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.     

Adsorption Kinetics of Some Polyethylene Glycol Octylphenyl Ethers Studied by the Fast Formed Drop Technique

The adsorption kinetics of Triton X-100 and Triton X-405 at solution/air and solution/hexane interfaces is studied by the recently developed fast formed drop technique. The dynamic interfacial tension of Triton X-100 and Triton X-405 solutions against hexane has been measured without preequilibration of the water and oil phases. It is found that the dynamic interfacial tension of Triton X-100 solutions passes through a minimum. This strange behavior is attributed to partial solubility of the surfactant in hexane. Such minima of the dynamic interfacial tension of Triton X-405 solutions have not been observed, which correlates well with the solubilities of both surfactants in hexane reported in the literature. The dynamic surface tension of solutions of both surfactants and the dynamic interfacial tension of Triton X-405 solutions are interpreted by the Ward and Tordai model for diffusion controlled adsorption. It is shown that proper interpretation of the experimental data depends on the type of isotherm used. More consistent results are obtained when the Temkin isotherm is used instead of the Langmuir isotherm. The results obtained with Triton X-100 at the solution/air interface confirm that the adsorption of this surfactant occurs under diffusion control. The adsorption of Triton X-405 at solution/air and at solution/hexane interfaces seems to occur under diffusion control at short periods of time, but under mixed (diffusion-kinetic) control at long periods of time. A hypothesis is drawn to explain this phenomenon by changes in the shape of the large hydrophilic heads of Triton X-405 molecules. Copyright 2000 Academic Press.     

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.     

A theoretical study on surfactant adsorption kinetics: effect of bubble shape on dynamic surface tension

A planar or spherical fluid-liquid interface was commonly assumed on studying the surfactant adsorption kinetics for a pendant bubble in surfactant solutions. However, the shape of a pendant bubble deviates from a sphere unless the bubble's capillary constant is close to zero. Up to date, the literature has no report about the shape effect on the relaxation of surface tension due to the shape difference between a pendant bubble and a sphere. The dynamic surface tension (DST), based on the actual shape of a pendant bubble with a needle, of the diffusion-controlled process is simulated using a time-dependent finite element method in this work. The shape effect and the existence of a needle on DST are investigated. This numerical simulation resolves also the time-dependent bulk surfactant concentration. The depth of solution needed to satisfy the classical Ward-Tordai infinite-solution assumption was also studied. For a diffusion-controlled adsorption process, bubble shape and needle size are two major factors affecting the DST. The existence of a needle accelerates the bulk diffusion for a small bubble; however, the shape of a large pendant bubble decelerates the bulk diffusion. An example using this method on the DST data of C12E4 is illustrated at the end of this work.     

A simplified method for predicting the dynamic surface tension of concentrated surfactant solutions

A simplified method for predicting the dynamic surface tension of concentrated surfactant solutions is proposed. It is implemented using the framework of the Henry's Law analytical solution to the Ward and Tordai equation for diffusion-controlled adsorption, with the necessary parameters being deduced from the measured equilibrium surface tension equation and a value for the surfactant monomer diffusivity. The method is tested by calculating the dynamic surface tension relaxations of aqueous C10E6 and C10E8 solutions over concentration ranges from well below to well above their critical micelle concentrations (cmc). Results are compared with measured relaxations over 0.001-50 s, and semiquantitative agreement is found, with the best results obtained for concentrations near the cmc. The predictive method may prove useful in such applications as the screening of candidate surfactants for inks used in inkjet printing.     

Dynamic Behavior of Surfactants in Solution

Adsorption of various surfactants at the gas liquid interface is studied with equilibrium and dynamic surface tension measurements. The Wilhelmey plate method and maximum bubble pressure method are used for this study. Dynamic surface tension of solutions of different surfactants, sodium lauryl sulfate (SLS), polyoxyethylene glycol 4‐tert‐octyl phenyl ether (Triton X 100), poly‐oxyethylene(20) cetyl ether (Brij 58), and tetraethylene glycol mono‐n‐dodecyl ether (Brij 30), is measured at different concentrations. Adsorption of different surfactants is compared on the basis of equilibrium and dynamic behavior. Effectiveness and efficiency of different surfactants is found from equilibrium surface tension measurement. A new parameter is defined to quantify the dynamic behavior of adsorption, which gives the concentration of surfactant needed to reduce surface tension to half of its maximum reduction within a defined time available for adsorption. The dynamics of surfactant solution is quantified by using this parameter.     

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.     

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.     

Adsorption Kinetics of Alkyl Polyglucoside at the Air—Solution Interface

The air-solution equilibrium tension γe and dynamic surface tension γt,of nonionic surfactant alkyl polyglucoside have been studied. γe was measured by the Wilhelmy method with Rruess K12 tensiometer. The CMC and the surface excesses T were determined from the surface tension vs. concentration curve. The γt decays were measured in the range 0.2-20s using a maximum bubble pressure instrument and analyzed with the Ward and Tordai equation.     

Fluorescence evidence that a phase transition causes the induction time in the reduction in dynamic tension during surfactant adsorption to a clean air/water interface and a kinetic-diffusive transport model for the phase-induced induction

An aqueous soluble surfactant adsorbing from solution onto an initially clean air/water interface often exhibits an induction period in the surface tension relaxation in which, as the adsorption begins, the tension remains near the clean interface value for an extended period of time before decreasing rapidly to the equilibrium value. In this study, using a model nonionic soluble surfactant, C14E6(CH3(CH2)13-(OCH2CH2)6-OH), we present direct fluorescence evidence that this induction is due to a first-order phase transition from a gaseous (G) to a liquid expanded (LE) phase that the assembling monolayer undergoes at constant surface pressure. An open channel flow cell is initially filled with water, and onto its air/water interface is spread an insoluble amphiphilic dye that fluoresces upon irradiation in the LE phase and whose fluorescence is quenched in the G phase. An aqueous solution of C14E(6) is then allowed to flow through the channel. We observe the immediate appearance of bright islands of the LE phase growing in a dark (G) background, confirming the presence of the G/LE phase transition. These islands eventually occupy the entire surface, after which the interface remains uniformly bright. We correlate this phase transition to the induction period by simultaneously measuring the tension of the interface of the open channel, and verifying that as the islands grow the tension remains at the clean value until the bright LE phase occupies the entire surface, whereupon the tension rapidly decreases. We further develop a phase transition surfactant transport model for the induction period in which surfactant diffuses toward and kinetically adsorbs onto the surface, and then rapidly equilibrates between the G and LE phases. For our model surfactant C14E6, we independently measure the surface concentration of the nucleating LE phase, the LE phase surfactant equation of state, the kinetic rate constants for adsorption into the LE phase, and the bulk diffusion coefficient. Using these measurements, we predict induction times for adsorption onto a clean surface without convection. We also measure these induction times in tension relaxation for adsorption onto a pendant bubble using axisymmetric shape analysis, and demonstrate agreement with the simulations with no adjustable constants.     

聚二甲基二烯丙基氯化铵及其与CTAB混合物水溶液 的吸附动力学

用最大泡压法分别测定了聚二甲基二烯丙基氯化铵,十六烷基三甲基溴化铵以及两者混合物水溶液的动表面张力。十六烷基三甲基溴化铵的吸附服从扩散-动力学控制机理。发现聚二甲基二烯丙基氯化铵水溶液的表面张力具有独特的时间相关性。吸附的前期服从扩散控制机理,而在吸附的后期,即接近吸附平衡时服从扩散-动力学控制机理。混合物水溶液的整个吸附过程受扩散控制。     

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.     

Dynamic surface tension and adsorption kinetics of beta-casein at the solution/air interface

A diffusion model is proposed to describe the adsorption kinetics of proteins at a liquid interface. The model is based on the simultaneous solution of the Ward-Tordai equation and a set of recently developed equations describing the equilibrium state of the adsorption layer: the adsorption isotherm, the surface layer equation of state, and the function of adsorption distribution over the states with different molar areas. The new kinetics model is compared with dynamic surface tensions of beta-casein solutions measured with the drop/bubble profile and maximum bubble pressure methods. The adsorption process for low concentrations is governed by the diffusion mechanism, while at large protein concentrations this is only the case in the initial stage. The effective diffusion coefficients agree fairly well with literature data. The adsorption values calculated from the dynamic surface tension data agree very well with the used equilibrium adsorption model.     

A review on experimental studies of surfactant adsorption at the hydrophilic solid-water interface

The progresses of understanding of the surfactant adsorption at the hydrophilic solid-liquid interface from extensive experimental studies are reviewed here. In this respect the kinetic and equilibrium studies involves anionic, cationic, non-ionic and mixed surfactants at the solid surface from the solution. Kinetics and equilibrium adsorption of surfactants at the solid-liquid interface depend on the nature of surfactants and the nature of the solid surface. Studies have been reported on adsorption kinetics at the solid-liquid interface primarily on the adsorption of non-ionic surfactant on silica and limited studies on cationic surfactant on silica and anionic surfactant on cotton and cellulose. The typical isotherm of surfactants in general, can be subdivided into four regions. Four-regime isotherm was mainly observed for adsorption of ionic surfactant on oppositely charged solid surface and adsorption of non-ionic surfactant on silica surface. Region IV of the adsorption isotherm is commonly a plateau region above the CMC, it may also show a maximum above the CMC. Isotherms of four different regions are discussed in detail. Influences of different parameters such as molecular structure, temperature, salt concentration that are very important in surfactant adsorption are reviewed here. Atomic force microscopy study of different surfactants show the self-assembly and mechanism of adsorption at the solid-liquid interface. Adsorption behaviour and mechanism of different mixed surfactant systems such as anionic-cationic, anionic-non-ionic and cationic-non-ionic are reviewed. Mixture of surface-active materials can show synergistic interactions, which can be manifested as enhanced surface activity, spreading, foaming, detergency and many other phenomena.     

Modeling of the adsorption kinetics of surfactants at the liquid-fluid interface of a pendant drop

This paper presents a theoretical model for simulating the adsorption kinetics of a surfactant at the liquid-fluid interface of a pendant drop. The diffusion equation is solved numerically by applying the semidiscrete Galerkin finite element method to obtain the time-dependent surfactant concentration distributions inside the pendant drop and inside the syringe needle that is used to form the pendant drop. With the obtained bulk surfactant concentration distributions, the adsorption at the interface is determined by using the conservation law of mass. It should be noted that the theoretical model developed in this study considers the actual geometry of the pendant drop, the depletion process of the surfactant inside the pendant drop, and the mass transfer of the surfactant from the syringe needle to the pendant drop. The present pendant-drop model is applied to study the adsorption kinetics of surfactant C10E8 (octaethylene glycol mono n-decyl ether) at the water-air interface of a pendant drop. The numerical results show that the Ward and Tordai equation, which was derived for adsorption from a semi-infinite surfactant solution to a planar interface, is unsuitable for interpreting the dynamic surface or interfacial tension data measured by using the pendant-drop-shape techniques, especially at low initial surfactant concentrations. The spherical-drop model, which assumes the pendant drop to be a perfectly spherical drop with the same drop volume, can be used to interpret the dynamic surface or interfacial tension data for pendant drops either with high initial surfactant concentrations or with low initial surfactant concentrations in short adsorption durations only. For pendant drops with low initial surfactant concentrations in long adsorption durations, the theoretical model developed in this study is strongly recommended.     

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 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.     

Monte Carlo simulations of Lennard-Jones nonionic surfactant adsorption at the liquid/vapor interface

We present Monte Carlo simulations of nonionic surfactant adsorption at the liquid/vapor interface of a monatomic solvent. All molecules in the system, solvent and surfactant, are characterized by the Lennard-Jones (LJ) potential using differing interaction parameters. Surfactant molecules consist of an amphiphilic chain with a solvophilic head and a solvophobic tail. Adjacent atoms along the surfactant chain are connected by finitely extensible harmonic springs. Solvent molecules move via the Metropolis random-walk algorithm, whereas surfactant molecules move according to the continuum configurational bias Monte Carlo (CBMC) method. We generate quantitative thermodynamic adsorption and surface tension isotherms in addition to surfactant radius of gyration, tilt angles, and potentials of mean force. Surface tension simulations compared to those calculated from the simulated adsorbed amounts and the Gibbs adsorption isotherm agree confirming equilibrium in our simulations. We find that the classical Langmuir isotherm is obeyed for our LJ surfactants over the range of head and tail lengths studied. Although simulated surfactant chains in the bulk solution exhibit random orientations, surfactant chains at the interface orient roughly perpendicular and the tails elongate compared to bulk chains even in the submonolayer adsorption regime. At a critical surfactant concentration, designated as the critical aggregation concentration (CAC), we find aggregates in the solution away from the interface. At higher concentrations, simulated surface tensions remain practically constant. Using the simulated potential of mean force in the submonolayer regime and an estimate of the surfactant footprint at the CAC, we predict a priori the Langmuir adsorption constant, KL, and the maximum monolayer adsorption, Gammam. Adsorption is driven not by proclivity of the surfactant for the interface, but by the dislike of the surfactant tails for the solvent, that is by a "solvophobic" effect. Accordingly, we establish that a coarse-grained LJ surfactant system mimics well the expected equilibrium behavior of aqueous nonionic surfactants adsorbing at the air/water interface.     

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。     

Dynamic Interfacial Behavior of Poly(oxyethylene) Lauryl Ether Based Surfactant Mixtures

The adsorption behavior of binary mixtures comprising nonionic surfactants at the air–water interface has been studied by bubble pressure tensiometry at concentrations above and below their critical micelle concentrations. Surfactants with the same hydrocarbon chains but different degree of ethoxylations were chosen as the components to understand their mixing behavior at equilibrium and dynamic conditions. At short times, the adsorption is found to be diffusion limited for individual components as well as for the mixtures, as predicted by the Ward and Tordai model. The effective diffusion coefficient of the monomers in the mixed state displays a dynamic synergism, consistent with the molecular thermodynamic model for dynamic surface tension. However, the equilibrium surface tension and micellar diffusion coefficient of the mixtures exhibit ideal behavior.