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Langmuir 2007, 23, 452-459
Ion Distribution around Electrostatically Stabilized Polystyrene Latex Particles Studied by Ellipsometric Light Scattering Andreas Erbe, Klaus Tauer, and Reinhard Sigel* Max Planck Institute of Colloids and Interfaces, D-14476 Golm, Germany ReceiVed July 13, 2006. In Final Form: October 3, 2006 The ion distribution around electrostatically stabilized polystyrene latex spheres for different ionic strengths is investigated by ellipsometric light scattering. This method is sensitive to the refractive index profile around colloidal particles, which is affected by the local salt content. At an average salt concentration of c* ) 10-4 mol L-1, the ion concentration at the particle interface increases discontinuously, and a layer of high salt content with 20-30 nm thickness is built up. The observation cannot be explained within the framework of the Poisson-Boltzmann equation; it rather resembles a prewetting transition. Interactions that could possibly lead to a stabilization of the observed layer of high salt content are discussed.
1. Introduction The central issue in the physical chemistry of colloids is the stability of the colloidal objects against precipitation. Two of the three stabilization mechanismsselectrostatic and electrosteric stabilizationsinvolve charges. To a large extent, the actual stability is governed by the counterions around the particles. Since stabilization and destabilization of the particles are the most important topics in all applications of colloids, the distribution of counterions around charged interfaces is the crucial point in all branches of the colloidal sciences.1 In addition to the central role in the field of colloids, it is very important in biological processes.2 While there are a number of workers performing numerical simulations of counterion distribution, experimental results on the counterion distribution in dispersions of colloidal particles, polymers, or molecular aggregates are much rarer. Many classic experiments yield only a single global quantity characterizing the particle including the counterions, e.g., the electrophoretic mobility.3 To obtain detailed information about the counterion shell, small-angle scattering techniques have been applied. Most experiments to date have been performed on systems with cylindrical geometry. These experiments can been compared with the predictions of the Poisson-Boltzmann (PB) cell model.4 Agreement is found for the distribution of counterions around cylindrical micelles, investigated with small-angle X-ray scattering (SAXS).4 These results have been verified for the distribution of counterions around rodlike polyelectrolytes by anomalous SAXS (ASAXS).5,6 Around DNA, the counterion distribution has been characterized by small-angle neutron scattering (SANS), and only for trivalent counterions have deviations from the predictions been found, which could be accounted for by assuming a finite size of the counterions.7 Das * Corresponding author. E-mail:
[email protected]. (1) Lyklema, J. Fundamentals of interface and colloid science. Volume II: Solid-liquid interfaces; Academic Press: London, 1995. (2) McLaughlin, S. Annu. ReV. Biophys. Biophys. Chem. 1989, 18, 113-136. (3) Hunter, R. J. Zeta potential in colloid science. Principles and application; Academic Press: London, 1988. (4) Wu, C. F.; Chen, S. H.; Shih, L. B.; Lin, J. S. Phys. ReV. Lett. 1988, 61, 645-648. (5) Guilleaume, B.; Blaul, J.; Ballauff, M.; Wittemann, M.; Rehahn, M.; Goerigk, G. Eur. Phys. J. E 2002, 8, 299-309. (6) Patel, M.; Rosenfeldt, S.; Ballauff, M.; Dingenouts, N.; Pontonic, D.; Narayanan, T. Phys. Chem. Chem. Phys. 2004, 6, 2962-2967. (7) Zakharova, S. S.; Egelhaaf, S. U.; Bhuiyan, L. B.; Outhwaite, C. W.; Bratko, D.; van der Maarel, J. R. C. J. Chem. Phys. 1999, 111, 10706-10716.
et al. use numerical solutions of the full nonlinear PB equation and compare the results with results from ASAXS studies of counterions around DNA to find quantitative agreement.8 Experimental evidence is found to indicate that the effect of finite ion sizes have to be taken into account at low separations.9 The counterions around cylindrical polyelectrolytes in shells of block copolymer micelles are reported to be confined to the corona of these particles by SAXS and ASAXS.10,11 Micelles of several shapes were studied by a combination of SAXS and SANS. The authors conclude that with a combination of these methods it is possible to evaluate the counterion distribution.12 This evaluation seems however to be difficult to perform in detail. Studies of the counterion distribution around colloidal spheres are almost completely absent from the literature, although most colloidal particles are spherically shaped. Sumaru et al. have used SANS experiments with contrast matching to investigate the ion distribution around spherical micelles of low molecular amphiphiles.13 They find agreement of their data with the predictions from numerical solutions of the PB equation assuming a certain interface potential. This interface potential is however difficult to access for micelles of low molecular amphiphiles, as those used by Sumaru and co-workers. In this work, we investigate the changes in the interface layer around spherical polystyrene (PS) latex particles stabilized with sodium perfluorooctanoate or sodium dodecyl sulfate (SDS) upon addition of sodium chloride. Reflection ellipsometry has been shown to be capable of probing the counterion distribution at planar interfaces.14 Here, we use ellipsometric light scattering (ELS), a technique that is the scattering analogue to reflection ellipsometry.15 It is therefore sensitive to changes at the interface of colloidal particles. (8) Das, R.; Mills, T. T.; Kwok, L. W.; Maskel, G. S.; Millett, I. S.; Doniach, S.; Finkelstein, K. D.; Herschlag, D.; Pollack, L. Phys. ReV. Lett. 2003, 90, 188103. (9) Andresen, K.; Das, R.; Park, H. Y.; Smith, H.; Kwok, L. W.; Lamb, J. S.; Kirkland, E. J.; Herschlag, D. Phys. ReV. Lett. 2004, 93, 248103. (10) Groenewegen, W.; Egelhaaf, S. U.; Lapp, A.; van der Maarel, J. R. C. Macromolecules 2000, 33, 4080-4086. (11) Dingenouts, N.; Merkle, R.; Guo, X.; Narayanan, R.; Goerigk, G.; Ballauff, M. J. Appl. Crystallogr. 2003, 36, 578-582. (12) Aswal, V. K.; Goyal, P. S.; De, S.; Bhattacharya, S.; Amenitsch, H.; Bernstorff, S. Chem. Phys. Lett. 2000, 329, 336-340. (13) Sumaru, K.; Matsuoka, H.; Yamaoka, H.; Wignall, G. D. Phys. ReV. E 1996, 53, 1744-1752. (14) Koelsch, P.; Motschmann, H. J. Phys. Chem. B 2004, 108, 18659-18664. (15) Erbe, A.; Tauer, K.; Sigel, R. Phys. ReV. E 2006, 73, 031406.
10.1021/la062033j CCC: $37.00 © 2007 American Chemical Society Published on Web 11/23/2006
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In comparison to the classical scattering methods employed by other authors, ELS has several advantages. In the classical SAXS and SANS experiments, it is hard to separate the information about the ion layer surrounding the particle from the information about the particle itself, since these techniques are not truly interface-sensitive. The information content about the shell usually has to be extracted by contrast variation. In ELS, no such change, which may affect the system, is needed. This problem of SAXS and SANS is to some extent overcome by ASAXS, where the contrast is varied by changing the wavelength of the X-rays used. In the results from ASAXS experiments, the information about the counterions still has to be extracted from scattering curves over a wide q-range in a complicated modeling procedure, while in ELS, all information about the shell is conveniently located in a narrow range of scattering angles. In this paper, the basic principle of ELS is introduced at first in section 2. After the Experimental Section 3, results for the counterion distribution around PS colloids are presented in section 4. The discussion in section 5 is divided into several subsections in order to address different implications of the results. After a comparison to the Gouy-Chapmann theory (5.1), a quantitative analysis section (5.2) yields a local salt concentration 2 orders of magnitude above the Gouy-Chapmann result, if the average salt concentration is above a given threshold. The change at this threshold is discussed as a prewetting transition on the colloidal interface (5.3). This thermodynamic argument is complemented by a discussion of involved interactions (5.4), before a final discussion on the properties of the salt-rich layer (5.5).
2. The MethodsEllipsometric Light Scattering (ELS) By applying the principles of null ellipsometry to scattered light, light scattering gets sensitive to information from the interface of colloidal particles.15 For the experiments, a polarizer, a compensator, and an analyzer were installed in the optical path of a light scattering goniometer. While reflection ellipsometry probes the change in the polarization of light upon reflection, ELS probes the change in the polarization of light upon scattering. This information is usually written in an amplitude scattering matrix with the elements S1, S2, S3, and S416
( ) EH EV
(f)
)
( )( )
exp[ik(r - x)] S2 S4 EH S3 S1 ‚ EV -ikr
(i)
(1)
paper.15 Ellipsometry measures the ratio of the two complex quantities S2 and S1.18 It is written as
S2 ) tan(Ψ)‚ei∆ S1
(2)
with the amplitude ratio tan(Ψ) and the phase difference ∆, similar to reflection ellipsometry. If the particles are small enough, there is a specific value ΘB of the scattering angle Θ where the parameters tan(Ψ) and ∆ behave like at the Brewster angle of reflection: tan(Ψ) shows a minimum, and ∆ steeply changes from 0° to 180° or -180°. Because of this similarity, ΘB is also called a Brewster angle. Like in reflection ellipsometry, the sensitivity to an interface layer is highest around the Brewster angle. Fitting to the model of a coated sphere is used for inversion of the data. Since Rc introduces a characteristic length scale to the experiment, it is advantageous to involve measurements at several wavelengths λ. In addition to the real sensitivity to layers on colloidal particles without the requirements on samples/solvent combinations common to contrast-matching techniques, there are other advantages of the ellipsometric scattering over classical scattering methods. These are mentioned only briefly here but will be addressed in upcoming publications. First, moderate polydispersity is not an issue in ELS. Basically, this is due to the nature of the measured quantity S2/S1 which is a complex number. Scattering contributions with a defined phase shift between EH and EV emerge from each individual scattering particle within the scattering volume. The superposition of all these generally elliptically polarized spherical waves can be split up in an incoherent, completely unpolarized part and a coherent, completely polarized contribution.17 One possible origin of the incoherent part is polydispersity, which results in slightly differing elliptical polarizations of scattered waves for particles of different size. The incoherent part reduces the signal-to-background ratio in the procedure of null ellipsometry but has no further effect. The ellipsometric measurement is sensitive to the coherent part, which contains the information on averaged particle properties (calculated by suitable weighting). As a second advantage, information about the absolute contrast of the particles can be extracted. The method is sensitive to effects of the Mie theory, which are beyond the simpler RayleighGans approximation of scattering based on the Fourier transform. The difference becomes obvious in the behavior of ∆, where the Rayleigh-Gans theory wrongly predicts the sharp step from ∆ ) 0° for Θ < 90° to ∆ ) 180° for Θ > 90°.
Here, x is the position of the scattering particle along the beam, r is its distance to the detector, and k ) 2πn/λ is the wave vector amplitude for light of vacuum wavelength λ within the medium of refractive index n. (EEVH)(f) and (EEVH)(i) are the polarizations of scattered and incident light, respectively. EV is the electric field component polarized perpendicular to the scattering plane, i.e., the plane spread by the incident beam and the direction of detection. EH denotes the component within the scattering plane. The designation follows the convention in light scattering. In ellipsometry, the standard nomenclature is Es instead of EV and Ep replacing EH. For a coated sphere with core radius Rc, shell thickness ds, and refractive index values nc and ns of core and shell, respectively, the diagonal elements S1 and S2 of the scattering matrix are calculated from the Mie theory and its modern supplements, while the off-diagonal elements S3 and S4 vanish.16,17 The required equations are listed in the Appendix of our previous
3.1. Polystyrene Latex Particles. Polystyrene particles were synthesized by emulsion polymerization according to standard procedures. In brief, 10 g of styrene monomer and 0.1 g of surfactant (either sodium dodecyl sulfate, Carl Roth, Karlsruhe, Germany, or sodium perfluorooctanoate, Lancaster Synthesis, Heysham, U.K.) were dissolved in 35 g of double-distilled water and heated to the polymerization temperature of 80 °C under stirring and purging with nitrogen. After thermal equilibrium was achieved, the polymerization was started by adding the initiator (0.32 g of potassium peroxodisulfate in the case of sodium dodecyl sulfate and 0.673 g of PEGA200 in the case of sodium perfluorooctanoate) dissolved in 5 g of double-distilled water. PEGA200 is a symmetrical poly(ethylene glycol)-azo initiator as described elsewhere.19 After 3 h,
(16) Bohren, C. F.; Huffman, D. R. Absorption and scattering of light by small particles; J. Wiley & Sons: New York, 1983. (17) Kerker, M. The scattering of light and other electromagnetic radiation; Academic Press: San Diego, 1969.
(18) Azzam, R. M. A.; Bazhara, N. M. Ellipsometry and polarized light; Elsevier: Amsterdam, 1977. (19) Tauer, K.; Antonietti, M.; Rosengarten, L.; Mu¨ller, H. Macromol. Chem. Phys. 1998, 199, 897-908.
3. Experimental Section
454 Langmuir, Vol. 23, No. 2, 2007 the polymerization was completed, and coagulum was removed from the latex by filtration through sintered glass frit. The latex particles were diluted to a final weight fraction of about 3‚10-5 with deionized water. The concentration is low enough to ensure that multiple scattering is negligible. NaCl (VWR International, Darmstadt, Germany) solution was added to adjust the NaCl concentration up to 5‚10-3mol/L. Deionized water with a conductivity of 0.055 µS cm-1 was used to prepare the samples. Because of the dissolution of atmospheric carbon dioxide in the samples and because the conductivity of water without salt is largely affected by dissolved gases, samples without excess NaCl are treated as having an excess salt concentration of ∼10-5mol L-1.20 This slightly arbitrary value is on the same order of magnitude as that used by other authors.21 In addition, it is consistent with the pH of the samples, which was determined as ∼5.5. All measurements were performed at 25 °C. 3.2. Ellipsometric Light Scattering. The experimental setup consists of a commercial ALV/SP-86 goniometer system (ALV GmbH, Langen, Germany) equipped with a polarizer, compensator, and analyzer (B. Halle & Nachf., Berlin, Germany). Measurements were done at λ ) 633 nm (HeNe Laser PL-3000, Polytec GmbH, Berlin, Germany) and λ ) 532 nm (frequency-doubled Nd:YAG Laser DPSS-532-400, Coherent, Inc., San Diego, CA). The measurements are performed in a way common in null ellipsometry in the PCSA (polarizer compensator sample analyzer) geometry.18 The compensator is fixed at an angle of +45° with respect to the scattering plane. The scattering intensities are scanned over a wide range of polarizer and analyzer angular settings. From a fit to the trigonometric equation, the parameters tan(Ψ) and ∆ are obtained. Because of the wide scanning range, the results correspond to twozone-averaged values.18 Data of tan(Ψ) and ∆ for two wavelengths and several Θ were fit simultaneously to the optical model described in section 2. Birefringence of the entrance window introducing a wavelength-dependent offset in ∆ was considered by two additional fixed parameters. The values of both of them are known from calibration measurements away from ΘB.15 The procedure used for fitting was a simulated annealing coupled to a downhill simplex method. By accepting a new set of parameters with a certain probability even if the χ2 increases, the simulated annealing is a method that is in principle able to find a global minimum even from parameter initializations far away from the best fit.22 The standard errors of the parameters were determined using the “bootstrap method”, where the experimental results are varied within their standard errors, and the data sets with varied data points fit to the same model. The resulting variations in the fitting parameters give the errors of the parameters.22 3.3. Additional Measurements. Static and dynamic light scattering measurements were performed with the same apparatus as described above, where the compensator was deactivated by turning it perpendicular to the scattering plane (and therefore parallel to the polarizer for V-polarization). The hydrodynamic radius Rh was determined from angular dependent polarized (VV) dynamic light scattering measurements with λ ) 633 nm. Data for the apparent diffusion coefficient were extrapolated to a scattering vector of magnitude q ) 0. The resulting diffusion coefficient was inserted in the Stokes-Einstein equation to calculate Rh.23 The angular dependence of the polarized (VV) scattering intensity was fit to the Rayleigh-Debye form factor P(q) of a sphere with radius Rf.17 The electrophoretic mobility was measured on a Malvern Zetamaster 5002 (Malvern Instruments, Malvern, U.K.). The refractive index of bulk water at λ ) 633 nm and λ ) 532 nm has been measured on an Abbe refractometer (Bellingham + Stanley, England, type 60/ED) which was illuminated via an optical fiber by light of the respective laser (see above). (20) Pashley, R. M.; Rzechowicz, M.; Pashley, L. R.; Francis, M. J. J. Phys. Chem. B 2005, 109, 1231-1238. (21) Fo¨rster, S. Ph.D. Thesis. University Mainz, Germany, 1991. (22) Press, W. H.; Teukolsky, W. T.; Vetterling, W. T.; Flannery, B. P. Numerical recipes in C; Cambridge University Press: New York, 1992. (23) Berne, B. J.; Pecora, R. Dynamic light scattering; Dover Publications: Mineola, 2000.
Erbe et al. Table 1. Hydrodynamic Radius Rh, Sphere Radius from the Form Factor Rf, and Electrophoretic Mobility u for the PS Latex Stabilized with Perfluorooctanoate at the Respective Excess NaCl Concentration cNaCla cNaCl/(mol/L) 1‚10-5
5.7‚10-5 1.9‚10-4 8.6‚10-4 4.5‚10-3 a
Rh/nm
Rf/nm
u/(108/(mVs))
63(2) 66(1) 64(1) 62(1) 66(1)
68(1) 67(1) 65(1) 66(1) 68(1)
-3.39(2) -3.34(8) -3.22(4) -3.40(6) -3.20(12)
The number in parentheses indicates the error in the last digit.
4. Results 4.1. Characterization with Static and Dynamic Light Scattering. For the hydrodynamic radius, Rh, as well as the radius derived from the form factor, Rf, no systematic change with the excess NaCl concentration cNaCl could be detected within the experimental error. The results for the particles stabilized with the sodium salt of perfluorooctanic acid are shown in Table 1. The form factor of the monodisperse sphere provides a good fit to the data at all salt concentrations. This shows (i) that the latex spheres exhibit negligible polydispersity and (ii) that there is no aggregation occurring at the higher salt concentration. Both conclusions are supported by the behavior of the apparent diffusion coefficient, which is independent of the scattering vector magnitude q. The electrophoretic mobility u shows no dependence on the salt concentration. For a dielectric sphere, u is linked to the ζ potential via Henry’s equation.1,3 Although u is independent of the salt concentration, the ζ potential itself changes with the changing Debye screening length κ-1. The relative change is less than 1/3.1 The values of u obtained here correspond to ζ potentials between -40 and -27 mV, with the latter value being the exact one in a salt-free dispersion.3 In the simulations in section 5.1, a value of -30 mV will be used. The results remain almost unchanged within the limits of ζ potentials obtained here. 4.2. Results from Ellipsometric Scattering. The results for the particles stabilized by perfluorooctanoate will be discussed in more detail at first. Data obtained with the SDS-stabilized particles will be briefly addressed subsequently. The advantage of perfluorooctanic acid as stabilizing agent is its refractive index, which is close to the refractive index of water. In addition, the stabilizing layer is very thin. Therefore, the effect of the surfactant can be neglected in the analysis of the ellipsometric scattering data. Figure 1 displays the behavior of tan(Ψ) for λ ) 532 nm for all investigated salt concentrations cNaCl. There is a distinct change around the salt concentration c* ) 10-4mol/L. For cNaCl > c*, the minimum of tan(Ψ) and the transition in ∆ are located at a lower scattering angle compared to the minimum position for cNaCl < c*. This difference is a clear indication of a change in the layer surrounding the particles. The ellipsometric data for both λ were fit simultaneously to the Mie theory for a coated sphere. An example for cNaCl ) 1.9‚10-4mol/L is shown in Figure 2. For c > c*, the free parameters in these fits were the core radius Rc, layer thickness ds, and layer refractive index ns, while the core refractive index nc was kept fixed at the bulk value of polystyrene, nc ) 1.59.24 For c < c*, the model of a solid sphere without an interface layer was sufficient to describe the data. Here, no information on the interface structure could be obtained. The only free parameter in these fits was the core radius Rc. The (24) Schrader D. Physical properties of poly(styrene). In Polymer Handbook, 4th ed.; Brandrup, J., Immergut, E. H., Grulke E. A., Eds.; Wiley-Interscience: New York, 1999.
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Figure 3. Results for the core radius Rc (cNaCl > c*; c* ) 10-4mol/L) or radius of a homogeneous sphere (cNaCl < c*) (b), layer thickness ds (9), and layer refractive index difference ∆ns (4). The dotted lines indicate power laws for an empirical description; arrows show the scale for the respective data set.
Figure 1. Data for tan(Ψ) (a) and ∆ (b) at 532 nm of PS latexes stabilized with perfluorooctanoate at all investigated NaCl excess concentrations. Filled symbols are data for cNaCl < 10-4mol/L (9, no added salt; 2, cNaCl ) 5.7‚10-5mol/L), and open symbols for cNaCl > 10-4mol/L (O, cNaCl ) 1.9‚10-4mol/L; 0, cNaCl ) 8.6‚10-4mol/L; 4, cNaCl ) 4.5‚10-3mol/L).
Figure 2. Data of tan(Ψ) (0) and ∆ (9) for a PS latex stabilized with perfluorooctanoate at cNaCl ) 1.9‚10-4mol/L at the wavelengths 532 nm (A) and 633 nm (B). Straight lines are the fits based on the Mie theory for a coated sphere.
refractive index values of the medium n633 medium ) 1.3317 (water ) 1.3345 (water at λ ) 532 nm) were at λ ) 633 nm) and n532 medium used for all salt concentrations. In the investigated range of cNaCl, a calculation based on the refractive index increment of NaCl indicates a variation below 10-4.25 The results for Rc, ds, and ∆ns (25) Lide, D. R., Ed. Handbook of Chemistry and Physics, 80th ed.; CRC Press: Boca Raton, 1999.
Figure 4. Ellipsometric parameter tan(Ψ) at 633 nm for PS latexes stabilized with dodecyl sulfate (9, no added salt; O, cNaCl ) 1.6‚ 10-3mol/L).
) ns - nmedium are shown in Figure 3. As expected for a polymer in the glassy state, the radius of the polystyrene core is independent of the salt concentration. On the other hand, size and refractive index of the layer depend on the salt concentration. The thickness ds of the layer detected at higher cNaCl is about 20 to 30 nm and increases with increasing cNaCl. An empirical description of this increase is a power law ds ∝ cNaCl1/8. The contrast ∆ns on the other hand takes a value around 0.03 and decreases with increasing cNaCl. Roughly, it follows the inverse power law ∆ns ∝ cNaCl-1/8, and the product ds∆ns is constant with the value ds∆ns = 0.9 nm. The sudden appearance of an interface layer is discussed in section 5.3. With the currently achieved quality of the data and measurements at only two λ, we cannot completely rule out another combination of fitting parameters obtained from the applied simulated annealing fitting procedure (described in the Experimental Section). In this combination, the layer refractive index is lower than the refractive index of the surrounding medium. These fits with a negative contrast are however not as good as the ones with positive contrast (as indicated by the χ2 value of the fit) and hardly give a consistent picture. If SDS-stabilized particles are used, there is a similar effect in tan(Ψ) as for the particles stabilized with perfluorooctanoate shown in Figure 1. Examples for two different salt concentrations are shown in Figure 4. As before, the data without added salt can be fit to a model of a sphere without any layer. At higher salt concentrations, the model of the simple sphere is no longer appropriate. In contrast to the measurements with perfluorooctanoate-stabilized particles, the simplified model of a sphere with homogeneous coating is also not sufficient to describe the data. Details of the refractive index profile become important for this sample. Even without a fit, the significant jump of the position where tan(Ψ) has its minimum indicates the appearance of a
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layer with a refractive index different from the solvent. A more structured refractive index profile needs however more parameters, making results unreliable with the currently achieved data quality and measurements at only two λ.
5. Discussion 5.1. Comparison with the Poisson-Boltzmann Theory. Usually, the PB equation is applied to calculate concentration profiles of low molecular weight ions around charged colloidal particles. It balances the electrostatic attraction versus the entropy of the ions.1,2 The concentration ci of ion species i as a function of distance r from the interface is given as
ci(r) ) c∞i exp
[
]
-zieΦ(r) kBT
(3)
with the elementary charge e, the valence zi of the ion species i, Boltzmann’s constant kB, the temperature T, and the bulk ion concentration c∞i . The distance dependence enters via the electric potential Φ(r). Solving the PB equation for a charged infinite plane and a symmetric electrolyte (|zi| ) z) yields the potential
(
)
2kBT 1 + Γ0 e-κr Φ(r) ) ln ze 1 - Γ0 e-κr
(4)
which is known as the Gouy-Chapman potential.1 The parameter Γ0 is determined by the potential Φ(0) directly at the interface
Γ0 ) tanh
(
)
zeΦ(0) 4kBT
(5)
The potential Φ(0) depends on the interface charge density, the dielectric constant of the medium, and the salt concentration in the bulk. Although the Gouy-Chapman theory describes the ion distribution at a charged planar interface, its results serve as an upper estimate for spherical interfaces. With the same bulk ion concentration and the same interface potential, the ion concentration profile calculated by eqs 3-5 is always higher than the ion concentration derived numerically from solutions of the PB equation for spherical particles or calculated from the DebyeHu¨ckel approximation for low potentials. In all these calculations, the value of the interface potential is essential. The current understanding of ion distribution around colloids permits several ways to approach the estimation of the interface potential of the particles. The first is the so-called Gouy-Chapman-Stern model.1 In this model, a static layer of adsorbed counterions know as the Stern layer is assumed to be present at the particle interface. It accounts for the finite ion size, which becomes important for ions close to the interface. The electric potential at the interface of the particle including the Stern layer is approximated by the ζ potential. Although not strictly correct, this approximation is often applied.1 Especially for large Debye lengths, i.e., low cNaCl, it is expected to give reasonable results. From the Stern layer outward, the ion distributions follows the laws of Gouy-Chapman.1 For our calculations, we assume a Stern layer with a uniform ion concentration. Figure 5 shows the comparison between the experimental results of tan(Ψ) and theoretical predictions. The dotted line shows calculations for tan(Ψ) for a sphere with Rc ) 65 nm and nc ) 1.59 in a medium with nmedium ) 1.3317. A layer with a thickness of 0.5 nm and a refractive index of 1.54 is used for the calculation of the dashed line in Figure 5, resembling a Stern
Figure 5. Comparison of measured tan(Ψ) (9, no added salt; O, cNaCl ) 4.5‚10-3mol/L) with theoretical predictions based on the Gouy-Chapman theory. Calculated curves for a sphere without any layer (‚‚‚), a sphere with a Gouy layer as modified by Smagala and Fawcett (-), and a sphere with a Stern layer and a Gouy layer (- -) are shown.
layer with the refractive index of crystalline NaCl.26 The numbers used here are supposed to overestimate the respective quantities in a “real” Stern layer. In addition, eqs 3-5 were applied to calculate the counter- and coion concentration profile around the particle with a Stern layer. With the approximation
dn dn 1 dn + (Cl ) ) (Na+) ) (Na Cl ) dc dc 2 dc one arrives at an interface refractive index of 1.3323 for Φ(0) ) -120 mV and 1.3318 for Φ(0) ) -30 mV.25 The resulting curves (e.g., the dashed line in Figure 5 for Φ(0) ) -30 mV) for tan(Ψ) show only a negligible deviation from the case for the sphere without layer. A more elaborate approach to relate the actual interface potential to the electrophoretic mobility is based on a combination of the O’Brian-White theory and the Gouy-Chapman theory.27 Here, the actual ion distribution is assumed to follow the GouyChapman laws. The ζ potential which governs the electrophoretic mobility is the potential of the decaying electric potential branch at the shear plane, i.e., at a certain distance away from the interface. Since the exact position of the shear plane is unknown, the numbers gained using this theory can serve only as a rough guideline. A potential of about -50 mV at the interface of the particle reproduces the actual measured electrophoretic mobility. Since this potential is comparable to the ones mentioned above, simulations on this basis (not shown) again lead to no measurable effect. Parametrizing the results obtained from Monte Carlo simulations, Smagala and Fawcett have come up with a modified GouyChapman equation which includes the effect of the finite ion size.28 Simulations based on their empirical model with the model parameters θ ) 0.13, R ) 1.5, and a surface potential of -50 mV are shown in Figure 5 as a continuous line. The results lead to lower ion concentrations compared to the unmodified GouyChapman equations, and therefore, the effect on tan(Ψ) is lower than with the unmodified equations. Another mechanism for an enhancement of the ion concentration at an interface is the specific binding of ions. Sodium ions are known to bind strongly to oligoethylene glycol units present in the perfluorooctanoate stabilized particles from the initiator, but also to surfactants as SDS.29,30 Since the binding happens (26) Elridge, J. E.; Palik, E. D. Sodium Chloride. In Handbook of Optical Constants of Solids; Palik, E. D., Ed.; Academic Press: Boston, 1985. (27) Antonietti, M.; Vorwerg, L. Colloid Polym. Sci. 1997, 275, 883-887. (28) Smagala, T. G.; Fawcett, W. R. Z. Phys. Chem. 2006, 220, 427-439. (29) Minatti, E.; Zanette, D. Colloids Surf., A 1996, 113, 237-246. (30) Tauer, K.; Mu¨ller, H. Colloid Polym. Sci. 2003, 281, 52-65.
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within about 1 nm from the interface, it changes the characteristics of the Stern layer. The calculations presented in Figure 5 show that just the presence of a Stern layer with a high refractive index does not change the signal to the extent observed. The same argument applies to the finite size of ions, which has recently been shown to play a role at short separations for ions around DNA.9 Also, ion correlations cannot account for the observed effect. Modern theories on ion distribution at a charged planar interface or around a cylindrical particle start considering ion correlations.31-33 As a result, a concentration profile is obtained which is modified but still comparable to the solution of the PB equation. Only close to the interface is there an enhanced counterion concentration. The comparison of the experimental results with predictions of the Gouy-Chapman theory is summarized as follows. The theory predicts values of ∆ns that are 2 orders of magnitude lower than the values observed for cNaCl > c*. As stated earlier, the Gouy-Chapman theory already overestimates the ion concentration compared to the numerical solution of the PB equation for spherical particles. Another fact worth noting is that the theories predict a decrease in the layer thickness with increasing cNaCl, while in the experiments, ds increased with increasing cNaCl. 5.2. Local Salt Concentration. Despite some shortcomings and simplified assumptions, the PB description in general and the Gouy-Chapman solution as a special case often describe experimental results quite well.1,34 The agreement is often achieved by fitting the unknown interface potential, which, however, is not a rigorous test.35 In this study, the results for cNaCl < c* are compatible with the Gouy-Chapman description, though, as shown by the simulations in Figure 5, deviations discussed in the literature hardly affect the experimental result. The observed change for cNaCl > c* has to be considered as an additional effect beyond the stoichiometric shielding of interface charges on the particle by low molecular weight counterions in solution. It is qualitatively different from the effects considered with the PB description. The magnitude of the additional effect is illustrated by a translation of the measured refractive index of the layer around the particles to an ion concentration. With the refractive index increment dn/dc ) 9.66 mL/mol of NaCl,25 the observed ∆ns ) 0.04 for cNaCl ) 1.9‚10-4 mol/L corresponds to a local NaCl concentration of 4 mol/L. The total number N of ion pairs in a spherical shell of the measured thickness ds ) 20 nm around a particle of Rc ) 63 nm is N ) 3.7‚106, or, in relation to the particle interface, 74 ion pairs per nm2. The term “ion pair” is used as an abbreviation, to count two oppositely charged ions as one neutral unit. More explicitly, 74 ion pairs per nm2 indicates 74 Na+ ions and 74 Cl- ions per nm2. The constant value of ds∆ns as mentioned in section 4.2 indicates that N hardly changes with cNaCl. The number of attracted ion pairs depends only weakly on the average salt concentration. An important parameter is the ratio of N to the number of charges on the particle interface. An upper limit of the charge density σ on the particle interface is derived from the formula of the particle synthesis described in section 3.1. Dividing the total number of charged surfactant molecules by the total interface area of all created particles yields σ ) 0.3 e/nm, where e is the elementary charge. This number is comparable to the charge density e/λB2 ) 0.5 e/nm. Here, λB is the Bjerrum length, the (31) Netz, R. R.; Orland, H. Eur. Phys. J. E 2000, 1, 203-214. (32) Naji, A.; Netz, R. R. Phys. ReV. Lett. 2005, 95, 185703. (33) Barbosa, M. C.; Deserno, M.; Holm, C. Europhys. Lett. 2000, 52, 80-86. (34) Bu, W.; Vaknin, D.; Travesset, A. Langmuir 2006, 22, 5673-5681. (35) Luo, G.; Malkova, S.; Yoon, J.; Schultz, D. G.; Lin, B.; Meron, M.; Benjamin, I. Vanysek, P.; Schlossman, M. L. Science 2006, 311, 216-218.
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distance of two elementary charges, where the electrostatic energy is equal to kBT. The Bjerrum length determines the maximum effective linear charge density of a linear polyelectrolyte, since a higher chemical linear charge density would be reduced by counterion condensation. For cNaCl > c*, these estimations yield a number of ion pairs around a particle which is more than 2 orders of magnitude higher than the number of interface charges on the particle. Another indicative number is the fraction of ion pairs in the solution which is trapped in the layers around the particles. This number decreases from 88% at cNaCl ) 1.9‚10-4mol/L to 4% at cNaCl ) 4.5‚10-3mol/L. As mentioned above, the total number of ions in the layer remains constant; however, their fraction diminishes on an increase of cNaCl. In other words, NaCl added after the formation of the layers is not incorporated in these layers but remains in solution. If the colloidal stability is maintained until high cNaCl, the PB description might finally get valid again, at the point where the decreasing concentration in the layer meets cNaCl. In conclusion, the result of this quantitative analysis of the refractive index values close to the particle interface is a rather high salt concentration of 4 mol/L for cNaCl > c*. On increasing cNaCl, the layer thickness increases and the local concentration decreases, leaving the total amount of ions in the layer constant. 5.3. Discussion as a Prewetting Transition. As a surprising result, these “back-of-the-envelope” calculations indicate a salt concentration around the particles comparable to the solubility. At c*, where the layer emerges, a significant fraction of the available ion pairs is bound to the particle interfaces. Such a behavior is uncommon within the framework of the PB equation. Instead, it resembles a prewetting transition at an interface (see, e.g., ref 36 for a recent review on wetting). A wetting transition can happen if a liquid and its vapor are in contact with a solid wall. Depending on the interface tensions among liquid, vapor, and solid, the contact angle of a liquid droplet might be finite (partial wetting) or zero (complete wetting). Although its denomination stems from the liquid/vapor coexistence at a wall, a wetting transition can also occur for the case of a binary mixture of two fluid constituents A and B in contact with a solid wall or the vapor. If A and B form a two-phase system in the bulk liquid, either the A-rich or the B-rich phase is attracted to the interface. Again, partial or complete wetting might occur, depending on the interface tensions and the respective contact angle. Even for a completely miscible A-B system, the interactions of A and B with an interface are generally not of similar magnitude. Therefore, either A or B is enriched at the interface. Without loss of generality, let us take A as the interface enriched component. The prewetting transition is a change of the A-rich layer from a thin to a thick interface film. Apart from the interface critical point, this transition is first-order, and the thickness of the A-film changes discontinuously. Since A and B are miscible in the bulk, the evolution of a macroscopic A-rich phase is suppressed, and the film thickness remains finite. The salt content in the observed layer increases drastically within a small step of cNaCl but stays finite for higher cNaCl. Therefore, the observed change can be classified as a first-order prewetting transition. The sharp transition in the experiment is in accordance with theoretical considerations, which indicate a negligible smearing out by finite size effects for colloidal particles of 100nm radius.37 The thickness of the salt-rich layer is comparable to the Debye length κ-1 ≈ 30 nm at cNaCl ) 10-4 mol/L, which determines (36) Bonn, D.; Ross, F. Rep. Prog. Phys. 2001, 64, 1085-1163. (37) Gelfand, M. P.; Lipowsky, R. Phys. ReV. B 1987, 36, 8725-8735.
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the thickness of the concentration profile below c*. At first glance, this comparison challenges the description of prewetting transitions, which are usually characterized by an increase in the layer thickness and not in the local concentration. However, ds and κ-1 have different physical basics. The relaxation of concentration fluctuations in salt solutions involves two time scales.23 For long wavelengths which contribute to light scattering, the relaxation consists of a fast local relaxation of the charge distribution and a slow diffusive decay. In a simplified view, these two mechanisms can be addressed as fluctuations involving a charge separation and unpolarized fluctuations. The coincidence of κ-1 as the screening length of charge effects with the thickness of an interface layer basically formed by an unpolarized concentration fluctuation exactly at c* is probably not accidental, but might be an essential ingredient of the mechanism building up the layer. Note that within the layer κ-1 is much smaller because of the high salt concentration. It is therefore not the local value of κ-1 which determines the extend of the layer. For a prewetting transition in general, it is the correlation length ξ of bulk concentration fluctuations which essentially determines the thickness of the prewetting layer.36 For solutions of NaCl in water, long-range concentration fluctuations in the range ξ ≈ 30-100 nm have been observed by static and dynamic light scattering.38,39 This size corresponds to the layer thickness detected in our measurements. The long range compared to molecular dimensions is surprising in both cases; however, it is consistent. The same physics seems to determine the long-range bulk fluctuations and the extent of the interface layer. The involved interactions will be discussed in the next section 5.4. They are partly long-range, and this property might be responsible for the large extent of bulk or interface fluctuations. Georgalis et al. as well as Sedlak discuss their results in terms of an aggregation phenomenon instead of a description as concentration fluctuations.38,39 However, their sample preparation shows that the respective scattering contributions cannot be removed, neither by centrifugation nor by filtering. In terms of aggregation phenomena, this property suggests a dynamic, reversible aggregation. Such a picture is equivalent to the discussion in terms of long-range concentration fluctuations. Here, the latter terminology is advantageous, since it connects the average size detected in bulk measurements with a correlation length, which in turn also determines the interface behavior. Sedlak mentions that his data cannot be analyzed with the Ornstein-Zernike function (see, e.g., ref 40). This functional form is expected for fluctuations in a system with pairwise additive interactions. The reported deviations can be interpreted as a hint to a significant contribution of nonpairwise additive interactions. The most prominent example for interactions with this property are van der Waals interactions (see, e.g., ref 41). An independent hint to a significant contribution of nonpairwise interactions from our measurements can be deduced from the high local salt concentration c in the interface layer. While the potential energy stored in pairwise interactions is proportional to c2, nonpairwise additive interactions have a stronger dependency on c. The observed high value of c favors nonpairwise additive interactions, which might be the origin for the high osmotic compression of the ions. As a summary of the discussion as a wetting behavior, the (38) Georgalis, Y.; Kierzek, A. M.; Saenger, W. J. Phys. Chem. B 2000, 104, 3405-3406. (39) Sedlak, M. J. Phys. Chem. B 2006, 110, 4329-4338. (40) Huang, K. Statistical Mechanics; Wiley: Singapore, 1987. (41) Israelachvili, J. N. Intermolecular and surface forces; Academic Press: London, 1992.
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picture of a first-order prewetting transition directly explains the sudden appearance of a thick interface layer at the colloidal interface. The picture also accounts for the layer thickness, which is comparable to the extent of bulk concentration fluctuations.38,39 This thermodynamic argument does not provide a physical picture of the involved interactions. They are discussed in the next section. 5.4. Interactions. The interactions leading to a stabilization of the layer are more complex than the simple Coulomb attraction due to the interface charge. The observed high concentration around the particles hints at a collective effect beyond a superposition of pairwise interactions between the colloid and a single ion. The interpretation as a prewetting transition implies a foundation within thermodynamic arguments, where the layer of high salt concentration is considered an interface-stabilized phase separation. Here, attractive interactions between ions become important. Attractive interactions beyond electrostatics are manifested also in other phenomena described in the literature. The existence of long-range concentration fluctuations mentioned above is one example.38,39 Similar effects have been seen in our lab. Previous to the experimental observation, a clustering of ions has been found in Monte Carlo simulations.42,43 Due to the limitations of the system size, only higher salt concentrations in a much smaller volume than investigated here were accessible to the simulations. Consequently, only small clusters of ions were detected. The driving force for clustering within the simulations relies exclusively on changes of the local water structure. The hydration structure of water around clusters with several ions involve fewer water molecules than the hydration of the same number of separated ions. Clustering therefore releases water molecules, and the mutual attraction of ions is of entropic origin. The relevance of entropic interactions of ions with polymers is discussed by Sinn et al., on the basis of measurements by isothermal titration calorimetry and an ion-selective electrode.44 The water-mediated interaction between ions contrasts with interface-induced structuring of water, usually discussed in connection with biological systems (see, e.g., ref 50). Water structuring can be induced by interface charges and results in a water layer with lower density and refractive index than bulk water. Na+ ions are expelled from this low-density layer. Since our experiments show an increase of the refractive index around the colloids compared to the bulk, a structuring of water as in biological systems can be excluded for cNaCl > c*. The water structure in an ion cluster with high content of Na+ ions is different from the interface-induced structure with low Na+ content. A different approach also resulting in attractive interactions between salt ions stems from theoretical considerations. Besides its electrostatic interactions, an ion also experiences van der Waals interactions, since in addition to its charge, the ion possesses also a polarizability.45,46 The possible effect of van der Waals attractions as nonpairwise interactions has been already outlined in the last section 5.3. In total, besides the Coulomb interactions, a short-range entropic interaction due to effects of the water structure and a long-range van der Waals interaction add up to the interaction (42) Degreve, L.; da Silva, F. L. B. J. Chem. Phys. 1999, 110, 3070-3078. (43) Degreve, L.; da Silva, F. L. B. J. Chem. Phys. 1999, 111, 5150-5156. (44) Sinn, C. G.; Dimova, R.; Antonietti, M. Macromolecules 2004, 37, 34443450. (45) Ninham, B. W.; Yaminsky, V. Langmuir 1997, 13, 2097-2107. (46) Bostro¨m, M.; Ninham, B. W. Langmuir 2004, 20, 7569-7574. (47) Buchner, R.; Hefter, G. T.; May, P. M. J. Phys. Chem. A 1999, 103, 1-9. (48) Mahanty, J.; Ninham, B. W. Dispersion forces; Academic Press: London, 1976. (49) Netz, R. R. Eur. Phys. J. E 2001, 5, 189-205. (50) Pollack, G. H. Cells, Gels and the Engines of Life; Ebner and Sons: Seattle, 2001.
Ion Distribution around Electrostatically Stabilized Particles
potential of ions. The approximation of this potential by the Coulomb interaction alone within the PB equation is not sufficient at high cNaCl. The van der Waals interaction has a further stabilizing effect on the layer, since it builds up a disjoining pressure on the salt-rich layer around the particles.41 The geometry-independent magnitude of the van der Waals forces is often expressed in terms of the Hamaker constant H. It consists of a low-frequency part determined by the dielectric constants of the involved materials and a high-frequency part, which depends on the refractive index values n. On the basis of bulk values of for salt solutions47 and the refractive index increment used above, both parts of H can be shown to favor a thick layer of salt solution around the particles. A quantitative calculation is difficult, since screening and retardation effects have to be taken into account.41 Further complications arise from the presence of surface charges and the presence of electrolytes.48,49 Still, the qualitative discussion illuminates which further contributions affect the observed prewetting phenomena in addition to the PB description and the underlying Coulomb interactions. It is not the mean field character of the PB equation that fails for a qualitative description of our results at high cNaCl, but an incomplete description of the relevant interactions resulting in an invalid mean field. By incorporation of van der Waals interactions between a single ion and the colloidal particle, Luo et al. succeeded in describing their measurements of the ion distribution on a planar interface without any adjustable parameter.35 The modeling of the observed prewetting transition requires one step further, where the attractive interactions between the ions are also taken into account. As a summary of the discussion of the involved interactions, attractive interactions between salt ions play an essential role for the prewetting transition, since they promote concentration fluctuations. 5.5. Layer Properties. The interpretation of the enhanced salt concentration around the colloids at high cNaCl as a soft structure is supported by the results for Rh reported in section 4.1. Rh is nonsensitive to the evolution of the thick interface layer and displays a constant value over the whole investigated range of cNaCl. In other words, the enhanced salt concentration around the colloids does not contribute to the hydrodynamic friction of the particles. Clearly, it is not a solid layer which is tightly fixed to the particle. In a microscopic picture of Brownian motion, a colloid suffers thermal kicks from the solvent molecules and performs a random walk. The steps of the random walk are much smaller than the dimension of the salt cloud which encloses the colloid, and therefore, the colloid remains within the cloud. In addition, the instantaneous shape of the cloud is not necessarily spherically symmetric, although the applied model of a spherically symmetric shell of higher refractive index describes well the temporal average of the fluctuations, at least for the latex particles stabilized with perfluorooctanoate. A more detailed discussion of DLS measurements is based on a dynamical picture. Shape fluctuations of the ion cloud should have a characteristic relaxation time τ, which has to be compared with the inverse of the relaxation rate Γ(q) ) Dq2 of the colloid diffusion, where D is the diffusion coefficient. For τ , Γ(q)-1, no effect on the colloidal diffusion (51) Beysens, D.; Narayanan, T. J. Stat. Phys. 1998, 95, 997-1008. (52) Rathke, B.; Grull, H.; Woermann, D. J. Colloid Interface Sci. 1997, 192, 334-337.
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is expected, while for q where τ ≈ Γ(q)-1, a crossover to q-independent behavior of the DLS correlation function should occur. The observed unperturbed diffusion within the experimental window yields the limit τ , 0.4 ms. The failure of the model to describe the results for the dodecyl sulfate stabilized particles might arise from nonspherical configurations. The microscopic picture for the measurements of the electrophoretic mobility u is more involved. In an electrophoresis experiment, the electric field pulls ions of opposite charges into opposite directions, and the resulting hydrodynamic flow field around the colloid gets quite complicated. The observed constant value of u indicates a negligible change in the net drag force within the experimental errors. The absence of a measurable effect could be explained by the partial destruction of the ion cloud around the colloids under the influence of an electric field. This destruction is a transient phenomenon occurring when the oscillating field of the experiment is switched on. Therefore, it does not contribute to the stationary state reflected in the measurements of u. In general, ELS in the presence of an electric field should allow for an investigation of the stability of the ion clouds around the colloids. As a conclusion of the lack of any effect in the hydrodynamic radius and in the electrophoretic mobility, the observed interface layer is a very soft structure. It can be pictured as a fluctuation of the salt concentration, trapped at the colloidal interface.
6. Conclusions and Outlook Ellipsometric light scattering measurements show an enhanced local salt concentration up to 4 mol L-1 around charge-stabilized colloids of radius 60 nm for an average salt concentration above c* ) 10-4 mol L-1. This layer extends over a thickness of 2030 nm. The effect cannot be explained within the PoissonBoltzmann theory, which fits the results for average salt concentrations below c*. Instead, the discontinuous popping up can be described as a first-order prewetting transition. The observation is discussed as a thermodynamic effect. Van der Waals forces and entropic interactions based on the local water structure are discussed as additional constituents of a modified mean field, which is expected to stabilize the prewetting layer. Further work is needed for a closer inspection of prewetting phenomena at colloidal interfaces. The location of the prewetting transition for different ions and its temperature dependence are of interest. In bulk concentration fluctuations, the formation of large clouds is not restricted to ions but occurs also for neutral water-soluble low molecular weight compounds.39 It is unclear what kind of wetting behavior of such compounds occurs on a colloidal interface. The wetting of the interface of colloids in a solvent mixture with a miscibility gap has been studied to some extent. An impact of the preferential adsorption of one component on the colloidal stability has been found.51,52 We may now have the possibility to extend this type of work to regions away from critical points and binodal curves. These questions will be addressed in future work. Acknowledgment. We acknowledge helpful discussions with Markus Antonietti. For technical assistance with the preparation of samples, we thank Birgit Schonert. Finally, the financial support of the Max Planck Society is gratefully acknowledged. LA062033J
