Frumkin Adsorption Isotherm Research Articles

A negative value of the interaction parameter for over-potentially deposited hydrogen at Pt, Ir, and Pt–Ir alloy electrode interfaces

The phase-shift method, i.e., the electrochemical impedance spectroscopy (EIS) technique for studying the linear relationship between the behavior (− φ vs. E) of the phase-shift (90° ⩾ − φ ⩾ 0°) for the optimum intermediate frequency and that ( θ vs. E) of the fractional surface coverage (0 ⩽ θ ⩽ 1), has been proposed and verified to determine the adsorption isotherms at the noble metal (alloy)|aqueous solution interfaces. At the Pt–Ir (90:10 wt%) alloy|0.1 M KOH aqueous solution interface, the Frumkin adsorption isotherm for over-potentially deposited hydrogen (OPD H) and related electrode kinetic and thermodynamic parameters are determined using the phase-shift method. The standard free energies ( Δ G θ 0 ), equilibrium constants ( K), and interaction parameters ( g) for OPD H and hydroxide (OH) at the noble metal (alloy)|aqueous solution interfaces are summarized. The negative values of g for OPD H at the Pt, Ir, and Pt–Ir alloy|aqueous solution interfaces imply a lateral attraction interaction between the adsorbed OPD H species, which is probably a unique feature of the Frumkin adsorption isotherms at these interfaces.

Some thiazole derivatives as corrosion inhibitors for carbon steel in acidic medium

The effect of some prepared compounds, namely thiazole derivatives on the corrosion of C-steel in 2 M HCl solutions has been studied using the weight loss and electrochemical polarization methods. The results showed that the inhibition efficiency of the investigated compounds was found to depend on the concentration and the nature of the inhibitor. The presence of the substituted donating group (–OCH 3) plays an important role in the inhibition percentage of these investigated compounds. The effect of temperature on the corrosion behavior of C-steel in 2 M HCl without and with the inhibitors was studied in the temperature range from 30 to 50°C. The activation and thermodynamic parameters E a ⁎, ΔH ⁎ and ΔS ⁎ were calculated. The inhibitors were adsorbed on the C-steel surface according to Frumkin's adsorption isotherm. The synergistic effect of some cations (Cu 2+, Ni 2+, Co 2+ and Ca 2+) on the corrosion inhibition of C-steel in 2 M HCl in the presence and absence of the investigated compounds was also studied by the weight loss method.

Adsorption phenomena on a mercury electrode in aqueous cryptand[2.2.2] solutions by the background of tetraalkylammonium salts

The electrochemical behavior of cryptand[2.2.2] (Cry) is studied on a mercury electrode in aqueous solutions of tetraalkylammonium tetrafluoroborates (Me4N+, Et4N+, and Bu4N+). Cryptand [2.2.2] is shown to exhibit high surface activity in Me4 NBF4 nd Et4NBF4 solutions. Based on the model of two parallel capacitors supplemented by the Frumkin adsorption isotherm, the adsorption parameters of Cry by the background of Me4BNF4 were calculated using the regression analysis methods. The calculated dependences of the differential capacitance on the potential adequately agree with experimental curves. The adsorption characteristics of Cry in the studied solutions are compared with those in MgSO4 solutions. By the background of Bu4NBF4, Cry molecules and Bu4N+ cations exhibit very close surface activity and form a mixed adsorption layer.

Adsorption and inhibitive properties of ethanol extracts of Musa sapientum peels as a green corrosion inhibitor for mild steel in H2SO4

The inhibition of the corrosion of mild steel by ethanol extract of Musa sapientumpeels in H2SO4 has been studied using gasometric and thermometric methods. The results of the study reveals that the different concentrations of ethanol extract of M. sapientum peels inhibit mild steel corrosion. Inhibition efficiency of the extract is found to vary with concentration, temperature, period of immersion and pH. Values of activation energy of the inhibited corrosion reaction of mild steel are greater than the value obtained for the blank. Thermodynamic consideration reveals that adsorption of M. sapientum peels extract on mild steel surface is spontaneous and occurred according to Langmuir and Frumkin adsorption isotherms. Physical adsorption mechanism has also been proposed for the adsorption of the inhibitor. Key words: Corrosion inhibition, mild steel, Inhibition, adsorption, Musa sapientum, peels.

The Inhibition of C-steel Corrosion in H3PO4Solution by Some Furfural Hydrazone Derivatives

The effect of some furfural hydrazone derivatives (I -V) as corrosion inhibitors for C-steel in 3M H 3 PO 4 solution in which M indicates mol/l has been studied using weight-loss and galvanostatic polarization techniques. A significant decrease in the corrosion rate of C-steel was observed in the presence of the investigated compounds. This study revealed that, the inhibition efficiency increases with increasing the inhibitor concentration, and the addition of iodide and thiocynate ions enhances it to a considerable extent. The effect of temperature on the corrosion behavior of C-steel was studied in the range from 30-60°C with and without 5×10 -6 M investigated compounds using weight-loss method. Apparent activation corrosion energy (E a *) and other thermodynamic parameters for the corrosion process were calculated and discussed. The polarization measurements show that the investigated compounds act as mixed-type inhibitors, but the cathode is more polarized when an external current was applied. The adsorption of these compounds on the surface of C-steel in 3 M H 3 PO 4 obeys Frumkin's adsorption isotherm. The mechanism of inhibition was discussed in the light of the chemical structure of the investigated compounds.

Model description of specific adsorption of two anions on electrodes

Simultaneous specific adsorption of two anions on the electrode surface is considered. Calculations involve the use of the Frumkin adsorption isotherm for each anion and a model that describes the electric double layer structure based on one or two Helmholtz planes. Dependences of the ion adsorption on both the electrode potential and the solution composition are calculated for different electrode charges, ion sizes, and capacitance characteristics of the electric double layer.

Determination of adsorption isotherms of hydrogen and hydroxide at Pt–Ir alloy electrode interfaces using the phase-shift method and correlation constants

At the Pt–Ir (Pt:Ir; 70:30 wt%) alloy/0.5 M H 2 SO 4 aqueous solution interface, the Frumkin and Temkin adsorption isotherms ( θ H vs. E ), equilibrium constants ( K H = 3.1 × 10 - 5 exp ( 2.5 θ H ) mol - 1 for the Frumkin and K H = 3.1 × 10 - 4 exp ( - 2.1 θ H ) mol - 1 for the Temkin adsorption isotherm), interaction parameters ( g = - 2.5 for the Frumkin and g = 2.1 for the Temkin adsorption isotherm), rates of change of the standard free energy of H with θ H ( r = - 6.2 kJ mol - 1 for g = - 2.5 and r = 5.2 kJ mol - 1 for g = 2.1 ), and standard free energies ( 25.7 ⩾ Δ G θ 0 ⩾ 19.5 kJ mol - 1 for K H = 3.1 × 10 - 5 exp ( 2.5 θ H ) mol - 1 and 0 ⩽ θ H ⩽ 1 and 21.1 < Δ G θ 0 < 24.2 kJ mol - 1 for K H = 3.1 × 10 - 4 exp ( - 2.1 θ H ) mol - 1 and 0.2 < θ H < 0.8 ) of H are determined using the phase-shift method and correlation constants. The negative value of g implies a lateral attraction interaction between the adsorbed H. The capability of two-dimensional phase formation is experimentally presented. At the Pt–Ir (Pt:Ir; 70:30 wt%) alloy/0.1 M KOH aqueous solution interface, the Frumkin and Temkin adsorption isotherms ( θ OH vs. E ), equilibrium constants ( K OH = 4.7 × 10 - 10 exp ( - 1.8 θ OH ) mol - 1 for the Frumkin and K OH = 4.7 × 10 - 9 exp ( - 6.4 θ OH ) mol - 1 for the Temkin adsorption isotherm), interaction parameters ( g = 1.8 for the Frumkin and g = 6.4 for the Temkin adsorption isotherm), rates of change of the standard free energy of OH with θ OH ( r = 4.5 kJ mol - 1 for g = 1.8 and r = 15.9 kJ mol - 1 for g = 6.4 ), and standard free energies ( 53.2 ⩽ Δ G θ 0 ⩽ 57.7 kJ mol - 1 for K OH = 4.7 × 10 - 10 exp ( - 1.8 θ OH ) mol - 1 and 0 ⩽ θ OH ⩽ 1 and 50.7 < Δ G θ 0 < 60.2 kJ mol - 1 for K OH = 4.7 × 10 - 9 exp ( - 6.4 θ OH ) mol - 1 and 0.2 < θ OH < 0.8 ) of OH are also determined using the phase-shift method and correlation constants. At the intermediate values of θ H , OH , i.e., 0.2 < θ H , OH < 0.8 , the Temkin adsorption isotherms correlating with the Frumkin adsorption isotherms, and vice versa, are readily determined using the correlation constants. The phase-shift method and correlation constants are accurate and reliable techniques to determine the Langmuir, Frumkin, and Temkin adsorption isotherms and related electrode kinetic and thermodynamic parameters.

New understanding of the nature of OH adsorption on Pt(1 1 1) electrodes

The adsorption of hydroxyl on Pt(1 1 1) single crystal electrodes from aqueous acidic solutions is carefully reinvestigated. The effect of small additions (10 −8–10 −5 M) of chloride and bisulphate anions on the OH adsorption region in perchloric acid solution has been studied. Two regions can be differentiated in the voltammetric profile, that behave differently after the addition of the foreign anion. The initial broad adsorption process is unaffected until the highest concentration is attained. However, the sharper peak at higher potentials is affected even at the lower anion concentration. Since mass transport limitations allow to discard the anion adsorption as the main process giving this peak, we propose that the two processes are due to the dissociative adsorption of two different kinds of water, that are affected by the anion in a different way. From this idea, a new model, based on the Frumkin adsorption isotherm, is proposed, which gives an excellent fit of the experimental results.

Determination of Adsorption Isotherms of Hydroxide ata Platinum Electrode Interface Using the Phase-Shift Method and Correlation Constants

The phase-shift method and correlation constants, i.e., the electrochemical impedance spectroscopy (EIS) techniques for studying linear relationships between the behaviors (<TEX>${\varphi}\;vs.\;E$</TEX>) of the phase shift (<TEX>$0^{\circ}{\leq}-{\varphi}{\leq}90^{\circ}$</TEX>) for the optimum intermediate frequency and those (<TEX>${\theta}\;vs.\;E$</TEX>) of the fractional surface coverage (<TEX>$1{\geq}{\theta}{\geq}0$</TEX>), have been proposed and verified to determine the Langmuir, Frumkin, and Temkin adsorption isotherms (<TEX>${\theta}\;vs.\;E$</TEX>) of H for the cathodic <TEX>$H_2$</TEX> evolution reaction (HER) at noble and transition-metal/aqueous solution interfaces. At the Pt/0.1 MKOH aqueous solution interface, the Langmuir, Frumkin, and Temkin adsorption isotherms (<TEX>${\theta}\;vs.\;E$</TEX>), equilibrium constants (<TEX>$K=5.6{\times}10^{-10}\;mol^{-1}\;at\;0{\leq}{\theta}<0.81$</TEX>, <TEX>$K=5.6{\times}10^{-9}{\exp}(-4.6{\theta})\;mol^{-1}\;at\;0.2<{\theta}<0.8$</TEX>, and <TEX>$K=5.6{\times}10^{-10}{\exp}(-12{\theta})\;mol^{-1}\;at\;0.919<{\theta}{\leq}1$</TEX>, interaction parameters (g = 4.6 for the Temkin and g = 12 for the Frumkin adsorption isotherm), rates of change of the standard free energy (<TEX>$r=11.4\;kJ\;mol^{-1}$</TEX> for g=4.6 and <TEX>$r=29.8\;kJ\;mol^{-1}$</TEX> for g=12), and standard free energies (<TEX>${\Delta}G_{ads}^0=52.8\;kJ\;mol^{-1}\;at\;0{\leq}{\theta}<0.81,\;49.4<{\Delta}G_{\theta}^0<56.2\;kJ\;mol^{-1}\;at\;0.2<{\theta}<0.8$</TEX> and <TEX>$80.1<{\Delta}_{\theta}^0{\leq}82.5\;kJ\;mol^{-1}\;at\;0.919<{\theta}{\leq}1$</TEX>) of OH for the anodic <TEX>$O_2$</TEX> evolution reaction (OER) are also determined using the phase-shift method and correlation constants. The adsorption of OH transits from the Langmuir to the Frumkin adsorption isotherm (<TEX>${\theta}\;vs.E$</TEX>), and vice versa, depending on the electrode potential (E) or the fractional surface coverage (<TEX>${\theta}$</TEX>). At the intermediate values of <TEX>${\theta}$</TEX>, i.e., <TEX>$0.2<{\theta}<0.8$</TEX>, the Temkin adsorption isotherm (<TEX>${\theta}\;vs.\;E$</TEX>) correlating with the Langmuir or the Frumkin adsorption isotherm (<TEX>${\theta}\;vs.\;E$</TEX>), and vice versa, is readily determined using the correlation constants. The phase-shift method and correlation constants are accurate and reliable techniques to determine the adsorption isotherms and related electrode kinetic and thermodynamic parameters. They are useful and effective ways to study the adsorptions of intermediates (H, OH) for the sequential reactions (HER, OER) at the interfaces.

Surfactant effects on the Rayleigh instability in capillary tubes – non-ideal systems

In a previous work, the instability of a liquid film deposited on the inner walls of a capillary under the presence of insoluble surfactant was analyzed; for that purpose the surface tension was related to the interfacial concentration of surfactant by a linear equation. In general, that assumption is valid when just trace amounts of surfactant are present. The present work extends previous analysis by considering a non-linear surface equation of state derived from the Frumkin adsorption isotherm. This equation of state account not only for the existing quantities of surfactant but also for non-ideal interactions between adsorbed molecules. Except for the equation of state, both the model and the numerical technique employed do not differ from those used in the preceding work. The new predictions here presented show that a linear surface equation of state gives reasonable results for strong surfactants. However, the action of weaker surfactants strongly depends on other parameters: the initial concentration and the type and strength of interaction between adsorbed molecules. Thus, the use of a linear equation of state in these circumstances might give erroneous results.

Self-Aggregation of Cationic Surfactants onto Oxidized Cellulose Fibers and Coadsorption of Organic Compounds

In this work, the adsorption of cationic surfactant and organic solutes on oxidized cellulose fibers bearing different amounts of carboxylic moieties was investigated. The increase in the amount of -COOH groups on cellulose fibers by TEMPO oxidation induced a general rise in surfactant adsorption. For all tested conditions, that is, cellulose oxidation level and surfactant alkyl chain length (C12 and C16), adsorption isotherms displayed a typical three-region shape with inversion of the substrate zeta-potential which was interpreted as reflecting surfactant adsorption and aggregation (admicelles and hemimicelles) on cellulose fibers. The addition of organic solutes in surfactant/cellulose systems induced a decrease in surfactant cac on the cellulose surface thus favoring surfactant aggregation and the formation of mixed surfactant/solute assemblies. Adsorption isotherms of organic solutes on cellulose in surfactant/cellulose/solute systems showed that solute adsorption is strictly correlated to (i) the surfactant concentration, solute adsorption increases up to the surfactant cmc, where solute partitioning between the cellulose surface and free micelles causes a drop in adsorption, and to (ii) solute solubility and functional groups. The specific shape of solutes adsorption isotherms at a fixed surfactant concentration was interpreted using a Frumkin adsorption isotherm, thus suggesting that solute uptake on cellulose fibers is a coadsorption and not a partitioning process. Results presented in this study were compared with those obtained in a previous work investigating solute adsorption in anionic surfactant/cationized cellulose systems to better understand the role of surfactant/solute interactions in the coadsorption process.

Inhibition of cobalt active dissolution by benzotriazole in slightly alkaline bicarbonate aqueous media

The efficiency of benzotriazole as inhibiting agent for the corrosion of cobalt was probed at pH ranging from 8.3 to 10.2 in a sodium bicarbonate solution, chosen to simulate mild natural environments. From electrochemical, Raman spectroscopy, atomic force microscopy and ellipsometry experiments, we have demonstrated that benzotriazole markedly affects the electrodissolution reactions, which become modeled by the formation of a [Co(II)(BTA) 2·H 2O] n film according to two different mechanisms. Surface-enhanced Raman spectroscopy has shown that the polarization of a cobalt electrode at cathodic potentials with respect to its potential of zero charge allows a mechanism of specific adsorption of the neutral form of benzotriazole to take place through a suspected metal-to-molecule electron transfer and which follows Frumkin's adsorption isotherms. At the onset of the anodic dissolution, some experimental evidence suggests that these adsorbed neutral benzotriazole molecules deprotonate to yield a very thin [Co(II)(BTA) 2·H 2O] n polymer-like and water-insoluble protective film, responsible for the inhibition of active dissolution processes occurring at slightly more anodic potentials. In the anodic dissolution region, deprotonated benzotriazole species present in the bulk solution favors the formation of a multilayered [Co(II)(BTA) 2·H 2O] n film, which also contributes to the inhibition of any further cobalt dissolution usually observed at higher electrode potentials.

Corrosion inhibition of iron by amphoteric surfactants in hydrochloric acid solutions

In the present work the corrosion inhibition of iron in 2 M HCl solution by amphoteric surfactants was studied. The techniques of measurements were (i) weight loss, (ii) linear polarization and (iii) electrochemical impedance spectroscopy. The investigated surfactants have the structure: where, R = –C14H29(I), –C15H31(II), –C16H33(III), –C17H35(IV), –C18H37(V). These surfactants have a high inhibitory effect on the corrosion of iron in HCl solution. The inhibiting effect of these surfactants may take place through the blocking effect resulting from their adsorption on the metallic surface and hydrophobic effect. The inhibition efficiency increases according to the order: I < II < III < IV < V. This is due to the increase of the length of the alkyl group (–R) in the surfactants. The adsorption of these surfactants via their adsorption centers on the metallic surface obeyed the Frumkin adsorption isotherm. The presence of these surfactants in the corrosive solution increases the activation energy of the corrosion process with that order of inhibition efficiency.

Effect of Deposition Time on the Orientation of [Ru(bpy)2Qbpy]2+ Adsorbed on Platinum

The mechanism for the formation of monolayers of [Ru(bpy)2 Qbpy]2+ on platinum microelectrodes has been investigated by monitoring the time evolution of the surface coverage using in-situ voltammetry; bpy is 2,2‘-bipyridyl and Qbpy is 2,2‘:4,4‘ ‘:4‘4‘ ‘-quarterpyridyl. The monolayers were formed in 2:1 H2O:DMF solution containing between 0.01 and 150 μM of the complex. The dependence of the surface coverage on the bulk concentration is described by the Frumkin adsorption isotherm in which the free energy of adsorption, ΔGads, is −33.8 ± 1.2 kJ mol-1 and the interaction parameter is 1.6 indicating destabilizing, surface coverage dependent, lateral interactions. Monolayer formation occurs on the 10−20 h time scale and is controlled by the dynamics of adsorption rather than bulk diffusion. The first-order rate constant for adsorption is 5.1 ± 0.2 × 10-6 s-1. Significantly, the heterogeneous electron-transfer rate constant measured at an overpotential of 50 mV, k50mV, is independent of surface coverage but in...

Some beta-Aminoketone Derivatives as Corrosion Inhibitors for Nickel in Hydrochloric Acid Solution

The use of some β-aminoketone derivatives as corrosion inhibitors for nickel in 2 M acid chloride solution was investigated by gravimetric and galvanostatic polarization techniques. It was found that the investigated compounds behave as inhibitors for nickel dissolution and their inhibition efficiencies increase by the addition of iodide ions. Also, the adsorption isotherms were studied and it was found to follow the Frumkin's adsorption isotherm. By increasing the temperature of the corrosive medium containing different concentrations of the tested inhibitors, the corrosion rate of nickel increases, indicating that these inhibitors are physically adsorbed on the metal surface. Finally, some quantum chemical quantities such as HOMO, LUMO and dipole moment of the inhibitors were calculated and correlated with the rate of corrosion.

Effect of halide ions on the corrosion inhibition of aluminium in alkaline medium using polyvinyl alcohol

AbstractThe corrosion inhibition of aluminum in NaOH in the presence of polyvinyl alcohol (PVA) at 30 and 40°C and the effect of addition of halides (KCl, KBr, KI) were studied using weight loss and hydrogen evolution methods. Results obtained showed that PVA acts as a corrosion inhibitor in the alkaline environment. The inhibition efficiency increased with increase in concentration of PVA and synergistically increased on the addition of the halides but however decreased with rise in temperature. The phenomenon of physical adsorption is proposed from the obtained Ea and Qads values. PVA was found to obey Freundlich and Frumkin adsorption isotherms. The increase in inhibition efficiency I (%), surface coverage θ values as well as synergistic parameter, SI were found to be in the order I− &gt; Br− &gt; Cl−, which clearly indicates that the radii and electronegativity of the halides play a significant role in the adsorption process. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 103: 2810–2816, 2007

Pyrazolone derivatives as corrosion inhibitors for C-steel in hydrochloric acid solution

4-phenylazo-3-methyl-2-pyrazolon-5-one and three of its derivatives have been investigated as corrosion inhibitors for C-steel in 2 M hydrochloric acid solution using weight-loss and galvanostatic polarization techniques. The efficiency of the inhibitors increases with the increase in the inhibitor concentration but decreases with a rise in temperature. The conjoint effect of the pyrozolone derivatives and KBr, KSCN and KI has also been studied. The apparent activation energy ( E a *) and other thermodynamic parameters for the corrosion process have also been calculated. The galvanostatic polarization data indicated that the inhibitors were of mixed-type, but the cathode is more polarized than the anode. The slopes of the cathodic and anodic Tafel lines (β c and β a ) are approximately constant and independent of the inhibitor concentration. The adsorption of these compounds on C-steel surface has been found to obey the Frumkin's adsorption isotherm. The mechanism of inhibition was discussed in the light of the chemical structure of the undertaken inhibitors.

Adsorption of kryptates of two-charged cations on a mercury electrode

The adsorption of complexes formed by cations of calcium, strontium, and barium with a macrobiocyclic ligand (Kryptofix® 222 of composition C18H36N2O6) is studied as a function of its concentration in 0.1 M solutions of corresponding chlorides by a method of measuring the differential capacitance on a stationary mercury drop. The adsorption parameters of the kryptates studied are determined by a method of regression analysis on the basis of the model of two parallel capacitors supplemented by the Frumkin adsorption isotherm. The differential-capacitance curves, calculated on the basis of these parameters, are compared with experimental data. The determined values of adsorption parameters for kryptates of single-and two-charged cations made it possible to confirm the earlier assumption about a deep mutual relation between the behavior of complexes under study and their structure.

Some Aromatic Hydrazone Derivatives as Inhibitors for the Corrosion of C‐Steel in Phosphoric Acid Solution

The effect of furfural benzoylhydrazone and its derivatives (I-VII) as corrosion inhibitors for C-steel in 1M phosphoric acid solution has been studied by weight-loss and galvanostatic polarization techniques. A significant decrease in the corrosion rate of C-steel was observed in the presence of the investigated inhibitors. This study revealed that, the inhibition efficiency increases with increasing the inhibitor concentration, and the addition of iodide ions enhances it to a considerable extent. The effect of temperature on the inhibition efficiency of these compounds was studied using weight-loss method. Activation energy (E(a)*) and other thermodynamic parameters for the corrosion process were calculated and discussed. The galvanostatic polarization data indicated that, the inhibitors were of mixed-type, but the cathode is more polarized than the anode. The adsorption of these compounds on C-steel surface has been found to obey Frumkin's adsorption isotherm. The mechanism of inhibition was discussed in the light of the chemical structure of the undertaken inhibitors.

Effect of the bubble size on the dynamic adsorption of frothers and collectors in flotation

This paper presents an analysis on the dynamic adsorption of frothers and collectors onto air bubbles in flotation. The analysis was carried out using the theory on the surfactant transfer by diffusion and adsorption. The bubble size was considered when solving the diffusion equation for the dynamic surface excess. The adsorption kinetics was modelled employing the Langmuir adsorption isotherm and its extension with the inclusion of the lateral interaction of the adsorbed molecules (the Frumkin adsorption isotherm). The numerical computation was applied to solve the governing equations. The dynamic adsorption of the surfactants was measured in terms of the dynamic surface tension for sodium dodecylbenzene sulphonate (SDBS) and Dowfroth 250, using pendant bubble tensiometry. The results show that, for air bubbles with diameter of the order of 1 mm, the bubble size effect on the dynamic surface tension is small and the solution of the diffusion equation for a planar gas–liquid surface can be used to describe the dynamic adsorption onto the air bubbles. The experimental data obtained with SDBS were well described by the Frumkin adsorption isotherm and the molecular diffusion. The model description for the experimental data obtained with Dowfroth 250 required the kinetic equations. The paper is relevant for the analysis of the role of the adsorbed surfactants in the bubble–particle attachment.