Surfactant Self-Assembly in Oppositely Charged Polymer Networks

J. Phys. Chem. B 2009, 113, 12903–12915

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Surfactant Self-Assembly in Oppositely Charged Polymer Networks. Theory Per Hansson* Department of Pharmacy, Biomedical Centre, Uppsala UniVersity, Box 580, SE-75123 Uppsala, Sweden ReceiVed: May 25, 2009; ReVised Manuscript ReceiVed: August 6, 2009

The interaction of ionic surfactants with polyion networks of opposite charge in an aqueous environment is analyzed theoretically by applying a recent theory of surfactant ion-polyion complex salts (J. Colloid. Int. Sci. 2009, 332, 183). The theory takes into account attractive and repulsive polyion-mediated interactions between the micelles, the deformation of the polymer network, the mixing of micelles, polyion chains, and simple ions with water, and the hydrophobic free energy at the micelle surface. The theory is used to calculate binding isotherms, swelling isotherms, surfactant aggregation numbers, compositions of complexes,and phase structure under various conditions. Factors controlling the gel volume transition and conditions for core/shell phase coexistence are investigated in detail, as well as the influence of salt. In particular, the interplay between electrostatic and elastic interactions is highlighted. Results from theory are compared with experimental data reported in the literature. The agreement is found to be semiquantitative or qualitative. The theory explains both the discrete volume transition observed in systems where the surfactant is in excess over the polyion and the core/shell phase coexistence in systems where the polyion is in excess. 1. Introduction Surfactant self-assembly in aqueous solution is driven by the hydrophobic effect.1 Ionic surfactants have in general higher cmc values than nonionic surfactants due to an entropic penalty of binding counterions to the micelle. Counterion binding is necessary to reduce the electrostatic repulsion between the surfactant head groups. A flexible polyion chain can accomplish the same effect by folding around the micelle.2-5 The association is strongly favored by the release of counterions bound to the polyion, explaining why the critical association concentration (cac) is often 1 or 2 orders of magnitude smaller than the cmc.2,6 Another important effect of polyion binding is to reduce the long-range electric double layer repulsion between micelles2 in favor of attractive interactions, in particular polyion-mediated forces, such as bridging.7 As a consequence, polyelectrolytesurfactant mixtures tend to phase separate at surfactant concentrations just above the cac.8,9 Recently, several aspects pertaining to the stability and structure of the dense complex phases formed have been clarified by systematic studies of phase diagrams7,10-13 and theoretical modeling.14 The interaction of surfactants with oppositely charged crosslinked gels is similar in many respects to the interaction with linear polyions, as pointed out in a recent review article.15 The associative phase separation in solutions of linear polyion has its counterpart in the gel Volume transition in which a swollen gel collapses at a critical surfactant concentration in the solution,16-20 here referred to as the critical collapse concentration (ccc). The ccc has been found to nearly coincide with the cac in solutions of linear polyions when determined at the same salt concentration and the microstructure of the cross-linked complexes in the collapsed state is similar to the linear counterparts.21-37 Dense complexes have been observed to form also when the polyion network is in excess over the surfactant in the system. In this case interesting core/shell structures are formed with the complexes making up a surface phase enclosing a swollen complex-free part of the gel network. Similar * Corresponding author. E-mail: [email protected].

structures have been observed also when polyions,38,39 proteins,40-45 and polypeptides46,47 bind to gels. Surfactant-rich shells were first described by Starodubtsev48,49 and the group of Zezin and Kabanov50 and later studied in detail by Hansson and co-workers.16,18,24-26,51-54 Core/shell structures appear as intermediates also during the volume transition in excess of surfactant16,18,52,53 and during temperature-triggered transitions in binary network-solvent systems.55 Theoretical investigations show that a large elastic energy barrier prevents a collapsed phase from nucleating in the bulk of a swollen macroscopic gel but not at the gel surface.56-58 However, there is an elastic energy phase coexistence cost involved since the network in the surface phase (shell) becomes anisotropically deformed. In the present paper, we investigate the phase behavior in surfactant-polyelectrolyte gel systems by means of a recent model of surfactant ion-polyion complex salts.14 The problem was addressed recently by Khokhlov and co-workers but without taking into account the elastic coupling between the coexisting gel phases.59,60 Furthermore, the micelles were not modeled explicitly, and their contribution to the osmotic pressure was neglected, as were also coulomb interactions, which means that swelling is only counteracted by the network elastic forces. They demonstrate in an elegant way that a collapsed gel state containing aggregated surfactant is stable at surfactant concentrations well below the cmc, and when there is not enough surfactant available for the whole gel to reach this state, the collapse structure may form in one part of the network in equilibrium with the swollen micelle-free part. However, their treatment is limited to weakly charged networks. Phase coexistence in highly charged networks is expected to require stronger “cohesive” forces than the network elastic energy can provide. The argument is the following: A higher charge density in the swollen phase requires a higher concentration of polyion counterions in the complex phase for the chemical potential of the network (polyion + counterions) to be the same in both phases and therefore leads to a higher swelling pressure. Thus, for highly charged networks, typical of gels displaying core/shell coexistence in the literature, attractive polyion-mediated forces between the micelles are expected to be of major importance.

10.1021/jp904866t CCC: $40.75  2009 American Chemical Society Published on Web 09/03/2009

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J. Phys. Chem. B, Vol. 113, No. 39, 2009

Figure 1. (a) Schematic of the transition from swollen micelle-free to fully collapsed gel state via a biphasic core/shell state (middle) observed when charged polymer networks interact with oppositely charge surfactants in aqueous systems. (b) Schematic of a section of a gel through the core/shell boundary defining the deformation ratio in the core, R, and the deformation ratios in the directions parallel to the interface in the shell, Rx and Ry. Vcore and Vshell are the volume per mole of polyion charges in the core and in the shell, respectively.

The present analysis61 is more detailed than the earlier ones including both repulsive and attractive electric double layer forces between the micelles, the contact energy between hydrocarbon and water at the micelle surface, excluded volume interactions between the micelles, and elastic network forces. The description takes into account that the collapsed phase forms a shell outside the core, and the explicit modeling of the micelles allows us to calculate the surfactant aggregation numbers and the phase structure (disordered/cubic). In a previous paper, the model was used to calculate phase diagrams of binary mixtures of linear polyion-surfactant ion complex salts and water with good agreement with experiments.14 Here we show that it reproduces many of the features observed when surfactant interacts with oppositely charged networks. 2. Theory 2.1. General. The system can be produced by adding a polyelectrolyte network M+P- to an aqueous solution of a surfactant S+X- and a simple salt M+X-. The surfactant ion S+ and a charged segment P- of a polyion chain can combine to form a complex salt S+P-. The system contains four components since, owing to the condition of electroneutrality, a combination of three salts and water is sufficient for a full description of any given composition of the system. Thus, at a given temperature and pressure, the state of the system is defined when three compositional variables are specified. In the model, there is no difference between S+ dissolved as unimers and M+ with respect to electrostatic interactions. Collectively, they will be referred to as G+. The gel states to be considered are shown in Figure 1a: swollen (monophasic; micelle-free), core/shell (biphasic), and collapsed (monophasic). In the core/shell state the shell contains

Hansson a fraction β′ of the polymer network together with surfactant micelles and an aqueous electrolyte solution. The core contains the remaining fraction of the network and an aqueous electrolyte solution but no micelles. The core and shell are assumed to be homogeneous and separated by a sharp boundary and treated as separate phases. The fact that the gel network is a single elastic body is no hindrance for the establishment of phase equilibrium between the two phases; all components, including those containing P-, can be transferred across the phase boundary. However, the states of the network in the two phases are related since the deformation of the network in directions parallel to the interface (x,y) must be continuous across the interface62 (Figure 1b). It follows that the composition of one phase depends on the volume of the other, in contrast to the case of fluid-fluid equilibrium. The state of the network can be characterized by the deformation ratios Rx, Ry, and Rz, defined as the length of the network in one direction relative to the length in a relaxed reference state. The continuity requirement can be core ) Rshell ) Rshell written: Rcore x x , Ry y . For spherically symmetric gels (the case considered here), the deformation of the network of a homogeneous core must be isotropic,62 i.e., it can be described by a single deformation ratio, R () Rx ) Ry ) Rz). The shell network is restricted to have the deformation ratios Rx ) Ry ) R at the core boundary but is allowed to swell freely in the radial (z-) direction. Rx and Ry are always equal but may change with the radial coordinate. However, for a gel in a given state, they will be assumed to be constant throughout the shell. This will simplify calculations of deformational free energy, as the problem reduces to that of a uniform deformation of a network with flat sides (Figure 1b). The approximation is excellent when the thickness of the shell is small. With R as an independent state variable in addition to temperature, pressure, and the three compositional variables, thermodynamic quantities can be derived independently for the core and the shell. Thus, the presence of the shell phase imposes a boundary condition that affects the chemical potentials, and hence the equilibrium composition, of the core phase (and vice versa). 2.2. Shell. The interactions in the shell are described using the model for complex salts derived earlier,14 extended to anisotropic deformation of the polymer network and nonstoichiometric S+/P- mixtures. The free energy is calculated assuming a uniform distribution of monodisperse spherical micelles of radius Rmic ) 3Vs/a and aggregation number N ) 36πVs2/a3, representing the optimal value for each composition. Here, a is the area per headgroup at the micelle surface, and Vs is the volume per hydrocarbon tail. The volume fraction of mic micelles is φ ) Nmic S+ Vs/V, where NS+ is the number of surfactant molecules in micelles and V is the volume of the phase. The free energy is given by

Gshell )

shell shell + Gshell ∑ niµi0 + Eelshell - TSmix def + Gsurf i

(1) where niµ0i is the standard contribution of the ith component. The other contributions will be described in turn below. The van der Waals interaction between micelles is neglected since it is comparatively small.7 Electrostatic Energy. Eelshell is calculated using a spherical capacitor model in which the inner charged surface represents the micelle surface and the outer surface the diffuse layer of counterions (P- and X-). The separation of the surfaces, regulated by a model parameter k, defines an effective double

Surfactant Self-Assembly in Polymer Networks

J. Phys. Chem. B, Vol. 113, No. 39, 2009 12905

layer thickness increasing with decreasing volume fraction of micelles. The contribution to the chemical potential of a surfactant ion in the micelle is14

µSm,el +

e2Vs

(

)

3 + kφ1/3(2 + φ - 3φ1/3) + φ4/3 - 4φ1/3 ) 2ε0εra2 (1 + kφ1/3)2 (2)

where e is the electronic charge and ε0εr is the permittivity of water. We use the superscript m to distinguish the expression from that for surfactant monomers in the aqueous regions in the shell. However, micelles are not allowed to form in the core. The corresponding contribution to the water chemical potential is

µwshell,el

)

Vwe2 2ε0εra2

·

(k + 1)φ4/3 (1 + kφ1/3)2

(3)

where Vw is the molecular volume of water. The contribution to the polyion chemical potential is obtained by replacing Vw in eq 3 by Vseg, the volume per segment in the polymer chain, each containing one unit charge. For polyions lacking bulky groups, the partial volume is close to zero in excess of water, and so this contribution can be neglected. (In this respect, the polyion is treated as simple counterions are treated in the PB-cell from the model.63) However, there is a contribution to µPshell,el interaction between charges along the chain, i.e., an energy that does not vanish even at infinite separation between polyion and micelles. It is assumed to have a constant value and therefore affects only the partitioning of the polyion between core and shell. shell has contributions from the Free Energy of Mixing. TSmix mixing of micelles, polyion, and monovalent ions with water. The excluded volume interaction between micelles is described using a hard sphere volume fraction, φHS. The contribution to the chemical potentials of water and surfactant ion in micelles in a disordered liquid phase is given by the Carnahan-Starling expressions64,65

µwmix,mic(disordered) -

µSmix,mic (cubic) +

36πVs3

µSmix,mic (disordered) ) +

kBTa3 36πVs2

·

2 3 1 + φHS + φHS - φHS

(1 - φHS)

3

(( ) ln

Vwa3φ

36πVs3

2 3 - φHS 7φHS - 3φHS

(1 - φHS)2

)

(5)

where kB is Boltzmann’s constant and T is the absolute temperature. The corresponding expressions for a micellar cubic liquid crystalline phase can be written14

µwmix,mic(cubic)

)-

VwkBTa3φZ 36πVs3

(6)

(( ) ln

36πVs3

+

∫φ′φ

HS

Z-1 dφ + φ

)

(7)

where c′ is an integration constant related to the lower integration limit φ′, as described in more detail elsewhere.14 For the compressibility factor, Z, we use Hall’s expression for an fcc lattice.66 The reduction of the thermodynamic free volume due to repulsive electric double layer forces is taken into account, approximately, by the following expression

φHS )

(

)

k + φ-1/3 3 φ k+1

(8)

Equation 8 considers a micelle to have an effective hard sphere radius equal to the micelle radius plus the effective double layer thickness (modeled by k).14 Mixing of polyion chains and water is described using the Flory-Huggins (FH) theory excluding the contribution from the translational entropy of the polymer chains since they are fixed in the network67

µPshell,FH )-

kBTVseg φ 1Vw 1-φ

((

µwshell,FH ) kBT ln 1 -

(

)

φ φ 1-φ 1-φ

)

(9)

)

(10)

where φ is the polymer volume fraction in the shell, given by + V ), where Nshell is the number of polymer φ ) NpshellVsegφ/(NSmic s p segments in the shell. φ/(1 - φ) is the polymer volume fraction in the aqueous subdomain. To extend the theory to nonstoichiometric complexes, we add the entropy of mixing simple ions in the aqueous regions. For a uniform distribution, we have

(4)

+

36πVs2

Vwa3φ

(1 - φHS)Z + c'

)

VwkBTa3φ

)

kBTa3

µishell ) kBT ln Cishell

(11)

shell µwshell,ion ) -RTVw(CGshell + + CX- )

(12)

is the concentration of the ith ionic species in the where Cshell i aqueous regions of the shell. In Supporting Information 1, results of the extended theory are compared with Monte Carlo simulations68 on a model network interacting with charged spheres in the presence of small ions. Elastic Free Energy. According to the classical theory, the free energy associated with a general deformation (Rx, Ry, Rz) of a network containing Np/p chains is67

Gdef )

NpkBT (ln RxRyRz - R2x - R2y - Rz2 + 3) 2p

(13) where p is an empirical constant related to the average length of chains between cross-links. For a uniform two-dimensional extension (R ) Rx ) Ry), R2Rz ) φref/φ, where φref is the volume

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Hansson

fraction of polymer in the relaxed reference state of the network. Insertion into eq 13 gives

Gshell def )

( ) )

(

Nshell p kBT φ 1 φref ln ref + 2R2 + 4 2p φ R φ

2

-3

(14)

The contribution to the chemical potentials of water, surfactant, and polymer becomes, respectively

µSm,def +

µPshell,def ) -

( )

dGshell def ) dNS+

Nj,R,T

(

( ))

kBTφVs 2 φref )1- 4 2pVseg R φ

(

kBT φ ln ref + 2R2 - 2 - φ + 2p φ 1 φref 2 (2φ - 1) R4 φ

)

( )

µwshell,def ) -

(

( ))

kBTφVw 2 φref 1- 4 2pVseg R φ

2

(15)

∑ Cicore(L)

(23)

The osmotic pressure is related to the water chemical potential by the definition

µw0 - µwcore ) Vw

(24)

(16)

2

(17)

The chemical potential of the polyion in the core will have contributions from the electrostatic free energy and from the deformation energy. In the PB-cell model, the electrostatic contribution per polyion unit charge is70

µPcore,el ) -eφ(0) - Eel + RTVw -

(19)

(20)

where R is the ideal gas constant. The second term in eq 19 is the contribution from the work of deforming the core network. In principle, it can be calculated from the theory of rubber elasticity. However, the agreement with experiments is poor for the type of networks we are interested in here. Instead we use a semiempirical expression26

(21)

where Vcore is the core volume per mole polyion charged groups, related to R by

∑ (Ci,av - Ci(L)) i

(25) with

(18)

i

2 πcore def ) k4 + k5Vcore + k6Vcore

( ( ) )

π

The first term is the osmotic pressure in the core due to the mobile ions, which is calculated using the cylindrical PoissonBoltzmann cell model. The osmotic pressure is then given by the concentration of ions at the cell border (L)69 core πion ) RT

(22)

2kBTβ′φref φref 2 1 13(1 - β′)pVsegR φ R6

core

2.3. Core. The micelle-free core will be modeled as in earlier publications.16,18,26 Three independent contributions to the osmotic pressure are taken into account core core πcore ) πion + πcore def + πshell

Vcore Vref

k4-k6 are constants that can be determined from a combination of swelling data and PB-calculations.26 The third term in eq 19 is the osmotic pressure in the core due to the work of deforming the shell network. It has been shown earlier26 that this can be calculated from (see also Supporting Information 2)

core πshell )-

For isotropic deformations, where R3 ) φref/φ, eqs 15-17 reduce to the standard ones for uniform network deformations.67 Surface Free Energy. The free energy due to the contact between hydrocarbon and water at the micelle surface gives a contribution to the surfactant chemical potential65

µSm,surf ) aγ +

R3 )

Eel )

ε0εr 2

∫ (∇φ)2dV

(26)

Here, φ(0) is the electrostatic potential at the surface of the charged cylinder; Eel is the electrostatic energy per cell; Vw is the volume of the aqueous part of one cell; Ci,av is the average molar concentration of mobile ionic species i in the aqueous part of the cell; and Ci(L) is its concentration at the cell border. Evaluation of eqs 25 and 26 requires a numerical solution of the PB equation. The deformation energy contribution to the chemical potential of the polymer is obtained from eq 21 using the Gibbs-Duhem equation (Supporting Information 2). The result is

{

µPcore,def ) NA-1 -

}

2k6 3 k5 2 2 3 )+ ) (Vcore - Vref (V - Vref 2 3 core

(27)

where NA is Avogadro’s number. In the FH theory, used to describe the mixing of polyion and water in the shell, the standard state is the pure components. Therefore, when eq 27 is used, it is necessary to add to the polyion chemical potential in the core the contribution from the free energy of mixing in the reference state (where the volume per polyion charge equals Vref)

µPcore,FH )-

(

kBTVseg Vseg 1Vw Vref

)

(28)



In this way, the polyion chemical potentials in the core and the shell are expressed on the same concentration scale. For the ≈ -kBT. systems in the present study µPcore,FH In the PB-cell model, the chemical potential of the polyion counterion is69 core µGcore + ) kBT ln CG+ (L)

(29)

2.4. Liquid Solution. The solution is restricted to contain water and the dissolved electrolytes S+X- and M+X-. For sufficiently dilute solutions, and below the cmc of the surfactant, the chemical potentials are 0,w liq liq µSliq+X- ) µS0,w + + µX- + kBT ln CS+CX-

(30)

liq 0,w 0,w liq liq µM +X- ) µM+ + µX- + kBT ln CM+CX-

(31)

µwliq ) µw0 - 2RTVwCXliq-

(32)

where CXliq- will always be equal to Csalt, the total electrolyte (G+X-) concentration in the liquid. 2.5. Computation of Phase Equilibrium. Equilibrium between phases requires that the chemical potential is the same for each electroneutral component in every phase. The ions in the present system can combine to form four different electrolytes. However, the chemical potentials of these are not all independent since

µS+P- ) µS+X- + µM+P- - µM+X-

(33)

Thus, adding water gives us four components. With all components present in three phases, there are eight phase equilibrium conditions. This reduces to seven by the fact that the polyion is absent from the solution. At constant temperature and pressure, one needs to know the value of three additional independent state variables for the state of the system to be determined. From a computational point of view, it is convenient in the present case to work at fixed osmotic pressure and to treat all phases as closed systems with respect to the polyion. It is then straightforward to establish equilibrium by solving, for specified values of Csalt and β′, the equation system liq shell represented by the following equalities: µcore ) µliq w ) µw , µw w, core liq shell liq core shell shell µG+X- ) µG+X-, µG+X- ) µG+X , µG+P- ) µG+P-, µS+X- ) µSliq+X-, core core liq and KexCSliq+CM + ) CS+ CM+. Here, Kex is a selectivity constant describing the mutual partitioning of surfactant monomers and sodium ions between the gel core and the solution, set equal to 1 if nothing else is stated. In all calculations, the aggregation number is optimized by minimizing the total free energy with respect to a as described in detail elsewhere.14 Supporting Information 3 contains an equation useful when solving the equation system and a note about a technical complication with computations of phase equilibrium for systems of polyelectrolyte networks placed in solutions containing surfactant and simple salt. 2.6. Model System. All calculations are made for a surfactant with a cationic headgroup and a linear tail with 12 carbons in the chain. The volume per hydrocarbon tail is calculated from the number of carbon atoms, nc: Vs ) 27 Å3(nc + 1).1 For the free energy of transferring the surfactant tail from water to the 0,w micelle interior, ∆µS0 + ) µS0,m + - µS+ , we use a value of -11kBT (molar scale), deduced earlier from PB-cell model calculations

on C12TA+ micelles in water71 but approximately valid also for a surfactant with a pyridinium ion headgroup (C12P+).71,72 The polymer is described using parameters relevant for polyacrylate (PA), with 2.5 Å between neighboring charges in the chain. In the PB calculations (core), PA is modeled as a charged cylinder of radius 2.0 Å with a volume per charge of 31 Å3. The latter measure is also the volume per segment assigned to the polyion when calculating the FH contribution to the free energy. For the constants k4-k6 describing the deformation energy of the network, we use the values determined earlier26 for PA networks with 1 mol % cross-linker (1%X): k4 ) 1.70 × 103 Pa, k5 ) -2.73 × 105 Pa mol/m3, k6 ) 0. The volume fraction φref of polymer in the relaxed reference state of the network is 0.026, relevant for a network formed by polymerization of a 10 weight percent (wt %) acrylic is acid solution. The polyion “self-energy” in the shell µPshell,el set to 2 kBT, representing the excess coulomb energy for ions arranged in an array over that of a uniform distribution of ions. The electrostatic double-layer parameter k is set to 0.8, a value determined earlier by fitting the model to experimental data of the linear complex salts C12TAPA and C16TAPA mixed with water.14 With this value, the model was found to be in reasonably good agreement with experiments both regarding the position of phase boundaries and the magnitude of surfactant aggregation numbers.14 In all calculations, the temperature is set to 25 °C. Only phase equilibria involving disordered micellar and micellar cubic phases are considered, in agreement with experiments.21,24,29,31,32,36 The surfactant binding ratio β is defined as the number of surfactants in the gel per polyion charge. Gel swelling is described either by the ratio between the gel volume in the presence and in the absence of surfactant (V/V0) or by the deformation ratio R. The terms nonreserVoir and reserVoir conditions are used to distinguish between, respectively, the cases when the binding lowers the surfactant concentration in the solution and when this is not the case. 3. Results and Discussion 3.1. Three Gel States. The theory considers three competing states: swollen micelle-free gels, core/shell gels with micelles in the shell only, and gels with a homogeneous distribution of micelles (Figure 1a). In this section, we treat the equilibrium between the liquid solution and each of theses gel states separately; the competition between them is dealt with in subsequent sections. Figure 2a shows binding isotherms calculated for each case in the presence of 0.01 M salt. As can be seen, the binding ratio (β) is small in the micelle-free case (dotted line), as expected for a statistical distribution of S+ and M+ between the gel and the liquid when M+ is in excess. In the case of a homogeneous distribution of micelles (dashed), equilibrium with the solution is found only when β > 0.6. For lower β values, the high swelling pressure from the simple ions (G+) in the aqueous domains between the micelles makes the collapsed homogeneous state unstable against a swollen homogeneous state. However, the latter state, also containing micelles, is only stable at very high surfactant concentrations in the liquid. This will be further discussed in connection to Figure 3b. The end point of the curve is where the osmotic modulus vanishes at zero osmotic pressure difference between gel and solution (dπ/dφ ) 0; ∆π ) 0). The shape of the curves in the neighborhood of β ) 1 has been discussed elsewhere.29 Further information about the collapsed homogeneous gel state is also given in Supporting Information 4. The solid lines in Figure 2a are binding isotherms calculated for the core/shell state with the elasticity parameter p set to 50,

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Figure 2. (a) Calculated binding isotherms for gels in the swollen (micelle-free), core/shell, and collapsed states calculated with different p as indicated. Csalt ) 0.01 M. Binding to collapsed state independent of p in the range 50-500. (b) Shell thickness normalized by gel radius plotted vs β. Csalt ) 0.01 M, p ) 150. (c) Deformation ratios in the core/shell (curves to the left) and collapsed homogeneous states as functions of β in the cases of p ) 50, 150, and 500. R: deformation ratio in core. Rxy, Rz: deformation ratios in shell.

150, and 500, respectively. The two major findings are: (1) the core/shell state is thermodynamically stable at sufficiently low β and (2) the binding isotherms have a negative slope. The latter means that the chemical potential of S+X- decreases with increasing β. (This follows from the equilibrium condition that the product CS+CX-, where CS+ is the concentration of surfactant unimers, is the same in all aqueous regions of the system and that the concentration of X- in the liquid is fixed.) The decrease reflects a lowering of the elastic deformation energy and shows that the binding of surfactant becomes more favorable with increasing β. This can be understood in the following way: The interaction with the micelles leads to a contraction of the gel network, not only in the shell but also in the core. In the core, the effect increases with increasing β (see below) since the pressure on it from the shell increases. This leads to an increase of the electrostatic free energy of the core polyelectrolyte

Hansson

Figure 3. (a) Chemical potential of the complex salt (µS+P-) in different gel states plotted vs log of the surfactant concentration in the solution. Csalt ) 10 mM, p ) 150. (b) Chemical potential of the complex salt (µS+P-) in different gel states plotted vs β. Csalt ) 0.01 M, p ) 150. Lines have the same meaning as in Figure 2a. Dash dotted line: phase transition between two homogeneous micellar states. (c) Gibbs free energy per mole of polyion charges (G/nP-) plotted against the surfactant to polyion ratio (nS+/nP-) for a system containing 1 × 10-3 M polyion, 0.01 M salt, and various concentrations of surfactant. Only the free energy in excess of a constant value common to homogeneous and core/shell states is shown (see text).

(mainly due to an increased M+ concentration) which, in turn, makes it more favorable for the surfactant and polyion to associate with each other. Strictly, the argument requires that the variation of the free energy of the surfactant-polyion complex (S+P-) in the shell is small. This is indeed the case, as is also the concentration of S+ unimers in the aqueous regions between the micelles (see Supporting Information 5). However, the excess of polyion over surfactant in the shell increases somewhat with increasing β. This is another effect associated with the increased chemical potential of M+P- which leads to an expulsion of X- from the shell (Donnan effect). Thus, the negative slope of the binding isotherms, equivalent to a lowering of the chemical potential of the surfactant salt in the system,

Surfactant Self-Assembly in Polymer Networks can also be seen as a lowering of CX- in the shell. Interestingly, in the shell the product CS+CX- is lowered by CX- at constant CS+, but in the liquid by CS+ at constant CX-. For a quantitative analysis of the free energy changes involved, see Supporting Information 5. When a high proportion of the network is in the shell, the model overestimates the elastic energy of the shell and therefore the compression of the core network. This gives rise to the peculiar shape of the binding isotherms in this region. The error stems from the approximation that the deformation ratio in the direction parallel to the gel’s surface (Rxy) is assumed to be the same throughout the shell. (In reality, Rxy needs only to be equal to R at the core interface.62) The error is negligible when the shell thickness is small in comparison with the curvature, which is the case except at the highest β values, as shown by Figure 2b where the thickness of the shell relative to the gel radius is plotted against β (p ) 150). Figure 2c shows how p influences the swelling ratio of the core network (R) as a function of β at 0.01 M salt. A comparison with Figure 2b reveals that even very thin shells have an effect on the core swelling. In the simple Wall theory of rubber elasticity employed here, p is the average number of segments between cross-links. In practice, it depends also on the topology and distribution of cross-links and should be treated as a parameter.73 Since the choice of p has no qualitative effect on the results, we shall for the rest of the paper use p ) 150 if nothing else is stated. 3.2. Reservoir Condition. 3.2.1. Volume Transition. The situation for gels in contact with reservoir solutions of gradually increasing surfactant concentration is illustrated in Figure 3a, showing a plot of the chemical potential of the complex salt (S+P-) as a function of the logarithm of the surfactant concentration at 0.01 M salt. The curves represent the chemical potential calculated for the three competing gel states. For a given surfactant concentration, the equilibrium state is the one where µS+P- is smallest. This follows from the fact that, at a given composition of the liquid, the chemical potentials of the other independent components are all fixed. As expected, the swollen micelle-free state is most favorable at sufficiently low concentrations. However, the chemical potential rises steeply and soon reaches the level of the curve for the collapsed homogeneous state at the point P. This is the point where phase transition would take place in a fluid system. This is not the case for gels. Instead the phase transition, i.e., the Volume transition, starts at Q where the chemical potential reaches the curve for the core/shell state and ends at R. (Q is exactly where the core/shell curve ends, i.e., where the shell thickness vanishes). The ccc is the corresponding surfactant concentration in the liquid. The argument for this, due to Sekimoto,57 will be described in detail below. Note, however, that ccc is slightly larger than the concentration at P where the free energy of the collapsed homogeneous state equals that of the swollen micelle-free state. This shows that there is a coexistence cost in elastic energy of having both swollen and collapsed phases in the gel.58 A similar “overshoot” is observed for temperature-controlled gel volume transitions where the threshold temperature for collapse (swelling) transition is higher (lower) than the Maxwell temperature, which is the temperature where the free energy is the same in the swollen and the collapsed states.55 The results presented so far are valid for 0.01 M salt. Calculations for 0.001 and 0.1 M salt give the same qualitative behavior, but the free energy difference between the swollen and collapsed states at the ccc decreases with increasing salt


Figure 4. Experimental data of relative gel volume (V/V0) and surfactant binding ratio (β) plotted vs the logarithm of the surfactant concentration in the liquid solution. System: PA gels and C12TAB in 0.01 M NaBr aqueous solutions. Data from ref 18.

concentration due to the decreasing gel swelling. At 1 M, the model predicts a stable shell at low β prior to the volume transition. However, the results for this salt concentration are mainly of academic interest since ccc > cmc. Sekimoto’s Argument. Onuki showed56 that nucleation of a new phase in the bulk of a gel is practically impossible as the activation energy for the process, due to shear deformation energy, scales with the volume of the gel and therefore becomes macroscopic. (This is in contrast to fluid systems where the nucleation is resisted only by the interfacial tension.) It means that for gels the distinction between stability and metastability is immaterial.57 Thus, if only isotropic network deformations were allowed, the volume transition would take place only under conditions of bulk instability (dπ/dφ ) 0; ∆π ) 0). However, Sekimoto demonstrated that before bulk instability sets in a thin layer of the new phase can be formed at the gel surface,57 in agreement with what has been observed experimentally.55 He suggested that the critical condition for volume transition can be determined from the condition that the uniaxially deformed surface layer is in equilibrium both with the solution and the bulk gel, at which the deformation ratio in the transversal direction (Rxy) in the surface layer is the same as in the isotropically deformed bulk gel (R). An investigation of the dynamics of the process carried out by Tomari and Doi58 confirmed not only that the appearance of the surface layer is determined accurately by these conditions but also that they ensure that the volume transition proceeds without the growth of the surface phase being terminated at any stage. The conditions determined by Sekimoto are identical to the ones used by us to establish core/shell equilibrium (section 2.1). Therefore, we conclude that ccc is the surfactant concentration in the liquid in equilibrium with the first shells formed. The volume transition at 0.01 M salt and p ) 150 is indicated by the dot-dashed vertical line in Figure 2a. Note that β is below 1 in the fully collapsed gel at the ccc. The reason for this is that the shell composition at the transition point is determined by the requirement of equilibrium with the swollen core in which the chemical potential of the polyelectrolyte is high, thus promoting an excess of polyion over surfactant in the complex salt. The deformational state of the shell changes dramatically during the collapse (Figure 2c); however, the elastic energy contributes very little to the chemical potential of the complex salt, and so the composition of the shell changes very little (see also Figure 10).

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Figure 5. Experimental swelling isotherms for the system C12PC/ PAgel/NaCl recorded during increase (open symbols) and decrease (filled) of the surfactant concentration (CS+) in reservoir solutions. Molar concentrations of salt as indicated. The arrows indicate the volume transitions. The volume ratios were calculated as V/Vref ) (d/dref)3 from gel diameters (d) reported by Sasaki et al.,19 where dref is the gel diameter directly after synthesis.

The energetics is further illustrated in Figure 3b, where the chemical potential of the complex salt is given as a function of β. Starting with a micelle-free gel and gradually increasing β, the system follows the dotted line up to the point Q where the surface phase appears in equilibrium with the bulk gel. Here, the system crosses over to the core/shell curve. As can be seen there is no energy barrier along the path of the core/shell curve. Therefore, the gel remains biphasic during the transition, until at high β a crossover to the collapsed homogeneous state takes place. The crossover will be discussed further below. Note, however, that P in Figure 3a would correspond to a “tie-line” in Figure 3b extending from a point just below Q to a point just prior to R. The “energy barrier” in the curve for gels with a homogeneous distribution of micelles in Figure 3b is not directly related to the elastic energy and should not be confused with Onuki’s activation energy for nucleation of the collapsed phase in the bulk of the gel which is due to shear deformation energy.56 Instead, it reflects that micelles are much less favorable in swollen than in collapsed gels. For more details, see Supporting Information 6. Role of Cross-Links. The elastic deformation energy is responsible for the negative slope of the binding isotherm in Figure 2a but not for the discreteness of the volume transition as such; it would have a first-order character even in the absence of cross-links. Thus, the volume transition is driven by the favorable association of polyion and surfactant. Instead, the cross-links are responsible for the coexistence cost in elastic energy corresponding to the shift of the transition point from P to Q, i.e., to higher surfactant concentrations. As can be seen from the deformation ratios in the transversal (Rxy) and longitudinal (Rz) directions in Figure 2c, the deformation of the network in the first formed shell is very anisotropic. In contrast, the isotropic network in the collapsed gel at high β is close to its relaxed reference state (R ) 1). Figure 3b shows that the elevated energy of the core/shell state due to deformations remains even at high β. 3.2.2. Comparison with Experiments. Figure 4 shows experimental swelling data for 1% cross-linked PA placed in reservoir solutions of C12TAB and 0.01 M NaBr18 (filled symbols). In agreement with the theoretical calculations for this

Hansson system, there is clearly a distinct surfactant concentration (ccc) separating the swollen and the collapsed regimes. Importantly, intermediate V/V0 values are not observed at equilibrium, indicating that the transition is of first-order with large collapse amplitude, in agreement with the large jump in β seen in Figure 2a. The influence of salt on gel swelling is exemplified in Figure 5 showing data from Sasaki’s lab for the neighboring system C12PC/PAgel/NaCl.19 Please note that the gel volumes have been normalized with the volume of the gel in the relaxed reference state (see figure legend). A volume transition occurs at all salt concentrations, but ccc increases and the collapse amplitude decreases with increasing salt concentration. Figure 6a shows, for comparison, theoretical swelling isotherms obtained for the model system at various concentrations of salt in the liquid. The slight volume decrease just prior to the transition point is not accounted for, presumably because the theory does not, in practice, allow micelles to form in the swollen network. Note that the calculations cover a broader range of salt concentrations than the experiments in Figure 5. Furthermore, the model network is slightly more cross-linked than that used in the experiments,19 explaining the slight discrepancy in gel volume below the ccc. The corresponding theoretical binding isotherms are shown in Figure 6b, where the dashed vertical lines indicate the jump in β at the ccc. Clearly, the theory reproduces the general behavior quite well. For the higher salt concentrations, the model predicts reswelling at elevated surfactant concentrations in agreement with experiments. This is due to “charge reversal” (β > 1), not dissolution of the complexes, as further discussed in section 3.5. The effect is difficult to see at the concentrations used in Figures 4 and 5 but is clearly seen at higher concentrations as shown in Supporting Information 7. The variation of ccc with salt concentration is shown in Figure 7 on a log-log plot, where the experimental points are for C12TAB/PAgel/NaBr18 and C12PC/PAgel/NaCl,19 respectively. Shown for comparison are also cac values in solutions of linear polyions. The increment of the theoretical curve is in very good agreement with experiments. This indicates that the variation of the change in counterion entropy associated with surfactant binding is described correctly but also that the theory accounts for the small variation of the composition of the complexes formed at the ccc. However, the absolute value of the calculated ccc is about 1 order of magnitude lower than that observed in experiments. One reason for the discrepancy can be ion-specific effects. Their importance is obvious when comparing the cac values obtained with different linear polyions (cf. the difference between PA and polyvinylsulfate in Figure 7). Since the volume collapse is directly related to the surfactant self-assembly, the difference in cac by an order of magnitude is expected to show up also in the ccc values, and hence a better agreement is not to be expected. It is common in the literature on ion exchange74 to introduce selectivity coefficients (Kex) to correct for differences between ions. The dashed line in Figure 7 is the theory calculated with Kex ) 0.1, in reasonable agreement with the value 0.2 reported for the competitive binding of C12TA+ and Na+ to 1%X PA gels below ccc.75 The discrepancy can also be attributed to other shortcomings in the description, e.g., an incorrect value of standard free energy of transferring the hydrocarbon tail from water to micelle. For further discussion and analysis of the factors determining ccc and cac, see Supporting Information 5. Shown in Figure 5 are also data for the reverse process of lowering the surfactant concentration in contact with precol-



Figure 6. Swelling (a) and binding (b) isotherms calculated for different molar salt concentrations as indicated: p ) 150, k ) 0.8, temperature ) 25 °C. All curves in (a) are for reservoir conditions. Vref is the volume of the network in the relaxed reference state. Dashed lines in (b): jump in β at gel volume transition.

Figure 7. Log-log plot of critical collapse concentration (ccc) vs salt concentration in the solution (Csalt). Dashed line is theory corrected with the selectivity coefficient Kex ) 0.1. Experimental data taken from Nilsson et al.18 (C12TAB/PAgel/NaBr), Sasaki et al.19 (C12PC/PAgel/ NaCl), Hansson et al.72 (C12TAB/linearPA/NaBr), and Hansson et al.86 (C12TAB/linearPVS/NaBr). Temperature: 25 °C.

lapsed gels.19 It is very clear that the critical swelling concentration (csc) is lower than ccc and that the hysteresis decreases with increasing salt concentration. The subject is beyond the scope of the present paper, but the theory can account for the hysteresis as shown in Supporting Information 8. 3.3. Nonreservior Condition. 3.3.1. Core/Shell Equilibrium. In reservoir solution, shells only appear as intermediate structures during volume transition. For a gel in a liquid of finite volume (nonreservoir) the total surfactant/polyion ratio in the system sets an upper limit to β. As can be inferred from the previous sections, at low β only the core/shell state is accessible. Thus, even for surfactant concentrations above ccc the gel is prohibited from reaching the fully collapsed state if there is not enough surfactant available. However, when β reaches approximately 0.6 in the present system, the homogeneous state becomes accessible and starts to compete with the core/shell state (see Figure 2a). Under nonreservoir conditions, the chemical potential of the complex salt alone does not indicate where the equilibrium state is. However, when the liquid behaves as a reservoir with respect to salt (M+X-) and water one can analyze the total Gibbs free energy per mole of polyion charges (G/nP-) as a function of the total polyion-to-surfactant ratio (nP-/nS+) (see Supporting Information 9). The result for a system containing 0.001 M polyion and 0.01 M salt (p ) 150)

Figure 8. Experimental data of relative gel volume (V/V0) and surfactant binding ratio (β) plotted vs the logarithm of the surfactant concentration in the solution. System: Sodium hyaluronate gels and cetylpyridinium chloride in 0.012 M TRIS buffer pH 7.4 (nonreservoir conditions). Data from ref 53.

is shown in Figure 3c. The solid line is for core/shell, and the dotted line is for the collapsed homogeneous state. In the range of compositions where there is a competition between the two states, the core/shell state has the lowest free energy and is thus thermodynamically stable. This is mainly an entropic effect: the polyion counterion M+ has larger entropy in the swollen core of biphasic gels than in the more compact homogeneous gel. An analysis shows that this is the case also for other total polyion concentrations and for salt concentrations between 0.001 and 1 M (not shown), as long as the polyion is in excess over the surfactant in the system. However, for polyion concentrations of the order of ccc the free energy difference between the two states is small and the behavior is more complex. The conclusions reached here allow us to construct from the data in Figure 2a, and the corresponding data for the other salt concentrations, the binding isotherms presented in Figure 6b (solid lines); only the results for p ) 150 are shown. The slope is negative at all salt concentrations as long as there is core left, but the effect decreases with increasing salt concentration. The corresponding effect is also seen in plots of the gel volume vs the free surfactant concentration, but in other respects these curves are almost identical to those for reservoirs in Figure 6a and are therefore not shown.

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Figure 9. V/V0 plotted as a function of β for gels with different p (indicated) under nonreservoir conditions at 0.01 M salt. Symbols are experimental data for the system C12TAB/PAgel/0.01 M NaBr.18 V0 is the equilibrium gel volume at zero surfactant concentration. Temperature: 25 °C.

Figure 10. Weight percent water (wt % H2O) and surfactant to polyion charge ratio (S+/P-) in complexes calculated from theory for the case of 0.01 M salt, p ) 150, and k ) 0.8.

3.3.2. Comparison with Experiments. Experimental binding and swelling isotherms for C12TAB/PAgel/0.01 M NaBr18 are shown in Figure 4 (open symbols). In agreement with the theoretical calculations, the volume decrease is directly coupled to surfactant binding and takes place within a very narrow range of surfactant concentrations in the liquid close to the reservoir ccc value. In contrast to the reservoir case, but in agreement with the theory, the core/shell state is stable when β < 0.8. Furthermore, in this range the gel volume is intermediate between that of fully swollen and collapsed gels. The negative slope of the core/shell part of the theoretical binding isotherms in Figure 6b is present in the experimental binding isotherm in Figure 4 at around β ) 0.35. Unfortunately, although the precision of the determination of the surfactant concentration in the liquid is high ((0.005 log units), the number of data points is too small to prove the effect. However, data reported for the related system of cetylpyridinium ions binding to cross-linked hyaluronate gels53 provide more compelling evidence. As can be seen in Figure 8, showing the results from two individually prepared sample series, the slope of the binding isotherm is clearly negative in the range 0.2 < β < 0.6 where core/shell coexistence was indeed observed.53 We have made no theoretical calculations with parameters relevant for this system, but it can be mentioned that the swelling of linear

Hansson complex salts of C16 tailed cationic surfactants, and linear polyions with the same charge density as hyaluronate are well described by the theory.14 Figure 9 shows plots of V/V0 vs β for the system in Figure 4 together with theoretical curves for p ) 50, 150, and 500. The agreement is reasonably good when β > 0.5 and for large p. For p ) 500 the swelling ratio of the core is little affected by the shell, and so the volume is nearly linear in β as long as there is core left. In the experimental system, the core disappears at around β ) 0.8, but the exact point is difficult to determine.18 Thus, the finding that the minimum volume is reached before β ) 1 correlates well with the experiments. It can be mentioned that the theoretical curves for 0.001 and 0.1 M salt are nearly identical to the one at 0.01 M salt, but V/V0 declines faster with β at 1 M salt. At low β, the agreement is poorer. In the experiments, significant deswelling takes place only when β > 0.2. A careful check of the literature shows that this appears to be a general effect for gels preswelled in the solution before surfactant is added.15 However, data for initially dry PA gel pieces simultaneously absorbing water and C16TA+ from solution do not show this behavior and are in much better agreement with the theory.26 Resolving these matters requires more knowledge about the processes taking place in the vicinity of the ccc, e.g., to what extent micelles form in the network prior to shell formation. For the system in Figure 9 (same as in Figure 4) shell formation was unmistakably confirmed only when β g 0.2.18 Recall that the negative slope of the theoretical binding isotherms is related to the compression of the core network by the shell. Clearly, if surfactant binds at low β without inducing a volume change or compression of the core, no negative slope should develop. This can explain the discrepancy between theory and experiment in this binding regime. An obvious flaw is, of course, the “cusp” on the part of the theoretical binding isotherm just before the core disappears, which is completely absent in the experimental isotherms. As already discussed, this is believed to reflect, mainly, an overestimation of the elastic energy of the shell network at high β. 3.4. Composition and Microstructure of Complexes. Composition. Figure 10 shows the water content and the S+/Pcharge ratio of complexes calculated for the case of 0.01 M salt and p ) 150. The result is almost identical at the other salt concentrations and p values. Shells are present for β < 0.85. When the core is consumed the charge ratio, now equal to β, increases. Once β > 1, surfactant counterions become incorporated, and therefore the water content increases. However, the reswelling is modest, due to strong polyion mediated forces. There are no experimental data on the water contents of shells in the C12TA+/PA system. However, 48 wt % H2O was found in a fully collapsed gel with β ) 0.94,24 in good agreement with the value of 46 wt % calculated here. The latter is also equal to the swelling limit calculated for the linear complex salt in pure water (p ) 300, k ) 0.8) in our previous paper.14 While the same swelling as for the noncross-linked system is expected for gels in the collapsed homogeneous state with β close to unity, deviations could have been anticipated in shells at low β where the polyion is in excess over the surfactant in the complexes. Apparently, a larger excess charge is needed to swell the structures. It should be pointed out that, apart from the indirect effect via the compression of the core, the elastic energy is too small to influence the swelling of the shells even in the highly deformed state at low β. Aggregation Numbers. In the model, N is directly related to the swelling of the complexes. Hence, for the system in Figure



Figure 11. Salt effect on surfactant binding under nonreservoir conditions. The plots show β as a function of total surfactant concentration in the systems at different molar concentrations of added salt (indicated). Total polyion concentration: 2.5 × 10-3 M. (a) Experimental data for a cationic gel (poly(diallyldimethylammonium chloride)) immersed in nonreservoir solutions of anionic surfactant (dodecylbenzenesulfonate) and NaCl. Data taken from ref 82. (b) Binding curves calculated from theory: p ) 150; k ) 0.8; ∆µS0 + ) -11.4kBT. (c) Same as in (b) except ∆µS0 + ) -16.1kBT.

10, it is constant at 55 in the shells and decreases during reswelling in the homogeneous state to reach 42 at β ) 2.5. The magnitude of N and the small variation in the shell are in reasonable agreement with experiments on C12TA+/PA gels where it was found to go from 54 at β ) 0.095 to 69 at β ) 0.83 where the fully collapsed state was reached.24 In further agreement with experiments, the calculated microstructure of all collapsed phases in the present work contains micelles ordered on a cubic lattice. 3.5. Excess Binding in the Presence of Salt. It has been reported for many systems that, at low salt, incubation of gels in solutions containing even a large excess of surfactant leads to the formation of nearly stoichiometric complexes.36,37,76-78 The model can account for that, as shown by the horizontal parts of the binding isotherms in Figure 6b. It is also clear that, depending on the conditions, additions of salt can either destabilize19,29,31,50,79,80 or stabilize81,82 gel/surfactant complexes. For systems of finite volume, the total surfactant concentration is typically much larger than cac (or ccc), even in the presence of substantial amounts of salt. Therefore, dissolution is often insignificant in practice. Rather, at high salt, β . 1 has been found even at moderate surfactant concentrations. An example taken from Mironov et al.82 is given in Figure 11a showing β as a function of total surfactant concentration for poly(diallyldimethylammonium chloride) gels in solutions of sodium dodecylbenzenesulfonate (SDBS) and 0-0.3 M NaCl. The total concentration of polymer charges in the system is 2.5 × 10-3 M. Within the theory, excess binding is driven by the hydrophobic effect but counteracted by the cobinding of surfactant counterions (Supporting Information 5). For a surfactant with a large hydrophobic group, like SDBS, salt is expected to have minor effects at low β, but the binding should be enhanced in the charge reversal regime. These expectations are borne out in Figure 11a. In general, the situation is a bit more complicated since salt also lowers the cmc, and so there is a competition between binding to the gel and forming micelles in the solution. In particular, when comparing surfactants with different tail lengths, it is not obvious that β is larger for a long-tailed one since one with a shorter tail may reach a higher β before exceeding its cmc. In Figure 11b we show the results for our model system at the same total polyion concentration as in Figure 11a. To highlight the effect of the hydrophobic driving force, we have in Figure 11c reduced ∆µS0 + to -16 kBT, a value

relevant for SDBS (and C16TA+).71 For simplicity, all other parameters are unchanged. The isotherms saturate when the free concentration reaches the cmc. In Figure 11b and 11c, we have used cmc values typical for C12TA+ and C16TA+ surfactants, respectively.83 The more hydrophobic surfactant reaches the highest β values, but the saturation limit at, e.g., 0.1 M salt, is not much larger than for the less hydrophobic one. At the lowest salt concentration, the calculated curve for the more hydrophobic surfactant first levels out and then increases again to finally reach a plateau at β ≈ 1.1. This is in agreement with the salt-free experiment. Note that (i) the experiment is not free from electrolyte since some salt accumulates in the solution during the course of binding and (ii) the first leveling out of the calculated curve is a true effect, not due to a too low number of data points. In conclusion, the theory captures the qualitative features of salt on binding seen in the experiments and explains that salt favors excess binding of surfactant by reducing the entropic penalty of cobinding surfactant counterions. 4. Conclusions We have analyzed in detail a complex four-component system where a number of interactions are at work on length scales ranging from molecular, via colloidal, to macroscopic. In particular, it appears that the equilibrium state of the gel depends on a delicate interplay between electrostatic and elastic interactions. The theory accounts (on a mean-field level) for the fact that substantial amounts of surfactant are never distributed homogeneously in a swollen network (except in systems where there is a hydrophobic interaction between micelles and polyion51,54,84,85 or, perhaps, when at least one of them have a low charge density21,76). The explanation provided is that the average separation between the micelle surface and its counterions (Pand X-) increases with increasing swelling of the complexes; the increased electrostatic energy makes micelle formation unfavorable. As a result, water-poor phases with micelles packed close together in ordered structures are more favorable alternatives but only at sufficiently high surfactant/polyion ratios in the complexes (at lower ratios, the pressure from small ions swells the structure and the micelles dissolve). When there is no surfactant available to reach this ratio throughout the gel,

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the gel separates into a micelle-free core and a shell with the required ratio, in agreement with experiments.18,24-26,49,50 At low average binding ratios, the shell structure is stabilized by attractive polyion-mediated forces between the micelles, included implicitly in the electrostatic model. At higher binding ratios, where the collapsed homogeneous state starts to compete with the core/shell state, there is a range where the latter state is favored by counterion entropy. The electrostatic model was used earlier in a theory of complex salts, where it was found to account for the stability against dissolution in water. The qualitative agreement with experiments obtained here shows that the extension of the model to nonstoichiometric complexes (shell) in equilibrium both with an excess polyion phase (core) and an electrolyte solution works reasonably well. Importantly, the compositions of all three phases depend on the relative amounts of core and shell. This is an effect of network elasticity. The calculations show that the formation of the shell is associated with a coexistence cost decreasing in magnitude as the shell grows, explaining the negative slope of binding isotherms seen in experiments. The theory accounts for the volume transition observed experimentally at a critical collapse concentration (ccc) in reservoir solutions of surfactant.16,18,19 The calculations verify Sekimoto’s condition that the transition takes place at the conditions where the first shells become stable. This explains why ccc is equal to the surfactant concentration required for the formation of the first shells. Similar arguments applied to the swelling transition upon lowering the surfactant concentration explain a hysteresis effect reported in the literature.19 The scaling of ccc with salt concentration is nearly in quantitative agreement with experiments, but the absolute values are too small. The increase of ccc (and cac) with increasing salt concentration explains why additions of simple salt tend to dissolve the surfactant-gel complex when the binding ratio is below 1. It is also concluded that salt enhances binding at binding ratios above 1 by reducing the entropic penalty of cobinding counterions. In general the results are in agreement with the consensus that the strong association between surfactants and oppositely charged polyions is a polyelectrolyteinduced surfactant self-assembly driven by the hydrophobic effect and favored by the entropic gain of releasing counterions bound to the polyelectrolyte. The theory can be improved in several ways, e.g., by correcting for the overestimation of the elastic deformations in the shell at high surfactant binding ratios and by implementing a rigorous description of polyion bridging, but explains already in its present form important properties of the systems. In particular, the results suggest that the major influence of the cross-links on the polyion surfactant interaction is to mediate elastic forces between complex-rich and complex-lean domains, thereby affecting the conditions of phase equilibrium and mechanisms of phase transition. In contrast, they have little direct influence on the stability of the polyion-surfactant complexes. Acknowledgment. This work has been financially supported by grants from the Swedish Research Council and the Swedish Foundation for Strategic Research. The author is thankful to Martin Malmsten for valuable comments on the manuscript and Lennart Piculell for fruitful discussions. Supporting Information Available: S1: Comparison with core and µPcore,def . S3: MC simulations. S2: Derivation of πshell Information useful to calculations of phase equilibrium. S4: Binding to collapsed homogeneous gels. S5: Critical collapse

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Surfactant Self-Assembly in Oppositely Charged Polymer Networks

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