Short-time behavior of mixed diffusion-barrier controlled adsorption

This paper focuses on the short-time adsorption kinetics of nonionic surfactants onto water/air surfaces, analyzed in the context of the mixed diffusion-barrier controlled adsorption modeling framework. Specifically, we reconcile the apparent contradiction between theoretical prediction and experimental observations on the adsorption kinetics mechanism at short times: while the mixed diffusion-barrier controlled model predicts a barrier-controlled adsorption, as well as the impossibility of a diffusion-controlled adsorption at asymptotic short times, the short-time experimental dynamic surface tension (DST) behavior of many nonionic surfactants has been interpreted to result from diffusion-controlled adsorption at asymptotic short times. This is because the short-time experimental DST of these surfactants displays a t variation, which is considered as a fingerprint for the existence of diffusion-controlled adsorption, based on the short-time asymptotic behavior of the diffusion-controlled adsorption model. As a result of this interpretation, the fundamental physical nature of the energy barrier has been proposed to be associated with high surfactant surface concentrations. In this paper, we derive a new nonasymptotic short-time formalism of the mixed diffusion-barrier controlled model to describe surfactant adsorption onto a spherical pendant-bubble surface, including determining the ranges of time and surfactant surface concentration values where the short-time formalism is applicable. Based on this formalism, we find that one can expect to observe an apparent t variation of the DST at short times even for the mixed diffusion-barrier controlled adsorption model. We analyze the consequence of this finding by re-evaluating the existing notions of the energy barrier. We conclude that the energy barrier is associated with the adsorption of a single surfactant molecule onto a clean surface.     

New methodology to determine the rate-limiting adsorption kinetics mechanism from experimental dynamic surface tension data

We present a new methodology to determine the rate-limiting adsorption kinetics mechanism (diffusion-controlled vs mixed diffusion-barrier controlled), including deducing the kinetics parameters (the diffusion coefficient, D, and the energy-barrier parameter, beta), from the experimental short-time dynamic surface tension (DST) data. The new methodology has the following advantages over the existing procedure used to analyze the experimental DST data: (a) it does not require using a model for the equilibrium adsorption isotherm, and (b) it only requires using the experimental short-time DST data measured at two initial surfactant bulk solution concentrations. We apply the new methodology to analyze the experimental short-time DST data of the following alkyl poly(ethylene oxide), CiEj, nonionic surfactants: C12E4, C12E6, C12E8, and C10E8 measured using the pendant-bubble apparatus. We find that for C12E4 and C12E6, the effect of the energy barrier on the overall rate of surfactant adsorption can be neglected for surfactant bulk solution concentrations below their respective critical micelle concentrations (CMCs), and therefore, that the rate-limiting adsorption kinetics mechanism for C12E4 and C12E6 is diffusion-controlled at any of their premicellar surfactant bulk solution concentrations. On the other hand, for C12E8 and C10E8, we find that their respective CMC values are large enough to observe a significant effect of the energy barrier on the overall rate of surfactant adsorption. In other words, for C12E8 and C10E8, the rate-limiting adsorption kinetics mechanism shifts from diffusion-controlled to mixed diffusion-barrier controlled as their premicellar surfactant bulk solution concentrations increase. We test the new methodology by predicting the short-time DST profiles at other initial surfactant bulk solution concentrations, and then comparing the predicted DST profiles with those measured experimentally. Very good agreement is obtained for the four CiEj nonionic surfactants considered. We also compare the results of implementing the new methodology with those of implementing the existing procedure, and conclude that using a model for the equilibrium adsorption isotherm can lead not only to different values of D and beta, but it can also lead to a completely different determination of the rate-limiting adsorption kinetics mechanism. Since the new methodology proposed here does not require using a model for the equilibrium adsorption isotherm, we conclude that it should provide a more reliable determination of the rate-limiting adsorption kinetics mechanism, including the deduced kinetics parameters, D and beta.     

Influence of electrolytes on the dynamic surface tension of ionic surfactant solutions: expanding and immobile interfaces

Here, we derive analytical asymptotic expressions for the dynamic surface tension of ionic surfactant solutions in the general case of nonstationary interfacial expansion. Because the diffusion layer is much wider than the electric double layer, the equations contain a small parameter. The resulting perturbation problem is singular and it is solved by means of the method of matched asymptotic expansions. The derived general expression for the dynamic surface tension is simplified for the special case of immobile interface and for the maximum bubble pressure method (MBPM). The case of stationary interfacial expansion is also considered. The effective diffusivity of the ionic surfactant essentially depends on the concentrations of surfactant and nonamphiphilic salt. To test the theory, the derived equations are applied to calculate the surfactant adsorption from MBPM experimental data. The results excellently agree with the adsorption determined independently from equilibrium surface-tension isotherms. The derived theoretical expressions could find application for interpreting data obtained by MBPM and other experimental methods for investigating interfacial dynamics.     

Dynamic Surface Tension and Adsorption Mechanism of Surfactant Benzyltrimethylammonium Bromide at the Air/Water Interface

The air‐solution equilibrium tension, γ_c and dynamic surface tension, γ_t, of aqueous solutions of a novel ionic surfactant benzyltrimethylammonium bromide (BTAB) were measured by Wilhelmy method and Maximum bubble pressure method (MBPM), respectively. Adsorption equilibrium and mechanism of BTAB at the air‐solution interface were studied. The CMC was determined to be 0.11 mol/L. The results show that at the start, the adsorption process is controlled by a diffusion step. Toward the end, it changes to a mixed kinetic‐diffusion controlled mechanism with the adsorption activation energy of about 11.0 KJ/mol. Effects of temperature, inorganic salts, and alcohols on adsorption kinetics also are discussed.     

Mass transport in micellar surfactant solutions: 2. Theoretical modeling of adsorption at a quiescent interface

Here, we apply the detailed theoretical model of micellar kinetics from part 1 of this study to the case of surfactant adsorption at a quiescent interface, i.e., to the relaxation of surface tension and adsorption after a small initial perturbation. Our goal is to understand why for some surfactant solutions the surface tension relaxes as inverse-square-root of time, 1/t(1/2), but two different expressions for the characteristic relaxation time are applicable to different cases. In addition, our aim is to clarify why for other surfactant solutions the surface tension relaxes exponentially. For this goal, we carried out a computer modeling of the adsorption process, based on the general system of equations derived in part 1. This analysis reveals the existence of four different consecutive relaxation regimes (stages) for a given micellar solution: two exponential regimes and two inverse-square-root regimes, following one after another in alternating order. Experimentally, depending on the specific surfactant and method, one usually registers only one of these regimes. Therefore, to interpret properly the data, one has to identify which of these four kinetic regimes is observed in the given experiment. Our numerical results for the relaxation of the surface tension, micelle concentration and aggregation number are presented in the form of kinetic diagrams, which reveal the stages of the relaxation process. At low micelle concentrations, "rudimentary" kinetic diagrams could be observed, which are characterized by merging of some stages. Thus, the theoretical modeling reveals a general and physically rich picture of the adsorption process. To facilitate the interpretation of experimental data, we have derived convenient theoretical expressions for the time dependence of surface tension and adsorption in each of the four regimes.     

Boltzmann distributions and slow relaxation in systems with spherical and cylindrical micelles

Equilibrium and nonequilibrium distributions of molecular aggregates in a solution of a nonionic surfactant are investigated at the total surfactant concentration above the second critical micelle concentration (CMC2). The investigation is not limited by the choice of a specific micellar model. Expressions for the direct and reverse fluxes of molecular aggregates over the potential humps of the aggregation work are derived. These aggregation work humps set up activation barriers for the formation of spherical and cylindrical micelles. With the aid of the expressions for molecular aggregate fluxes, a set of two kinetic equations of micellization is derived. This set, along with the material balance equation, describes the molecular mechanism of the slow relaxation of micellar solution above the CMC2. A realistic situation has been analyzed when the CMC2 exceeds the first critical micelle concentration, CMC1, by an order of magnitude, and the total surfactant concentration varies within the range lying markedly above the CMC2 but not by more than 2 orders of magnitude. For such conditions, an equation relating the parameters of the aggregation work of a cylindrical micelle to the observable ratio of the total surfactant concentration and the monomer concentration is found for an equilibrium solution. For the same conditions, but in the nonequilibrium state of materially isolated surfactant solution, a closed set of linearized relaxation equations for total concentrations of spherical and cylindrical micelles is derived. These equations determine the time development of two modes of slow relaxation in micellar solutions markedly above the CMC2. Solving the set of equations yields two rates and two times of slow relaxation.     

Diffusion-Controlled Adsorption at the Liquid-Air Interface: The Long-Time Limit

The applicability of the Hansen and Joos long-time limits for the dynamic surface tension of solutions is investigated by regressing diffusion coefficients from numerical solutions to the Ward and Tordai equation. The Hansen limit is found to correctly describe the dynamic surface tension evolutions at long times. However, both the surfactant concentration and the adsorption time affect the accuracy of the long-time limit. The study also indicates that, because the reduction in surface tension (at long times) may be smaller than can be measured by current tensiometry methods, the application of the Hansen limit to long-time data may not always be feasible. Copyright 2001 Academic Press.     

Dynamic Interfacial Tension of Surfactant Mixtures at Liquid-Liquid Interfaces

Dynamic interfacial tensions for surfactant mixtures at liquid-liquid interfaces were obtained with a drop volume tensiometer. The surfactants tested were Triton X-100, palmitic acid, and Span 80 at both the water-hexadecane and water-mineral oil interfaces. Two-surfactant mixtures were examined with the surfactants initially dissolved in different phases to minimize bulk-phase interactions. For concentrations below the CMC, it was found that the adsorption kinetics of palmitic acid and Triton X-100 mixtures were dominated by the latter surfactant. Apparent diffusion coefficients were obtained for Triton X-100 both in the absence and in the presence of palmitic acid. These values were largely insensitive to the presence of palmitic acid. For mixtures of Span 80 and Triton X-100, the adsorption kinetics were found to be influenced significantly by both surfactants. In this case, relative changes in surfactant concentrations affected the dynamic interfacial tension of the mixed system. A previously proposed multicomponent adsorption model described the dynamic interfacial tension adequately at low concentrations of Triton X-100, when desorption could be neglected. At higher concentrations, modifications were needed to account for solubilization into the oil phase. These corrections allowed the model to describe the long time adsorption quite well. However, predicted values of short time interfacial tensions were overestimated, likely due to a synergistic interaction of the two surfactants. Copyright 1999 Academic Press.     

Curvature Effects in the Analysis of Pendant Bubble Data: Comparison of Numerical Solutions, Asymptotic Arguments, and Data

The pendant bubble method is commonly used to measure the evolution of the surface tension of surfactant solutions. Initially, the bubble interface is free of adsorbed surfactant. As time progresses, surfactant diffuses to the interface, adsorbs, and reduces the surface tension. The surface tension is assumed to be in equilibrium with the instantaneous surface concentration. Therefore, surface tension data are analyzed in terms of interfacial thermodynamics and mass transfer models in order to infer the mechanisms which determine the surfactant transport. Diffusion from the bulk solution to the bubble can be approximated as diffusion to a spherical interface. Approximating this process as diffusion to a plane introduces significant errors into the data analysis. Mass transfer to a sphere differs from that to a plane; the equilibration of the spherical interface is more rapid simply because of geometry. The failure to account for this effect in the interpretation of pendant bubble data can lead to serious errors in the transport coefficients for the surfactants. In the diffusion-controlled limit, surfactant diffuses to the sublayer immediately adjacent to the interface and adsorbs in local equilibrium according to the adsorption isotherm. There is a closed-form solution for Fick's law describing adsorption to a sphere in an infinite solution which reduces to the Ward and Tordai solution when the bubble radius is large. This equation, along with the adsorption isotherm relating the surface concentration and the sublayer concentration, must be solved numerically in order to solve for the time evolution of the surface concentration. At early times, the adsorption isotherm can be expanded about the clean interface state. At long times, small departures from the equilibrium state can be assumed. In these limits, asymptotic expansions can be obtained. The short- and long-time expansions are found in this study for adsorption to a sphere and compared to those obtained previously for adsorption to a planar interface. In particular, the long-time asymptote for adsorption to a sphere is proportional to t(-3/2); this asymptote differs significantly from that for adsorption to a plane, which goes as t(-1/2). The full solution for adsorption to a sphere is compared to the Ward and Tordai solution for adsorption to a planar interface. From a comparison of the full solutions, it is established that curvature cannot be neglected unless the ratio of the adsorption depth to the bubble radius is negligible. This ratio can be calculated a priori from equilibrium isotherm parameters. Using constants which describe the surfactant C(12)E(8), for which curvature plays a strong role in the surfactant adsorption dynamics, the short- and long-time solutions for adsorption to the interface are compared to the full solutions and to dynamic surface tension data to infer the range of validity of the approximations. Copyright 2001 Academic Press.     

十八烷基二甲基氯化铵水溶液动态表面张力及吸附动力学研究

使用最大气泡法测定了十八烷基二甲基氯化铵（C_(18)DAC）水溶液的动态表 面张力,考察了浓度、温度等对其DST的影响,详细表征了DST随时间的变化过程, 计算了动态表面张力的各种参数（n,t_i,t~*,t_m,R_(1/2)）。结合Word- Tordai方程计算了表观扩散系数（D_a）和吸附势垒（E_a）,对其吸附动力学模式 进行了研究,探讨了DST参数的物理意义。结果表明,t~*值越小,吸附势垒E_a越 大,宏观扩散系数D_a越小,表面活性剂分子越不易吸附在溶液表面;C_(18)DAC低 浓度时吸附属于扩散控制模式,高浓度时属于混合控制模式;高浓度时,在吸附初 期（t → 0）为扩散控制模式,吸附后期（t → ∞）为混合控制模式。     

Kinetics of the surface tension of micellar solutions: Comparison of different experimental techniques

The kinetics of the surface tension of micellar solutions of nonionic surfactant Triton X-100 is measured experimentally by means of three different techniques: oscillating jet, maximum bubble pressure and inclined plate. They allow to study the micellization kinetics at various time scales (from a few milliseconds to a few seconds) in fairly large concentration region up to 50 times CMC. The experimental data are satisfactorily explained by a theoretical model accounting for the kinetics of micellization, diffusion of surfactant species and expansion of the bubble interface. By this model are computed the characteristic times of diffusion and micellization, which are of comparable magnitude (about 5 to 200 ms), and the Gibbs' elasticity. The micellization time constant corresponds to the slow relaxation process known to coincide with the disintegration of micelles. Comparing our data with other data from literature one can conclude that more realistic information for the micellization kinetics is obtained by the maximum bubble pressure and the oscillating jet method. The inclined plate seems too slow to measure the relaxation processes in micellar solutions of this surfactant.     

Dynamic surface tensions of micellar Triton X-100 solutions

Considering surfactant solutions at concentrations exceeding the CMC, another relaxation process besides diffusion occurs, also affecting the dynamic surface tension. The latter equilibration process concerns a micellisation/demicellisation process, representing the disintegration of micelles into monomers. The micellisation kinetics are accounted for by adding a single source term to the diffusion equation of the free monomers.

In the present paper the integration of the diffusion equation is avoided by using the concept of the diffusion penetration depth. Nevertheless, when this approximation is made, good agreement is achieved between experiment and theory for micellar Triton X-100 solutions. Moreover, it follows that diffusion of micelles may not be neglected.     

Dynamic surface tension of micellar solutions studied by the maximum bubble pressure method

A theoretical model for the dynamic surface tension of an air bubble expanding in surfactant solution is proposed. The model accounts for the effect of convection on the surfactant diffusion and the effect of expansion of the bubble surface during the adsorption of surfactant molecules. Assuming small deviation from equilibrium and constant rate of expansion, an analytical solution for the surface tension and the subsurface concentration as a function of time is derived. The parameters of the model are computed from experimental data for sodium dodecyl sulfate obtained by the maximum bubble pressure method.     

Equilibrium and dynamic surface properties of long-chain quaternary ammonium hydroxides with different chain length and number

The equilibrium and dynamic surface tensions of five long-chain alkyl ammonium hydroxides (AAH) at the air/aqueous solution interface were investigated, and the effects of the length and number of alkyl chain on surface tensions had been discussed. With the increase of the length, the equilibrium surface tension (EST) increased from 28.65 to 40.52?mN/m. While, for the double chains at the critical micelle concentration (CMC), the EST decreased from 32.71 to 26.61?mN/m with the length increasing. In addition, the adsorption behaviors of the AAH were analyzed and the effective diffusion coefficients (D_eff) were calculated on basis of the Ward–Tordai equation. Moreover, the time required to attain the EST decreases with the increase of surfactant concentration. The longer the C–H chain is, the lower surface tension at initial concentration is. What’s more, the diffusion processing of the AAH to air/water interface mainly depends on the surfactant concentration, and the adsorption is controlled by diffusion mechanism in a dilute concentration, while under a high concentration the adsorption is controlled by mixed diffusion–kinetic mechanism.     

Power-law stage of slow relaxation in solutions with spherical micelles

General (independent of models selected for surfactant molecular aggregates) analytical relations are derived to describe the initial stage of slow relaxation in micellar solutions with spherical micelles. This stage precedes the final stage of the relaxation occurring via an exponential decay of disturbances with time. The relations obtained are applicable throughout the interval of micellar solution concentrations from the first to the second critical micellization concentration. It is shown that the initial stage is characterized by power laws of variations in the concentrations of monomers and micelles with time, these laws being different for the relaxation processes proceeding from above and below toward equilibrium values of micellar solution parameters. Relations are derived for the duration of this stage, and the effect of initial conditions is studied. Characteristic times of the power-law stage are determined and compared with the characteristic time of the final exponent-law relaxation stage. The behavior of these times is investigated at surfactant solution concentrations in the vicinity of, and noticeably above, the first critical micellization concentration. On the basis of the droplet and quasi-droplet thermodynamic models of surfactant molecular aggregates, numerical solutions are found for nonlinearized equations of slow relaxation for the time dependence of surfactant monomer concentrations at all stages of the slow relaxation. Numerical results obtained from the models are compared with the results of a general analytical study.     

Which surfactants reduce surface tension faster? A scaling argument for diffusion-controlled adsorption

Consider the example of surfactant adsorbing from an infinite solution to a freshly formed planar interface. There is an implicit length scale in this problem, the adsorption depth h, which is the depth depleted to supply the interface with the absorbed surfactant. From a mass balance, h can be shown to be the ratio of the equilibrium surface concentration gamma eq to the bulk concentration C infinity. The characteristic time scale for diffusion to the interface is tau D = h2/D, where D is the diffusivity of the surfactant in solution. The significance of this time scale is demonstrated by numerically integrating the equations governing diffusion-controlled adsorption to a planar interface. The surface tension equilibrates within 1-10 times tau D regardless of bulk concentration, even for surfactants with strong interactions. Dynamic surface tension data obtained by pendant bubble method are rescaled using tau D to scale time. For high enough bulk concentrations, the re-normalized surface tension evolutions nearly superpose, demonstrating that tau D is indeed the relevant time scale for this process. Surface tension evolutions for a variety of surfactants are compared. Those with the smallest values for tau D equilibrate fastest. Since diffusion coefficients vary only weakly for surfactants of similar size, the differences in the equilibration times for various surfactant solutions can be attributed to their differing adsorption depths. These depth are determined by the equilibrium adsorption isotherms, allowing tau D to be calculated a priori from equilibrium surface tension data, and surfactant solutions to be sorted in terms of which will reduce the surface tension more rapidly. Finally, trends predicted by tau D to gauge what surfactant properties are required for rapid surface tension reduction are discussed. These trends are shown to be in agreement with guiding principles that have been suggested from prior structure-property studies.     

Dynamics of protein and mixed protein/surfactant adsorption layers at the water/fluid interface

The adsorption behaviour of proteins and systems mixed with surfactants of different nature is described. In the absence of surfactants the proteins mainly adsorb in a diffusion controlled manner. Due to lack of quantitative models the experimental results are discussed partly qualitatively. There are different types of interaction between proteins and surfactant molecules. These interactions lead to protein/surfactant complexes the surface activity and conformation of which are different from those of the pure protein. Complexes formed with ionic surfactants via electrostatic interaction have usually a higher surface activity, which becomes evident from the more than additive surface pressure increase. The presence of only small amounts of ionic surfactants can significantly modify the structure of adsorbed proteins. With increasing amounts of ionic surfactants, however, an opposite effect is reached as due to hydrophobic interaction and the complexes become less surface active and can be displaced from the interface due to competitive adsorption. In the presence of non-ionic surfactants the adsorption layer is mainly formed by competitive adsorption between the compounds and the only interaction is of hydrophobic nature. Such complexes are typically less surface active than the pure protein. From a certain surfactant concentration of the interface is covered almost exclusively by the non-ionic surfactant. Mixed layers of proteins and lipids formed by penetration at the water/air or by competitive adsorption at the water/chloroform interface are formed such that at a certain pressure the components start to separate. Using Brewster angle microscopy in penetration experiments of proteins into lipid monolayers this interfacial separation can be visualised. A brief comparison of the protein adsorption at the water/air and water/n-tetradecane shows that the adsorbed amount at the water/oil interface is much stronger and the change in interfacial tension much larger than at the water/air interface. Also some experimental data on the dilational elasticity of proteins at both interfaces measured by a transient relaxation technique are discussed on the basis of the derived thermodynamic model. As a fast developing field of application the use of surface tensiometry and rheometry of mixed protein/surfactant mixed layers is demonstrated as a new tool in the diagnostics of various diseases and for monitoring the progress of therapies.     

Adsorption of nonionic surfactants on cellulose surfaces: adsorbed amounts and kinetics

Kinetic and equilibrium aspects of three different poly(ethylene oxide) alkylethers (C12E5, C12E7, C14E7) near a flat cellulose surface are studied. The equilibrium adsorption isotherms look very similar for these surfactants, each showing three different regions with increasing surfactant concentration. At low surfactant content both the headgroup and the tail contribute to the adsorption. At higher surface concentrations, lateral attraction becomes prominent and leads to the formation of aggregates on the surface. The general shape of the isotherms and the magnitude of the adsorption resemble mostly those for hydrophilic surfaces, but both the ethylene oxide and the aliphatic segments determine affinity for the surface. The adsorption and desorption kinetics are strongly dependent on surfactant composition. At bulk concentrations below the CMC, the initial adsorption rate is attachment-controlled. Above the CMC, the micellar diffusion coefficient and the micellar dissociation rate play a crucial role. For the most hydrophilic surfactant, C12E7, both parameters are relatively large. In this case, the initial adsorption rate increases with increasing surfactant concentration, also above the CMC. For C12E5 and C14E7 there is no micellar contribution to the initial adsorption rate. The initial desorption kinetics are governed by monomer detachment from the surface aggregates. The desorption rate constants scale with the CMC, indicating an analogy between the surface aggregates and those formed in solution.     

Diffusion-controlled adsorption kinetics at the interface between air and aqueous micellar solution of heptaethylene glycol monododecyl ether

The diffusion-controlled adsorption kinetics of micellar surfactant C₁₂E₇ (heptaethylene glycol monododecyl ether) solutions was studied theoretically and experimentally. The corrected diffusion equation, which was used to describe the diffusion of the monomers in the micellar solutions, was solved under the initial and boundary conditions by means of Laplace transformation. The dynamic surface adsorption γ(t) as a function of surface lifetime t, monomer diffusion coefficient D and the demicellization constant was derived. The dynamic surface tensions γ(t) of aqueous submicellar and micellar solutions were measured via maximal bubble pressure method. By analyzing the experimental data, the determined demicellization constant of C₁₂E₇ at 25°C was between 100–116 s^?1.     

Dynamic surface tension of micellar solutions studied by the maximum bubble pressure method

A theoretical model for the dynamic surface tension of an air bubble expanding in micellar surfactant solution is proposed. The model accounts for the effect of expansion of the bubble surface during the adsorption of surfactant molecules (monomers) and the effect of disintegration of polydisperse micelles on the surfactant diffusion. Assuming small deviations from equilibrium and constant rate of expansion analytical expression for the surface tension and the subsurface concentration of monomers as a function of time is derived. The characteristic time of micellization is computed from the experimental data for two surfactants (sodium dodecyl sulfate and nonylphenol polyglycol ether) obtained by the maximum bubble pressure method.