Colloidal Stability Influenced by Inhomogeneous ... - ACS Publications

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Colloidal Stability Influenced by Inhomogeneous Surfactant Assemblies in Confined Spaces A. B. Jo´dar-Reyes*,† and F. A. M. Leermakers‡ Departamento de Fı´sica Aplicada, Facultad de Veterinaria, UniVersidad de Extremadura, AVda. de la UniVersidad s/n, Ca´ceres, Spain, and Laboratory of Physical Chemistry and Colloid Science, Wageningen UniVersity, Dreijenplein 6, 6703 HB Wageningen, The Netherlands ReceiVed: April 23, 2009; ReVised Manuscript ReceiVed: June 8, 2009

Recently, a molecular-level self-consistent field approach was used to show that some surfactants assemblies (with local cylindrical structure) can bridge between two surfaces that in turn are covered by surfactant bilayers. The stability of such a connection is related to a higher end-cap (free) energy of the worm-like micelle (in solution) than the connection (free) energy of the micelle with the surface layer. This preliminary study has been extended here to know the viability of this connection as a function of different parameters related to the surfactant structure and the interaction between the surfaces and the different moieties composing the surfactant. The effect of such parameters on the structure of the connection, the thermodynamic stability of such a formation, and the interaction curve between such surfaces has been analyzed in case the connection was possible. A secondary minimum has been found, which could give rise to surfactant-induced flocculation. This minimum is strongly affected by the surfactant properties. Other interesting surfactant structures have been predicted to form in confined spaces under certain parameter regimes. Introduction The important role of surfactants as stabilizers of different systems (i.e., colloidal dispersions) in several fields (industry, medicine, biology) is well-known.1-16 Examples are found in textile industry (dyeing of fibers, water resistance5), paper manufacturing and recycling of wastepaper,6 paints,7 adhesives, photographic material,7 cleaners,9 cosmetics10 and pharmaceuticals,11 immunoassays,12,13 agriculture (pesticides), and food technology. In mineral processing, the adsorption of surfactants and polyphosphates improves the grinding of minerals and it makes possible the flotation process.14 In oilfield applications, they prevent or retard the formation of precipitates and their deposition on piping.15 Other interesting examples are groundwater and wastewater treatment, or electronics industry.16 Adsorption of surfactants onto the surfaces at or near the critical micellization concentration (cmc) of the surfactant usually guarantees the stability of the system.17 However, such a system may become completely unstable at very high micelle concentrations after saturated adsorption, far above the cmc.17-19 This destabilization is found to be reversible. The main explanation for this phenomenon is that micelles that are taken not to adsorb onto the surfactant-coated surfaces induce attraction between particles by a depletion mechanism. However, there exists an alternative explanation. In a previous paper, we have shown the possibility of attraction between two surfaces due to the presence of linear micelles connected to both of them.20 This idea is inspired from known results in polymer solutions. In these systems a scenario exists where attraction between particles is made possible by adsorbing chains that form bridges between them. The viability of these structures will trigger an attraction between surfaces at distances of the mean radius of gyration of the linear micelles. * Corresponding author. E-mail: [email protected]. † Universidad de Extremadura. ‡ Wageningen University.

Connections between lipid bilayers have been reported elsewhere.21,22 In addition, several authors23-25 have detected weak attractive forces between lipid bilayers at distances larger than the thickness of two bilayers by using atomic force microscopy. They suggest this phenomenon could be due to some type of connection between bilayers. Such connections are known as stalks and could be modeled as linear micelles of lipids connecting two bilayers. Studies on linear micelles between two adsorbed surfactant layers could be a novel approach to the stalk problem. In the present paper, we search for the conditions at which given surfactants can form linear micelles, and can connect two surfaces at close proximity. For doing that, we use a statistical mechanical approach that allows us to analyze the problem in a molecularly realistic way. This approach is called self-consistentfield theory for adsorption and/or association (SCF-A).26,27 As stated previously, this approach was recently used to elucidate, for one particular surfactant, that the connection of linear micelles is possible when the surfaces are hydrophilic, i.e., when there exists an adsorbed surfactant bilayer on the surfaces.20 This phenomenon is responsible for a secondary minimum in the interaction free energy as a function of the distance between the layers. A sufficient number of these connections between surfaces can provoke the flocculation of such a colloidal system. The free energy of interaction between surfaces can be obtained as the SCF-A theory permits carrying out a thermodynamic analysis of the system. The results of this first study on the connection phenomenon20 have given rise to many questions, for instance, about the effect of the surfactant characteristics and its affinity for the surface and the solvent on the stability of the connector, on the depth of the secondary minimum, etc. To help answer these questions is one of the main aims of this paper. The self-consistent field approach used for the present analysis has been presented in detail in ref 20. Here, we will briefly review the main characteristics and summarize the parameters of the model. We focus on the surface-surfactant interaction

10.1021/jp9037599 CCC: $40.75  2009 American Chemical Society Published on Web 07/14/2009

Surfactants as Stabilizers

Figure 1. Representation of the two-gradient cylindrical lattice between two (parallel) surfaces. Lattice layers between the surfaces are numbered as z ) 1, 2, ..., H. Each layer presents concentric shells of lattice sites referred to with R ) 1, 2, ..., M (only seven rings are drawn). The different intensities are only for illustration purposes.

parameters and on the surfactant properties that allow bridging to occur. For the latter, different commercial nonionic surfactants of the alkylethylene oxide CnEm family have been studied. For analyzing the effect of the length of the hydrophilic head (m), surfactants C12Em with m ) 4, 5, 6, 7 were used, whereas the effect of the hydrophobic tail length was studied for the chains CnE6 with n ) 12, 14, 16. First, conditions (i.e., free surfactant bulk volume fraction) at which linear micelles of such surfactants are present in the system have to be found.28 We need to work at that surfactant concentration in order to make sure that linear micelles can connect two surfaces. For each type of surfactant, the thermodynamic stability of the connection was analyzed as a function of the relevant interactions with the substrate. Calculations have been done for systems presenting different interaction parameters between surface-hydrophilic head and surface-hydrophobic tail of the surfactant. Therefore, a wide range from strongly hydrophilic to strongly hydrophobic surfaces has been analyzed. Besides a linear micelle connecting two adsorbed surfactant bilayers, other structures have been observed at the different interaction regimes studied in this work. In case of connection, the interaction curve between the surfaces as a function of the distance between them is obtained as well as the depth of the secondary minimum. Our results are summarized in the end of this paper. Theoretical Approach A statistical thermodynamic approach that implements meanfield and lattice approximations is used in this work. It is known as self-consistent-field theory for adsorption and/or association, SCF-A.26,27 One and two gradients in molecular distribution can be obtained as it is possible to apply a mean-field approximation in two, or in one dimension(s), respectively. Here, we use a two-gradient cylindrical coordinate system (see Figure 1). The z-coordinate runs from z ) 1, 2, ..., H, where z refers to lattice layers parallel to the two surfaces. Within each layer there are rings of lattice sites numbered R ) 1, 2, ..., M. Below, the generalized coordinate r is used to point to a particular ring of lattice sites r ) (z,R). Reflecting boundary conditions in the R-direction are applied. The surface has homogeneous surface properties. Nonionic surfactants of the alkylethylene oxide are represented by a chain of segments with either polar or apolar nature, that is, CnEm with n C united atoms (CH2 or CH3) connected to m ethylene oxide E ) O1C2 units capped by a terminal O group (representing the terminal alcohol OH). The water molecules are modeled by a pair of polar segments, that is, W2. Details

J. Phys. Chem. B, Vol. 113, No. 32, 2009 11187 on the statistical weight of all possible and allowed conformations (freely jointed chains in a potential field) of the chainlike molecules are found in the literature, i.e., for the one-gradient SCF-A27,29-32 and for the two-gradient SCF-A33-35 approaches. Each lattice site r ) (z,R) is assumed to be filled by one of the segments X ) W, C, O. The volume fraction of segments in a layer r (φX(r)) is defined as the number of segments nX(r) in this layer divided by the number of lattice sites at r, L(r). The corresponding values in the bulk are φXb. In each ring of lattice sites r ) (z,R) a mean-field approximation is applied and for each segment type X there is a given self-consistent potential uX(r) conjugated to the volume fraction φX(r).33-35 To study the formation of linear (cylindrical) micelles in solution, the coordinate system shown in Figure 1 can be used, but with the modification that the surfaces are replaced by mirrorlike boundary conditions. Micelle long axis are along the z-direction. Segment density gradients in the radial direction, R, and in the direction along the axis of the cylinder, z, are possible. The constraint that linear micelles, in solution or between the surfaces, are not allowed to curve or bend, has some entropic consequences. As the same entropic consequences are neglected in the freely floating linear micelles as well as in the case that the linear micelle bridges between the surfaces, we may argue that this error cancels. There may be secondary entropic effects of semiflexible linear micelles in confined spaces, leading for example to depletion forces. Such contributions can be estimated from knowledge about the average length of the linear micelles and the persistence length. Typically, depletion forces are expected to be smaller than the bridging forces and therefore we believe that our approach is reasonable. The segment potentials uX(r) include segment-surface interactions (when a surface is present) and excluded-volume contributions such as the short-range nearest-neighbor contact energies that feature Flory-Huggins interaction parameters. Consistency between the segment potentials and the segment volume fractions is generated numerically,26,32 taking into account the following incompressibility constraint, ∑XφX(r) ) 1 ∀ r ) (z,R). How to obtain the adsorption isotherms is described in detail elsewhere.20,29 From calculations, the equilibrium volume fraction profiles (normal to the surface; assuming lateral homogeneity) are obtained. The adsorbed amount of surfactants is calculated by integration over these profiles, and it is expressed as the excess amount of chains per surface site (Γ ≡ nexcN) (where N is the number of segments in the surfactant). This quantity is computed for a large set of volume fractions of surfactant in the bulk to get the adsorption isotherms, i.e., Γ(φb). This SCF approach also provides the grand potential per unit area Ω, which is interpreted as (minus) the surface pressure γ in the adsorption problem, (γ for surface next to a pure solvent phase is close to zero). Volume fractions are typically obtained with 7 significant numbers and the grand potential has at least 5 significant numbers. Parameters. The size of each segment in the surfactant is supposed to be of a lattice site l. All linear lengths are given in units l, with approximately l ≈ 0.2 nm (corresponding to the size of a united atom CH2) to convert lattice units to real dimensions. Different nonionic surfactants of the alkylethylene oxide CnEm family will be studied. Surfactants with the same hydrophobic tail size, C12Em with m ) 4, 5, 6, 7, and chains with the same hydrophilic head CnE6 with n ) 12, 14, 16. We have considered here the reasonable set of interaction parameters used in previous studies.20,28 For the hydrocarbon

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TABLE 1: Characteristics of Linear Micelles Formed from Different Surfactants under the Interaction Conditions: χCW ) 1.5, χOW ) -0.5, χCO ) 2a C12E4 φsb*(10-4) c

E

C12E5

C12E6

C12E7

C14E6

C16E6

1.44596 1.52776 1.60739 1.68291 0.25991 0.04082 16.19 13.18 10.87 9.08 14.84 19.57

a The free surfactant bulk volume fraction (φsb*) is expressed in segments per site, and the end-cap (free) energy, Ec, in kBT units.

(C)-water (W) interaction, we have, χCW ) 1.5. For ensuring the hydrophilicity of the headgroup, we choose χOW ) -0.5. Finally, repulsive interaction between the hydrophobic tail and the hydrophilic headgroup is guaranteed with χCO ) 2.2. When a surface is present, we have to include interaction parameters between segments and surface, that is, χSX with X ) O, C, W. In line with common practice we have set χSW ) 0. The affinity of the surface for the polar and apolar moiety of the surfactant molecule will be varied, considering the χSO in range χSO ) (-6, ..., 2), and χSC in range χSC ) (-2, ..., 2). The most hydrophilic surface corresponds to χSO ) -6, χSC ) 2. Results and Discussion First, some properties of the linear micelles in solution formed from the surfactants under study are presented. In the subsequent subsection we will show the structural properties of stable connections formed by the different surfactants. The effects of the surfactant properties and their affinities for the surface on such a structure are analyzed. The following subsection is focused on the thermodynamic stability of the connections for the different surfactant chains under different interaction conditions (varying χSO and χSC). Then, when the connection is viable, the interaction curve between two connected surfaces as a function of the distance between them is analyzed. Finally, other surfactant structures found between the two surfaces in the wide parameter regime analyzed are shown. Linear Micelles in Solution. As a prerequisite for the connection problem, it is necessary to establish the system conditions at which linear connectors (linear micelles) are stable. More specifically, we have to work at the chemical potential (free surfactant bulk volume fraction) for which long linear micelles can exist for each surfactant, φsb* (here and below the asterisk is reserved to the condition where long linear micelles are present). In a previous paper,28 we showed how to obtain detailed structural and thermodynamical data for linear dumbbell-like micelles composed of CnEm surfactants by means of the SCF-A approach. The end-cap energy, Ec, the (free) energy required to create two end caps from a semiinfinite cylinder, can be also obtained from the calculations on linear micellization.28 As will be mentioned below, this is an important parameter for discussing the thermodynamic stability of the connection of linear micelles to surfaces. For the current set of parameters these end-cap (free) energy values are presented in Table 1. In Table 1, we see that the surfactant concentration at which stable linear micelles start forming (known as the second cmc) increases slightly by increasing the head size, but decreases strongly by increasing the tail length. Indeed, these trends follow the tendency of the (first) cmc. Regarding the end-cap energy, its value increases approximately quadratic with the tail length and is approximately inversely proportional to the headgroup size. These trends are in agreement with data presented elsewhere.28 Molecules with a high end-cap energy are very likely to form long linear micelles. Indeed for such systems it might be difficult to find experimental conditions where still

spherical micelles form (only very close to the true cmc). Indeed, the surfactants used in the present study were selected to be likely to have a large part in their phase diagram where the aggregation form is the wormlike micelle (they all have a reasonably large Ec value). Structural Analysis on Connection. If two identical surfaces are near each other and placed in the presence of surfactants at the chemical potential at which long linear micelles exist in the bulk (φsb*), a connection of the surfaces through a linear micelle is possible. As stated previously, the segment distribution of each type of segment as well as the overall surfactant volume fraction profile in the system can be calculated from the SCF-A theory. Examples for such profiles of thermodynamically stable connections for the different surfactants considered in this work are shown in Figures 2 and 3, where the effect of the head and the tail size, respectively, are shown. An analysis of the head and tail segment distributions allows us to know the type of surfactant layer (e.g., bilayer, monolayer) formed on the surfaces. Interaction parameters are fixed to the values given in the legend. Linear lengths are given in units l, with l ≈ 0.2 nm. Discussion on the analysis of the thermodynamic stability of those systems can be found in next section. From the profiles showed in Figures 2 and 3, and the analysis of the corresponding head and tail segment distributions, we prove the feasibility of a connection by way of a linear micelle to surfactant bilayers adsorbed on each surface. We also see that the linear connector is narrower just near the connection point. This is called the neck.20 Far from the stalk (another name for the linear micelles connecting bilayers), the adsorbed bilayer is homogeneous. However, we appreciate that the size of the polar head of the surfactant (see Figure 2) clearly affects the structure of the bilayer near to the connector. For m ) 4, 5, 6, less segment density is found in the bilayer just under the connector, which is pulling somehow. In addition, the bilayer becomes thinner around the stalk forming a collar when increasing m. This collar causes the breaking of the bilayer for m > 7 leading to the instability of the connection. In this situation, there is a hole in the surface bilayer whose size depends on the system, and, therefore, we consider it is less relevant to show. Just slight differences are observed when comparing the profiles for surfactants with different tail length (Figure 3). However, the connection becomes thermodynamically unstable for C16E6 as the corresponding linear micelles prefer to stay in solution instead of bridging two surfaces (see next section). The effect of the interaction parameters on the connection profile can be studied by taking a given surfactant as an example (here C12E5). Results are shown in Figures 4 and 5 for different χSO and χSC values, respectively. In Figure 4 we present profiles for C12E5 at a fixed χSC ) -0.5, which means that just a slight attraction between the surface and the apolar segments is considered. We show profiles for surfaces with different affinity for the polar segments, from very strong attraction (χSO ) -6) to slight repulsion (χSO ) 0.5). Again, the linear micelle is connected to adsorbed bilayers on the surface, but a detachment of the bilayers starting under the connection point and extending to the remainder of the bilayer by increasing χSO is observed. Higher repulsion (higher χSO) leads to the complete detachment of the bilayer. Figure 5 shows profiles for C12E5 and a high attraction between the surface and the polar segments χSO ) -3.4. Different affinity of the surface for the apolar segments is considered, from high attraction (χSC ) -2) to high repulsion (χSC ) 2). Detachment of the connected bilayers starting under

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Figure 2. 2D “equal density” contour plot of the overall surfactant volume fraction profile for a linear micelle in the presence of two surfaces for different C12Em surfactants (χSO ) -5.7, χSC ) -1.5) at φsb*. Dark gray: φs g 0.875, white: φs < 0.125, ∆φs ) 0.125. The surfaces are placed at z ) 0 and z ) 100. The surfactants used are indicated.

tensionless at the conditions under study, the latter contribution is only due to the end-cap energy (higher than 9 kBT for the surfactants we are analyzing). In our analysis we ignore other possible contributions to the total free energy of interaction between the surfaces (van der Waals forces or electrostatic forces (if the surfaces are charged)). They should be added on top of the effects discussed below. Due to the symmetry in the system, computations can be done using only half of the system, that is, connection of half a linear micelle to one surface. To calculate the connection free energy, Econ, we use the following expression

E con ) Ω - πR2γ*

Figure 3. 2D “equal density” contour plot of the overall surfactant volume fraction profile for a linear micelle in the presence of two surfaces for different CnE6 surfactants (χSO ) -5.4, χSC ) -0.5) at φsb*. Dark gray: φs g 0.875, white: φs < 0.125, ∆φs ) 0.125. The surfaces are placed at z ) 0 and z ) 100. The surfactants used are indicated.

the connection point and extending to the remainder of the bilayer by increasing the repulsion is also observed, but the connection of the linear micelle to the adsorbed bilayers is still stable even at repulsion (χSC ) 2) conditions. As a conclusion, the stable connection state for C12E5 surfactant needs surfaces that present attraction for polar surfactant segments (or just slight repulsion), and a certain repulsion for the apolar segments is allowed. Besides profiles corresponding to the state where a linear micelle is connected to two adsorbed surfactant bilayers (stalklike connections), other structures have been observed at the different interaction regimes studied in this work. We will comment on them in the last subsection. In next subsections the focus is on stalk-like connections. Thermodynamic Stability Analysis. Two contributions to the free energy of the system are compared in order to establish the thermodynamic stability of the connected state. On the one hand, the free energy involved on the connection of the linear micelle to the surfaces, and on the other hand, the free energy of a linear micelle in solution.20 As the linear micelles are

(1)

where Ω is the grand potential obtained from 2G-SCF computations when the connection is formed. Again, as we are working at φsb*, the central part of the wormlike micelle is tensionless and therefore there is no contribution due to some tension in the linear micelle. The connection itself and the inhomogeneities appeared in the adsorbed bilayer as a consequence of such a connection are contributing to Ω. In the absence of the connector, we have a laterally homogeneous surfactant layer on the surface, that is why we have to subtract γ* · area ) γ*πR2. Here, γ* is the surface tension of the adsorbed surfactant film at φsb* obtained from 1G-SCF calculations. More details of this procedure can be found in ref 20. The stability criterion is the following: if Ec/2 > Econ, the spontaneous formation of stalks is expected, whereas for Ec/2 < Econ the wormlike micelles rather remain freely dispersed in solution. However, if Econ - Ec/2 is very small ( 0, other structures different from the stalklike connection are more stable. However, when the repulsion between the surface and the apolar segments is stronger, the stalklike connected state becomes more stable but only at a very strong attraction between the surface and polar segments. For C12E7 only very strong attractive surface-polar segments interaction makes the connection stable. The tail length also strongly influences the stability of the connected state. By increasing n, the stability region becomes smaller and even disappears. Interaction between Surfaces in the Stalklike Connection State. In this section we consider the contribution of the bridging of linear micelles to the free energy of interaction between two surfaces. The effect of the surfactant characteristics and the affinity of the surfactant segments for the surfaces are again analyzed. The free energy of interaction is defined as

Fint(H) ) 2(E con(H/2) - E con(∞))

(2)

where H is the total distance between the two surfaces. The corresponding force between the particles is given by f ) -∂Fint/ ∂H, which is attractive (negative) when the free energy of interaction decreases with decreasing distance, and repulsive (positive) when Fint increases with decreasing distance. In a recent paper,20 we found that for a given surfactant under certain interaction parameter conditions, the interaction between two surfaces with a bridging micelle between them showed an oscillatory behavior. The wavelength of the oscillations λ was intrinsic to the surfactant system and not a property influenced by the surfaces. When surfaces were far apart, they did not feel each other and the free energy of interaction remained zero. At intermediate distances, we observed the oscillatory behavior, with attractive regions (an integer number of wavelengths fit) and repulsive regions (the wavelength did not fit between the surfaces). At smaller distances between the surfaces, the amplitude of the oscillation grew exponentially, similarly to structural forces. At very close proximity when less than one wavelength did fit onto the stalk, a huge attraction was observed (H ≈ 30). At the corresponding absolute minimum the linear micelle disappeared between the adsorbed bilayers. Decreasing H even further led to repulsion due to compression of the adsorbed bilayers. The clue to understand the contribution of the micelle bridging to the interaction between the surfaces and the viability of influencing the colloidal stability of the system is therefore

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J. Phys. Chem. B, Vol. 113, No. 32, 2009 11191 TABLE 2: Energetic Barrier (kBT Units) in the Interaction Curve between Two Surfaces Due to Bridging by a Linear Micelle χSC ) -1, χSO ) -5.5 barrier C12E5, χSC ) -1 barrier C12E5, χSO ) -3.5 barrier

C12E4

C12E5

C12E6

C12E7

C14E6

0.5086 χSO ) -6 0.7322 χSC ) -2 0.3990

0.7080 χSO ) -5 0.6808 χSC ) -1 0.5748

0.7854 χSO ) -4 0.6148 χSC ) 0 0.6526

0.4794 χSO ) -3 0.5278 χSC ) 1 0.6888

1.3116 χSO ) -2 0.4042 χSC ) 2 0.7048

surfactant (see Figure 7). However, by increasing the tail length, the primary maximum is shifted to higher values. Correspondingly, the position of the secondary minimum shifts. If this secondary minimum is sufficiently deep, it defines the equilibrium distance between two surfaces. Therefore, at equilibrium, the (preferred) distance between surfaces increases by increasing the tail length. The influence of the headgroup length m on the barrier height is also small. The corresponding effect of the tail length n is strong. For ease of comparison, we have collected these relevant values for the primary maximum in Table 2. The effect of the affinity of the polar and apolar surfactant segments for the surface on the interaction curve can be discussed from data presented in Figures 8 and 9, respectively. The positions of the maximum and secondary minimum are not affected by the affinity of the surface for the polar segments. However, the barrier decreases by decreasing the attraction between surface and segments O (see Table 2). Flocculation by bridging is more favorable when a surface strongly attracts the polar segments. Figure 6. Phase diagrams of stable states for different CnEm surfactants by changing interaction conditions (χSC and χSO). Stalklike connection (gray); detachment of connected bilayers (light gray); inhomogeneous connection (lines); monolayer connection (dark gray). Labels in white spaces indicate the nature of the surfactant layers on the surfaces in absence of connection. The different panels are for different surfactants as indicated.

Figure 8. Interaction free energy Fint in units of kBT versus the separation distance H in lattice units between two flat surfaces connected by a linear micelle for C12E5 surfactant at χSC ) -1, and φb* s at different χSO values: χSO ) -6 (solid), χSO ) -5 (dash), χSO ) -4 (dot), χSO ) -3 (dash dot), and χSO ) -2 (dash dot dot). Figure 7. Interaction free energy Fint in units of kBT versus the separation distance H (normalized by the lattice site l) between two flat surfaces connected by a linear micelle for different surfactants at χSO ) -5.5, χSC ) -1 and φb* s : C12E4 (solid), C12E5 (dash), C12E6 (dot), C12E7 (dash dot), and C14E6 (dash dot dot).

analyzing the barrier and the minimum (called as secondary minimum) that appear at distances slightly larger than that of the absolute minimum (H > 30). Results for the interaction curve between surfaces in the case of a stable connected state for the different surfactants under study are shown in Figure 7. To observe only the effect of the surfactant characteristics, the curves are obtained by using the same set of interaction parameters. The position of the maximum and the secondary minimum are just slightly affected by the size of the headgroup of the

Figure 9. Interaction free energy Fint in units of kBT versus the separation distance H in lattice units between two flat surfaces connected by a linear micelle for C12E5 surfactant at χSO ) -3.5, and φsb* at different χSC values: χSC ) -2 (solid), χSC ) -1 (dash), χSC ) 0 (dot), χSC ) 1 (dash dot), and χSC ) 1 (dash dot dot).

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Figure 10. 2D “equal density” contour plot of the volume fraction profile for a linear micelle composed of C12E7 surfactant in the presence of two surfaces at φsb* and χSC ) -1, χSO ) -1. Dark gray: φs g 0.875, white: φs < 0.125, ∆φs ) 0.125. The surfaces are placed at z ) 0 and z ) 100.

An affinity between apolar segments and surface does affect the position of the primary maximum and the secondary minimum. Both quantities shift to longer distances by increasing χSC from negative to positive values. The barrier also increases in this case (see Table 2). The equilibrium position between surfaces connected by a linear micelle is mainly determined by the size of the surfactant tail and its affinity for the surface. The barrier is also strongly affected by the tail size, but slightly influenced by the affinity between surface and surfactant segments. We now can summarize our findings. Very few stalks per particle are needed to overcome the kinetic energy of the particles completely. The range of interactions is expected to be large, as this range is set by the characteristic length of the wormlike micelles in solution. Due to the presence of the secondary minimum, the flocculation of colloidal particles (seen as flat surfaces in comparison with linear micelle dimensions) is reversible. At separation distances between surfaces in the order of the secondary minimum of the interaction curve, there remains a surfactant layer on each of the two surfaces. If the surfactant concentration is reduced to below the second cmc, the connection becomes unstable and the particles redisperse. As stated previously, if there are other confinement effects, these should be added. It may be a potentially interesting avenue to explore entropic confinement effects which, e.g., lead to depletion, rather than to bridging. Other Surfactant Structures in Confined Spaces. Besides the stalklike connection state analyzed in previous sections, other interesting surfactant structures can be found between two surfaces depending on the type of surfactant, and its interaction with the surfaces. In this section, we comment on the profiles corresponding to the main structures observed and on the phase diagrams of all the states for the different CnEm surfactants used in this work. We have already presented the profiles of some of the main surfactant structures found. On the one hand, in Figure 2 we could observe that under certain conditions for C12E7 surfactants a collar started forming in the adsorbed bilayer just around the connector. Let us call this situation as the “inhomogeneous connection” state. On the other hand, in Figure 3, for C12E5 surfactants at χSC ) -1 and χSO ) 0.5, we observed the detachment of the bilayer just under the connected linear micelle, this is, the “detachment of connected bilayers” state. Other structures involving adsorbed monolayers have been also predicted. This is the case for the “monolayer connection” state. The corresponding profile is given in Figure 10. A neck in the linear micelle and a collar in the adsorbed monolayer around the connector are also observed. In the following we will describe the phase diagrams of stable states presented previously in Figure 6 for each of the surfactants.

Jo´dar-Reyes and Leermakers For C12E4 surfactant (top left graph), in the case of very high hydrophilic surfaces, surfactant bilayers are expected on the surfaces, whereas linear micelles remain in solution. For χSC > 0, when decreasing the surface-polar segment attraction the “detachment of connected bilayers” state is predicted. The connection is possible, but it is different from the “stalklike” state as the connector pulls the bilayer causing its detachment. By decreasing such an attraction even further, the connection is lost and a completely detached bilayer is found. As it was mentioned, attraction between the surface and the polar segments (χSO < 0) is necessary in order to obtain the “stalklike” state, but only a small attraction for the apolar segments is allowed. Otherwise, when the surface becomes highly hydrophobic, the “monolayer connection” state is expected. Other structures in which linear micelles are absent are more stable for χSC e -2: adsorbed bilayers at very strong attraction between the surface and the polar segments, and adsorbed monolayers when this attraction decreases down a certain value. This is due to the fact that it becomes more difficult to have linear micelles for surfactants with a small head. In the case of C12E5 (top right graph), we see a wide regime corresponding to the “stalklike connection” state. This state is always found for all the χSC values analyzed, but it is lost when χSO increases from a certain value. For instance, at χSC g -0.5, when increasing χSO the “detachment of connected bilayers” state can be observed. A completely detached bilayer is found when increasing such a value even further. Other scenarios are expected for χSC < -0.5: up to certain χSO value, the adsorbed bilayer starts breaking through the collar, and we enter in the “inhomogeneous connection” state; By making the surface less attractive or even a little repulsive for the polar segments, the system prefers to reach the “monolayer connection” state. Finally, for the more positive values of χSO, an adsorbed monolayer is predicted on each surface, while linear micelles remain in solution. From the phase diagram of C12E6 (middle left graph), for χSC > 0, only when the repulsion between the surface and the apolar segments is very high the stalklike connection state becomes more stable at a very strong attraction between the surface and polar segments. However, at less negative χSO values, the breaking of the bilayer through the collar is observed, that is, the “inhomogeneous connection” state. If repulsion between the surface and the apolar segments becomes lower, but still χSC > 0, an adsorbed bilayer is expected instead of the stalklike connection state at highly negative χSO values. For χSC e 0, this is, when apolar segments feel attraction for the surface, the stalklike connection state is always observed at a very strong attraction between the surface and polar segments. However, other states are found when decreasing such an attraction or even by turning to repulsion: “detachment of connected bilayers” state and detached bilayer at χSC ) 0; inhomogeneous connection state at χSC e - 0.5; and monolayer connection state or only monolayer adsorption at χSC e - 1, that is, when the surface is highly hydrophobic. For C12E7 (middle right graph), we already commented that only at a very strong attractive surface-polar segments interaction the stalklike connection state was found. The formation of a collar leads to the instability of the connection at higher χSO values for all the χSC values analyzed, this is, to the inhomogeneous connection state until the linear micelle is not expected to connect to the inhomogeneous adsorbed bilayer anymore. This is the case when there is no attraction of the apolar segments for the surface, that is, for χSC > 0. Otherwise, when this attraction is allowed, the inhomogeneous connection state

Surfactants as Stabilizers can be followed by the monolayer connection state and only monolayer adsorption by increasing the χSO value even further. When analyzing the phase diagrams of surfactants with a higher tail length (bottom graphs), we see that the stalklike connection state is limited to surfaces that present high attraction for the apolar segments, and very strong attraction for the polar segments (in the C14E6 case) or is not even expected (C16E6 case) in the whole range of interaction parameters studied. For C14E6 (bottom left graph), at χSC g - 0.5, and very strong hydrophilic surfaces, surfactant bilayers are expected on the surfaces. By decreasing the surface-polar segment attraction the “detachment of connected bilayers” state is predicted until a completely detached bilayer is found by decreasing such an attraction even further. For χSC e -1, at higher χSO values than those corresponding to the stalklike connection state, the inhomogeneous connection state, the monolayer connection state and the scenario in which there are only monolayers on the surfaces can be observed. As stated previously, C16E6 surfactant (bottom right graph) is unable to reach the stalklike connection state at the wide surface interaction regime analyzed in this work. The adsorption of bilayers is always more favorable for the system at the most negative χSO values. At χSC g -1, the “detachment of connected bilayers” state followed by the total detachment of the bilayer are found by making χSO less negative or even positive. For surfaces which present higher attraction for the apolar segments, that is, for χSC e -1.5, at higher χSO values than those corresponding to adsorption of bilayers, the inhomogeneous connection state, the monolayer connection state, and the monolayer adsorption can be observed. Conclusions The molecularly realistic SCF-A theory predicts different surfactant structures between two surfaces depending on the interaction between the surface and the polar and apolar part of the surfactant. The thermodynamic stability of such a state depends also on the surfactant head and tail length. One of these states consists of the connection between surfaces by means of a linear micelle (stalklike state). The interaction curve between the surfaces connected by the linear micelle shows a secondary minimum and a barrier which are mainly affected by the surfactant tail properties. The flocculation of the system is already possible when just a few of such connecting micelles between the surfaces. Therefore, this bridging mechanism is a serious alternative for depletion flocculation. Other interesting surfactant structures (i.e., the detachment of adsorbed bilayers connected by linear micelles, the connection of linear micelles to adsorbed monolayers) have been observed. Besides the importance of these results regarding the role of surfactants as stabilizing agents in industry, medicine, or biology, they could also help research in the field of lipid vesicles, lipid bilayers, and other systems of interest in nanotechnology. Acknowledgment. A.B.J. thanks the University of Extremadura for financial support through the “Plan de Iniciacio´n a la Investigacio´n, Desarrollo Tecnolo´gico e Innovacio´n. Project

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