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Langmuir 2008, 24, 3960-3969
Self-Consistent Field Modeling of Non-ionic Surfactants at the Silica-Water Interface: Incorporating Molecular Detail Bart R. Postmus,*,†,‡ Frans A. M. Leermakers,† and Martien A. Cohen Stuart† Laboratory of Physical Chemistry and Colloid Science, Wageningen UniVersity, Dreijenplein 6, 6703 HB Wageningen, The Netherlands, and Dutch Polymer Institute (DPI), P.O. Box 902, 5600 AX, EindhoVen, The Netherlands ReceiVed December 7, 2007. In Final Form: January 15, 2008 We have constructed a model to predict the properties of non-ionic (alkyl-ethylene oxide) (CnEm) surfactants, both in aqueous solutions and near a silica surface, based upon the self-consistent field theory using the Scheutjens-Fleer discretisation scheme. The system has the pH and the ionic strength as additional control parameters. At high ionic strength, the solvent quality for the surfactant head groups is affected, which changes both the bulk and the adsorption behavior of the surfactant. For example, with increasing ionic strength, the CMC drops and the aggregation increases. Surfactants adsorb above the critical surface association concentration (CSAC). The CSAC is a function of the surfactant and the surface properties. Therefore, the CSAC varies with both the ionic strength and the pH. We predict that with increasing ionic strength, the CSAC will first slightly increase but then drop substantially. The charge on the surface is pH dependent, and as the head groups bind through H-bonding to the silanol groups, the CSAC increases with increasing pH. We focus on adsorption/desorption transitions for the surfactants and compare these to the experimental data. Both the equilibrium predictions and the consequences for the kinetics of adsorption follow experimental findings. Our results show that molecularly realistic models can reveal a much richer interfacial behavior than anticipated from more generic models.
Introduction Surfactants are essential in technology. Good examples are various detergency applications, their use as fabric softeners, their presence in formulations of herbicides, their occurrence as additives in oil recovery, and they serve as compatibilizers in paints.1 In many of these applications one makes use of the ability of these molecules to adsorb onto surfaces or to reside at interfaces. For such applications, non-ionic surfactants are often preferred over other types of surfactants because they are not so sensitive for salts and they adsorb both onto cationic and anionic surfaces. In the scientific literature on surfactant adsorption, silica is a frequently used substrate. This is due to the technological relevance of silica. For instance, pigment particles in paints are often coated with silica. The popularity of silica is also due to the availability of highly reflective silica wafers which have a natural silica coating. These wafers can easily be utilized in techniques such as reflectometry and ellipsometry.2,3 Now almost 25 years ago, Levitz et al. performed fluorescent decay experiments to study the adsorption of non-ionic surfactants on silica.4,5 They adsorbed polydisperse surfactants of the Triton series on polydisperse silica particles. Using a depletion method, it was found that surfactants adsorb following a sigmoidally shaped isotherm. They added pyrene and determined its average diffusion path by recording the fluorescent decay. At low surface
coverage, short diffusion paths were found, indicating that the surfactants form individual surface aggregates. At high surface coverage, the pyrene had a large average diffusion path which can indicate a complete 2-D layer. A decade later, Tiberg et al. used ellipsometry to study the adsorption of a series of homodisperse, linear, non-ionic surfactants of the CnEm-type.3,6 They also found that surfactants adsorb according to a sigmoidally shaped isotherm. The concentration where the adsorption plateau sharply increased was defined as the critical surface aggregation concentration, or CSAC, and was, for their systems, always found to be slightly lower than the CMC. Furthermore, they reported that the surface excess of surfactants with a small head group is larger than the excess of surfactants with a bigger head group. Also, they observed that surfactants with a longer tail adsorb somewhat more than surfactants with a short tail. Because Tiberg et al. managed to control the surfactant transport to some extent, they successfully related the kinetics of adsorption to the kinetics of desorption and to the equilibrium adsorption isotherm in a very elegant model.7 This model is very similar to the polymer adsorption model by Dijt et al.2 Both models are based on the assumption that there is a local equilibrium between the adsorbed phase and the aqueous phase just next to the surface. Therefore, one can look up the local concentration, cl, that coexists with the current surface excess in the adsorption isotherm. If this local concentration is not equal to the bulk concentration, cb, there will be a gradient for mass transport. The flux, J, is then described by
* To whom correspondence should be addressed. E-mail: bart.postmus@ wur.nl. † Wageningen University. ‡ Dutch Polymer Institute.
J ) ktr(cb - cl)
(1) Jo¨nsson, B.; Lindman, B.; Holmberg, K.; Kronberg, B. Surfactants and polymers in aqueous solution; John Wiley and Sons: Chichester, 1998. (2) Dijt, J. C.; Cohen Stuart, M. A.; Hofman, J. E.; Fleer, G. J. Colloids Surf. 1990, 51, 141-158. (3) Tiberg, F.; Landgren, M. Langmuir 1993, 9, 927-932. (4) Levitz, P.; Van Damme, H.; Keravis, D. J. Phys. Chem. 1984, 88, 22282235. (5) Levitz, P.; Van Damme, H. J. Phys. Chem. 1986, 90, 1302-1310.
(1)
where ktr describes the transport conditions. For solutions that have cb > CMC, it is important to correct for the lower diffusion coefficient of the micelles compared to single surfactant (6) Tiberg, F.; Jo¨nsson, B.; Lindman, B. Langmuir 1994, 10, 2294-2300. (7) Tiberg, F.; Jo¨nsson, B.; Lindman, B. Langmuir 1994, 10, 3714-3722.
10.1021/la703827m CCC: $40.75 © 2008 American Chemical Society Published on Web 03/04/2008
SCF Modeling of Non-ionic Surfactants
molecules. This idea works for adsorption, which is straightforward, and for desorption. For desorption, we set cb ) 0 and cl ) CSAC. In a number of studies the powerful technique of neutron reflection was used to examine the structure of a surfactant layer.8-10 It was found that at the CMC, the non-ionic surfactant C12E6 adsorbs in an islandlike fashion also called a fragmented bilayer. The thickness of this layer is ∼5 nm, and the surface coverage is about 60%. There are good reasons to believe that the degree of fragmentation in a layer depends on the type of surfactant and the strength of the adsorption. In this paper we will assume that the layer remains homogeneous. The existence of surface micelles has also been shown using an AFM setup. Grant et al. showed for graphite and for silica with different hydrophobicities that the structure of the surfactant layer depends both on the surfactant architecture and on the surface properties.11 On hydrophillic silica, most surfactants adsorb forming small globular objects (admicelles). On hydrophibized silica, the surfactant forms a flat layer, which can frustrate the preferential packing of the surfactant monomers. The energy it costs to disrupt the preferential packing is supplied by the shielding of the surface from the water. On a graphite surface, most surfactants bind epitaxially. This is probably caused by matching the distance between the carbon hexagons and the surfactant tail groups. This binding in this system results in long hemi-cylinders on the surface. There is also a lot of literature about the adsorption of nonionic surfactants on a hydrophobic surface using a variety of techniques such as neutron reflection,12-14 optical ellipsometry,15 and reflectometry.16,17 For a more general overview on surfactant adsorption at the air-water interface, the reader is referred to the review by Lu et al.,18 and for an overview of surfactant adsorption on the solid-water interface to Paria and Khilar19 or to Zhang and Somasundaran.20 Various theoretical approaches have been used to model the behavior of surfactants. For instance, the bulk behavior can be modeled using a number of approaches.21-23 To understand the adsorption of non-ionic surfactants, one can look at the two-step model24 or, in the case of a hydrophobic surface, to the model by Kumar and Tilton.25 A very effective and convenient approach to look at both the bulk and the adsorption behavior in one model is the self-consistent field (SCF) theory. For instance, this theory has been used to better understand industrially relevant applica(8) Lee, E. M.; Thomas, R. K.; Cummins, P. G.; Staples, E. J.; Penfold, J.; Rennie, A. R. Chem. Phys. Lett. 1989, 162, 196-202. (9) McDermott, D. C.; Lu, J. R.; Lee, E. M.; Thomas, R. K. Langmuir 1992, 8, 1204-1210. (10) Bo¨hmer, M.; Koopal, L. K.; Jannsen, R.; Lee, E. M.; Thomas, R. K.; Rennie, A. R. Langmuir 1992, 8, 2228-2239. (11) Grant, L. M.; Tiberg, F.; Ducker, W. A. J. Phys. Chem. B 1998, 102, 4288-4294. (12) Gilchrist, V. A.; Lu, J. R.; Staples, E.; Garret, P.; Penfold, J. Langmuir 1999, 15, 250-258. (13) Howse, J. R.; Steitz, R.; Pannek, M.; Simon, P.; Schubert, D. W.; Findenegg, G. H. Phys. Chem. Chem. Phys. 2001, 3, 4044-4051. (14) Thirtle, P. N.; Li, Z. X.; Thomas, R. K.; Rennie, A. R.; Satija, S. K.; Sung, L. P. Langmuir 1997, 13, 5451-5458. (15) Gilchrist, V. A.; Lu, J. R.; Keddie, J. L. Langmuir 2000, 16, 740-748. (16) Kumar, N.; Garoff, S.; Tilton, R. D. Langmuir 2004, 20, 4446-4451. (17) Geffroy, C.; Cohen, Stuart, M. A.; Wong, K.; Cabane, B.; Bergeron, V. Langmuir 2000, 16, 6422-6430. (18) Lu, J. R.; Thomas, R. K.; Penfold, J. AdV. Colloid Interface Sci. 2000, 84, 143-304. (19) Paria, S.; Khilar, K. C. AdV. Colloid Interface Sci. 2004, 110, 75-95. (20) Zhang, R.; Somasundaran, P. AdV. Colloid Interface Sci. 2006, 121-123, 213-229. (21) Puvvada, S.; Blankschtein, D. J. Chem. Phys 1990, 92, 3710-3724. (22) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934-2969. (23) Johnson, R. A.; Nagarajan, R. Colloids Surf. A. 2000, 167, 21-30. (24) Zhu, B.-Y.; Gu, T. AdV. Colloid Interface Sci. 1991, 37, 1-32. (25) Kumar, N.; Tilton, R. D. Langmuir 2004, 20, 4452-4464.
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tions such as colloidal (de)stabilization of non-ionic surfactants.26 Also, the complex bulk behavior of non-ionic surfactant can be described using SCF theory.27 Most of the articles describing SCF adsorption calculations are using a mean field approximation where there is just one concentration gradient (1G), giving rise to a surfactant profile perpendicular to the surface. In this case the adsorbed layer is smeared out. However, it is also possible to allow for gradients in two directions (2G) and study the inhomogeneous adsorbed state of surfactants at interfaces.28 These type of calculations also give an indication about the fragmentation of the layer. The mayor disadvantage of 2G calculations is that they require substantially more computer time. In this paper we focus on the 1G theoretical description of the adsorption of linear non-ionic surfactants onto silica. Recently, we reported on the self-consistent field modeling of the nonionic polymer PEO at the silica-water interface in a molecular realistic manner.29 The idea was that the classical polymer adsorption theory can explain many experimental observations qualitatively, but each particular system invariably has many specific features that are not dealt with by the general polymer adsorption theory.30-32 Such molecular specific features can only be included when the properties on a monomeric scale are accounted for. We now argue that a similar reasoning also applies to surfactant adsorption. In this article, we therefore try to incorporate some molecular details into SCF calculations for non-ionic surfactant systems in combination with a silica surface. With this model we will try to reproduce the experimentally measured response of the system upon increasing ionic strength or changing pH. In this study, we have chosen to fix the temperature. Our parameters are chosen such that they describe surfactant behavior at approximately room temperature. There exists a convenient method to include the temperature in a SCF approach.33 In this method EO groups can occur in two internal states, each with its own χ parameters. The temperature can be changed by allowing segments to change their internal state. Here, we want to describe the surfactants on a molecular realistic level but also keep things as simple as possible. Therefore, we have chosen not to include any temperature effects. Currently, there is ample attention for the adsorption of molecules from complex mixtures.34,35 Often such systems are not well understood due to the large number of interactions between the species in the bulk of mixed solutions and on the adsorbing surface. In a forthcoming publication, we will combine the knowledge from our previous paper on the molecularly realistic modeling of PEO onto silica and the present one which focuses on non-ionic surfactants at the same interface, with the goal to explain experimental observations for the adsorption phenomena found in mixtures of PEO and non-ionic surfactants. (26) Jo´dar-Reyes, A. B.; Leermakers, F. A. M. J. Phys. Chem. B 2006, 110, 18415-18423. (27) Jo´dar-Reyes, A. B.; Leermakers, F. A. M. J. Phys. Chem. B 2006, 110, 6300-6311. (28) Jo´dar-Reyes, A. B.; Ortega-Vinuesa, J. L.; Martin-Rodriguez, A.; Leermakers, F. A. M. Langmuir 2003, 19, 878-887. (29) Postmus, B. R.; Leermakers, F. A. M.; Cohen Stuart, M. A. Langmuir 2008, in press. (30) Fleer, G.; Stuart, M. C.; Scheutjens, J.; Cosgrove, T.; Vincent, B. Polymers at interfaces; Chapman and Hall: London, 1993. (31) de Gennes, P. Scaling concepts in polymer physics; Cornell University Press: Ithaca, NY and London, 1979. (32) Doi, M.; Edwards, S. F. The theory of polymer dynamics; Clarendon Press: Oxford, 1989. (33) Linse, P. Macromolecules 1993, 26, 4437-4449. (34) Postmus, B. R.; Leermakers, F. A. M.; Koopal, L. K.; Cohen Stuart, M. A. Langmuir 2007, 23, 5532-5540. (35) Taylor, D. J. F.; Thomas, R. K.; Penfold, J. AdV. Colloid Interface Sci. 2007, 132, 69-110.
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∑ φ (z) ) 1
Theory
X
SCF Theory and the Molecular Model. The aim is to model the behavior of non-ionic (alkyl-ethylene oxide) surfactants (CnEm) in solution (i.e., the micellization) and on silica-like surfaces as a function of the pH and the ionic strength. The reason for choosing SCF theory is that this technique allows for the incorporation of molecular details into the model, while keeping the computational efforts within reasonable limits. The latter is important as our systems typically have a large number of parameters. Below we will briefly mention some approximations that are intimately associated with the mean field approach. Nevertheless, despite its approximations, this method is believed to accurately describe a wide range of systems. The model used in this paper is, apart from the molecular architecture, the same as the one used in a previous publication on the behavior of PEO.29 Because the non-ionic surfactants are built up from the same atoms as PEO, we require the same input parameters, such as the short-range Flory-Huggins χ parameters, the values for the permittivity, etc. We do need a more detailed thermodynamical analysis to interpret the outcomes of the calculations. More specifically, for the surfactant micellization we will apply the thermodynamics of small systems of Hill.36 The SCF theory has been quite extensively described in our previous publication. Therefore, we will only mention its most important features here. The central idea of a SCF calculation is to place a test molecule in a potential field. The molecule can assume many spatial conformations, and the procedure essentially considers the statistical weight of all possible conformations in this potential field. These weights are collected into a single-chain partition function. From this partition function it is possible to extract the thermodynamics, as well as measurable quantities such as the volume fraction profiles. The potential fields are chosen such that they represent the interactions of all other molecules with the test molecule, that is, they are taken to be a function of the volume fraction profiles. As a result there exists a complication that the volume fractions, on the one hand determine the potentials and on the other hand follow from them. To make a long story short, the calculations proceed as follows. The start is to somehow choose suitable values for the segment potentials. We label these with the superscript (i) to remind that they are an input guess for the potentials, u(i) X (z). Here z is a spatial coordinate and X refers to a segment type. Using these potentials, we can generate the statistical weights of all possible conformations using a freely jointed chain model. The set of all conformations results subsequently in the composition of the system expressed in the so-called volume fraction profile, φX(z). It is important to mention that the volume fractions follow (for a given chain model) uniquely from the segment potentials. With these volume fraction profiles, which are dimensionless concentrations, it is possible to evaluate the distribution of the charged species q(z). These charge density profiles are used in the Poisson equation to generate electrostatic potentials ψ(z). Now, all relevant quantities are available to recompute the potentials using neighbor uX(z) ) uvol(z) + uelec (z) X (z) + uX
(2)
It is important to mention that the segment potentials follow uniquely from the volume fraction profiles. In somewhat more detail, eq 2 shows that there are several aspects accounted for in the segment potentials. The first contribution uvol(z) is needed to generate space at position z for a segment. As all segments are equally large (occupy one lattice site), this contribution does not depend on the segment type X. Its value is coupled to the constraint that the volume fractions add up to unity, i.e. (36) Hill, T. L., Ed. Thermodynamics of small systems; Dover Publications Inc.: New York, 1991.
(3)
X
Then, there is the electrostatic contribution uelec X (z) which features the electrostatic potential as in the Gouy-Chapman theory, as well as a contribution due to the polarization of segments. Finally, uneighbor (z) expresses the short-range interactions wherein a BraggX Williams approximation is implemented and contact interactions are given by the Flory-Huggins χ parameters. The model needs a χ parameter for all unlike segment contacts. Using eq 2, we thus recompute the potentials from the volume fractions. To remind ourselves that it is an output potential, we add the superscript (o), i.e., u(o) X (z). For an acceptable self-consistent solution, the output potentials u(o) X (z) should be numerically equal (z). Typically, this is not the case for the to the input potentials u(i) X first calculation because the first input potentials u(i) X (z) are simply guessed. Therefore, we iteratively adjust the input potentials such that the output potentials converge to the input ones. If this is the case we call such fixed point the SCF solution.37 An acceptable SCF solution also obeys the incompressibility constraint (eq 3) for each coordinate z. When we find such a solution, we have the most likely composition of the system and, as a bonus, the mean field free energy. The SCF machinery is implemented using a discrete set of coordinates, better known as a lattice. This lattice facilitates the evaluation of the statistical weights of the various conformations of our molecules, as was first suggested by Scheutjens and Fleer for the case of homopolymers.38,39,30 This method was later extended to account for co-polymers and surfactants by Leermakers and Scheutjens.40 The type of lattice used depends on the problem at hand. More specifically, the calculations on bulk properties of surfactant solutions calls for the evaluation of spherical micelles. This is possible using a spherical lattice. Such a lattice consists of consecutive spherical shells of thickness l. The volume, V, of shell z increases as the distance to the center increases: V(z) ) 4/3π[(z + 1)3 - z3]. Because the volume of a lattice site is fixed at l3, the number of lattice sites also increases as the distance to the center increases. In this paper we use a spherical lattice to calculate the surfactant CMC. The use of this spherical lattice pre-imposes the formation of spherical micelles. We know that some surfactants like to form other types of aggregates in bulk solution.27 However, the formation of these alternative shapes usually occurs a surfactant concentration higher than the CMC, i.e., the surfactants first form spherical micelles and upon increasing the concentration, they may form alternative shapes of aggregates. In this paper we primarily use the CMC to establish the maximum chemical potential of surfactants in solution. Hence, the use of the spherical lattice does not impose any further approximations here. Adsorption studies typically call for a flat lattice, where we assume that the curvature of the surface is negligible. The use of a flat lattice in surfactant systems has some important implications. For instance, it is not possible to have spherical micelles on a flat lattice. In fact, the bulk self-assembly is suppressed in this lattice geometry. Therefore, it is possible to do calculations in which the surfactant monomer concentrations exceeds the (bulk) CMC (as found using the spherical coordinate system). These results are valid, as long as we keep in mind that in real systems it is very hard to increase the monomer concentration above the CMC. In this paper we will focus on the results below and up to the bulk CMC. One lattice site can hold one molecular segment. Recall that we want to reproduce experimental results, and therefore, we model the molecules in a molecularly realistic manner. A CnEm molecule has (37) Evers, O. A.; Scheutjens, J. M. H. M.; Fleer, G. J. Macromolecules 1990, 23, 5221-5233. (38) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 16191635. (39) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1980, 84, 178-190. (40) Leermakers, F. A. M.; Scheutjens, J. M. H. M. J. Colloid Interface Science 1990, 136, 231-241.
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n united carbon segments (CH2 or CH3), one O segment, and m times a three-unit segment C-C-O. If, for instance, the C segment at the end of the molecule is placed on a given lattice site, the next segment (number two) will be on an adjacent lattice site which can be in a previous layer, in the same layer, or in a next layer (freely jointed chain). To account for these effects, lattice parameters, λ-1, λ0, and λ+1, are used. The a priori probability that this second segment is in the previous layer is given by λ-1, that it is in the same layer is given by λ0, and in the next layer is given by λ+1. For an isotropic cubic lattice, λ0 ) 4/6 and λ+1 ) λ-1 ) 1/6.30 For the spherical coordinate system, the λ parameters become a function of the coordinate. They are chosen such that they obey the detailed balance equation L(z)λ+1(z) ) L(z + 1)λ-1(z + 1). The calculations presented in this paper are 1G calculations, which means that the potential field is taken to be a function of one spatial coordinate only. Therefore, the use of u ) u(z) is equivalent to replacing all variations in the xy plane by averaged homogeneous properties. As mentioned, it is possible to perform 2G calculations, which can account for lateral inhomogeneities.28 However, such computations require substantially more computer time. As far as we know, this is the first attempt to try and model the bulk and the surface behavior of polymers and surfactants using exactly the same model. The system is described in a molecularly realistic manner. Using this level of detail, we have included the solvent quality as a function of the ionic strength and the surface charge as a function of both the pH and ionic strength. Thermodynamics of Small Systems. For the interpretation of our results on micellization, we have to use the thermodynamics of small systems, as developed by Hill and later applied to micellar solutions by Pethica and Hall.36,41,42 The thermodynamics of small systems assumes on generic physical grounds that in the system there is a hidden variable called the number of subdivisions, N,, which divides the system in a number of subsystems. In our case, each subsystem contains one micelle, which means that the number of subdivisions is equal to the number of micelles, N. The intensive variable that is conjugated to the extensive N is the subdivision potential, . We can incorporate these variables into thermodynamics by writing down the change of the Gibbs energy as dG ) - S dT + V dP +
∑ µ dn + dN i
i
(4)
i
Here S is the total entropy, T is the absolute temperature, V is the volume, P is the pressure, µi is the chemical potential of species i, ni is the total number of molecules of i. In line with classical thermodynamics, equilibrium implies that the system has minimized its free energy with respect to every degree of freedom it has. Therefore, thermodynamics of small systems implies that at equilibrium ∂G (∂N )
T,P,ni
))0
(5)
Moreover, to have stable micelles, the Gibbs energy should be minimal. This implies ∂2G ∂ >0 ) ∂N 2 ∂N
(6)
If the system does not comply with eq 6, it can lower its free energy by either increasing or decreasing the number of micelles. In the SCF calculations, we effectively fix the center of mass of a micelle to the center of the spherical coordinate system and predict its properties as a function of the number of surfactants in the micelle (aggregation number g). For an accepted SCF solution, we can evaluate the grand potential that is associated with our pinned micelle, (41) Hall, D.; Pethica, B. Thermodynamics of micelle formation. In Surfactant science series; Schick, M., Ed.; Marcel Dekker: New York, 1967; Chapter 15, pp 516-557. (42) Lyklema, J., Ed. Fundamentals of interface and colloid science, Volume V: Soft colloids; Elsevier Academic Press: Amsterdam, 2004.
m. This differs from the ‘true’ grand potential or subdivision potential, , as the collective degrees of freedom of the micelle as a whole are neglected. This means that the mixing entropy on a micellar level gives the difference between the true and the computed m. For dilute micellar systems we may write m ) - kT ln φms
(7)
where kT is the thermal energy and φms is the volume fraction of micelles. Let us assume that there exists some relevant micellar-like structure. The first issue is to compute the aggregation number g. For this we take the excess number of surfactants in the micelle, or mathematically g)
1 Ns
∑ L(z)(φ (z) - φ ) b s
s
(8)
z
where the subindex refers to the surfactant molecule, N is the number of segments that make up one surfactant molecule and L(z) is the number of lattice sites in layer z. Hence, φs(z) is the volume fraction of surfactant in layer z, and φbs is the volume fraction of surfactant in the absence of a field (which corresponds with the bulk). As explained, for such a micelle it is possible to compute the translationally restricted grand potential (m(g)). Combining the equilibrium condition, eq 5, with eq 7 thus gives the volume fraction of micelles with aggregation numbers g:
(
φms (g) ) exp -
)
m(g) kT
(9)
As the surfactants in these micelles are densely packed, we can estimate the volume of the micelle by Vm ) gNsl3. Recall that the monomer concentration that is in equilibrium with the micelles with size g, φbs (g) is also available from the computations. We can formulate the mass balance for the surfactant as
( )
φts ) φbs + φms ) φbs + exp -
m kT
(10)
This equation is used to estimate the CMC. Below the CMC, the volume fraction of micelles is very small and φts ≈ φbs . Above the CMC the majority of the surfactants is in micelles and φts ≈ φms . At the CMC the two terms are of the same magnitude, that is, φbs ) φms . However, it is quite elaborate to calculate the CMC according to this definition. Therefore, we have made a more pragmatical choice. Below we will implement an operational CMC by selecting the micellar system that is characterized by a grand potential m ) 10kT. This basically implies that we require a certain concentration of micelles to be present at the CMC. The required concentration of micelles in units of volume fraction is φms ≈ 4.5 × 10-5, which is close to the condition φbs ) φms for most surfactant systems. The System and Its Parameters. As is the case for many experimentally relevant systems, there is a relatively large number of parameters in our model. We will first discuss the parameters relevant for the bulk and then give some details how we modeled the silica substrate. Bulk Components. There are two types of segments that make up the surfactant molecule. The first segment is an apolar united carbon atom, C, that is present both in the tail and in the head group. The second segment is the polar oxygen, O, which is exclusively in the head group. The water, W, that surrounds the surfactant molecules is modeled as a four-armed star (and occupies five lattice sites). As compared to monomeric water, the advantage of such a compact object is that on a volume base the star has less translational freedom. Water is a strongly associative liquid in which the free mobility of water molecules is also somewhat hindered. Using such a star incorporates some aspects of the self-associating properties of water. The result of the reduced translational degrees of freedom is that the
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micellar cores are virtually free of water, which is consistent with experimental facts.42 To model the silica surface in a realistic manner, we should include its response to the pH. The first issue is that the water may loose or to take up a proton. In our model this is incorporated by defining three internal states for water, W: (i) OH-, (ii) H2O, and (iii) H3O+. The concentration of these states is controlled by the water equilibrium 2H2O h H3O+ + OH-
(11)
which has an equilibrium constant of pK ) 14. All states of water have equal interactions (χ parameters) with the other segments, but the states obviously respond differently to the presence of an electric field. Next, it is necessary to introduce an 1:1 electrolyte, and we call the ions Na and Cl. We hasten to mention that these names are generic; in fact they may represent a set of different salts. The issue is that for a particular pair of ions one must specify unique interactions with the surface and all other components. Needless to say, the values of such set of parameters are largely unknown. Below we will take idealized choices for these parameters. The interactions between the polymer segments with water are the same as those used in a previous publication, where we focused on PEO adsorption on silica.34 In fact, the parameters used in that model were taken such that the modeling represents the known behavior of non-ionic surfactants in water (i.e., the present system). More specifically, we select a set of parameters that reproduces in first order the CMC as a function of the tail and head group lengths. Here we have used χC,W ) 1.1, χO,W ) -0.6, and χC,O ) 2.42 Naturally, we need to define also the corresponding Flory-Huggins parameters for the salt ions, for which we have chosen χNa,C ) χCl,C ) 2. All other interaction parameters were set to zero. Note that the two ions behave completely symmetrically in the sense that the two ions have identical interactions with any other segment. Any asymmetry in these interactions will, e.g., charge up the non-ionic surfactant. Although it may not be directly obvious from this set of interaction parameters, increasing the ionic strength will decrease the overall solvent quality for EO in water. Below we will examine in detail what the consequences are for the micellization and adsorption characteristics. We further need to specify the relative permittivity, X, for all our molecular species X, to calculate the local value of the dielectric permittivity. In more detail, in the calculations we have implemented the volume fraction-weighted average over the segment permittivities. Again, for simplicity we have used the relative dielectric permittivities X ) 80 for all molecular species, except for the hydrocarbon, where we used C ) 2.43 This choice will ensure that in the bulk the dielectric constant is high whereas it is low in the core of the micelle. To evaluate the Poisson equation, we need to specify the unit length, i.e., the size of the lattice sites, l. We have set l ) 0.3 nm. This is approximately the average size of the segments used in the model. In this paper, we have chosen to use λ-1 ) λ0 ) λ+1 ) 1/3. This choice minimizes so-called lattice artifacts, as it allows easy placing of a sharp interface onto the lattice. We do note that this corresponds to an anisotropic lattice and that some results need a trivial rescaling to apply to an isotropic solution. In this paper, this issue is of minor importance, as we focus mainly on the trends. The Silica Surface. In this paper the focus is on the adsorption of surfactants onto a silica surface. Of course, the true silica surface is very complex and we can only grasp its main features and thus the name ‘silica’ stands for a range of metallic oxides. In its simplest description, a silica oxide surface has two kinds of surface groups: silanol and siloxane. An essential feature is that silanol groups can dissociate and that therefore the surface charge responds to the salt concentration and the pH. To model this we allow the silanol groups to exist in different states, namely associated groups called HA and dissociated groups called A. An HA group (43) Sastry, N. V.; George, A.; Jain, N. J.; Bahadur, P. Chem. Eng. Data 1999, 44, 456-464.
Figure 1. Translationally restricted grand potential, m, plotted versus the aggregation number, g, for C14Em where m ) 4, 5, ..., 12. The background electrolyte concentration was set at φbCl ) 0.1. In the graph the line m ) 10 is exemplified by the horizontal dotted line. The vertical dotted lines point to the corresponding aggregation number, g, that are present at the experimentally relevant CMC. can react with water to form an A group (or visa versa) AH + H2O h A- + H3O +
(12)
where the equilibrium constant pK ) 6. Siloxane groups can only exist in one state called Si. In a previous article we calculated titration curves for this silica surface and validated these with experimental data from literature.29 Again, all interaction parameters needed for the molecular species with the surface segments were taken from our previous publication. We have set a moderate repulsion between the C groups and all types of surface groups: χC,HA ) χC,A ) χC,Si ) 2. The oxygen atoms are attracted by all surface groups. For the A groups and the Si groups there is a moderate attraction: χO,A ) χO,Si ) -2.5. The oxygen groups can form H bonds with the silanol groups. Therefore, we modeled a strong attraction between O and HA groups, χO,HA ) -6. This choice of χ parameters is chosen such that for pH ≈ 7 and low salt, CSAC ≈ 0.7CMC, which corresponds with literature values.6 For simplicity we will assume that between the salt ions and the surface groups, there is a slight symmetric repulsion, i.e., χNa,Si ) χNa,A ) χNa,HA ) χCl,Si ) χCl,A ) χCl,HA ) 1. This slight repulsion models the breaking of the hydration shell around the ions. Again, the ions behave symmetrically with respect to the bulk and the surface groups. This choice was made to make sure that our results are not unnecessarily complicated by ion specificity effects. In all our calculations, we have taken half of the surface groups as silanol, and the other half as siloxane groups. Since we have set the segment size at 0.3 nm, this implies that there are 1/(2‚0.32) ) 5.6 silanol groups per nm2. This corresponds with literature values.44 In the theory described above, all concentrations are expressed in volume fractions. To convert the volume fractions to moles per liter, we use the fact that the concentration of pure water is 1000/18 ) 56 mol/L. We can use this conversion factor directly for the monomeric species in the system. For the other molecules, which are composed of multiple segments, and each segments occupies a single lattice space, we should additionally divide by the chain length to account for the molecular volume in the conversion from volume fractions to molar quantities.
Results and Discussion Bulk Study: CMC. In Figure 1 we plot the work of formation of a micelle, or equivalently the translationally restricted grand potential, m, as a function of the aggregation number g for C14Em where m ) 4, 5, ...,12. Every curve in this figure clearly features a maximum. The micelle that exists exactly on this maximum is the smallest thermodynamically stable micelle (according to eq 6), which is also referred to as the first micelle. The surfactant bulk concentration where these micelles occur is termed the theoretical CMC. Again, it is often the case that near the first (44) Iler, R. The chemistry of silica; John Wiley and Sons: New York, 1979.
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Figure 2. CMC as a function of the ionic strength for eight different CnEm surfactants. (a) For fixed tail length (n ) 14) and for a number of head group lengths, m, as indicated and (b) for fixed head group length (m ) 7) and a number of tail segments n as indicated.
appearance of micelles, the micelle concentration is too low to be noticed in experiments. However, the estimate of the experimentally more relevant CMC given above is typically very close to the theoretical value. All micelles that are smaller than the first micelle are rejected based on the stability criterium, eq 6. This is why we have dashed these parts of the curves in Figure 1. Also, at high aggregation numbers there is typically a part of the curve where m starts to increase as g increases. These parts are also dashed since they do not obey eq 6. As discussed above, a pragmatic choice for the CMC is to compute the volume fraction of surfactants in the bulk that is in equilibrium with micelles that have a work of formation m ) 10kT. This horizontal dotted line is drawn for this value of m and the vertical lines point to the aggregation numbers g for each case. For some surfactants, m never drops below10kT. In such a system the surfactant has a limited solubility or, equivalently, is close to the cloud point. Such systems will also have the tendency to form cylindrical micelles. Indeed, this occurs for surfactants with a small head group (data not presented). In Figure 2a we plot the volume fraction of surfactants in the bulk that is present at the experimentally relevant CMC as a function of the ionic strength in the system for a number of non-ionic surfactants. We recall that, with increasing ionic strength, the solvent quality for the EO head group deteriorates. So, as the ionic strength increases, the surfactant molecule becomes progressively more ‘hydrophobic’. As a result, the molecules have a larger tendency to self-assemble, and the CMC will therefore be lower. This is seen in all curves of Figure 2a. Moreover, in line with experimental data it is found that the CMC is a weak function of the head group size. In Figure 2a, the head group size was varied from 5 to 11 EO units, and the CMC changed only by a factor of about two. When we plot the CMC as a function of the head group size for a fixed tail length and ionic strength, we find a linear dependence (graph not shown). The effect of the length of the tail on the CMC is much larger, which is illustrated by Figure 2b. Varying the tail length from 10 to 22 C atoms, lowers the CMC by 4 orders of magnitude. Plotting the tail length as a function of the CMC, shows that the CMC varies exponentially as a function of the tail length (graph not shown). This is according to Traube’s rule.45 The ionic strength not only affects the CMC, but the aggregation number is affected as well. In Figure 3 we plot the aggregation numbers for surfactants at the experimentally relevant CMC for three values of the ionic strength as a function of the length of the head group. The lower the ionic strength, the better the solvent quality and the better the head group can stop the self-assembly process. This means that with increasing ionic strength the (45) Tanford, C. The hydrophobic effect; John Wiley and Sons: New York, 1973.
Figure 3. Aggregation number, g, plotted versus the number of EO groups in the surfactant head group m for surfactants with a C14 tail for three different ionic strengths (φbCl) 0.1, 0.01, and 0.001).
stopping force becomes less and the number of surfactants in the micelle, g, increases. As the ionic strength effects only become important for relatively high ionic strengths, we find very little effects of salt on g in the range of salt volume fractions of 0.0010.01, but much larger effects when the concentration is increased by another factor of 10. The stopping force also increases with increasing length of the head group. A larger head group will oppose to the driving force for aggregation more efficiently which leads to lower aggregation numbers. From Figure 3 it can be seen that the increase of ionic strength to the molar range can compensate for a head group size change of two to four ethylene oxide units. Adsorption Study: CSAC. For the evaluation of surfactant adsorption, we used a flat lattice with a surface positioned at z ) 0. The presence of the surface implies an adsorbing boundary condition: all conformations that cross this boundary obtain an infinitely small statistical weight. On the other side of the system, i.e., far from the surface, we have the bulk solution. In the last lattice layer we implemented reflecting boundary conditions to prevent conformational restrictions and to minimize the effects of a finite system size. In Figure 4 we show a set of three adsorption isotherms of C12E5 on silica for the same three ionic strengths as used in Figure 3. Due to the mean field approximation, which averages the volume fractions parallel to the surface, one finds surfactant adsorption isotherms that have a so-called van der Waals loop. Such a loop indicates a surface phase transition. In more detail, this means that there is a window of surfactant bulk concentrations for which there exists multiple SCF solutions. Each of these differ with respect to the adsorbed amounts, the structure of the surfactant layer, the charge distribution on the surface, etc. Two of these adsorbed states correspond with a (local) minimum of the free energy (meta-stability). The third one is a local maximum (instability). Below the transition point the free energy is lowest for the state of low adsorbed amounts (the so-called ‘gas’ state
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Figure 4. Adsorption isotherms of C12E5, i.e., the adsorbed amount as a function of the logarithm of the equilibrium bulk volume fraction for three values of the ionic strength as indicated. The isotherms feature a van der Waals loop. The transition from the gas to the bilayer state occurs at the CSAC, which is indicated by the jump in the isotherm. The metastable parts in the isotherm are dashed; the instable parts are dotted. The isotherm terminate at the CMC.
because here the surfactants are in low adsorbed amounts far separated from each other). Above the transition point the layer with the highest adsorbed amount has the lowest free energy (a well-defined surfactant layer, more specifically a bilayer, is at the surface). At the transition point the two minima of the surface free energy are equally deep and the ‘gas’ and bilayer phase coexists. The point of the transition can be found by plotting the grand potential as a function of the bulk concentration. This gives a cusped figure. The transition point is at the crossing point because only for this point, the two phases have the same grand potential and the same chemical potentials for all components. We can identify the transition point as the CSAC. On a hydrophilic surface, the non-ionic surfactant will adsorb with head groups facing the surface, and an oppositely oriented monolayer is placed on top of this (bilayer configuration). The head groups will shield the tails from the water by closely packing them inside the aggregate. In Figure 5 we plotted a volume fraction profile of adsorbed C12E5 at pH ) 7 and φCl ) 0.01. We chose a surfactant concentration just higher than the CSAC. Figure 5a shows that the surfactant layer is ∼4 nm thick. We indeed see that all tail groups are found sandwiched between the head groups in a layer that hardly contains any water. However, despite the fact that most head groups are found on the outside of the surfactant layer, there are head groups mixed with the tails in the center. Also, the separation between head groups and tail is not very sharp. This is due to the schizophrenic character of the EO head groups. On the average they are hydrophillic, but they do also contain C groups that dislike water, and hence like the tails. Just next to the surface, there is a relatively high amount of O segments. This is due to high adsorption energy (i.e., the low χ parameter between oxygen and the surface). Because every O segment in the PEO is connected to two C segments on both sides, there is a low amount of O segments and a high amount of C segments at 0.5 nm from the surface. The central part of the bilayer is approximately filled 75% with tails, 20% with head groups, and 5% with water. Since the head groups also contain nonpolar segments, there is no sharp transition between the hydrophobic part and the hydrophillic parts of the bilayer. Furthermore, it very well is possible that there are some head groups mixed in between the tail groups in the hydrophobic domain of the bilayer. However, in our 1G approximation, all lateral surface structures are smeared out. In Figure 5b we plot the volume fraction profiles of the salt. The hydrophobic part of the bilayer has a volume fraction of salt
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that is approximately five times lower than the bulk volume fraction. This is due to the repulsion between the C segments and the salt. Near the surface, the concentration of the counterion is rather high. The bulk components are either water, surfactant, or salt. Because the sum of their volume fractions is unity and the salt is only 1%, we can say that the part above the surfactant curve in Figure 5 corresponds to the local volume fraction of water (φW(z)). Hence, near the surface there is quite a large amount of water. The presence of this water and the favorable electrostatic potential, cause a rather high amount of counterions near the surface. Furthermore, the dielectric constant of the C groups is rather low and it is energetically rather costly to have a high potential inside the hydrophobic part of the bilayer. Therefore, it is favorable that the potential drops rather fast and this causes a high concentration of counterions near the surface. We have evaluated the CSAC for a number of different surfactants at varying pH and ionic strength for six surfactants of varying head and tail lengths. In Figure 6a we focus on the CSAC as a function of the ionic strength for a given pH 7. In first order, all trends of the CSAC follow those of the CMC. As for the CMC, the effect of the tail length on the CSAC is much larger than the effect of the head group length. Upon a closer inspection, however, it is found that the curves in Figure 6b all have a small maximum. This maximum is found in the range 10-3 < φbCl < 10-2. At ionic strengths lower than that where the maximum is found, the CSAC is suppressed due to the surface charge that increases as the ionic strength increases. As mentioned above, the surfactant layer adsorbed onto the surface is of the bilayer type where the core is densely packed with alkyl chains. However, these hydrophobic cores have a low dielectric constant and thus do not like to be near a charged surface. Hence, charging the surface adds an extra adsorption barrier for the surfactant and the CSAC increases with increasing ionic strength. At higher salt concentrationsshigher than that corresponding to the maximumswe enter the regime where the EO head groups gradually experience a decrease of the solvent quality. This causes the surfactant molecules to become more hydrophobic and this reduced solubility will promote the adhesion of the surfactant to the surface. In Figure 6b we present the predictions on how the CSAC responds to changes of the pH at fixed ionic strength φCl ) 0.001. Increasing the pH causes the silanol groups on the surface to dissociate. Since dissociated silanol groups cannot form H bonds with the EO head groups, the net attraction between these groups decreases. As a result the surfactant only starts adsorbing at higher concentrations and the CSAC increases. Combining Bulk and Surface Properties. We present the CSAC/CMC ratio as a function of the ionic strength for a number of surfactants at pH 7 in Figure 7a. It is observed that this ratio increases with ionic strength φbCl in the low ionic strength regime, 10-5-10-3. At these ionic strengths, the CMC is almost constant, as we can see in Figure 2a,b. This rise is thus entirely due to the rise in the CSAC found in Figure 6. In the high ionic strength regime the trend is reversed (Figure 7a). From both Figures 2a,b and 6a we concluded that in this regime both the CMC and the CSAC decrease. However, the CSAC decreases more rapidly, and hence, the ratio also decreases. Part of the reason is that the electrostatic potential generated by the electrified interface drops more quickly at increasing ionic strength so that the electrostatic potential in the vicinity of the hydrophobic core of the adsorbed bilayer is relatively low. So the penalty to have a hydrophobic region near the interface decreases with increasing ionic strength.
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Figure 5. Volume fraction of different segments plotted versus the distance from the surface. (a) The volume fraction of C12E5 both as a whole and split up in different segments. (b) Plot of the profile of the salt.
Figure 6. (a) CSAC as a function of ionic strength in double logarithmic coordinates at pH 7. (b) The logarithm of the CSAC as a function of the pH for a fixed ionic strength φbCl ) 0.001. The surfactants that are used are indicated.
Figure 7. Ratio between the CSAC and the CMC plotted versus the logarithm of the ionic strength, φbCl, for a number of surfactants. (a) At pH 7 and (b) at pH 9. The parts of the curve with CSAC > CMC are dotted because experimentally it is difficult to increase the concentration of surfactants above the CMC.
Both the CMC and the CSAC are strong functions of the surfactant tail length. Interestingly though, their ratio is hardly a function of the tail length anymore. Apparently, the shielding of the tails by the hydrophilic head groups is similar in the bulk micelles and near the surface layers. As the adsorption mechanism of the surfactants onto silica involves the affinity of O for the silanol groups, the ratio CSAC/CMC remains a stronger function of the size of the head groups. Kumar et al. considered the adsorption of a number of CnEm surfactants on a hydrophobic surface using reflectometry.16 They found that, on a hydrophobic surface, all adsorption isotherms could be collapsed onto a single curve by scaling the concentration using the CMC and the adsorbed amount, Γ, by its plateau value, Γ(CMC). On the basis of this observation, they argue that there is a direct relationship between the bulk and surface behavior of these surfactants. If we compare our predictions with their
results, we see that there is a match if we compare surfactants with a fixed head group size. Our predictions do show that there is a head group effect that is not measured by Kumar et al. We attribute this to the different substratum. In our case, the surfactant has to adsorb using its head group, whereas on a hydrophobic surface, the surfactant adsorbs with its tail group. Therefore, the head group length plays a more prominent role in our case. At pH 7 the CSAC < CMC for all ionic strengths and for all surfactants. This implies that, if the surfactant concentration is high enough, eventually the surface will be covered by a dense surfactant layer irrespective of the ionic strength. This changes when the pH increases. In Figure 7b we consider the basic solution pH 9. At this pH the surface is charged more, and as a result the number of undissociated silanol groups has decreased. As a result the attraction between the EO groups and the surface decreases with respect to the neutral pH case. The main features of Figure
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7b are similar to those in Figure 7a. However, the curves are shifted to ratios closer to unity. Indeed, parts of some curves show the special situation that the CSAC > CMC. It is well known that the chemical potential of a surfactant solution has its intricacies. That is, it is a function of the 42 concentration of only the monomers cmon s
µs ) µ0s + kT ln cmon s
(13)
In a first-order approximation, the monomer concentration is fixed above the CMC. Therefore, the chemical potential of surfactants in solution effectively has the chemical potential at the CMC as its maximum. If the CSAC is higher than the CMC, it actually implies that the surfactant chemical potential needed for the surfactants to adsorb should be larger than the chemical potential at the CMC. Since this is not possible, the surfactant cannot adsorb. The point were CSAC/CMC ) 1 in Figure 7b signals the adsorption/desorption transition. This is the reason why we have dotted these parts in Figure 7b. Further inspection of Figure 7b shows that it is possible to have an adsorption/ desorption transition at relatively low ionic strength, followed by an desorption/adsorption transition at high ionic strength. Such complex behavior may be of importance for practical applications. The data shown above point to the possibility to induce an adsorption transition at fixed ionic strength by varying the pH only. Figure 8 illustrates this by presenting the CSAC/CMC ratio as a function of the pH at a fixed ionic strength of φCl ) 0.001. In this graph we find that the conditions CSAC > CMC is found at a relatively high pH. This is natural because in this limit the driving force force for adsorption vanishes simply because the number of available silanol groups decreases. Comparing the Model with Experiments. As stated above, the idea is to advance molecularly realistic SCF calculations which describe experimental systems accurately, that is, at least
Figure 8. CSAC/CMC ratio as a function of the pH for a number of surfactants at φbCl ) 0.001.
semiquantitatively. Therefore, it is necessary to critically compare prediction with experimental data. We have studied the adsorption of C12E5 using reflectometry in large detail. As the present paper is mainly theoretical, we cannot go into full detail about the experimental conditions, but we may refer to one of our previous papers for this.34 It suffices to mention that the surfactant C12E5 was dissolved in deionized water at a concentration of 6.5 × 10-5 mol/L (which is near the CMC).46 When appropriate, we adjusted the pH using HCl or NaOH and used NaCl to increase the ionic strength. The adsorption of this surfactant was studied using a dynamic reflectometer technique in which the adsorbed amount can be followed quantitatively in real time. In Figure 9a we present three adsorption isotherms of C12E5 at 0, 1, and 10 mmol NaCl/L as found by a set of reflectometry experiments. It is seen that the adsorbed amount is rather insensitive for the pH < 7. However, at a higher, more basic pH, the adsorption becomes a strong function of pH. In this regime the adsorption isotherms become a strong function of the salt concentration as well. Indeed, in the absence of salt, the surfactants adsorb for all pH < 11. The addition of only 1 mmol NaCl/L causes a shift of the adsorption/desorption transition to pH ≈ 9, while the transition shifts to even lower pH ≈ 7 values when the salt concentration is 10 mmol/L. In Figure 9b we tried to reproduce Figure 9a using the numerical SCF model. By doing so, we can find significant adsorbed amounts, that is, ΓC12E5, for a wide range of pH and ionic strengths. However, as we explained above, an adsorption/desorption transition takes places when CSAC > CMC. In Figure 9b we indicate the occurrence of this transition by the sudden drop of the isotherms. In line with the experimental data, an increase in the ionic strength causes the transition to shift to lower pH values, whereas the adsorbed amount is rather insensitive to pH and ionic strength the remaining part of the isotherm. In the theoretical analysis the drop of the adsorbed amount is jumplike, whereas this is clearly not the case in experiments. This must be attributed to the irrealistic assumption of the mean field model wherein the lateral averaging of the adsorbed layer is implemented. Instead the adsorption, especially near the CSAC, is necessarily inhomogeneous and smooth. Therefore, the adsorption/desorption condition taking place at CSAC ) CMC is also artificial. In reality, we must expect a smooth transition similarly as found experimentally. In a reflectometry experiment we not only have access to the equilibrium adsorbed amount (obtained by waiting long enough for the adsorption to stabilize) but can analyze the kinetics of adsorption and desorption as well. From these experiments, we know that the kinetics of adsorption is also influenced by the addition of salt. In Figure 10 we plotted the initial rate of adsorption (solid curves) and the inital rate of desorption (dashed curves).
Figure 9. (a) Adsorbed amount in mg/m2 plotted versus the pH for C12E5 as measured by reflectometry for three values of the ionic strength as indicated. (b) Adsorbed amount in equivalent monolayers as calculated by our model for a number of ionic strengths as indicated.
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Figure 10. (a) Initial rate of adsorption (solid lines) and the initial rate of desorption (dashed lines) of C12E5 as a function of the pH for three different ionic strengths as measured by reflectometry. The surfactant concentration is fixed to be very close to the CMC. (b) The corresponding adsorption flux, J˜ads, and desorption flux, -J˜des, as a function of pH computed from the theoretical data using the CSAC and CMC for different values of the ionic strength as indicated. The vertical dotted line presents the adsorption/desorption conditions.
We see that the kinetics of adsorption slows down when the pH increases. The effect of different ionic strengths is also clear. Without added salt, i.e., for 0 mmol/L salt, the adsorption rate is higher than for the other two curves at all pH values. For a salt concentration of 1 and 10 mmol/L the adsorption kinetics is the same for low values of the pH, but at higher pH the 10 mmol/L curve drops faster than the other one. The desorption kinetics show the trend that the initial rates increase slightly with increasing pH, and the addition of salt seems to increase the rate of desorption even more. Using the kinetic model by Tiberg et al. (see eq 1), we may compute the kinetics of adsorption from knowing the position of the CSAC and the CMC.47 With our model we exactly predict these quantities, but it is not possible to extract a value for the transport coefficient, ktr. We therefore introduce two new variables. For adsorption we have J˜ads ) CMC - CSAC ∝ Jads, and for desorption we write J˜des ) -CSAC ∝ Jdes. Obviously, these fluxes are related to the initial adsorption/desorption rates. In Figure 10b we present theoretical predictions for the adsorption and desorption of C12E5 using our model and eq 1. It can be seen that the kinetics of adsorption does not depend on pH at low ionic strength provided that the pH is acidic. At higher pH the adsorption kinetics becomes both a strong function of pH and ionic strength. Similarly, as found experimentally (see Figure 10a), the speed of adsorption drops to zero above a threshold pH value. This value increases strongly with decreasing ionic strength. The desorption kinetics, which is only a function of the CSAC, also follow the experimental trends. With increasing pH the desorption rate increases and the rate is further enhanced by the addition of salt. The adsorption and desorption curves for the ionic strength of φ ) 10-2 appear to behave exceptionally. Both the adsorption and desorption rates are lower than the other curves at low pH. (46) van Os, N.; Haak, J.; Rupert, L. Physico-chemical properties of selected anionic, cationic and nonionic surfactants; Elsevier: Amsterdam, 1993. (47) Tiberg, F. J. Chem. Soc., Faraday Trans. 1996, 92 (4), 531-538.
In Figure 6, where the CSAC was plotted as a function of the ionic strength φCl showed that the CSAC started to become significantly influenced by the poorer solvent quality already for φbCl ) 10-2. The rate of adsorption is a function of the CMC and the CSAC. In Figure 2a,b we further showed that the CMC also becomes a strong function of the salt strength above φCl ) 10-2. The decrease of the adsorption and desorption rates for the rather high ionic strength system must therefore be attributed to these more dramatic solvency effects.
Conclusion We have applied the molecularly realistic model that was used previously for the adsorption of PEO onto silica to model the adsorption of non-ionic surfactants onto the silica as a function of pH and ionic strength. Using this model we predicted the CMC and the aggregation number of the surfactants, which are a function of the ionic strength. We focused on the sudden adsorption of the non-ionic surfactants that occurs above the so-called CSAC and note that the jumplike behavior is due to the mean-field assumption that we use. In reality the CSAC is a sharp, yet smooth transition. We predicted how the CSAC responds to changes in the pH and ionic strength. By combining the bulk and adsorption results, we find an adsorption/desorption transition as a function of the pH, which depends on the ionic strength. This transition is also found experimentally. By combining results of the theoretical model with a simple kinetic model, we can also predict the initial rate of adsorption and desorption. Again, these predictions compare favorably to the rates measured experimentally. In summary, by introducing molecular detail into SCF calculations, we have found rich bulk and interface behavior that largely follows experimental findings. Acknowledgment. This research forms part of the research program of the Dutch Polymer Institute (DPI), Project No. 292. LA703827M
