Kinetics of Nonionic Surfactant Adsorption and Desorption at the Silica

Langmuir , 1998, 14 (5), pp 1058–1071. DOI: 10.1021/ ... Cite this:Langmuir 14, 5, 1058-1071 ..... The Journal of Chemical Physics 2001 115 (14), 66...
0 downloads 0 Views 181KB Size
Download PDF
1058

Langmuir 1998, 14, 1058-1071

Kinetics of Nonionic Surfactant Adsorption and Desorption at the Silica-Water Interface: One Component Johanna Brinck* and Bengt Jo¨nsson Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University, PO Box 124, SE-221 00 Lund, Sweden

Fredrik Tiberg Institute for Surface Chemistry, PO Box 5607, SE-114 86 Stockholm, Sweden Received July 14, 1997. In Final Form: November 14, 1997 In this paper we present a theoretical model which describes the kinetics of adsorption and desorption from a micellar solution of nonionic surfactants at a silica surface. Numerical calculations based on this model have been compared with experimental results of CnEm surfactant adsorption, obtained by ellipsometry, and show good agreement. The aim of this work was to develop a model for adsorption through a stagnant layer onto a solid, hydrophilic surface. The surface is considered to be planar and homogeneous. Outside the surface there is a micellar solution of a pure nonionic surfactant. Both monomers and micelles are considered to be able to adsorb. To facilitate the evaluation of the model, a computer program was written which solves the mathematical equations numerically. The course of adsorption and desorption of a number of short-chain CnEm surfactants has been simulated with this program. The results obtained, in terms of amounts adsorbed as a function of time, were compared with experimental data determined by time-resolved null ellipsometry. The same program was used to calculate concentration profiles outside the silica surface. Not only has this model made it possible for us to explain and better understand experimental results, but it has also allowed us to gain an understanding of how the course of adsorption and desorption is affected by parameters which are difficult to vary experimentally in a controlled way. Two examples of this, which will be discussed in this paper, are the effects of stagnant layer thickness and the relation between critical surface aggregation concentration (csac) and critical micelle concentration (cmc).

Introduction Due to the many methods available for the determination of dynamic surface tension at gas-liquid interfaces, adsorption kinetics have so far mainly been studied at the air-water interface (cf. literature reviews1-3). By using ellipsometry, it has, for some time, been possible to study adsorption and desorption kinetics with a relevant time resolution at solid-liquid interfaces.4-8 Several of these studies have treated the case of adsorption from water to a hydrophilic solid surface, namely silicon dioxide (silica) (e.g., refs 5-7). This surface is interesting not only because of its qualities as a model surface for other hydrophilic solid surfaces, which may be difficult to study for experimental reasons, but also due to the presence of silica in glass and glazing and its widespread occurrence in the earth’s crust. A few years ago, two of the authors used the ellipsometric method, to study both equilibrium adsorption and the kinetics of adsorption and desorption of a series of nonionic * To whom correspondence should be addressed. E-mail: [email protected] (1) Chang, C. H.; Franses, E. I. Colloids Surf., A 1995, 100, 1. (2) Miller, R.; Kretzschmar, G. Adv. Colloid Interface Sci. 1991, 37, 97. (3) Dukhin, S. S.; Kretzschmar, G.; Miller, R. Dynamics of adsorption at liquid interfaces: theory, experiment, application; 1st ed.; Elsevier Science B. V.: Amsterdam, 1995; Vol. 1. (4) Tiberg, F.; Landgren, M. Langmuir 1993, 9, 927. (5) Tiberg, F.; Jo¨nsson, B.; Lindman, B. Langmuir 1994, 10, 3714. (6) Wahlgren, M.; Arnebrant, T.; Lundstro¨m, I. J. Colloid Interface Sci. 1995, 175, 506. (7) Amiel, C.; Sikka, M.; Schneider, J. W.; Tsao, Y.; Tirrell, M.; Mays, J. W. Macromolecules 1995, 28, 3125. (8) Eskilsson, K.; Tiberg, F. Macromolecules 1997, 30, 6323.

surfactants of the type poly(ethylene glycol) monoalkyl ether (CnEm) at the silica-water interface.5,9 The equilibrium amounts adsorbed as a function of concentration of these surfactants at this particular surface is now well documented (e.g., refs 9-12 and further references therein). The monomer contribution to the adsorption is very small. When the critical surface aggregation concentration (csac ≈ 0.7 × cmc) is reached in the bulk, the adsorption increases rapidly within a narrow concentration interval. At this concentration, aggregates begin to form at the surface. Around the cmc, the adsorption levels off and reaches a plateau. For surfactants with short polar chains a step isotherm is observed, whereas Langmuirshaped isotherms are observed for surfactants with longer chains. Only a few studies have dealt with the kinetics of nonionic surfactant adsorption at the silica-water interface.5,12-14 The first, to our knowledge, is the work by Klimenko et al., who used light scattering to study the adsorption kinetics of three different nonionic surfactants on dispersed silica particles.13 When formulating a model for surfactant adsorption kinetics on silica surfaces, it is necessary to bear in mind some special features of the interaction between the surfactants and the surface. The process of adsorption to (9) Tiberg, F.; Jo¨nsson, B.; Tang, J.; Lindman, B. Langmuir 1994, 10, 2294. (10) Bo¨hmer, M. R.; Koopal, L. K.; Janssen, R.; Lee, E. M.; Thomas, R. K.; Rennie, A. R. Langmuir 1992, 8, 2228. (11) Levitz, P. Langmuir 1991, 7, 1595. (12) Partyka, S.; Zaini, S.; Lindheimer, M.; Brun, B. Colloids Surf. 1984, 12, 255. (13) Klimenko, N. A.; Permilovskaya, A. A.; Tryasorukova, A. A.; Koganovskii, A. M. Kolloidn. Zh. 1975, 37, 972. (14) Brinck, J.; Tiberg, F. Langmuir 1996, 12, 5042.

S0743-7463(97)00784-1 CCC: $15.00 © 1998 American Chemical Society Published on Web 02/07/1998

Kinetics of Nonionic Surfactant Adsorption

silica is initially driven by the interactions between the ethylene oxide headgroups and the silica surface. A great deal of research has been devoted to the very origin of these interactions, which is indeed a difficult subject.15,16 These interactions are weak, as seen by the very low monomer adsorption. However, close to the cmc, lateral interactions induce the formation of aggregates at the surface through a cooperative process.11,17 The fact that the csac is of the same order of magnitude as the cmc shows that the interaction between silica and ethylene oxide is of the same order as that between silica and water. When a surface comes into contact with a micellar solution, one way of achieving aggregates at the surface is by monomer adsorption and the consequent formation of surface aggregates at the surface. It can, however, also be expected that micelles in the immediate vicinity of the surface adsorb directly. The importance of micellar adsorption in a particular system depends on the strength of the attraction between the surface and the headgroups. If the interaction is strong, the adsorption is initiated at concentrations far below the cmc and micellar adsorption can then be expected to be of limited interest. If the interaction between the headgroup of the surfactant and the surface is weak, as is the case with poly(ethylene glycol) monoalkyl ethers at a silica surface, then the monomer adsorption is low and micellar adsorption must be taken into consideration. The first attempt to formulate a physically based model for the time dependence of adsorption was made by Ward and Tordai.18 The model treated adsorption at the airwater interface, which was described as a planar surface in contact with an infinite stagnant medium. This medium contained a single surface-active substance. The initial concentration of surfactant was assumed to be completely homogeneous, i.e., independent of the distance from the surface. Adsorption was described as a process consisting of two steps. The first was diffusion from the bulk to a plane immediately outside the surface. The second was the transfer of the substance from the dissolved to the adsorbed state. The diffusion was driven by the concentration gradient produced as surfactants in the immediate vicinity of the surface were adsorbed. The transport from the bulk was described by Fick’s second law, while the flow through the sublayer was obtained from Fick’s first law. In this work, an attempt was made to systematize the influence of different fundamental parameters, such as the cmc and surfactant concentration, as well as surfactant-independent parameters like the stagnant layer thickness outside the surface, on the adsorption and desorption processes. In a previous publication, the first steps were taken in the process of developing such a model.5 The adsorption was described as a two-step process, where the first step was diffusion from the bulk solution to a subsurface, and the second step was transport from the subsurface to the surface and the concomitant adsorption. The stagnant layer outside the surface was assumed to be finite due to convection caused by stirring during the measurements. The adsorption was observed to be diffusion controlled, and the concentration immediately outside the surface was determined by a local equilibrium in the sublayer region. The micelles were assumed to contribute to the adsorption only by releasing monomers during diffusive transport and not by direct adsorption. (15) Rubio, J.; Kitchener, J. A. J. Colloid Interface Sci. 1976, 57, 132. (16) Behl, S.; Moudgil, B. M. J. Colloid Interface Sci. 1993, 161, 443. (17) Cases, J. M.; Villieras, F. Langmuir 1992, 8, 1251. (18) Ward, A. F. H.; Tordai, L. J. Chem. Phys. 1946, 14, 453.

Langmuir, Vol. 14, No. 5, 1998 1059

The model quantitatively described several characteristics of the adsorption and desorption process. The initially linear increase in the amount adsorbed with time was ascribed steady-state conditions as referring to the concentration gradient within the stagnant layer. The rate of adsorption within this linear region was, for premicellar solutions, shown to be a linear function of bulk concentration, the csac, the thickness of the stagnant layer and the diffusion coefficient of the monomers. Similar relations were found for concentrations above the cmc. Steady-state conditions were also assumed during the linear part of the desorption, and the rate of desorption was predicted from knowledge of the csac, the thickness of the stagnant layer, and the monomer diffusion constant. The description of the conditions prevailing in the other kinetic regimes was, however, less straightforward. The theoretical model presented in this paper is intended to help in the understanding of how different parameters affect the processes of adsorption and desorption as a whole. We make no assumptions about which process is rate determining (the diffusion or the adsorption step). Any surfactants present in the micellar form in the system can contribute to the process of adsorption both by releasing monomers and by being able to adsorb to the surface. Numerical calculations of concentration profiles outside the surface during both adsorption and desorption processes are discussed. Finally, the results obtained by simulating changes in the amount adsorbed with time are compared with experimental data. Materials and Methods Materials. A series of monodisperse poly(ethylene glycol) monoalkyl ethers (CnEm; C10E6, C12E5, C12E8, and C14E6) was purchased from Nikko Chemicals, Tokyo, Japan, and used without further purification. Polished silicon wafers, thermally oxidized to produce a SiO2 layer thickness of ≈300 Å and then cut into slides with a width of 12.5 mm, were provided by Dr Stefan Klintstro¨m, University of Linko¨ping, Linko¨ping, Sweden. The slides were cleaned in a mixture of NH4OH, H2O2, and H2O, followed by cleaning in a mixture of HCl, H2O2, and H2O. The wafers were then stored in ethanol until used. Just before being placed in the ellipsometer cuvette, the slides were cleaned in a plasma cleaner. Methods. The optical properties of the adsorbed surfactant layer were determined by means of in situ null ellipsometry. The instrument used in this study was a modified, automated Rudolph Research thin-film ellipsometer, model 43603-200E. It is equipped with five-phase stepper motors from Berger-Lahr, model VRDM 566, and controlled by a personal computer. The experiments were performed at a wavelength of 4015 Å and with an angle of incidence of Φ ≈ 67.7°. The water used in these experiments was doubly distilled Millipore water with a pH of approximately 5.8. The surfactant was added to the temperaturecontrolled cuvette (25.0 ( 0.1 °C). The contents where stirred at 300 rpm. Rinsing was performed at a continuous flow of 10 mL/min. The mean optical thickness (df) and the refractive index (nf) of the film were calculated from the ellipsometric angles Ψ and ∆. By using de Feijter’s formula19

Γ)

(nf - n0) df dn/dc

with dn/dc as determined by Chiu et al.20 the amount adsorbed could be calculated. The parameter n0 in this formula is the refractive index in the ambient bulk solution. More detailed descriptions of the silica surface preparation, the instrument and the procedure for characterization of thin (19) de Feijter, J. A.; Benjamins, J.; Veer, F. A. Biopolymers 1978, 17, 1759. (20) Chiu, Y. C.; Chen, L. J. Colloids Surf. 1989, 41, 239.

1060 Langmuir, Vol. 14, No. 5, 1998

Figure 1. Adsorption isotherm for C14E6 at T ) 25 °C. The bulk cmc is indicated by an arrow. Note that the bulk concentration of surfactant is presented on a logarithmic scale. The solid line is drawn only to guide the eye.

Brinck et al.

Figure 3. Change in the amount of C14E6 adsorbed with time at different surfactant concentrations ecmc, to the left, and concentrations >cmc, to the right. The initial adsorption rate increases with increasing surfactant concentration: 0.007, 0.009, and 0.01 mM in the graph on the left. The corresponding values in the plot on the right are 0.02, 0.025, 0.05, 0.25, and 1 mM.

Figure 2. Adsorption-desorption kinetics of C16E6 at a bulk concentration of 0.025 mM. The surfactant is injected just prior to t ) 0, while rinsing is initiated at around t ) 3000 s. films adsorbed on layered substrates can be found in previous publications.4,9,21

Experimental Results The aim of this section is to present the characteristic features of poly(ethylene glycol) monoalkyl ether adsorption at the silica-water interface. The results will be further discussed during the presentation of the theoretical model and the comparison of experimental and simulated data. The experimental data in Figures 1-4 have been published in previous papers,5,9 where more information can be found. The C14E6 isotherm shown in Figure 1 serves as a good example of the dependence of the equilibrium adsorption on the total surfactant concentration. The bulk cmc is indicated by an arrow. The plateau adsorption, Γp, (expressed as µmol/m2) increases with decreasing hydrophilic chain length and/or increasing hydrophobic chain length. A typical example of the time dependence of the adsorption is shown in Figure 2. As has been shown previously,5 an adsorption-desorption cycle can be divided into a number of distinguishable regions, each with its own characteristic time dependence. The initial increase in adsorption is approximately linear with time. The slope in this region is proportional to the bulk concentration for concentrations below the cmc. At higher concentrations, the dependence is weaker, but still noticeable. As the amount adsorbed approaches the plateau value, the adsorption rate begins to decrease and finally becomes zero. When rinsing is initiated, there is a time delay before the adsorption starts to decrease. Desorption then begins and proceeds linearly with time until most of the material has been transported away from the surface. The slope in this region is always the same for a given surfactant; i.e., it is independent of the initial concentration but proportional to the csac (see Figure 4) and the inverse of the thickness of the stagnant layer. (21) Landgren, M.; Jo¨nsson, B. J. Phys. Chem. 1993, 97, 1656.

Figure 4. Desorption kinetics for three different hexa(ethylene glycol) monoalkyl ethers: C16E6 (filled diamonds); C14E6 (open diamonds); C12E6 (filled triangles). The insert shows the corresponding process for three different poly(ethylene glycol) monododecyl ethers: C12E5 (open triangles); C12E6 (filled triangles); C12E8 (open circles).

As described in an earlier paper,5 the thickness of the stagnant layer can be estimated from the rate of desorption if the diffusion coefficient of the monomers and the csac are known. The initial adsorption rates illustrated in Figure 3 have been normalized to a stagnant layer thickness of 1 × 10-4 m. Similar previous estimations of stagnant layer thicknesses for ellipsometric setups of the type used here have resulted in values ranging from 2.5 to 20 µm.22,23 It is likely that details such as the exact position of the surface in the cuvette and the point of measurement on the surface, together with several factors connected with the stirrer, are of importance for the thickness of the stagnant layer and can therefore be expected to cause differences in thickness even between experimental setups of the same type. Values of cmc, csac, and Γp for the surfactants discussed in this paper can be found in Table 1. Description of the Model The model is formulated to describe the conditions in the experimental ellipsometric setup. In this, the silica surface is placed vertically in a cuvette with a magnetic stirrer on the bottom of the cuvette. This renders a stagnant layer outside the surface with a thickness, δ, on the order of 1 × 10-4 m.5 Within the stagnant layer, the transport of surfactants in monomer and micellar form occurs by means of diffusive flow, see Figure 5. The concentrations of both components outside the stagnant (22) Corsel, J. W.; Willems, G. M.; Kop, J. M. M.; Cuypers, P. A.; Hermens, W. T. J. Colloid Interface Sci. 1986, 111, 544. (23) Kop, J. M. M.; Corsel, J. W.; Janssen, M. P.; Cuypers, P. A.; Hermens, W. T. J. Phys. 1983, C10, 491.

Kinetics of Nonionic Surfactant Adsorption

Langmuir, Vol. 14, No. 5, 1998 1061

Table 1. Refractive Index Increment (dn/dc),20 Critical Micellar Concentration (Cmc),24 Critical Surface Aggregation Concentration (Csac),9 Plateau Adsorption (Γp),9 and Values of r Determined by Fitting Eq 14 to the Experimental Adsorption Isotherms (See End of the Section Description of the Model) surfactant

dn/dc

cmc, mmol L-1

csac, mmol L-1

Γp, µmol/m2

r

C10E6 C12E5 C12E6 C12E8 C14E6 C16E6

0.129 0.131 0.136 0.142 0.135 0.133

0.90 0.057 0.087 0.092 0.010 0.0017

0.70 0.050 0.065 0.060 0.0060 0.0012

2.5 5.7 4.1 1.8 5.2 5.4

15 80 18 15 15 8

Lucassen was the first to address the problem of describing monomer-micelle kinetics in connection with adsorption kinetics.25 The model he presented described adsorption at the air-water interface. Three processes were considered to take place in solution: monomer diffusion, micellar diffusion, and micellar dissociation. It was assumed that micelles did not adsorb to the hydrophobic surface. The micellar kinetics were treated according to the model of Kreschek et al.31 in which micelles are assumed to be monodisperse with an aggregation number N. To maintain local equilibrium between micelles and surrounding monomers after a perturbation, complete dissolution and formation of micelles is required. The mechanism for this can be described as

NS T SN where S represents the free monomer and SN an aggregate formed by N surfactant molecules. Under these assumptions, one single relaxation time should be observed upon perturbation. A much more elaborate theory, suggesting stepwise aggregation and disintegration, was later developed by Aniansson and Wall32

Si-1 + S T Si Figure 5. Schematic picture of the solution profile outside the silica surface. The thickness of the stagnant layer is strongly reduced relative to the rest of the picture.

layer are considered to be uniform. Transport in the bulk occurs predominantly by convection. Since the surfaceto-volume ratio in the experimental setup is very low, the concentrations in the bulk can be assumed to be constant during the experiment. The silica surface, as it is presented in this model, is planar and homogeneous. The amount adsorbed per unit area is an average value; we are not considering the effects of adsorption in domains at the surface. The adsorption and desorption processes are considered to proceed in two consecutive steps. During adsorption, the first step is the diffusion of the surfactant in monomer and micellar form from the bulk to the subsurface located immediately outside the silica-water interface. The location of the subsurface is determined by the range of the surface-surfactant interactions. Outside the subsurface, the stagnant layer can be considered to be homogeneous. Inside the subsurface, the surface interactions become important. We assumed that the distance between the subsurface and the surface is between 10 and 100 Å. All surfactants inside the sublayer are regarded as being adsorbed. The second step in the adsorption process is the passage into this layer. Both steps occur simultaneously. Parallel with these transport processes, there is continuous equilibration between surfactants in monomer and micellar forms, see Figure 5. The flow of surfactants in monomer and micellar forms at a certain point in the diffusive region, depends on their respective concentration gradients (or to be exact, their chemical potential gradients) at that point. These flows can be calculated using Fick’s laws. The parallel continuous equilibration between surfactants in monomer and micellar forms, has been described in many different ways in reports on dynamic surface tension and adsorption.25-30 (24) Mukerjee, P.; Mysels, K. J. Critical Micelle Concentration; National Bureau of Standards (U.S.): Washington, DC, 1971.

In this model, each step was given its own rate constant. Assuming a micellar size distribution expected from experiments, with two maxima, where one of these represents the monomers (i ) 1) and the other the average micellar aggregation number (i ) n), two relaxation times should be observed upon perturbation of the system. The shorter of these corresponds to monomer exchange, while the longer is associated with the process of micelle formation-dissolution. In our work, we would have preferred to use the theory of Aniansson and Wall to describe the micellization kinetics, but it is far too complex for our purposes. Not only is there a continuous flow between aggregates of different sizes but also each micellar aggregation number corresponds to its own individual diffusion constant. To couple this with diffusion through a stagnant layer and adsorption is too difficult a task, especially since the majority of the characteristic parameters are not known. In presenting the approach we have utilized, we will begin by describing the way in which we handled the kinetics of this process in bulk, where no concentration gradient exists. We then will continue with the way in which this affects the transport through the stagnant layer. In a micellar solution, we can expect a continuous size distribution of the type shown in Figure 6. Amphiphiles in aggregates with an aggregation number of m or less are considered to be in the monomer form (cmon), while amphiphiles in larger aggregates are considered to be in the micellar form (cinmic). The value of m is chosen to be the aggregation number at the minimum in the size distribution curve. It is important to note that cinmic is equivalent to the total concentration of individual amphiphiles in the micelles and not to the concentration of micellar aggregates. (25) Lucassen, J. Faraday Discuss. Chem. Soc. 1976, 59, 76. (26) Miller, R. Colloid Polym. Sci. 1981, 259, 1124. (27) Rillaerts, E.; Joos, P. J. Phys. Chem. 1982, 86, 3417. (28) Dushkin, C. D.; Ivanov, I. B.; Kralchevsky, P. A. Colloids Surf. 1991, 60, 235. (29) Fainerman, V. B. Kolloidn. Zh. 1981, 43, 94. (30) Fainerman, V. B. Zh. Fiz. Khim. 1984, 58, 2006. (31) Kresheck, G. C.; Hamori, E.; Davenport, G.; Scheraga, H. A. J. Am. Chem. Soc. 1966, 88, 246.

1062 Langmuir, Vol. 14, No. 5, 1998

Brinck et al.

correlation time, τslow,.32,37

dNmic Nmic - Nmic,eq )dt τslow

(2)

The left-hand side of this equation can be replaced by the corresponding terms in eq 1 to give Figure 6. Schematic illustration of the equilibrium size distribution in a micellar solution. The average aggregation number of the micelles is denoted n, while the aggregation number at the minimum is denoted m. Surfactants in aggregates with an aggregation number i e m are considered to be in monomer form, while those in larger aggregates are considered to be in micellar form.

If a micellar solution is exposed to a perturbation, surfactants will flow from monomer to micellar form or vice versa. When the stepwise model is applied, the concentration of amphiphiles in the micelles can increase in two different ways; either by a monomer encountering a premicellar aggregate, resulting in a micelle with aggregation number m + 1, or by a monomer encountering a micelle, thereby increasing its original aggregation number (i) by one. The association reaction is very close to being diffusion controlled,32-36 and we therefore assumed the rate of association to be proportional to the concentration of free monomers, c1, in both cases. Similarly, the buildup of amphiphiles in aggregates with aggregation number i, can be expected to be proportional to the concentration of amphiphiles in micelles of size i - 1. Another way to say this is that the probability of a monomer encountering a micellar surface increases with the size of the surface. The rate of formation of amphiphiles in micellar form is then proportional to c1 and cm in the first case and c1 and ci (i > m) in the second case. Each step occurs with a characteristic rate constant kon,i. Let us now turn our attention to micellar decay, which is determined by the rate constant koff,i. The release of surfactants from the micelles is assumed to be proportional to the concentration of surfactants in micellar form ci (i > m). The total change in the concentration of amphiphiles in micellar form induced by the perturbation can then be written

-(m + 1)koff,m+1cm+1 + (m + 1)kon,mc1cm ) cinmic - cinmic,eq (3) τslow We will assume that τslow can be assigned a constant value; i.e., the concentration dependence on the correlation time is negligible. The next step is to rewrite the sums in eq 1 to give an effective rate constant multiplied by the total concentration of amphiphiles in micellar form:

∑

kon,ici ) koncinmic

(4a)

koff,ici ) koffcinmic - koff,m+1cm+1

(4b)

i ) m+1

∑

i ) m+2

The rate constants defined here will depend on the size distribution of the aggregates. We will, however, assume that they are constant. By recognizing that cmon ≈ c1 and using eqs 3 and 4 to rewrite eq 1, we find that the time dependence of the amphiphile concentration in micellar form, can be written

cinmic - cinmic,eq dcinmic )- koffcinmic + koff,m+1cm+1 + dt τslow cmonkoncinmic (5a) At equilibrium, the following relation prevails

0 ) -koffcinmic,eq + koff,m+1cm+1,eq + cmonkoncinmic,eq (5b) which can also be written as

cinmic,eq ) dcinmic dt

) -(m + 1)koff,m+1cm+1 + (m + 1)kon,mc1cm -

∑

i ) m+2

koff,ici + c1

∑

i ) m+1

kon,ici (1)

It can be noted that the two sums give the change in concentration of amphiphiles in micellar form corresponding to the fast monomer-micelle equilibrium described by Aniansson and Wall.32 The number of micelles does not change during this fast step. The first two terms on the right-hand side in eq 1 correspond to the slow step, during which micelles are formed and disintegrated. According to Aniansson, the change in the number of micelles (Nmic) can be described by a characteristic (32) Aniansson, E. A. G.; Wall, S. N.; Almgren, M.; Hoffman, H.; Kielmann, I.; Ulbricht, W.; Zana, R.; Lang, J.; Tondre, C. J. Phys. Chem. 1976, 80, 905. (33) Frindi, M.; Michels, B.; Zana, R. J. Phys. Chem. 1992, 96, 6095. (34) Frindi, M.; Michels, B.; Zana, R. J. Phys. Chem. 1992, 96, 8137. (35) Hoffmann, H.; Kielman, H. S.; Pavlovic, D.; Platz, G.; Ulbricht, W. J. Colloid Interface Sci. 1981, 80, 237. (36) Kato, S.; Harada, S.; Sahara, H. J. Phys. Chem. 1995, 99, 12570.

koff,m+1cm+1,eq koff - koncmon

(5c)

Subtracting eq 5b from eq 5a gives

(

(

))

koff dcinmic 1 ) kon cmon + (cinmic - cinmic,eq) + dt kon τslowkon koff,m+1(cm+1 - cm+1,eq) (6a) In most cases, the concentration of surfactants in aggregates of size (m + 1) is very small in comparison with the total concentration of surfactants in aggregates. We will therefore assume that the last term in eq 6a is negligible in comparison with the first term, in the systems we will study.

(

(

))

koff dcinmic 1 ≈ kon cmon + (cinmic - cinmic,eq) dt kon τslowkon (6b) If we define the cmc as the free monomer concentration in a solution with a total concentration much greater than (37) Aniansson, E. A. G. Prog. Colloid. Polym. Sci. 1985, 70, 2.

Kinetics of Nonionic Surfactant Adsorption

Langmuir, Vol. 14, No. 5, 1998 1063

Figure 7. Equilibrium concentrations of surfactants in monomer and in micellar form as a function of total concentration (Ctot ) Cmon + Cinmic). The concentrations were calculated from eq 10 with R ) 0.01 and m ) 20. Note that Cinmic is equivalent to the concentration of individual surfactants in the micelles and not to the concentration of micellar aggregates. All the concentrations are expressed as fractions of the cmc.

the cmc, eq 6b can be written as

dcinmic ) kon(cmon - cmc)(cinmic - cinmic,eq) dt

(7)

To facilitate further calculations, we now introduce the following relative concentrations

Cmon ) concentration of amphiphiles in monomer form cmc (8a) Cinmic ) concentration of amphiphiles in micellar form (8b) cmc Using these expressions we can rewrite eq 7

dCinmic ) koncmc(Cmon - 1)(Cinmic - Cinmic,eq) dt

(9)

If Cinmic, eq is not known for all monomer concentrations, the following equation can generally be used to describe this relation

Cinmic,eq )

RCm+1 mon 1 - Cmon

(10)

where R and m are related to the cooperativity of the micellization process. In Figure 7, it can be seen how eq 10 describes the variations of Cinmic,eq and Cmon,eq with total concentration. By using Fick’s second law and eqs 9 and 10, the following expressions for the variation in the surfactant concentrations in monomer and micellar forms with time and distance from the surface in the stagnant layer can be obtained: 2

d Cinmic dCinmic ) Dmic + koncmc(RCm+1 mon + 2 dt dx Cinmic(Cmon - 1)) (11a) d2Cmon dCmon ) Dmon - koncmc(RCm+1 mon + 2 dt dx Cinmic(Cmon - 1)) (11b)

Figure 8. Adsorption isotherms calculated from eqs 14 and A12a (A refers to Appendix A). The amount adsorbed (Γ) is presented as a function of the total concentration (Ctot ) Cmon + Cinmic) in the bulk. The isotherms have been calculated for two different fractions of csac/cmc and for each of these for two different values of r: csac/cmc ) 0.7, r ) 20 (open squares); csac/cmc ) 0.7, r ) 80 (filled triangles); csac/cmc ) 0.85, r ) 20 (open circles); csac/cmc ) 0.85, r ) 80 (filled diamonds). Ctot is normalized with respect to the cmc, while Γ is normalized with respect to Γp.

These equations can be solved numerically if the boundary conditions at the interfaces toward the bulk (x ) δ) and toward the sublayer (x ) δ1) immediately outside the surface are known. At x ) δ, the concentration of amphiphiles in monomer and micellar form are determined by their equivalents in the bulk solution. The boundary conditions at the subsurface will depend on the concentrations of surfactants in monomer and micellar forms in the sublayer and the change in these concentrations with time. If the same type of relation between buildup and disintegration of aggregates is used at the subsurface as was previously assumed for the bulk, the following boundary conditions result:

dCmon konΓp ) (Γ + Rsurf(Cmon))(Cmon - Cmon,eq(Γ)) dx Dmon (12) dCinmic kon,mic,s ) (1 - Γ)(Cinmic - Cinmic,eq(Γ)) dx Dmic

(13)

In these equations, we have introduced Γ, the amount of adsorbed surfactant (normalized to Γp), and the function Rsurf(Cmon), which is related to the cooperativity of the aggregation process at the surface. The flow of surfactants in micelles across the subsurface is assumed to be determined by a characteristic rate constant kon,mic,s. Cmon,eq and Cinmic,eq are the equilibrium concentrations at a given value of Γ. The relation between the two concentrations and the amount adsorbed is given by the experimental adsorption isotherm. If the complete adsorption isotherm, as a function of concentration of amphiphiles in monomer and micellar form, is not known, the following relation can generally be used

Γ)

1 r

(

1 csac 1+ -1 cmcr - csacr Crmon,eq

)

(14)

together with eq 10. Csac is defined as the monomer concentration at which Γ ) 0.5, and r is used as a fitting parameter to describe the adsorption isotherm. The effect of changes in r on the adsorption isotherm can be seen in Figure 8. A detailed description of the steps leading to eqs 12 and 13 can be found in Appendix A.

1064 Langmuir, Vol. 14, No. 5, 1998

Brinck et al.

By using the boundary conditions presented above, we can now calculate the total flow of amphiphiles through the subsurface at x ) δ1. All surfactants within the sublayer are considered to be adsorbed and this flow is therefore equal to the change in the amount adsorbed with time:

dCinmic dCmon dΓ + ADmoncmc AΓp ≈ ADmiccmc dt dx dx

(15)

To find a numerical solution to the problem defined by eqs 11-13 and 15 we used a finite-element method. A description of this method can be found in ref 38. The model presented here can be divided into two parts; one related to the transport of surfactants within the stagnant layer and the other related to the surface processes. In this paper numerical calculations based on the model will be compared with experimental results of adsorption of surfactants to a planar silica surface. The model is however not limited to this field. A change in geometry of the surface enters in the diffusion equations, and by exchanging those for the ones related to for example spherical symmetry, the case of particles can be investigated. If a different surface is used, the effect of this can easily be studied by exchanging the old equilibrium adsorption isotherm for the new one. Returning to the field of the experimental results considered here, we will continue with a short discussion of the numerical values used in the calculations presented in this paper. Beginning with those related to the equilibrium conditions in the bulk solution, m was set to 20 while R was set to 0.01. These values correspond to the distribution between surfactants in monomer and micellar forms presented in Figure 7. An increase in m or decrease in R leads to increased cooperativity of the micellization process (sharper cmc). The effects of changes in m and R can easily be studied further by using eq 10. If we assume that the deviations from equilibrium concentrations are small, which in most cases can be considered to be the case, then the characteristic rate constants, kon and koff, of this model, will be similar to the rate constants corresponding to the fast relaxation as described by Aniansson. Some investigations of the fast relaxation times for nonionic surfactants have been reported.33-36 Two of these studies concern CnEm surfactants.33,36 The association rate constant (k+) for ionic surfactants decreases slightly with decreasing cmc, due to electrostatic effects, but for nonionic surfactants k+ does not vary with the cmc. It is important to recognize the discrimination between the concentration of micelles and, as it is defined in this model, the concentration of amphiphiles in micellar form, when determining the rate constants. Usually, the dissociation rate constant, k-, is expressed as the number of monomers leaving one micelle per second, while k+ represents the number of monomers entering one micelle per second. We therefore need to use kon ) k+/n and koff ) k-/n when relating experimentally determined parameters to the rate constants of this model. One should not give any physical significance to the fraction k+/n; it merely reflects the difference in concentration units. When transferring the rate constants from the unit (per micelle) to the unit (per amphiphile in micellar form) we used the data in all four publications on nonionic surfactant kinetics, since their unanimous results did not give us any reason to discriminate between them. The publications include data which indicate that the association of monomers to a micelle is a diffusion(38) Jo¨nsson, K. A.-S.; Jo¨nsson, B. T. L. AIChE J. 1992, 38, 1349.

controlled process with k+ approximately equal to 7 × 109 M-1 s-1. Each value of k+ (at 25 °C) was divided by its corresponding experimentally determined value of n to give an average value of kon equal to 7 × 107 M-1 s-1. This was then used in eqs 11 and 12. Values of r were obtained by fitting the experimental isotherms of the individual surfactants to the relationship between Ctot and Γ given by eqs 14 and 10 together with the fact that Ctot ) Cmon + Cinmic. The results of these fits can be seen in Table 1, while the experimental isotherms can be found in a previous publication.9 In the simulations, the function Rsurf(Cmon) was described by eq 16 (see Appendix A). The definition of m, as the

Rsurf(Cmon) )

δ1(cmc)R(m + 1) m Cmon Γp(csac/cmc)m

(16)

aggregate number at the minimum in the size distribution curve, is the same both in the bulk and at the surface. The numerical values, however, cannot, without further analysis, be assumed to be the same. As a preliminary value for m at the surface we used m ) r where r is given in Table 1. The practical significance of Rsurf is to initiate nucleation of surface micelles. We found that Rsurf is only of minute importance in the adsorption processes in the systems we have studied. The reason for this seems to be that in almost all cases a very small, but still sufficient, amount of micelles diffuse in from the stagnant layer to the surface in time to serve as nuclei for the further adsorption of material. The monomer diffusion coefficient used in the simulations presented in this paper is 4 × 10-10 m2/s. This is the same as the value reported by Kamenka et al. for C12E8 in H2O at 33 °C.39 Another report presents measurements on C12E5 in D2O at 25 °C.40 After having taken into account the viscosity differences, the resulting coefficient is equal to 3.9 × 10-10 m2/s in H2O. The micellar diffusion coefficient used in the simulations presented in this paper is 10 × 10-11 m2/s. This value has been determined experimentally for C14E6 for conditions identical to those under investigation here.5 As can be seen in the section “Dependence on the Diffusion Coefficient of Amphiphiles in Micellar Form”, Dmic is of very little importance for the desorption process. This made it possible for us to use this value of Dmic throughout the comparison between simulations and experimental results. Discussion We begin this section by demonstrating how the model describes a typical adsorption and desorption experiment of the kind shown in Figure 2. When the general features have been pointed out, we will show how the flow through the stagnant layer and the adsorption are influenced by factors such as the diffusion constant of surfactants in micellar form (Dmic) and the total surfactant concentration. From the knowledge gained from this, we are able to draw some conclusions about the dependence of the transport through the stagnant layer on the cmc of the surfactant. Another parameter which has a considerable influence on the rate of adsorption is the thickness of the stagnant layer (δ). We will use the model to compare the course of adsorption for two different thicknesses of this layer. (39) Kamenka, N.; Puyal, M.; Brun, B.; Haouche, G.; Lindman, B. In Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1984; Vol. 1, p 359. (40) Scho¨nhoff, M.; So¨derman, O. J. Phys. Chem. B 1997, 101, 8237.

Kinetics of Nonionic Surfactant Adsorption

Figure 9. Concentration profiles demonstrating the flow of surfactants in monomer (lower graph) and in micellar (upper graph) forms in the stagnant layer during adsorption from a 0.1 mM C12E5 solution. Initially, at t ) 0, the concentration is zero within the whole layer. The concentration profiles during the course of adsorption are presented at six different times after adsorption is initiated. When the system has reached equilibrium, the concentrations of surfactants in the monomer and micellar forms within the layer equal those in the bulk, i.e., 0.056 and 0.044 mM. The mathematical relation between the two concentrations is described by eq 10.

Figure 10. The amount adsorbed as a function of time during adsorption (left graph) and desorption (right graph) from a 0.1 mM C12E5 solution. In the left graph the total amount adsorbed is shown by the open squares. The total adsorption is the result of both adsorption of surfactants in monomer form (open circles) and surfactants in micellar form (open diamonds). Note that the time during which the adsorbed amount actually changes is small in comparison with the total time required to fill and to empty the stagnant layer (cf. with Figures 9 and 11).

Finally, results obtained with the model will be compared with those measured using ellipsometry. General Features during Adsorption. When the simulation described in Figure 9 and the left part of Figure 10 begins, a micellar solution of nonionic surfactant has been put into contact with a planar silica surface. Outside the surface, there is a stagnant layer which the surfactants have not yet begun to penetrate. The parameters used in this simulation are those of a 0.1 mM C12E5 solution and a thickness of the stagnant layer corresponding to that of the setup used in the ellipsometric measurements (i.e. δ ) 1 × 10-4 m). As the micelles diffuse toward the surface, they experience a monomer concentration lower than the cmc and therefore release monomers. When the concentration immediately outside the surface has reached ≈0.9csac,

Langmuir, Vol. 14, No. 5, 1998 1065

Figure 11. Concentration profiles demonstrating the flow of surfactants in monomer (lower graph) and in micellar (upper graph) form through the stagnant layer during desorption from a 0.1 mM C12E5 solution. Initially, at t ) 0, the bulk concentration is zero, the monomer concentration within the layer is 0.056 mM while the concentration of surfactants in micellar form is 0.044 mM.

the adsorption of surfactants at the surface becomes noticeable. At this point, the micellar concentration outside the surface is very low (about 2 × 10-5 mM). After a short while, the amount adsorbed increases almost linearly with time. During this time, the concentration profiles of the surfactant in monomer and micellar form in the stagnant layer change rather slowly. As the adsorption approaches its plateau value (Γp), there is a slight decrease in the adsorption rate before the adsorption finally reaches its final value. Although the net flow of surfactants to the surface now is zero, the system is not yet in equilibrium. The concentration of surfactants within the stagnant layer will continue to increase until the concentration in the stagnant layer and in the bulk are the same. Looking at the whole adsorption process, it can be noted that the main contribution to the adsorption is the addition of monomers to surface micelles. General Features during Desorption. When the simulation described in the right part of Figure 10 and Figure 11 starts, the bulk surfactant solution has been exchanged to pure water in one step. The stagnant layer is still filled with the original bulk solution, i.e., 0.1 mM C12E5. Monomers and micelles begin to diffuse out through the stagnant layer. The micelles experience a monomer concentration lower than the cmc when they diffuse out of the layer, and hence, they release monomers. As the surfactants reach the bulk solution, they are immediately removed by convection. Desorption from the surface does not begin until most (but not all) of the surfactants in micellar form have been transported away from the stagnant layer. Soon, the amount adsorbed decreases almost linearly with time. As we can see in Figures 10 and 11, most of the material is transported away from the surface while the concentration outside the surface is close to the csac, which gives a concentration gradient across the stagnant layer equal to csac (since Cbulk ≈ 0). This gives a cmc (csac ≈ 0.7cmc) dependence of the slope of desorption. Emptying the stagnant layer of surfactants is a slow process, which proceeds long after the experi-

1066 Langmuir, Vol. 14, No. 5, 1998

mentally measured amount adsorbed has passed below the limit of detection. Dependence on Cooperativity. First we would like to comment on the effect of the cooperativity of micelle formation in the bulk solution on the types of concentration profiles seen in Figures 9 and 11. These figures show that a narrow tongue of amphiphiles is also present in micellar form where the monomer concentration is below the cmc, during both adsorption and desorption. This is partly a kinetic effect due to the fact that the dissociation of micelles is not immediate, and local equilibrium conditions between the amphiphiles in monomer and in micellar forms do not prevail. It is also an effect of the equilibrium properties of the system. The critical micelle concentration is not considered to be infinitely sharp. As can be seen in Figure 7, there is a finite concentration of surfactants in micellar form also at total concentrations below the cmc. A narrower transition region, a sharper cmc, is obtained in this model by decreasing R or increasing m (see eq 10). Cooperativity is also discussed when studying adsorption isotherms and the process of interest is then micelle formation at the surface. To increase the cooperativity and thereby approach a step isotherm, r is increased. The effect on the isotherm of a change of r can be seen in Figure 8. If a step isotherm had been used, there would have been no adsorption until the concentration outside the surface had reached the csac. This can be compared with the simulation discussed above, where the concentration outside the surface was around 90% of the csac at the onset of adsorption. Furthermore, instead of the concentration outside the surface increasing slowly while the adsorption increases, the concentration would be constant and equal to the csac until the adsorbed layer is close to saturation. The same trend can be seen in the desorption behavior. Desorption would not begin until the csac was reached outside the surface, and the concentration outside the surface would not change during the linear decrease in the amount adsorbed from Γp to 0. The changes described above imply that local equilibrium conditions prevail at the surface. A more thorough discussion of the kinetic effects due to the two simultaneous processes, one of which is the flow of material to the subsurface and the other the accommodation of material at the surface, will follow later. The simulations presented below describe the course of adsorption and desorption through a stagnant layer for a surfactant with the same characteristics as C14E6. The cmc of C14E6 is about 6 times lower than that of C12E5 (see Table 1). It should also be borne in mind that the dissociation rate constant, koff, scales with the cmc (see discussion in the section Description of the Model). These are not the only properties that differ between the two surfactants, which makes a direct comparison between their concentration profiles nontrivial. We will therefore begin by illustrating how three other factors; the csac/ cmc ratio, the overall concentration, and the micellar diffusion coefficient, affect the profiles using C14E6 as an example. We will then return to the question of the importance of the cmc. Dependence on the Relation between Csac and Cmc. Experimental data indicate that the ratio csac/ cmc for C14E6 is equal to 0.6, which is small compared with some other nonionic surfactants (see Table 1). We wish to study the effect of an increasing csac/cmc ratio on the adsorption-desorption process. Since the concentration profiles do not change significantly during the time when most of the adsorption occurs (see Figures 9 and 10), we will use so-called “half-time” profiles when

Brinck et al.

Figure 12. Concentration profiles for surfactants in monomer (lower graph) and in micellar (upper graph) forms during adsorption from a C14E6 solution. The profiles have been determined at the time when the amount adsorbed at the surface is equal to half Γp for each of the individual simulations. Four different cases are shown: reference conditions with csac ) 0.006 mM, total concentration ) 0.02 mM and Dmic ) 10 × 10-11 m2/s with solid lines (s); the csac was then increased to 0.0085 mM (r ) 22) (- - -); the total concentration increased to 0.10 mM (- s); and Dmic decreased to 2.5 × 10-11 m2/s (- -).

comparing the major features of the different adsorption processes. The term “half-time” is used to denote the concentration profiles in the stagnant layer determined at the instant at which the amount adsorbed is equal to half the plateau adsorption. First of all, we will show the adsorption behavior under reference conditions. The half-time concentration profiles (after 105 s) during adsorption from a 0.02 mM C14E6 solution (2cmc) are shown as the two solid lines (s) in Figure 12. Here, csac is set to the experimentally determined value of 0.6cmc (0.006 mM). We now wish to increase csac, but an unwanted effect of doing this is that the steepness of the isotherm changes (see eq 14 and Figure 8). We therefore determined a new value of r, which gave the same steepness as the original isotherm. In Figure 12, it can be seen that increasing csac from 0.6cmc to 0.85cmc, while keeping other conditions (including cooperativity) constant, strongly affects the concentration profiles at half the plateau adsorption (csac v (- - -)). The amount adsorbed versus time curves during both adsorption and desorption for the two values of csac are shown in the two upper graphs in Figure 13. From the concentration profiles it is clear that the penetration depths of the surfactants in micellar form increase with increasing csac. This can intuitively be understood from the fact that surfactants in micellar form in a system with a high csac/cmc ratio experience a slower drop in monomer concentration as they diffuse in toward the surface. It can also be viewed as the result of the system equilibrating the two flows of amphiphiles in monomer and in micellar forms. To illustrate this, we will use a simplified picture in which the flows are approximately given by

Jmon ) -Dmoncmc and

dCmon cmc - csac ) -Dmon (17a) dxmon δmon

Kinetics of Nonionic Surfactant Adsorption

Langmuir, Vol. 14, No. 5, 1998 1067

where they have a lower diffusion coefficient also contributes. It can also be noted that in the case of csac ) 0.85cmc, as much as 7% of the total amount adsorbed at the plateau is adsorbed in micellar form, while the micellar contribution in the case of csac ) 0.6cmc is negligible. Turning to the dependence of the desorption on the csac we see the opposite trend. As can be seen from the upper right graph in Figure 13, desorption is faster when the csac is higher. This is to be expected since we concluded in the discussion regarding C12E5 that most of the material transport away from the surface occurs while the monomer concentration outside the surface is around the csac. A higher csac implies a steeper concentration gradient and a faster flow

dΓ csac ≈ -Dmon dt δ

Figure 13. Amounts adsorbed as a function of time during adsorption (graphs to the left) and desorption (graphs to the right) from a C14E6 solution. In the two upper graphs the effect of csac on the course of adsorption and desorption can be seen, the two middle graphs show the effect of total concentration, while simulations for two different values of Dmic can be compared in the lower graphs. The adsorption and desorption curves obtained under reference conditions, are the same in all the plots (filled circles) (csac)0.006 mM, total concentration)0.02 mM and Dmic ) 10 × 10-11 m2/s).

Jinmic ) -Dmiccmc

dCinmic Ctotcmc - cmc ) -Dmic dxinmic δinmic (17b)

The sum of the two parameters δmon and δinmic is the thickness of the stagnant layer. δmon is the thickness of the layer through which the amphiphiles pass in monomer form while δinmic is the part of the layer through which the amphiphiles pass in micellar form. If the two flows can be considered equal, these equations can be combined to give

Dmon 1 - csac/cmc δmon ) δinmic Dmic Ctot - 1

(18)

Everything on the right-hand side is kept constant except the csac. Since the difference (1 - csac/cmc) decreases when we change the fraction csac/cmc from 0.6 to 0.85, the fraction δmon/δinmic must decrease. A larger δinmic is seen in the concentration profiles in Figure 12 as the amphiphiles in micellar form being transported a greater distance through the stagnant layer. The consequences of this can be seen in the upper left graph in Figure 13, where the rate of adsorption is shown to decrease as the csac is increased. The adsorption is slowed partly due to a lower total concentration difference over the stagnant layer (Ctot - 0.85) instead of (Ctot - 0.6). However, the fact that the amphiphiles are transported a relatively speaking longer distance in micellar form

(19)

An increased monomer concentration close to the surface slows down the dissolution of micelles in that area. Micelles are therefore present outside the surface for a much longer time when csac is increased. The increased monomer concentration gradient has, however, the greatest effect on desorption. This is evident from the fact that the ratio of the two slopes (dΓ/dt) is 0.73 while the ratio of the two csacs, 0.6cmc divided by 0.85cmc, is equal to 0.71. Dependence on Total Concentration. We will now discuss the effects of an increase in the total bulk concentration from 0.02 to 0.1 mM, i.e., from two to 10 times cmc, keeping all other parameters constant. The solid lines (s) in Figure 12 show the curve obtained for 0.02 mM C14E6 as a reference, while the 0.1 mM concentration profiles are depicted as (Ctot v (- -)). The concentration profiles clearly show the increased importance of transport in micellar form. By using eq 18 we can also interpret this as a result of the system equating the monomer and micellar flows through the stagnant layer. An increase in the total bulk concentration makes the difference (Ctot - 1) larger which in turn increases δinmic at the expense of δmon. Turning our attention to the amount adsorbed as a function of time, presented in the middle graphs in Figure 13, we see that the effect of increasing the total concentration gradient during adsorption is an increased rate of adsorption. After rinsing has been initiated, it takes a long time until the actual decrease in the amount adsorbed is seen. However, once the desorption from the surface has started, the rate of desorption is the same. The change in time before onset is due to the difference in time required to transport enough material from the stagnant layer to decrease the concentration outside the surface to ≈csac. When desorption begins, the concentration profiles are so similar for the different concentrations, so that the slopes of the amount adsorbed as a function of time become approximately equal. Dependence on the Diffusion Coefficient of Amphiphiles in Micellar Form (Dmic). It is important to note that the model presented in this paper does not take into account different micellar sizes in any other way than what is reflected in the average micellar diffusion constant. In the simulations of adsorption and desorption of C14E6 we have used a value of 10 × 10-11 m2/s, which has been experimentally determined from adsorption data in a previous paper.5 This is an effective value averaged over the amphiphiles in micellar form in the stagnant layer

1068 Langmuir, Vol. 14, No. 5, 1998

during adsorption. In cases where one wants to use this model to simulate a certain adsorption and desorption process where it is not possible to obtain the corresponding information, it is necessary to bear in mind how variations in Dmic affect the course of adsorption and desorption. Here, we give an example of how a decrease in Dmic influences the concentration profiles and the rate of adsorption and desorption. In Figure 12, the solid lines (s) show the concentration profiles at half Γp when the experimentally determined value of Dmic ) 10 × 10-11 m2/s is used. A reduction in the diffusion coefficient to 2.5 × 10-11 m2/s gives profiles drawn as (Dmic V (- -). The amount adsorbed as a function of time during both adsorption and desorption for the two different values of Dmic can be seen in the two lower graphs in Figure 13. If we begin by examining the amount adsorbed as a function of time, we see that the adsorption slows down as Dmic is decreased. A lower value of Dmic means that the transport of surfactants in micellar flow is slowed compared with our reference case, but the system requires this flow to be equal to the monomer flow and, therefore, responds by adjusting the gradients. When the monomer gradient is decreased and the micellar gradient is increased, the flows of surfactants in monomer and in micellar form again become equal, but the total transport to the surface is slowed. To get a rough idea of how the profiles will change due to a decrease in Dmic, eq 18 can once more be employed. The desorption on the other hand, is hardly affected by changes in Dmic, as can be seen in Figure 13. It takes a slightly longer time for the desorption to begin, but this is negligible in comparison with the total time of desorption. Dependence on Cmc. We have now reached a point where we can say which differences exist in the adsorption and desorption behavior of two surfactants with different cmcs, according to the model presented in this paper. From the above, we know that in order to compare the effects of cmc alone, the parameters csac/cmc, Ctot and Dmic must be the same in the two simulations. However, two surfactants with cmcs that differ by, for example, a factor of 5, will, in cases when the equilibrium relationship koff ) koncmc is valid, have dissociation rate constants, koff, which also differ by a factor of 5, since kon is constant (see end of the section Description of the Model). If there are no other differences than cmc and koff between the two cases, the model indicates that there will be no difference between the concentration profiles during adsorption and desorption. The following discussion will show why. The surfactant with the higher cmc is called SH while the other is called SL. To facilitate the comparison we assume some parameter values: cmc(SH) ) 0.05 mM, cmc(SL) ) 0.01 mM, and total concentrations of 0.1 mM SH and 0.02 mM SL. Csac/cmc is equal to 0.6 for both surfactants. If one starts with adsorption, the monomer flow to the surface is five times faster for SH than for SL, due to the higher gradient, which is proportional to (cmc - csac). The same is true for the flow of surfactants in micellar form. The gradient in this case is proportional to (total concentration - cmc). If the flow is much faster for the surfactant with the higher cmc, why then does the penetration depth of surfactants in micellar form not increase? The reason is that the dissociation of micelles is 5 times faster since koff scales with cmc for this type of surfactant. The net result of all this is that the concentration profiles (with reduced concentrations on the y-axes) becomes identical. The same reasoning holds for the desorption process. Large differences are, however, observed in the amount adsorbed as a function of time for the two different surfactants, SH and SL. However, due to the concomitant changes in the

Brinck et al.

Figure 14. Concentration profiles for surfactants in monomer and micellar form during adsorption from a 0.05 mM C14E6 solution. The figure shows diffusion through a stagnant layer for two different thicknesses of the layer; δ ) 100 µm (s) and δ ) 10 µm (- -). The profiles were determined at the time when the amount adsorbed at the surface was equal to half Γp for each of the simulations.

Figure 15. Amounts adsorbed as a function of time during adsorption (left graph) and desorption (right graph) from a 0.05 mM C14E6 solution. The figure shows adsorption and desorption through a stagnant layer for two different thicknesses of the layer: δ ) 100 µm (filled circles) and δ ) 10 µm (open squares).

cmc and the total concentration, these simulation results have not been included. Importance of Stagnant Layer Thickness (δ). In all the simulations presented so far we have used a stagnant layer thickness equal to the average thickness obtained in our experimental setup ≈ 100 µm. Since this can vary between a few µm and, in principle an infinite thickness in other types of experiments, we will give an example of the importance of the size of δ. In Figure 14, we can study the concentration profiles at half the plateau adsorption during adsorption from a 0.05 mM C14E6 solution (5cmc) in systems with two different stagnant layer thicknesses; 10 and 100 µm. The amount adsorbed as a function of time is shown in Figure 15. It is clear from the concentration profiles that the surfactants in micellar form come much closer to the surface when δ is reduced by a factor of 10. Intuitively, we can also understand the large increase in adsorption and desorption rates due to the larger concentration gradients (proportional to the inverse of δ). If we focus on the adsorption from a 0.05 mM solution of C14E6 through the thin stagnant layer (δ ) 10 µm) an interesting phenomenon related to the formation of the very first micelles at the surface can be seen in Figure 16. This figure shows three monomer concentration profiles at very short times. The first profile determined after 0.01 s shows how the monomers begin to enter the stagnant layer; 0.07

Kinetics of Nonionic Surfactant Adsorption

Figure 16. Monomer concentration profiles during adsorption from a 0.05 mM C14E6 solution through a stagnant layer of thickness δ ) 10 µm. Due to the very thin stagnant layer, transport of material to the surface is faster than the initial adsorption. This is seen in that the monomer concentration outside the surface initially increases beyond csac which is equal to 0.006 mM (t ) 0.08 s) and is then reduced again as the rate of adsorption increases (t ) 0.13 s).

Figure 17. Evolution of the monomer concentration immediately outside the surface during adsorption from a 0.05 mM C14E6 solution. The thickness of the stagnant layer is 10 µm. The nucleation step at the surface results in a maximum in the monomer concentration curve.

s later the concentration immediately outside the surface has increased beyond csac (csac ) 0.006 mM), but there is still no adsorption. When 0.13 s have lapsed since the start of the simulation, the monomer concentration has decreased well below the csac and adsorption has begun. In Figure 17, the monomer concentration immediately outside the surface is shown as a function of time. What happens is that, initially, the flow of surfactant to the surface is so fast that the surface cannot accommodate all the incoming material. The maximum in the curve in Figure 17 represents a nucleation step. From the simulations we can see that the main reason for the relaxation of the monomer concentration after ≈0.08 s is not the formation of micelles from monomers at the surface, but it is the diffusional flow of bulk micelles to the surface that provide nuclei for incoming monomers to build on. Here we have a case where, according to the model, we no longer have a purely diffusion-controlled process. For reasons discussed at the end of the section “Description of the Model”, one should not place too much significance on the absolute values in this case but instead should regard it as an example of how a change in the thickness of the stagnant layer can change the qualitative behavior of the adsorption process. Comparison between Model and Experiments. Below, some features of nonionic surfactant adsorption and desorption kinetics are discussed which a model of this type can explain, and some which cannot be accommodated within this approach. Adsorption Kinetics. In Figure 18, the evolution of the amount adsorbed with time is presented for some concentrations of C14E6 below and above the cmc. The left-hand part of Figure 18 shows the experimental data

Langmuir, Vol. 14, No. 5, 1998 1069

Figure 18. Evolution of the amount adsorbed with time at different C14E6 concentrations according to the following: experimental results (left graph); results from simulations (right graph). The concentrations in the two graphs are, in order of increasing initial rate of adsorption as follows: 0.009, 0.025, 0.05, and 0.25 mM.

Figure 19. Desorption kinetics for three different hexaethylene glycol monoalkyl ethers: C16E6 (filled diamonds); C14E6 (open diamonds); C12E6 (filled triangles). The main graph shows experimental data while the insert shows corresponding results from simulations.

while the right shows the simulated data. Under the subtitle “Dependence on Total Concentration” earlier in this section, we gave examples of how this model describes the dependence of the adsorption rate on the bulk concentration. Here, we have the possibility to determine the agreement between simulations and experiments. Beginning with the qualitative features, the experimental adsorption initially increases linearly and then levels off, and finally, an adsorption plateau is reached. Data obtained from simulations display the same initial linear behavior, but the transition between the linear increase in the amount adsorbed and the constant amount adsorbed is more abrupt. At very high bulk concentrations, simulations indicate a small bend before the increase in adsorption with time assumes a more constant value of zero. Due to the experimental time resolution, it is not possible to determine whether there is a similar bend before the linear region in the experimental adsorption curves or not. A quantitative comparison between the initial slopes of the experimental and the theoretical adsorption curves reveals good agreement between the two sets of data. Desorption Kinetics. In Figure 19 the evolution of the amount adsorbed as a function of time during desorption is presented for three surfactants with large differences between their cmcs (and csacs); see Table 1. The main plot shows experimental data, while the insert shows equivalent simulation data. If we look first at the qualitative features of the experimental desorption curves, we see that soon after desorption is initiated, the decrease in the amount adsorbed is linear with time. In the section “Experimental Results”, it was pointed out that the slope in this region seems to be independent of the initial concentration for a given surfactant. As zero adsorption

1070 Langmuir, Vol. 14, No. 5, 1998

is approached, the absolute value of the slope decreases slightly. Regarding the simulations presented in the insert, we can see that they have the same linear region during which most of the surfactant is transported out, but there is no slow bend at the end of the desorption curve. The effect of initial concentration on the desorption according to the model was discussed in “Dependence on Total Concentration” and will therefore not be further discussed here, but in short, the simulations display the same behavior as the experimental data. The change in slope with csac (csac ≈ 0.7cmc) is very clear when going from C12E6 to C16E6. The origin of this behavior is that the aggregates at the surface maintain the monomer concentration immediately outside the surface approximately equal to csac, which produces a concentration gradient equal to csac across the stagnant layer (see “General Features during Desorption”). A quantitative comparison between the equivalent slopes reveals relatively good agreement. Some discrepancy can be expected due to the variation in stagnant layer thickness between the different experimental runs. Concluding Remarks The formulation of a model that describes the adsorption from a micellar solution through a stagnant layer to a silica surface has allowed calculations of the amounts adsorbed as a function of time and concentration profiles within the stagnant layer. Results from the simulations have been compared with experimental results obtained from short-chain CnEm surfactant adsorption studies by in situ null ellipsometry. The increase in adsorption rate with increasing total concentration and the decrease in desorption rate with decreasing cmc are well described by the model. Other experimentally observed phenomena, for instance, the fact that an increase in the initial total concentration does not affect the rate of desorption once desorption is started but only delays its onset, are also reproduced and explained by the simulations. What cannot be explained by a model of this type, however, are the soft bends noticed during adsorption just before the adsorbed amount reaches the plateau and during desorption as the amount adsorbed approaches zero. One way to use this model is, as already discussed, to interpret experimental data. Another is to study the effects of changes in conditions which cannot be varied in a sufficiently controlled way experimentally. An example of this, which we have discussed in this paper, is the variation in the thickness of the stagnant layer (δ). A decrease from 100 to 10 µm in thickness speeds up the rates of adsorption and desorption, but there is also another effect which perhaps is more interesting. Upon going from the thicker to the thinner layer, we may leave diffusion-controlled kinetics and enter a regime of mixed kinetics. A nucleation step is then observed, during which the transport to the surface is, for a short while, faster than the accommodation of new material at the surface. A third field of application for this model may be in studies of the effects of changes in one of a set of parameters which in experiments cannot be varied independently. An example mentioned here is the dependence of the adsorption and desorption behavior on the csac in relation to the cmc. A high ratio of csac/cmc makes it possible for the surfactants in micellar form to penetrate further into the stagnant layer, since the monomer concentrations they experience during their transport through the stagnant layer are higher than they otherwise would be. The reduced total concentration gradient (total concentration - csac) decreases the rate of adsorption but the rate of

Brinck et al.

desorption increases, since the concentration gradient when most of the material is transported away from the surface is equal to the csac. An interesting reflection brought about by the computer simulations concerns the relation between the cmc and the penetration depth of surfactants in micellar form into the stagnant layer during adsorption. Since surfactants with a low cmc have a low dissociation rate constant (koff) the micelles in these systems could be expected to reach further into the stagnant layer before they reach a critical size, when a transition from micellar to monomer form is imminent. Such a conclusion cannot be drawn without further analysis. If we compare two systems with the same reduced total concentrations (Ctot ) (total concentration)/cmc) we find that the surfactant with the low cmc has a low koff, but since the total flow is also slower, there will be only minor differences in the concentration profiles. When using simulations to gain an understanding of real systems, it is of course necessary to bear possible shortcomings of the model in mind. When the simulations are used with care, however, studies on phenomena which previously might not have been considered or have not received any attention can be initiated and perhaps result in a fruitful discussion on the subject. Acknowledgment. This work was sponsored by the Swedish National Board for Technical Development (NUTEK) and the Swedish Research Council for Engineering Sciences (TFR). Appendix A In this appendix we will clarify the steps that led us to the boundary conditions presented in eqs 12 and 13. First of all, we begin by looking at how the monomer concentration within the sublayer varies with time. Two processes can cause a change in the monomer concentration: diffusional flow of monomers through the subsurface and micellar buildup or disintegration within the sublayer.

Aδ1cmc

dC h mon dCmon ) ADmoncmc - Qmic from mon (A1) dt dx

The concentration within the sublayer, denoted C h mon, is averaged over the volume of the layer. Cmon is the monomer concentration immediately outside the sublayer. Qmic from mon gives the number of monomers within the sublayer, which go from monomer to micellar form per unit time. We assume all surfactants within the sublayer to be adsorbed which means that eq A1 also can be written as

dΓmon dCmon ) ADmoncmc - Qmic from mon AΓp dt dx

(A2)

where Γmon is equal to the amount adsorbed in monomer form (normalized with respect to Γp). We will now describe the equilibrium in the sublayer between amphiphiles in monomer and in micellar form in the same way as we did in the bulk (see eq 5a). However, instead of describing Cinmic as a function of Cmon, we write Cmon as a function of the amount adsorbed in micellar form, Γinmic (cf. Γmon). The number of surfactant molecules in the sublayer that go from monomer to micellar form

Kinetics of Nonionic Surfactant Adsorption

Langmuir, Vol. 14, No. 5, 1998 1071

per unit time can then be written

Qmic from mon A(cmc)Γp

)

(Cmon - Cmon,eq)δ1 koff,s Γ + τslowΓp cmc inmic koff,s,m+1 Γ + Cmonkon,sΓinmic (A3a) cmc m+1

where AΓpΓ is the number of moles of surfactant in micellar form in the sublayer. Γm+1 represents the aggregates within the sublayer with i ) m + 1. At equilibrium the following condition applies:

0)-

koff,s koff,s,m+1 Γ Γ + + Cmon,eqkon,sΓinmic cmc inmic cmc m+1,eq (A3b)

Combination of eqs A3a and A3b results in

koff,s,m+1 (Γm+1 - Γm+1,eq) + cmc A(cmc)Γp δ1 kon,sΓinmic + (Cmon - Cmon,eq) (A4) τslowΓp

Qmic from mon

)

(

( ))

We now assume that the relation between the concentration of amphiphiles in aggregates with i ) m + 1 within the sublayer and Cmon can be described in the same way as in the bulk

δ1(cmc)Rs m+1 Cmon Γm+1 ) Γp

dCmon konΓp ) (Γ + Rsurf(Cmon))(Cmon - Cmon,eq(Γ)) (A8) dx Dmon where Rsurf(Cmon) and Cmon,eq(Γ) are characteristic functions of the particular adsorption process being studied. The rate constant of micellar buildup at the surface (kon,s) has in eq A8 been assumed to be equal to the equivalent rate constant in the bulk (kon). The exact relation between Rsurf and Cmon is not known. However, by using the same approach that lead us to eq A8, eq 16 results. The variation of Cmon,eq with Γ is uniquely given by the experimental adsorption isotherm. If the complete adsorption isotherm as a function of monomer concentration is not available, eq 14 can generally be used. Not only is there a monomer flow through the subsurface at x ) δ1 but there is also micellar transport across this interface. Micelles can pass through the subsurface at places where no micelle is adsorbed to the surface and the flow through the subsurface is therefore defined to be proportional to the available surface area or A(1 - Γ). We can write the flow of amphiphiles in micellar form through the subsurface as follows

dCinmic koff,mic,s ) kon,mic,s(1 - Γ)Cinmic ΓΓ Dmic dx cmc p (A9a) At equilibrium, there is no net flow of amphiphiles in micellar form across the subsurface

koff,mic,s ΓΓ cmc p

(A9b)

dCinmic kon,mic,s ) (1 - Γ)(Cinmic - Cinmic,eq(Γ)) dx Dmic

(A10)

kon,mic,s(1 - Γ)Cinmic,eq(Γ) )

(A5)

From eqs A9a,b we obtain If the difference between the monomer concentration outside the surface and the equivalent equilibrium concentration is small, the two terms within the first parenthesis in eq A4 can be linearized to give

koff,s,m+1 (Γm+1 - Γm+1,eq) ≈ cmc koff,s,m+1δ1Rs(m + 1) m Cmon(Cmon - Cmon,eq) (A6) Γp Combining eqs A4 and A6, we obtain

Qmic from mon

(

) kon,sΓinmic + A(cmc)Γp δ1 1 + koff,s,m+1Rs(m + 1)Cm mon (Cmon - Cmon,eq) Γp τslow (A7)

(

))

Since the thickness of the sublayer in most cases is less than 100 Å and the maximum monomer concentration is low (approximately equal to the cmc), the monomer contribution to the adsorption in these cases is negligible in comparison with the total adsorbed amount. The consequences of this are that the left-hand side of eq A2 can be set to zero and the total adsorbed amount is Γ ≈ Γinmic. Equation A2 can then be written as

We have used the following value for kon,mic,s

kon,mic,s ≈

Dmic δ1

(A11)

The definition of the rate constant kon,mic,s reflects that the micellar adsorption is assumed to be diffusion controlled. The variation of Cinmic,eq with Γ in a certain system is uniquely given by the experimental adsorption isotherm if eq 10 is used to describe the relationship between Cmon and Cinmic at equilibrium. If eq 14 is used to describe the adsorption isotherm, the following relation between Cinmic,eq and Γ results:

Cinmic,eq )

m+1 RCmon,eq 1 - Cmon,eq

where

Cmon,eq ) LA9707843

1

(

1 + ((cmc/csac)r - 1)

1 - Γ 1/r Γ

)

(A12)