Complexation and Phase Behavior of Oppositely Charged

Langmuir 2004, 20, 1997-2009

1997

Complexation and Phase Behavior of Oppositely Charged Polyelectrolyte/Macroion Systems Rosalind J. Allen University Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW, United Kingdom

Patrick B. Warren* Unilever Research and Development Port Sunlight, Quarry Road East, Bebington, Wirral CH63 3JW, United Kingdom Received September 5, 2003. In Final Form: December 2, 2003 The complexation and phase behavior of oppositely charged polyelectrolyte/macroion mixtures is investigated using a Poisson-Boltzmann cell model, extended to include the polyelectrolyte component at the level of a mean field treatment. We find that most of the observed features of associative phase separation in oppositely charged polymer/surfactant mixtures are recovered by this model. In addition, the model makes predictions for trends in the critical aggregation concentration in these systems which are in reasonable agreement with experiment.

1. Introduction Mixtures of polyelectrolytes and oppositely charged macroions are prototypes for a wide class of systems which are interesting both from a fundamental point of view and in applications. These include polymer/surfactant, polymer/colloid, and polymer/protein mixtures, where surfactant micelles, charged colloids, and globular proteins, respectively, play the role of macroions. There are also biologically important examples, such as chromatin formed from the complexation of negatively charged DNA with positively charged histone assemblies.1 In this paper, we focus mainly on polymer/surfactant mixtures, for which there is an abundance of experimental information.2,3 Much experimental and theoretical interest has targeted the structure of polyelectrolyte/surfactant complexes, and it has long been noted that micelles associated with the polymer form at much lower surfactant concentrations than free micelles. From the surfactant/polymer binding isotherm, one can clearly identify a critical aggregation concentration (CAC) where the polymer-associated micelles appear, which is below the critical micellization concentration (CMC) for the formation of free micelles. The smallness of the ratio CAC/CMC in oppositely charged mixtures is generally attributed to a reduction in the free energy of the double layer surrounding the micelle, as small counterions are replaced by a “poly-counterion”. A number of theoretical and simulation methods have been used to model this effect, ranging from Monte Carlo simulations,4-6 through self-consistent field theory,7 to more or less elaborate molecular thermodynamic theories.8-11 Alternative models, such as the Satake-Yang (1) Twyman, R. M. Advanced molecular biology; BIOS Scientific: Oxford, 1998. (2) Kwak, J. C. T. Polymer-surfactant systems; Surfactant Science Series Vol. 77; Marcel Dekker: New York, 1998. See especially Chapter 3 (Picullel, L.; Lindman, B.; Karlstro¨m, G.) in this reference. (3) Hansson, P.; Lindman, B. Curr. Opin. Colloid Interface Sci. 1996, 1, 604. (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. B 1997, 101, 5506. (7) Wallin, T.; Linse, P. Langmuir 1998, 14, 2940. (8) Nikas, Y. J.; Blankschtein, D. Langmuir 1994, 10, 3512. (9) Konop, A. J.; Colby, R. H. Langmuir 1999, 15, 58.

model,12 or more recent work by Kuhn et al.,13 postulate that the surfactants aggregate on the surface of the polyelectrolyte, tails outward, like a molecular “bottlebrush”. The resulting structures are highly hydrophobic but may perhaps occur for stiff polyelectrolytes such as DNA.13 For flexible polyelectrolytes, it seems that the experimental evidence strongly supports models in which polymer-associated micelles form at the CAC.2,14 Another common experimental observation in oppositely charged polymer/surfactant systems is a strong tendency for coacervation or associative phase separation, although this aspect has received much less theoretical attention than the problem of the CAC. However, extensive experimental studies of the phase behavior of oppositely charged polymer/surfactant mixtures have revealed a number of interesting generic features.15-23 We note that the volume collapse transition of polyelectrolyte gels in the presence of oppositely charged surfactants is a related problem.24,14 Here we summarize the general experimental observations. Associative phase separation in the form of precipitation of a polymer/surfactant coacervate occurs on mixing polymer and surfactant.25 The coacervate is a dense phase rich in surfactant and polymer which may be described (10) Hansson, P. Langmuir 2001, 17, 4167. (11) Jiang, J.; Prausnitz, J. M. J. Phys. Chem. B 1999, 103, 5560. (12) Satake, I.; Yang, J. T. Biopolymers 1976, 15, 2263. (13) Kuhn, P. S.; Levin, Y.; Barbosa, M. C. Chem. Phys. Lett. 1998, 298, 51. (14) Hansson, P. Langmuir 1998, 14, 2269. (15) Thalberg, K.; Lindman, B. J. Phys. Chem. 1989, 93, 1478. (16) Thalberg, K.; Lindman, B.; Karlstro¨m, G. J. Phys. Chem. 1991, 95, 6004. (17) Thalberg, K.; Lindman, B.; Bergfeldt, K. Langmuir 1991, 7, 2893. (18) Carnali, J. O. Langmuir 1993, 9, 2933. (19) Hansson, P.; Almgren, M. Langmuir 1994, 10, 2115. (20) Li, Y.; Xia, J.; Dubin, P. L. Macromolecules 1994, 27, 7049. (21) Li, Y.; Dubin, P. L.; Havel, H. A.; Edwards, S. L.; Dautzenberg, H. Langmuir 1995, 11, 2486. (22) Ranganathan, S.; Kwak, J. C. T. Langmuir 1996, 12, 1381. (23) Ilekti, P.; Piculell, L.; Tournilhac, F.; Cabane, B. J. Phys. Chem. B 1998, 102, 344. (24) Khokhlov, A. R.; Kramarenko, E. Y.; Makhaeva, E. E.; Starodubtzev, S. G. Macromolecules 1992, 25, 4779. (25) Segregative phase separation has also been reported under high salt concentrations in some systems; see refs 16, 18, and 19.

10.1021/la0356553 CCC: $27.50 © 2004 American Chemical Society Published on Web 01/29/2004

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as a polymer-enriched surfactant mesophase.18,23 The structure in this dense phase is distinct from the stable solution structures in the dilute phase which can be characterized, as already mentioned, as polymer-dressed micelles (or micelle-decorated polymers). The phase separation region is often shown in the polymer/surfactant mixing plane or in a ternary (solvent, polymer, surfactant) representation. However, it is now clear that the behavior can only be properly represented in a quaternary (solvent, polymer, surfactant, salt) phase diagram, because the phase separation occurs between the dense polymer/surfactant complex phase and, essentially, salt solution.16,23 The tie lines are more or less orthogonal to the usual mixing plane. A quaternary phase diagram is the minimum required to represent this behavior. The ternary representations mentioned above are therefore projections or slices through the full quaternary phase diagram, and the boundaries of the phase separation region in these representations are cloud curves. A striking consequence of the quaternary nature of the system was noted by Ilekti et al. who found that it is possible to concentrate the polymer/surfactant coacervate in the phase coexistence region by adding more water.23 Going into more detail, phase separation tends to be restricted to the vicinity of the charge stoichiometry plane, where the average charge density in the system due to the polymer balances that due to the surfactant. Phase separation is suppressed by addition of salt, and there is often a critical electrolyte concentration (CEC) above which there is no phase separation (hence the importance of the salt axis in the phase diagram). Finally, the phase separation region often shrinks and disappears at sufficiently high concentrations of polymer and surfactant. We seek to reproduce these features in the present paper. Previously, elaborations of Flory-Huggins theory have been used to generate possible phase diagrams for ternary mixtures.26 For suitable choices of Flory χ-parameters, one can recover associative phase separation in a ternary system. While this is important from a didactic point of view, such approaches have little predictive capability and omit the essential quaternary nature of the problem. Moreover, experimentally observed features such as confinement to the vicinity of the charge stoichiometry plane and the existence of a CEC point to the importance of electrostatic interactions. Electrostatics can be included in Flory-Huggins theory at the mean field level by requiring electroneutrality in all bulk phases, and models based on this can recover some of the general observations mentioned above.27,28 However, the interpretation of the Flory χ-parameters in such models is unclear for polymer/ surfactant mixtures. Jiang and Prausnitz have introduced a “molecular thermodynamic” theory for precipitation in oppositely charged protein/polyelectrolyte mixtures, which is clearly also applicable to the present systems.11 This is an important attempt to identify the underlying mechanisms at work, but it restricts phase coexistence to the ternary mixing plane, thereby suppressing the role of the salt axis.29 More recently, Hansson outlined a model in which phase separation is attributed to a long-range attraction between micelles, although the origin of such an attraction (26) Bergfeldt, K.; Piculell, L.; Linse, P. J. Phys. Chem. 1996, 100, 3680. (27) Warren, P. B. J. Phys. II (France) 1997, 7, 343. (28) Gottschalk, M.; Linse, P.; Piculell, L. Macromolecules 1998, 31, 8407. (29) See for example eq 1 and Figure 2 in ref 11.

Allen and Warren

was not elaborated.10 Integral equation methods can also be used to investigate these systems; see for example Harnau and Hansen.30 A theory for the complexation between much more strongly charged DNA and histones has been developed by Nguyen and Shklovskii.31 Recently, Linse and co-workers have undertaken Monte Carlo simulations of protein/polyelectrolyte cluster formation, which indirectly address the phase behavior problem.32,33 Even more recently, Svensson et al. have presented an extensive experimental study, accompanied by Monte Carlo calculations of the interactions between surfactant aggregates, which suggest that the driving force for phase separation is polymer bridging.34 These results, which complement our own calculations, will be discussed in greater depth later. For both the phase behavior problem and the computation of the CAC, we need to calculate the free energy of a polyelectrolyte/macroion mixture. In the present work, this is done by computing the free energy per macroion in a cell model. The polyelectrolyte is treated using mean field methods, and the whole approach can be characterized as a Poisson-Boltzmann cell model generalized to include a polyelectrolyte component. The free energy per macroion in the limit of a large cell radius is used to calculate the CAC. For some intermediate cell radii, the free energy per macroion depends on the macroion concentration in such a way as to be thermodynamically unstable. This instability is manifested as a region of phase separation in the phase diagram with virtually all the features observed in the experimental systems. This, we argue, is the essential origin of the phase instability in these systems: it is due to a nontrivial concentration dependence of the free energy of micellization. A brief account of our findings has already been published elsewhere.35 We now turn to a detailed description of the cell model and the associated polyelectrolyte-Poisson-Boltzmann free energy, before presenting the predictions of the theory for the CAC and for the phase behavior. In the final section, we discuss the limitations of our model and draw conclusions. 2. The Cell Model The cell model of the surfactant system in the presence of oppositely charged polymer and salt is based on several simplifying assumptions. First, we treat the micelles as macroions, that is, charged hard spheres, of radius Rm and charge Z. This assumes that the surfactants form spherical micelles with an internal structure which is unaffected by the presence of the polymer and salt. Moreover, any phase separation that occurs is taken to involve dilute and dense suspensions of these micelles, thus neglecting the likely complex mesophase structure of the dense phase. If we are dealing with a single univalent surfactant species, we would expect Z ) Nagg where Nagg is the aggregation number. We consider a suspension of such macroions at concentration nm in a dielectric continuum of relative permittivity ) 80, in the presence of polyelectrolyte chains and positive and negative microions. For concreteness, we take the macroions to be positively charged and the polymer (30) Harnau, L.; Hansen, J.-P. J. Chem. Phys. 2002, 116, 9051. (31) Nguyen, T. T.; Shklovskii, B. I. J. Chem. Phys. 2001, 115, 7298. (32) Skepo¨, M.; Linse, P. Macromolecules 2003, 36, 508. (33) Carlsson, F.; Malmsten, M.; Linse, P. J. Am. Chem. Soc. 2003, 125, 3140. (34) Svensson, A.; Piculell, L.; Karlsson, L.; Cabane, B.; Jo¨nsson, B. J. Phys. Chem. B 2003, 107, 8119. (35) Allen, R. J.; Warren, P. B. Europhys. Lett. 2003, 64, 468.

Polyelectrolyte/Macroion Systems

to be negatively charged. Image charges arising from the dielectric interface between the inside and outside of a macroion are ignored. The polyelectrolytes are modeled as Gaussian chains with a continuous and uniform charge density -p per segment along the backbone. This is the simplest choice of a charge distribution; others, including those that are influenced by local conditions of pH and electrostatic potential, have been considered by other authors. The differences are small provided the charge density is not too large.36-38 To facilitate construction of the cell model, we suppose that the macroion suspension is in equilibrium with a notional polyelectrolyte-salt reservoir containing polymers, with segment concentration nRp , and salt at concentration nRs . Our model is therefore semi-grand canonical in the sense that the macroions are treated in a canonical ensemble framework (fixed density), whereas the microions and polymer segments are maintained at fixed chemical potential µ+, µ-, and µp. One could of course work directly with the chemical potentials µ( and µp as control variables, but it is more convenient to use nRp and nRs as control variables, since this eliminates considerations of reference states. It should be emphasized that the system does not have to be in contact with a real reservoir; rather, in our calculations we can compute the concentrations of salt and polyelectrolyte in the real system from eqs 13-15 as described later. These concentrations will in general be different from the reservoir concentrations and may also be different in coexisting phases (although, by construction, the chemical potentials are the same between coexisting phases). We divide the suspension into electrically neutral cells, each containing one macroion. We then make the basic approximation that all cells are identical and spherical with the macroion in the center (spherical Wigner-Seitz cells). The cell radius Rc is determined by 4πRc3/3 ) 1/nm. Such an approach is frequently used in studying colloidal suspensions and in particular has been used to investigate the phase behavior of highly charged colloids at very low salt concentration, a problem closely related to the present one.39 Interactions between macroions are implicitly taken into account through boundary conditions at the cell surface. 2.1. Free Energy Expression. We now develop an expression for the semi-grand potential of the system as a functional of the polymer segment and microion concentration profiles in the cell. Since we use spherical cells, these profiles are functions only of the radial coordinate r. The semi-grand potential is minimized with respect to the profiles, subject to suitable conditions at the macroion surface r ) Rm and at the cell boundary r ) Rc. From this, one can compute all the thermodynamic properties of the suspension. Although this may seem rather intricate, in the absence of polymer it reduces to the PoissonBoltzmann formalism for the distribution of microions in the cell. The semi-grand potential of the system, here denoted by X, is defined relative to the grand potential of the polymer segment and microion reservoir. It includes the mixing free energy of the macroions and a contribution, (36) Borukhov, I.; Andelman, D.; Orland, H. Europhys. Lett. 1995, 32, 499. (37) Borukhov, I.; Andelman, D.; Orland, H. Eur. Phys. J. B 1998, 5, 869. (38) Chaˆtellier, X.; Joanny, J.-F. J. Phys. II France 1996, 6, 1669. (39) von Gru¨nberg, H. H.; van Roij, R.; Klein, G. Europhys. Lett. 2001, 55, 580.

Langmuir, Vol. 20, No. 5, 2004 1999

Ω, from each cell:

X ) nm(log nm - 1) + nmΩ(nm,µ(,µp) VkBT

(1)

where V is the volume of the system, kB is Boltzmann’s constant, and T is the temperature. Unimportant constants and terms linear in nm have been omitted here.40 Note that Ω depends on nm through the cell radius. Our starting point in computing Ω is a “self-consistent mean field” approach that has been used by a number of workers for related problems.36-38,41-44 We split Ω (which is relative to the reservoir) into three local contributions:

Ω ) Ωions + Ωpol + Ωel )

∫cell d3r [ωions(r) + ωpol(r) + ωel(r)]

(2)

where the integral ∫cell d3r denotes integration over the volume of the cell, excluding the region occupied by the macroion. Given the spherical symmetry of the problem,

∫cell d3r ) 4π∫R R r2 dr c

m

The microion free energy density ωions is given by

ωions )

[ni log ni - ni - nRi log nRi + nRi - µi(ni - nRi )] ∑ i)(

(3)

where n((r) are the positive and negative microion concentration profiles and nR( are the (bulk) reservoir concentrations nR+ ) nRs + pnRp and nR- ) nRs . The theory assumes that the counterions to the polymer are identical to the salt anions and that all microions are univalent. It does not include any effects of the finite size of the microions or take account of specific interactions between the microions and the polymer or the macroion, although microions cannot enter the region occupied by the macroion at the center of the cell. Multivalent microions are expected to show significant correlations which are not considered here. The contribution ωpol incorporates into the theory the effect of the polymer chain connectivity, as well as including short-ranged interactions between the chain segments. It is derived from a model in which the charged polymer is approximated as a infinitely long random coil, with electrostatic and short-ranged interactions being included as a self-consistent external field.45,46 A calculation (described in refs 45 and 46) involving the Green’s function for the random coil leads to an expression for ωpol in terms of a “polymer order parameter” φ(r), where the local concentration of polymer segments is given by np(r) ) φ2(r):

ωpol )

a2 1 |∇φ|2 + v(φ4 - (nRp )2) - µp(φ2 - nRp ) (4) 6 2

(40) Since µm ) kBT log nm plus nonideal terms, we have implicitly set the standard state chemical potential such that µm ) 0 when nm ) 1 in whatever units are being used. Note that the introduction of the de Broglie thermal wavelength into the entropy expressions is neither necessary nor strictly speaking correct from the point of view of solution thermodynamics; see for example ref 66. (41) Varoqui, R.; Johner, A.; Elaisseri, A. J. Chem. Phys. 1991, 94, 6873. (42) Varoqui, R. J. Phys. II France 1993, 3, 1097. (43) Borukhov, I.; Andelman, D. Macromolecules 1998, 31, 1665. (44) Borukhov, I.; Andelman, D.; Orland, H. J. Phys. Chem. B 1999, 103, 5042. (45) de Gennes, P. G. Scaling concepts in polymer physics; Cornell: Ithaca, NY, 1979. (46) Doi, M.; Edwards, S. F. The theory of polymer dynamics; Oxford University Press: Oxford, 1986.

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Allen and Warren

Here, a is the Kuhn segment length of the random coil and v is the Edwards excluded volume parameter. The first term in eq 4 accounts for the connectivity of the polymer chains. Increasing the parameter a corresponds to increasing the chain stiffness. The second term accounts for short-ranged interactions between polymer segments. The parameter v reflects the solvent quality and may be positive (good solvent) or negative (poor solvent). Some polyelectrolytes are in poor solvent conditions in water due to their organic backbones.38 Many cationic polyelectrolytes, on the other hand, are in neutral or good solvent conditions in water. In this work, we investigate the effects of changing the parameter v. The derivation of eq 4 assumes that the polymer remains Gaussian in the presence of the external field. Stiffening of the polymer chain due to electrostatic interactions between monomers is not included; however, this is expected to be a small effect if the polymer concentration is large enough.38 Additional effects such as the formation of bead-necklace structures for highly charged polyelectrolytes in poor solvents are also neglected. The final part, ωel, is the electrostatic contribution. In some previous works,36,37,43,44 this has been written in terms of the dimensionless electrostatic potential ψ(r) and the electric field E(r) ) -∇ψ(r) as

ωel ) -pφ2ψ + n+ψ - n-ψ -

1 |E|2 8πlB

(5)

where lB ) e2/(4π0kBT) is the Bjerrum length of the solvent. The last term in eq 5 functions as a Lagrange multiplier, ensuring that Poisson’s equation

∇‚E + 4πlB(pφ2 - n+ + n-) ) 0

(6)

is satisfied at the extremum of eq 5 with respect to ψ(r).38 We have found, however, that this extremum is a maximum rather than a minimum. Not only does this present numerical problems, but eq 5 cannot be a true free energy except at the extremum. We prefer to constrain E to satisfy Poisson’s equation at all stages of the minimization procedure, in which case we can use a simpler expression for ωel,

ωel )

1 |E|2 8πlB

(7)

This is equivalent to the extremum of eq 5 and is a bona fide contribution to the free energy. In practice, we seek the variational minimum of Ω with respect to φ(r) and E(r), ensuring that Poisson’s law is satisfied by computing the microion profiles n((r) from φ(r) and E(r) using Gauss’s law: 2 1 d(r E(r)) ) -pφ2(r) - n-(r) + n+(r) ) dr 4πlBr2

-pφ2(r) -

nR+ nRn+(r)

+ n+(r) (8)

The final result in eq 8 is obtained by using the Boltzmann distribution for the microions (which follows from minimizing the free energy with respect to n+ and n-): n((r) ) nR( exp[-ψ(r)], so that n+(r)n-(r) ) nR+ nR-. The chemical potentials required in eqs 3 and 4 are obtained by setting Ω to a minimum with respect to variations of φ, n+, and n- in the reservoir, where ni ) nRi ,

φ ) (nRp )1/2 and E ) 0 (the polymer in the reservoir is assumed to be simply a uniform solution of monomers42). This leads to

µ+ ) log nR+

µ- ) log nR-

µp ) vnRp

(9)

2.2. Boundary Conditions. The conditions on the electric field E(r) at the surface of the macroion r ) Rm and at the cell boundary r ) Rc are fixed by the macroion charge and by the requirement for electroneutrality of the cell:

E(Rm) ) ZlB/Rm2

E(Rc) ) 0

(10)

The polymer order parameter φ(r) must be symmetrical across the boundary between two identical cells:

|

dφ )0 dr r)Rc

(11)

The boundary condition on φ(r) at the micellar surface is determined by the physical behavior of the polymer close to the macroion. Our aim here is to model the micelle by a nonadsorbing (charged) hard sphere, and so we seek the appropriate boundary condition for a continuous description of a polymer next to a hard wall. This has been determined by diMarzio,47 using the statistics of random walks, to be φ(Rm) ) 0, as used in studies by some other authors.43,44 Such a boundary condition may seem rather counterintuitive, since it implies that neutral polymer segments will be depleted near a wall, even in the absence of any wall-polymer interaction. The depletion arises from a detailed consideration of the number of configurations available to a Gaussian chain near a hard wall. Some authors have used an alternative boundary condition,36,38 equivalent to dφ/dr|r)Rm ) D-1φ(Rm) where D is the polymer “adsorption length” which is inversely proportional to the strength of the interaction between the polymer and the surface. This can be derived by including a contribution due to the polymer-surface interaction in the expression for the total free energy. This boundary condition does not include the correct effects of polymer statistics near a noninteracting wall, since when there is no short-ranged interaction (D-1 ) 0), it reduces to dφ/dr|r)Rm ) 0 rather than φ(Rm) ) 0. In fact, one obtains φ(Rm) ) 0 by taking the opposite limit, D ) 0. We have found that the nature of the boundary condition for φ has a very significant effect on the behavior of the polymer-surfactant system. While we have performed the majority of our calculations using φ(Rm) ) 0, we have also investigated the effect of changing the polymermicelle interactions by testing the alternative boundary condition dφ/dr|r)Rm ) 0. The conditions on the electric field and polymer order parameter profiles at the micellar surface incorporate into the theory the effect of the charge of the micelle as well as the physics of the polymer near the surface. Those at the cell boundary ensure electroneutrality and smoothness of φ(r) between cells. This introduces a nontrivial dependence of Ω on the micellar concentration (which determines Rc) and so accounts for interactions between micelles in a dense suspension. 2.3. The CAC in the Cell Model. The cell model can be used to predict the ratio CAC/CMC for oppositely charged surfactant/polymer mixtures, as well as the (47) diMarzio, E. A. J. Chem. Phys. 1965, 42, 2101.


Langmuir, Vol. 20, No. 5, 2004 2001

variation in the CAC due to changes in the polymer properties and in the salt concentration. Our model is very close in spirit to the self-consistent field (SCF) calculations of Wallin and Linse7 (and suffers from the same defects). Later work by Konop and Colby9 and by Hansson10 shows the importance of counterion condensation on the polyelectrolyte chains. This effect is absent from our theory, which neglects correlations between the microions and the polyelectrolyte chains. Our model is suitable for weakly charged polyelectrolytes for which counterion condensation effects are small (this restriction also makes the model insensitive to the precise distribution of the charges on the polymer). For two systems (labeled 1 and 2), for which we assume the internal structure of the micelles is identical, we find

CAC(1)/CAC(2) ) exp[-∆Ω/Nagg] ∆Ω ) Ω(2) - Ω(1)

polymer segments and microions adsorbed on the micelle, measured by Γpol and Γ(, where

∫RR r2 dr [φ2(r) - nRp ] R Γ( ) 4π∫R r2 dr [n((r) - nR(] Γpol ) 4π

c

m

3. Numerical Results: Structure In this section, the theory described above is used first to study the structure and free energy of dilute micellar systems and then to predict the changes that occur when the systems become more concentrated. Calculations were performed using the parameter set Rm ) 3 nm, Z ) 40, nRs ) 100 mM, nRp ) 10 mM, a ) 1 nm, v ) 0.2 nm3, and p ) 1. The charge on the micelle is the bare charge, since ion condensation effects are taken into account in the variation of the microion density profiles. However, the charge must remain rather low, since the polymer theory is strictly valid only for small electric fields. We note, however, that micelles of reduced charge can be formed from mixtures of charged and uncharged surfactants.48 We investigate the effects of varying the concentration of salt, nRs , and of polymer segments, nRp , in the reservoir, the effective excluded volume parameter v, the monomer charge p, the polymer stiffness parameter a (keeping the linear charge density fixed), and finally the micelle concentration nm. We maintain the monomer charge p low enough so that, according to the Manning criterion p < a/lB, counterion condensation on the polymer is unimportant, as discussed above.49,50 To study dilute micelles, we use a cell radius large enough that all profiles have reached their bulk values before r ) Rc and the value of Ω does not depend on nm. We expect this to be the case for the CAC. A suitable value in this regime using our parameter set is nm ) 0.5 mM or Rc ) 9.25 nm. Experimentally the CAC is smaller than the CMC, which is generally ∼10 mM surfactant.51 In addition to the concentration profiles of the various solution components, we are interested in the number of (48) Dubin, P. L.; Oteri, R. J. Colloid Interface Sci. 1983, 95, 453. (49) Manning, G. S. J. Chem. Phys. 1969, 51, 954. (50) Barrat, J.-L.; Joanny, J.-F. Adv. Chem. Phys. 1996, 94, 1. (51) Evans, D. F.; Wennerstro¨m, H. The colloidal domain: Where physics, chemistry, biology and technology meet; Wiley: New York, 1999.

(13)

Electroneutrality of the cell ensures that pΓpol + Γ- - Γ+ ) Z. Experiments are usually carried out at fixed average concentrations of polymer segments and of salt (i.e., in the canonical ensemble with respect to all the components) rather than at fixed chemical potential, as in these semigrand canonical calculations. We therefore compute the average concentrations of polymer segments and of positive and negative microions, np and n(:

(12)

where Ω(1) and Ω(2) are the cell contributions to the free energy calculated using eq 2, and the micelles have a common aggregation number Nagg. This is a standard result which is derived in the context of the cell model free energy in Appendix A (note that in the absence of polymer, the CAC is the CMC, so that we find the ratio CAC/CMC by setting the polymer concentration to zero in system 2).

c

m

np )

Γpol + nRp Vcell

n( )

Γ( + nR( Vcell

(14)

where Vcell is the volume of the cell. An important experimental parameter is the average salt concentration within the cell, ns. This is given by

ns )

Γ1 [Γ+ - pΓpol] + nRs ) - Z + nRs (15) Vcell Vcell

Negative values of ns are not unphysical but merely correspond to situations where the counterions of the polymer and surfactant are excluded from the cell. In very dilute micellar suspensions, where the cell is large, the average concentrations of all species are expected to be close to the reservoir concentrations. However, this may not be the case for more concentrated suspensions. To compare the numerical results for the polyelectrolyemicelle systems to those of surfactant solutions in the absence of polymer, we note again that the polyelectrolyte mean field theory presented here is an extension of nonlinear Poisson-Boltzmann theory, to which it reduces if the polymer part of the semi-grand canonical potential is neglected and the polymer profile is set to zero. Results for systems in the absence of polymer are therefore obtained by minimizing eq 2 with respect to the electric field under these conditions. 3.1. Structure around Micelle and Grand Potential Values. Profiles for the polymer segment concentration φ2(r), the microion concentrations n((r), and the radial electric field E(r), using the parameter set given above, with nm ) 0.5 mM and φ(Rm) ) 0, are shown in Figure 1. The dotted lines show results obtained for the equivalent model in the absence of polymer (Poisson-Boltzmann theory). Numerical results were obtained using the method described in Appendix B. Figure 1a shows that the negatively charged polymer segments form a layer of high density around the positively charged micelle, due to the attractive electrostatic interaction. The formation of this layer is opposed by the polymer connectivity, the electrostatic repulsion between the like-charged monomers, and the short-ranged monomer-monomer interactions, which are repulsive in this case (v > 0). The polymer adsorption Γpol ) 29.5 indicates that the polymer layer is sufficient to screen about threequarters of the micellar charge (Z ) 40). Figure 1b shows the microion concentration profiles in the presence and absence of polymer. In both cases, anions are attracted to the micelle and cations are repelled, but these effects are less pronounced in the presence of the

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Allen and Warren Table 1. Components of βΩ (where β ) 1/kBT) and Adsorbed Amounts for Various Casesa

no polymer with polymer no polymer with polymer no polymer with polymer dφ/dr|Rm ) 0

nRs / mM

βΩ

100 100 200 200 300 300 100

42.9 38.8 33.3 32.9 28.5 28.7 21.1

βΩpol βΩions βΩel Γpol 0.0 5.6 0.0 2.1 0.0 0.7 6.9

19.2 9.0 15.0 11.8 12.9 12.2 1.3

Γ+

Γ-

23.6 0.0 -16.0 25.4 24.2 29.5 -1.8 8.8 18.3 0.0 -13.7 26.3 18.9 19.3 -5.6 15.2 15.6 0.0 -14.3 25.7 15.8 9.9 -9.9 20.2 13.0 37.7 -0.6 1.7

a For all systems, R ) 3 nm, n ) 0.5 mM, Z ) 40, nR ) 10 mM, m m p p ) 1, v ) 0.2 nm3, a ) 1 nm, and φ(Rm) ) 0 except where indicated otherwise.

Figure 1. Results for the parameter set Rm ) 3 nm, nm ) 0.5 mM (Rc ) 9.25 nm), Z ) 40, nRs ) 100 mM, nRp ) 10 mM, p ) 1, v ) 0.2 nm3, and a ) 1 nm, with φ(Rm) ) 0. (a) Polymer segment concentration φ2(r)/nRp . (b) Microion concentrations n((r)/nR(. (c) Electric field E(r)/E(Rm) (the inset shows a magnified version of the same profile). Solid lines show the results in the presence of polymer; dotted lines are in the absence of polymer (Poisson-Boltzmann theory).

polymer. With polymer, Γ+ ) -1.8 and Γ- ) 8.8, while without polymer, Γ+ ) -12.6 and Γ- ) 27.4, showing that when polymer is present, far less counterion localization is required to screen the charge of the micelle. The wobbles in these curves are numerical artifacts due to the choice of basis functions for the minimization; see Appendix B. The efficiency of the charge screening with and without polymer is reflected in the electric field profiles of Figure 1c. The field decays more sharply in the presence of polymer. An interesting feature is the reversal of sign of E(r), magnified in the inset to Figure 1c, which occurs in the presence but not in the absence of polymer. Gauss’s law means that E(r) is proportional to the total electrical charge inside a sphere of radius r, so that the combined effect of the accumulation of polymer segments and anions and the depletion of cations is in fact an overcompensation of the charge of the micelle beyond r ∼ 5.3 nm. Several theoretical studies of isolated colloid/polyelectrolyte complexes have predicted that the total charge of the complex should be opposite to that of the colloid.52-55 (52) Mateescu, E. M.; Jeppesen, C.; Pincus, P. Europhys. Lett. 1999, 46, 493.

However, given that here the whole cell is electrically neutral, the definition of the effective charge of the complex is not straightforward. This overcompensation phenomenon was not observed in the simulation studies of Wallin and Linse4-6 and is somewhat surprising given our rather weakly charged polyelectrolyte, which adsorbs only enough to neutralize about three-quarters of the micelle charge. The results for Ω/kBT and its components, in the presence and absence of polymer, are shown in Table 1. The grand potential of the cell is reduced in the presence of polymer, despite the fact that the polymer term Ωpol is unfavorable. The main contribution to this reduction is a decrease in the microion entropy Ωions. Equation 12 predicts that CAC/CMC is 0.90 for our system (taking the surfactants to be univalent, so that Nagg ) Z): the CAC is indeed reduced compared to the CMC, but rather weakly. This reflects the low charge of our micelle and low polymer linear charge density, as compared to many experimental systems. 3.2. Effect of Salt. Adding salt to a polyelectrolyte adsorbed on an oppositely charged micelle is expected to have several effects. The salt screens the repulsive electrostatic interactions between monomers, but the favorable interaction between the polymer and the micelle is also reduced.38 In addition, the entropic driving force for the adsorption of polymer rather than microions will be reduced, since the entropy of free microions is lower when they are at higher concentration. Figure 2 shows the effect on the structure around the micelle of changing the reservoir salt concentration between nRs ) 100 mM and nRs ) 500 mM. The polymer adsorption is greatly reduced and finally abolished as nRs increases: Γpol ) 29.5, 19.3, 9.9, and -1.3 for nRs ) 100, 200, 300, and 500 mM. The grand potential Ω and its components are given in Table 1 for nRs ) 100, 200 and 300 mM, showing that Ω decreases as salt is added, both in the presence and in the absence of polymer. This indicates that both the CAC and the CMC decrease on addition of salt, although the CAC decreases more slowly. This result is similar to that of SCF calculations7 but is contrary to most experimental evidence, which suggests that the CAC actually increases when salt is added.2 The reversed trend is most likely due to counterion release effects from condensed counterions associated with the polymer,9,10 an effect omitted from the present theory. Alternative models of the CAC such as that by Kuhn et al. may correctly predict this trend.13 3.3. Polymer Properties. Changing the properties of the polyelectrolyte also affects the adsorption behavior. (53) Park, S. Y.; Bruinsma, R. F.; Gelbart, W. M. Europhys. Lett. 1999, 46, 454. (54) Gurovitch, E.; Sens, P. Phys. Rev. Lett. 1999, 82, 339. (55) Kunze, K.-K.; Netz, R. R. Phys. Rev. Lett. 2000, 85, 4389.


Langmuir, Vol. 20, No. 5, 2004 2003

Figure 3. Polymer segment concentration φ2(r)/nRp . (a) Monomer charge fraction p is varied, with a ) 1 nm. Solid lines, p ) 1; dotted lines, p ) 0.75; dashed lines, p ) 0.5. (b) Polymer flexibility a is varied, keeping p/a fixed. Solid lines, a ) 1 nm, p ) 1; dotted lines, a ) 1.5 nm, p ) 1.5; dashed lines, a ) 2 nm, p ) 2. For all curves, Rm ) 3 nm, nm ) 0.5 mM, nRs ) 100 mM, Z ) 40, nRp ) 10 mM, v ) 0.2 nm3, and φ(Rm) ) 0.

Figure 2. Effect of changing reservoir salt concentration nRs : Rc ) 3 nm, nm ) 0.5 mM, Z ) 40, nRp ) 10 mM, p ) 1, v ) 0.2 nm3, a ) 1 nm, and φ(Rm) ) 0. (a) Polymer segment concentration φ2(r)/nRp . (b) Microion concentrations n((r)/mM. (c) Electric field E(r)/E(Rm) (the inset shows a magnified version of the same profile). Solid lines, nRs ) 100 mM; dotted lines, nRs ) 200 mM; dashed lines, nRs ) 300 mM; dot-dashed lines, nRs ) 500 mM.

Figure 3a shows the polymer segment concentration profiles when the fraction of charged monomers p is varied. As p decreases, the adsorbed polymer layer broadens. The number of adsorbed segments increases: Γpol ) 29.5, 35.7, and 42.2 for p ) 1.0, 0.75, and 0.5. However, the total charge carried by the polymer, pΓpol, decreases slightly, indicating that the polymer becomes less effective at neutralizing the micelle charge. This results in greater microion localization and an increase in Ωions, which causes the grand potential Ω and thus the CAC to increase. Figure 3b shows the effect on φ2(r) of reducing the polymer intrinsic flexibility a, while keeping the linear charge density p/a fixed. This greatly reduces polymer adsorption (Γpol ) 29.5, 16.7, and 10.4 for a ) 1, 1.5, and 2 nm), and there is also a decrease in pΓpol. Again, increased counterion localization causes Ωions to increase, which leads to an increase in the CAC. These trends are in agreement with experimental and simulation results.4,5,56 Changing the polymer segment concentration in the reservoir, nRp , between 10 and 100 mM has little effect on the adsorption characteristics. This observation agrees with experimental findings that the polymer concentration has only a minor effect on the CAC, suggesting a physical (56) Hansson, P.; Almgren, M. J. Phys. Chem. 1996, 100, 9038.

picture in which micelles are attached to polymer chains such that only a relatively small proportion of the segments of any one chain are in contact with a micelle. Such a picture is supported, for example, by the observation that micellization in a polyelectrolyte gel is similar to that in a dilute polyelectrolyte solution.14 Interesting effects are obtained on varying the effective excluded volume parameter v in our model, which describes the short-ranged interactions between the chains and may be positive or negative. As shown in Figure 4, when v becomes large and negative, there is a dramatic increase in polymer adsorption. The total charge carried by the adsorbed polymer becomes much greater than Z, and the electric field and microion profiles are strongly affected. The polymer component of the grand potential, Ωpol, varies nonmonotonically, with a maximum at v ≈ -1 nm3, due to the competing effects of the short-ranged interactions and the chain connectivity. Ωions decreases with v, as does the total Ω. The observation of strongly enhanced adsorption for large negative values of v agrees well with previous work by Chaˆtellier and Joanny.38 Using a linearized version of the same theory to study adsorption on a planar surface, these authors predicted a wetting transition with decreasing v, at v ) -p2/(2nRs + pnRp ); that is, v ≈ -8 nm3 for our parameter set. 3.4. Polymer Boundary Condition. In this section, we investigate the effect of replacing the polymer boundary condition at the micelle surface φ(Rm) ) 0 by the alternative boundary condition dφ/dr|r)Rm ) 0, which is the limit of the general condition dφ/dr|r)Rm ) D-1φ(Rm) when the polymer is not adsorbed (D-1 ) 0). The method used for the numerical minimization of eq 2 in this case is described in Appendix B. The polymer segment concentration φ2(r), microion concentrations n((r), and electric field E(r) are shown in Figure 5 for both polymer boundary conditions. The grand potential values and adsorption of the various system

2004


Figure 4. Effect of changing polymer effective excluded volume v: Rm ) 3 nm, nm ) 0.5 mM, nRs ) 100 mM, Z ) 40, nRp ) 10 mM, a ) 1 nm, p ) 1, and φ(Rm) ) 0. (a) Polymer segment concentration φ2(r)/nRp . (b) Microion concentrations n((r)/nR(. (c) Electric field E(r)/E(Rm) (the inset shows a magnified version of the same profile). Solid lines, v ) -2.5 nm3; squares, v ) -2 nm3; circles, v ) -1 nm3; dotted lines, v ) 0; dashed lines, v ) 1 nm3; dot-dashed lines, v ) 2 nm3.

components are given in Table 1. It is clear that the choice of polymer boundary condition is very significant. The polymer adsorption is much higher for the alternative boundary condition, with φ2(r) having a very different form. The screening of the electric field is much more effective (E(r) decays more sharply), and consequently the electrostatic energy Ωel is much lower. In addition, due to the strong polymer adsorption, very little counterion localization occurs and Ωions is greatly reduced. These results suggest that the nature of the short-ranged interactions between the polymer chain and the surfactant headgroups may be a very important factor in influencing the behavior of such oppositely charged systems. 3.5. Effect of Changing Micelle Concentration. As discussed in section 2.2, changing the micelle concentration nm has a nontrivial effect on Ω and on the concentration profiles, through the conditions on φ(r) and E(r) at the cell boundary, r ) Rc. Figure 6 shows the polymer concentration φ2(r) and the electric field E(r) for nm varying between 1.5 and 6.5 mM. The theory becomes inappropriate for Rc - Rm < v1/3 when the space available for the polymer becomes smaller than the approximate size of a monomer. This occurs for nm > 8.5 mM for our parameter set.

Allen and Warren

Figure 5. Effect of changing polymer boundary condition: Rm ) 3 nm, nm ) 0.5 mM, nRs ) 100 mM, Z ) 40, nRp ) 10 mM, a ) 1 nm, p ) 1, and v ) 0.2 nm3. (a) Polymer segment concentration φ2(r)/nRp . (b) Microion concentrations n((r)/nR(. (c) Electric field E(r)/E(Rm) (the inset shows a magnified version of the same profile). Solid lines, φ(Rm) ) 0; dotted lines, dφ/ dr|r)Rm ) 0.

The structure of the adsorbed polymer undergoes a qualitative change as the cell radius decreases. Instead of forming a layer around the micelle which decays toward the reservoir value nRp near the edge of the cell, the polymer starts to form a “bridging” layer with the maximum concentration at the edge of the cell. The electric field profile E(r) is also affected, becoming more linear as nm increases. Similar behavior has been obtained using the same theory for polymers between two planar surfaces at fixed electrical potential.44 Podgornik et al. have carried out a Monte Carlo simulation study of the force between two charged spherical colloids in the presence of an oppositely charged polymer.57 They observe long-ranged attraction between the colloids, which corresponds to bridging configurations in which the polymer is complexed to both colloids. The force is mainly due to the stretching of harmonic bonds between the monomers of the bridging chain. As the intercolloidal distance increases, there is an abrupt change in the polymer configuration so that just one colloid is complexed and the force changes suddenly to a weak electrostatic attraction, due to the imbalance of charge (57) Podgornik, R.; Åkesson, T.; Jo¨nsson, B. J. Chem. Phys. 1995, 102, 9423.


Langmuir, Vol. 20, No. 5, 2004 2005

components. This reflects the fact that the electric field is more effectively screened, as can be seen in Figure 6b, and that some microions are expelled from the cell. For large values of nm, Ω shows a strong increase, due to the polymer component Ωpol. This reflects the “squeezing” of the polymer layer that can be seen in Figure 6 for very large nm, leading to increases in both contributions to Ωpol. The variation of Ω with nm leads to a significant phase instability, to which we turn in the next section. 4. Phase Behavior We have already described in the Introduction the general features of associative phase separation seen in oppositely charged polymer/surfactant mixtures. In this section, we show that the cell model of section 2 predicts phase behavior which shares many of these characteristics. The semi-grand potential per micelle X/Nm of a suspension of Nm micelles at concentration nm ) Nm/V is given by

X ) log nm - 1 + Ω NmkBT Figure 6. Effect of changing micellar concentration: Rm ) 3 nm, nRs ) 100 mM, Z ) 40, nRp ) 10 mM, a ) 1 nm, p ) 1, v ) 0.2 nm3, and φ(Rm) ) 0. (a) Polymer segment concentration φ2(r)/nRp , (b) Electric field E(r)/E(Rm). Solid lines: nm ) 1.5 mM (Rc ) 6.42 nm), dotted lines: nm ) 4.5 mM (Rc ) 4.45 nm), dashed lines: nm ) 5.5 mM (Rc ) 4.16 nm), dot-dashed lines: nm ) 6.5 mM (Rc ) 3.94 nm).

(16)

Since, as we have seen, Ω is a rather nontrivial function of nm, interesting phase behavior is likely to arise from eq 16. 4.1. Spinodal Instability. We first consider possible spinodal instabilities, indicative of the phase behavior, that may be predicted by eq 16. A spinodal instability occurs when the (osmotic) compressibility κ becomes negative. The compressibility is κ-1 ) nm(∂Π/∂nm), where Π is the osmotic pressure given by Π ) nm2(∂(X/NmkBT)/ ∂nm). The partial derivatives here are at constant µs, µp, and T. From eq 16, we obtain

[

κ-1 ) nm 1 +

) ]

(

∂ ∂Ω n 2 ∂nm m ∂nm

(17)

µs,µp,T

We now define a function g ) 1 + gpol + gions + gel ) κ-1/nm, with

gpol )

Figure 7. Grand potential Ω of the cell as a function of micellar concentration nm: Rm ) 3 nm, nRs ) 100 mM, Z ) 40, nRp ) 10 mM, a ) 1 nm, p ) 1, v ) 0.2 nm3, and φ(Rm) ) 0. (a) Ω(nm). (b) Components of Ω. Dotted line, Ωpol; dashed line, Ωions; dotdashed line, Ωel.

between the two colloids. The present mean field model also predicts the existence of two different stable polymer configurations at different intercolloidal distances, but with a gradual rather than sudden change between them on varying Rc. Figure 7 shows the grand potential Ω of the cell and its components Ωpol, Ωions, and Ωel as functions of the micelle concentration nm. As nm increases, Ω first decreases, due to reductions in both the electrostatic and entropic

(

)

∂Ωpol ∂ n 2 ∂nm m ∂nm

etc.

(18)

µs,µp,T

The condition g < 0 implies spinodal instability, such that the system must be in a phase-separated region of the parameter space (nm, µs, µp). Furthermore, by examining gpol, gions, and gel individually as functions of nm, we may deduce the physical origin of the phase separation. Note that by using a semi-grand ensemble in which only the density of micelles (macroions) is controlled directly, we have an effective one-component system, which greatly simplifies the analysis. 4.2. Phase Coexistence. Two different phases can coexist in equilibrium if their osmotic pressure Π is the same, as well as the chemical potentials of all the system components.58 This may be solved using the “double tangent” construction on the function X/VkBT ) nmµm Π plotted as a function of nm at constant µ+, µ-, and µp (i.e., at fixed reservoir concentrations nRs and nRp ). If a straight line can be drawn that forms a tangent to this curve at two or more values of nm, these can coexist in equilibrium. Although the reservoir concentrations, and hence µs and µp, are the same in the coexisting phases, (58) Hansen, J.-P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: New York, 1990.

2006


Figure 8. The function g(nm) for the parameter set Rm ) 3 nm, nRs ) 200 mM, Z ) 40, nRp ) 10 mM, a ) 1 nm, p ) 1, v ) 0.2 nm3, and φ(Rm) ) 0. (a) g(nm). (b) Dotted line, gpol; dashed line, gions; dot-dashed line, gel.

they may have very different values of the average microion and polymer segment concentrations n+, n-, and np. 5. Numerical Results: Phase Behavior 5.1. Spinodal Instability. The function g, as defined in section 4.1, and its components gpol, gions, and gel are plotted as functions of the micelle concentration nm in Figure 8, for the same set of parameters: Rm ) 3 nm, Z ) 40, nRp ) 10 mM, a ) 1 nm, p ) 1, and v ) 0.2 nm3, now with nRs ) 200 mM, again using the polymer boundary condition φ(Rm) ) 0. Figure 8a shows that the system is unstable (g < 0) when nm is in the range 1.8 mM < nm < 3.4 mM. The existence of such a region of spinodal instability points to a tendency to form two phases having high and low micelle concentrations. Figure 8b shows the functions gpol, gions, and gel. The observed spinodal instability is due mainly to the entropic component gions. The electrostatic part gel tends toward instability for larger nm values, but this is counteracted by the steep rise in the polymer component gpol. Thus the polymer connectivity and excluded volume contributions are mainly responsible for the stability of the dense phase, although the entropic component also rises for very large nm values. Figure 8 shows that the entropic mechanism (microion release on binding of the polymer to the micelle) is sufficient to account for spinodal instability and phase separation and that it is indeed the entropic component of the free energy which is mainly responsible for the instability in this model. Repulsive forces between micelles provided by compression of the adsorbed polymer layers are responsible for the stability of the dense phase. We find that when the alternative polymer boundary condition dφ/dr|r)Rm ) 0 is used, this repulsion is much reduced, especially for negative or small positive values of the effective excluded volume parameter v. In this case, the model tends to predict a very high concentration of micelles in the dense phase: the micelle-micelle excluded volume might need to be included in the free energy expression to stabilize the dense phase if using such a boundary condition.

Allen and Warren

Figure 9. (a) Binodal (solid line) and spinodal (dotted line) curves in the (nRs , nm) plane. (b) Replotted curves showing the average salt concentration ns instead of nRs . The appended numbers show the average polymer segment concentration np in the systems at the ends of the tieline.

5.2. Phase Diagram. Although our system contains four solution species (positive and negative microions, polymer segments, and micelles), the requirement for electroneutrality means that the concentrations of only three of these can be varied independently. We formulate phase diagrams as a function of three concentration variables: nm, the micelle concentration, nRp , the concentration in the reservoir of polymer segments, and nRs , the reservoir salt concentration. Since experimental measurements are generally not carried out in the presence of such a reservoir, the phase diagrams are also replotted as functions of the average concentrations in the cell, np and ns, as defined in eqs 14 and 15. Figure 9a shows the binodal (with tielines) and spinodal curves in the nm, nRs plane, when nRp is fixed at 10 mM. The phase separation is suppressed on increasing the reservoir salt concentration and disappears for nRs larger than a value of about 250 mM. Thus we recover a CEC in our model, with an order of magnitude that is in good agreement with experiment.15,17,19 In Figure 9b, the data are replotted as a function of the average salt concentration ns. Because the calculations were performed at fixed reservoir rather than average polymer segment concentration, Figure 9b is not a true slice through an “experimental” (nm, ns, np) phase diagram. However, np values are shown next to some of the tielines. As expected, the average polymer segment concentration is higher in the dense phase. The average salt concentration is lower in the dense than in the dilute phase, which is consistent with the idea that polymer binding allows microions to be expelled from the dense phase. The salt concentration in the dense phase is not, however, reduced to very low or negative values, and likewise the polymer concentration in the dilute phase is far from negligible. This reflects the fact that the parameter set used for the calculations gives only fairly weak complexation between the polymer and micelle (in particular, the polymer is rather stiff). We would expect


Langmuir, Vol. 20, No. 5, 2004 2007

6. Discussion

Figure 10. (a) Binodal (solid line) and spinodal (dotted line) curves in the (nRp , nm) plane. (b) Replotted curves showing the average polymer segment concentration np instead of nRp . The dashed line shows the charge stoichiometry line: Znm ) pφ2.

the partitioning of salt into the dilute phase and polymer into the dense phase to be enhanced for systems with stronger complexation. Figure 10a shows the binodal and spinodal curves in the nRp , nm plane, when the reservoir salt concentration nRs is fixed at 200 mM. The phase separation is abolished for large values of nRp and nm, in agreement with experimental observations. Experimentally, the phase separation also disappears for very low polymer concentrations, since of course in that limit the system is simply a solution of micelles plus electrolyte. However, because the polymers in our model are assumed to be infinitely long, the polyelectrolyte mean field theory does not reduce to Poisson-Boltzmann theory in the limit of very small nRp , but only when the polymer concentration in the cell, φ2(r), is set to zero. If the finite length of the polymers were to be taken into account in the theory then, at some point as nRp f 0, the polymers would dissociate from the micelles. There would then be no polymer in the cell. There is no phase separation in such a Poisson-Boltzmann cell model,39 so that the phase boundary in Figure 10a would be expected to close up again as nRp f 0, in effect forming a closed-loop miscibility gap. The data are replotted in Figure 10b as a function of the average polymer segment concentration np. This shows that the dense phase contains more concentrated polymer than the dilute phase, but this effect is rather small. The “charge stoichiometry line”, where Znm ) pφ2, is marked in Figure 10b. It is generally observed in experimental systems that phase separation is confined to the vicinity of this line, but this effect is not particularly marked in our model. This is because the complexation of our polymer is rather weak, so that some counterions are always required in the dense phase to aid the polymer in neutralizing the charge of the micelle. For systems complexing more strongly, phase separation closer to the charge stoichiometry line is expected.

In this paper, we have described a cell model for oppositely charged polyelectrolyte/macroion mixtures. We have applied this model to polymer/surfactant systems for which there is a large experimental literature. This leads to predictions for trends in the CAC (for the formation of polymer-associated micelles) in reasonable agreement with the experimental results. The main new observation though is that such a cell model predicts a region of phase separation between low and high density macroion phases which shares many of the features seen experimentally for associative phase separation in polymer/surfactant mixtures. Since the cell model computes a solution free energy per macroion (micelle), we can succinctly attribute the phase instability to a nontrivial concentration dependence of the free energy of micellization. Our cell model naturally suffers from a number of drawbacks, which we now enumerate. First, it is in the nature of all cell models that they neglect macroion fluctuations and correlations. This is a potentially important omission which is often overlooked; for instance, the macroion structure factor in a cell model cannot show the divergence at long wavelengths that one would expect near a critical point. Thus it is likely that the cell model underestimates the extent of phase separation. Second, we use a number of mean field approximations. As in Poisson-Boltzmann theory, we neglect all correlations between microions. More crucially, we also neglect correlations between microions and the polymer backbone. This means the model as currently formulated does not include counterion-polymer condensation phenomena and is restricted to weakly charged polyelectrolytes. Since counterion condensation and in particular counterion release are recognized as important effects for many polyelectrolytes, this limits the wider applicability of our model. Some of the discrepancies between our model and experiment (such as the effect of salt on the CAC) could be attributed to this simplification. Third, we use a Gaussian model for the polymer chain which neglects electrostatic chain stiffening effects and restricts our calculations to relatively small electric fields, and hence relatively weakly charged macroions (micelles). Despite these drawbacks, we believe the cell model demonstrates the basic mechanism driving associative phase separation in these oppositely charged mixtures. Let us discuss briefly the dense phase, which in the present theory is a dense micellar suspension. This is stabilized, in part at least, by excluded volume effects arising either from polymer-polymer interactions or from macroion-macroion interactions. The short-range polymer-macroion interaction also has a very significant effect on the distribution of polymer and strongly influences the stability of the dense phase. It is known experimentally that the dense phase in associative polymer/surfactant phase separation is more like a polymer-stabilized surfactant mesophase (e.g., a hexagonal phase). Such a phase could be accommodated within a cell model rather easily by changing the shape of the cell. Difficulties arise, however, in obtaining a free energy expression for the change in the micellar environment (e.g., from spheres to cylinders). Without a robust theory for this aspect (in other words for the underlying micelle-mesophase transition), one cannot make detailed calculations. We now compare our calculations with the recent work by Svensson et al.34 These authors qualitatively reproduce the experimental trends for phase separation in their system by Monte Carlo simulations of the force between macroion surfaces, separated by a polyelectrolyte solution.

2008


Phase separation is attributed to an attractive force acting between macroion surfaces, where the dominant contribution arises from polymer bridging. This is apparently in contrast to our conclusions; for instance, Figure 8b in our work can be contrasted with Figure 10b of ref 34. Nevertheless, the cell model also predicts phase behavior which is in reasonable accord with experimental observations. The essential difference between the two approaches is that Svensson et al. calculate an accurate pair interaction between macroion surfaces, whereas our cell model free energy is an approximation to the full many-body interaction, which includes contributions beyond the pair interaction level. It seems likely that aspects of both mechanisms may operate, but a detailed analysis must be left to future work. We note though that Monte Carlo calculation of the exact free energy in a cell model is in principle possible. The major difficulty that must be addressed is either fixing or measuring the chemical potentials of the polyelectrolyte and salt components. This would be necessary to make definitive statements about the phase behavior. Finally, we place our calculations in a wider context. It has been apparent for a long time that ion correlation effects in electrolyte solutions generate a tendency for phase separation.59,60 The archetype for this is the much studied restricted primitive model (RPM) for simple electrolytes.61 This shows phase separation between dense and dilute ionic solutions, driven essentially by the electrostatic correlation free energy but limited by excluded volume effects. A related problem is the phase transition in simple macroion suspensions. It has been demonstrated that such a system shows phase coexistence between dense and dilute suspensions for sufficiently strong electrostatic coupling.62 Whether the transition can be seen for colloidal suspensions in an aqueous 1:1 electrolyte solution has been the subject of great debate, both experimentally and theoretically. It is not our intention to enter this debate here, as it has been reviewed extensively;60,63-65 rather, we wish only to point out a connection between our calculation and this problem. We use a Poisson-Boltzmann cell model for a macroion suspension, extended to include a polyelectrolyte component. We find, for some parameters, quite extensive phase separation between macroion-rich and macroionpoor phases. It has been determined though that the corresponding cell model for a macroion suspension in the presence of simple electrolyte contains no phase transition.39 Thus, adding an oppositely charged polyelectrolyte to the supporting 1:1 electrolyte in a chargestabilized suspension dramatically amplifies an existing tendency for phase separation into an easily observable effect. However, our calculations also indicate the importance of short-range interactions and excluded volume effects in determining the nature of the dense phase and the overall extent of phase separation. Acknowledgment. The authors thank J.-F. Dufreˆche, A. S. Ferrante, J.-P. Hansen, J.-F. Joanny, A. A. Louis, and T. O. White for their help with this work. R.A. is grateful to EPSRC and to Unilever for funding. (59) Langmuir, I. J. Chem. Phys. 1938, 6, 873. (60) Levin, Y. Rep. Prog. Phys. 2002, 65, 1577. (61) Fisher, M. E. J. Stat. Phys. 1994, 75, 1. (62) Resˇcˇicˇ, J.; Linse, P. J. Chem. Phys. 2001, 114, 10131. (63) Arora, A. K.; Tata, B. V. R. Adv. Colloid Interface Sci. 1998, 78, 49. (64) Belloni, L. J. Phys.: Condens. Matter 2000, 12, R549. (65) Hansen, J.-P.; Lo¨wen, H. Annu. Rev. Phys. Chem. 2000, 51, 209.

Allen and Warren

Appendix A. Critical Aggregation Concentration in the Cell Model In this appendix, we derive eq 12 in the main text directly from our cell model free energy eq 2. To do this, we imagine a system, described by the cell model, which can contain micelles of concentration nm, free surfactant molecules of concentration nsf, polymer segments, and both positive and negative counterions. The system is in equilibrium with a reservoir containing microions and polymer segments, at chemical potentials µ( and µp. As in section 2, each cell is electrically neutral, contains one micelle at its center, and has radius Rc given by 4πRc3/3 ) 1/nm. The system is grand canonical with respect to the polymer segments and microions but canonical with respect to the surfactant molecules and the micelles. The semi-grand canonical potential of this system X(nm, nsf, µp, µ() can be written as

X ) nm(log nm - 1) + nsf(log nsf - 1) + nmΩ(Rc,nsf,µ(,µp) + nsffsf (19) The free energy difference between a surfactant molecule that is free and one that is inside a micelle is given by fsf. Several effects make contributions to fsf, such as loss of hydration of the surfactant tails on entering the micelle, and these will not be evaluated in detail. Also, contributions associated with the definition of standard states are absorbed into this term.40,66 Here Ω(Rc, nsf, µ(, µp) is the nonideal contribution to the free energy from the cell. We now imagine that Nagg free surfactant molecules are able to convert into a single micelle. To find the equilibrium concentrations of free surfactants and micelles, we must minimize X with respect to nsf and to nm, subject to the constraint that the total density of surfactants ntot ) Naggnm + nsf is conserved. Introducing a Lagrange multiplier λ (which turns out to be equal to the surfactant chemical potential) and writing Y ) X - λ(Naggnm + nsf - ntot), the minimization is found by solving ∂Y/∂nm ) ∂Y/∂nsf ) 0. Performing these calculations leads to the relation

[

∂Ω ∂Ω agg ) nm exp Ω - Naggfsf + nm - Naggnm nN sf ∂nm ∂nsf

]

(20)

We now define the concentration of surfactants at the CAC to be that at which half of the surfactants are located in micelles: nsf ) Naggnm ) ntot/2. Substituting this into eq 20, we obtain agg-1 ) nN sf

[

1 × Nagg

exp Ω - Naggfsf +

( )

nsf ∂Ω Nagg ∂nm

CAC

- nsf

( ) ] ∂Ω ∂nsf

(21)

CAC

The third term in the exponential of eq 21 is negligible, since nsf is small at the CAC and so the cell radius is large enough to be in the regime where Ω is independent of nm (no interactions between micelles). The fourth term is also assumed to be negligible, since the concentration of the free surfactants and their counterions inside the cell is much smaller than that of the salt. For the same reason, (66) Smith, E. B. Basic chemical thermodynamics, 4th ed.; Oxford University Press: Oxford, 1990.


Langmuir, Vol. 20, No. 5, 2004 2009

we can identify the value of Ω(Rc, nsf, µ(, µp) in eq 21, with that of Ω(Rc, µ(, µp), defined in eq 2. This leads to

[

nsf ) (Nagg)1/(1-Nagg) exp

]

Ω - Naggfsf Nagg - 1

[

Ω - fsf Nagg

]

(23)

The choice of units for nsf is bound up with the definition of fsf, but if we restrict ourselves to the case where the micelles are unchanged, so that fsf and Nagg must be the same, eq 23 allows us to calculate the relative magnitudes of the CAC under different conditions. Application of eq 23 under these conditions leads directly to eq 12 in the main text. Appendix B. Numerical Minimization of Free Energy The numerical minimization of the grand potential of the cell, eq 2, with respect to the polymer segment order parameter profile φ(r) and the radial electric field E(r) (the microion profiles n((r) being obtained from eq 8) is done using an expansion in spherical Bessel functions j1(x), when the boundary conditions are as in eq 10 and eq 11, with φ(Rm) ) 0: N

φ(r) ) (

x ∑c nRp )

((

n j1

n)1

β1n

r - Rm

))

Rc - Rm

r - Rm

)

Rc - R m

( (

N

+

r - Rm

∑ dn j1 R1n R n)1

c

))]

- Rm

(25)

(22)

In the limit Nagg . 1, the prefactor in this tends to unity and we can write Nagg - 1 ≈ Nagg. We conclude that the concentration of free surfactants at the CAC is given by

nsf ) exp

[ (

E(r) ) E(Rm) 1 -

Here, Rmn is the nth root of jm(x) and βmn is the nth root of the derivative j ′m(x). The expansion is terminated after N terms. For the calculations reported here, we find N ) 25 to be sufficient. The grand potential eq 2 was minimized with respect to variations in the expansion coefficients {cn} and {dn} using Powell’s method.67 Spherical Bessel functions j1(x) were chosen as a suitable basis because they allow the expansions eq 24 and eq 25 to satisfy the required boundary conditions automatically. Alternatively, the grand potential eq 2 can be minimized using a radial grid, on which the functions log(φ2(r)/nRp ) and E(r)/E(Rm) are discretized. We find this to be less convenient than the expansion method since the number of variables for the minimization is generally larger, the derivatives required in the calculation of the grand potential must be found numerically, and the radial grid must be re-evaluated if the cell radius is changed, so as to keep the grid spacing the same. The two methods were found to produce identical results. For calculations using the alternative boundary condition, dφ/dr|r)Rm ) 0, we continued to use eq 25, but the expansion of the polymer profile eq 24 was replaced by

φ(r) ) (

((

N R np ) cn j0 n)1

x ∑

β0n

r - Rm

))

Rc - Rm

(26)

LA0356553

(24)

(67) Press, W. H.; Flannery, B. P.; Teucholsky, S. A.; Vetterling, W. T. Numerical Recipes; Cambridge University Press: Cambridge, 1989.

Complexation and Phase Behavior of Oppositely Charged

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