/air surface, γ sl -the surface tension of the solid/liquid interface, γ l -the surface tension of a measuring liquid/air surface, and θ-the contact angle between the solid and the liquid. Only γ l and θ can be directly measured while the other two quantities cannot be determined directly. The effect of surfactant adsorption on the liquid/solid interface should be considered [29,30]. Nevertheless, many researchers neglect this phenomenon because its effect is small, although bilayer is often formed on that area. Yet, this adsorption is important for the effect of super-spreading as reported by Venzmer et al. [31][32][33]. Therefore, in order to solve Young equation, some additional assumptions regarding the relations between γ s , γ l , and γ sl should be made as well.
As mentioned above, there exist semi-empirical methods for calculating the values of and γ sl , γ s under certain restrictive conditions: • Berthelot made the assumption W sl = √ W ss W ll where W sl is adhesion work and W ss , W ll are cohesion works of the corresponding phases [34].

•
Other methods: Zisman method [51,52]. It is used for determination of so-called critical surface free energy. Neumann method [39,40]. This method consists in suitable transformation of Young Equation and equation of state.

•
Contact angle hysteresis method. It is a resent method for calculating the surface tension of polymeric materials [53,54].
Below we will use the method Girifalco and Good [24] as based on the most general thermodynamic considerations. They modified equation of state by introducing the parameter Φ, characterizing the interfacial interactions, as follows: with Young Equation (2). The Equations (2) and (3) form a system of equations for the unknown γ s and γ l . The most important problem is determining the value of parameter Φ for the system under study. Parameter Φ is defined as: Coatings 2020, 10, 573 where ∆G a ls is the free energy of adhesion for the interface between the phases l and s, ∆G c l is the free energy of cohesion for phase l, and ∆G c s is the free energy of cohesion for phases. For many systems, Φ lies between 0.5 and 1.2 [21]. If the two phases are composed of spherical or nearly spherical molecules, Φ can be represented in the form [24]: Here V l and V s are the molar volumes of the liquid (water) and the substrate. Molar volumes can be formally calculated from the formula: where M is the molar mass and ρ is the density of corresponding phase.

Results and Discussion
Figures 1 and 2 present the results for the surface tension of the cationic surfactant, anionic surfactants, and anionic-cationic surfactant mixtures. One can see that the increase of the concentration of the cationic and the three anionic surfactants cause decrease of the surface tension. It is well-known that surfactants with longer chains are stronger, resulting in lower values of the surface tension of their aqueous solutions and their CMC values. Yet, when cationic and anionic surfactants are mixed they exhibit synergetic effect expressed in lowering the surface tension value (see Figure 1). G  is the free energy of cohesion for phases. For many systems, Ф lies between 0.5 and 1.2 [21]. If the two phases are composed of spherical or nearly spherical molecules, Ф can be represented in the form [24]: Here Vl and Vs are the molar volumes of the liquid (water) and the substrate. Molar volumes can be formally calculated from the formula: where М is the molar mass and ρ is the density of corresponding phase.

Results and Discussion
Figures 1 and 2 present the results for the surface tension of the cationic surfactant, anionic surfactants, and anionic-cationic surfactant mixtures. One can see that the increase of the concentration of the cationic and the three anionic surfactants cause decrease of the surface tension. It is well-known that surfactants with longer chains are stronger, resulting in lower values of the surface tension of their aqueous solutions and their CMC values. Yet, when cationic and anionic surfactants are mixed they exhibit synergetic effect expressed in lowering the surface tension value (see Figure 1).  of complex aggregates of both types of surfactants on the air/water interface. Shah et al. [17] report about such formations of complexes as well. They form because of the electrostatic attraction at long distance and van der Waals attraction at short distance. The higher concentration favors the formation of precipitate aggregates [56]. Overall, the strongest synergistic effect is exhibited by C10 amine/C10 sulfonate, followed by C10 amine/C8 sulfonate and C10 amine/C12 sulfonate (the weakest). To our understanding, the chain compatibility plays role here. [55]. Reprinted from Ref. [55] with permission. Copyright Elsevier 2009. Figure 3 presents the receding contact angle of the particles versus concentration of the cationic, and the mixed cationic/anionic surfactant systems. One can see that in all of the cases θ values increase with the concentration (the particles become more hydrophobic) until reaching a maximum, beyond which θ drops, which means that the particles become more hydrophilic. The dependence for the case of the cationic surfactant alone (C10amine) is the strongest one. Its maximum of θ is at C = 10 −2 mol/L C10 amine. Completely hydrophilic surface is obtained at CMC of C10amine. The presence of sulphonate surfactants weakens the above-mentioned dependences, although their shapes are similar. The weakening of the dependence increases in the order C8sulphonate, C10sulphonate, and C12sulphonate. The optimal concentrations for maximal level of hydrophobicity shifts to smaller concentrations in the same order of the added anionic surfactant-C8sulphonate, C10sulphonate, and C12sulphonate. As seen in Figure 1 all the mixtures exhibit several orders of magnitude lower CMC values as compared the CMC value of the C10 amine. The lower the mixed surfactants, significantly stronger the surface tension compared to the pure systems, but the synergistic effect is strongest when their chain lengths are equal. When the total concentration of the mixtures increases, the surface tension value increases as well, thus reducing their surface activity. This can be explained with the formation of complex aggregates of both types of surfactants on the air/water interface. Shah et al. [17] report about such formations of complexes as well. They form because of the electrostatic attraction at long distance and van der Waals attraction at short distance. The higher concentration favors the formation of precipitate aggregates [56]. Overall, the strongest synergistic effect is exhibited by C10 amine/C10 sulfonate, followed by C10 amine/C8 sulfonate and C10 amine/C12 sulfonate (the weakest). To our understanding, the chain compatibility plays role here. Figure 3 presents the receding contact angle of the particles versus concentration of the cationic, and the mixed cationic/anionic surfactant systems. One can see that in all of the cases θ values increase with the concentration (the particles become more hydrophobic) until reaching a maximum, beyond which θ drops, which means that the particles become more hydrophilic. The dependence for the case of the cationic surfactant alone (C10amine) is the strongest one. Its maximum of θ is at C = 10 −2 mol/L C10 amine. Completely hydrophilic surface is obtained at CMC of C10amine. The presence of sulphonate surfactants weakens the above-mentioned dependences, although their shapes are similar. The weakening of the dependence increases in the order C8sulphonate, C10sulphonate, and C12sulphonate. The optimal concentrations for maximal level of hydrophobicity shifts to smaller concentrations in the same order of the added anionic surfactant-C8sulphonate, C10sulphonate, and C12sulphonate.  Because Ф changes slightly in the interval (0.5-1.2), we neglect the influence of the adsorption layer on the value of Ф. Its value will be calculated with the data for amorphous silicon dioxide (composition of ballotini) and water: MH₂O = 18 g/mol, ρH₂O = 1 g/cm 3 , VH₂O = 18 cm 3 , MSiO₂ = 60.08 g/mol ρSiO₂ = 2.196 g/cm 3 , and VSiO₂ = 27.36 cm 3 . With this values we get Ф = 0.995. In Figure 4 the calculated surface tension values for the solid-air surface are shown. It is noticeable that the values for C10 amine and mixed systems are close to those for the liquid-air surface, but are slightly lower. The reason for this effect should be sought in the mechanism of forming the contact angle in the pendant drop method. The ballotini (the colloidal glass sphere) penetrates the meniscus and forms a balanced three-phase contact. In this, part or the entire adsorption layer is applied to the surface of the glass sphere by the so-called zip effect [57]. In the case of mixed systems, the graph has a structure similar to that of liquid-air surface dependence. Here again, a synergistic effect is observed: the lowest surface tension values for the C10 amine/ In Figure 4 the calculated surface tension values for the solid-air surface are shown. It is noticeable that the values for C10 amine and mixed systems are close to those for the liquid-air surface, but are slightly lower. The reason for this effect should be sought in the mechanism of forming the contact angle in the pendant drop method. The ballotini (the colloidal glass sphere) penetrates the meniscus and forms a balanced three-phase contact. In this, part or the entire adsorption layer is applied to the surface of the glass sphere by the so-called zip effect [57]. In the case of mixed systems, the graph has a structure similar to that of liquid-air surface dependence. Here again, a synergistic effect is observed: the lowest surface tension values for the C10 amine/ C10sulfonate system. There are also qualitatively different dependencies in the case of mixtures and C10 amine. While C10 amine vs. concentration of surface tension has a classical behavior, it is highly specific in mixtures because of the strong Coulomb's interaction. When anionic and cationic surfactants are mixed in equivalent quantities, precipitation aggregates are formed [58][59][60][61]. Yet, ref. [62] reports on the formation of structured multilayer of polyelectrolytes on both the air/liquid and the solid/liquid interface. The surface tension decreases at surfactant concentration below the critical aggregation concentration (C < CAC). The electrostatic attraction between the anionic and the cationic surface active ions starts playing a role at small concentration (see Figures 1 and 4). Figures show that the dependence "surface tension vs. log C" of mixed systems is much more complex.

Figure 2. Measured surface tension versus surfactant concentration for different sulfonates
Conditionally, this dependence can be divided into three areas (see Figure 5). First (I) section, in which the dependence has a conventional character, second (II) in which, as the concentration of the surfactants mixture increases, the surface tension also enhances, and a third section (III), in which the surface tension remains practically constant. Such behavior is relatively unusual, although it has been observed in the case of other cationic-anionic surfactants mixtures [63]. These facts give us the reason to assume that in area (I) there are cationic and anionic monomers, neutral dimmers, and eventually neutral precipitates. In region (II) unilamellar vesicles and micelles, probably of spherical shape, arise. Micelles with spherical shape would most easily fulfill the requirements of electro neutrality as a state with the lowest energy. We should note that the vesicles may not be electrically neutral, which would lead to a gradual diversion from the equimolar surfactant ratio in the bulk phase. As a result, the phase surface adsorbs more monomers of a type whose adsorption is favored primarily by the structure of the charged hydrophilic head [64,65]. Thus, the adsorption layer appears to be electrically charged, and because of repulsion between same-charge ions, the surface tension increases. In area (III) the surface tension remains constant, because with the packed adsorption layer, the increase in the concentration of monomers and dimmers only increases the number of vesicles and micelles. Figure 3. Measure of the contact angle vs. the C10 amine concentration for different systems [55]. Reprinted from Ref. [55] with permission. Copyright Elsevier 2009. Because Ф changes slightly in the interval (0.5-1.2), we neglect the influence of the adsorption layer on the value of Ф. Its value will be calculated with the data for amorphous silicon dioxide (composition of ballotini) and water: MH₂O = 18 g/mol, ρH₂O = 1 g/cm 3 , VH₂O = 18 cm 3 , MSiO₂ = 60.08 g/mol ρSiO₂ = 2.196 g/cm 3 , and VSiO₂ = 27.36 cm 3 . With this values we get Ф = 0.995. In Figure 4 the calculated surface tension values for the solid-air surface are shown. It is noticeable that the values for C10 amine and mixed systems are close to those for the liquid-air surface, but are slightly lower. The reason for this effect should be sought in the mechanism of forming the contact angle in the pendant drop method. The ballotini (the colloidal glass sphere) penetrates the meniscus and forms a balanced three-phase contact. In this, part or the entire adsorption layer is applied to the surface of the glass sphere by the so-called zip effect [57]. In the case of mixed systems, the graph has a structure similar to that of liquid-air surface dependence. Here again, a synergistic effect is observed: the lowest surface tension values for the C10 amine/ C10sulfonate system. There are also qualitatively different dependencies in the case of mixtures and C10 amine. While C10 amine vs. concentration of surface tension has a classical behavior, it is highly specific in mixtures because of the strong Coulomb's interaction. When anionic and cationic surfactants are mixed in equivalent quantities, precipitation aggregates are formed [58][59][60][61]. Yet, ref.
[62] reports on the formation of structured multilayer of polyelectrolytes on both the air/liquid and the solid/liquid interface. The surface tension decreases at surfactant concentration below the critical aggregation concentration (C < CAC). The electrostatic attraction between the anionic and the cationic surface active ions starts playing a role at small concentration (see Figures 1 and 4). Figures show that the dependence "surface tension vs. log C" of mixed systems is much more complex. Conditionally, this dependence can be divided into three areas (see Figure 5). First (I) section, in which the dependence has a conventional character, second (II) in which, as the concentration of the surfactants mixture increases, the surface tension also enhances, and a third section (III), in which the surface tension remains practically constant. Such behavior is relatively unusual, although it has been observed in the case of other cationic-anionic surfactants mixtures [63]. These facts give us the reason to assume that in area (I) there are cationic and anionic monomers, neutral dimmers, and eventually neutral precipitates. In region (II) unilamellar vesicles and micelles, probably of spherical shape, arise. Micelles with spherical shape would most easily fulfill the requirements of electro neutrality as a state with the lowest energy. We should note that the vesicles may not be electrically neutral, which would lead to a gradual diversion from the equimolar surfactant ratio in the bulk phase. As a result, the phase surface adsorbs more monomers of a type whose adsorption is favored primarily by the structure of the charged hydrophilic head [64,65]. Thus, the adsorption layer In contrast to the solid-air interface results, the solid-liquid surface tension values are significantly lower as shown in Figure 6. The drop after C = 10 -3 mol/L of the solid-liquid interface for c10 amine represents an increase in hydrophilicity (see Figures 3 and 6) and can be explained by the formation of a bi-layer (see Figure  7). The effect of line tension is neglected here. Unlike the C10 amine, mixtures do not exhibit such behavior as they are adsorbed as neutral complexes. It is of clear interest to compare the results obtained for the different phase ranges for the C10 amine and mixtures of equimolar amounts of cation and anion active surfactants. Figure 8 shows the results for the C10 amine for the three phase ranges. Clearly, as already noted, the behavior of the solid-air interface is similar to that of liquid/air as the values are somewhat lower. At the same time, the values of the solid-liquid phase boundary are significantly lower. The drop after C = 10 -3 mol/L of the solid-liquid interface for c10 amine represents an increase in hydrophilicity (see Figures 3 and 6) and can be explained by the formation of a bi-layer (see Figure 7). The effect of line tension is neglected here. Unlike the C10 amine, mixtures do not exhibit such behavior as they are adsorbed as neutral complexes. The drop after C = 10 -3 mol/L of the solid-liquid interface for c10 amine represents an increase in hydrophilicity (see Figures 3 and 6) and can be explained by the formation of a bi-layer (see Figure  7). The effect of line tension is neglected here. Unlike the C10 amine, mixtures do not exhibit such behavior as they are adsorbed as neutral complexes. It is of clear interest to compare the results obtained for the different phase ranges for the C10 amine and mixtures of equimolar amounts of cation and anion active surfactants. Figure 8 shows the results for the C10 amine for the three phase ranges. Clearly, as already noted, the behavior of the solid-air interface is similar to that of liquid/air as the values are somewhat lower. At the same time, the values of the solid-liquid phase boundary are significantly lower. It is of clear interest to compare the results obtained for the different phase ranges for the C10 amine and mixtures of equimolar amounts of cation and anion active surfactants. Figure 8 shows the results for the C10 amine for the three phase ranges. Clearly, as already noted, the behavior of the solid-air interface is similar to that of liquid/air as the values are somewhat lower. At the same time, the values of the solid-liquid phase boundary are significantly lower. A similar nature of dependence ( Figure 9) is also observed in mixtures of cationic and anionic surfactants: near surface tension values for the liquid/air and solid-air phase interfaces and significantly lower values of the solid-liquid interface. For the sake of example, the commented dependence is shown for the C10 amine/C10sulfonate mixture in Figure 9. The minima in Figure 9 are due to the formation of complexes of both surfactants with larger surface activity. A precipitation starts occurring at higher concentrations, thus embracing the latter in larger more hydrophilic precipitates. The results obtained from the free energy calculations of the phase interfaces have reasonable values. Like any semi-empire model, this puts the question of the accuracy of the picture it describes. As noted above, the adsorption layer was not taken into account. This non-homogeneous structure, according to ref. [24], can approximately treat the complex surfactant molecules as structures composed of the molecular segments. Hence, if phase A is buildup of molecules with segments of types m and n (type m being in the majority) and phase B is buildup of molecules with segments of a single type, b, then formula for Ф takes the form as ref. [24]: A similar nature of dependence ( Figure 9) is also observed in mixtures of cationic and anionic surfactants: near surface tension values for the liquid/air and solid-air phase interfaces and significantly lower values of the solid-liquid interface. For the sake of example, the commented dependence is shown for the C10 amine/C10sulfonate mixture in Figure 9. A similar nature of dependence ( Figure 9) is also observed in mixtures of cationic and anionic surfactants: near surface tension values for the liquid/air and solid-air phase interfaces and significantly lower values of the solid-liquid interface. For the sake of example, the commented dependence is shown for the C10 amine/C10sulfonate mixture in Figure 9. The minima in Figure 9 are due to the formation of complexes of both surfactants with larger surface activity. A precipitation starts occurring at higher concentrations, thus embracing the latter in larger more hydrophilic precipitates. The results obtained from the free energy calculations of the phase interfaces have reasonable values. Like any semi-empire model, this puts the question of the accuracy of the picture it describes. As noted above, the adsorption layer was not taken into account. This non-homogeneous structure, according to ref. [24], can approximately treat the complex surfactant molecules as structures composed of the molecular segments. Hence, if phase A is buildup of molecules with segments of types m and n (type m being in the majority) and phase B is buildup of molecules with segments of a single type, b, then formula for Ф takes the form as ref. [24]: The minima in Figure 9 are due to the formation of complexes of both surfactants with larger surface activity. A precipitation starts occurring at higher concentrations, thus embracing the latter in larger more hydrophilic precipitates. The results obtained from the free energy calculations of the phase interfaces have reasonable values. Like any semi-empire model, this puts the question of the accuracy of the picture it describes. As noted above, the adsorption layer was not taken into account. This non-homogeneous structure, according to ref. [24], can approximately treat the complex surfactant molecules as structures composed of the molecular segments. Hence, if phase A is buildup of molecules with segments of types m and n (type m being in the majority) and phase B is buildup of molecules with segments of a single type, b, then formula for Φ takes the form as ref. [24]: provides a correction term of about 5% of unity in most cases. The most important result here is the replacement of the molar volumes in equation with the group-molar volumes [24]. Therefore, in the opinion of the authors Girifalco and Good, the possible mistakenness of the contribution of the adsorption layer in the parameter Φ does not exceed 5% [24]. For this reason, the calculations made here lead to reasonable results describing with sufficient precision the free interface energy of the system under consideration. The wetting of adsorbed surfactants solids is important for various technological applications and in particular for the process of foam flotation. The presence of surfactants in water is often accompanied by the requirement that the solid surface be well wetted. Since water has a high surface tension (72.8 mN/m), it does not spontaneously spread over solids that have surface free energy of less than 72.8 mN/m [65]. The work per unit area for spreading the aqueous solution at the solid surface can be defined.
If we combine the spreading parameter with the Young relation we obtain the Young-Dupré Equation: W s = γ l (cos θ − 1) When Ws > 0, we have complete wetting. If Ws < 0, we have partial wetting. Figure 10 shows the dependence of Ws vs. concentration. This shape is not a surprise, because according to the Young-Dupré Equation, the dependence of concentration follows that of the contact angle vs. concentration. It shows that in the whole area of concentrations the value of Ws < 0. This result indicates that complete wetting does not occur. The worst are the values for C10 amine while the increase in the alkyl chain length of the sulfonates increases (algebraically) the values to positive. The best wetting is in the absence of adsorption layer (C = 0) or in a dense bi-layer. This is a natural result because the molecules of the second adsorption layer are directed to the water with their polar or charged heads.
provides a correction term of about 5% of unity in most cases. The most important result here is the replacement of the molar volumes in equation with the group-molar volumes [24]. Therefore, in the opinion of the authors Girifalco and Good, the possible mistakenness of the contribution of the adsorption layer in the parameter Ф does not exceed 5% [24]. For this reason, the calculations made here lead to reasonable results describing with sufficient precision the free interface energy of the system under consideration. The wetting of adsorbed surfactants solids is important for various technological applications and in particular for the process of foam flotation. The presence of surfactants in water is often accompanied by the requirement that the solid surface be well wetted. Since water has a high surface tension (72.8 mN/m), it does not spontaneously spread over solids that have surface free energy of less than 72.8 mN/m [65]. The work per unit area for spreading the aqueous solution at the solid surface can be defined.
When Ws > 0, we have complete wetting. If Ws < 0, we have partial wetting.  Figure 10 shows the dependence of Ws vs. concentration. This shape is not a surprise, because according to the Young-Dupré Equation, the dependence of concentration follows that of the contact angle vs. concentration. It shows that in the whole area of concentrations the value of Ws < 0. This result indicates that complete wetting does not occur. The worst are the values for C10 amine while the increase in the alkyl chain length of the sulfonates increases (algebraically) the values to positive. The best wetting is in the absence of adsorption layer (C = 0) or in a dense bi-layer. This is a natural

Conclusions
The main objective of the present work was to calculate the surface tensions of the three phase interfaces as a function of the surfactant concentration by the Girifalco and Good method [24]. For this purpose, the surface tension and contact angle of the test substances amines and sulfonates were measured. Based on the results obtained, the following conclusions can be drawn: • The surface tension values of aqueous solutions of cationic surfactant, anionic surfactants, and anionic/cationic surfactant mixtures are measured. The surface tension decreases upon the increase of the concentration of each one of the surfactants. The length of the hydrocarbon chain determined the hydrophobicity of the surfactant molecule. When cationic and anionic surfactants are mixed a synergistic effect of additional lowering of the surface tension emerges. The effect is the strongest when the lengths of the hydrocarbon tails of the cationic and anionic surfactants are equal. This is explained with the formation of precipitation complexes, whose hydrophobicity is maximal when the hydrocarbon chains of the two types of surfactants are equal-C10amine/C10sulphonate form the most surface-active complexes, followed by C10amine/C8sulphonate and finally C10amine/C12sulphopnate; • The silica particles increase their hydrophobicity with the increase of the surfactant concentration in both cases-the cationic surfactant only and cationic/anionic surfactant mixtures until reaching maximum, after which the silica particles becomes more hydrophilic with the increase of the surfactant concentration. This dependence is the strongest in the case of cationic surfactant and weakens with the mixture in the order of C10amine/C8sulphonate, C10amine/C10sulphonate, and C10amine/C12sulphopnate. The optimal concentration of the maxima of θ for each one of the cases shifts toward the smaller concentrations in the order of C10amine/C8sulphonate, C10amine/C10sulphonate, and C10amine/C12sulphopnate, and their maximal value of θ decreases in the same order; • Surface tensions of solid-air and solid-liquid phase surfaces are calculated for different surfactant concentrations by the method proposed by L.A. Girifalco and R.J. Good [24]. In the opinion of the authors Girifalco and Good, the possible mistakenness of the contribution of the adsorption layer in the parameter Φ does not exceed 5%; • Calculated surface tension of solid-air interface vs. concentration for C10 amine and mixed systems are close to those for the liquid-air surface, but are slightly lower. In the case of mixed systems, the graph has a specific structure similar to that of liquid-air surface dependence. Here again a synergistic effect is observed-the lowest surface tension values for the C10 amine/C10sulfonate system. At the same time, the values of the solid-liquid phase interface are significantly lower; • It is shown that in the whole area of concentrations the value of the work for spreading is negative. This result indicates that complete wetting does not occur. The worst are the values for C10 amine while the increase in the alkyl chain length of the sulfonates increases (algebraically) the values to positive. The best wetting is in the absence of adsorption layer (C = 0) or in a dense bi-layer.
The behavior of the contact angle and the phase interfacial energy in the case of mixtures is determined primarily by the strong coulombic interaction between the opposite ions. In the case of ionic surfactant with a unique ionic charge (for example C10 amine), behavior is determined primarily by the coulomb interaction (glass-liquid) or hydrophobic interaction of the surface of air. The properties of the solid air phase surface are determined by the surfactant adsorption according to the so-called zip effect.

Conflicts of Interest:
The authors declare no conflict of interest.