Counterion Localization in Solutions of Starlike Polyelectrolytes and

Exact numerical solutions are obtained for starlike polyelectrolyte molecules composed of ... (2, 9) Strong support for this effect was found by Monte...
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Langmuir 2008, 24, 10026-10034

Counterion Localization in Solutions of Starlike Polyelectrolytes and Colloidal Polyelectrolyte Brushes: A Self-Consistent Field Theory Frans A. M. Leermakers,† Matthias Ballauff,‡ and Oleg V. Borisov*,§ Laboratory of Physical Chemistry and Colloid Science, Wageningen UniVersity, The Netherlands, Physikalische Chimie I, UniVersita¨t Bayreuth, Bayreuth, Germany, Institut Pluridisciplinaire de Recherche sur l’EnVironnement et les Mate´riaux, UMR 5254 CNRS/UPPA, Pau, France ReceiVed April 28, 2008. ReVised Manuscript ReceiVed June 12, 2008 A quantitative analysis of the distribution of counterions in salt-free solutions of colloidal polyelectrolyte brushes and starlike polyelectrolytes is performed on the level of the Poisson-Boltzmann approximation. Exact numerical solutions are obtained for starlike polyelectrolyte molecules composed of f ) 20, . . ., 50 arms with a fixed fractional charge R per segment by applying the self-consistent field method of Scheutjens and Fleer (SF-SCF). The Wigner-Seitz cell dimension defines the concentration of polyelectrolyte stars in the system. The numerical results are compared to predictions of an analytical mean field theory and related to experimental observations on the osmotic pressure in solutions of starlike polyelectrolytes and colloidal polyelectrolyte brushes.

1. Introduction Ionic polymers, or polyelectrolytes, constitute an important class of functional polymers, particularly for aqueous solutions. Indeed, most biomacromolecules, including proteins, nucleic acids, and polysaccharides, carry ionizable groups such as carboxylate, amino, phosphate, and so forth. These, and also synthetic polyelectrolytes (PEs) such as sodium poly(acrylate), find many technological applications, from oil recovery to nanomedicine. Both experimental and theoretical studies aim to understand the conformational properties of polyelectrolytes in aqueous solutions. Of key importance is the ability of these polymers to respond, by conformational changes, to variations in their environmental conditions. Truly intriguing and highly specific properties of polyelectrolyte solutions arise from the interplay between long-ranged electrostatic interactions and the chemical connectivity of their ionic monomers that form the polymer chains. Even though significant progress in the understanding of properties of linear polyelectrolytes has been achieved in the past few years, the corresponding behavior of more complex macro- and supramolecular architectures that comprise ionic macromolecular chains as one of its structural elements remains poorly understood. Examples of such architectures are given by colloidal polyelectrolyte brushes, randomly or regularly branched polyelectrolytes, and nanostructures formed in aqueous environments via the selforganization of amphiphilic ionic block copolymers.1–3 In living organisms, there are many examples of topologically complex structures composed of biopolymers. A few examples may suffice: branched polysaccharides decorate the extracellular bacterial surfaces, which is important for recognition; the presence of the aggrecan protein in articular cartilage provides an almost frictionless weight-bearing surface in joints; rodlike neurofila* Corresponding author. E-mail: [email protected]. † Wageningen University. ‡ Universita¨t Bayreuth. § Institut Pluridisciplinaire de Recherche sur l’Environnement et les Mate´riaux. (1) Ballauff, M.; Borisov, O. V. Curr. Opin. Colloid Interface Sci. 2006, 11, 316–323. (2) (a) Borisov, O. V. J. Phys. II (France) 1996, 6, 1–19. (b) Borisov, O. V.; Zhulina, E. B. Eur. Phys. J. B 1998, 4, 205–217. (3) Borisov, O. V.; Daoud, M. Macromolecules 2001, 34, 8286–8293.

ments give an axon its mechanical strength through the brushlike action of the projection domains of a triplet of neurofilament proteins. Indeed, many of these biological functions depend on suprabiomolecular structures and rely heavily on their ability to respond to small variations in the local environmental conditions.4 A thorough, on the biomacromolecular level, understanding of the physical mechanisms and their corresponding regulatory functions is of key importance. The molecular structure of biopolymers is often much more complex than polymers that can be made in the chemical laboratory. However, recent progress in controlled radical polymerization has made it possible to synthesize ionic macromolecules with a controlled topological complexity. As a result, ionic block-copolymers, starlike polyelectrolytes, and colloidal and molecular polyelectrolyte brushes have become available.5,6 On the one hand, these molecules provide an excellent chance to study model systems systematically. On the other hand, these macromolecules are expected to find numerous applications in emerging domains of nanomedicine and nanotechnology. Some of the most distinctive features of branched polyelectrolytes arise as a result of a strong localization of small mobile counterions that invariably are present in the aqueous solution to ensure the total electroneutrality. It has been proven that, similarly to strongly charged colloidal particles,7 branched PE architectures are capable of maintaining a high local electrostatic potential. This electrostatic potential attracts the counterions and is able to overcome the translational entropy of these ions. As a result, the ions remain preferentially localized in the vicinity of the macroion.2,3,8,9 This effect is most pronounced in salt-free solutions, that is, in the absence of added electrolyte when the concentration of mobile ions in the bulk of the solution is extremely low. In contrast to classical hard (4) Zhulina, E. B.; Leermakers, F. A. M. Biophys. J. 2007, 93, 1421. Zhulina, E. B.; Leermakers, F. A. M. Biophys. J. 2007, 93, 1452. (5) Zhang, M.; Mu¨ller, A. H. E. J. Polym. Sci., Part A: Polym. Chem. 2005, 43, 3461–3481. (6) Fo¨rster, S.; Abetz, V.; Mu¨ller, A. H. E. AdV. Polym. Sci. 2004, 166, 173– 210. (7) Alexander, S.; Chaikin, P. M.; Grant, P.; Morales, G. J.; Pincus, P.; Hone, D. J. Chem. Phys. 1984, 80, 5776. (8) Pincus, P. Macromolecules 1991, 24, 2912–2919. (9) Klein Wolterink, J.; Leermakers, F. A. M.; Fleer, G. J.; Koopal, L. K.; Zhulina, E. B.; Borisov, O. V. Macromolecules 1999, 32, 2365–2377.

10.1021/la8013249 CCC: $40.75  2008 American Chemical Society Published on Web 08/13/2008

Counterion Localization in Solutions

colloidal particles, the branched polyelectrolyes can accommodate a huge fraction of its counterions in the interior volume of the macromolecule where, because of the swelling the monomer concentration is typically relatively low. Remarkably, these “trapped” counterions retain some translational freedom inside the volume occupied by the macroion, whereas the strong Coulomb attraction inside the highly branched (and, as a result, heavily charged) macroion prevents them from escaping from the interior and/or immediate vicinity of the macroion. As a result, branched ionic macromolecules swell strongly because of the osmotic pressure exerted by the counterions localized in their interior volume. This property of branched macroions to maintain a virtually constant ionic strength, ensured by the counterions trapped in this interior volume, is of special interest for potential applications where a controlled (buffered) microenvironment is essential. The concept of counterion localization in starlike PE was first proposed on the basis of scaling-type arguments8,2 and was found from a Poisson-Boltzmann approach in the two-phase approximation.2,9 Strong support for this effect was found by Monte Carlo10 and molecular dynamics simulations.11–13 Unambiguous evidence of the strong localization of counterions is found from osmotic pressure measurements in dilute salt-free solutions of colloidal PE brushes14,15 and PE stars.16 The main objective of this article is to perform an analysis of counterion localization in salt-free solutions of colloidal PE brushes and starlike PEs on the basis of a quantitative Poisson-Boltzmann approach. We focus on the effects of the degree of polymerization of the star branches and the fraction of ionic monomer units and give a systematic comparison with the predictions of the approximate analytical scaling-type theory. We aim to prove and to quantify the applicability of the concept of charge renormalization, as developed for colloids, to PE stars, which exemplify “soft” charged colloidal objects. Following the classical approach,7 this is done here on the basis of an exact numerical solution of the corresponding Poisson-Boltzmann (PB) problem, where the conformational degrees of freedom of the flexible polyelectrolyte branches are accounted for within the Scheutjens-Fleer self-consistent field method. Because of the computational efficiency, the latter method enables us to study highly branched ionic macromolecules, which are far beyond the capacity of contemporary molecular dynamics simulations. The remainder of the article is organized as follows. We start with a review of the classical theory of charge renormalization in salt-free solutions of charged colloids and discuss its applicability to solutions of colloidal PE brushes with an arbitrary ratio of the core to the corona dimensions. Then we present systematic results of our numerical solution of the PB problem for selected starlike PEs, corresponding to the vanishing core size limit. The approximate analytical solution of the corresponding PB problem in the two-phase approximation and some general aspects of its applicability to colloidal polyelectrolyte brushes are directed to the Appendix. (10) Roger, M.; Guenoun, P.; Muller, F.; et al. Eur. Phys. J. E 2002, 9, 313– 326. (11) Jusufi, A.; Likos, C. N.; Ballauff, M. Colloid Polym. Sci. 2004, 282, 910–917. (12) Mei, Y.; Hoffmann, M.; Ballauff, M.; Jusufi, A. Phys. ReV. E 2008, 77, 031805. (13) Sandberg, D. J.; Carillo, J. Y.; Dobrynin, A. V. Langmuir 2007, 23, 12716–12728. (14) Guo, X.; Ballauff, M. Phys. ReV. E 2001, 64, 015406. (15) Ballauff, M. Prog. Polym. Sci. 2007, 32, 1135–1151. (16) Plamper, F. A.; Becker, H.; Lanzendo¨rfer, M.; Patel, M.; Wittemann, A.; Ballauff, M.; Mu¨ller, A. H. E. Macromol. Chem. Phys. 2005, 206, 1813–1825.

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2. On the Osmotic Pressure of a Salt-Free Solution of Charged Colloids In this section, we briefly review what is known about the electrostatic potential and corresponding counterion distribution in salt-free solutions of charged colloidal dispersions, mostly following the lines of Alexander et al.7 Let us consider a dilute solution of charged colloidal particles with radius Rc that have a uniform charge density on their surface; that is, each particle has a (positive) charge eQ. We focus on salt-free solutions. This means that we have only mobile (negative) counterions in the system. The number of these counterions is such that the system is electroneutral overall. The counterions are distributed inhomogeneously in the solution with obviously a higher concentration near the surface of the particles. The osmotic pressure of the solution is determined (for Q . 1) by the concentration of counterions in the intermediate space between the particles, that is, far away from the colloids (macroions). If one applies the cell model for the solution, where each macroion is placed in the center of the regular Wigner-Seitz cell (which is usually approximated as a spherical volume), then the osmotic pressure is determined by the counterion concentration at the cell boundary.7 (See also section 4 below.) In general, the distribution of the electrostatic potential and that of the counterions in the cell (in the range Rc e r e D, where r is the radial distance from the center of the particle and D is the cell radius) is determined by the Poisson-Boltzmann equation, which in spherical coordinates has the form

1 ∂2rψ(r) ) 4πlBn0 exp(ψ(r)) r ∂r2

(1)

where n(r) ) n0 exp(ψ(r)) is the local number density of counterions. In eq 1, we have taken into account that the counterions are monovalent and negatively charged. Here, ψ(r) ≡ eΨ(r)/kBT is the dimensionless potential, lB ) e2/εkBT is the Bjerrum length, and n0 is a constant that depends on the calibration of the electrostatic potential. If we set ψ(D) ) 0, then n0 is the concentration of counterions at the cell edge, r ) D. Equation 1 has to be complemented by two boundary conditions. One at the outer boundary

( ∂ψ(r) ∂r )

D

)0

(2)

that reflects the total electroneutrality of the cell as a whole and one at the inner boundary

( ∂ψ(r) ∂r )

Rc

)-

2 λ

(3)

wherein the Gouy-Chapman length λ is a new important length scale

λ)

1 ≡ 2Rc2/lBQ 2πlBσ

(4)

that basically depends on the charge per unit area of the particle σ. 2.1. Weakly Charged Particles, λ.Rc. The condition λ . Rc ensures that the electrostatic potential at the surface of the particle is ψ(Rc) ≈ lBQ/Rc , 1. Indeed, the electrostatic potential at r g Rc is even lower, and this enables us to apply the linearized form of the PB equation (the Debye-Hu¨ckel (DH) approximation) safely

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1 ∂2rψ(r) ) κ2(1 + ψ(r)) r ∂r2

Leermakers et al.

(5)

in the space Rc e r e D. In other words, eq 5 together with the boundary conditions, eqs 3 and 2, correctly describe the distribution of the electrostatic potential and that of the number concentration of counterions n(x) ) k2(1 + ψ(x))/4πlB in the whole cell. The value of κ has to be found from the boundary conditions (and the calibration of the potential, ψ(D) ) 0) together with the solution of eq 5. In the dilute regime (i.e., for D . Rc), κ2 ≈ 4Rc2/D3λ. The counterion concentration at r ) D, which determines the osmotic pressure, is n(D) ) κ2/4πlB. Hence, we find that n(D) ≈ Q/2πD3 (i.e., counterions are distributed fairly uniformly throughout the cell). This is what one has to expect from the Boltzmann law, n(r) ) n(D) exp(ψ(r)), and the fact that the variation of the reduced potential within the whole cell is ∆ψ e 1. 2.2. Strongly Charged Particles, λ , Rc. Now, the particle is sufficiently strongly charged to create a large surface potential ψ(Rc) . 1, and in the vicinity of the surface, it is not allowed to use the DH approximation. However, the DH approximation can be applied far away from the particle, that is, close to the cell boundary r ) D. For strongly charged particles, λ is a characteristic distance away from the charged surface, within which most of the counterions are retained as a result of the significant Coulombic attraction. Indeed, in the range r - Rc ≈ λ , Rc the curvature of the surface is not important, and the distribution of the electrostatic potential and that of the counterions are well approximated by the solution of the PB equation for a charged flat plane with the same surface charge density σ as we have on the sphere: σ ) Q/4πRc2 ) 1/2πlBλ and

n(r) ≈ [2πlB(r - Rc + λ)2]-1, r - Rc e Rc

(6)

By integrating the counterion distribution (eq 6), we find that the number of counterions escaping to distances much larger than Rc from the surface of the particle is ∼Rc/lB , Q (where the last inequality is ensured by the assumed condition λ , Rc). At this stage, we are coming to the concept of charge renormalization.7 According to this concept, when λ , Rc (i.e., when the electrostatic potential at the surface of the charged particle is high, ψ(Rc).1), and then a certain number of counterions gets confined (“condensed”) in the vicinity of the surface, which reduces the apparent surface potential down to the level of ψ ≈ 1. The last condition determines the number of noncondensed (osmotically free) counterions as ∼Rc/lB. Then, in the range r . Rc, where the potential is sufficiently small, one can safely apply the DH approximation and describe the radial distribution of the potential and that of the “noncondensed” counterions using the equations presented in the previous subsection with respect to the actual charge of the particle Q substituted by a renormalized charge Q/≈ Rc/lB. Similarly to the case of weakly charged particles, now the noncondensed (osmotically active) counterions are distributed fairly uniformly in the range of Rc , r e D so that their concentration at r ) D and the osmotic pressure coincide (with the accuracy of the numerical factor on the order of unity) with their average concentration in the cell, ∼Q//D3. At this point, three remarks are appropriate: (1) The distribution of the concentration of counterions and that of the electrostatic potential in the range of Rc , r e D are virtually independent of the bare charge Q of the particle but depend only on its radius.

(2) The proportionality coefficient between Q/ and Rc/lB is to be found from a matching of the PB and DH equations in the range of r - Rc g Rc. This coefficient is a weak function of Q, D, and Rc. (3) Although one can never unambiguously distinguish between a condensed and a noncondensed state for a counterion, the value of Q/ has a clear physical meaning; it appears as a parameter in the DH solution, which properly matches the exact distribution of the potential (and counterions) at the peripheral (and also intermediate) regions of the cell.

3. Colloidal Polyelectrolyte Brushes and Starlike Polyelectrolytes Now we shall apply similar arguments to evaluate the osmotic pressure in dilute salt-free solutions of colloidal and starlike PE brushes. Colloidal polyelectrolyte brushes are core-shell particles, where end-grafted polyelectrolytes create a (soft) charged shell around a solid core. Let Lc ) Nl be the contour length of the grafted chains (l is the segment length, and N is the degree of polymerization). The fraction of charged monomers is R e 1, and thus Q1 ) RN is the charge of one chain. As long as the segment length is on the order of the Bjerrum length, the condition R e 1 implies that effects of Manning condensation of the counterions on the branches can be safely neglected. The area on the particle available per chain is s, and thus f ) 4πRc2/s is the total number of chains that are grafted onto the core. Again, Rc is the radius of the core, L is the thickness of the brush, and R ) Rc + L is the outer radius of the full core-shell particle. We shall consider cases L,Rc and LgRc separately. 3.1. Thin Brush, L , Rc. We start with the case in which the shell (brush) thickness is small compared to the core size (i.e., L , Rc. In this situation, the structure of the brush and the distribution of the electrostatic potential and that of the counterions in the vicinity of the particle surface can be well approximated by those known for planar PE brushes.20 Per unit surface area, the “bare” charge immobilized as a result of the grafted polyelectrolytes is equal to σp ) Q1/s. It is convenient to introduce the length λp ) 1/2πlBσp. When λp , L, a majority of the counterions are localized inside the brush (the osmotic regime). Because of the osmotic pressure of the counterions that are trapped inside the brush, the polymer layer swells up to a thickness of L = R1/2Lc ≈ Lc. Still some counterions escape from the brush. From large distances (i.e., r g Rc + L), the upper edge of the brush appears as a weakly charged surface with a surface charge density of

σ ≈ L/2πlBL02

(7)

where L0 ) √(8 ⁄3π 2) LcR1/2 is a new characteristic length. The osmotic regime is found when L0/λp . 1. In the osmotic regime,

the brush thickness L ≈ L0√ln(2L0 ⁄ λpπ1 ⁄ 2) ≈ L0 if we neglect the logarithmic dependence. Hence, the characteristic length scale for the decay of the counterion concentration above the brush edge is

λ ) 1/2πlBσ ) L02/L = L0

(8)

In this range, r g Rc + L, the counterion concentration obeys the usual PB equation for a charged surface with a surface charge density σ given by eq 7. As follows from eqs 7 and 8, the effective (uncompensated) charge per unit area of the brush decreases and λ increases with increasing chain length. Because λ = L , Rc, the solid core including its grafted osmotic brush presents itself as a strongly charged colloidal particle,

Counterion Localization in Solutions

according to the definition given in the previous section. In other words, the electrostatic potential at the edge of the brush (i.e., at r ) Rc + L ≡ R) is large. Therefore, a significant number of counterions are released from the brush but will remain localized in the range of ∼R ≈ Rc around the core-shell particle. This reduces the charge that is “visible” from large distances down to the value ∼R/lB. The latter also equals the number of osmotically active counterions whose (average) concentration determines the osmotic pressure, ∼R/lBD3. 3.2. Starlike Polyelectrolyte (Thick Brush), L g Rc. Let us next consider the case in which the chain length is comparable to or larger than the core size. This case corresponds to strongly curved spherical PE brushes or to starlike PEs, where long PE arms are grafted to17,18 or, in the most recent experimental approach,16,19 grafted from a particle with a small core. This situation, L g Rc, is also common for starlike micelles formed in aqueous solutions by ionic/hydrophobic diblock copolymers, where the size L of the PE corona significantly exceeds the size Rc of the collapsed hydrophobic core.6 In this limit, the effects of the curvature on the structure of the spherical PE brush22 and the distribution of its counterions are most important. Unfortunately, no exact analytical solution is available for the set of PB-SCF equations that describe the curved polyelectrolyte brush. However, as follows from previous numerical results,9,24 the structure of a strongly curved polymer or PE brush can be reasonably well described by the crossover of two solutions. (i) A power-law decay of the polymer volume fraction describes the central region. (ii) A quasi-planar brush structure describes the outer region of the bush (e.g., key properties such as the bimodal radial distribution of the end segments and the parabolic profile for the electrostatic potentialstypical results for a planar PE brushscan clearly be found in the peripheral regions of strongly curved brushes or stars). Because the outer radius of the core-shell particle R coincides in this case with the brush thickness, the marginal situation, λ ≈ L ≈ R occurs (i.e., the potential at the brush edge is close to unity). As a result, the total uncompensated charge of the polyelectrolyte brush coincides with the “renormalized” charge Q/ with an accuracy of a factor on the order of unity. Therefore, we expect that the electrostatic potential and the counterion distribution outside the corona of the PE star can be described by the DH equations with a renormalized charge, ∼R/lB ≈ L/lB. As has been discussed in refs 2, 8, and 22, the extension of arms in curved PE brushessswollen by the osmotic pressure of entrapped counterionss(in scaling terms) does not depend on the radius of curvature of the grafting surface or on the number of arms, L = R1/2Lc. (The exact numerical coefficient can be obtained under the assumption of equal stretching of all the arms and a local compensation of charges of the arms by mobile counterions22 or for the opposite case of barely charged PE stars23). We also mention that whereas the linear dependence L ≈ Lc is quite universal the exponent of the R dependence of the star size is slightly smaller if the excluded volume effects between monomer units are taken into account.2 The renormalized charge, (17) Mays, J. W. Polym. Commun. 1990, 31, 170. (18) Heinrich, M.; Rawiso, M.; Zilliox, J. G.; Lesieur, P.; Simon, J. P. Eur. Phys. J. E 2000, 131–142. (19) (a) Plamper, F. A.; Ruppel, M.; Schmalz, A.; Borisov, O. V.; Ballauff, M.; Mu¨ller, A. H. E. Macromolecules 2007, 40, 8361–8366. (b) Plamper, F. A.; Walther, A.; Mu¨ller, A. H. E.; Ballauff, M. Nano Lett. 2007, 7, 167–170. (20) Zhulina, E. B.; Borisov, O. V. J. Chem. Phys. 1997, 107, 5952. (21) Klein Wolterink, J.; van Male, J.; Cohen Stuart, M. A.; Koopal, L. K.; Zhulina, E. B.; Borisov, O. V. Macromolecules 2002, 35, 9176–9190. (22) Zhulina, E. B.; Birshtein, T. M.; Borisov, O. V. Eur. Phys. J. E 2006, 20, 243–256. (23) Borisov, O. V.; Zhulina, E. B. J. Phys. II (France) 1997, 7, 449–458. (24) Wijmans, C. M.; Zhulina, E. B. Macromolecules 1993, 26, 7214–7224.

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therefore, is much smaller than the total bare charge of the starlike PE brush provided that the total number of arms is sufficiently large (i.e., f . f/ ≈ R-1/2(lB/l)-1). The latter inequality provides an estimate of the smaller number of arms necessary to ensure the entrapment of a major fraction of its counterions inside the corona. As we demonstrate below, an increase in the number of arms leads to a progressive increase in the fraction of entrapped and thus osmotically inactive counterions, though at f . f/ this dependence practically levels off. For experimentally relevant values of R g 0.1, PE stars with f g 10 arms are reliably found in the regime of a strong localization of their counterions.

4. Osmotic Pressure The osmotic pressure Π of a salt-free solution of colloidal PE brushes or PE stars can be calculated by differentiating the free energy of the system at a fixed number of colloidal particles to the volume

∂F ( ∂V )

Π)-

N,T

(9)

Within a cell model, the change in the total volume at a fixed number of PE stars changes the cell volume available per star Vc. An infinitesimal change in the volume pushes the counterions of the PE star into a smaller volume. As at the cell boundary where the electric field is zero, we only have to pay a free-energy penalty proportional to the osmotic pressure generated by the counterions at the cell boundary. The dimensional osmotic pressure of the counterions at the cell boundary is given by the volume fraction at the boundary φc(D). In many cases, this contribution to the counterions exceeds the ideal term due to the star alone, which is given by Π/kBT ) 1/Vc. When all of the counterions in the solution are osmotically active, we would expect the osmotic pressure to be given by

Nstars + NstarsfNR Π ) kBT V

(10)

As explained above, it is expected that in reality the osmotic pressure (in the limit in which the PE stars are not overlapping) is lower than that given by eq 10 because a large number of counterions will be trapped inside the stars and are therefore not osmotically active (i.e., they are not able to reach the edge of the cell). Only a fraction of the counterions can do this. Our interest is in the evaluation of the fraction of the counterions that are osmotically active.

5. Numerical SCF Approach We solve the PB equation using the discretization scheme of Scheutjens and Fleer in spherical coordinate systems for PE stars in low concentrations of 1:1 electrolyte. The cell dimension is fixed to the value l ) 0.5 nm (approximately the Bjerrum length), which implies that the conversion factor for the volume fraction of salt φs to a molar concentration is approximately 10 (i.e., cs ≈ 10φs M). All lengths are normalized with the cell dimension. We consider PE stars with a fixed linear charge density along the contour of the arms. The fractional charge is given by R, which is varied between 0.2 and 1.0. Because the monomer unit size is chosen to be slightly smaller than the Bjerrum length, the effect of Manning condensation, which is not explicitly taken into account in our model for symmetry reasons, is negligible at low values of the fractional charge R but may become important for R approaching unity. The length of each arm is given by N, and the default value is N ) 100. The number of arms per star is given by f, and we typically varied this parameter in the range of f ) 20-50. The arms are restricted with their first segment

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Figure 1. (a) Volume fraction of counterions at the system boundary in excess of the volume fraction of ions in the bulk as a function of the bulk volume fraction of salt in double-logarithmic coordinates. The degree of ionization of the star segments is indicated. Parameters: f ) 50, N ) 100, and D ) 500. (b) Limiting value of the excess counterions accumulated at the boundary as a function of the degree of charging.

Figure 2. (a) Excess counterions at the system boundary as a function of the volume fraction of salt in the bulk for various box sizes as indicated. The degree of charging is R ) 0.2, N ) 100, and f ) 50. (b) Corresponding limiting value (at low ionic strength) of the excess counterions at the edge of the cell ∆φc# (line with symbols) and the ideal gas value for the dimensionless osmotic pressure (line without symbols) as a function of the cell size D in double-logarithmic coordinates.

to be near the center of the coordinate system. More precisely, we pinned the units to the volume given by the first five lattice layers. The volume in these five lattice layers is given by V ) 4/3π53 ≈ 500l3 so that the volume fraction of grafting segments in the center does not exceed 0.1. The repulsion between the arms forces most of the grafting segments to be at the r ) 5 coordinate. All nearest-neighbor interactions except the excluded volume effects are neglected (athermal solvent conditions). The step probability for the flat lattice (large r coordinate) is 1/6 corresponding to a simple cubic lattice.

6. Results and Discussion PE stars and brushes in the limit of no added salt are extremely CPU-intensive because of slow convergence and the large number of iterations required. The strategy that is used below is to find good initial guesses for the PE star in the presence of salt and then decrease the ionic strength to such low values that one enters the counterion-only limit asymptotically. The high costs in terms of CPU time limit our investigations in terms of both system size (D) and values for the chain length N used. Very large values of the system size also give numerically accuracy problems when the concentration of counterions is calculated at the system boundary. As a result, the following default parameters are used: N ) 100, f ) 50, and R ) 0.2. When the ionic strength is not mentioned, we have the no-salt limit. Let us first analyze how many counterions are found at the cell boundary when we systematically change the concentration of salt in solution. In Figure 1, the volume fraction of counterions at the boundary in excess of the bulk concentration is plotted as a function of the bulk concentration for a star with f ) 50 arms each with length N ) 100 and a cell size of D ) 500. The degree of charging along the chains is varied from 0.2 in steps of 0.2 to 1.0. It is quite obvious that at high ionic strength the electrostatic

potential at the system boundary has faded away and we do not find a significant excess of ions at the system boundary with respect to the bulk concentration. Below the salt volume fraction of φb ) 10-5, we start to notice the finite size of the cell volume, and there is an excess of counterions at the system boundary. This excess grows approximately as a power law to a plateau that is reached below φb ) 10-7. With decreasing degree of charging R of the polymer segments, the onset of the buildup of excess counterions at the boundary occurs at lower ionic strengths. The plateau is also a decreasing function of the ionic strength. The ideal gas term to the osmotic pressure for the D ) 500 case is given by Π/kBT ) 1/Vc ≈ 2 × 10-9. With decreasing ionic strength, the interaction contribution easily becomes larger than the ideal term. From Figure 1b, it is concluded that in the limit of no added salt the excess number of counterions at the system boundary ∆φ#c ≡ φc(D) - φbc ≈ φc(D) shows a power law dependence of R0.32. The dependences on the box size are also of interest. The larger the box (i.e., the lower the concentration of PE stars in solution), the more space there is for the electrostatic potential to decay. We therefore also expect a decrease in the number of counterions at the system boundary. In Figure 2, we present a selection of the results for a given case (R ) 0.2, N ) 100, f ) 50) where we have varied the box size systematically. In Figure 2a, we have varied the ionic strength and found qualitatively the same results as in Figure 1a. In the high-ionic-strength regime, there is no measurable excess of counterions at the system boundary. Depending on the box size, the volume fraction of osmotically active ions increases sharply below some salt concentration. The larger the box size, the lower this threshold ionic strength. Again for low-ionic-strength situations, the concentration of counterions at the box edge saturates to some limiting value. Although the data for the D ) 1000 curve are

Counterion Localization in Solutions

Langmuir, Vol. 24, No. 18, 2008 10031

Figure 3. The no-added-salt result for the volume fraction of counterions at the system boundary (a) as a function of the number of arms in the PE star (with parameters R ) 0.2, N ) 100, and D ) 500) and (b) as a function of the length of the arms N for fixed fN ) 5000 (---) and f ) 50 (-), R ) 0.2, and system size D ) 500. The graph in part b is in double-logarithmic coordinates.

noisy, it is clear that with increasing box size the buildup of charge at the system boundary decreases rapidly as expected. According to theory (section 2.1), the scaling with the box size in the limit of no salt follows the ∆φ#c ∝ D-3 dependence as is easily seen in Figure 2b. In this graph, we have also plotted the ideal gas value for the dimensionless osmotic pressure. It is evident that the ideal gas contribution due to PE stars is 2 orders of magnitude smaller than the contribution originating from the concentration of counterions at the system boundary. Above, we found the concentration of counterions in the limit of zero added salt by extrapolating the dependence as a function of the volume fraction of salt toward zero. In the remainder we will approximate this by plotting the predicted value for the salt volume fraction of φb ) 10-9. Although it is expected that this low concentration is a very good measure for the no-salt limit, we note that the results are also somewhat less accurate because in contradiction to the results of Figures 1b and 2b we no longer average the value in the plateau region. The results are therefore somewhat more scattered due to numerical noise. In Figure 3a we show that the volume fraction of counterions at the edge of the system is, in the no-salt limit, a very weak function of the number of arms f. This is a reasonable result because most of the counterions get trapped in the PE star and therefore an increase in the number of arms hardly results in an increase in the number of osmotically active counterions in the system. In this result the PE star number density was fixed (D ) 500) and also the charge density and the arm lengths remained constant. As discussed in Section 3.2 and also demonstrated below by our calculations, the size of a many arm star is predicted to be a very weak function of the number of branches, f. Hence, one can expect a virtual independence of the effective (renormalized) charge of the PE star of the number of arms. How, in the limit of slat-free solutions, the ion contribution to the osmotic pressure for a given concentration of stars (D ) 500) depends on the length of the arms is shown in Figure 3b. Here, the charge density R ) 0.2 is constant, and the length of the arms is increased either at a fixed total number of segments in the star (fN ) 5000, ---) or at a fixed number of arms (f ) 50, -). The increase in the length of the chains has an effect on the radius of the star. Hence, according to the charge renormalization concept, the effective charge of the star should also increase as a function of N. Moreover, because the system size is fixed, the distance between the edge of the star and the edge of the cell becomes smaller. Both of these effects lead to an increase in the number of counterions found at the system boundary. This is exactly what is seen in Figure 3b. It appears that the number of counterions at the system boundary increases sublinearly as a

function of N and a slightly stronger N dependence is found when the number of arms is fixed. We will now try to establish a more systematic relationship between the distribution of counterions and the conformations of the branches of the PE star. In Figure 4, we show the ion concentration at the system boundary together with the average size of the PE stars as a function of the degree of charge in the polymer chains. Here, we fixed the architecural parameters N ) 100, f ) 50, and the cell volume to D ) 500 (cf also Figure 1b.) We have marked the free end of the polymer chains with label G and recorded the radial volume fraction profiles: φG(r). We may find a measure for the PE star size by taking moments over this endpoint distribution:

R)

{

D

rL(r) φG(r) f r)1

∑

( (∑

r2L(r) φG(r) f r)1 D

∑

r3L(r) φG(r) f r)1 D

) )

1⁄2

(11) 1⁄3

The closer the endpoints are distributed as a δ function at the periphery of the stars, the closer these three values of the radius are to each another. In Figure 4a, we show that these three measures of the PE star size are very close to each other, especially when the star is highly charged and the chains are almost fully extended. Both the size of the star and its effective uncompensated charge (measured by the number of osmotically active counterions) increase sublinearly as a function of R. The latter observation implies that upon an increase in R the fraction of counterions that are osmotically active decreases. Indeed, as follows from Figure 4b, Q//Q = 0.2 and 0.06 for R ) 0.2 and 1, respectively. The apparent power law exponent for the star size versus R is well in line with that expected from the scaling argument R ≈ R1/2 dependence. The slightly smaller value of the observed exponent is related to the finite core size and to non-negligible effects of excluded volume interactions on the radial stretching of the star arms. From Figure 4a, it becomes possible to estimate the volume occupied by the star and the volume outside the star that is available for the counterions. In this case, the volume available for each star and its counterions may be 1000 times larger than that occupied by the star itself. Finally, in Figure 4c a set of volume fraction profiles of the counterions is presented for the degree of dissociation ranging from 0.2 to unity. In this plot, the peak positioned at r ) 5

10032 Langmuir, Vol. 24, No. 18, 2008

Leermakers et al.

Figure 4. (a) Size of the PE star as a function of the degree of dissociation of the polymer segments in double-logarithmic coordinates. The three graphs represent different moments over the end-point distribution of the chains (points to the right, first moment; points up, second moment; points to the left, third moment). Parameters: N ) 100, f ) 50. and D ) 500. (b) Corresponding excess counterions at the system boundary (complementary data to Figure 1b). The line is a power-law fit with exponent 0.28. (c) Radial volume fraction profiles of the counterions in log-linear coordinates for the degree of dissociation R ) 0.2, with steps of 0.1 to 1.0. With increasing degree of dissociation, the volume fraction of counterions is larger (for a given radial coordinate r).

represents the grafting point of the arms of the star. The inner space is free for the ions to penetrate into, and the ion concentration in the center is a weak function of the degree of dissociation. The counterion profile decreases as the radial coordinate increases. Inside the star, the decrease differs significantly from that outside the star. As a result, the “edge” of the star can be estimated from the “kink” in the curve. In line with scaling theory predictions8,2 and with the result presented in Figure 4b, the size of the star increases with increasing degree of dissociation. Outside the star, the counterion concentration drops quickly to the values near those found at the edge of the system. In other words, far outside the edge of the corona of the star the ion distribution is more or less homogeneous, as expected from section 3.2. In Figure 5, a set of graphs similar to those in Figure 4 are presented. In this case, the chain length N is the variable, and the number of arms is fixed to f ) 50, the degree of dissociation is fixed to R ) 0.2, and again, the box size is given by D ) 500. As follows from Figure 5a, the radius of the PE star increases approximately as a power-law function of the length of the arms. The observed power-law exponent tends to increase with the chain length and is slightly less than unity (0.9 for the R ) 0.2 and 0.94 for the R ) 1.0 case). We remind the reader that the approximate analytical theory2,8,22 predicts a linear increase in the PE star size in infinitely dilute salt-free solutions as a function of N. The fact that for finite N the slope of the R versus N curves is slightly smaller than unity may have something to do with the grafting of the chains (in the range of r < 5) and with the finite box size. In Figure 5b, it is shown that the concentration of counterions at the system boundary scales sublinearly with the chain length. (The power-law exponent is, to good approximation, 2/3; see Figure 3b for N < 100.) The slope of the dependence progressively increases for larger chain lengths but remains smaller than the expected scaling prediction of unity. This deviation is clearly attributed to finite chain length N and limited box size effects. The radial volume fraction profiles of the counterions reveal that far outside the star the volume fraction of ions is rather

homogeneous. This observation correlates with theoretical arguments presented in section 3.2 that the reduced electrostatic potential at the edge of the corona of the PE star is on the order of unity and decays further toward the computational cell boundary. The endpoint distribution for the PE star arms is presented in Figure 5d. In this graph, one can see that the ends have their highest probability at the periphery of the star and that there exists a quasi-plateau of endpoint distributions in the outskirts of the star and a dead zone in the central part. For this type of endpoint distribution, we do not expect a large difference between the various moments of the endpoint distribution. As can be seen in Figure 6, the size of the PE star is an increasing function of the number of arms f. This increase is more pronounced for a small number of arms, and the dependence levels off as soon as the number of arms exceeds f ≈ 10 (for a chosen value of R). As expected from theory2,8 and illustrated by our calculation results presented in Figure 6b, a pronounced initial increase in the star size as a function of f is accompanied by the progressive localization of the counterions inside the star corona. In the quasiplateau region (corresponding to large f), the fraction of entrapped counterions asymptotically approaches unity. Consequently, for stars with a sufficient number of arms, the counterion concentration at the system edge grows very slowly as a function of the number of arms. Because the number of counterions for each PE star is exactly known, it is trivial to evaluate the expected volume fraction of counterions in the system if the ions distribute homogeneously. This would be the expected result when all counterions are osmotically active. This concentration of counterions is presented by a dotted line in Figure 6b. It is of some interest to point to the fact that the two-armed star (which is a linear chain with the middle segment fixed near the center of the coordinate system) cannot confine the counterions in its surroundings and most of the counterions are osmotically active when the charge density along the chain is sufficiently low (i.e., R ) 0.2). However, when the linear chain is highly charged (R ) 1), we find that 40% of the counterions are osmotically inactive.

Counterion Localization in Solutions

Langmuir, Vol. 24, No. 18, 2008 10033

Figure 5. (a) Radius of the PE star as a function of the chain length as measured by the moments over the endpoint distribution (point to the right, first moment; point up, second moment; point to the left, third moment) and (b) the concentration of counterions at the system boundary as a function of the length of the arms in double-logarithmic coordinates. The linear parts follow a power law with an exponent close to 2/3. The degree of charging of the polymer chains is indicated. (c) Radial volume fraction profile of the counterions for R ) 0.2. The longer the chain length, the higher the counterion volume fraction. The chain lengths are N ) 50 with steps of 10 to 200 for D ) 500. (d) Endpoint distribution for N ) 100, 200, 500, and 1000 with D ) 1000. In all graphs, f ) 50.

Figure 6. (a) First moment of the radial endpoint distribution as a function of the number of arms in the PE stars. Parameters: N ) 100, D ) 500, and R. (b) Corresponding excess number of ions at the system boundary. The dotted lines are the expected results when all counterions of the PE star would have been distributed homogeneously in the system.

7. Conclusions We have presented an analysis of the counterion distribution in salt-free solutions of spherical PE brushes and PE stars on the basis of a systematic treatment of the corresponding PB problem. Because an exact analytical solution of the PB equation for spherical PE brushes (or stars) is not available, we had to combine different approaches for thin (weakly curved) PE brushes, where we made use of the available exact solution for a planar PE brush. In the limiting case of strongly curved PE brushes (or PE stars), we are able to solve the PB problem exactly by employing the SF-SCF approach to evaluate the partition function of stars in the self-consistent electrical field created by its charged arms with mobile counterions. Our analysis proves that in both cases a majority of its counterions are localized inside the PE brush (or star) provided that the grafting density is sufficiently high. Furthermore, in the case of a weakly curved PE brush, the electrostatic potential at the edge of the brush is large enough to set up a strongly inhomogeneous distribution of “escaped” counterions outside the brush. Hence, in this case a significant number of its

counterions are not exactly confined in the brush but do not contribute to the osmotic pressure either because of their localization in the vicinity of the brush (Gouy-Chapman layer). This explains why a simple two-phase PB approximation (Appendix) provides, in the case of thin brushes, an overestimate of the osmotic pressure because it presumes a uniform distribution of counterions outside the brush. Remarkably, for PE brushes with L = Rc, the experimentally measured osmotic coefficient15 appears to be almost an order of magnitude lower than that predicted by the two-phase model. In contrast, for PE stars (or strongly curved spherical PE brushes), the dimensionless electrostatic potential at the edge of the corona drops almost to unity. Consequently, the counterions outside the corona are distributed fairly uniformly and the effective (renormalized) charge of the star, which determines the counterion concentration at the boundary of the computational cell, approximately coincides with the total uncompensated charge within the star corona. In this case, the two-phase approximation for the PB problem9 produces more accurate results.

10034 Langmuir, Vol. 24, No. 18, 2008

Interestingly, the two-phase approximation (Appendix) predicts a decrease in the proportionality coefficient between the effective charge of the brush (controlling the number of osmotically active counterions) and the outermost radius of the corona upon an increase in the radius of the corona (at fixed core size). As follows from the results presented in Figure 6, assuming Q/ ≈ 2π D3∆φ#c , we find Q/lB/R ≈ 1, which is much smaller than for charged hard colloids.7 This can be attributed again to the relatively large space available for counterions inside the PE star corona. Acknowledgment. This work has been supported by the European Union within the Marie Curie Research and Training Network POLYAMPHY.

Appendix: Two-Phase Core-Shell Model Here we summarize the results obtained by solving the PB equation for the curved polyelectrolyte brush in the so-called two-phase approximation. We basically follow the approach suggested in ref 9 (and further elaborated in ref 11) and apply a simplified scheme, presuming a uniform counterion density inside the brush, Rc e r e R (shell), and another uniform counterion concentration in the outer space, R e r e D. As we have discussed above, in reality the decay of the electrostatic potential profile and gradients in the counterion concentration in the vicinity of the outer edge of the brush (i.e., for r g Rc + L) are rather steep. Indeed, this is the range wherein the counterions that are renormalizing the net particle charge are confined and reduce the effective charge of the particle down to the level Q/ ≈ R/lB). In our scheme, we effectively push these counterions into the brush and identify the effective charge Q/, which equals the number of osmotically active counterions, with the uncompensated charge of the region r e Rc + L ) R. Minimizing the free energy of the cell while taking the free energy of the Coulomb interactions and the translational entropy of the counterions into account, we obtain the following equation for the uncompensated charge Q/

Leermakers et al.

Q/ )

[(

)

Q (D/R)3 - 1 R 1 ln lBχ Q/ 1 - (Rc/R)3

]

(12)

where

χ(x, y) )

1/5 - x3 + 9x5/5 - x6 1 - 9y/5 + y3 - y6/5 + (1 - x3)2 (1 - y3)2 (13)

is a function of x ) Rc/R and y ) R/D. The particular form of χ(x, y) is related to the assumption of uniform charge density within each of the regions, Rc e r e R and R e r e D. In eq 12, Q is the overall bare charge of the brush. It is reasonable to rewrite eq 12 by introducing reduced variables q/ ) Q/lB/R and q ) QlB/R, which leads to

q/ )

[( )

q (1/y)3 - 1) 1 ln / - 1 χ(x, y) q 1 - x3

]

(14)

where x ) Rc/R and y ) R/D. As follows from the analysis of eq 14, q/ ≈ q for q e 1, and q/ grows logarithmically as a function of q in the range of q.1. It is important, however, that q/ varies relatively weakly as a function of x or other parameters. With the parameters D/lB ) 200, Q ) 4000, and x ) Rc/R that vary in the range between 0.1 (thick brush) and 0.9 (thin brush), we find that q/ varies in the range between 9.5 and 12.5. The value of q/ monotonically decreases with decreasing x. This can be explained by an increase in the volume available to the counterions inside the brush. Because the value of q/ provides an estimate of the electrostatic potential at r = R, we can conclude that the two-phase model that assumes a uniform distribution of the counterions in the exterior of the corona is a poor approximation for thin brushes but becomes asymptotically more accurate for stars with long arms. LA8013249