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Langmuir 2002, 18, 3524-3527
Interactions between Charge-Regulating Surface Layers Nily Dan† Department of Chemical Engineering, Drexel University, Philadelphia, Pennsylvania 19104 Received July 23, 2001. In Final Form: February 14, 2002 In this study we use the Poisson-Boltzmann analysis in the Debye-Huckel limit to examine the effect of charge regulation on the interactions between similarly charged ion-penetrable adsorbed layers. We find that the magnitude of the interactions decays rapidly with the degree of charge regulation, although the decay length remains the Debye screening length. As a result, the interactions between charge-regulating layers are much weaker than those between strongly dissociated ones.
Introduction The inherent tendency of colloidal suspensions to flocculate may be arrested through incorporation of surface charges or adsorbed polymeric layers. The former provides repulsive electrostatic interactions, while the latter gives rise to osmotic forces.1-3 Numerous studies have shown that the electrostatic interactions between charged surfaces can adequately be described using the Poisson-Boltzmann mean-field model.1-3 In this approach, the repulsion between similarly charged surfaces is given by the osmotic pressure of the surface counterions which are trapped in the gap between the surfaces by the condition of electroneutrality.1-4 Like any chemical reaction, the dissociation of ionic species is governed by an equilibrium between the combined and dissociated states. Strongly charged ionic salts (e.g., NaCl) tend to be fully dissociated, regardless of system conditions. However, the degree of dissociation of weakly charged ionic salts (e.g., weak acids or bases) depends on the electrochemical potential. As a result, surfaces carrying weak ionic groups are charge regulated; namely, their surface charge density varies as a function of system parameters such as salt concentration or pH. Various studies examined the effect of charge regulation on the interactions between charged surfaces (see, for example, refs 5-8). They find that charge regulation significantly softens the repulsive interactions between surfaces, since the increase in counterion osmotic pressure is relieved by a shift in the dissociation equilibrium toward the combined state. Electrostatic stabilization of uncharged colloidal particles is obtainable through the adsorption of charged polymeric surface layers. Adsorbed polyelectrolyte layers are used to control surface properties and interactions in a variety of colloidal systems, such as water filtration, paper making, paints, and inks. The polymers may be attached to the surface by one end, thereby forming a † Phone 215 895 6624; Fax 215 895 5837; e-mail dan@coe. drexel.edu.
(1) Israelachvili, J. N. Intermolecular and Surface Forces: with Applications to Colloidal and Biological Systems; Academic Press: New York, 1985. (2) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989. (3) Safran, S. Statistical Thermodynamics of Surfaces, Interfaces and Membranes; Addison-Wiley: Reading, MA, 1994. (4) Parsegian, V. A.; Gingel, D. Biophys. J. 1972, 12, 1192. (5) Ninham, B. W.; Parsegian, V. A. J. Theor. Biol. 1971, 31, 405. (6) Chan, D. Y. C.; Perram, J. W.; White, L. R.; Healy, T. W. J. Chem. Soc., Faraday Trans. 1976, 72, 2844. (7) Prieve, D. C.; Ruckenstein, E. J. Theor. Biol. 1976, 56, 205. (8) Behrens, S. H.; Borkovec, M. Phys. Rev. E 1999, 60, 7040.
grafted brush, or by multiple attachments throughout the chain length, thereby forming adsorbed layers. The effect of charge regulation on end-grafted, charged polymer brushes has been studied theoretically by Zhulina et al.9 They find that, due to charge regulation, the brush layer thickness varies nonmonotonically with the solution salt concentration, and the interactions between overlapping or compressed brushes are softer than those between uncharged polymers. These calculations are in qualitative agreement with experimental observations.10 Although of theoretical interest, the use of polymer brushes in colloidal stabilization is limited. Adsorbed layers, on the other hand, are ubiquitous. The adsorption and interactions between fully dissociated, charged polymers on oppositely charged surfaces have been studied11-16 as well as the adsorption of charge-regulating polymers on charge-regulating surfaces.17,18 To date, little is known regarding the effect of layer-layer interactions on the long-range electrostatic forces and layer properties in charge-regulating, adsorbed layers. In fact, the only analysis of such systems, by Tsao,19 focused on the limits of the Poisson-Boltzmann model validity. Neither the relationship between charge regulation and the surfacesurface interactions nor the effect of surface separation on the degree of dissociation in the layer was examined.19 Why would the interactions between charge-regulating layers differ from those of fully dissociated ones? The dissociation of charge-regulating species is controlled by the system electrostatic potential.1-4 In noninteracting surfaces (namely, widely separated ones), the electrostatic potential varies as a function of the solution salt concentration or pH. In interacting surfaces, however, the electrostatic potential is affected also by the separation between surfaces. As a result, the degree of layer charging varies as surfaces are brought together or taken apart.5 In this paper we apply the Poisson-Boltzmann model in its low electrostatic potential limit (i.e., Debye-Huckel1) (9) Zhulina, E. B.; Borisov, O. V.; Birshtein, T. M. Mcromolcules 2000, 33, 3488. (10) Currie, E. P. K.; Cohen-Stuart, M. A. Langmuir 1999, 15, 7116. (11) Chatellier, X.; Joanny, J. F. J. Phys. II 1996, 6, 1669. (12) Borukhov, I.; Andelman, D.; Orland, H. Macromolecules 1998, 31, 1665. (13) Filippova, N. L. Chem. Eng. Commun. 1998, 167, 181. (14) Gurovitch, E.; Sens, P. PRL 1999, 82, 339. (15) Joanny, J. F. Eur. Phys. J. B 1998, 9, 117. (16) Andelman, D.; Joanny, J. F. C. R. Acad. Sci. IV: Phys. 2000, 1, 1153. (17) Vermeer, A. W. P.; Leermakers, F. A. M.; Koopal, L. K. Langmuir 1997, 13, 4413. (18) Shubin, V; Linse, P. Macromolecules 1997, 30, 5944. (19) Tsao, H. K. J. Colloid Interface Sci. 1998, 205, 538.
10.1021/la011147m CCC: $22.00 © 2002 American Chemical Society Published on Web 03/28/2002
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Langmuir, Vol. 18, No. 9, 2002 3525
Fg(x) ) ne(-eeβψ + e-eβψ) ≈ 2ne2βψ
(2b)
where Fp is the charge density in the polymer layer, Fg is the ion distribution in the gap, n is the overall salt ion concentration, e is the electron charge, Ff is the normalized density of fixed polymeric charges in the adsorbed layer, and β is the inverse of the entropic energy kT (where k is the Boltzmann constant and T the temperature). The pressure between two surfaces is given, in the Debye-Huckel limit, by1-3
p/kT ) n(eβψ*)2
Figure 1. Schematic of the system. The small circles denote mobile co- and counterions, which are free to distribute between the gap and the adsorbed layer.
to examine the interactions between charge-regulating layers of polyelectrolytes adsorbed on uncharged and charged surfaces in monovalent electrolyte solutions. We find that the range of the electrostatic interactions is unaffected by charge regulation. However, the magnitude of the interactions decreases rapidly with the degree of charge regulation. As a result, the repulsion between charge-regulating surfaces may be orders of magnitude lower than those expected from strongly dissociated ones. Model The system examined is composed of two identical layers of charged polymers, adsorbed on either charged or uncharged surfaces (see Figure 1) in solutions of monovalent salt. We discuss only weak polyelectrolytes where the fraction of charged groups on the chain is relatively low and take the solution salt concentration to be high. These allow us to use the Debye-Huckel limit of the Poisson-Boltzmann model,1-3 where the electrostatic potential is taken to be smaller than the entropic energy.We focus here only on electrostatic interactions, neglecting all chain effects such as configurational entropy. For simplicity, we assume that the adsorbed layer thickness is constant (independent of the degree of charge dissociation in the layer) and take the polymer density profile in the layer to be uniform. The limits of validity for all our assumptions, which clearly fail in the case of strongly charged polymers, are defined in the Appendix. The electrostatic potential ψ is defined by the Poisson relationship1-3
∇2ψ ) -4πF(r)/
(1)
where r is the three-dimensional spatial position vector, F is the density of charges, and is the dielectric constant of the medium. In our case (assuming that the adsorbing surface is large) the potential varies only as a function of distance from the adsorbing surface, x, as shown in Figure 1. Our system is composed of two separate regions, at equilibrium with each other: the gap between the adsorbed layers, which contains only mobile ions, and the adsorbed polymer layer, which contains a combination of mobile ions and fixed (polymeric) charges. Solution salt ions and dissociated polymer counterions contribute to the mobile ion population in either region. Applying the Boltzmann distribution, in a one-dimensional system where x is the relevant spatial vector,1-3
Fp(x) ) ne(2eβFf - eeβψ + e-eβψ) ≈ 2ne2β(ψ - Ff) (2a)
(3)
where ψ* is the potential at the point where dψ/dx ) 0. In our symmetric case, this is the midpoint of the gap between the two adsorbed layers. It should be noted that, since the value of the potential at the midpoint can vary as a function of separation between surfaces, so would the pressure. Obviously, eq 1 must be solved for each region separately, using the appropriate ion density, i.e., eq 2a or 2b. Boundary conditions include symmetry in the gap midpoint (x ) 0) and continuity of the electrostatic potential and its derivative across the boundary between the polymer layer and the gap (x ) D). The last condition fixes the derivative of the potential at the solid surface1-3
(dψ/dx)x)D+L ) σ*
(4)
where σ* is the normalized surface charge. Note that the value of the potential at the surface is not fixed but is consistently determined as a function of the system parameters and the separation between surfaces (see, for example, the discussion of interactions between solid charged surfaces in refs 1-3). In the case of charge-regulating polymers, the ionic groups on the polymer chains are subject to an equilibrium between “condensed” and dissociated states: PH T P- + H+. The fraction of dissociated charges, R, depends on the electrostatic potential,1,3,5 given in the Debye-Huckel limit by an expansion up to second order
R ) 1/(1 + H/kd) + eβΨ(H/kd)/(1 + H/kd)2 ) R0 + R1eβψ (5) where kd is the polymer dissociation constant and H is the reservoir concentration of H+ ions. The density of dissociated ionic groups in the adsorbed layer is thus given by an effective density RFf, where Ff is the maximal, or nominal, density of ionic groups in the layer.Note that R1 defines the sensitivity of the dissociation equilibrium to the electrostatic potential; the greater the value, the greater the degree of charge regulation. Thus, R1 ) 0 defines a strongly dissociating polymer which is not charge-regulated. In many cases the solid surface charges will be charge-regulating as well (i.e., σ* is not fixed but dependent on the electrostatic potential). However, for simplicity, we assume that the solid surface is strongly charged so that σ* is indeed a constant. Results Uncharged Surfaces. Calculating the pressure, as a function of separation between the surfaces for σ* ) 0, we find
p/kT ) n(2R0βeFfκ/K)2e2kD(e2KL - 1)2 {(e2kD - 1)(e2KL + 1) + K/κ(e2kD + 1)(e2KL - 1)}2
(6)
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Dan
Figure 2. Effect of H/Kd on the pressure between chargeregulating layers adsorbed on uncharged surfaces (eq 6). The pressure is normalized by n(βeFf)2. The adsorbed layer thickness, L, is equal to 1/κ. As H/Kd increases, so does the sensitivity to charge regulation (i.e., R1). The main figure displays the exponential relationship between p and D, while the inset shows it in linear coordinates.
Figure 3. Effect of layer thickness L on the pressure between charge-regulating layers adsorbed on uncharged surfaces (as derived from eq 6). The pressure is normalized by n(βeFf)2, and H/Kd ) 1. The dashed line denotes κL ) 1, and the double dash line κL ) 5.
where κ ) (8πβne2/)1/2 and K ) κ(1 + R1βeFf)1/2. 1/κ, the Debye screening length, defines the effective range of electrostatic interactions in the system. In should be emphasized that the expression derived for the pressure is valid only when the layers are not compressed, namely, until the two polymer layers start to overlap (D ) 0, as sketched in Figure 1). This is because the interactions between compressed layers are dominated by characteristics not accounted for in our model such as the polymer chain configurational entropy. The maximal repulsion between the two layers, below compression, is given by the pressure at D ) 0. In this limit the potential at the midplane is given by -2R0Ff/(1 + R1eβFf) and
p/kT )
n(R0βeFf)2 (1 + R1βeFf)2
(7)
which clearly decreases with increasing degree of charge regulation, as given by an increasing value of R1. In the limit of large separations between the surfaces so that e2KD . 1, the pressure is given by
p/kT )
n(2R0βeFf)2(e2KL - 1)2e-2kD (1 + R1βeFp)2(e2KL - 1 + κ/K(e2KL + 1))2
(8)
As expected, the pressure between the surfaces decays exponentially with the characteristic Debye screening length,1-3 1/κ, going to zero in the limit of infinite D where the surfaces do not interact. However, the prefactor determining the magnitude of the interactions is sensitive to characteristics of the polymer layer. In Figure 2 we plot the interactions between surfaces as a function of separation as a function of the ratio between H, the system pH, and Kd, the polymer dissociation constant. Thus, the different lines correspond to (for a given polymer layer) different pH values. We see that the magnitude of the interactions decreases significantly with the degree of charge regulation, as may be expected. In Figure 3 we explore the effect of layer thickness on the pressure between moderately charge regulating layers (where H/Kd ) 1). We see that increasing the layer thickness, and thus the overall number of charges in the layer, by a factor of 5 hardly affects the strength of the repulsive interactions between the layers.
Figure 4. Effect of surface separation on the degree of dissociation, R (as calculated in eq 9), for polymer layers adsorbed on uncharged surfaces. H/Kd ) 1; κL ) 1.
It is interesting to examine how the effective degree of charging in the polyelectrolyte layer changes as a function of separation. When D ) 0, namely, when the adsorbed layers begin to overlap, the fraction of dissociated charges is also independent of position in the layer, given by
R(D)0) ) R0(1 - R1βeFf)/(1 + R1βeFf)
(9)
In Figure 4 we plot the degree of dissociation as a function of the separation, D, for H/Kd ) 1. As expected, the degree of dissociation decreases significantly with the distance between the layers. Charged Surfaces. For simplicity, we focus in this section on strongly charged surfaces where σ* is not charge regulating. Examining the pressure between layers of charged polymers adsorbed on charged surfaces where σ* is fixed, we find that, for D ) 0,
p/kT ) nβ2e2R02Ff2{1 - 2σ*eKL(1 + R1βeFf)/R0FfK(e2KL - 1)}2 (1 + R1βeFf)2 (10) In the limit of large separation, the pressure is given by
Charge-Regulating Surface Layers
Langmuir, Vol. 18, No. 9, 2002 3527
p/kT ) n(2R0βeFf)2{e2KL - 1 - 2eKL(1 + R1βeFp)σr}e-2kD (1 + R1βeFp)2(e2KL - 1 + κ/K(e2KL + 1))2
(11)
where σr is given by σ*/R0KFf. In Figure 5 we plot the pressure between surfaces as a function of separation, for non-charge-regulating layers (H/Kd ) 0) and for moderately charge-regulating layers (H/Kd ) 1). We compare layers adsorbed on similarly charged surfaces (where the normalized density is given by σ* ) -0.5) and oppositely charged surfaces (σ* ) 0.5). Note that we may consider the former case since we assume that the layers are adsorbed via nonelectrostatic interactions. We see that the force between layers where the surface charge is similar to that of the polymer is higher than in the case when the surface charge is opposite to, and partially neutralizes, the adsorbed layer (σ* ) 0.5). Charge regulation reduces the strength and range of the interactions. As may be expected, at some specific value of the substrate charge the substrate and the adsorbed layer will neutralize each other, leading to an effective net charge of zero and, as a result, to zero pressure between the layers. We find that this critical surface charge is given by 2KL
σ* )
R0KFf(e
2 -2KL
- 1) e
4nβe(1 + R1Fp/2n)
(12)
Discussion and Conclusions In this paper we examine the interactions between adsorbed layers of charge-regulating polyelectrolytes. Because of the large number of possible parameters and the complexity of the system, we focused on interactions between uncompressed layers, assuming that the polyelectrolytes are weakly charged. The first point of interest is the effect of charge regulation on the long-range electrostatic interactions between polymer-stabilized surfaces.As may be expected, we find that (eq 8) the long-range decay length for the electrostatic interactions is 1/κ, identical to that of noncharge-regulating systems. However, the magnitude of the interactions depends on the parameters of charge regulation and the surface charge (see Figures 2 and 4). Quite surprisingly, we find that the interactions are mostly insensitive to the layer thickness, despite the fact that the overall number of charges in the layer (at fixed charge density Ff) is proportional to L. Since our model assumes that the thickness and density of the polymeric layer are fixed, we cannot directly address the issue of polymer desorption or adsorption as a function of surface separation. However, it is likely that in systems where the adsorption is driven by interactions between charged polymers and oppositely charged surfaces, a reduction in the effective charge density within the layer will lead to desorption of some polymeric chains. On the other hand, in layers adsorbed via nonelectrostatic interactions, a reduction in the effective charge density in the layer will reduce the repulsive interactions within, thereby enabling adsorption of free chains and an increase in the layer density and/or thickness. As discussed earlier, adsorbed polyelectrolytes are commonly used to stabilize colloidal suspensions. Our analysis shows that the barriers for particle coagulation imparted by charge-regulating layers may be quite low, even if the nominal surface charge (measured via titration) indicates a high charge density.
Figure 5. Effect of H/Kd and the surface charge density on the pressure between charge-regulating layers adsorbed on charged surfaces. The pressure is normalized by n(βeFf)2. The adsorbed layer thickness, L, is equal to 1/κ. The solid lines denote H/Kd ) 0, and the dashed lines H/Kd ) 1. Lines without symbols are for σ* ) 0.5, and lines with full diamonds are for σ* ) -0.5.
Acknowledgment. Thanks to S. A. Safran and J. Klein for helpful discussions. This project was funded y NSF Grant 0049076. Appendix Several assumptions were made during the derivation of the model presented above. We now examine the limits of validity for each one for the case of uncharged surfaces. Similar arguments hold for the case when σ* * 0. 1. The electrostatic potential ψ was assumed to be small when compared to the entropic energy kT. Examining the potential, we find that it reaches a maximal value when D ) 0, given by -R0Ff/(1 + R1eβFf). Since R0 ) 1/(1 + H/kd) is always smaller than unity, to satisfy our low potential assumption the density of polymeric charges Ff in the layer must be low. The density of polymeric charges is given by the overall polymer density in the layer, Fp, times the (maximal) degree of charge, f. In adsorbed layers, the density of segments may be high, namely, close to unity. However, in weakly charged chains f is low, so our assumption (maximal) fraction of charge is relatively low is likely to hold. 2. The layer thickness, L, was taken to be constant. The layer thickness is set by the internal interactions between polymeric segments within the adsorbed layer. Therefore, assuming that the layer thickness does not vary with the degree of separation D, we must require that the overall degree of dissociation in the adsorbed layer, R, does not vary much when the separation is changed from D f ∞ to D ) 0. Examining eq 9, we see that the change in R is given by 1/(1 + R1βeFf)1/2. As noted in above, Ff is relatively low. Thus, to satisfy our requirement, the value of R1 ) H/kd/(1 + H/kd) must be moderate; namely, the solution pH must be of the same order of magnitude of the polymer’s pK. 3. The polymer density profile in the layer was assumed to be uniform. As shown, for example, by Vermeer et al.,17 the density profile of the polymer in the layer is, in fact, a decaying function of distance from the solid surface. Therefore, our assumption is satisfied only if we focus on cases where the layer thickness, L, is of the same order of magnitude as the Debye screening length 1/κ, which applies to systems where the polymers are strongly adsorbing. LA011147M
