End-Grafted Polymers with Surfactants: A Theoretical Model

5740

Langmuir 1998, 14, 5740-5750

End-Grafted Polymers with Surfactants: A Theoretical Model E. P. K. Currie,* J. van der Gucht, O. V. Borisov,† and M. A. Cohen Stuart Department of Physical and Colloid Chemistry, Agricultural University Wageningen, Dreijenplein 6, 6703 HB Wageningen, The Netherlands Received February 10, 1998. In Final Form: July 8, 1998 A mean-field analytical model is developed for end-grafted polymers in contact with a solution containing surfactants. At concentrations above the critical association concentration, the surfactants are capable of cooperative association in a micellar form with the polymer chains. For single-grafted coils it is found that the coverage of the polymer chain by micelles increases continuously with increasing adsorption strength or bulk surfactant concentration. The size of the coils can increase, decrease, or have a maximum with increasing micellar coverage, depending on the relative strength of the monomer-monomer, monomermicelle, and micelle-micelle excluded volume interactions. The amount of surfactants adsorbed on densely grafted polymer chains (brushes) is found to increase either continuously or discontinuously with increasing adsorption strength or decreasing grafting density. The character of the transition is determined by the interplay of the adsorption energy and the excluded volume interactions. When the adsorbed amount decreases continuously with increasing grafting density, the brush height increases at small and large grafting densities and has a local maximum in the intermediate range. In the case of an abrupt transition in the adsorbed amount the coexistence of stretched and collapsed chains is predicted in the transition regime. Critical adsorption strengths and grafting densities are discussed. Where possible, the numerical results are compared to experimental data.

Introduction Polymers adsorbed on or grafted to a surface add stability to colloidal dispersions and are applied in, for instance, food technology and medicine delivery.1-5 A system of special interest is that of densely end-grafted neutral polymers, so-called brushes. Due to repulsive excluded volume interactions between polymer segments, the polymers in a brush are stretched normal to the surface and form a continuous layer. Polymer brushes on a planar surface were first studied theoretically by Alexander6 and de Gennes7 using scaling arguments. In these models it is assumed that the polymer density is uniform (boxlike) throughout the brush layer and all the chains are equally stretched. The Alexander-de Gennes model gives scaling relationships for such quantities as the surface pressure and the height of the brush layer as a function of the length of the grafted chains and the grafting density. More advanced self-consistent-field models by several authors showed that in the case of a good solvent the polymer density decreases as a parabolic function of the distance to the grafting surface while the chain ends are distributed throughout the brush.8-11 The scaling relations for the * To whom correspondence should be addressed. Tel.: 00-31317-482277. Fax: 00-31-317-483777. E-mail: [email protected]. † Institute of Macromolecular Compounds of the Russian Academy of Sciences, 199004 St. Petersburg, Russia. (1) Halperin, A.; Tirrell, M.; Lodge, T. P. Adv. Polym. Sci. 1992, 100, 31. (2) Kuhl, T. L.; Leckband, D. E.; Lasic, D. D.; Israelachvili, J. N. Biophys. J. 1994, 66, 1479. (3) Lasic, D. D.; Martin, F. J.; Gabizon, A.; Huang, S. K.; Papahadjopoulos, D. Biochem. Biophys. Acta 1991, 1070, 187. (4) Lasic, D.; Martin, F. E. Stealth Liposomes; CRC Press: Boca Raton-London-Tokyo, 1995. (5) Napper, D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: London, 1985. (6) Alexander, S. J. Phys. (Paris) 1977, 38, 983. (7) de Gennes, P.-G. Macromolecules 1980, 13, 1069. (8) Semenov, A. N. Sov. Phys. JETP 1985, 61, 733. (9) Milner, S. T.; Witten, T. A.; Cates, M. Macromolecules 1988, 21, 2610.

height and the surface pressure, however, are identical to those of the box model. There are several experimental studies on polymer brushes,12-18 but the scaling picture has only recently been confirmed convincingly.13,19 The interaction of polymers and surfactants in a bulk phase is also a rich area of study. It is well-known that at concentrations below the critical micelle concentration (CMC) certain surfactants form polymer-micelle complexes with neutral polymers.20 Such complexes are predominantly formed by anionic surfactants; cationic and neutral surfactants show little affinity for the polymer chains. NMR studies suggest that the polymer segments coat the micellar surface and do not penetrate the micellar interior.21 It is found by some authors that the size of the micelles adsorbed on the polymer is less than that of free micelles in the bulk and does not depend on the degree of coverage (i.e., on the surfactant concentration).22-24 Other authors suggest that the number of surfactants per (10) Skvortsov, A. M.; Pavlushkov, I. V.; Gorbunov, A. A.; Zhulina, E. B.; Borisov, O. V.; Priamitsyn, V. A. Polym. Sci. U.S.S.R. 1988, 30, 1706. (11) Zhulina, E. B.; Borisov, O. V.; Priamitsyn, V. A.; Birshtein, T. M. Macromolecules 1991, 24, 140. (12) Bijsterbosch, H. D.; de Haan, V. O.; de Graaf, A. W.; Leermakers, F. A. M.; Cohen Stuart, M. A.; van Well, A. A. Langmuir 1995, 11, 4467. (13) Currie, E. P. K.; Leermakers, F. A. M.; Cohen Stuart, M. A.; Fleer, G. J. Submitted to Macromolecules. (14) Kent, M. S.; Lee, L. T.; Factor, B. J.; Rondelez, F.; Smith, G. H. J. Chem. Phys. 1995, 103, 2320. (15) Richards, R. W.; Rochford, B. R.; Webster, J. R. P. Polymer 1997, 38, 1169. (16) Patel, S.; Tirrell, M. Colloids Surf. 1988, 31, 157. (17) Kent, M. S.; Lee, L. T.; Farnoux, B.; Rondelz, F. Macromolecules 1992, 25, 6240. (18) Kent, M. S.; Lee, L. T.; Farnoux, B.; Rondelez, F. Macromolecules 1992, 25, 6240. (19) Marra, J.; Hair, M. L. Colloids Surf. 1988, 34, 215. (20) Hansson, P.; Lindman, B. Curr. Opin. Colloid Interface Sci. 1996, 1, 604. (21) Cabane, B. J. Phys. Chem. 1977, 81, 1639. (22) Gilanyi, T.; Wolfram, E. Colloids Surf. 1981, 3, 181. (23) Shirahama, K. Colloids Polym. Sci. 1974, 252, 978. (24) Cabane, B.; Duplessix, R. Colloids Surf. 1985, 13, 19.

S0743-7463(98)00168-1 CCC: $15.00 © 1998 American Chemical Society Published on Web 09/12/1998

End-Grafted Polymers with Surfactants

adsorbed micelle does depend on the surfactant concentration.25 The micelles are found to be of spherical shape: the complex of a polymer chain loaded with micelles has been described as the “necklace” model.21,23,24 A widely studied experimental model system for polymer-micelle complexes is poly(ethylene oxide) (PEO) and sodium dodecyl sulfate (SDS). The critical association concentration (CAC) at which complexes are formed is 10-3 M for a 0.1 M NaCl concentration, whereas the CMC under these ionic conditions is approximately 1.5 10-3 M.22,23 The micelles bound to the PEO at this ionic strength were found to consist of 50 surfactants, which is significantly smaller than the aggregation number for unbound micelles.26 Similar values were also obtained for SDS in combination with poly(vinyl alcohol) and poly(vinylpyrrolidone).22 The number of PEO segments surrounding one micelle is estimated to be on the order of 100.22,23 The formation of polymer-micelle complexes strongly enhances the viscoelasticity of a bulk solution of PEO/SDS solutions.27 Polymer-micelle complexes have various applications, for instance, in paints28 and in detergents.29 The driving force for complex formation is suggested to be the shielding of hydrophobic areas on the micellar surface by the polymer chain.30 Several models are available that describe the behavior of a single polymer chain in contact with a surfactant solution. Nagarajan et al.31,32 and Ruckenstein et al.33 proposed models that determine the CAC’s and the aggregation number of the polymer-micelle complexes as a function of molecular parameters of the surfactant and polymer, that is, the size of the headgroup, the length of the alkyl chain, and so forth. Nagarajan et al. assume the polymer segments to adsorb directly on the hydrophobic areas on the micellar surface. This results both in a decrease of the hydrophobic interactions and an increase in the steric repulsion between the charged heads. Ruckenstein et al. assume the polymer adsorption to make the area around the micelle less hydrophilic. This is advantageous for the apolar micellar core, but disadvantageous for the hydrophilic headgroups. Recently, an elaborate molecularthermodynamic model was suggested by Nikas and Blankstein for polymer chains covered by spherical micelles.34 In this model the elasticity of the polymer, solvent quality, electrostatic forces, and specific polymersurfactant interactions are considered. This model calculates the CAC, amount of micelles adsorbed per polymer, aggregation number, and size of the complex as a function of several molecular input parameters. The above models all consider the behavior of a single polymer chain dissolved in a reservoir containing surfactants. None of them consider intra- and intermolecular excluded volume interactions between polymers covered by adsorbed micelles. At high polymer concentrations such intermolecular interactions are of major importance and largely determine the properties of polymer-micelle complexes. This is the case for end-grafted polymers at high grafting densities in a solution containing surfactants. The aim of this paper is to develop a simple (25) Van Stam, J.; Almgren, M.; Lindman, C. Prog. Colloid Polym. Sci. 1991, 84, 13. (26) Witte, F. M.; Engberts, J. B. F. N. Colloids Surf. 1989, 36, 417. (27) Brackman, J. C. Langmuir 1991, 7, 469. (28) Dulog, L. Angew. Macromol. Chem. 1984, 124, 437. (29) Vogel, F. Chem. Uns. Zeit. 1986, 20, 156. (30) Linse, P.; Piculell, L.; Hansson, P. In Surfactant Science Series; Kwak, J. C. T., Ed.; Marcel Dekker: New York, 1998; Vol. 77. (31) Nagarajan, R. Colloids Surf. 1985, 13, 1. (32) Nagarajan, R. J. Chem. Phys. 1989, 90, 1980. (33) Ruckenstein, E.; Huber, G.; Hoffman, H. Langmuir 1987, 3, 382. (34) Nikas, Y. J.; Blankschtein, D. Langmuir 1994, 10, 3512.

Langmuir, Vol. 14, No. 20, 1998 5741

theoretical model that describes the behavior of polymer brushes in the presence of associating surfactants. Such a model can give insight into phenomena as colloidal stabilization by grafted polymers and detergents. We therefore analyze conformations of grafted polymers in contact with a reservoir of a solvent containing surfactants. Both grafting density regimes (i.e., individual grafted coils, mushrooms, and densely grafted chains, brushes) are considered. The analysis is based on the Flory-type mean-field approach. For the micellar-binding process, we rely on existing models and an average adsorption energy is assigned to a polymer-micelle complex of a fixed size and number of adsorbed segments. Our model predicts the amount of adsorbed surfactants and the dimensions of the grafted chains (brush thickness) as a function of adsorption energy, surfactant concentration (chemical potential), grafting density, and effective excluded volume parameters of micelle-micelle and micelle-monomer interactions. Model We consider a system of neutral polymer chains end-grafted onto a planar (inert) surface and immersed in a solution. Each polymer chain is composed of N monomer units of length l, which is the unit length in the system. All lengths are expressed in units l and all energy terms in units kT. The polymer chains are tethered to a surface with a grafting density σ, expressed in number of chains per unit area. The solvent is assumed to be good (athermal) for monomers of the grafted chains. Depending on the grafting density, two different regimes can be distinguished. If the grafting spacing is much larger than the characteristic size of an individual coil, then the polymer conformation is not affected by the interaction with other chains and is equivalent (within the accuracy of steric repulsion’s introduced by the impermeability of the grafting surface) to that of a swollen coil in the bulk of the solution. This regime is often referred to as the “mushroom regime”. In the opposite limit of dense grafting, the neighboring chains strongly overlap and form a polymer brush. In this brush regime the excluded volume repulsion between overlapping chains results in their extension perpendicular to the surface. In our analysis of the brush regime, we shall utilize the socalled boxlike Alexander-de Gennes model, according to which all the chains in the brush are stretched equally and uniformly.6,7 This model disregards the inhomogeneous distribution of the monomer density and unequal extension of chains in the brush, but provides a qualitatively correct description of the large-scale behavior of the brush and gives correct power dependencies of the average properties, such as the brush thickness or lateral pressure, on the main parameters.13 The grafted polymer layer is in contact with a bulk reservoir of a solvent containing surfactants and added electrolyte. We assume that the surfactant concentration is larger than the CAC, which marks the onset of cooperative association of surfactants in micellar form on the polymer.20,30 The surfactant micelles have affinity for the polymer chain due to screening of the hydrophobic areas on the micellar surface. The concentration of added electrolyte is assumed to be large enough to screen Coulombic repulsion between charged headgroups of surfactants on the scale of order of the micelle size. The chemical potential of a single surfactant is constant throughout the system and equals

µ0b + ln Fb

(1)

where Fb is the bulk concentration of surfactants and µ0b is the standard chemical potential of the surfactants in the bulk. Adsorption of micelles on the polymer chains effectively modifies both intramolecular and (in the brush regime) intermolecular interactions and affects the conformation of grafted chains. An equilibrium adsorbed amount of surfactant per chain as well as the equilibrium chain extension can be obtained on

5742 Langmuir, Vol. 14, No. 20, 1998

Currie et al.

the basis of analysis of the grand canonical free energy (in units kT per chain):

The total contribution of the micelle adsorption to the free energy of a polymer chain is

Ω ) Ωad + Ωmix + Ωelas + Ωosm

Ωad ) mNadUad ) NθUad

(2)

which includes four main contributions. The first term describes the gain in the free energy due to the adsorption of m micelles on the polymer chain. When surfactants aggregate on a polymer, a polymer strand of length Nad associates with one micelle, consisting of p surfactants. The free energy of adsorption per unit chain length equals U0ad. If there are m adsorbed micelles per chain, we thus have

Ωad ) mNadU0ad + mp∆µ

(3)

∆µ ) µ0b - (µ0b + ln Fb)

(4)

where

and µ0p is the standard chemical potential of a surfactant in a micelle of aggregation number p. We note that both polymer segments forming “trains” (sequences of monomers in direct contact with the micellar surface) as well as the segments forming adsorption loops contribute to Nad.35 The micelles are assumed to be spheres with a uniform radius Rm, which is large compared to the segment length l. The aggregation number p and the number of polymer units associated with one micelle Nad (the adsorbance capacity of one micelle) depend on the contact energy between polymer segments and the micellar surface. Our model of the polymer-micelle complex is based on existing theoretical models describing the adsorption of a flexible polymer chain on small colloidal particles.36-40 These give the structure and the free energy of the complex (polymer chain and adsorbing spherical particle) in terms of the particle radius and the energy of the monomer attraction to the surface. Here, we assume that the affinity of polymer segments with the micellar surface is strong enough so that the thickness D of the adsorbed layer (i.e., the characteristic size of loops), given by D ≈ (| - cri|)-1, is much smaller than the micellar size Rm.36-39 The critical contact energy cri, corresponding to the onset of adsorption of a polymer onto a particle surface, is of the order kT and provides compensation for local steric restrictions imposed by the localization of a monomer at the surface. In the framework of simple mean-field arguments37 the adsorbance capacity Nad per micelle, for good solvents, is proportional to the area of the micellar surface, Nad ≈ | - cri|Rm2. The free energy of adsorption of a polymer strand of length Nad scales as -Nad(| - cri|)2. The area per surfactant in an adsorbed micelle, a, is given as a ) 4πRm2p-1. Exact exponents of | - cri| dependencies may be different under different conditions, but this is not important for further analysis. Omitting numerical prefactors, we write the free energy of adsorption per polymer segment in an adsorbed strand, Ωad(mNad)-1, as

Ωad(mNad)-1 ≈ -( - cri)2 + (( - cri)a)-1 (µ0p - (µ0b + ln Fb)) ≡ Uad (5) In the following it is convenient to define the fraction of adsorbed polymer segments (both in trains and loops) as θ ) mNadN-1. (35) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, T. Polymers at Interfaces; Chapman & Hall: London, 1993. (36) Alexander, S. J. Phys. (Paris) 1977, 38, 977. (37) Pincus, P. A.; Sandroff, C. J.; Witten, T. A. J. Phys. (Paris) 1984, 45, 725. (38) Birshtein, T. M.; Borisov, O. V. Polym. Sci. U.S.S.R. 1986, 28, 2265. (39) Pincus, P. A. In Lectures on Thermodynamics and Statistical Mechanics XVII; Gonzalez, A. C., Varea, C., Eds.; World Publishing: Singapore, 1988. (40) Birshtein, T. M.; Borisov, O. V. Polymer 1991, 32, 923.

(6)

Detailed models which give µ0p and µ0b in terms of molecular parameters are available in the literature, which consider only the case of diluted polymer phases.34 As the behavior of closely packed micelles and polymers is investigated in this paper, an overall value for the adsorption energy is used as a starting point. Hence, we describe the strength of adsorption of micelles onto a polymer chain by an effective adsorption energy per monomer, Uad, which depends not only on the monomer-surface contact energy but also on the chemical potential of the surfactant in the bulk. The adsorption strength can therefore be tuned by variation of the temperature or the surfactant concentration. The second term in the free energy of a polymer chain, Ωmix, describes the translational entropy of m adsorbed micelles along the chain. The polymer contains NN-1 ad sites for the micelles, and thus

Ωmix )

{

N θ ln θ + (1 - θ) ln(1 - θ) Nad

}

(7)

The third term, Ωelas, accounts for the elastic free energy of stretching, due to the conformational entropy loss of the extended polymer. As a simplest approximation we assume the elasticity of the grafted chains to be Gaussian, regardless of the degree of coverage by micelles. The adsorption of micelles modifies the unperturbed dimension of a grafted chain and thus renormalizes its elasticity. The necklace chain consists of m micelles of diameter 2Rm (the adsorbed layer included) connected by bridges formed by the N(1 - θ) nonadsorbed segments of the chain (of unit length). The elastic free energy of the extended chain is given by

Ωelas ≈

3 H2 2 R02

{

R02 ) N(1 - θ) + m(2Rm)2 ) N (1 - θ) +

}

θ (2Rm)2 Nad

(8)

where H denotes either the size of the coil at low grafting densities (mushroom regime) or the height of the layer at high grafting densities (brush regime). We thus renormalize the elastic modulus of a necklace chain (i.e., account for the effective increase in the average segmental length). For simplicity we can write (2Rm)2N-1 ad as (β +1), so that

Ωelas )

H2 3 2 N(1 + βθ)

(9)

Hence, we have introduced the factor β to account for the renormalization of the chain elastic modulus due to the adsorption of micelles. As follows from the definition, β depends on the contact energy of a monomer with the micellar surface and, generally speaking, on the solvent strength for the polymer. The last contribution to the free energy, Ωosm, is the osmotic contribution, resulting from excluded volume effects. In the framework of our mean-field model we characterize the grafted chain by the average concentration of nonadsorbed segments N(1 - θ)V -1 (forming bridges) and that of micelles mV -1 in the volume V occupied by a chain. In the case of grafted coils the volume V is approximately H3. At high grafting density in the Alexander-de Gennes box model we have V ) Hσ-1, where σ denotes the number of grafted chains per unit area. The osmotic term Ωosm accounts for three types of binary (repulsive) interactions: (i) between two nonadsorbed monomers, (ii) between two micelles covered by adsorbed polymer, and (iii) between nonadsorbed monomers and covered micelles. The excluded


Langmuir, Vol. 14, No. 20, 1998 5743

volume parameters are denoted as ν˜ 0, ν˜ 1, and ν˜ 2 respectively. The osmotic free energy is given as

Ωosm )

{

1 ν˜ {N(1 - θ)}2 + ν˜ 1m2 + ν˜ 2m{N(1 - θ)} V 0

}

(10)

It is handy to write the osmotic energy in terms of an effective excluded volume parameter per monomer, νeff, which is a function of the degree of coverage θ.

Ωosm )

N2 N2 νeff(θ) ) {ν0(1 - θ)2 + ν1θ2 + ν2θ(1 - θ)} V V

(11)

˜ 2N -1 where we define ν0 ) ν˜ 0, ν1 ) ν˜ 1N -2 ad and ν2 ) ν ad . The equilibrium conformation for a certain set of parameters (N, σ, Uad, νi) is determined from the condition of the minimum of the free energy per chain. For a bare brush (no adsorption of micelles) the free energy of grafted polymers is dominated by two terms: the elastic energy and the excluded volume interactions between polymer segments. Minimization with respect to the height of the layer yields scaling relationships for the brush height H and the surface pressure π:6,7

H ∼ Nσ1/3 (12)

π ∼ Nσ5/3

In the case of adsorbed micelles we must minimize the free energy with respect to both the height and the degree of coverage (i.e., the number of adsorbed micelles per chain). This gives us two equations for θ and H:

(∂Ω ∂θ )

H

)0

and

∂Ω (∂H ) )0 θ

which must be solved simultaneously to obtain the equilibrium degree of coverage and the brush height. Mushroom Regime. At low grafting densities minimization with respect to H gives the coil radius as a function of θ:

H ≈ N3/5{(1 + βθ)νeff}1/5

(13)

where νeff is given by eq 11 and numerical prefactors are omitted. Combining this with the first equilibrium condition yields

( )

ln

{

ν′eff θ + NadUad + NadN -4/5 (1 - θ) {(1 + βθ)νeff}3/5 βν2/5 eff

}

2(1 + βθ)8/5

) 0 (14)

where we define

ν′eff )

∂νeff(θ) ∂θ

From this equation θ can be calculated for a given set of excluded volume parameters as a function of the effective adsorption energy Uad. In this way a binding isotherm is obtained because the effective adsorption energy depends on the surfactant concentration in the bulk (eq 5). Brush Regime. In the brush regime, the height and degree of coverage for a given adsorption energy are dependent on the grafting density σ. The brush height as a function of θ and σ follows from minimization of the free energy with respect to H:

H ≈ Nσ1/3{(1 + βθ)νeff}1/3

(15)

It is evident that for a constant degree of coverage the scaling behavior of the height as a function of the chain length and grafting density is equivalent to that of conventional polymer brushes, with an effective excluded volume parameter that depends on the degree of coverage. Minimizing the free energy

with respect to the degree of coverage gives an expression for θ as a function σ, Uad, νi:

ln

( )

{

ν′eff θ + NadUad + Nadσ2/3 (1 - θ) {(1 + βθ)νeff}1/3 βν2/3 eff

}

2(1 + βθ)4/3

) 0 (16)

Equations 15 and 16 describe (parametrically) the dependence of the brush height H on the grafting density σ and the effective adsorption energy Uad. These equations are solved numerically.

Estimation of Parameters To proceed further and to make a basis for comparison between the predictions of the above model and experimental results, it is necessary to estimate the molecular parameters that appear in the model, namely the radius of a micelle (Rm), the number of adsorbed segments per micelle (Nad), and the excluded volume parameters (ν0, ν1, ν2). This is done with help from experimental data, available for several surfactant and polymer systems.22-24,26,41 The system predominantly used experimentally is that of poly(ethylene oxide) (PEO) and SDS. The radius of the adsorbed micelle is typically between 2 and 5 nm as the aggregation number of SDS in the presence of PEO increases with increasing salt concentration.24,26,41 For a given salt concentration the number of surfactants per polymer-micelle complex is reported to be constant (i.e., independent of the degree of coverage).24 We therefore use a constant value for the micelle radius of 8 times the polymer segment length (l) in our model. In the case of PEO the value for l is 0.33 nm;42 the radius is therefore approximately 3 nm. As the micelle radius is large compared to the segment length and the affinity of the polymer segments to the micellar surface is sufficiently high, the curvature of the micellar surface plays no significant role in the adsorption of the polymer.37 As the effect of the curvature of the micellar surface is negligible, the number of adsorbed segments per micelle, Nad, is proportional to the surface area of a micelle. Furthermore, it is independent of the degree of coverage, as the micelle radius is also assumed constant. The number of adsorbed polymer PEO segments per SDS micelle reported by Gilańyi et al. is of the order of 120; Cabane gives a value of 60 and Shirahama’s value is close to 100.21-23 We therefore set Nad equal to 100 for all values of θ and Uad. We thus assume that Uad is tuned by variation of the surfactant bulk concentration Fb at constant . The polymer segments are treated as hard spheres without any long-range repulsion or attraction between themselves. The excluded volume parameter ν0 (the second virial coefficient of monomer-monomer interactions) under athermal solvent conditions then equals

ν0 )

1 4π 3 4π 8 r ) 2 3 monomer 6

(

)

(17)

as the radius of a monomer is one-half the unit length. Water at 298 K is a good solvent for PEO and therefore unity seems a reasonable value for ν0.43 In theta or poor solvent conditions the parameter ν0 becomes zero or negative, respectively. (41) Brown, W.; Fundin, J.; da Graca Miguel, M. Macromolecules 1992, 25, 7192. (42) Cao, B. H.; Kim, M. W. Faraday Discuss. 1994, 98, 245. (43) Molyneux, P. Water-Soluble Synthetic Polymers: Properties and Behaviour; CRC Press: New York, 1983.

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Currie et al.

For the micelle-micelle (ν1) and monomer-micelle (ν2) excluded volume interactions we distinguish between micelles comprised of charged surfactants and noncharged micelles. In the case of neutral surfactants the micellemicelle interactions are determined by the effective excluded volume of a micelle. This is larger than that of a bare micelle because of the swollen corona of adsorbed polymer segments. As a simplest approximation the interactions are treated as hard-sphere repulsions. This gives for the parameter of excluded volume micellemicelle interactions:

ν˜ 1 )

1 4π 16π (Rm + D)3 8 (R + D)3 ) 2 3 m 3

(

)

(18)

where D is again the thickness of the adsorbed layer. In the case of strong adsorption D can be neglected in comparison with Rm and ν1 can be written as

ν1 )

ν˜ 1

≈ 2

Nad

3 16π Rm ∼ Rm-1 3 N 2

(19)

ad

as Nad is proportional to Rm2. There is also a negative contribution to ν1 due to a bridging attraction (i.e., loops in a corona adsorbing on another micelle). Such an attraction can be important at intermicellar distances between centers of micelles of the order 2(Rm + D). Hence, in the absence of long-range Coulomb repulsion (adsorbed, uncharged micelles) the necklace of micelles may collapse. We remark that as the formation of bridges is not instantaneous, the swollen necklace conformation may exist as a metastable state. As stated, experimental results show surfactants that adsorb onto neutral polymers to be predominantly anionic. Several explanations have been suggested for this; the exact reason remains to be determined.26,31,33,34 It is therefore essential to include the micellar charge in the model and to estimate its effect on the parameters. We use the mean-field approximation and consider the effective excluded volume of a micelle to be equal to that of a sphere of radius Rm + κ-1 where κ-1 is the Debye length, defined as

e2 κ2 )

∑i zi2Fi kT

(20)

and Fi is the number density of ion i of charge zie. The Debye length is typically between 1 and 10 nm in a solution of ionic strength 10-1-10-3 M. In the case that the Debye length is large compared to the thickness of the adsorbed layer and is comparable to the micellar radius, the effective interaction parameter can be estimated as

ν1 ≈

16(Rm + κ-1)3 Nad2

(21)

and may exceed significantly the “geometrical” micellemicelle excluded volume given by eq 19. Furthermore, for most systems κ -1 . D so that bridging effects are neglected in further analysis. The second virial coefficient for monomer-micelle interactions is determined by the effective excluded volume of a micelle covered by an adsorbed polymer for a monomer unit. As the micellar radius is large compared to the

polymer segment length l and to the thickness of the adsorbed layer D, it follows that

ν2 )

ν˜ 2 1 4π 3 1 ) ∼ Rm R Nad 2 3 m Nad

(

)

(22)

This value is correct in the limit of total coverage of the micellar surface by the polymer. For partial coverage the above value is overestimated as there are attractive interactions between the uncovered hydrophobic patches on the micelle and the polymer segments.44 Because we restrict ourselves to neutral polymers, charges play no role in the polymer-micelle interactions. Therefore, ν2 is the same as for the neutral surfactants and is independent of the salt concentration. The last parameter which must be estimated is that of the mean adsorption energy per polymer segment, Uad. Shirahama,23 Cabane,24 and Gilańyi and Wolfram22 report adsorption of SDS on PEO in 0.1 M inert electrolyte (Cabane 0.4 M) to occur at approximately 1 mM SDS. The number of adsorbed segments is of the order 100. The free energy of adsorption is estimated to be 13kT per micelle by Shirahama, 10-20kT by Cabane, and 16kT by Gilańyi et al. This gives a value for the effective adsorption energy per polymer segment, Uad, of approximately 0.15kT. Naturally, this value depends on the surfactant concentration, as is evident in the definition of Uad (eq 5). Therefore, Uad is a variable in our model, but the values used are of the order 0.15kT. We note that when Uad is tuned by variation of the surfactant concentration, Nad remains unaffected, which is assumed in our model. This is not the case if the temperature or solvent composition is varied. Results Mushroom Regime. We first consider the case of single-grafted coils in a good solvent in the presence of adsorbing surfactants. Under the condition that the grafting spacing is much larger than the unperturbed coil radius, the coil radius is given by eq 13. In this case the osmotic contribution to the free energy, Ωosm, accounts for intramolecular excluded volume interactions modified due to adsorption of micelles. Minimization of the grand canonical potential with respect to the degree of coverage yields an equation for θ as a function of the adsorption energy and the excluded volume parameters, eq 14. We note that our results should also apply to free coils in solutions. For long chains (N . Nad) the third term in eq 14 is negligible compared to the first two terms, as a result of the factor N -4/5. In this case the adsorption is described by

ln

(

)

θ ) -NadUad (1 - θ)

(23)

that is, a Langmuir-like isotherm. In Figure 1 the degree of coverage θ is given as a function of the mean adsorption energy per segment for three different values of ν1. The chain length is set 10000, ν0 is unity, ν2 is set 20, Nad is 100, and Rm is 8. The values for the chain length, ν0, Nad, and Rm are default for all the following results. The (44) In the case of strong adsorption the dense corona of swollen loops ensures strong monomer-micelle repulsion at distances of an order (Rm + D). The corona, therefore, prevents extra unadsorbed monomers approaching the micellar surface, where attraction plays the dominant role. Hence, as a first approximation we neglect the attractive part of interaction between nonadsorbed monomers and micelles..


Figure 1. Coverage of a single coil as a function of the adsorption energy per monomer for three values of ν1 (0.1, 5, 100). N ) 10000, Nad ) 100, ν0 ) 1, and ν2 ) 20 (default values).

Figure 2. Reduced size of a single coil (H/H0) as a function of the adsorption energy per monomer for the three values of ν1 used in Figure 1. H0 is the size of a bare coil without adsorbed micelles. Other parameters are the same as those in Figure 1.

adsorption as shown in Figure 1 is continuous and virtually independent of the excluded volume parameters. This proves that the contribution of intramolecular excluded volume interactions is negligible in comparison to the adsorption contribution for long chains. The shape of the isotherm is in qualitative agreement with the experimental isotherms of PEO and SDS in solution. Thus, the degree of coverage of single-grafted coils depends only on the mean adsorption energy (i.e., the chemical potential of the surfactant). Whereas the degree of coverage depends solely on Uad and is independent of the excluded volume parameters, the size of the grafted polymer is affected by the adsorption of micelles and depends on θ via the effective virial coefficient νeff(θ). In Figure 2 the size of the coil normalized with respect to the size of a bare coil, H/H0, is plotted as a function of Uad for a constant ν2 parameter (ν2 ) 20) and varying ν1. When ν1 is much larger than ν0 and ν2, the coil radius increases monotonically with increasing adsorption strength, as the excluded volume interactions between the micelles are large compared to the segment-segment and segment-micelle interactions. The strong micellemicelle interactions provide strong swelling of the coil at large degrees of coverage. When both ν1 and ν2 are smaller than ν0, the opposite is the case. The coil wraps itself around the micelles and effectively shrinks with increasing adsorption. This behavior may occur for adsorbing neutral micelles or charged micelles at high ionic strength. A interesting case is that of ν2 larger than both ν0 and ν1. Under such conditions, a maximum occurs in the coil radius at an intermediate coverage, as shown in Figure 2. At low coverage the polymer has wrapped itself around

Langmuir, Vol. 14, No. 20, 1998 5745

a few micelles. This has effectively shortened the chain, but the osmotic interactions between uncovered segments and the adsorbed micelles result in an overall swelling of the chain with increasing coverage. At high coverage the chain is nearly totally adsorbed on micelles, and the complex consists effectively of NN-1 ad segments of length 2Rm. In this regime the size decreases with increasing coverage. As the ν2θ(1 - θ) term dominates the osmotic interactions, the maximum in size occurs at Uad ≈ 0 (i.e., half coverage). It is interesting to compare the above results with experimental data. In the literature viscosity measurements on hydrophilic polymers in the presence of aggregating surfactants have been presented. At low polymer concentrations, the reduced viscosity of a polymer solution is a measure for the size of the polymer coils. Wang and Olofsson found that the reduced viscosity of EHEC, a hydrophilic cellulose polymer, at high SDS concentrations is lower than that of the bare polymer.45 They conclude that adsorption leads to a more compact polymer complex, albeit the electrostatic repulsion between the micelles. In contrast, various authors find the relative viscosity of PEO with SDS to have a maximum as a function of the SDS concentration at a constant high ionic strength.41,46 Such a maximum in the viscosity corresponds to a maximum in the coil size. The decrease in coil size is ascribed to increasing electrostatic screening due to increasing surfactant concentration, with a subsequent decrease in coil size. However, Brown et al. found distinct maxima in the relative viscosity of PEO/SDS at constant high electrolyte concentrations.41 Hence, the interpretation in terms of modified screening is questionable. We suggest that such experimental maxima correlate with the maxima in the coil size as predicted by our model. The above examples show that the behavior of polymer coils with adsorbed micelles varies with the choice of system and that such variation is well-modeled with different sets of ν parameters. Brush Regime. When the spacing between grafted chains is considerably smaller than the undisturbed coil radius, the chains are stretched normal to the surface and form brushes. The height depends on the length of the chain, the grafting density, and the effective excluded volume parameter νeff, as given in eqs 11 and 15. When no adsorbed micelles are present on the chain, νeff reduces to the segmental excluded volume, ν0. In the case of isolated grafted coils the degree of coverage depends solely on Uad and therefore νeff (and the coil size) are determined by the adsorption strength. This is a result of the weakness of the excluded volume interactions as compared to the energy of micelle adsorption in a dilute coil. On the contrary, if grafted chains overlap strongly and form a brush, both excluded volume interactions and the adsorption energy per chain scale linearly with the chain length N. In this regime the interplay of these interactions determines the degree of coverage. Hence, for brushes the degree of coverage is also a function of the grafting density σ. Equation 16 gives the equilibrium value for θ as a function of σ. It is the change in θ as a function of the grafting density and the subsequent change in νeff, that determines the overall behavior of brushes with adsorbed micelles. In the following we investigate several regimes of varying grafting densities and excluded volume parameters. We start with charged micelles adsorbed with a strength comparable to that of SDS and PEO at relatively low (45) Wang, G.; Olofsson, G. J. Phys. Chem. 1995, 99, 5588. (46) Chari, K.; Antalek, B.; Lin, M. Y.; Sinha, S. K. J. Chem. Phys. 1994, 100, 5294.

5746 Langmuir, Vol. 14, No. 20, 1998

Currie et al.

(a)

Figure 3. Coverage of a polymer brush as a function of the grafting density σ for four values of ν1 (15, 25, 50, 100), Uad ) -0.05, and ν2 ) 10.

electrolyte concentrations. In this case ν1 is large compared to ν2 due to electrostatic repulsion. Intuitively, the behavior in the asymptotic grafting regimes is clear. At low grafting densities the coverage and height of the polymer layer is equal to that of isolated coils; in the limit of high grafting densities the micelles are “squeezed” out of the polymer layer and a bare brush remains. It is the desorption process of the micelles in the intermediate density regime that is of interest to us. The degree of coverage of the chain by micelles as a function of the grafting density for several values of ν1 is shown in Figure 3. The coverage decreases with increasing grafting density from approximately unity at low grafting densities to zero at high grafting densities. The limit at low grafting densities is apparent from eq 23.

θ|σf0 ≈ (1 + eNadUad)-1

(24)

The decrease in coverage with increasing grafting density is continuous. At a fixed grafting density the coverage decreases with increasing ν1 (i.e., decreasing electrolyte concentration). Thus, the adsorbed micelles gradually leave the polymer layer due to the increase in the micellemicelle electrostatic repulsion. The height of the polymer layer depends strongly on the degree of coverage and on the values of ν0, ν1, and ν2. In Figure 4a the height is plotted as a function of the grafting density for several values of ν1 using the same adsorption strength Uad ) -0.05 and ν2 ) 10 as in Figure 3. In Figure 4b the same results are plotted on a double logarithmic scale. At high degrees of coverage (i.e., relatively low grafting densities) the height scales as (σν1)1/3, as shown in Figure 4b. A line corresponding to this power law (i.e., slope 1/3) is drawn, by way of illustration. The brush with adsorbed micelles thus behaves as a neutral brush with a renormalized excluded volume parameter. At high grafting densities the coverage is close to zero (Figure 3) and the scaling behavior of bare brushes is recovered (i.e., H/N ∼ (σν0)1/3). In the intermediate density regime the height reaches a (local) maximum and thereafter the brush shrinks with increasing grafting density. The physical interpretation of this maximum in the height is straightforward: the loss of repulsive micelle-micelle interactions due to desorption is not compensated for by the increase in segmental osmotic interactions. The maximum in the brush height is shifted to lower grafting densities with increasing ν1 (i.e., decreasing salt concentration). This is shown in Figure 5. The maximum in the brush height is reminiscent to the maximum found for weak polyacid or

(b)

Figure 4. (a) The height of the brush as a function of the grafting density σ for four values of ν1. The parameters are the same as those in Figure 3. (b) Same results as those in Figure 4a plotted double logarithmically. A line corresponding with a power law H ∼ σ1/3 is drawn as an illustration.

Figure 5. The grafting density σmax at which a (local) maximum in the brush height is found as a function of the micelle-micelle excluded volume parameter. The plot is on a double logarithmic scale. Three adsorption strengths are shown (-0.05, -0.2, -0.5), ν2 ) 10; other parameters are default. A line corresponding with the power law σmax ∼ ν1-1 is drawn as an illustration.

polybase brushes at low salt concentrations, so-called annealed polyelectrolyte brushes.47-49 In this case the charge density of the brushes decreases with increasing grafting density. As the chains are stretched by electrostatic interactions of the dissociated segments, the brush height decreases with increasing grafting density. (47) Israe¨ls, R.; Leermakers, F. A. M.; Fleer, G. J. Macromolecules 1994, 27, 3087. (48) Zhulina, E. B.; Birshtein, T. M.; Borisov, O. V. Macromolecules 1995, 28, 1491. (49) Lyatskaya, Y. V.; Leermakers, F. A. M.; Fleer, G. J.; Zhulina, E. B.; Birshtein, T. M. Macromolecules 1995, 28, 3562.


Langmuir, Vol. 14, No. 20, 1998 5747

It is interesting to evaluate the relationship between the density at which this local maximum in the brush height occurs and the value of ν1. The minimalization of the grand canonical potential Ω with respect to H yields H ∼ N(σνeff)1/3. As we are considering grafting densities around the maximum in H, the elastic free energy in this grafting regime is approximately constant. Therefore, the fourth term in eq 16 is neglected. The micelle-micelle excluded volume interactions are strong compared to the polymer-micelle and polymer-polymer interactions so that the effective volume parameter is reduced to νeff(θ) ≈ ν1θ2. We write σ in eq 16 as a function of the (absolute) adsorption strength, degree of coverage, and excluded volume interactions and insert the result in the expression for the brush height.

(| |

H ≈ N Uad -

(

1 θ ln Nad (1 - θ)

))

1/2

(θ(1 + βθ))1/2

(25)

Figure 6. The coverage of a brush as a function of the adsorption strength for three values of ν1 (20, 40, 100). The grafting density equals 1.5 × 10-3; other parameters are default. Dotted lines show the abrupt transition with decreasing adsorption strength from a covered brush to a bare brush for small values of ν1.

In Figure 3 it is shown that the coverage decreases monotonically with increasing grafting density. Therefore, the maximum in the height as a function of σ can also be found by setting the derivative of H with respect to θ zero. For simplicity β is set to zero and Nad to unity; this affects the results only in a quantitative manner. This yields for the coverage at which the maximum is found

{ (| | ( θ Uad - ln

θ (1 - θ)

))} (| | ( -1/2

)

θ (1 - θ) 1 ) 0 (26) 1-θ

Uad - ln

)

This shows that, in a first approximation, the coverage at which the maximum in the brush height occurs depends solely on the adsorption strength. The grafting density corresponding with the maximum for a given adsorption strength is then found to scale as σmax ∼ ν1-1. In Figure 5 the density at which the maximum in the height is found is shown in a double logarithmic plot as a function of ν1 for constant values of Uad. The total analytical expressions give an effective exponent for σmax as a function of ν1 between -1.07 and -1.12 for the used adsorption energies. A straight line corresponding to the power law is drawn as an illustration in Figure 5. The results agree well with the above power law. Moreover, the conclusion that the grafting density at which the osmotic interactions between micelles are at a maximum increases with decreasing ν1 (i.e., increasing salt concentration) is intuitively sound. A different situation arises when the polymer-micelle excluded volume interactions, denoted by ν2, are comparable or larger than the micelle-micelle excluded volume interactions, denoted by ν1. This is the case for adsorbed micelles at high electrolyte concentrations when the osmotic interactions between charged micelles are close to those of uncharged micelles. We remind the reader that ν1 and ν2 are “effective” excluded volumes, renormalized by Nad2 and Nad, respectively. In Figure 6 the coverage θ is shown as a function of the adsorption strength at constant grafting density σ ) 1.5 × 10-3 and ν2 ) 20. The coverage no longer decreases monotonically with increasing Uad, but shows a van der Waals loop. This indicates a first-order phase transition in which the coverage jumps abruptly from a high value at high adsorption strength to a low value at low adsorption strength. For values of ν1 significantly larger than ν2 the

Figure 7. The coverage of a brush for three values of ν1 (20, 40, 100) as a function of the grafting density. The adsorption strength is -0.15; other parameters are default. Dotted lines show the abrupt transition with increasing grafting density from a covered brush to a bare brush.

coverage increases continuously with the adsorption strength, for values of ν1 comparable to or smaller than ν2 the coverage is discontinuous. In Figure 6 the van der Waals loop of the coverage is plotted with a line, the abrupt transition from strongly stretched, highly covered to a collapsed, lowly covered polymer layer is denoted by a dotted line.50 The same behavior is found for varying grafting density at fixed adsorption strength. In this case a grafting density can be found at which an abrupt transition from a covered chain to a bare, collapsed chain occurs. This is shown in Figure 7, where the coverage is plotted as a function of the grafting density for various values of ν1. At high values of ν1 (i.e., low electrolyte concentration) the transition is continuous, at low values discontinuous. Some caution is necessary; the model assumes the aggregation number of surfactants in a micelle and amount of polymer segments adsorbed per micelle to be constant, irrespective of the electrolyte concentration. Experimentally, it is found that these values can shift somewhat for low and high salt concentrations.26 Nevertheless, this shift is expected not to change the qualitative picture of an abrupt deswelling of the polymer layer. (50) The value of the adsorption energy at which the transition from strongly stretched, highly covered, to collapsed, lowly covered chains occurs is found by using the Maxwell equal area construction or by equating the two minima in the free energy of the highly and lowly covered conformation.

5748 Langmuir, Vol. 14, No. 20, 1998

Figure 8. The coverage as a function of the adsorption strength for four values of grafting density (10-4, 5 × 10-4, 10-3, 2 × 10-3). The micelle-micelle interaction ν1 is set at 40; other parameters are default.

The character of the adsorption-desorption transition induced by the variation of Uad for a fixed grafting density is determined by the relative values of the excluded volume parameters. The reverse also applies: for a fixed set of ν parameters the grafting density σ determines whether the transition induced by the variation of Uad is continuous or not. Figure 8 shows the coverage as a function of the adsorption energy for several grafting densities. At low densities the transition is continuous (the asymptotic case is naturally that of a single-grafted coil); at high densities the transition is abrupt. The underlying reason of this discontinuity in the coverage is the dominant effect of the polymer-micelle excluded volume interactions. These interactions depend as ν2θ(1 - θ) on the coverage and clearly are the strongest at 50% coverage. When the grafting density at given adsorption strength increases, the excluded volume interactions become increasingly important. As shown above, strong osmotic interactions between the adsorbed micelles result in a gradual desorption. If the micellemicelle interactions are weak in comparison to the polymer-micelle interactions, then the overall excluded volume interactions effectively increase with decreasing coverage. This results in an instability of the conformation corresponding to intermediate degrees of loading, θ ∼ 1 - θ, and to an abrupt transition from densely covered polymers to lowly covered polymers. As the osmotic interactions are increasingly important with increasing grafting density, the abrupt transition for fixed ν parameters occurs at high densities whereas at low grafting densities it is continuous. For a given set of ν parameters the grafting density σ ) σtr at which the free energies of the loaded and unloaded phase are equal (the transition point) depends on Uad. The function σtr(Uad) corresponds to the line of the firstorder phase transition in the σ,Uad-plane (dashed line in Figure 9). The spinodal lines, σ1 and σ2 defined by (∂σ/ ∂θ)σ1,σ2 ) 0, are also shown as functions of Uad. Both spinodal lines and σtr(Uad) end in the critical point (σtr,Ucr). The value for Ucr for the parameter set in Figure 9 is approximately -0.05. In the case of adsorption energies above the critical value the deswelling of the brush due to the desorption of micelles occurs continuously with increasing grafting density or adsorption energy. In contrast, if the adsorption energy is below its critical value, then an increase of the grafting density results in a crossing of the σtr line at which a jumpwise transition occurs. The position of the critical point (σtr,Ucr) and shape of the phase diagram are determined by the set of ν parameters. With increasing ν1 (i.e., decreasing ionic

Currie et al.

Figure 9. The phase diagram of a brush with adsorbed micelles in the σ, Uad-plane. The ν parameters used are the same as those for Figure 8. The upper and lower spinodal density are shown with straight lines and the transition density with a dotted one. The critical adsorption energy is found to be approximately -0.05.

Figure 10. The height of a brush in the box model as a function of the adsorption strength for three values of ν1. The values used for all parameters are the same as those in Figure 6.

strength) the (absolute) value of Ucr becomes larger (not shown). As expected, the desorption of micelles with increasing grafting density occurs continuously over a larger range of adsorption energies. Also, for a given Uad < Ucr, both the spinodal densities, as the transition grafting density, decrease with increasing ν1. When ν1 remains constant and ν2 increases, the transition density remains relatively unaffected (not shown). The spinodal curve, however, widens greatly: the grafting densities at which the layer is unstable are more extreme. In this case the unfavorable monomer-micelle osmotic interactions in the regime of intermediate coverage are stronger, and the first-order transition in the coverage is more pronounced. Brush Height. In the box model the discontinuity in the coverage is reflected in the behavior of the brush height. In Figure 10 the height is shown as a function of the adsorption energy for a fixed grafting density above the critical value. The brush height is clearly seen to follow a van der Waals loop similar to the coverage for small values of ν1. The transition occurs as a collapse from a strongly stretched polymer layer loaded with micelles to a weakly stretched layer of bare chains. Nevertheless, the conclusion that the overall height of the grafted layer decreases abruptly with increasing adsorption energy (or grafting density) appears unrealistic. As we apply a box model, the abrupt transition from a densely covered conformation to a sparsely covered one occurs simultaneously in all chains forming the brush. One would, however, expect a transition regime with a coexistence of strongly stretched, covered chains and less


Langmuir, Vol. 14, No. 20, 1998 5749

Figure 11. Illustration showing the two-phase regime: the stretched and densely covered phase has a brush height H, whereas the collapsed, sparsely covered phase has a height H′.

stretched, sparsely covered chains to be physically more realistic. This should correspond to a situation as shown in Figure 11. In a first-order approximation such a coexistence of collapsed and stretched chains in the grafted layer can be modeled with a two-box model. The fraction R of chains is strongly loaded with adsorbed micelles, and the corresponding grafting density of the extended phase is Rσ. The fraction (1 - R) of chains is sparsely covered by micelles and the corresponding grafting density of the collapsed phase is given (1 - R)σ. The model thus assumes a bimodal distribution of chain extension in the brush. This distribution allows minimalization of the total free energy of the brush with respect to the fraction of extended chains. We stress that the brush remains laterally homogeneous on a scale larger than (σ(max(R,1-R)))-1/2. In the two-box model the assumption is made that the contribution of monomers belonging to extended chains to the overall monomer density in the collapsed phase is negligibly small. This assumption is reasonable for large differences in height, and thus coverage, of both phases. The two-box model is, therefore, only a reasonable approach for intermediate values of R. However, the qualitative behavior of the grafted polymers is expected to be correctly modeled with the two-box model. The same expressions as those used previously are applied to the two-box model (eqs 15 and 16). We must correct the total grafting density for the fraction of chains in both phases. For a given set of input parameters (σ, Uad, νi) we now have three conditions which must be obeyed. The free energy of both the stretched and the collapsed phases must be minimized with respect to the corresponding height and coverage of the phases. Also, to have equilibrium between both phases the free energies of the chains in both phases must be equal, that is, st col ΩRσ,H,θ ) Ω(1-R)σ,H′,θ′

(27)

st is the free energy of a chain in the stretched where ΩRσ,H, col the free phase of height H and coverage θ, and Ω(1-R)σ,H′,θ′ energy of a chain in the collapsed phase of height H′ and coverage θ′. The overall brush height is an average of both phases. Experimentally, it depends on the means of investigation: a technique sensitive to the polymer density measures a value close to H′, as the polymer density is low in the stretched phase. A technique sensitive to the adsorbed micelles measures H, as the coverage in the collapsed phase is close to zero. Here, we use the first

Figure 12. The fraction R of chains in the extended phase in the two-box model as a function of the grafting density. The adsorption strength is -0.15 and ν1 ) 20; other parameters are default.

Figure 13. The average height H h in the two-box model as a function of the grafting density. Parameters are the same as those of Figure 12. The dashed line is the value of H in the box model.

moment of the monomer density distribution to follow its behavior, that is,

H h ) RH + (1 - R)H′

(28)

In Figure 12 the behavior of the fraction of extended chains, R, is shown as a function of the grafting density at a given adsorption strength Uad ) -0.15. The values for the input parameters are the same as those for Figure 7, with ν1 equal to 20. The fraction of extended chains drops from unity at low grafting densities, that is, σtr ) 2.7 × 10-3 to close to zero at high grafting densities. In Figure 13 the average height of the brush in the two-box model is plotted as a function of the grafting density. To compare the results, the height as found with the one-box model is also shown. It is evident that the variation of the height changes from abrupt to continuous when the coexistence of both phases is taken into account. The average height at large grafting densities is close to that of the uncovered phase in the one-box model. It must be emphasized that the two-box model is only valid when the two phases differ strongly (i.e., far from the critical point). Also, both limits of R or (1 - R) approaching zero are not well-defined: in such cases the condition of overlapping of chains in the extended (or collapsed) phase is violated, and thus our expressions for the free energy are no longer correct. Moreover, as the grafting density of the collapsed chains increases the difference in height of both phases decreases and the assumption of noninteracting phases is not valid. Neglecting the interactions between both phases underestimates the total free energy. The underestimation of the

5750 Langmuir, Vol. 14, No. 20, 1998

free energy of chains in phase a increases in magnitude as the fraction of chains in phase a approaches zero. In more advanced models the interactions between both phases should be taken into account. The polymer density of the collapsed phase near the interface should naturally affect the coverage of the stretched phase in that region. Nevertheless, this simple approach does show that a bimodal distribution of chains between a collapsed and extended phase is found, which shifts toward the collapsed phase with increasing grafting density. The same applies naturally when the grafting density is constant and the adsorption strength is varied; see Figure 6. We thus find that with increasing adsorption energy or grafting density the polymer brush transforms from a densely covered to a bare brush. The character of the transition is determined by the extent of the various excluded volume interactions. Although the height of a single chain in the brush exhibits a bimodal distribution in the transition region, the overall height is found to change continuously with increasing grafting density or adsorption strength. One can ask whether the assumptions made in the model merely modify the quantitative results, or also modify the qualitative picture of the grafted polymer-micelle complexes. The box model assumes a constant segment density throughout the grafted layer, whereas more extensive models show the monomer density to decrease with increasing distance from the grafting surface. As a result, one can expect a nonuniform coverage of grafted chains by micelles. More detailed calculations are necessary to determine how this density gradient changes the behavior of the brushes. The two-box model is a crude representation of the polymer layer; in a more advanced model the interactions between the two phases must be taken into account. Also, a nonuniform coverage of the extended phase is necessary. Such problems may be answered in future work. Conclusions We have investigated the behavior of grafted polymers in contact with a surfactant solution in a good solvent. The surfactants are adsorbed on the polymer chain in micellar form. In the model the size of the micelles and number of adsorbed polymer segments are assumed to be independent of the grafting density or the degree of coverage of the polymers. An average adsorption energy per adsorbed polymer segment is used to describe the polymer-micelle affinity. The monomer-monomer, monomer-micelle, and micelle-micelle excluded volume interactions are described with mean-field excluded volume parameters. The degree of coverage of the polymers with micelles and dimensions of the polymer chains (at high grafting densities the height of the polymer brush) is investigated as a function of the adsorption strength, grafting density, and excluded volume parameters. The coverage of single coils with surfactants is found to be virtually independent of the excluded volume interactions and to have a Langmuir-like isotherm, that is, increases monotonically with increasing adsorption strength (surfactant concentration). This agrees qualitatively with experimental adsorption isotherms for hydrophilic polymers and associating surfactants. The size of the coils depends strongly on the strength of the excluded volume interactions. In the case of large micellemicelle interactions (charged micelles at low electrolyte concentrations) the coil swells with increasing coverage. In the case of relatively large monomer-monomer and small micelle-micelle excluded volume interactions, the coil shrinks. If the monomer-micelle interactions are dominant, then the size of the coil shows a maximum as a function of the adsorption strength at a coverage around

Currie et al.

one-half. Experimentally polymers in a surfactant solution are found to swell, shrink, or have a maximum, depending on the choice of system. At large grafting densities the polymers form brushes in which the chains are extended normal to the surface due to osmotic interactions. The coverage of the polymers by micelles is found to decrease as a function of the grafting density; the nature of the transition from high coverage at low grafting densities to low coverage at high densities depends on the relative strength of the excluded volume interactions. Large excluded volume interactions between the micelles result in a continuous decrease in coverage. At comparatively low densities the classical scaling of the height of brushes as a function of the grafting density is found, with a renormalized excluded volume parameter. The height of the polymer brush exhibits a maximum in this regime because the effective osmotic interactions in the brush decrease due to the progressive desorption of the micelles and the brush shrinks. When the coverage is close to zero, the classical scaling of the height of “bare” brushes as a function of the grafting density is retrieved. The grafting density at which the maximum is found at a given adsorption strength scales approximately inversely proportional to the micelle-micelle excluded volume parameter. No experimental reports of these effects seem to exist; these theoretical predictions are to be confirmed experimentally. For the case that the osmotic interactions between micelles are comparable or smaller than the monomermicelle interactions, our model exhibits a van der Waals loop in the dependence of the coverage on the grafting density. This indicates a first-order phase transition between a highly covered and a lowly covered brush conformation. This transition occurs either with increasing grafting density at a fixed adsorption strength or with decreasing adsorption strength at a fixed grafting density. The abrupt transition, induced by variation of the adsorption strength, is only found for polymers grafted at high densities; for a given set of excluded volume parameters a critical grafting density is found, above which the transition is of the first-order and below which it is continuous. If the constraint of equal stretching of the chains in the brush (the box model) is released, then an extended (loaded) and collapsed (unloaded) phase coexist in the brush, due to unequal stretching of the chains forming the brush. In the coexistence regime the brush is characterized by a bimodal distribution of the chain ends with respect to the distance from the grafting surface. As a result, the average chain extension (brush height) varies continuously with varying grafting density and/or adsorption strength in the coexistence regime, due to the gradual repartitioning of chains between the extended and collapsed phase. The general conclusion is that within the limits of our model brushes in contact with surfactants show a rich phase behavior. It remains to be seen if the results of our model are validated experimentally. To this end wellcontrolled brush-surfactant systems are necessary. The coexistence of the collapsed and swollen phases in a brush is an interesting issue which may be investigated by a technique sensitive to the intrinsic brush structure (e.g., SANS). Clearly, polymer-surfactant systems are a promising field for investigation and more studies are necessary to examine the properties fully. Acknowledgment. O. V. Borisov acknowledges G. J. Fleer and M. A. Cohen Stuart, University of Wageningen, for hospitality, and NWO for financial support. LA980168M

End-Grafted Polymers with Surfactants: A Theoretical Model

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