Ionizable polyelectrolyte brushes: brush height and electrosteric interaction

Journal of Colloid and Interface Science 275 (2004) 97–106 www.elsevier.com/locate/jcis

Ionizable polyelectrolyte brushes: brush height and electrosteric interaction P. Maarten Biesheuvel Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, 6703 HB Wageningen, The Netherlands Received 8 September 2003; accepted 6 February 2004 Available online 19 March 2004

Abstract Semi-analytical scaling theory is used to describe quenched and annealed (weakly charged, ionizable, charge-regulating) polyelectrolyte brushes in electrolyte solutions of arbitrary salt concentration. An Alexander-De Gennes box model with homogeneous distribution of polymer segments and the free ends located at the edge of the brush is assumed, as is local electroneutrality in the brush. For annealed polyelectrolyte and in the low-salt regime, the theory predicts that for sufficiently dense brushes, the salt concentration has a small influence on brush height, while the brush expands with increasing grafting density, in agreement with experiment. Expressions are presented for the interaction free energy of compressed ionizable and quenched polyelectrolyte brushes (proportional to the force between particles or curved surfaces). In all cases, the required prefactors are explicitly stated. The theory is compared directly with published experiments on the influence of salt concentration, pH, and grafting density on the thickness and interaction force of polystyrene sulfonate (quenched) and poly(meth)acrylic acid (annealed) brushes. In general, trends are well reproduced but significant deviations remain.  2004 Elsevier Inc. All rights reserved. Keywords: Polyelectrolyte; Brush; Colloid; Interaction; Electrosteric; Electrostatic; Modeling; Theory; Donnan; Electroneutrality

1. Introduction A polyelectrolyte brush is formed when polyelectrolytes are adsorbed or chemically attached to a solid or liquid interface. In aqueous solutions, polyelectrolyte brushes are able to electrosterically stabilize colloidal systems by a combination of steric and electrostatic forces. To describe the nature and magnitude of these forces two lines of approach can be found in the literature. In an approach originally used for charged membranes, the distribution of monomer segments is fixed in space, as is the brush height (thickness) [1,2]. Such models are based on the Poisson–Boltzmann equation and describe the electrostatic potential across the brush and in the double layer outside the brush, as well as the electrostatic interaction forces between two (nontouching) brushes [3–5]. Charge regulation due to cation adsorption [1] and due to pH changes [2,4,5] can be incorporated. Several analytical solutions are available, often based on the Donnan assumption of local electroneutrality (assumed either in the entire brush or only at the grafting interface) and for a fixed degree of dissociation. For chargeE-mail address: [email protected]. 0021-9797/$ – see front matter  2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2004.02.024

regulating polyelectrolyte brushes an analytical expression (for the potential profile and the electrostatic repulsion of nontouching brushes) was derived by Tsao [6] based on the Debye–Hückel approximation, the Donnan assumption, and a linearization of the charge regulation expression [7]. A second line of approach is the extension of brush models from polymer theory that were originally developed for noncharged polymer brushes [8]. In the past decade and a half such models have been extended to describe quenched polyelectrolyte brushes [9–11] and annealed polyelectrolyte brushes [12–14]. Numerical models can be found, often based on the Scheutjens–Fleer formalism and analytical theory based on solving the diffusion equation in the strong stretching limit or based on scaling arguments [12–14]. Analytical expressions are valid in the limit of a very high or very low salt concentration. A semi-analytical theory for an ionizable brush and arbitrary salt concentration is presented in Ref. [13] and is discussed later. In (semi-)analytical scaling theory, local electroneutrality within the brush is assumed, as is a constant segment distribution with the free ends located at the edge of the brush, resulting in the so-called AlexanderDe Gennes (AdG) box model. The equilibrium brush height is obtained from minimization of the total free energy (per

98

P.M. Biesheuvel / Journal of Colloid and Interface Science 275 (2004) 97–106

polyion or per unit surface area) or, equivalently, by setting the sum of forces on a polyion equal to zero. In this work a semi-analytical (scaling) theory is used for ionizable polyelectrolyte brushes that basically takes the same approach as in Refs. [12–14]. However, some additional elements are that the prefactors in the expressions are explicitly stated, the focus is on the theory required for arbitrary salt concentration, and a direct comparison with experiments of the brush height and electrosteric interaction force (of both quenched and ionizable brushes) is attempted. The prefactors are stated explicitly (in expressions for forces, free energies, and height) to show that they are not unknown [15] and certainly are provided [16] by the (scaling) theory. By stating prefactors, it is easier to compare the relative importance of different contributions to the forces and free energy (conformational, mixing, electric, chemical). However, the use of prefactors should not distract from the obvious fact that the box model approach that underlies the scaling theory is a significant simplification, regardless of the use of prefactors. Though the analytical expressions are reiterated for the osmotic brush and the salted brush regimes, the objective is to show the use of semi-analytical theory for the intermediate regime of arbitrary salt concentration. Indeed, as remarked by Israëls et al. [17], in major parts of the parameter space the experimental conditions are intermediate between the limiting regimes of high and low salt concentration and the analytical expressions for these regimes are in some cases recovered only for relatively extreme conditions. In the theory, the grafting density of the polyelectrolyte chains on the surface is assumed fixed, with all chains attached at their ends (end-grafted), either chemically or due to the adsorption of the “anchor” block of a block copolymer. The surface is assumed neutral, and all polyelectrolytes are equal in composition and extension with their free ends located at the edge of the brush. Brushes with fixed degrees of dissociation (quenched brushes) are discussed, as are ionizable (annealed) brushes in which the dissociation degree is a function of local pH (charge regulation). Expressions are given for the equilibrium brush height and the interaction force and free energy of overlapping polyelectrolyte (quenched and ionizable) brushes (electrosteric repulsion) and compared with experiments from the literature.

thereby neglecting electrostatic stiffening of the chain. This is formally true only when, compared with the Debye length, the charges on the chain are spaced sufficiently far apart. Corrections can be implemented, as discussed for instance in Ref. [18]. For each of m segment, there is one ionizable group; thus the distance between two ionizable groups along the fully stretched chain is ma. The electrostatic part of the theory is set up for polyacid and polybase brushes in which the ionizable segments can be in either a neutral or a charged form by the adsorption (polybase) or release (polyacid) of a proton. The free energy of a polyion F consists of electrostatic aspects, F el (including free energies related to the small ions, and the surface dissociation process; also called “coulomb,” “ion,” or “osmotic”), conformational terms, F conf (“elastic” or “stretching”), and concentration terms F conc (“mixing”, “steric” or “excluded volume”). 2.1. Conformational entropy and excluded volume To describe the conformational free energy (all free energies have dimension kT ) a Gaussian chain of N/k Kuhn segments each of length ak = ak is assumed, resulting for the free energy per polyion in 3H 2 , (1) 2kNa 2 where H is the brush height where all free ends are assumed to be located, and Na is the contour length. Equation (1) for a Gaussian chain can be derived from the classic Rayleigh random flight problem [19] for N freely jointed segments in free space and, assuming that the free ends are located at the edge of the brush, is valid for x = H /Na smaller than ∼0.5 (see Fig. 1). A second condition for Eq. (1) is Nx 2  1, which (for a given degree of stretching x) requires that N is sufficiently large. However, for polyelectrolytes, x = H /Na can be larger than 0.5, in which case for sufficiently high N , Fconf =

2. Theory Each polyion consists of N groups (monomers or segments). The segment length, a, depends on the number of atoms in the backbone of the monomer, but typically a = 0.25 nm for a monomer consisting of two C atoms in the backbone. The segment length a is not equal to the statistical or Kuhn length, which is a multiple k of a; thus, a Kuhn segment has length ka. As is common in the AdG box model approach, a constant Kuhn length is assumed independent of salt concentration and charge density of the chain,

Fig. 1. Conformational free energy (per segment) of freely jointed chain using Eqs. (1), (2), and (4) and exact calculations based on numerical integration of the inverse Langevin function (k = 1).

P.M. Biesheuvel / Journal of Colloid and Interface Science 275 (2004) 97–106

the exact solution is obtained from integration of the inverse Langevin equation (which is x = 1/ tanh(af ) − 1/(af ), with f the force exerted on a polyion) which can be expanded in a Taylor series [19], resulting in Fconf =

N
ai x 2i , k

(2)

i=1...∞

with the first coefficients being a1 = 3/2, a2 = 9/20, a3 = 99/350, a4 = 1539/7000, and a5 = 126117/673750. For an expansion up to i = 3, Eq. (2) can be used up to x ∼ 0.75 and up to x ∼ 0.80 for i = 5. However, due to the slow convergence this expansion is not very practical at higher relative extensions, x, and always predicts a finite F conf at x = H /Na = 1 (whereas it should diverge). The correct limit for F at high degrees of stretching is [20]  N Fconf = − ln(1 − x) + C1 , k

(3)

with C1 ∼ 0.307 to match the exact solution (if F conf,exact = 0 at x = 0). Eq. (3) can be used for x > 0.5. The empirical expression Fconf =

 3N  2 x − ln(1 − x 2 ) 4k

(4)

remains close to the exact solution, diverges when x = H /Na → 1, and converges to Eq. (1) for low enough H /Na (see Fig. 1). Though Eq. (4) is not exact at high extensions, it might be useful for certain practical calculations in which a single expression for F conf is desired with the correct limiting Gaussian behavior at low extensions but in which H cannot exceed Na. From this point forward Eq. (1) is used, though for large extensions the conformational free energy is underestimated. Recently Lyulin et al. [40] derived for the conformational free energy the following expression that includes a non-Gaussian overstretching term,   N 3 2 9 Fconf = x − x 3 ln(1 − x) , k 2 20 which is a natural interpolation between the Gaussian limit and the strongly-stretched regime. As an alternative to Eq. (4) it is closer to the exact solution. The description of mixing free energy per (Kuhn) segment is based on the Flory–Huggins theory which, for high enough N and low enough φ, simplifies to [21] φ FFH = (1 − 2χ) , 2

(5)

where v = 1 − 2χ is the second virial coefficient. For an athermal mixture χ = 0 and v = 1. In that case the remaining term is based on entropy only and can be considered to be due to excluded volume (steric effects). Expressed per polyion the mixing free energy becomes [22,23] Fconc = v

N 2σ υ N φ= , 2k 2kH

(6)

99

where use is made of the fact that within the AdG box model φ is given by φ=

Nσ υm , H

(7)

where σ is the grafting density (m−2 ), υm the volume per segment (m3; equal to the molar volume Vm of segments (m3 /mol) divided by N av ), and N av = 6.02 × 1023 mol−1 ; the excluded volume parameter is υ = vυm [21]. The force exerted on a single polyion, f , is given by f = −dF /dH (negative f is a contractive force). For the conformational and concentration aspects, f is given by summation of Eqs. (1) and (6) and taking the derivative with respect to H to obtain fconf + fconc = −

3H N 2σ υ + . kNa 2 2kH 2

(8)

The first term is the familiar expression for the conformational force on a grafted polymer [11,13,15,24,25]. Refs. [11,25] multiply by σ . Setting f conf + f conc = 0 in Eq. (8) results, for the neutral brush regime, in [11,12,22,26,27] HNB = γ Na 2/3 σ 1/3 υ 1/3

(9)

with γ = 6−1/3 ∼ 0.55. 2.2. Electrostatics Within the brush local electroneutrality is given by φα = 2n∞ υm m sinh zy,

(10)

in which α is the degree of ionization, n∞ the salt concentration in the bulk solution (in m−3 , = c∞ N av with c∞ the salt concentration in mol/m3 = mM), z the charge number, and y the dimensionless electrostatic potential. In this work, all ions (fixed and free) are monovalent; z is thus either +1 (for a brush of polycations) or −1 (for a polyanion brush). Equation (10) is a straightforward statement that the fixed charges are compensated by the ions in the solution phase inbetween the polymer. An assumption in Eq. (10) is that the electrostatic potential is constant throughout the brush, right up to the edge of the brush. The double layer extending from the brush edge is neglected. Eq. (10) is equivalent to the Donnan assumption that is often made in electrostatic calculations of brushes, gels (e.g., humic acid), and membranes. Eq. (10) furthermore assumes that the entire brush volume is available for water and ions (else, a factor 1 − φ should be added to the right hand side to account for the fact that the polymer blocks out part of the volume). Also, it assumes that y is constant at the microscale. In reality this is approximately true only for sufficient double layer overlap, when the Debye length κ −1 is much larger than the interchain distance ∼ σ −1/2 , σ −1/2 κ  1. For more dilute brushes (compared with κ −1 ), y at the polymer chain decays away from the polymer into the solution phase inbetween the chains.

100

P.M. Biesheuvel / Journal of Colloid and Interface Science 275 (2004) 97–106

The ionization degree α is fixed for a quenched brush while for an ionizable brush it is given by α=

1 , 1 + Qezy

(11)

in which Q = 10z(pH−pK)

(12)

for a polyacid (charge number z = −1) and polybase (z = 1), with pH defined in the bulk solution. The parameter αB which will be used is given by αB = 1/(1 + Q), and vice versa Q = (1 − αB )/αB . The electrostatic force on a polyion is given by (for both quenched and ionizable materials) fel =

2n∞ (cosh zy − 1), σ

(13)

as suggested in Ref. [13] and references therein. Equation (13) is equivalent to the osmotic pressure due to the difference in ion concentration in the brush and in the bulk solution. In Appendix A the Chan and Mitchell [28] formalism is used to derive Eq. (13) for both quenched and annealed brushes. Equation (10) can also be expressed in φ, α, and αB without using zy, which results in [29]  2  α 1 − αB αφ αφ = 1+ − 1 − α αB 2n∞ υm m 2n∞ υm m  2  αNσ αNσ = 1+ (14) − 2n∞ mH 2n∞ mH or [13] αφ αNσ αB 1 − α 1 − αB α − = = . 1 − αB α αB 1 − α n∞ υm m n∞ mH

(15)

For a quenched brush α = αB in Eq. (14) and (15). Similarly, the force fel can be expressed as [13]   2 2n∞  1 + 2n∞αφυm m − 1 fel = σ  2n∞  = (16) 1 + (Z/H )2 − 1 , σ with Z = ασ N/(2mn∞ ). 2.3. Quenched brush For a fixed α, the equilibrium brush height is obtained from combination of Eqs. (8) and (16) (fconf + fconc + fel = 0), which results in the implicit  3H N 2 σ υ 2n∞  2 − 1 = 0. − (17) + + 1 + (Z/H ) σ kNa 2 2kH 2 Equation (17) can be solved by iteration, e.g., using a Newton routine, to obtain the brush height H as function of k, m, α, n∞ , υ, σ , and a, after which φ and zy are obtained from Eqs. (7) and (10).

For a high salt concentration (salted brush regime), Z is low and Eq. (16) results in [11,15,24,25,31] q2 , (18) 4n∞ H 2 with q the charge per polyion = Nα/m. (In some references, pressure Π is given, which is f σ kT , while σ equals 1/d 2 with d the interchain distance for a square lattice). Combining Eq. (8) with Eq. (18) leads to the scaling law for the salted brush regime [11–13,15,16,24–26,30]  kα 2 3 + υ. HSB = γ Na 2/3 σ 1/3 (19) 2m2 n∞

fel,SB = σ

Equation (19) is based on the assumption that Z  H , −1/3 which, because Z ∝ n−1 ∞ and H (at most) ∝ n∞ , becomes increasingly accurate with increasing salt concentration, n∞ . For a low salt concentration and a fixed α (osmotic brush regime) Eq. (16) simplifies to [11,15,24,25,31] q αN = . (20) mH H Combination of Eqs. (8) and (20) results with υ = 0 in the height of a quenched brush in the osmotic regime [11–13,15, 25,27,32]

kα q . HOB = Na (21) 3m q

fel,OB =

2.4. Ionizable brush For charge-regulating brushes, Eq. (16) cannot be used directly because α is a function of zy and Eq. (13) is used instead. Also, it is not possible to obtain a single expression in H , but only in zy (or α). Combination of Eqs. (8) and (13) results in N 2 σ υ 2n∞ 3H (cosh zy − 1) = 0. + + (22) 2 kNa 2kH 2 σ H and zy are related through electroneutrality, which follows from Eqs. (10) and (11):

−

H=

Nσ . 2n∞ m sinh zy(1 + Qezy )

(23)

Equations (22) and (23) can be combined into a single implicit expression in zy, but can just as easily be solved jointly. In the osmotic brush regime for annealed brushes, α is low and zy high and α is approximated by 1 , Qezy while electroneutrality can be expressed as α=

(24)

φ = n∞ υm mQe2zy ,

(25)

resulting in ann fel,OB

   1 φ 2n∞ cosh ln −1 . = σ 2 n∞ υm mQ

(26)

P.M. Biesheuvel / Journal of Colloid and Interface Science 275 (2004) 97–106

For n∞ sufficiently small, Eq. (26) becomes [31]   φn Nn∞ 1 ∞ ann fel,OB = . = σ υm mQ σ H mQ

(27)

ann As required, for n∞ equal to zero, fel,OB goes to zero and the neutral brush is recovered. Combining Eq. (27) with Eq. (8) results in  1/3

ann = γ Na 2/3 σ 1/3 + υ + ( + υ)2 − υ 2 , (28) HOB

with =

n∞ k 2 a 2 . 3mQσ 2

(29)

For n∞ = 0, Eq. (28) correctly reduces to HNB , as given by Eq. (9), while setting υ = 0 results in [13,14]   n∞ k 2 1/3 ann 4/3 HOB = a N (30) . 9mQσ The anomalous behavior in this regime—H increases with increasing n∞ and decreasing σ —occurs when the excluded volume term can be neglected, e.g., for very dilute brushes or for theta conditions (χ = 1/2). 2.5. Electrosteric stabilization In the above sections the equilibrium height, H ∗ , of an undisturbed brush was calculated (from this point forward ∗ denotes the undisturbed brush). In this section, pressure Π (Pa) is calculated as is the interaction free energy V (J/m2 ) of two equal brushes that are compressed beyond contact, as a function of the height H with H < H ∗ and H half of the separation of two surfaces. The interaction free energy V is obtained from the free energy change per polyion, F = F − F ∗ using V = 2kT σ F . Multiplying V by 2π results in “F /R” which is measured in force experiments between two crossed cylinders or a plate and a sphere (N/m); multiplying V by πa results in the electrosteric repulsive force between two equal spheres of radius a (N). The pressure Π is the interaction force between two plane-parallel brushes, Π = −1/2dV /dH = (f conf + f conc + f el )kT σ . 2.5.1. Pressure For a quenched brush, Eqs. (8) and (16) are combined to obtain the explicit 3H σ N 2σ 2υ Πq = σftot = − + kT kNa 2  kH 2  + 2n∞ 1 + (Z/H )2 − 1 ,

(31)

which can be used for all salt concentrations (as long as α is fixed). To obtain Π for an ionizable brush, the third term of Eq. (31) must be replaced by Πelann = σfel = 2n∞ (cosh zy − 1). kT

(32)

101

Unfortunately, zy cannot be expressed explicitly in H (only the reverse, see Eq. (23)). However, in the salted brush limit where α = αB = 1/(1 + Q), Eq. (31) can still be used. In the osmotic brush regime for an ionizable brush, based on Eq. (27) the third term in Eq. (31) is replaced by [31]   ann Πel,OB φn Nσ n∞ ∞ ann = σfel,OB = . (33) = kT υm mQ H mQ 2.5.2. Free energy Having obtained H ∗ from Eq. (17) the free energy change per polyion F for a quenched brush is obtained by combining Eqs. (1), (6), and (A.5):   α Z 3H 2 N 2σ υ q +N arcsinh + F = 2 2kH m H 2kNa  H   H − (34) 1 + (Z/H )2 − 1 . Z ∗ H

[F (H )]H H∗

F (H ∗ ).

Here = F (H ) − For a quenched brush, the third, electrostatic term of Eq. (34) can be simplified: for the salted brush regime in   1 n∞ Z 2 1 q Fel,SB = (35) − ∗ σ H H and for the osmotic brush regime in αN H ∗ (36) ln . m H For an ionizable brush, Eq. (A.3) replaces Eq. (A.5) and Eq. (34) becomes  3H 2 N 2σ υ N + ln(1 − α) F ann = + 2 2kNa 2kH m  H   Nσ α 2 2n∞ H − (37) 1 + 2n∞ mH − 1 . σ H∗ q

Fel,OB =

Because α is not an explicit function of H , Eq. (37) cannot be solved directly in H . However, both H and F ann are explicit functions of α; thus, for each value of α (< α ∗ ) height H is directly obtained from Eq. (15) and F ann from Eq. (37), after which F ann can be analyzed as a function of H . For the limiting regimes explicit expressions in H are obtained. In the salted brush regime, α = αB = 1/(1 + Q) and Eq. (35) replaces the last two terms in Eq. (37). In the osmotic brush regime Eqs. (24) and (25) apply and the last two terms of Eq. (37) can be replaced by   √  √ √ Ψ− H H − H∗ N ann ln , − Fel,OB = (38) √ m Ψ Ψ − H∗ where  Ψ=

Nσ Q . n∞ m

(39)

102

P.M. Biesheuvel / Journal of Colloid and Interface Science 275 (2004) 97–106

(A)

(B)

Fig. 2. Polyacid brush height as function of pH, salt concentration, and grafting density. (A) σ = 0.1 nm−2 , (B) c∞ = 100 mM.

3. Results and discussion Results for brush height and interaction force are presented for a polyacid brush as a function of pH–pK. Note that these results at pH∗ also apply to a polybase for pH = pKacid + pKbase − pH∗ . Calculations are based on segment length a = 0.25 nm, molar volume Vm = 40 cm3 /mol (υm ∼ 0.066 nm3 , corresponding to a cubic cell with sides of ∼0.4 nm), polymerization degree N = 100 (contour length Na = 25 nm), k = 1, m = 1, and v = 1 (χ = 0, υ = υm ), unless otherwise stated. Within reasonable bounds, pK, υ, k, and m are used as fitting parameters. In Fig. 2 the brush height H /Na is plotted as function of pH, salt concentration, and grafting density. Fig. 2A shows that with increasing pH the brush height increases. For a fixed degree of ionization (quenched brush, α = 1) the brush thickness increases when the salt concentration is decreased, reaching a plateau value. This is the typical behavior for a quenched brush, e.g., made of polystyrene sulfonate [33]. However, for an ionizable brush, there exists (for each pH value) a salt concentration at which the brush is most extended [13,14]. Fig. 2B shows that with increasing pH (and for a given salt concentration), the brush height increases steadily and reaches a plateau value (that is dependent on salt concentration), resulting in the typical S shape that is experimentally observed, e.g., for σ = 0.022 nm−2 in Ref. [35]. With increasing grafting density, the general behavior is that the brush height increases. However, a reversal of this trend is observed for the lower grafting densities (0.3 nm−2 and below) and for intermediate pH values, in the range 0.5 < pH − pK < 2. This counterintuitive behavior has been reported before for weakly charged brushes [14]. At sufficiently low pH, the minimum brush height, H NB , is obtained, as given by Eq. (9). Fig. 3 is based on experiments by Ahrens et al. [24] in which grafting density and salt concentration were varied for polystyrene sulfonate brushes and the height obtained from the electron density profile as measured by X-ray re-

Fig. 3. Polystyrene sulfonate brush height: experiments [24] and model (N = 83, k = 1.25, m = 1, υ = 0.133 nm3 , α = 1).

flectivity. The model is based on a quenched brush, Eq. (17) (α = 1), and reproduces the trends well, though the influence of added salt on the reduction of thickness, especially at low grafting densities, is underestimated. Based on full ionization (α = 1) and a charge on each segment (m = 1), a best fit was obtained for k = 1.25 (ka = 0.31 nm). It must be noted that such a low Kuhn length is quite unrealistic for PSS. Ref. [34] reports a persistence length of 1.2 nm; thus, k = 2.4/0.25 nm = 9.6. If k = 9.6 is used, it is required to increase the distance between charges to m ∼ 4 to bring the calculated values in the range of the experimental data, but the fit is much less accurate (not shown). Note that m = 4 implies that three of each four sulfonate groups are effectively uncharged, e.g., due to ion condensation. The subject of Kuhn length and charging degree of PSS is further discussed in conjunction with Fig. 6. The ionizable character is more pronounced for PMAA brushes (see Fig. 4). Brushes in these experiments [35] have

P.M. Biesheuvel / Journal of Colloid and Interface Science 275 (2004) 97–106

a high polymerization degree of N = 6400 and a grafting density of 0.022 nm−2 (pH 4, salt: NaNO3 ). The brush height is measured by multiple-angle ellipsometry and calculated as twice the first moment of the segment density profile described by a complementary error function. A protonation constant of pK = 3 was used to obtain an optimum fit despite the fact that for PMAA, pK = 4.7 is more often suggested in the literature. As function of salt concentration, a maximum in brush height is measured around 1–3 mM. A similar maximum is obtained in the calculations, but unfortunately is located at a salt concentration ∼10 times larger than experimentally observed, just as was noted by Biesalski et al. [35]. Otherwise, the model gives a good description of the experiments: if the theoretical curve is shifted by an order of 10 to the left the agreement becomes very acceptable, showing the same limiting dependencies of brush height on salt concentration as experimentally ob1/3 −1/3 served (H ∝ c∞ in the low salt regime and H ∝ c∞ in the

Fig. 4. PMAA brush height: experiments [35] and model (solid line: pH 4, pK = 3, k = 2, N = 6400, υ = 0.133 nm3 , σ = 0.022 nm−2 ). The dashed line is the theoretical curve shifted an order of 10 to the left.

(A)

103

high salt regime). Though adjusting Q to reflect adsorption of cations to charged sites (Qmod = 10pK−pH + 10pKC c∞ ) did shift the maximum somewhat to the left, it did not give a much better descriptions. Deviations observed at relatively high salt concentrations and lower grafting densities (e.g., in Figs. 3 and 4) might be related to electrostatic stiffening, but may also be due to the assumed lateral homogeneity in the Donnan model: the ions in the brush as well as the charged polymer groups all feel the same electrostatic potential. In reality, for a low enough Debye length compared to the spacing of chains, the electrostatic potential changes significantly across the space between the chains. Such lateral inhomogeneities have been considered, e.g., by Seidel [36] and Dean et al. [37]. In experiments by Currie et al. [30] using PAA polyelectrolytes, much higher grafting densities were used than in Ref. [35] (up to 0.4 nm−2 in Currie et al.) with the consequence that the excluded volume term becomes more relevant. The experimental data from ellipsometry (fixed incidence angle, assumed homogeneous refractive index/ monomer density) [30] are reproduced in Fig. 5. To obtain a best fit v = 0.4 (χ = 0.3) is used (υ = 0.027 nm3 ) and pK = 4.1 (∼4.7 more typical). In Fig. 5A, brush height is plotted as function of salt concentration (NaCl) at different pH (1/σ = 3.9 nm2). Trends are well reproduced, especially the fact that the maximum thickness is found in the range c∞ = 100 mM to 1 M. However, the calculations underestimate the maximum thickness for pH 4 and overestimate the thickness for pH 5.8. Fig. 5B shows similar data as well as model calculations for several grafting densities. Because the absolute changes in thickness are underestimated quite significantly by the model, the results of the calculations are plotted on a different y-axis. However, trends are well reproduced: the thickness increases with increasing grafting density, and the thickness slowly increases with increasing salt concentration up to a maximum thickness (at c∗ ), after which the thickness again decreases. Interestingly, for the

(B)

Fig. 5. PAA brush height: experiments [30] and model (pH 4, pK = 4.1, k = 1, υ = 0.027 nm3 , N = 368). Dashed lines in (B) are based on Eq. (28). Note that in (B) the theoretical calculations are plotted on the right y-axis.

104

P.M. Biesheuvel / Journal of Colloid and Interface Science 275 (2004) 97–106

most dilute brush (highest value of 1/σ ), the experimental ∗ ∼ c∗ . and theoretical values for c∗ match rather well: cexp th However, with increasing grafting density (decreasing 1/σ ), ∗ decreases (from ∼ 0.2 to ∼ 0.1 M) while c ∗ increases cexp th (from ∼0.3 to ∼0.9 M). For the osmotic brush regime, the analytical expression (28) is plotted in Fig. 5B which is correct up to ∼10 mM. Currie et al. [30] find for the osmotic regime that the brush height scales with salt concentration with an exponent ∼1/10, in contrast to the theoretical value of 1/3 based on Eq. (30). They [30] explain the difference as due to the absence in the scaling theory of steric effects (excluded volume). In the present calculations that include excluded volume—using Eqs. (22) and (23), or Eq. (28), and the parameter set given in Fig. 5—a very small influence of salt concentration on brush height in the osmotic regime is found as well. Indeed, Eq. (28) shows that for a sufficiently low salt concentration and a sufficiently high grafting density, the brush is quasi-neutral and the salt concentration does not influence brush height. Currie et al. [30] also observe that in the osmotic regime brush height increases with increasing grafting density, in strong contrast with Eq. (30), which predicts that in the osmotic brush regime for annealed polyelectrolyte the brush height scales with a negative exponent of −1/3 (H ∝ σ −1/3 ). Currie et al. [30] suggest that the absence of excluded volume (“steric effects”) in the scaling theory is again the cause of the difference. Here we arrive at the same conclusion and find that the brush expands when the grafting density is increased when using Eq. (28) or Eqs. (22) and (23) (Fig. 5B). Interaction between polystyrene sulfonate brushes is measured by Balastre et al. [15] at various salt concentrations (NaNO3 ). The experiments are reproduced in Fig. 6. Fig. 6A shows the separation when the force starts to increase (F /R = 50 µN/m) as well as the calculated equilibrium separation, 2H ∗ . In Fig. 6B, the force/radius F /R is presented as a function of separation 2H . Polystyrene sulfonate brushes have fixed charge (α = 1) and thus Eq. (34) is used to obtain the interaction free energy V (= F /(2πR)). Because of the low grafting density, the second, concentration, term in Eq. (34) is neglected. Based on Ref. [34] a Kuhn length for polystyrene sulfonate of k = 2.4/0.25 nm = 9.6 is used. To fit the model to the data, it is assumed that only one in three segments carries a charge (m = 3). This is much lower than the sulfonation degree reported in [15] and may be due to counterion condensation [15]. Using these data and Eq. (17) for a quenched brush, the brush extends at low salt almost to its full contour length. Therefore the correction discussed in Eq. (4) is implemented which implies that the first term of Eq. (17) is multiplied by (2 − x 2)/(2 − 2x 2) and the first term of Eq. (34) replaced by 3N/4k(x 2 − ln(1 − x 2 )). The agreement between model and experiment is quite reasonable, with the influence of salt concentration and separation on the repulsion well repro-

(A)

(B) Fig. 6. Interaction between polystyrene sulfonate brushes as function of the separation of the surfaces 2H : experiments [15] and model (N = 612, 1/σ = 105 nm2 , α = 1, k = 9.6 (2.4 nm), m = 3). (A) Separation where the force reaches 50 µN/m (experiments) and twice the equilibrium brush height, 2H ∗ (model). (B) Interaction force F /R.

duced by the model, as well as the point where the repulsion starts to increase (“contact”). In Fig. 7A results are presented for the interaction free energy V of compressed ionizable brushes and compared with the case of two brushes with fixed charge (α = 1). The quenched brush with α = 1 is the upper limit, and with decreasing pH the interaction free energy V decreases (for a polyacid). In all cases, both the total interaction free energy V and the electrostatic component V el increase steadily with decreasing separation. Also, the force f ∝ Π ∝ −dV /dH and f el ∝ Π el ∝ −dV el /dH increase continuously (the curves for V bend concave upward toward the y axis). Though during compression of ionizable brushes the ionization degree α decreases and gradually approaches zero, the

P.M. Biesheuvel / Journal of Colloid and Interface Science 275 (2004) 97–106

(A)

105

(B)

Fig. 7. Interaction between ionizable brushes (N = 612, υ = 0.133 nm3 , 1/σ = 105 nm2 , k = 1.75). pH = pH − pKacid . V = 2kT σ F (solid lines) is the free energy of interaction; V el is the electrostatic component (dashed lines). The difference V − V el is due to conformational aspects and excluded volume. (A) c∞ = 10 mM, (B) pH = 1.

electrostatic potential zy continuously increases, resulting in the fact that the electrostatic force, f el , also increases with deformation, see Eq. (13). Finally, Fig. 7B shows similar calculations for varying salt concentrations. Interestingly, in contrast to the case of quenched brushes (Fig. 6) the interaction free energy (both V el and V ) is at a maximum for an intermediate salt concentration (in this case ∼10–30 mM), with a decreasing interaction free energy above and below that range. This behavior was observed by Kurihara et al. [38] (their Fig. 2) and, in terms of the pressure Π , is discussed in Ref. [31]. The semi-analytical scaling theory is able to reproduce observed trends in the height and interaction force between ionizable brushes, but significant deviations remain when the theory is used for a quantitative description, e.g., of the (maximum) brush height in Figs. 4 and 5. There are several causes for the discrepancy, related to the assumptions made in the model. First, as discussed in the Introduction, the AdG box model approach significantly simplifies the relevant polymer physics (free ends at brush edge, constant Kuhn length). Furthermore, ion adsorption is important, as is the local inhomogeneity of the electrostatic potential, and, especially for the denser brushes used in Figs. 3–5 (up to 2.5 chains per nm2 ), Eq. (6) may underestimate the contribution due to mixing because a dilute brush is assumed.

4. Summary (Semi-)analytical expressions are summarized for endgrafted, quenched, and annealed, polyelectrolyte brushes having a single ionizable group (e.g., polyacid) that can be used irrespective of the salt concentration. The prefactors are stated explicitly which allows more quantitative comparison

of the different contributions to the free energy. A new analytical expression is derived for the osmotic brush regime that includes excluded volume which, in agreement with experiments, predicts that in the osmotic brush regime for a sufficiently dense brush, height becomes independent of salt concentration and increases with grafting density. Experiments from the literature in which the height and interaction force of polystyrene sulfonate and poly(meth)acrylic acid brushes were measured could be reproduced with varying degrees of accuracy using the protonation constant pK and the Kuhn length k as the most important fitting parameters. Due to the high dissociation degree of sulfonate in water, polystyrene sulfonate brushes are quenched, in contrast to the annealed poly(meth)acrylic acid brushes. Analytical expressions for the electrosteric repulsive force and the interaction free energy of overlapping quenched and ionizable brushes are summarized. For both quenched and ionizable brushes the electrosteric (restoring) force and the interaction free energy increase steadily when brushes are increasingly compressed. For ionizable brushes, the electrosteric repulsion does not monotonously increase with decreasing salt concentration (as for quenched brushes) but is at a maximum at an intermediate salt concentration.

Acknowledgment This research was financially supported by the Netherlands Organisation for Scientific Research (NWO). Very helpful suggestions were made by Y. Lauw, J. Klein Wolterink, J. van der Gucht, F.A.M. Leermakers, M.A. Cohen Stuart, C. Seidel and E.B. Zhulina.

106

P.M. Biesheuvel / Journal of Colloid and Interface Science 275 (2004) 97–106

Appendix A. Electrostatic free energy of ionizable polyions

References

The electrostatic free energy (in units kT ) of an ionizable polyion is given by [28] N Fel = m

α

(zyD − zyS ) dα ,

(A.1)

0

which combines electric work related to the formation of the double layer with chemical work related to the ionization of the ionic polymer groups. D and S refer to a calculation based on the free ions (diffuse layer) and a calculation based on the dissociation of the ionizable group at the polymer “surface.” Equation (A.1) can be rewritten as N Fel = − m

 ∞



zy

αS dzy + zy

αD dzy .

[14] [15] [16] [17]

0

2Nn∞ υm N (cosh zy − 1) = ln(1 − α) − m φ 2Nn∞ υm  N = ln(1 − α) − 1 + ( 2n∞φαυm m )2 − 1 . m φ (A.3)

Using Eqs. (7), (11), and (23), the electrostatic force f el of an ionizable polyion in a brush modeled by electroneutrality is finally obtained from Eq. (A.3) as fel = −

[13]

(A.2)

Using Eq. (11) for αS and Eq. (10) for αD , F el for an ionizable polyion is given by [29] Felann

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

dFel 2n∞ = (cosh zy − 1). dH σ

(A.4)

Note that though Eq. (A.4) is valid for a quenched polyelectrolyte as well, Eq. (A.3) is not. An analysis similar to that above shows that for a quenched brush, Eq. (A.3) must be replaced by [39]:   α cosh zy − 1 q Fel = N zy − m sinh zy    α Z H =N 1 + (Z/H )2 − 1 . (A.5) arcsinh − m H Z (Note that Z/H equals f by Lifson [39] when α = 1 and m = 1.)

[18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

H. Ohshima, S. Ohki, Bioelectrochem. Bioenergetics 15 (1986) 173. H. Ohshima, T. Kondo, J. Theor. Biol. 124 (1987) 191. H. Ohshima, T. Kondo, J. Theor. Biol. 128 (1987) 187. H. Ohshima, T. Kondo, Biophys. Chem. 32 (1988) 161. H. Ohshima, Colloid Polym. Sci. 277 (1999) 535. H.-K. Tsao, J. Colloid Interface Sci. 205 (1998) 538. N. Dan, Langmuir 18 (2002) 3524. P.G. De Gennes, Macromolecules 13 (1980) 1069. S.J. Miklavic, S. Marˇcelja, J. Phys. Chem. 92 (1988) 6718. S. Misra, S. Varanasi, P.P. Varanasi, Macromolecules 22 (1989) 4173. P. Pincus, Macromolecules 24 (1991) 2912. R. Israëls, F.A.M. Leermakers, G.J. Fleer, Macromolecules 27 (1994) 3087. E.B. Zhulina, T.M. Birshtein, O.V. Borisov, Macromolecules 28 (1995) 1491. Yu.V. Lyatskaya, F.A.M. Leermakers, G.J. Fleer, E.B. Zhulina, T.M. Birshtein, Macromolecules 28 (1995) 3562. M. Balastre, F. Li, P. Schorr, J. Yang, J.W. Mays, M.V. Tirrell, Macromolecules 35 (2002) 9480. T. Abraham, S. Giasson, J.F. Gohy, R. Jérôme, Langmuir 16 (2000) 4286. R. Israëls, F.A.M. Leermakers, G.J. Fleer, E.B. Zhulina, Macromolecules 27 (1994) 3249. X. Guo, M. Ballauff, Langmuir 16 (2000) 8719. P.J. Flory, Statistical Mechanics of Chain Molecules, Wiley, New York, 1969. A. Naji, R.R. Netz, C. Seidel, Eur. Phys. J. E 12 (2003) 223. P.G. De Gennes, Scaling Concepts in Polymer Physics, Cornell, Ithaca, NY, 1979. S.T. Milner, Science 251 (1991) 905. E.P.K. Currie, W. Norde, M.A. Cohen Stuart, Adv. Colloid Interface Sci. 100 (2003) 205. H. Ahrens, S. Förster, C.A. Helm, Phys. Rev. Lett. 81 (1998) 4172. C. Prinz, P. Muller, M. Maaloum, Macromolecules 33 (2000) 4896. G.J. Fleer, Ber. Bunsen-Ges. Phys. Chem. 100 (1996) 936. F.S. Csajka, R.R. Netz, C. Seidel, J.-F. Joanny, Eur. Phys. J. E 4 (2001) 505. D.Y.C. Chan, D.J. Mitchell, J. Colloid Interface Sci. 95 (1983) 193. E.B. Zhulina, O.V. Borisov, Macromolecules 35 (2002) 9191. E.P.K. Currie, A.B. Sieval, G.J. Fleer, M.A. Cohen Stuart, Langmuir 16 (2000) 8324. E.B. Zhulina, O.V. Borisov, T.M. Birshstein, Macromolecules 33 (2000) 3488. E.B. Zhulina, O.V. Borisov, J. Chem. Phys. 107 (1997) 5952. P. Guenoun, A. Schlachli, D. Sentenac, J.W. Mays, J.J. Benattar, Phys. Rev. Lett. 74 (1995) 3628. L. Wang, H. Yu, Macromolecules 21 (1988) 3498. M. Biesalski, D. Johannsmann, J. Rühe, J. Chem. Phys. 117 (2002) 4988. C. Seidel, Macromolecules 36 (2003) 2536. D. Dean, J. Seog, C. Ortiz, A.J. Grodzinsky, Langmuir 19 (2003) 5526. K. Kurihara, T. Kunitake, N. Higashi, M. Niwa, Langmuir 8 (1992) 2087. S. Lifson, J. Polym. Sci. 23 (1957) 431. S.V. Lyulin, L.J. Evers, P. van der Schoot, A.A. Darinskii, A.V. Lyulin, M.A.J. Michels, Macromolecules 37 (2004), in press.