J. Phys. Chem. B 2007, 111, 979-981
979
Surface Conductivity Reveals Counterion Condensation within Grafted Polyelectrolyte Layers Stanislav S. Dukhin,† Ralf Zimmermann,*,‡ and Carsten Werner‡,§ New Jersey Institute of Technology, Newark, New Jersey 07102-1982, Leibniz Institute of Polymer Research Dresden, Max Bergmann Center of Biomaterials Dresden, Hohe Strasse 6, 01069 Dresden, Germany, and UniVersity of Toronto, Institute of Biomaterials and Biomedical Engineering, 5 King’s College Road, M5S 3G8 Toronto, Canada ReceiVed: August 29, 2006; In Final Form: NoVember 29, 2006
Surface conductivity (SC) has been demonstrated to be a valuable parameter for the characterization of surfacebound polyelectrolyte layers (PLs). The measurement of the SC in dependence of the pH and solution concentration yields information about the Donnan potential, ΨD, the intrinsic charge, the potential of the PL electrolyte interface, Ψ0, the pK of the ionizable groups within the PLs, and the concentration of segments, n. We discuss herein that SC measurements may additionally provide information about counterion condensation. The mobility of the counterions within grafted poly(acrylic acid) (PAA) layers was estimated from the density of COOH groups and SC data to be only 14% of that of free ions (Zimmermann, et al. Langmuir 2005, 21, 5108). In view of this large deviation and the limited sterical constraints within the brushes, we conclude that the number of freely moving counterions is decreased due to counterion condensation. This interpretation agrees well with the measurement of the osmotic pressure for PAA solution (Boisvert, et al. Polymer 2002, 43, 141), which can be exclusively attributed to the remaining mobile counterions of the polyelectrolyte.
Introduction Polymer brushes received great attention during the past decade. A polymer brush can be considered as a dense array of end-attached polymer chains on a surface. The applications of these materials comprise affinity biosensors,3 microanalytical devices,4 chemical microreactors,5 protein resistant surfaces,6,7 thin film waveguides,8 optical data storage,9 and liquid displays.10 Theories describing the static structure of brushes, both for neutral and for charged polymers, are well developed, and most predictions have been experimentally verified in quite some detail.11 The influence of chain length, grafting density, and solvency on, for example, the thickness (height) and the segment density distribution profile normal to the substrate are now relatively well understood. However, much less is known on the mobility of solvent and small solute species inside polyelectrolyte brushes. To probe such dynamic properties, surface conductivity measurements are attractive, since they can provide information on the number, distribution, and mobility of mobile ions at the interface. A theory of surface conductivity (SC) has been recently elaborated12 and confirmed quantitatively in experiments13 with sodium acrylate gels of low volume fraction (0.05). Agreement between theory and experiments was achieved with the assumption that no big difference exists between the counterion mobilities in the inner polyelectrolyte layer and those in the bulk electrolyte. The application of the same theory to SC data obtained for grafted poly(acrylic acid) (PAA) layers revealed a strong decrease of the counterion mobility inside the * To whom correspondence should be addressed. E-mail: zimmermn@ ipfdd.de. Phone: 0049 351 4658 258. Fax: 0049 351 4658 533. † New Jersey Institute of Technology. ‡ Max Bergmann Center of Biomaterials Dresden. § University of Toronto.
brush.1 The volume fraction of polymer, φ, in the PAA layers was about 0.24. This exceeds the φ value of the sodium acrylate almost 5 times. As the surface conductivity measures the number of freely moving small ions, an alternative interpretation may be that the number of freely moving counterions is decreased due to counterion condensation. Therefore, we discuss below the likeliness of counterion condensation based on SC data for grafted PLs. Theory Donnan Potential and Surface Conductivity of Thick PLs at Any Degree of Dissociation. The Donnan potential depends on the ratio of the fixed charge density, F, to the bulk electrolyte concentration, C0. The value of ΨD is determined by the condition of local electroneutrality:
F 1 + 10
pK-pH
+ exp(-ΨD) C0F( exp(-ΨD) - exp(ΨD)) ) 0 (1)
where the dimensionless Donnan potential is ΨD ) FΨ ˜ D/(RT), F is the Faraday number, T is the absolute temperature, and R is the gas constant. An approximate analytical equation for ΨD as a function of the dimensionless group B ) 10pK-pH|F|/(C0F) is derived and verified by numerical calculation:
exp(-ΨD) )
10pH-pK (x4B + 1 - 1) 2
(|ΨD| > 2) (2a)
exp(-ΨD) ) xB + 1
(|ΨD| < 2) (2b)
10.1021/jp065585j CCC: $37.00 © 2007 American Chemical Society Published on Web 01/12/2007
980 J. Phys. Chem. B, Vol. 111, No. 5, 2007
Dukhin et al.
The determination of ΨD at low potentials (|ΨD| < 2) yields the value of B according to eq 2b, whereas ΨD measured at high potentials (|ΨD| > 2) yields the pK, taking into account eq 2a and that B is measured already. The combination of the measured pK and B values yields the fixed charge density, F, or n ) |F|/F. Hence, the main characteristics n and pK may be determined if ΨD is measured within a broad range of pH values, which comprises |ΨD| < 2 and |ΨD| > 2. Surface conductiVity measurements within a broad pH range are Valuable, since the Donnan potential can be determined and afterwards the pK and n may be calculated. This opportunity exists for thick PLs, whose main portion is occupied by the electroneutral zone with the potential ΨD. Accordingly, the thickness, d, of the PLs has to exceed the length of the transition zone, κi-1, that is, d > κi-1 ) κ-1 exp(ΨD/2). This condition is mainly satisfied if the PL thickness is larger than 1-3 nm. The theory developed for thick PLs12 permits the calculation of the pH dependence of the fractional PL charge, R, as well: -1 1 R ) 1 + (-1 + x1 + 4B) 2
[
]
(3)
The SC is caused by the excessive concentration of counterions, C+(x) - C0, and by the deficit of coions, C-(x) - C0, within the double layer (DL). For a binary electrolyte, the equation for the surface conductivity is
∫-d∞ (C+(x) - C0) dx + u-∫-d∞ (C-(x) - C0) dx]
Kσm ) F[u+
(4) where u+ and u- are the cation and anion mobility. The equation for the surface conductivity is extremely simplified, when the segment distribution is uniform (n(x) ) const), because almost the entire PL is isopotential (Ψ(x) ) ΨD):
Kσm ) FC0d[u+(exp(-ΨD) - 1) + u-(exp ΨD - 1)]
(5)
The substitution of exp(-ΨD) in eq 5 according to eqs 2a and b yields a simple system of algebraic equations for the calculation of n and the pK from surface conductivity data (Kσm) measured at low and high pH. The contribution of the diffuse layer to the measured SC is rather small12 in comparison to the conductivity, Kσ,i m , caused by the electroneutral portion of the PL, characterized by ΨD. Good agreement between the theory described above, data of numerical simulation, and experimental data was recently accomplished by Yezek for sodium acrylate gels.13 Ion Condensation. The description of the double layer by the solution of the linearized Poisson-Boltzmann equation is known to become inadequate once the electrostatic energy of small ions near the surface exceeds the thermal energy. The small ions accumulate close to the surface and become coupled to the macroion. Manning14 treats the distribution of counterions around highly charged polyelectrolytes in terms of the linear charge density parameter, ξ ) lB/b, where lB ) e2/4π0kT is the Bjerrum length (lB ) 0.71 nm at 25 °C in water) and b is the axial distance between charged groups fixed along the polyelectrolyte chain. Manning considered highly charged polyelectrolytes, ξ > 1, and counterions with the valency zc. He argued that the PB ion atmosphere around the polyion is unstable for ξ > 1/zc and proposed that, as a result, the fraction 1 - 1/zcξ of the fixed charges on the rod becomes completely neutralized by counterions that condense onto them until ξ reaches the value 1/zc. In the theory of the counterion condensation, it is then assumed
that the noncondensed counterions are distributed according to the linearized PB equation. Hence, the critical value of ξ is ξc ) |zc|-1. The theory of ionic condensation is developed elsewhere15,16 and predicts an effective charge number, Zeff, which increases with the bare charge number, Zstr, up to the point where the thermal energy of the ions balances the reversible work necessary to remove ions from the charged species. From that point on, the effective charge of the polymer remains constant. Thus, a maximum effective charge can be expected and all additional charges added to the polyelectrolyte above this maximum value will condense (ionic condensation), leading to a constant Zeff value. Ion Condensation and Conductivity. Conductivity studies provide information on the number of ionic species in a solution and their mobilities. As the association of ions reduces the number of charge carriers, the conductivity of a solution also depends on the degree of association. In the case of polyelectrolytes, it depends on the extent of counterion association. Lyklema17 introduced the notion of conductometrically bound counterions and their fraction 1 - fcond. He suggested to restrict a first approximation solely on the distinction of immobile bound counterions and free counterions (having bulk mobility). However, no experimental studies18,19 on the conductometry of polyelectrolyte solutions have been interpreted on the basis of this oversimplified assumption. Analysis of such data, in terms of ionic concentrations and mobility, is ambiguous because both fcond and the mobility of bound counterions are unknown. The possibility of a nonzero mobility of bound counterions is mentioned in ref 17 as a possible reason for the fact that the measured fraction of bound counterions, f, is larger than that predicted according to the Manning theory. Counterion Condensation and Osmotic Pressure. Osmometry is a simple experimental method to access the effective charge via an equation of state. In this method, a Donnan equilibrium is established by the distribution of a diffusible electrolyte in two compartments separated by a semipermeable membrane when a nondiffusible polyion (charged particle) is localized in one of the compartments. The charge of the polymer creates an uneven distribution of the small ions across the membrane which is the origin of the net osmotic pressure, Π, experienced by this system. An equation for the osmotic pressure was derived by Boisvert et al.2
Π = ΠID + Πχ + Πexc + ΠD
(6)
where ΠID is the limiting osmotic pressure (ideal gas law), Πχ is the pressure coming from the solvency effect, Πexc is the excluded volume contribution to the net osmotic pressure, and ΠD is the contribution of the polymer charge via the Donnan equilibrium. After correction of eq 6 for the polymer contribution to the total osmotic pressure (solvency and excluded volume), the remaining pressure can be attributed to the counterions of the polyelectrolyte and used to compute the effective charge number per polymeric chain, Zeff. The behavior of Zeff against the chain concentration and pH was investigated in ref 2 after neutralization of poly(acrylic acid) with LiOH, NaOH, and N(CH3)4OH. The results clearly indicate that the nature of the monovalent counterions has no effect on Zeff, leading to the conclusion that the interaction between monovalent counterions and the acrylate gel is purely electrostatic in agreement with conductometric and potentiometric results reported in the literature.2 The behavior of Zeff at varying degrees of ionization of the polymer and its
Ion Condensation within Polyelectrolyte Layers concentration is also in good agreement with the theoretical expectations obtained by the theory of ionic condensation.2 Results and Discussion The notion of counterion condensation has been found to be useful for the interpretation of conductivity data of polyelectrolyte solutions.17 Accordingly, counterion condensation has to be accounted for in the investigation of grafted PLs by surface conductivity measurements as well. However, to our knowledge, this topic has not yet attracted attention so far. Ion condensation may manifest itself in SC measurements if the following conditions are fulfilled: (i) The distance between the charged groups, b, has to be shorter than the Bjerrum length, lB. (ii) The monomer concentration, n, or the polymer volume fraction has to be sufficiently high. If this is not the case, the manifestation of counterion condensation is weak. This follows from eq 2.2.8 for fcond in ref 17, derived according to the Oosawa two-phase model.20 In ref 2, the dependence of the effective charge, Zeff, on the chain concentration was determined by Boisvert et al. for poly(acrylic acid) solutions: Zeff was found to be maximal at high dilution of the polyelectrolyte solution and decreases monotonously with increasing chain concentration. In addition to conditions i and ii, the nature of the ionic sites has to be accounted for. The theory of ion condensation was originally developed for strong polyelectrolytes.20 Therefore, the question of whether this phenomenon can also be observed for brushes of weak polyelectrolytes arises. In order to estimate the ion mobility within grafted poly(acrylic acid) layers, the surface conductivity was determined by us for the case of complete dissociation in 10-3 M KCl solutions.1 Although the brushes show a high surface conductivity of 67 nS under the experimental conditions, the counterion (K+) mobility was calculated to be only about 14% of the corresponding bulk mobility. The volume fraction of the polymer was about 0.26. Taking into account hydrodynamic and steric effects, the ion mobility can be reduced to a maximum of 50%; that is, additional phenomena contribute to the reduced ion mobility. Interestingly, a similar low ion mobility was recently found for grafted poly(glutamic acid) layers.21 In the case of the grafted PAA, the axial distance between the charged groups is about 0.14 nm. This is much smaller than the Bjerrum length (lB ) 0.71 nm); that is, condition i is fulfilled for the PAA brush. The grafting density of the polymer chains was 1013 cm-2, and the number of repeating units was 768 (corresponding to a molecular weight of 55.3 kg/mol). Taking these values and the layer thickness of 30 nm, a chain concentration of 5.6 mmol/L is obtained. Since Boisvert et al.2 have observed ion condensation at lower chain concentrations of PAA solutions (MW ) 35 kg/mol), the monomer concentration in the brushes can be considered as sufficiently high. Taking this all together, we may assume that under the experimental conditions used in ref 1 the decreased surface conductivity (as compared to the value calculated from the total concentration of ionic sites) is caused by the counterion condensation. It is noteworthy that Lyklema recommends working at a sufficiently high pH to observe the manifestation of counterion condensation in the case of weak polyelectrolytes.17 The quantification of the effect has not yet been achieved as fcond, and the mobility of bound counterions is unknown.17 Although the latter complicates the quantification, the assumption of any nonzero mobility of bound counterions corresponds to a stronger counterion condensation.
J. Phys. Chem. B, Vol. 111, No. 5, 2007 981 if it is calculated with the assumption Let us nominate fcond 0 that the mobility of bound counterions is zero. After the ion mobility is accounted for, fcond smaller than fcond corresponds to 0 the same measured SC. It is obvious that a nonzero mobility of bound counterions does not complicate the detection of the counterion condensation, although the quantification becomes more difficult. The interpretation of the conductivity of polyelectrolyte solutions is complicated17 because the macroions contribute to it, which is not the case for surface-bound polyelectrolyte (layers) and their (surface) conductivity. Consequently, SC measurements at grafted PLs may offer a new opportunity for the investigation of counterion condensation. The parameter b increases with decreasing degree of dissociation, or fractional PL charge, R, and may be expressed quantitatively through R. As an analytical expression for R was derived in ref 12 and as fcond may be expressed through it as well, the counterion condensation may be incorporated in the theory of the SC at any degree of dissociation.12 However, this result does not provide a basis for the quantification of the effect yet. The Oosawa or Manning theories were developed for strong electrolytes only. As counterion condensation influences many phenomena (osmotic pressure, polyelectrolyte adsorption, electrostatic repulsion, surface conductivity, etc.), the interest in a clarification of the effect increases.22-26 The proposed evaluation of SC data may become a useful means in that context. Future work will be devoted to the quantification of the effect. References and Notes (1) Zimmermann, R.; Norde, W.; Cohen Stuart, M. A.; Werner, C. Langmuir 2005, 21, 5108. (2) Boisvert, J.-Ph.; Malgat, A.; Pochard, I.; Daneault, C. Polymer 2002, 43, 141. (3) Riepl, M.; Mirsky, V. M.; Novotny, I.; Tvarozek, V.; Rehacek, V.; Wolfbeis, O. S. Anal. Chim. Acta 1999, 392, 77. (4) Fodor, S. P. A.; Rava, R. P.; Huang, X. C.; Pease, A. C.; Holmes, C. P.; Adams, C. L. Nature 1993, 364, 555. (5) Braun, H.-G.; Meyer, E.; Kratzmu¨ller, T. In Micro Total Analysis Systems ’98, Proceedings of the µTAS ’98 Workshop; Harrison, D. J., van den Berg, A., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1998. (6) Jeon, S. I.; Andrade, J. D. J. Colloid Interface Sci. 1991, 142, 159. (7) Uyama, Y.; Kato, K.; Ikada, Y. AdV. Polym. Sci. 1998, 137, 1. (8) Mathy, A.; Mathauer, K.; Wegner, G.; Bubeck, C. Thin Solid Films 1992, 215, 98. (9) Sawodny, M.; Schmidt, A.; Urban, C.; Ringsdorf, H.; Knoll, W. Makromol. Chem., Macromol. Symp. 1991, 46, 217. (10) Machida, S.; Urano, T. I.; Sano, K.; Kawata, Y.; Sunohara, K.; Sasaki, H.; Yoshiki, M.; Mori, Y. Langmuir 1995, 11, 4838. (11) Currie, E. P. K.; Norde, W.; Cohen Stuart, M. A. AdV. Colloid Interface Sci. 2003, 100-102, 205. (12) Dukhin, S. S.; Zimmermann, R.; Werner, C. J. Colloid Interface Sci. 2004, 274, 309. (13) Yezek, L. P. Langmuir 2005, 21, 10054. (14) Manning, G. J. Phys. Chem. 1975, 79, 262. (15) Belloni, L. Colloids Surf., A 1998, 140, 227. (16) Nyquist, R.; Ha, B.; Liu, A. Macromolecules 1999, 32, 3481. (17) Cohen Stuart, M. A.; De Vries, R.; Lyklema, J. Polyelectrolytes. In Fundamentals of Interface and Colloid Science: Soft Colloids; Lyklema, J., Ed.; Academic Press: London, 2005; Vol. V. (18) Vink, H. Macromol. Chem. 1982, 183, 2273. (19) Mandel, M. Polyelectrolytes. In Encyclopaedia of Polymer Science and Engineering, 2nd ed.; Mark, H. F., Bikales, N. M., Overberger, C. G., Mendes, G., Eds.; Wiley: New York, 1988. (20) Oosawa, F. Polyelectrolytes; Marcel Dekker: New York, 1971. (21) Zimmermann, R.; Osaki, T.; Kratzmu¨ller, T.; Gauglitz, G.; Dukhin, S. S.; Werner, C. Anal. Chem. 2006, 78, 5851. (22) Ohshima, H. J. Colloid Interface Sci. 2002, 248, 499. (23) Ohshima, H. Colloids Surf., A 2003, 222, 207. (24) Balastre, M.; Li, F.; Schorr, P.; Yang, J.; Mays, J. W.; Tirrell, M. V. Macromolecules 2002, 35, 9480. (25) Das, B.; Guo, B.; Ballauff, M. Prog. Colloid Polym. Sci. 2002, 121, 34. (26) Mei, Y.; Ballauff, M. Eur. Phys. J. E 2005, 16, 341.
