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Langmuir 1998, 14, 2940-2949
Articles Polyelectrolyte-Induced Micellization of Charged Surfactants. Calculations Based on a Self-Consistent Field Lattice Model Torsten Wallin and Per Linse* Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University, P.O. Box 124, S-221 00 Lund, Sweden Received November 25, 1997. In Final Form: March 9, 1998 The complexation of charged surfactant molecules and oppositely charged polyelectrolytes was studied by the use of a mean-field lattice theory for flexible polyelectrolytes and surfactants in solution. The effect on the critical aggregation concentration, cac, and the surfactant aggregation number, Nagg, of (i) salt concentration, (ii) polyelectrolyte concentration, (iii) linear charge density of the polyelectrolyte, and (iv) hydrophobicity of the polyelectrolyte was investigated. The changes in the cac and Nagg upon changes of polyelectrolyte properties were found to qualitatively agree with experimental results, but the predicted cac was slightly decreased at salt addition instead of being increased as experimentally observed. The results showed that the solvency of the polymer backbone affected the structure of the aggregate. A polyelectrolyte with a hydrophilic backbone was accumulated in the headgroup region and slightly outside it but did not otherwise penetrate the micelle. For the case with a hydrophobic backbone, a substantial penetration of the micellar core was found, which also was accompanied by a reduction of the cac and, at a low linear charge density of the polyelectrolyte, a diminished aggregation number.
I. Introduction Considerable amount of experimental effort has been devoted to mixtures of polymers and surfactants in aqueous solution. The experimental work from the late 80s and early 90s have been summarized by Lindman and Thalberg,1 and a comprehensive overview of more recent developments are given in a monograph edited by Kwak.2 A noticeable feature found in aqueous solutions of mixtures of hydrophilic polyelectrolytes and charged surfactants is that (i) the polyelectrolyte strongly reduces the critical micellization concentration (cmc) of oppositely charged surfactants. In the case of added polymer, this concentration is normally referred to as the critical aggregation concentration (cac). In addition, other general features are (ii) an increased linear charge density of the polyelectrolyte reduces the cac/cmc ratio, (iii) an increased length of the hydrophobic surfactant tail reduces the cac/ cmc ratio, (iv) an increased salt concentration increases in the cac but decreases the cmc, and (v) none (sometimes with exception of (i) and (ii)) of these changes has any essential effect on the size of the aggregate (aggregation number). The polyelectrolyte-surfactant complexation has also been subjected to theoretical investigations and this area has also been recently summarized.3 We have recently investigated the complexation between one hydrophilic polyelectrolyte and one micelle composed of oppositely charged surfactants by performing Monte Carlo (MC) (1) Lindman, B.; Thalberg, K. In Interactions of Surfactants with Polymers and Proteins; Goddard, E. D., Ananthapadmanabhan, K. P., Eds.; CRC Press: Boca Raton, FL, 1993. (2) Kwak, J. C. T., Ed.; Polymer-Surfactant Systems; In Surfactant Science Series; Marcel Dekker: New York, 1998. (3) Linse, P.; Piculell, L.; Hansson, P. In Polymer-Surfactant Systems; Kwak, J. C. T., Ed.; Marcel Dekker: New York, 1998.
simulations of simple model systems.4-6 The micelle was treated as a charged hard sphere and the polyelectrolyte as a chain of charged beads connected with harmonic bonds. We found that the electrostatic interaction was sufficient to form strong complexes between the polyelectrolyte and the oppositely charged micelle. The complexation was associated with (i) a strong reduction of the electrostatic energy of the system and (ii) a release of counterions of the micelle and the polyelectrolyte. Both aspects are important for the difference between the cmc and the cac values. Moreover, we found that the cac/cmc ratio decreases with (i) increasing flexibility of the polyelectrolyte,4 (ii) increasing linear charged density of the polyelectrolyte,5 and (iii) increasing surfactant tail length.6 Finally, we found that the value of the cac/cmc ratio, which typically was between 0.01 and 0.1, compares favorably with experimental data for matching systems. Hence, the model seems to give a good picture of the polyelectrolyte-charged surfactant complexation at dilute solutions for hydrophilic polyelectrolytes. In many experimental systems, however, the polymer is more hydrophobic (but still water soluble). In the presence of surfactant micelles, the polymers associate to the headgroup region of the micelle or in the case of hydrophobically modified polymers, the hydrophobic side chains are involved in the self-aggregation in a fashion similar to that for surfactant molecules.3,7-14 (4) Wallin, T.; Linse, P. Langmuir 1996, 12, 305. (5) Wallin, T.; Linse, P. J. Phys. Chem. 1996, 100, 17873. (6) Wallin, T.; Linse, P. J. Phys. Chem. 1997, 101, 5506. (7) Effing, J. J.; McLennan, I. J.; Kwak, J. C. T. J. Phys. Chem. 1994, 98, 2499. (8) Iliopoulos, I.; Olsson, U. J. Phys. Chem. 1994, 98, 1500. (9) Magny, B.; Iliopoulos, I.; Zana, R.; Audebert, R. Langmuir 1994, 10, 3180. (10) Guillemet, F.; Piculell, L. J. Phys. Chem. 1995, 99, 9201.
S0743-7463(97)01291-2 CCC: $15.00 © 1998 American Chemical Society Published on Web 05/06/1998
Polyelectrolyte-Induced Micellization
In this study, we continue to model the complexation between surfactant micelles and polyelectrolytes. We extend our scope to involve hydrophobic polyelectrolytes as well, and we have employed a more detailed model system composed of surfactant, polyelectrolyte, salt, and solvent. The more detailed model system makes it possible to relax some of the constraints in our previous work. The three most important ones are as follows: (i) the cmc and the cac are, at least in principle, directly accessible, (ii) there is a possibility of the polyelectrolyte to penetrate the micelle, which is crucial for more hydrophobic polyelectrolytes, and (iii) the micellar structure and aggregation number is able to respond to the different conditions. However, the more complex model system makes it possible to solve it only approximately, and we have used a self-consistent lattice mean-field theory, initially developed by Scheutjens and Fleer15,16 and later developed in many different directions.17 Our work has been inspired by a study Bo¨hmer et al.,18 where they modeled the micellization of charged surfactant molecules in the presence of salt, but without polyelectrolyte, also using a mean-field lattice theory. The same theory has recently also been applied to model the adsorption of polyelectrolytes on solid surfaces.19-23 The paper is organized as follows. A brief overview of the theoretical model is given in the following section. In section III we present the results of our investigation. The first part deals with the micellization of charged surfactants in the absence of the polyelectrolyte. In the second part, the influence of the polyelectrolyte on the micellization is considered, and finally results from systematic changes of parameters characterizing the system are presented. The main observations are extracted and discussed in section IV, and the paper ends with a summary given in section V. II. Method Theoretical Model. The adsorption of polyelectrolytes on a micelle (or on a colloidal particle in general) of the opposite charge can modeled on the basis of a selfconsistent field theory, initially developed by Scheutjens and Fleer15,16 and later extended to charged surfactants and polyelectrolytes by Bo¨hmer et al.24 and by Israe¨ls.25 We will here only give the main features of the theory for adsorption on micelles; for further details the reader is referred to the original publications. Briefly, the space is divided into spherical shells, i ) 1, 2, ..., and each shell is further divided into lattice cells of (11) Kevelam, J.; van Breemen, J. F. L.; Blokzij, W.; Engberts, J. B. F. N. Langmuir 1996, 12, 4709. (12) Anthony, O.; Zana, R. Langmuir 1996, 12, 3590. (13) Petit, F.; Iliopoulos, I.; Audebert, R.; Szo¨nyi, S. Langmuir 1997, 13, 4229. (14) Piculell, L.; Lindman, B.; Karlstro¨m, G. In Polymer-Surfactant Systems; Kwak, J. C. T., Ed.; Marcel Dekker: New York, 1998. (15) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619. (16) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1980, 84, 178. (17) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; B., V. Polymers at Interfaces; Chapman & Hall: London, 1993. (18) Bo¨hmer, M. R.; Koopal, L. K.; Lyklema, J. J. Phys. Chem. 1991, 95, 9569. (19) van de Steeg, H. G. M.; Cohen Stuart, M. A.; de Keizer, A.; Bijsterbosch, B. Langmuir 1992, 8, 2538. (20) Shubin, V.; Linse, P. J. Phys. Chem. 1995, 99, 1285. (21) Linse, P. Macromolecules 1996, 29, 326. (22) Shubin, V.; Linse, P. Macromolecules 1997, 30, 5944. (23) Dahlgren, M. A.; Leermakers, F. A. M. Langmuir 1995, 11, 2996. (24) Bo¨hmer, M. R.; Evers, O. A.; Scheutjens, J. M. H. M. Macromolecules 1990, 23, 2288. (25) Israe¨ls, R. Thesis, Wageningen, 1994.
Langmuir, Vol. 14, No. 11, 1998 2941
equal volume. The conformations of a polymer chain or a surfactant molecule are described as random walks on the spherical lattice. Within each shell the BraggWilliams approximation of random mixing is applied, and thus all sites in a shell are equivalent. Each lattice cell contains either solvent (water), one solvated ion, one surfactant segment, or one polymer segment. The possibility of obtaining radial concentrations profiles is coupled to the existence of radial dependent potentials. The potential is species dependent and is defined to be zero in a homogeneous bulk solution far away from the center of the lattice. The potential can be expressed as a sum of three terms according to
uSi ) u′i + uSiint + uSiel
(1)
where u′i is a “hard core” contribution, uSiint a contribution from short-range interactions, and uSiel a contribution from long-range Coulomb interactions. The species-independent term u′i ensures that the space is completely filled in layer i according to ΣSφSi ) 1, i ) 1, 2, ..., where φSi is the volume fraction of species S in layer i. u′i is related to the lateral pressure in a continuous model, and in bulk u′i becomes zero. The short-range contribution, uSiint, is expressed by
uSiint ) kT
χSS′(〈φS′i〉 - φS′b) ∑ S′
(2)
where χSS′ is Flory-Huggins interaction parameter for the S-S′ pair.26 The angular brackets in eq 2 denote an averaging over three adjacent layers and φS′b is the free (bulk) volume fraction of species S′. In bulk, uSiint ) 0. Finally, the long-range Coulomb interaction energy is given by
uSiel ) qSψi
(3)
where qS is the charge of species S and ψi the electrostatic potential of mean force. In line with the random mixing approximation of the short-range interaction, we let the potential depend only on the radial distance (layer number) and moreover we assume (arbitrarily) ψb ) 0. We relate the potential of mean force to the charge density through Poisson’s equation
0r∇2ψi ) -Fi
(4)
where 0r is the dielectric permittivity of the medium, ∇2 is the Laplacian, and Fi ) ∑SqSφSi the charge density in layer i. The charges of the species are located to surfaces in the middle of each lattice shell, and the space between the charged surfaces is free of charge. To calculate the volume fractions of a species in layer i, it is convenient to introduce weighting factors, which are Boltzmann weights of the species potentials according to
GSi ) e-uSi/kT
(5)
If uSi and thus GSi are known, the relative weight of all the possible conformations can be calculated, and hence also the concentration profiles can be evaluated. The species volume fraction, φSi, is simply related to nxsi, the number of sites in layer i occupied by segments of rank s (the sth segment in a chain) belonging to component x (26) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953.
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Wallin and Linse
according to
φSi )
1 Li
rx
∑x s)1 ∑δS,t(x,s)nxsi
(6)
where Li is the number of lattice sites in a layer i, and rx is the number of segments in component x (rx ) 1 for salt components and water and rx > 1 for the surfactant and polyelectrolyte). The Kronecker δ selects only segments of rank s of component x if they are of type S. The expression for the segment distribution is more complex. The correct weights of all conformations, as well as the connectivity of the chains, have to be taken into account. With the partition function as origin, nxsi is obtained by the use of a matrix method and is then given by27 s
s+1
∏ s′)r
nxsi ) Cx{∆Ti ‚[
x
(Wt(x,s′))T]‚s}{∆Ti ‚[
Wt(x,s′)]‚p(x,1)} ∏ s′)2
(7)
where Cx is a normalization factor related to the bulk volume fraction of component x, Wt(x,s′) is a tridiagonal matrix comprising elements which contains factors describing the lattice topology as well as weighting factors for each segment of rank s belonging to component x, and p(x,1) is a vector describing the distribution of the first segment of component x among the layers, ∆ and s being elementary column vectors. Since uSi is needed for obtaining φSi using eqs 5-7 and since uSi depends in turn on φSi according to eqs 1 and 2, eqs 1, 2, and 5-7 need to be solved self-consistently. In addition the electrostatic potential, which enters in eq 1 via eq 3, has to fulfill Poisson’s equation, eq 4. Equations 1-7 have two classes of numerical solutions, one corresponding to homogeneous concentration profiles throughout the entire lattice and one to a single micelle formed at the center of the lattice and in equilibrium with specified bulk concentration of the components. The free energy of forming the micelle at a fixed position, Aσ, is readily calculated from the volume fraction distributions of the species and from the chemical potentials of the components in bulk according to eqs 2-5, 14, and 15 in ref 28. For charged systems, we have also the electrostatic contribution. At equilibrium, Aσ (>0) is balanced by a negative contribution from the mixing entropy kT ln(Vm/ Vs), where Vm is the volume of the micelle and Vs is the volume of a subsystem containing one micelle and its accompanying solution.29 The total volume fraction of component x in the subsystem (equal to the stoichiometric concentration of x in the micellar solution) is the sum of the excess and bulk volume fractions according to φxtot ) Γx/Vs + φxb, where Γx is the excess number of segments of component x in the subsystem (given by eq 16 in ref 28). The critical micellization concentration is usually defined as the maximum of Aσ corresponding to the lowest possible total surfactant volume fraction and smallest micellar aggregation number, in the concentration interval where we find a numerical solution representing a thermodynamic stable nonhomogeneous system. However, in systems containing large volume fraction gradients (preferentially occurring in systems with short chain molecules and with large interaction parameters), lattice (27) Linse, P.; Bjo¨rling, M. Macromolecules 1991, 24, 6700. (28) Linse, P. Macromolecules 1993, 26, 4437. (29) Van Lent, B.; Scheutjens, J. M. H. M. Macromolecules 1989, 22, 1931.
artifacts causes Aσ to display local extreme points, which have been discussed by Bo¨hmer et al.18 Instead of smoothing the oscillating free energy curve for obtaining the cmc18 and the cac, we have chosen to compare solutions at fixed total volume fractions of surfactants and make our conclusions from the free surfactant volume fractions and the aggregation numbers. A smaller volume fraction of free surfactant molecules, which normally is accompanied with a larger aggregation number, implies a lower cmc or cac. As so far the free surfactant volume fraction is nearly constant above the cmc, or the cac (which as such is a good approximation), our free surfactant volume fractions are good approximations of the cmc and the cac of the systems. The micellar aggregation number, Nagg, is evaluated from the excess of surfactant segments arising from layers with a volume fraction larger than the free (bulk) surfactant volume fraction φbsurf according to Nagg ) (1/rsurf)∑i Li max[0,(φsurf,i - φbsurf)]). Model System. The polyelectrolyte is considered to be a completely flexible chain consisting of rpolymer segments and with τrpolymer elementary charges. Two different distributions of the total polyelectrolyte charge will be considered: (i) a smooth charge distribution, i.e., a fractional charge |e|τ on each segment and (ii) a discrete and regular charge distribution, i.e., τrpolymer charged segments evenly distributed and separated by blocks of neutral segments. The latter distribution is the most realistic one, whereas the former one was also considered since this distribution has been used in previous modeling of polyelectrolyte systems.19,21 In the smooth charge distribution polymer segments are denoted (P), whereas in the discrete charge distribution (Pc) for the charged and (Pn) for the uncharged ones. The other species are referred to as (A) neutral surfactant segment, (B) charged surfactant segment, (C) cation, (D) anion, and (W) solvent. The surfactant AnBm is also treated as a flexible chain consisting of n neutral and m charged segments. Our approach implies that the segment length of the polyelectrolyte is the same as that of the surfactant. The choice of mapping one CH2 unit of the surfactant on one segment leads to an overestimation of the flexibility of the polyelectrolyte as compared to real systems. However, our previous MC simulations showed4 that the effect of the flexibility of the polyelectrolyte on the cac is very modest provided that the bare persistent length is smaller than the dimension of the micelle. Hence, the approximation of making a flexible polyelectrolyte even more flexible should be of minor importance. The system parameters pertaining to the micellar system (without polyelectrolyte) are the same as in the investigation by Bo¨hmer et al.,18 except that we are using the same dielectric permittivity for all the components. All calculations were performed for a model system at 298 K using a relative permittivity r ) 80 and a lattice spacing d ) 3.1 Å. A rather large set of parameters is needed to characterize a model system, and we will therefore use one particular system as a reference system. The reference system is characterized by an ionic surfactant A14B3 consisting of 14 aliphatic and 3 charged hydrophilic segments with -0.33 elementary charges each, a positively charged polyelectrolyte with rpolymer ) 1000 segments with τ ) 0.5 and a total volume fraction φtotpolymer ) 3.58 × 10-4, and a salt volume fraction φtotsalt ) 1.79 × 10-4. To facilitate comparisons with other investigations we often convert the volume fraction into molar concentration, where φsite ) 0.0179 corresponds to csite ) 1 M. Thus, for the reference system we get cpolymer ) 2.0 mM (segment) and csalt ) 10.0 mM. For simplicity, all
Polyelectrolyte-Induced Micellization
Langmuir, Vol. 14, No. 11, 1998 2943
Figure 1. Volume fraction profiles of A, B, C, and D for a surfactant system without polyelectrolyte at φtotsurf ) 0.095 and csalt ) 10 mM. The profiles start from the center of the micelles: in the left panel φSi values are given on a logarithmic scale, and in the right panel the vertical scale is expanded and species D is omitted. Table 1. Surfactant Bulk Volume Fractions, Obsurf, and Micellar Aggregation Numbers, Nagg, for A14B3 Surfactant-Salt Mixtures at Various Salt Concentrationsa surfactant
salt c (mM)
104φb
c (mM)
Nagg
figure
1 10 100
2.559 2.336 1.411
0.839 0.766 0.463
55.22 56.49 61.41
1
a φtot surf ) 0.095. χSS′ ) 2 for S ) {A} and S′ ) {B, C, D, W}; otherwise, χSS′ ) 0.
interaction parameters between hydrophobic and hydrophilic species are assumed to be equal and set to 2 and all other interaction parameters are 0. Hence, the FloryHuggins interaction parameters are χSS′ ) 2 for S ) {A} and S′ ) {B, C, D, P, Pc, Pn, W} and otherwise 0. Bo¨hmer et al. found that χAW ) 2 reproduced the cmc dependence on the aliphatic tail length for nonionic surfactants.18 Departing from the reference system, we have varied (i) the salt concentration, (ii) the polyelectrolyte concentration, (iii) the polyelectrolyte length, (iv) the polyelectrolyte linear charge density, and (v) the hydrophobicity of the polyelectrolyte by altering the χ parameters. Most results are for the smooth linear charge density, but some aspects of the discrete charge distribution will also be presented. In addition, calculations for the pure micellar solution without polyelectrolyte have also been made at the same conditions (when appropriate) as in the polyelectrolytesurfactant reference system. III. Results No Polyelectrolyte. The micelle formation of the A14B3 surfactant molecules without polyelectrolyte at 1, 10, and 100 mM 1:1 electrolyte has been modeled. Figure 1 shows the volume fraction profiles at 10 mM salt. The surfactant molecules form a charged micelle with their tails (A) forming the micellar core, their headgroups (B) making the corona of the micelle and in contact with the water, and their counterions (C) located in the vicinity of the headgroups. The micelle co-ions (D) are depleted from the neighborhood of the micelle, and the concentrations of the cations (C) and anions (D) coincide (within 1%) ca. 25 layers from the center of the micelle. Thus, the model reproduces the general structure of a surfactant micelle. Table 1 gives the bulk (free) surfactant concentration and the aggregation numbers at the different salt concentra-
Figure 2. Volume fraction profiles of A, B, C, D, and P for the polyelectrolyte-surfactant reference system at csurf ) 0.525 mM, csalt ) 10 mM, and cpolymer ) 2 mM. The polyelectrolyte is smoothly charged with τ ) 0.5 and entirely hydrophilic (χPW ) 0). The profiles start from the center of the micelles: in the left panel φSi values are given on a logarithmic scale, and in the right panel the vertical scale is expanded and species C and D are omitted.
tions for the same total surfactant concentration. We see that φbsurf and Nagg are nearly the same at 1 and 10 mM salt concentration, but φbsurf decreases and Nagg increases when csalt is increased to 100 mM. The standard explanation of the salt dependence of the cmc is that the electrostatic repulsion among the headgroups are screened by the salt and therefore the micellization is facilitated. However, an alternative and sometimes a more preferred view is that the entropy cost of bringing the counterions close to the micelle is reduced at increasing salt concentration. A comparison with the model calculations performed by Bo¨hmer et al. (A12B3 surfactants) shows that the changes in structure, cmc, and aggregation number upon addition of salt is qualitatively equal to the results found in their study.18 However, the aggregation number varies less in our study. Obviously, the lower dielectric permittivity of the aliphatic surfactant tail in their study makes the aggregation number more sensitive to the salt concentration. Hydrophilic Polyelectrolytes. Smooth Charge Distribution. Volume fraction profiles for the polyelectrolytesurfactant reference system, where water is an athermal solvent for the polyelectrolyte (χPW ) 0), are shown in Figure 2. The structure of the system resembles that of the plain surfactant solution. Again, the surfactant tails (A) form the micellar core and the headgroups (B) constitute the corona which is in contact with water. However, near the micelle the counterions (C) are almost entirely replaced by polyelectrolyte (P) segments, and φPi displays a maximum only slightly shifted outward as compared to φAi. The maximal τφPi is ca. 1.5 times larger than the maximal φCi in the corresponding system without polyelectrolyte (more polyelectrolyte charges than counterion charges are brought in to the micelle). The cation (C) volume fraction still displays a local maximum, but the maximum is weaker, and a small local minimum appears ca. 12 layers from the center of the aggregate. The anions (D) are completely depleted from the micellar core and their distribution displays a weak maximum at the same radial distance as the minimum of the cation distribution. These two small extremes exist due to a surplus of polyelectrolyte charges relative to the surfactant charges in the vicinity of the micelle/water interface (further discussed below). The appearance of a weak minimum of the counterion distribution outside the
2944 Langmuir, Vol. 14, No. 11, 1998
Wallin and Linse
Figure 3. Volume fraction profiles of polyelectrolyte segments, P, for polyelectrolyte-surfactant systems at csurf ) 0.525 mM: csalt ) 10 mM, cpolymer ) 2 mM, τ ) 0.5, and rpolymer ) 1000 (reference system) (curve a); same as for part a but with csalt ) 100 mM (curve b); same as for part a but with cpolymer ) 10 mM (curve c); same as for part a but with τ ) 0.2 (curve d); same as for part a but with rpolymer ) 100 (curve e). The polyelectrolyte is smoothly charged and entirely hydrophilic (χPW ) 0). The profiles start from the center of the micelles: in the left panel φPi is given on a logarithmic scale and in the right panel the vertical scale, is expanded. Table 2. Surfactant Bulk Volume Fractions, Obsurf, and Micellar Aggregation Numbers, Nagg, for A14B3 Surfactant-Hydrophilic Polyelectrolyte-Salt Mixtures at Various Salt and Polyelectrolyte Concentrations and Length, Linear Charge Density, and Linear Charge Distribution of the Polyelectrolytea polyelectrolyte c (mM) 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 10.0 10.0 2.0 2.0
length
τ
salt c (mM)
surfactant 104φb
c (mM)
Nagg
100 100 100 1000 1000 1000 1000 1000 1000 100 1000
Smooth Charge Distribution 0.5 1 0.853 0.280 0.5 10 0.849 0.278 0.5 100 0.816 0.268 0.5 1 0.768 0.252 0.5 10 0.765 0.251 0.5 100 0.742 0.243 0.4 10 0.869 0.285 0.3 10 1.054 0.346 0.2 10 1.451 0.476 0.5 10 0.817 0.268 0.5 10 0.760 0.249
64.00 64.02 64.24 64.31 64.33 64.55 62.75 60.42 53.14 64.02 64.30
1000 1000
Discrete Charge Distribution 0.5 10 0.761 0.250 0.2 10 1.409 0.462
64.39 55.40
figure and curve
3e 3a 3b 3d 3c
tot ) 1.6 × 10-4. χ a φ surf SS′ ) 2 for S ) {A} and S′ ) {B, C, D, P, Pc, Pn, W}; otherwise, χSS′ ) 0.
micellar surface was also obtained in previous MC simulations.4-6 The distribution of the polyelectrolyte segments shows a local minimum outside the micelle. Similar depletion of polyelectrolyte outside charged surfaces adsorbed with oppositely charged polyelectrolytes has been observed at low electrolyte concentrations.21-23 Volume fraction profiles of hydrophilic polyelectrolytes complexing a surfactant micelle at different conditions are shown in Figure 3. The profile for the reference system is shown (curve a) together with the profiles at an increased salt concentration, csalt ) 100 mM (curve b); an increased polyelectrolyte concentration, cpolymer ) 10 mM (curve c); a decreased linear charge density, τ ) 0.2 (curve d); and a decreased polyelectrolyte length, rpolymer ) 100 (curve e). Common to all changes is that the pronounced maximum of the polyelectrolyte volume fraction remains in layer 8. The micellar aggregation numbers are given in Table 2 and no drastic changes are discerned. Thus, the picture of an essentially regular micelle decorated by polyelec-
Figure 4. Concentration of free surfactant molecules (filled circles) and micellar aggregation number (open circles) as a function of the linear charge density of the polyelectrolyte at csurf ) 0.525 mM, csalt ) 10 mM, cpolymer ) 2 mM, and rpolymer ) 1000. The polyelectrolyte is smoothly charged and entirely hydrophilic (χPW ) 0). The corresponding values in a polyelectrolyte-free solution are also given (squares on the abscissa).
trolyte segments, instead of being decorated by counterions as in the absence of polyelectrolyte, remains. However, a closer inspection reveals some distinct effects. Most prominent is the increasing amount of polyelectrolyte segment complexing the micelle upon a reduction of the linear charge density of the polyelectrolyte. To the leading order, this is due to the tendency of matching the micellar charge with polyelectrolyte charges; a lower τ implies a large number of complexing polyelectrolyte segments. Previous lattice modeling of the adsorption of polyelectrolyte at oppositely charged surfaces showed that the adsorbed amount Γ was inversely proportional to τ over a broad τ range. However, at sufficiently low τ, the polyelectrolyte cannot any longer compete with the simple salt and is not adsorbed any longer.19-21 In the present system there is, however, a second-order effect, via a regulation of the micellar aggregation number. Figure 4 and Table 2 show that Nagg decreases continuously from 64 to 53, as τ is reduced from 0.5 to 0.2. The micellar aggregation number at τ ) 0.2 is indeed smaller than that for the corresponding polyelectrolyte-free solution (see Figure 4). Hence, at low linear charge density, the packing issue becomes important. Figure 4 also shows the reduction of the cac with increasing linear charge density. At τ ) 0.5 the cac becomes a third of the cmc for a corresponding polyelectrolyte-free solution. The other modifications discussed in connection with Figure 3 lead to much smaller changes of the polyelectrolyte volume fraction profiles in the vicinity of the maximum close to the headgroup region. The increased salt concentration (curve b) screens out the depletion minimum, as in the case of polyelectrolyte adsorbed at oppositely charge and solid surfaces.20,21,23 The increase of the polyelectrolyte concentration shifts the depletion minimum closer to the micelle, and a shortening of the polyelectrolyte has only a minor effect. These three changes hardly affect the micellar aggregation number or the free surfactant concentration (the cac) (see Table 2). The simultaneous change of two parameters (chain length of the polyelectrolyte, salt concentration, or polyelectrolyte concentration) starting from the reference system has been modeled, but no large cooperative effects were found for the parameter ranges considered (see Table 2). Discrete Charge Distribution. The polyelectrolytesurfactant complexation with polyelectrolytes modeled with a discrete charge distribution has also been examined.
Polyelectrolyte-Induced Micellization
Figure 5. Same as in Figure 2, but for a hydrophobic polyelectrolyte (χPW ) 1) and at csurf ) 0.118 mM.
Figure 6. Same as in Figure 2, but for a hydrophobic polyelectrolyte (χPW ) 1.6) with τ ) 0.2 and at csurf ) 0.023 mM.
Table 2 shows result for the reference system with τ ) 0.5 and for the case with τ ) 0.2. For the reference system (with the higher linear charge density), the smooth and the discrete charge distributions give virtually identical results. However, for the smaller τ, the discrete representation gives a larger Nagg and a smaller cac. Moreover, an inspection of the volume fraction profiles for the charged and uncharged polyelectrolyte segments separately shows that the distribution of the uncharged segments peaks one layer further away from the micelle as compared to the charged ones (data not shown). Hydrophobic Polyelectrolyte. The complexation between hydrophobic polyelectrolytes and charge surfactant micelles will now be considered. The change of the solvency of the polyelectrolyte is easiest envisioned with the discrete charge distribution. In this representation, the character of the neutral polyelectrolyte segments is changed from being B-like to be A-like, i.e., to the same hydrophobicity as the surfactant tail segments. Obviously, since the neutral segments are hydrophobic and the charged ones hydrophilic, the hydrophobicity of the polyelectrolyte molecule increases with decreasing linear charge density. This is, e.g., the situation where a polyelectrolyte is synthesized from a mixture of neutral hydrophobic segments and charged hydrophilic ones. For the smooth charge distribution, the χ parameters used are obtained by averaging over all polyelectrolyte segments, thus retaining the homopolymer description. For τ ) 0.5 and 0.2, we get, e.g., χPW ) 1 and 1.6, respectively. The hydrophobic polyelectrolytes are still water soluble (at the concentrations used) due to the effect of the counterion entropy.30,31 (30) Khokhlov, A. R.; Nyrkova, I. A. Macromolecules 1992, 25, 1493.
Langmuir, Vol. 14, No. 11, 1998 2945
Figure 7. Volume fraction profiles of polyelectrolyte segments, P, for polyelectrolyte-surfactant systems at rpolymer ) 1000: csalt ) 10 mM, cpolymer ) 2 mM, csurf ) 0.118 mM, and τ ) 0.5 (curve a); same as for part a but with csalt ) 100 mM (curve b); same as for part a but with cpolymer ) 10 mM (curve c); csalt ) 10 mM, cpolymer ) 2 mM, csurf ) 0.023 mM, and τ ) 0.2 (χPW ) 1.6) (curve d); same as for part d but with csalt ) 100 mM (curve e); same as for part d but with cpolymer ) 10 mM (curve f). The polyelectrolyte is smoothly charged and hydrophobic.
Smooth Charge Distribution. Figure 5 shows volume fraction profiles for the hydrophobic polyelectrolyte, corresponding to the profiles for the hydrophilic polyelectrolyte given in Figure 2, and Figure 6 shows the corresponding profiles for the hydrophobic polyelectrolyte with τ ) 0.2. Again, on a logarithmic volume fraction scale, the structures of the systems resemble each other and represent a micelle decorated with polymer segments. However, for the hydrophobic polyelectrolyte with τ ) 0.5, the micellar counterions (C) are no longer accumulated close to the micellar surface (cf. Figures 2 and 5), and for the hydrophobic polyelectrolyte with τ ) 0.2 the counterions are depleted from the micellar surface region and the micellar counterions (D) are now accumulated in the headgroup region (see Figure 6). For the hydrophobic polyelectrolytes, the peaks of the distributions of the polyelectrolyte segments (P) are shifted to smaller radii and are higher, and this effect is more accentuated for τ ) 0.2 (cf. Figures 2, 5, and 6, right panel). Figure 7 shows volume fraction profiles of the polyelectrolyte segments for the hydrophobic polyelectrolytes complexing the charged surfactant micelle under different conditions. First, the polyelectrolyte profiles given in Figures 5 and 6 are reproduced in Figure 7 (curves a and d, respectively). Moreover, Figure 7 shows the polyelectrolyte profiles at an increased salt concentration, csalt ) 100 mM (curves b and e), and at an increased polyelectrolyte concentration, cpolymer ) 10 mM (curves c and f). For τ ) 0.5 (curves a-c), the changes in the salt or polyelectrolyte concentrations do not lead to any changes in the volume fraction profiles in the micellar core or in the headgroup region, whereas the changes in the aqueous domain are similar to those for the corresponding hydrophilic polyelectrolyte system. However, the complexation involving the less charged and more hydrophobic polyelectrolyte (curves d-f) is more sensitive to the external conditions. An increase in csalt increases the amount of complexed polyelectrolyte, whereas an increase in the polyelectrolyte concentration shifts the polyelectrolyte distribution inward (Figure 7, right panel). Tables 3 and 4 gives the micellar aggregation number and the cac for the hydrophobic polyelectrolytes. A (31) Piculell, L.; Iliopoulos, I.; Linse, P.; Nilsson, S.; Turquois, T.; Viebke, C.; Zhang, W. In Gums and Stabilizers for the Food Industry; Phillips, G. O., Williams, P. A., Wedlock, D. J., Eds.; Oxford University Press: Oxford, England, 1994; Vol. 7.
2946 Langmuir, Vol. 14, No. 11, 1998
Wallin and Linse
Table 3. Surfactant Bulk Volume Fractions, Obsurf, and Micellar Aggregation Numbers, Nagg, for A14B3 Surfactant-Hydrophobic Polyelectrolyte-Salt Mixtures at Various Salt and Polyelectrolyte Concentrations for Different Length and Linear Charge Distributions of the Polyelectrolytea polyelectrolyte τ
salt c (mM)
surfactant 104φb
c (mM)
length
2.0 2.0 2.0 10.0 2.0 2.0
1000 1000 1000 1000 100 1000
Smooth Charge Distribution 0.5 1 0.360 0.118 0.5 10 0.360 0.118 0.5 100 0.360 0.118 0.5 10 0.360 0.118 0.5 10 0.360 0.118 0.2 10 0.361 0.118
62.50 62.53 62.65 62.64 56.22 53.94
1000 1000 1000
Discrete Charge Distribution 0.5 10 0.360 0.118 0.5 100 0.360 0.118 0.5 10 0.360 0.118
60.11 60.26 60.22
2.0 2.0 10.0
c (mM)
Nagg
figure and curve
7a 7b 7c
Figure 8. Same as in Figure 7, but for the discrete charge distribution. 8a 8b 8c
a φ tot ) 0.36 × 10-4. χ surf SS′ ) 2 for S ) {A, Pn} and S′ ) {B, C, D, Pc, W} and χSS′ ) 1 for S ) {P} and S′ ) {A, B, C, D, W}; otherwise, χSS′ ) 0.
Table 4. Surfactant Bulk Volume Fractions, Obsurf, and Micellar Aggregation Numbers, Nagg, for A14B3 Surfactant-Hydrophobic Polyelectrolyte-Salt Mixtures at Various Salt and Polyelectrolyte Concentrations for Different Length and Linear Charge Distributions of the Polyelectrolytea
c (mM)
Nagg
figure and curve
1000 1000 1000
Smooth Charge Distribution 0.2 10 0.0700 0.0229b 0.2 100 0.0617 0.0202c 0.2 10 0.0700 0.0229b
71.80 73.33 57.84
7d 7e 7f
1000 1000 1000
Discrete Charge Distribution 0.2 10 0.0750 0.0246d 0.2 100 0.0750 0.0246d 0.2 10 0.0750 0.0246d
46.57 60.97 47.27
8d 8e 8f
polyelectrolyte c (mM)
length
2.0 2.0 10.0 2.0 2.0 10.0
τ
salt c (mM)
surfactant 104φb
a χ SS′ ) 2 for S ) {A, Pn} and S′ ) {B, C, D, Pc, W}, χSS′ ) 1.6 for S ) {P} and S′ ) {B, C, D, W}, and χAP ) 0.4; otherwise, χSS′ ) 0. b φsurftot ) 7.00 × 10-6. c φsurftot ) 6.20 × 10-6. d φsurftot ) 7.50 × 10-6.
comparison with the corresponding data for the hydrophilic polyelectrolyte given in Table 2 shows that (i), at τ ) 0.5, Nagg is slightly reduced and the cac is reduced by a factor of 2 and (ii), at τ ) 0.2, Nagg increases by ca. 40% and the cac decreases by a factor of 20 as the neutral hydrophilic segments are replaced by hydrophobic ones. Moreover, as for the hydrophilic polyelectrolyte, the effects of the salt and the polyelectrolyte concentrations on Nagg are weak. As a consequence of our procedure, which was motivated by the lattice effects, our free (bulk) concentrations of the surfactants are only approximations to the real cmc and cac values of the model systems. For the hydrophobic polyelectrolyte, this approach gives very similar surfactant bulk volume fractions at a given total surfactant volume fraction. This makes it difficult to examine the effects of the salt and polyelectrolyte concentrations on the cac in great detail. However, for a qualitative use, conditions giving a larger Nagg are normally associated with a smaller cac. The obtained differences between the two hydrophobic polyelectrolytes are attributed to both a change of the linear charge density and a change of the interaction parameters. To discriminate between the effects of τ and χ parameters, we have also briefly considered a hydrophobic polyelectrolyte with τ ) 0.2, but with interaction
parameters as for the previous hydrophobic polyelectrolyte with τ ) 0.5, i.e., χPW ) 1, etc. (see note a in Table 3). Table 3 shows that a reduction from τ ) 0.5 to τ ) 0.2 at constant χ parameters reduces Nagg in a manner similar to that for the hydrophilic polyelectrolyte. Such comparison would correspond to experimental conditions where polyelectrolytes are synthesized from uncharged and charged segments with equal hydrophobicity. Discrete Charge Distribution. The corresponding model calculations have been performed with the discrete charge distribution, and selected results are given in Figure 8 and in Tables 3 and 4. As for the hydrophilic polyelectrolyte, the use of the discrete charge distribution gives qualitatively the same results. For τ ) 0.5 the changes are small, but for the smaller linear charge density (τ ) 0.2), the peaks of the polyelectrolyte volume fraction distributions are shifted inward and Nagg is substantially reduced. When viewing the uncharged and charged polyelectrolyte segments separately, we find that these distributions are split up again, now with the neutral and hydrophobic segments closer to the micellar center as compared to the charged and hydrophilic ones. IV. Analysis and Discussions In the previous section we have described volume fraction profiles, cmc, cac, and micellar aggregation numbers for the micellization without polyelectrolyte, with hydrophilic polyelectrolyte, and with hydrophobic polyelectrolyte present. We will now make a further analysis of the results and make connections with experimental data. Headgroup and Polyelectrolyte Location. The volume fraction profiles of the surfactant head, surfactant tail, and polyelectrolyte segments for both high and low charged hydrophilic and hydrophobic polyelectrolytes are collected in Figure 9. We will in the following focus on the relative locations of the surfactant headgroups and the polyelectrolyte segments. Starting with the hydrophilic polyelectrolyte and τ ) 0.5 (panel a), we find that the distribution of the headgroups and the polyelectrolyte segments overlap considerably, but the polyelectrolyte segments are slightly displaced toward the solution. An increased hydrophobicity shifts the location of the polyelectrolyte segments inward and the φBi and φPi peaks match each other completely (panel b). The difference in magnitude of the peaks is mainly reflected by the difference in charge, -1/3 and +1/2 elementary charges, respectively. What happens if the polyelectrolyte becomes less charged? For the hydrophilic polyelectrolyte the φPi peak is shifted outward (panel c). The larger fraction of neutral polyelectrolyte segments makes the anchoring of the
Polyelectrolyte-Induced Micellization
Langmuir, Vol. 14, No. 11, 1998 2947
Figure 10. Net charge density profiles for surfactant solution at csalt ) 10 mM and csurf ) 0.766 mM (curve a), and polyelectrolyte-surfactant solutions at rpolymer ) 1000, cpolymer ) 2 mM, and csalt ) 10 mM (curves b-f): csurf ) 0.5258 mM, χPW ) 0, and τ ) 0.5 (curve b); same as for part b but with τ ) 0.2 (curve c); csurf ) 0.118 mM, χPW ) 1.0, and τ ) 0.5 (curve d); same as for part d but with τ ) 0.2 (curve e); same as for part e but with csurf ) 0.023 mM and χPW ) 1.6 (curve f). The polyelectrolyte is smoothly charged.
Figure 9. Volume fraction profiles of A, B, and P for polyelectrolyte-surfactant systems at csalt ) 10 mM, cpolymer ) 2 mM, and rpolymer ) 1000: hydrophilic polyelectrolyte at csurf ) 0.525 mM (panels a and c) and hydrophobic polyelectrolyte at csurf ) 0.118 mM (panel b) and csurf ) 0.023 mM (panel d). The polyelectrolyte is smoothly charged and has the linear charge density τ ) 0.5 (panels a and b) and τ ) 0.2 (panels c and d). P denotes the sum of Pc and Pn (panels b and d).
polyelectrolyte to the charged micelle weaker and short loops protrude into the water region. On the other side, for the hydrophobic polyelectrolyte the change is the opposite; the φPi peak is moved toward the micellar center and is now located within the peak of the headgroups (panel d). Moreover, an appreciable amount of polyelectrolyte segments are also residing in the micellar core and these are electrostatically neutralized by charged headgroups also appearing in the core. Hence, whereas for the hydrophilic polyelectrolyte the complexation becomes weaker upon a reduction of the linear charge density of the polyelectrolyte, the complexation becomes stronger for the hydrophobic polyelectrolyte. When the linear charge density is reduced from τ ) 0.5 to 0.2 at constant interaction parameters, the maximum of the distribution of headgroups and polyelectrolyte segments remains at the same location (as in panel b), but the distribution of polyelectrolyte segments displays a wing on the aqueous side and, of course, the relative peak heights of the B and P distributions are changed. Hence, the polyelectrolyte segments are shifted away from the micelle at reducing linear charge density (in particular for the hydrophilic polyelectrolyte) and the segments are shifted toward the micelle at increasing hydrophobicity. Charge Distribution. The radial charge distribution is directly available from the volume fraction profiles. We will consider the net charge density profile, zi ) ∑SzSφSi, i Ljzj, the former and the integrated net charge profile ∑j)1 giving the charge density in layer i and the latter giving the total number of elementary charges summed up to and including layer i. The net charge density profiles for the surfactant solution are shown in Figure 10 (curve a). As already anticipated from Figure 1, the net charge density is small in the micellar core; it becomes negative
Figure 11. Same as in Figure 10, but for integrated net charge profiles.
around i ) 8 (headgroup region) and positive 3-4 layers further out (counterion region). The appearance of a region with a net negative charge density followed by one with a positive net charge density is a consequence of the separation of the headgroup and the counterion distributions. The integrated net charge profile displays a prominent minimum arising from the charges in the headgroup region (Figure 11, curve a). The minimum corresponds to an excess of 20 negative elementary charges; the limit -56 would indicate no counterion penetration into the headgroup region and 0 would indicate a perfect radial matching. The net charge profiles for the polyelectrolyte-surfactant reference system resembles qualitatively those for the surfactant solution (cf. curve b with a in Figures 10 and 11). However, the region with a negative net charge density centered at i ) 8 has a smaller extension for i > 8 and the magnitude of the positive net charge density outside the micelle is larger. This results in a smaller integrated excess of negative charges at the micellar surface and the integrated charge density (and hence the electrical field) levels out more quickly outside the micelle. In fact a charge reversal occurs in layer i ) 12, i.e., the total charge of the micelle-polyelectrolyte complex for i > 12 has a charge opposite from that the pure surfactant micelle. The integrated net charge profile for the hydrophilic polyelectrolyte and τ ) 0.2 (Figure 11, curve c) has
2948 Langmuir, Vol. 14, No. 11, 1998
the same excess of charges at the micellar surface as for the polyelectrolyte-free solution but the integrated charge density levels out nearly as fast as for the polyelectrolytesurfactant reference system. The charge distributions for systems containing hydrophobic polyelectrolytes shown in Figures 10 and 11, curves d-f, differ qualitatively from those with hydrophilic polyelectrolytes. For χPW ) 1 and τ ) 0.2 or 0.5 (curves d and e) the headgroups and polyelectrolyte profiles nearly match each other radially (cf. Figure 9b) leading to only a small net charge density at all distances. However, for χPW ) 1.6 and τ ) 0.2 (curve f), there is a large surplus of positive polyelectrolyte charges in the headgroup region, causing a positive net charge density. The integrated net charge profile shows that for this case the micellepolyelectrolyte complex is recharged as compared to the bare micelle (Figure 11, curve f). The hydrophobic effect acting between the surfactant tails and the hydrophobic polyelectrolyte is obviously sufficiently strong to overcome the net repulsive electrostatic interaction within the complex (similar to the fact that the hydrophobic effect keeps micelles formed by charged surfactants together against the repulsion among the charged headgroups). In the case of χPW ) 1, the hydrophobic effect is not sufficiently strong for such a recharging of the complex at any of the linear charge densities considered. Smooth vs Discrete Charge Distribution. When comparing results from the discrete charge distribution with those from the smooth charge distribution, we found the following. (i) For the hydrophilic polyelectrolyte the discrete charge distribution gave larger Nagg and smaller cac. (ii) For the hydrophobic polyelectrolyte the discrete charge distribution gave smaller Nagg and higher cac. (iii) The difference of Nagg and cac between the two distributions models became larger for the hydrophobic polyelectrolyte than for the hydrophilic one. (iv) For the hydrophilic polyelectrolyte the uncharged segments in the discrete charge distribution were displaced outward as compared to the charged segments. (v) For the hydrophobic polyelectrolyte the uncharged segments in the discrete charge distribution were displaced inward as compared to the charged segments. (vi) The magnitude of the charge displacements in observations iv and v increases with reducing linear charge density of the polyelectrolyte. Observations i and iv lead us to the conclusion that, for the hydrophilic polyelectrolyte and τ ) 0.2, the packing problem in the headgroup region is slightly exaggerated by the smooth charge distribution. Obviously the possibility of forming short loops of neutral polymer segments protruding out in the solution between the charge segments is neglected, leading to smaller Nagg and higher cac. However, for the hydrophobic polyelectrolyte, the smearing out of the polyelectrolyte charges had the opposite effect (observations ii and v). Moreover, the different ordering of the uncharged and charged segments (observations iv and v) reflects the different nature of the neutral segments in the two types of polyelectrolytes. Finally, observation vi is a consequence of the fact that at decreasing τ the loop lengths becomes longer. Previous mean-field lattice calculations of the adsorption of polyelectrolyte at oppositely charged surfaces showed that the adsorption of polyelectrolyte with a discrete charge distribution was larger than for a smooth distribution at conditions where the adsorption was weak.21 This is in line with the larger effect on the cmc for the discrete charge distribution as compared to the smooth charge distribution. At strong adsorption conditions, the adsorption is controlled by charge compensation,
Wallin and Linse Table 5. Critical Micellization Concentrations and Micellar Aggregation Numbers, Nagg, for C14 Surfactant-Salt Mixtures at Various Salt Concentrations at 25 °Ca surfactant
salt
CH3(CH2)13N(CH3)3Br NaBr NaBr NaBr CH3(CH2)13N(CH3)3NO3 CH3(CH2)13SO3Na CH3(CH2)13NC5H5Br KBr KBr a
salt surfactant c (mM) cmc (mM) 0 5 10 80 0 0 0 25 50
3.3-3.8 2.4 1.5 0.39 2.7 3.2-5.0 2.65-4.1 2.0 1.5
Nagg 63-97
80 (60 °C) 79
From ref 32.
but such condition of charge compensation is obviously not applicable here where Nagg is not fixed. Comparison with Experimental Data. As mentioned, the surfactant tail-water interaction parameter was adjusted to reproduce the cmc dependence on the length of the aliphatic tail of nonionic surfactants. In Table 5 we give cmc data for different C14 surfactants at various salt content compiled by van Os et al.32 A comparison shows that the predicted cmc at low salt concentration is 3-5 times too small and that the salt dependence is qualitatively correct but too weak. Our results are in sufficient accordance with the experimental data for our purpose (investigation of the effect of adding polyelectrolyte), although further refinement of the model including a worse sophisticated choice of interaction parameters would likely bring the model predictions into closer agreement with the experimental data. For polyelectrolyte-surfactant solutions, we first notice that the model predicts a weak reduction of the cac at increasing salt concentration in conflict with experimental data,33 which showed an increase of the cac. At the present conditions, we cannot point to a single explanation. Apart from the response to changes in the salt concentration, there is good qualitative agreement with experimental results. The experimental and modeling results show similar effect of the hydrophobicity of the polyelectrolyte backbone on the cmc and the micellar aggregation number. Hansson et al. have given the cac’s for highly charged NaCMC, NaPA, and NaPSS under similar conditions.34 The cac’s were determined for C12TAB at the concentrations of ca. 10 mM salt and 0.5 mM polyelectrolyte. The hydrophobicity order for these polyelectrolytes is NaCMC < NaPA < NaPSS. The experimental results show that the cac is reduced by increasing backbone hydrophobicity; the cac for NaPSS is ca. 50 times lower than that for NaCMC. The predicted reduction of the cac with increasing hydrophobicity is much smaller, but no attempts have been made to compare the experimental variation in chain hydrophobicity with the interaction parameters used. The cac in the presence of NaCMC at different linear charge densities has also been examined experimentally,34 and a 5-fold increase of the cac was obtained for a 4-fold decrease in the linear charge density. In our model, we obtained an increase of the cac with a factor 2.5 upon a reduction of the linear charge density by the factor 2.5. Thus, the magnitude of the response on the change in the linear charge density is comparable. (32) van Os, N. M.; Haak, J. R.; Rupert, L. A. M. Physico-Chemical Properties of Selected Anionic, Cationic and Nonionic Surfactants; Elsevier Science Publishers B. V.: Amsterdam, 1993. (33) Hansson, P.; Almgren, M. J. Phys. Chem. 1995, 99, 16684. (34) Hansson, P.; Almgren, M. J. Phys. Chem. 1996, 100, 9038.
Polyelectrolyte-Induced Micellization
V. Summary The formation of surfactant micelles and surfactantpolyelectrolyte complexes were modeled on the basis of a mean-field lattice theory for solutions of chain molecules in heterogeneous systems. Both the surfactant and the polyelectrolyte were modeled as a sequence of linked segments. The surfactant chain carried in total one elementary charge located at one end and two charge distributions of the polyelectrolyte were considered. In the discrete charge representation, only two types of segments were used in the system, one hydrophilic, and one hydrophobic and hence only one interaction parameter needed to be specified. Given the specification of the components, the model is able to predict central quantities as the cmc, cac, the micellar aggregation number, and various density profiles characterizing the complexation. From the simplicity of the lattice model and the rather coarse approximations involved as well as the use of only one free interaction parameter, it is evident that such a model cannot be used for quantitative description of the complexation between charge surfactants and polyelectrolytes. Nevertheless, a quantitative discrepancy between experimental and modeling data should not restrain one from using the model for qualitative studies to examine the effects of changing conditions in the system, and it is on this level that the results should be viewed. With this background, the major conclusions from our study are as follows: (1) In the absence of polyelectrolyte, the model is able to predict the formation of surfactant micelles and to show how the cmc depends on the salt concentration in a qualitative agreement with experimental data. (2) In the presence of a hydrophilic polyelectrolyte, the gross features of the micellar structure remains. The aggregation number is only weakly dependent on the external conditions, except that it is reduced as the linear charge density of the polyelectrolyte decreases. The cac is lower than the cmc, but the reduction is smaller than in comparable experiments and previous MC simulations. The cac increases with decreasing linear charge density in agreement with experiments and MC simulations. Structurally, there is a considerable spatial overlap of the
Langmuir, Vol. 14, No. 11, 1998 2949
headgroup charges and the charges of the polyelectrolytes. The electrostatic potential outside the complex decays faster than the potential outside a micelle due to the stronger accumulation of polyelectrolyte segments as compared to the accumulation of counterions in the polyelectrolyte-free case. (3) An increasing hydrophobicity of the polyelectrolyte leads to a stronger penetration of the polyelectrolyte segments into the headgroup region and eventually into the micellar core. In the intermediate case, χPW ) 1, the cac is reduced as compared to the hydrophilic polyelectrolyte, and there is a nearly complete overlap between the headgroup charges and the polyelectrolyte charges, leading to only a small (in absolute sense) electrostatic potential outside the micelle. A reduction of the linear charge density of the polyelectrolyte, keeping the interaction unchanged, leads to a slight weakening of the complexation, demonstrating that the electrostatic effect plays a role for the complexation for intermediate hydrophobic polyelectrolytes. (4) At an even stronger hydrophobicity of the polyelectrolyte, χPW ) 1.6 (but still having a water soluble polyelectrolyte), the cac is further reduced and the hydrophobic effect is so strong that the complex becomes recharged by the polyelectrolyte. Hence, when one starts with a hydrophilic polyelectrolyte, the polyelectrolyte segments are shifted toward the micelle at increasing hydrophobicity. (5) At the lower linear charge density, τ ) 0.2, the smooth charge model displayed differences as compared to the discrete charge model. Such differences are likely to increase at even lower linear charge densities. (6) The results of the model system agreed qualitatively with experimental data with the exception that cac did not increase with increased salt concentrations as experimentally observed. Acknowledgment. This work has been supported by the Swedish National Board for Industrial and Technical Development (NUTEK) and the Swedish National Research Council (NFR). LA9712911
