Langmuir 1996, 12, 305-314
305
Monte Carlo Simulations of Polyelectrolytes at Charged Micelles. 1. Effects of Chain Flexibility Torsten Wallin* and Per Linse Physical Chemistry 1, Chemical Center, Lund University, P.O. Box 124, S-221 00 Lund, Sweden Received May 10, 1995. In Final Form: September 13, 1995X The complexation between a charged micelle and an oppositely charged polyelectrolyte was studied by using a simple model system with a particular emphasis on the electrostatic interaction and the polyelectrolyte rigidity. Structural data of the micelle-polyelectrolyte complex and thermodynamic quantities of the complexation as a function of the flexibility of the polyelectrolyte were obtained by using Monte Carlo simulation and thermodynamic integration. Moreover, the ratio of the critical aggregation concentration, cac, and the critical micellization concentration, cmc, was calculated, cac being the lowest surfactant concentration at which the surfactants self-aggregate in the presence of polyelectrolyte. The cac was found to be ca. 4 (rigid polyelectrolyte) to ca. 60 (flexible polyelectrolyte) times lower than the cmc. The latter value is in line with experimental data of similar systems. Hence, it seems that the electrostatic interactions as well as the rigidity of the polyelectrolyte are important factors for controlling the reduction of the cmc.
Introduction For some years now, aqueous solutions of mixtures of polyelectrolytes and charged surfactants have received great attention.1-11 The most notable features of polyelectrolytes, compared to nonionic polymers, are their high solubility in water and self-repulsion in solution, yielding a larger end-to-end distance and radius of gyration. Regarding charge surfactants, they are more hydrophilic than their nonionic relatives, which makes the critical micellization concentration (cmc) relative high. However, this repulsive electrostatic interaction among the headgroups within the micelles can be lowered, by adding either salt or small amounts of an oppositely charged polyelectrolyte, which hence yields a lower concentration at which aggregates start to form. In the case of added polyelectrolyte, this concentration is normally referred to as the critical aggregation concentration (cac). The cac of solutions containing charged surfactants with added oppositely charged polyelectrolyte as well as the cmc of similar solutions without added polyelectrolyte has been determined for various systems.12-15 Figure 1 shows the cmc of a cationic surfactant of different length (CnTAB) and the corresponding cac’s in the presence of different anionic polyelectrolytes. We see that the surfactant X Abstract published in Advance ACS Abstracts, December 1, 1995.
(1) Hayakawa, K.; Kwak, J. J. Phys. Chem. 1982, 86, 3866. (2) Santerre, J. P.; Hayakawa, K.; Kwak, J. Colloids Surf. 1985, 13, 35. (3) Abuin, E.; Scaiano, J. P. J. Am. Chem. Soc. 1984, 106, 6274. (4) Chu, D.; Thomas, J. K. J. Am. Chem. Soc. 1986, 108, 6270. (5) Goddard, E. D. Colloids Surf. 1986, 19, 301. (6) Gao, Z.; Kwak, J.; Wasylishen, R. E. J. Colloid Interface Sci. 1988, 126, 371. (7) Skerjanc, S.; Kogej, K.; Vesnaver, G. J. Phys. Chem. 1988, 92, 6382. (8) Dubin, P. L.; The´, S. S.; McQuigg, D. W.; Chaw, C. H.; Gan, L. M. Langmuir 1989, 5, 89. (9) Thalberg, K.; Lindman, B.; Karlstro¨m, G. J. Phys. Chem. 1990, 94, 4289. (10) Wong, T. S.; Thalberg, K.; Lindman, B.; Gracz, H. J. Phys. Chem. 1991, 95, 8850. (11) Lindman, B.; Thalberg, K. Polymer-Surfactant Interactions Recent Developments. In Interactions of Surfactants with Polymers and Proteins; Goddard, E. D., Ananthapadmanabhan, K. P., Eds; CRC Press: Boca Raton, 1993. (12) Hayakawa, K.; Santerre, J. P.; Kwak, J. C. T. Macromolecules 1983, 16, 1642. (13) Thalberg, K.; Lindman, B. J. Phys. Chem. 1989, 93, 1478. (14) Thalberg, K.; Lindman, B.; Bergfeldt, K. Langmuir 1991, 7, 2893. (15) Hansson, P.; Almgren, M. Langmuir 1994, 10, 2115.
0743-7463/96/2412-0305$12.00/0
Figure 1. Cmc values without and cac values in the presence of different anionic polyelectrolytes for alkyltrimethylammonium bromides (CnTAB) as a function of the number of alkyl carbon atoms. Data from ref 12 (hyaluronate), ref 12 (alginate), and refs 12 and 13 (polyacrylate).
concentration at which micelles start to form can be reduced up to 1000 times. The above mentioned experimental results constitute the experimental background for our interest in how different factors affect the reduction of the cmc. In the first study, we report results from an investigation of the complexation between a micelle and an oppositely charged polyelectrolyte using a simple model system. We present structural information obtained from Monte Carlo simulations, and by thermodynamic integration techniques, we have obtained free energy quantities that are related to the ratio cac/cmc. We have in particular been interested in the determination of how the flexibility of the polyelectrolyte affects the ratio cac/cmc. Related Monte Carlo simulations have been performed by Granfeldt et al.16 They investigated the interaction between two charged micelles carrying adsorbed polyelectrolytes. These results show that there is no noticeable interaction between the micelles at distances larger than twice the micelle radius (even for highly charged micelles), which means that for low concentrations of micelles and polyelectrolyte we can regard the micelles as independent of each other. The results also indicated that for higher micelle and polyelectrolyte concentrations these aggregates would form large clusters (macroscopically this (16) Granfeldt, M.; Jo¨nsson, B.; Woodward, C. E. J. Phys. Chem. 1991, 95, 4819.
© 1996 American Chemical Society
306
Langmuir, Vol. 12, No. 2, 1996
Wallin and Linse
is seen as gels or phase separations), which has been confirmed by experimental results.17 Also related to our work is the molecular thermodynamic theory of the complexation of nonionic polymers and surfactants in dilute aqueous solutions by Nikas and Blankschtein.18 The theory contains explicit factors describing the solvent quality, the polymer hydrophobicity and flexibility, and specific interactions between polymer segments and surfactant headgroups. The cmc and cac as well as the micellar aggregation number and the number of micelles bound per polymer chain were predicted. The remainder of the paper is organized as follows. In the next section, the method and model system are described. First we relate the cac/cmc ratio to the free energy of complexation between a micelle and a polyelectrolyte in solution, and then we describe the model system used. Thereafter, we account for the free energy evaluation path employed and give some details of the Monte Carlo simulations. In the following section, structural and thermodynamic results of the complexation are presented and discussed. We also calculate the cac/ cmc ratio, discuss some approximations involved in the investigation, and relate the cac/cmc ratio of the model system to experimental data. In the last section, we conclude our main findings. Method and Model System Background. We divide the change in Gibbs free energy of forming a micelle in the presence of a polyelectrolyte, ∆Gmic, into two parts according to
∆Gmic ) ∆Gmic0 + ∆Gpe
(1)
where ∆Gmic0 is the free energy of forming a micelle without the polyelectrolyte and ∆Gpe is the change in the free energy of the micellization due to the presence of the polyelectrolyte and its counterions. Using the approximate relations19
∆Gmic0 ) NaggRT ln(cmc)
(2)
∆Gmic ) Nagg RT ln(cac)
(3)
and
where Nagg is the micellar aggregation number, R the gas constant, and T the absolute temperature, we find that the factor with which the cmc is reduced is given by
[
]
∆Gpe cac ) exp cmc NaggRT
(4)
Here we assume that the aggregation number is not affected by the presence of the polyelectrolyte. Thus, to obtain the factor with which the cmc is reduced by, we need to calculate ∆Gpe. Since G is a state function, we can identify ∆Gpe as the free energy change of mixing a micellar solution with a polyelectrolyte solution giving a mixed micellarpolyelectrolyte solution with the desired final composition. The initial micellar and polyelectrolyte solutions to be mixed are referred to as the M and the P system, respectively, whereas the final one is referred to as the MP system. In the calculations we neglect the volume-pressure work caused by the addition of polyelectrolyte and we identify ∆Gpe with ∆Ape calculated from the canonical ensemble. Model System. To be able to calculate ∆Ape, the initial and final model systems have to be specified. We have used a number of approximations and used a very simplified model, which contains only items which we believe are the essence of describing (17) Thalberg, K.; Lindman, B. Langmuir 1991, 7, 277. (18) Nikas, Y. J.; Blankschtein, D. Langmuir 1994, 10, 3512. (19) Evans, D. F.; Wennerstro¨m, H. The Colloidal Domain; VCH Publishers: New York, 1994.
Figure 2. Illustration of the initial micellar (M) and polyelectrolyte (P) systems (state I), the final mixed micellepolyelectrolyte (MP) system (state IV), and the two intermediate and decoupled systems (states II and III). The pathway for the calculation of the Helmholtz free energy of mixing, ∆Ape, is also indicated (see text for details). the change in the cac of the system. Moreover, by using a simple model system, we are in principle able to solve it exactly. The main approximations are as follows: (1) The so-called primitive model is used, i.e. (i) the water is treated as a dielectric medium and it enters the model only through its dielectric permittivity and (ii) all other constituents are described in terms of hard spheres with point charges in the centers of the spheres. (2) The cell model is applied; one micelle and/or one short polyelectrolyte plus counterions are enclosed in a spherical cell. (3) The concentration of free surfactants is neglected. (4) The polyelectrolyte is modeled as a chain of charged hard spheres (beads) jointed by harmonic bonds with the flexibility controlled by harmonic angular energy terms. (5) The micelle is described as a hard sphere with fixed charge and radius, and hence the aggregation number is assumed not to change upon addition of the polyelectrolyte. Recent experiments support that the aggregation number does not change appreciably when polyelectrolyte is added, provided that the polyelectrolyte is not incorporated into the micelle.15 Later, we will consider the effects of approximations 2 and 3 by using the Poisson-Boltzmann equation. With this model the essential contributions to ∆Ape arise from changes in (i) the electrostatic interaction energy, (ii) the bond and angular energies of the polyelectrolyte, (iii) the conformational entropy of the polyelectrolyte, and (iv) the configurational entropy of the small ions. Figure 2 illustrates the cell model of the initially separated micellar (M) and polyelectrolyte (P) solutions and the final mixed micelle-polyelectrolyte (MP) solution as well as two decoupled intermediate states (further described below). The radii of the cells are selected so that the volume is conserved upon mixing
Polyelectrolytes at Charged Micelles
Langmuir, Vol. 12, No. 2, 1996 307 Table 1. Data of the Simulated System
Figure 3. Illustration of the bond and angular energy terms of the polyelectrolyte. and that they can accommodate the fully stretched polyelectrolyte. The M and MP systems contain one micelle, the position of which is fixed in the center of the cell, and freely moving counterions, and the P and MP systems contain one polyelectrolyte with nbead beads and its counterions. In the P system, the position of one central bead of the chain is fixed at the center of the cell, whereas in the MP system, the same bead is fixed at the surface of the micelle. In the former case, the constraint keeps the polyelectrolyte in the center of the cell and hence minimizes the effect of the cell boundary, whereas in the latter one the constraint enforces the polyelectrolyte to be close to the micelle. Even without the constraint, a flexible polyelectrolyte forms a strong complex with the micelle, but at some increased rigidity of the polyelectrolyte, the attractive interaction between the micelle and the polyelectrolyte is not sufficient to keep them together. In a real system and at surfactant concentrations below the cmc, but above the cac, such a micelle-polyelectrolyte complex would exist even at higher rigidity of the polyelectrolyte, since a separation would imply a disassembling of the micelle (since the surfactant concentration is below the cmc) with an accompanied high free energy penalty. In that respect our constraint would formally correspond to an infinitely large cost of disassembling the micelle. With the model described above, the total energy of a system (M, P, or MP) becomes
U ) Unonbond + Ubond + Uangle
(5)
The nonbonded energy is given by
Unonbond )
∑u + ∑ u i
i
ij
(6)
i Rcell - Ri
ui ) 0,
ri e Rcell - Ri
(7)
and
uij ) ∞, uij )
zizje2 , 4π0rrij
rij < Ri + Rj rij g Ri + Rj
cell radius, M system cell radius, P system cell radius, MP system micelle radius bead radius small ion radius micelle charge bead charge small ion charge no. of beads no. of cations no. of anions temperature relative dielectric permittivity nbead-1
Ubond )
∑
kbondf(ri,i+1,r0)(ri,i+1 - r0)2
(9)
i)1
with an equilibrium separation r0 ) 5 Å, a force constant kbond ) 0.2 N m-1, and
f(ri,i+1,r0) ) 1,
ri,i+1 e 2r0
f(ri,i+1,r0) ) [1 - 0.0023(ri,i+1 - 2r0)Å-1],
ri,i+1 > 2r0 (10)
The reason for using the factor f(ri,i+1,r0) < 1 at large separations is given below. The angular energy term is represented by nbead-1
Uangle )
∑
kang(Ri - R0)2
(11)
i)2
where Ri is the angle formed by the vectors ri+1 - ri and ri-1 ri (Ri is later referred to as the angle between consecutive beads) with an equilibrium angle R0 ) 180° and a force constant kang. Five different values of kang [0, 1.72, 4.30, 18.0, and 164 × 10-24 J/(deg)2] were used to construct different backbone flexibilities. This model parameter controls the inherent flexibility of the chain. In addition to the angle force constant, the electrostatic interaction among the charged beads also contributes to the rigidity of the polyelectrolyte, and this contribution is dependent on the presence of other ionic species. Simulations of the P system but with no electrostatic or hard core interactions, i.e., with only bond and angular energy terms (referred to as the P0 system), were performed to map the angular force constants on more physically relevant parameters. In particular, the obtained average value of the angle between consecutive beads of the P0 system will be used to describe the inherent chain flexibility, and it is denoted by 〈R〉. Calculation of the Helmholtz Free Energy. The calculation of ∆Ape for the mixing of the M and P systems to obtain the MP system is divided into three steps by introducing two intermediate states. Beside the initial and final systems described above (states I and IV), Figure 2 shows the two intermediate states. These are related to states I and IV, respectively, but the interactions are decoupled except for the hard sphere repulsion, i.e., there are no electrostatic, no bond, and no angular energy terms. Hence, using the four states, ∆Ape can be expressed as a sum of three terms according to
∆Ape ) ∆A1 + ∆A2 + ∆A3
(8)
respectively, where 0 is the dielectric permittivity of vacuum, r the relative dielectric permittivity of water, e the elementary charge, ri the location of particle i (i is either the micelle, a polyelectrolyte bead, or a small ion), rij the distance between the centers of particles i and j, Rcell the radius of the cell, Ri the radius of particle i, and zi the charge of particle i. The dielectric permittivity is constant throughout the system and hence surface polarization is neglected.20 Data of the model systems are compiled in Table 1. The other two terms in eq 5, the bond and the angular energy terms, arise from the polyelectrolyte only, and they are illustrated in Figure 3. The bond energy term is given by
Rcell,M ) 79.5 Å Rcell,P ) 160.0 Å Rcell,MP ) 166.3 Å Rmic ) 15.0 Å Rbead ) 2.0 Å Rion ) 2.0 Å zmic ) 20 zbead ) -1 zion ) (1 nbead ) 40 ncation ) 40 nanion ) 20 T ) 298 K r ) 78.3
(12)
with
∆A1 ) AII - AI ∆A2 ) AIII - AII
(13)
∆A3 ) AIV - AIII Since states I and II consist of two independent systems, we have AI ) AI,M + AI,P and AII ) AII,M + AII,P, and we obtain (20) Linse, P. J. Phys. Chem. 1986, 90, 6821.
308
Langmuir, Vol. 12, No. 2, 1996
Wallin and Linse (14)
∆A1 ) -∆AM - ∆AP ∆A3 ) ∆AMP
where ∆AM ≡ AI,M - AII,M, ∆AP ≡ AI,P - AII,P, and ∆AMP ≡ AIV,MP - AIII,MP denote the free energy differences between the coupled and decoupled M, P, and MP systems, respectively. The free energy contributions from the first and last steps, ∆A1 and ∆A3, are calculated from Monte Carlo simulations and thermodynamic integration, whereas ∆A2 is calculated from analytic theory. The calculations of ∆AM, ∆AP, and ∆AMP needed in eq 14 are performed by introducing a coupling parameter λ that continuously brings the decoupled systems into the interacting ones as λ varies from 0 to 1. The λ-dependence in U, given by eqs 5-11, is achieved by replacing e2 by (λe)2, kbond by λ7kbond, and kang by λ6kang in eqs 8, 9, and 11, respectively. The free energy difference between the states described by λ ) 0 and λ ) 1 is then given by
∆A ≡ A(1) - A(0) ≡
∫ ∂λ∂ A(λ) dλ ) ∫ 〈∂λ∂ U(λ)〉 dλ 1
0
1
0
(15)
where 〈 〉 denotes a canonical ensemble average. The integral was discretized by a summation using the trapezoidal rule and up to 15 terms. Figure 4 shows the typical behavior of the integrand for the three systems and the components of the integrand for the MP system as well as the λ-values employed. The rather high λ-power of the bond term was chosen to suppress the height of the peak of the integrand (cf. Figure 4b). A smaller power gives a higher and narrower peak at small λ and produces a larger uncertainty of the value of the integrand. The factor f < 1 for ri,i+1 > 2r0 in eq 10 was introduced to make the force constant progressively smaller at larger bead-bead separations. This is only effective at small λ and reduces the uncertainty of the integral by making the integrand less peaked. The free energy change of mixing the two hard sphere systems to form one system, ∆A2, is evaluated analytically. For simplicity, we treated the position of the micelle as fixed, thus neglecting its change in mixing entropy. That makes it possible to employ the free energy expression of hard sphere systems obtained by integrating the Carnahan-Starling compressibility along an isotherm.21,22 The choice of additive cell volumes and not too dissimilar particle concentrations makes |∆A2| < 0.01RT, which is negligible compared with ∆A1 and ∆A3. Monte Carlo Simulation. To obtain ensemble averages, as needed in eq 15, Monte Carlo (MC) simulations according to the Metropolis algorithm23 were performed for the M, P, and MP systems. The examination of the configuration space was enhanced by using the Pivot algorithm.24 In this algorithm global polyelectrolyte rearrangements are achieved by rotating a part of the polyelectrolyte, which includes one end, at a randomly selected bead. The improvement was less drastic than for a simulation of a single, isolated, and neutral polymer, but still considerable. Our MC-code was checked by comparing results with data given by Granfeldt et al.25 and Jo¨nsson et al.26 The simulation length is in general 106 trial moves per particle (polyelectrolyte beads and small ions). For cases where the accuracy is worse, the simulation length was increased 3-5 times. That occurred for the P and MP systems at kang ) 1.72 × 10-24 J/(deg)2 and λ ) 0.9-1.0, at kang ) 4.30 × 10-24 J/(deg)2 and λ ) 0.7-1.0, at kang ) 18.0 × 10-24 J/(deg)2 and λ ) 0.6-1.0, and at kang ) 164 × 10-24 J/(deg)2 and λ ) 0.5-1.0. To improve the precision of the radial distribution functions also examined, the simulation length was also increased 5 times for the remaining MP system at λ ) 1.0. We also performed a complementary simulation (2 × 106 trial moves per particle) with a polyelectrolyte of 60 segments with maximal flexibility (kang ) 0). The ∆Ape for this longer polyelectrolyte did not differ significantly ( 250 corresponding to a local bead number density ρ > 10-3 Å3) is largest for the most flexible polyelectrolyte and decreases gradually as the chain becomes stiffer. Figure 5b shows that the distributions of the anions (counterions of the micelle) vary drastically with the flexibility of the polyelectrolyte. For 〈R〉 e 150°, we obtain gmic,anion(r) < 1 near the micelle, i.e., these anions are depleted from the neighborhood of the micelle by the strong accumulation of the polyelectrolyte. However, the radial distribution function still displays a local maximum at micellar contact and a global minimum occurs at r ≈ 25 Å. We attribute the “penetration” of the counterion as a correlation effect, since in a mean-field treatment such (28) Stevens, M. J.; Kremer, K. J. Chem. Phys. 1995, 103, 1669.
310
Langmuir, Vol. 12, No. 2, 1996
Figure 5. (a) Micelle-polyelectrolyte bead, (b) micelle-anion, and (c) micelle-cation radial distribution function (gmic,i(r)) versus the separation (r) at indicated average angles between consecutive beads (〈R〉). Since there is only one micelle and its position is fixed, the local concentration of particle i is given by Fi(r) ) Figmic,i(r), where Fi is the average number density of particle i in the system. In part a the δ-contribution from the polyelectrolyte bead at fixed r ) 17 Å is not displayed.
behavior is unlikely. For 〈R〉 ) 165°, gmic,anion(r) > 1 close to micelle, and the local minimum is moved farther away from the micellar surface. These two observations are in line with the conclusion that the polyelectrolyte is less tightly complexed with the micelle and hence allows the micellar counterions to come close to the micellar surface to a larger extent. For 〈R〉 ) 175°, the distribution of micellar counterions is very close to that found in absence of polyelectrolyte (cf. Figure 5b), and hence the polyelectrolyte hardly screens the micellar charge any more. [The reason for the smaller contact value of gmic,anion for the M
Wallin and Linse
system, as compared to the MP system with 〈R〉 ) 175° (the opposite would be expected), is the different cell volumes (cf. Table 2). A small cell volume gives a smaller contact value of the radical distribution function, since the average concentration is higher and the dependence of the contact concentration on the mean concentration is weak.] Also the distribution of the cations (counterions of the polyelectrolyte) depends on the flexibility of the polyelectrolyte. Figure 5c shows that for 〈R〉 e 150° there is a pronounced maximum in the local concentration at r ) 22 Å, i.e., 5 Å outside the distance of closest approach. As 〈R〉 is increased from 90° to 165°, the maximum in the local concentration is reduced by a factor of 5, and at 〈R〉 ) 165° the maximum is displaced to r ) 30 Å. This decrease in the cation concentration is obviously related to the reduction of the local concentration of the polyelectrolyte beads close to the micelle. For 〈R〉 ) 175°, the cations are depleted from the micelle and its distribution is diffuse, again consistent with only a few polyelectrolyte beads being close to the micelle. Finally, Figure 6 shows some snapshots of the micellepolyelectrolyte complex at different values of 〈R〉. These pictures illustrate graphically how the inherent flexibility of the polyelectrolyte affects its structural behavior close to the oppositely charged micelle and visualizes the results discussed. Thus, the picture that emerges is the following. For the flexible case, essentially the whole polyelectrolyte is adsorbed. The chain is locally rather folded and forms small loops extending from the micellar surface, resulting in a fairly extended adsorbed layer (cf. Figures 5a and 6a). The net charge of the micelle-polyelectrolyte complex is opposite that of the micelle. This reverse net charge is screened by cations which form a diffuse layer, and this layer is partly overlapping the outer part of the adsorbed polyelectrolyte layer. At larger chain rigidity, the polyelectrolyte encounters larger difficulties to be close to the micelle. At 〈R〉 ) 150°, the polyelectrolyte is partly detached, and those beads which are near the micelle form a thinner adsorbed layer. This becomes even more accentuated at 〈R〉 ) 165°, where at least one end of the polyelectrolyte has left the micelle, and at 〈R〉 ) 175°, where only a few beads remain in close contact with the micelle. At this stage (〈R〉 g 150°), lP has exceeded lP* ) 27 Å. Finally, as discussed in the previous section, one central polyelectrolyte bead is attached at the micellar surface to model the situation in which a polyelectrolyte-micelle separation would imply a disassembling of the micelle, which is highly unfavorable. When simulations were carried without the attachment, corresponding to the case of a complexation between a stable charged particle and a polyelectrolyte (which is of interest as such), the structural results became essentially the same as those for 〈R〉 e 135°, but the polyelectrolyte tends to separate from the micelle at 〈R〉 g 150°. Thermodynamics. In the following we will deal with changes in energy, entropy, and free energies for the coupling process of the M, P, and MP systems separately, and then we will consider the corresponding quantities for the overall mixing process as described in Figure 2. We recall that the contribution from step 2 is negligible. The obtained differences in the thermodynamic quantities between the coupled and decoupled micellar systems were ∆UM/RT ) -67.1, -∆SM/R ) 25.5, and ∆AM/RT ) -41.5, respectively. The reduction of the electrostatic energy and the partly counteracting entropy increase are both intimately related to the accumulation of the counterions to the micelle (cf. Figure 5b). These results are
Polyelectrolytes at Charged Micelles
Langmuir, Vol. 12, No. 2, 1996 311
Figure 6. Images of the micelle and the polyelectrolyte for the MP system at different chain flexibility corresponding to an average angle between consecutive segments for the P0 system of (a) 90°, (b) 135°, (c) 150°, (d) 165°, and (e) 175°. The small ions are omitted for clarity.
similar to those obtained in previous MC simulations of related micellar systems.29 Figure 7a shows the corresponding differences in energy for the P (dashed curves) and MP (solid curves) systems as a function of 〈R〉. Included also is the division of ∆U into contributions from nonbonded (i.e., electrostatic), bond, and angular energy terms. The corresponding differences in entropy and free energy for the P and MP (29) Linse, P.; Gunnarsson, G.; Jo¨nsson, B. J. Phys. Chem. 1982, 86, 413.
systems are given in parts b and c of Figure 7, respectively, where ∆A is evaluated according to eq 15 and ∆S from the difference between ∆U and ∆A. For the P system, Figure 7a shows that all energy contributions are positive, thus making ∆UP positive. However, their dependencies on the chain rigidity are unequal; ∆UP,elec remains essentially constant, ∆UP,bond decreases slightly, and ∆UP,angle increases steadily with increasing 〈R〉. (The latter is expected since the increased rigidity is obtained by increasing the angular force
312
Langmuir, Vol. 12, No. 2, 1996
Figure 7. (a) Difference in total energy (∆U) and its components (∆Uelec, ∆Ubond, and ∆Uangle), (b) difference in entropy (∆S), and (c) difference in free energy (∆A) between the coupled and decoupled P system (dashed curves) and similar data for the MP system (solid curves) as a function of the average angle between consecutive beads (〈R〉). The ordinate is in reduced units per system. The largest estimated uncertainties according to eqs 16 and 17 are σ(∆U/RT) ) 0.4, σ(∆S/R) ) 0.5, and σ(∆A/RT) ) 0.4.
constant.) This makes ∆UP/RT increase by 35 (ca 0.9 per bead) as 〈R〉 is raised from 90° to 175°. Thus, the increase of the stiffness of the chain occurs without too large a penalty in the electrostatic interaction, with a reduction of the bond energy, and with an increase of the angular energy. The reduction of the bond energy is in line with a less skewed (cf. Table 2), and hence probably more peaked, bond distance distribution about r0. Regarding the MP system, the major dependence of ∆UMP and its components on 〈R〉 is the same as for the P system. The main difference occurs in the electrostatic energy, which is ca. 100RT lower for the MP system due to favorable electrostatic interaction between the micelle
Wallin and Linse
and the polyelectrolyte. This favorable electrostatic interaction causes (i) a more repulsive angular energy and (ii) a less repulsive bonding energy of the polyelectrolyte. The former is in agreement with a smaller persistence length and the latter with an average beadbead separation closer to its equilibrium value r0 ) 5 Å (cf. Table 2). Both these conformational changes facilitate the possibility of the polyelectrolyte to increase its favorable electrostatic interaction with the micelle. However, at large chain rigidity, 〈R〉 > 165°, this picture is changed. At 〈R〉 ) 175°, the electrostatic energy is strongly enhanced, whereas the angular energy is reduced and the bond energy increased as compared with 〈R〉 ) 165°. Obviously, since lP . lP*, the very strong gain in electrostatic energy by firm complexation is not possible any longer (cf. parts d and e of Figure 6). The driving force for a conformational adjustment to increase the attractive micelle-polyelectrolyte energy, as discussed above, is not present any more. Thus, from the energetic point of view, the polyelectrolyte sacrifices angular energy to gain electrostatic energy up to some value of the angular force constant. At larger values the sacrifice becomes too large, and the excitation of the angular degrees of freedom relaxes back. Figure 7b shows that the P system displays a large negative ∆S, which originates from the confinements of the beads into a chain and the accumulation of the counterions close to the polyelectrolyte as the coupling is turned on. Moreover, ∆S is further reduced as the chain becomes stiffer, and the main contribution to this reduction is the loss in chain conformational entropy. The MP system displays very similar entropy differences and dependence on 〈R〉. The difference between the P and the MP systems amounts to ca. 15R for 〈R〉 ) 90° and decreases to ca. 2R for 〈R〉 > 165°. We attribute the higher entropy of the MP system to the overall larger release of counterions from the macroions. At 〈R〉 ) 90°, the anions and half of the cations are relatively equally distributed throughout the system, whereas at 〈R〉 ) 175°, the anions are accumulated close to the micellar surface and the cations close to the polyelectrolyte (cf. Figure 5). There is of course an opposite but weaker contribution to the free energy, from the restriction of the chain conformations upon complexation in the MP system. Finally, we arrive at the free energy given in Figure 7c. Since both ∆UP and -∆SP increase with 〈R〉, ∆AP is bound to increase with increased rigidity. The increase in ∆AP is largely dominated by the entropy. Similar logic is applied to the MP system. The difference in ∆A between the M and MP systems is dominated by their energy difference (cf. parts a and b of Figure 7), which originated from their difference in electrostatic energy. We are now at the stage where the change in the thermodynamic quantities for the mixing process of the micellar and the polyelectrolyte solutions, as described in Figure 2, can be calculated according to eqs 12-14. Figure 8 shows that this mixing is energetically as well as entropically favored, making ∆Ape strongly negative. The favorable ∆Upe/RT amounts to -45 at 〈R〉 ) 90°, increases slowly with 〈R〉 up to ≈150°, and rapidly approaches zero at larger 〈R〉. The entropy of mixing is less sensitive to the chain flexibility; -∆Spe/R increases from -40 to -25, for the same interval. Hence, for flexible chains the largest contribution to the mixing free energy comes from the energy, but for the more rigid chains the entropy contribution dominates. To summarize, the driving force for the negative mixing energy originates from the attractive micelle-polyelectrolyte electrostatic interaction. This interaction leads to (i) a large reduction of the electrostatic energy for the complexation process, despite the already large attractive
Polyelectrolytes at Charged Micelles
Langmuir, Vol. 12, No. 2, 1996 313 Table 3. Excess Free Energy of the M and MP Systems at Different Salt Concentrations and Dilution As Obtained from the Cell Model Using the Poisson-Boltzmann Equationa
Figure 8. (a) Change in energy (∆Upe), entropy (∆Spe), and free energy (∆Ape) for the mixing process given in Figure 2 as a function of the average angle between consecutive beads (〈R〉). The ordinate is in reduced units per system. The largest estimated uncertainties are σ(∆U/RT) ) 0.5, σ(∆S/R) ) 0.5, and σ(∆A/RT) ) 0.6.
Figure 9. Ratio cac/cmc as a function of the average angle between consecutive beads (〈R〉) calculated according to eq 4. The error bars denote the 95% confidence limit using Student’s t-distribution with 9 degrees of freedom and σ(∆Ape) evaluated as described in the Method and Model System section.
electrostatic energy between the micelle and its counterions and similar relationship between the polyelectrolyte and its counterions, and (ii) an increase in entropy by the release of the ions accumulated close to the micelle and the polyelectrolyte. Thus, the micelle-polyelectrolyte complexation is favored both energetically and entropically. At intermediate chain flexibility, these favorable contributions are partly counteracted by an increase in angular energy and loss in chain conformational entropy. Both these effects are driven by the inclination of the complex to maintain the favorable micelle-polyelectrolyte electrostatic interaction. When the rigidity of the chain is increased even more, there are no further possibilities for a strongly attractive electrostatic micelle-polyelectrolyte interaction; the electrostatic mixing energy increases, and the gain in the counterion configurational entropy is reduced. But the effect on the total energy and entropy becomes smaller, since the chain stress that appeared at intermediate flexibility is now absent. Cac/cmc Ratio. From the calculated ∆Ape, which describes the change in free energy of mixing a micellar and a polyelectrolyte solution, we are able to estimate the reduction of the cmc due to the presence of the polyelectrolyte according to eq 4. Figure 9 shows the cac/cmc ratio as a function of the chain flexibility. At high flexibility, the reduction amounts to almost 2 orders of magnitude, whereas the reduction becomes smaller as the chain becomes more rigid. Before making connections with experimental data, we will consider some of the approximations made in our
csalt (M)
Rcell (Å)
Aex/RTb
0 0.010 0.010 0.050 0.050
M System 79.5 79.5 ∞ 79.5 ∞
2.18 1.98 2.38 1.58 1.66
0.002 0.002 0.010 0.010
MP System 166.3 ∞ 166.3 ∞
2.07 2.30 1.69 1.74
a Model: Charged spherical aggregate with radius R agg and charge zagg enclosed in a spherical cell with radius Rcell with intervening dielectric continuum containing counterions of the aggregate and a 1:1 electrolyte. Parameters: Ragg ) Rmic + Rion ) 17 Å and zagg ) zmic ) 20 (M system), Ragg ) 22 Å (radial maximum of the cation distribution) and zagg ) zmic + nbeadzbead ) -20 (MP system), T ) 298 K, and r ) 78.3. b Per micellar charge.
model in more detail. As mentioned, the calculation of ∆Ape is made for a system where the micelle and the micelle-polyelectrolyte complex occur at unequal and finite concentrations (cell model) and the concentration of free surfactants is neglected (zero salt concentration). The effects of these two approximations have been estimated at the Poisson-Boltzmann (PB) level by calculating the excess free energy (i) of the M and MP systems and (ii) of the corresponding systems but at infinite dilution with appropriate salt concentrations. In the M system the salt concentration should correspond to the cmc and in the MP system to the cac. However, experimentally, the polyelectrolyte concentration often exceeds the cac, and hence the salt concentration to be used in the PB calculation is determined by the polyelectrolyte concentration. Table 3 gives the conditions and the results of the PB calculations. The applied corrections to ∆Ape were evaluated according to -[Aex,M(csalt ) cmc, Rcell f ∞) - Aex,M(csalt ) 0, Rcell ) Rcell,M)] + [Aex,MP(csalt ) cac + cpolyelectrolyte, Rcell f ∞) - Aex,MP(csalt ) cpolyelectrolyte, Rcell ) Rcell,MP)], where the first square bracket denotes the free energy of bringing the M system from csalt ) 0 and Rcell ) Rcell,M to csalt ) cmc and Rcell f ∞ and the second square bracket denotes the similar quantity for the MP system. Obviously, the correction depends on the actual values of the cmc, the cac, and the polyelectrolyte concentration. The choice cmc ) 10 mM and cpolyelectrolyte ) 2 mM (cpolyelectrolyte . cac) gives a correction of 0.03RT per surfactant or an increase of the ratio cac/cmc by a factor of 1.03 (cac/cmc becoming closer to one). Thus, the salt and finite concentration corrections are negligible as compared to the complexation as such. This also validates the treatment of the salt and concentration effects by using a simpler model and a more approximate statistical mechanical treatment. Moreover, a weak point in our model is the short polyelectrolyte, 40 beads. The use of 60 beads did not change the results at one specific flexibility, but preferentially much longer polyelectrolytes should be used. At the present state, it is, however, computationally too demanding to treat longer chains, since free energies need to be computed. Despite the short chain, the main effects of the reduction of the cmc, as we believe, the favorable electrostatic micelle-polyelectrolyte interaction and the related release of the counterions of the micelle and polyelectrolyte are captured by the model. A direct comparison between the experimental and the calculated cac/cmc ratio is not trivial since the model
314
Langmuir, Vol. 12, No. 2, 1996
system does not completely match (and was not meant to do so) a particular experimental system. For example, the micellar radius of the model corresponds roughly to that found for micelles formed by C8TAB,30 but the charge of the model micelle is only ca. 2/3 of the experimental one. Figure 1 showed the cmc and the cac of CnTAB in the presence of some polyelectrolytes, and our flexible model polyelectrolyte corresponds reasonably well to polyacrylate. An extrapolation of the surfactant chain length to C8TAB would give the ratio cac/cmc ≈ 10-2. Hence, our calculated ratio 1.7 × 10-2 is in an astonishingly good agreement with the extrapolated experimental data, but a more thorough comparison has to be postponed until experimental and calculated data for these systems are available. Moreover, the experimental cac depends on the polyelectrolyte concentration and that has also to be addressed more properly in the model. Conclusion On the basis of Monte Carlo simulation and thermodynamic integration of model systems containing charged micelles and polyelectrolytes, we have investigated the structure of the micelle-polyelectrolyte complex and changes in thermodynamic quantities for the corresponding complexation. Our results clearly show that the flexibility of the polyelectrolyte affects the structure of the complex formed. Highly flexible polyelectrolytes form complexes without any significant increase in internal stress, whereas larger rigidity causes the internal stress to increase to maintain optimal electrostatic interaction with the micelle. When the persistence length of the polyelectrolyte becomes larger than ca. half of the effective circumference of the micelle, a much weaker electrostatic (30) Tanford, C. J. Phys. Chem. 1972, 76, 3020.
Wallin and Linse
interaction with the micelle is obtained and the chain stress is released. Moreover, the cac was found to be up to 2 orders of magnitude lower than the cmc but increase with chain rigidity. Both the strong attractive electrostatic interaction between the micelle and the polyelectrolyte and the release of their counterions were found to be important for the reduction of the cmc. Preliminary comparison with experimental data shows that the obtained reduction of the cmc is reasonable and hence indicates that despite the simplicity of the model, it correctly describes the essential parts of the sytem. Thus, still keeping the shortcomings of the model used in mind, we believe that the present approach is an important step for increasing our understanding of the micelle-polyelectrolyte complexation and its implications on the self-aggregation of charged surfactants. Obvious extensions in future investigations are to consider the effect (i) of the linear charge density of the polyelectrolyte, (ii) of the micellar charge and size, and (iii) of added salt. It would also be interesting to consider cases where favorable short range interactions between the surfactants and the polyelectrolyte lead to a penetration of the polyelectrolyte into the micelle, with an accompanying reduction of the aggregation number at the cac.31 Acknowledgment. This work has been supported by grants from the Swedish National Board for Industrial and Technical Development (NUTEK). LA950362Y (31) Magny, B.; Iliopoulos, I.; Zana, R.; Audebert, R. Langmuir 1994, 10, 3180. (32) Flory, P. J. Statistical Mechanics of Chain Molecules; Interscience: New York, 1969.
