Methods for Determining the Intrinsic and Effective Charges on

An Experimental Test of the Ion Condensation Theory for Spherical Colloidal Particles. M. Quesada-Pérez , J. Callejas-Fernández , R. Hidalgo-Álvare...
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Anal. Chem. 1998, 70, 3667-3673

Methods for Determining the Intrinsic and Effective Charges on Spherical Macroions James M. Roberts, John J. O’Dea, and Janet G. Osteryoung*

Department of Chemistry, North Carolina State University, Raleigh, North Carolina 27695-8204

The intrinsic number of charges per particle, Z, on particles in a suspension of monodisperse sulfonated polystyrene latex is found by measuring steady-state voltammetric transport-limited currents for the reduction of hydrogen counterion at a Pt disk microelectrode in a suspension containing excess supporting electrolyte. Limiting currents measured in deionized latex suspensions yield a corresponding effective charge, Z*. Electrostatic binding of an inner layer of counterions to the particle renders Z* < Z. Voltammetrically determined charges agree with the intrinsic and effective Stokes charges as determined by titration and electrophoresis, respectively. ζ- Potentials calculated from the measured electrophoretic mobility of the particles yield the Loeb charge number, which agrees more closely with Z than with Z*. For existing data on spherically charged macroions, Z*/Z decreases with increasing ratio of intrinsic charge to macroion radius, Z/a. This finding is supported by effective charge values calculated from the cell model using the nonlinear Poisson-Boltzmann equation with the convention that Z* is the charge found between the c(r)/ 〈c〉 ) 1 boundary and the cell wall. Colloidal dispersions containing electrically charged particles are widely found in commercial products (e.g., paint, glue, suspension formulations of drugs) and in natural systems (e.g., blood, natural waters, milk, liposomes).1-5 In most applications, it is desirable to keep the particles in suspension. Latex suspensions similar to the one used in this study, for example, are useful as paints only if the polymeric particles do not aggregate. The thermodynamically preferred state of all charge-stabilized colloidal dispersions is a phase-separated aggregate. Paint coagulates and milk curdles because of insufficient charge stabilization. Aggregation is due to attractive dispersion interactions between the particles, which always overcome electrostatic repulsion between the particles at sufficiently short distances (typically 10 nm or less). * Corresponding author. E-mail: [email protected]. (1) Prost, J.; Rondelez, F. Nature 1991, 350 (Suppl.), 11. (2) Ross, S.; Morrison, I. D. Colloidal Systems and Interfaces; John Wiley & Sons: New York, 1988. (3) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. (4) Evans, D. F.; Wennersto¨m, H. The Colloidal Domain; VCH Publishers: New York, 1994. (5) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: San Diego, 1992. S0003-2700(97)01364-4 CCC: $15.00 Published on Web 08/05/1998

© 1998 American Chemical Society

Higher surface charge density on the particles, attained by either incorporation of ionogenic groups or adsorption of ions (e.g., ionic surfactants), delays aggregation by increasing the spacing between particles. Electrolyte screens the electrostatic repulsion between particles; thus, even highly charged particles will aggregate rapidly if the concentration of salt is sufficiently high. In short, the macroscopic properties of the dispersion (e.g., rheological, optical) and its usefulness for a given application are determined, to a large extent, by the electrostatic interactions between the microscopic particles.2 Control of dispersions depends on a theoretical understanding of electrostatic interactions in colloidal dispersions as well as on experimental techniques that characterize these interactions. The electrostatic stabilization of colloidal dispersions against aggregation is usually described in terms of the DLVO theory.1-6 The DLVO theory is said to characterize completely the colloidal dispersion with respect to its stability given the particle size and size distribution, the amount and valency of added electrolyte, the magnitude of the dispersion interaction, and either the ζ-potential or the number of charges on the surface of the particle. The intrinsic charge is a directly measurable thermodynamic quantity, as discussed below. However, the DLVO theory requires an effective charge, which accounts for screening of the intrinsic charge by counterions. The effective charge is not a thermodynamic quantity and is, therefore, not measurable directly. It must be inferred from measurement by invoking a model. In short, in terms of electrostatic stabilization, neither the potential nor charge formulations of the DLVO theory are cast in terms of directly measurable quantities. The main objective of this work was to develop a method for determining effective charge and, therefore, to predict, by means of the DLVO theory, electrostatic interactions between spherical macroions. Determination of Intrinsic and Effective Charges. We define the intrinsic charge number, Z, as the total number of elementary charges on the surface of the particle. This is a thermodynamic quantity accessible by well-defined experiment. Typical intrinsic charge numbers range from 50 to 100 000. In dispersions of charged lyophobic colloids, the intrinsic charge number on a particle with volume VP is found by quantifying the concentrations, ci, of all of its counterions (assuming here singly charged ions): (6) Schmitz, K. S. Langmuir 1997, 13, 5849.

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Z ) [VPNA(1 - φ)/φ]

∑c

i

(1)

i

where NA is Avogadro’s number and φ is the volume fraction of particles (eqs 10.6.4-10.6.6 in ref 3). Here, ci is expressed in terms of the accessible volume, VT(1 - φ), where VT is the total volume of the suspension. Equation 1 is simply a statement that the suspension is electrically neutral. A specific example of the use of eq 1 illustrates its utility. The latex particles used in this study are monodisperse spheres, each having a radius of 44 nm (i.e., VP ) 3.57 × 10-22 m3). The total number of particles in a given macroscopic volume is known directly from the volume fraction, φ (measured gravimetrically), and the volume, VP, of each particle. Determination of the total number of elementary charges on each particle, Z, then amounts to quantifying the total number of counterions in the system. The intrinsic charge on these latex particles arises from the dissociation of, primarily, sulfonic acid and, to a far lesser extent, sulfonate salts. As the sulfonate moiety is a very strong acid (and electrolyte), all of the hydrogen ions (and alkali metal ions) are completely dissociated and exist as solvated species in the aqueous phase. Thus, measuring the concentration of hydrogen and alkali metal ions provides the value of Z. Each of these quantities, VP, φ, and ci, is a directly measurable thermodynamic quantity. The counterions exist in a diffuse double layer around each particle and accumulate near its surface. As a result, micelles and latex particles exhibit effective charge numbers, Z*, that are less than Z, due to partial neutralization of Z by these accumulated, solvated counterions.7-14 This association of oppositely charged ions in the solution or suspension is the mechanism by which the system minimizes its electrical energy, and it is common to all globally electrically neutral systems.5 The Gouy-ChapmanStern theory of the electrical double layer at the electrodesolution interface,15,16 Bjerrum’s theory of ion pairing in salt solutions,17 Manning’s theory of ion condensation in solutions containing linear polyelectrolytes,18 and Alexander’s theory of charge renormalization in spherical (latex) macroionic suspensions8 are examples of attempts to quantify this tendency. The effective charge is not a thermodynamic quantity: a model is required to infer the effective charge from experiment. The basic idea underlying the inference is that the counterions exist in two states: those associated with the macroion and those that do not interact significantly with the macroion. The bound, or (7) Moroi, Y. Micelles: Theoretical and Applied Aspects; Plenum Press: New York, 1992. (8) Alexander, S.; Chaikin, P. M.; Grant, P.; Morales, G. J.; Pincus, P.; Hone, D. J. Chem. Phys. 1984, 80, 5776. (9) Ito, K.; Ise, N.; Okubo, T. J. Chem. Phys. 1985, 82, 5732. (10) Roberts, J. M.; Osteryoung, J. G. Langmuir 1998, 14, 204. (11) Stilbs, P. Prog. Nucl. Magn. Reson. Spectrosc. 1987, 19, 1. (12) Bell, G. M.; Dunning, A. J. Trans. Faraday Soc. 1970, 66, 500. (13) Larsen, A. E.; Grier, D. G. Nature 1997, 385, 230. (14) Osteryoung, J. G.; Ciszkowska, M. Transport of Counterions in Solutions of Biological Polyelectrolytes. Presented at the 214th ACS National Meeting, Las Vegas, NV, September 7-11, 1997. (15) Bard, A. J.; Faulkner, L. R. Electrochemical Methods; John Wiley & Sons: New York, 1980. (16) Schmickler, W. Interfacial Electrochemistry; Oxford University Press: New York, 1996. (17) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry; Plenum Press: New York, 1970; Vol. 1. (18) Manning, G. S. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 909.

3668 Analytical Chemistry, Vol. 70, No. 17, September 1, 1998

Figure 1. Contours of equal counterion concentration around the latex particles used in this work, as predicted by the nonlinear Poisson-Boltzmann equation. Parameters: a ) 44 nm, φ ) 0.237%, Z ) 10 750, T ) 25 °C, r ) 78.3. The mean concentration of counterions, 〈c〉, is 0.130 mM (from eq 1). Contours are labeled with normalized counterion concentrations, c(r)/〈c〉. The outer bold circle is the cell wall, and the dashed circle designates the radius at which the local concentration is equal to the mean value.

strongly interacting, counterions neutralize surface sites on the particle, giving rise to a value for the effective charge number, Z*, which is lower than that for Z. The validity of this two-state approximation is supported by observations that a large fraction of counterions exist in a layer at the surface of highly charged latex particles.19 The analogous phenomenon occurs in solutions containing linear polyelectrolytes, as supported by NMR relaxation measurements.20 We can predict a value for the effective charge, Z*, by employing the following argument. Figure 1 shows selected contours of equal concentration (equipotential surfaces) calculated from the nonlinear Poisson-Boltzmann equation for spherical symmetry17 using a cell model12 for the latex used in this study.10 Details of the calculation will be discussed in a forthcoming paper. The latex particle shown at the center of the figure (radius 44 nm) has Z ) 10 750 negative charges on its surface and is neutralized by an equal number of singly charged counterions (there is no added electrolyte). The bold outer contour, 286 nm from the surface of the particle, is the radius of the cell based on the convention that the cell volume, Vc, is the volume of the suspension per particle, given by VP/φ. The numerical value assigned to each contour is the local concentration of counterions, c(r), normalized with respect to the measured mean concentration, 〈c〉 ) ∑ci (eq 1). The dashed contour, 32 nm from the surface of the particle, is the distance at which the local concentration of counterions equals the mean value (i.e., c(r)/〈c〉 ) 1, or, equivalently, the distance at which the electrostatic potential is zero). By integration, approximately 94.2% of the counterions reside between the surface of the particle and this dashed contour. Also note that the concentration of counterions increases sharply from the mean value at distances closer to the surface of the particle. (19) Midmore, B. R.; Hunter, R. J. J. Colloid Interface Sci. 1988, 122, 521. (20) Delville, A.; Laszlo, P. Biophys. Chem. 1983, 17, 119.

By contrast, for r such that c(r)/〈c〉 < 1, c(r) is relatively constant. In short, most of the counterions reside close to the particle and, thereby, lower its intrinsic charge to an effective value. Using a two-state model, and assuming that the counterions between the surface of the particle and the contour c(r)/〈c〉 ) 1 neutralize an equivalent number of anionic sites on the particle, the value for the effective charge number for this latex is predicted to be Z* ) 0.058Z ) 620. Differences in Z* predicted thusly from the Poisson-Boltzmann equation and from that determined from experiment might be due to the arbitrary definition of “bound” charge or to features not accounted for in the Poisson-Boltzmann formulation. Several experimental methods have been used to infer a value of the effective charge: conductivity or conductance,9,21-23 video microscopy,13,24 shear modulus,25,26 electrophoresis,27 torsional resonance,27 light scattering,28,29 and neutron scattering.30 In some experiments, the effective charge is treated as an adjustable parameter which is fit to a theoretical model.13,24,25,30 In this work, we use voltammetry at a microelectrode to measure the intrinsic charge and to infer an effective charge on polystyrene sulfonate latex particles suspended in water. The intrinsic charge number is found in the following way. The latex is ion-exchanged to the acid form so that the concentration term in eq 1 is predominantly that of hydrogen ion. The potential at the Pt microelectrode is scanned through the region where hydrogen ion is reduced according to the reaction

2H3O+ + 2e- f 2H2O + H2

(2)

At sufficiently negative potentials, the current due to reaction 2 is limited by the rate at which hydrogen ion diffuses to the surface of the electrode.15 For a disk microelectrode of radius ro, this steady-state diffusion-limited current is given by31

il ) 4nFDcro

(3)

where n is the stoichiometric number of electrons transferred (unity for hydrogen ion reduction), F is Faraday’s constant (96 485 C/mol e-), c is the mean concentration of hydrogen ion, and D is the diffusion coefficient of hydrogen ion. Note that the mean concentration, c, in eq 3 corresponds to ci in eq 1, where i corresponds to hydrogen ion. Equation 3 is valid if there is (21) Matsuoka, H.; Harada, T.; Yamaoka, H. Langmuir 1994, 10, 4423. (22) Palberg, T.; Mo ¨nch, W.; Bitzer, F.; Piazza, R.; Bellini, T. Phys. Rev. Lett. 1995, 74, 4555. (23) Sumaru, K.; Yamaoka, H.; Ito, K. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 1176. (24) Grier, D. G.; Murray, C. A. In Ordering and Phase Transitions in Charged Colloids; Arora, A. K., Tata, B. V. R., Eds.; VCH Publishers: New York, 1996; p 69. (25) Lindsay, H. M.; Chaikin, P. M. J. Chem. Phys. 1982, 76, 3774. (26) Palberg, T.; Kottal, J.; Bitzer, F.; Simon, R.; Wu ¨ rth, M.; Leiderer, P. J. Colloid Interface Sci. 1995, 169, 85. (27) Palberg, T.; Ha¨rtl, W.; Wittig, U.; Versmold, H.; Wu ¨ rth, M.; Simnacher, E. J. Phys. Chem. 1992, 96, 8180. (28) Versmold, H.; Wittig, U.; Ha¨rtl, W. J. Phys. Chem. 1991, 95, 9937. (29) Corti, M.; Degiorgio, V. J. Phys. Chem. 1981, 85, 711. (30) Chen, S.-H.; Sheu, E. Y. In Micellar Solutions and Microemulsions; Chen, S.-H., Rajagopalan, R., Eds.; Springer-Verlag: New York, 1990; p 21. (31) Montenegro, M. I., Quieros, J. L., Daschbach, J. L., Eds. Microelectrodes: Theory and Applications; Kluwer Academic Publishers: Dordrecht, 1991.

sufficient excess of supporting electrolyte to suppress migrational transport. For 100-fold excess, the migrational contribution should be less than 0.3%.15,32 In addition, cations of the electrolyte exchange with hydrogen counterions near the particle in proportion to their mole fraction. Thus, the concentration of hydrogen ion is uniform. The concentration of hydrogen ion follows directly from eq 3, the measured limiting current, and the known value of diffusion coefficient. The intrinsic charge number follows from eq 1, the concentration of hydrogen ion determined voltammetrically, and the concentration of residual alkali metal ions determined from atomic emission spectroscopy. The effective charge was determined in suspensions without added electrolyte. Without added electrolyte, strong electrostatic interaction between hydrogen ions and the particles lowers the apparent diffusion coefficient of the former, which decreases the limiting current (see Figure 1). The effective charge on the particle is inferred from these measured limiting currents by assuming a two-state model in which strongly interacting ions (electrostatic potential, ψ < 0) have the diffusion coefficient of the particle and weakly interacting ions (ψ g 0) have the known diffusion coefficient of hydrogen ion. The diffusion coefficient of the particles in this case is negligibly small. As a corollary, we compare these voltammetric results with the intrinsic charge number determined by conductometric titration and the effective charge number determined by measurement of the electrophoretic mobility of the particle. Finally, the voltammetric results are compared with existing data on ion-binding in micellar solutions and in other latex suspensions.

EXPERIMENTAL SECTION Latex. The synthesis and characterization of the latex has been described in detail.10 Briefly, the particles are monodisperse, with a radius of 44 ( 2 nm determined by microscopy. A deionized suspension of particles is obtained by treating the latex with ion-exchange resins. The deionized suspension has hydrogen ion as counterion and is brilliantly iridescent, indicating a colloidal crystalline state.33,34 Conductometric titration of the hydrogen ion in the latex suspension with standard base yielded an equivalence of 11 700 negative charges per particle. An additional 450 negative charges are compensated by residual alkali metal counterions, as determined by atomic emission spectroscopy. Electrochemical Experiments. Platinum disk microelectrodes with radii of 5.4 and 10.5 µm were used as the working electrodes. The electrode radii were measured optically using an inverted metallurgical microscope (Leitz, Diavert). A twoelectrode cell was used, with a platinum wire serving as both quasireference and counter electrodes. The platinum quasi-reference electrode was employed to avoid contamination of the deionized latex with adventitious electrolyte from a liquid junction.35 This procedure is permissible because the potential of the working electrode need not be well specified in this experiment. The electrochemical cell was enclosed in a Faraday cage to prevent (32) Myland, K. J.; Oldham, K. B. J. Electroanal. Chem. 1993, 347, 49. (33) Hiltner, P. A.; Krieger, I. M. J. Phys. Chem. 1969, 73, 2386. (34) Arora, A. K.; Tata, B. V. R. Ordering and Phase Transitions in Charged Colloids; VCH Publishers: New York, 1996. (35) Ciszkowska, M.; Osteryoung, J. G. J. Phys. Chem. 1994, 98, 3194.

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immediately before the measurement. For these measurements, particle volume fractions were much less than 0.01%. The ζ-potential values were obtained from the measured electrophoretic mobilities using Figures 3 and 4 from ref 37.

Figure 2. Transport-limited currents for the reduction of hydrogen ion at a Pt disk microelectrode (ro ) 5.4 µm) in latex suspension at initial volume fractions, φ/%, and [LiClO4]/mM of (3) 0.309, 17.4; (/) 0.237, 12.5; and (O) 0.119, 6.4. Successive dilutions are made with ultrapure water. The abscissa is the reciprocal accessible volume of the suspension.

distortion of the small currents by external electric fields. Diamond suspension (0.25 µm, Buehler) was used to polish the electrodes on a soft, wet pad (Microcloth, Buehler) between scans. After polishing, the electrodes were rinsed vigorously with water and inspected under a microscope to ensure that the surface of the electrode was smooth and free of debris. Solutions of LiClO4 were made with 99.99% pure LiClO4 (Aldrich). All water used in these experiments had a resistivity of 18 MΩ‚cm (Milli-Q). Deionized latex was added to 5 mL of an aqueous solution of LiClO4 in a polycarbonate container. The 100-fold concentration excess of LiClO4 (relative to hydrogen counterion) required to eliminate migrational transport was calculated from the concentration of hydrogen ion determined from the known volume fraction (measured gravimetrically using a polystyrene density of 1.05 g/cm3) and the titrated amount of hydrogen ion in the suspension. Voltammetry was carried out by means of a PAR 273 potentiostat controlled with in-house software by means of a PC computer. Potential was programmed as a staircase waveform with a 5-mV step height and a frequency of 1 Hz. This frequency ensured that the limiting current was within 3% of the predicted steady-state value.36 Currents were measured using a Keithley 427 current amplifier connected to the potentiostat. One voltammetric scan required about 3 min. Limiting currents were measured with respect to the baseline current. The latex suspension was sparged with water-saturated argon between voltammetric scans and blanketed with argon during each scan to remove dissolved oxygen (an electrochemical interference) from the suspension. Complete removal of the oxygen interference was achieved within 20 min of sparging. Determination of the intrinsic charge (Figure 2) was carried out by successive dilutions of the suspension with pure water (Milli-Q). Experiments were carried out at 25 °C. Electrophoretic Mobility and ζ-Potential. Measurements of particle electrophoretic mobilities were carried out using a Malvern ZetaMaster IV. Observation was done at the stationary level. The strength of the applied electric field was 10 V/cm. Deionized latex was added to prepared electrolyte solutions (36) Aoki, K.; Akimoto, K.; Tokuda, K.; Matsuda, H.; Osteryoung, J. G. J. Electroanal. Chem. 1984, 171, 219.

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RESULTS AND DISCUSSION Measurement of the Intrinsic Charge. In acidic suspensions, potentiometric determination of the pH is complicated by the suspension effect.38-40 Furthermore, it is difficult to relate the activity of hydrogen ion (measured potentiometrically) to the concentration of hydrogen ion (and hence to the particle surface charge via eq 1), as the definition of mean activity coefficient is obscure.41 Thus, from both theoretical and experimental perspectives, potentiometry is unsatisfactory for determining the intrinsic charge. In this case, voltammetric determination of the intrinsic charge by quantifying the concentration of hydrogen ion is preferred. From eq 3, the slope of a plot of limiting current against reciprocal accessible volume of the suspension, [VT(1 - φ)]-1, gives the total amount of hydrogen ion in the sample. Measurement of limiting currents in suspensions of known initial volume and particle volume fraction and in suspensions prepared from them by serial dilution gives the concentration of hydrogen ion and permits calculation of the intrinsic charge using eq 1. Figure 2 shows the results of three such experiments at different initial volume fractions of particles. The diffusion coefficient of hydrogen ion is expected to change slightly from dilution to dilution due to changes in the concentration of supporting electrolyte. This change in D° during the experiment was small (1.5% at most, estimated from the Debye-Hu¨ckel theory),10 and we use the mean value of D in eq 3 to obtain the amount of hydrogen ion and, thus, the intrinsic charge number from eq 1. The slopes of the lines (r2 > 0.9986) from four such experiments gave 10 300 ( 600 equivalent charges on each particle (-6.78 ( 0.40 µC/cm2). Adding the charge equivalent to alkali metal ion concentrations gave a total intrinsic charge of 10 750 ( 600 (-7.08 ( 0.40 µC/ cm2). The repeatability of limiting current values in suspensions containing excess supporting electrolyte was determined by carrying out five voltammograms for the reduction of hydrogen counterion at a 5.4-µm-radius disk electrode in the latex (φ ) 0.43%; [LiClO4] ) 30 mM). The limiting current value was 3.787 ( 0.033 nA (rsd ) 0.9%). The reproducibility of the method with respect to the intrinsic charge value determined from four concentration calibration plots was 6%. This is comparable with the reproducibility of Z values determined by conductometric titration of similar polymer latexes.42 The value of the intrinsic charge number determined here is 12% lower than the Z value of 12 150 (-7.73 µC/cm2) determined previously by conductometric titration of hydrogen ion in the (37) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (38) Bates, R. G. Determination of pH: Theory and Practice; John Wiley & Sons: New York, 1973. (39) Oman, S.; Godec, A. Electrochim. Acta 1990, 36, 59. (40) Oman, S. Electrochim. Acta 1991, 36, 943. (41) James, R. O.; Davis, J. A.; Leckie, J. O. J. Colloid Interface Sci. 1978, 65, 331. (42) Hearn, J.; Wilkinson, M. C.; Goodall, A. R. Adv. Colloid Interface Sci. 1981, 14, 173.

Table 2. Effective Charge and ζ-Potential Characterization by Electrophoresis

Table 1. Charge Characterization by Voltammetry [LiClO4] (mM) 100-fold excess relative to 0.001

Z H3O+

Z*

10 750 260

latex.10 The two most likely explanations for this discrepancy are error in the titration42 and aggregation of the latex under the condition of excess electrolyte required to maintain diffusional control. We have observed aggregation in samples that gave decreased limiting current after the addition of electrolyte. For this reason, the experiments shown in Figure 1 were carried out at low volume fractions to minimize the concentration of electrolyte required to eliminate migrational transport. Aggregation under these conditions is evidence that the intrinsic charge is not the charge responsible for the interaction between particles, since the repulsive energy barrier to aggregation of the latex predicted by the DLVO theory based on the intrinsic charge value of 10 750 at the salt concentrations used in this experiment is at almost 2000 kT.6 Inference of an Effective Charge from Voltammetry. Electrodes of very small size permit voltammetric measurements in highly resistive solutions.31 This property can be exploited to infer effective charge on the latex particles. In the deionized latex suspension, without added electrolyte, strong electrostatic interaction between the counterions and the particles decreases the apparent diffusion of the counterions. Diffusion coefficients have been measured in the latex suspension and in perchloric acid with no supporting electrolyte.10 The current measured in the deionized latex is only 2.4% of the value measured in perchloric acid at an equivalent concentration of hydrogen ion. According to our two-state model, the current arises only from ions that do not interact strongly with the latex (i.e., ψ g 0; c(r) e 〈c〉), and the effective charge on the particle is Z* ) 0.024Z ) 260 (-0.17 µC/ cm2). This is consistent with the observed aggregation of the latex, since the salt concentration used in these experiments is larger than the critical coagulation concentration calculated from the DLVO theory using an effective charge of 260.3 These results are summarized in Table 1. Inference of an Effective Charge from Electrophoresis. The electrophoretic mobility, u, of a colloidal particle is a transport property that can be used to infer an effective charge number. Under the influence of an applied electric field, a spherical particle is transported through a viscous medium with a velocity proportional to its charge, eZS. The Stokes charge number, ZS, is defined by Stokes’s law,

ZS ) 6πηahu/e

(4)

In eq 4, ah is the hydrodynamic radius of the particle (ah = a ) 44 nm), η is the viscosity of water (0.001 kg/(m‚s)), and e is the elementary charge.2 Table 2 presents experimental results for mobility, together with parameters derived therefrom. The Stokes charge numbers determined from mobility measurements (Table 2) are similar to the effective charge numbers inferred from voltammetry (Table 1). The observation that ZS is significantly less than Z is in accord with the model of Figure 1; that is, r at

[LiClO4](mM)

κa

u (×108 m2 V-1 s-1)

ZS

ζ (mV)

ZL

0.001 0.030 0.078 0.200

0.14 25 40 65

7.7 6.2 5.4 3.5

400 320 280 180

-210 -130 -82 -46

19 500 11 900 8 200

c(r)/〈c〉 ) 1 is roughly equivalent to ah. Addition of electrolyte changes the composition of this inner layer in proportion to the mole fraction of electrolyte but does not change significantly the net negative charge within this inner region (i.e., the effective charge). The mobility can be used to calculate also the ζ-potential (i.e., the electrostatic potential at the surface of shear between the particle and the solvent). The ζ-potential is not measurable directly; it can only be calculated from a model and a measurable property of the system (e.g., mobility). The calculated value of the ζ-potential can be used directly in the DLVO theory to predict the stability of the dispersion if the interaction between particles occurs at constant potential for all distances of separation. In reality, however, most colloidal dispersions exhibit mixed control at short distances, for which neither the potential nor the charge is constant.2,5 Calculations made assuming constant charge tend to overestimate the stability of the dispersion; constant potential calculations underestimate the stability. For this reason, it seems useful to know both the potential and the charge, in order to estimate a range of stability. Table 2 summarizes ζ-potential values calculated from the measured mobility according to ref 37. Values of the dimensionless Debye screening parameter, κa, were calculated from the particle radius and the amount of added LiClO4 (see eq 9 in ref 10). The Loeb charge number, ZL, is a quantity used often to characterize charged colloids and can be calculated from the ζ-potential (eq 4.8.7 in ref 3). Comparison of the calculated values of ZL shown in Table 2 with the charge numbers in Table 1 shows that ZL is closer to the intrinsic charge number than to the effective charge. Studies on liposomes43 and on latexes similar to this one yield the same result, indicating that the charge calculated from the Loeb expression is quite different from the effective charge responsible for the interaction between the particles.44 In light of the method presented here for determining Z directly, the Loeb charge number seems of limited utility. Furthermore, the physical origin of ZL is not clear, as it is calculated from an empirical expression involving the ζ-potential, which is not measurable directly. The data presented thus far characterize the surface charge on particles in a single polystyrene latex suspension. To put properties of this latex into perspective, we compare our results with those for other well-characterized, spherical macroionic systems. Correlation of the Effective Charge on Spherical Macroions with Measurable Parameters. Prediction of electrostatic properties of spherical macroions may be guided by similar work carried out in solutions containing linear macroions (polyelectro(43) Carmona-Ribeiro, A. M.; Midmore, B. R. J. Phys. Chem. 1992, 96, 3542. (44) Ha¨rtl, W.; Zhang-Heider, X. J. Colloid Interface Sci. 1997, 185, 398.

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Table 3. Effective Charge Ratios of Spherical Macroions

Figure 3. Fractional effective charge, Z*/Z, as a function of Z/a. The solid line is as predicted from the nonlinear Poisson-Boltzmann equation according to the model of Figure 1, in which Z* is the integrated charge number between c(r)/〈c〉 ) 1 and the cell wall. Points represent the data of Table 3.

lytes). A large amount of literature exists in this area, 45-56 and it is generally well accepted that the extent of counterion binding to highly charged linear polyelectrolytes (and therefore the effective charge) is predicted reasonably well by Manning’s theory of counterion condensation. An advantage of Manning’s theory is that it predicts D/D° values (identified here as Z*/Z values) using a single, measurable parameter, the linear charge density. We therefore surveyed the data on effective charge values in wellcharacterized, spherical macroionic systems with the goal of finding a parameter that would correlate Z*/Z values with a measurable parameter of the system. Figure 3 shows the correlation of Z*/Z with Z/a for the data shown in Table 3. The effective charge (and, therefore, the extent of ion binding) correlates with the ratio of the intrinsic charge to macroion radius, Z/a, in solutions of micelles and in latex suspensions. The data point for the latex used in this study (244 nm-1, 0.024) is somewhat low, but it is consistent with the general trend of decreasing effective charge with increasing Z/a. The solid line is the theoretical prediction of Z*/Z calculated from the nonlinear Poisson-Boltzmann equation for spherical geometry, using the criterion that Z* is the charge found between the c(r)/〈c〉 ) 1 contour and the cell wall (see Figure 1). Several observations may be made. First, the criterion of c(r)/ 〈c〉 < 1, applied to the theoretical ion distribution, yields values of effective charge that agree reasonably well with experiment. This suggests that the Poisson-Boltzmann model as here employed (45) Ciszkowska, M.; Osteryoung, J. G. J. Phys. Chem., in press. (46) Schmitz, K. S. Macroions in Solution and Colloidal Suspension; VCH Publishers: New York, 1993. (47) Piculell, L.; Rymden, R. Macromolecules 1989, 22, 2376. (48) Yuryev, V. P.; Plashchina, I. G.; Braudo, E. E.; Tolstoguzov, V. B. Carbohydr. Polym. 1981, 1, 139. (49) Rinaudo, M.; Karimian, A.; Milas, M. Biopolymers 1979, 18, 1673. (50) Kowblansky, M.; Tomasula, M.; Ander, P. J. Phys. Chem. 1978, 82, 1491. (51) Ander, P. In Water Soluble Polymers; Shalaby, S. W., McCormic, C. L., Butler, G. B., Eds.; ACS Symposium Series 476; American Chemical Society: Washington, DC, 1991. (52) Huizenga, J. R.; Griegor, P. F.; Wall, F. F. J. Am. Chem. Soc. 1950, 72, 4228. (53) Nilsson, L. G.; Nordenskio ¨ld, L.; Stilbs, P. J. Phys. Chem. 1987, 91, 6210. (54) Scordilis-Kelley, C.; Osteryoung, J. G. J. Phys. Chem. 1996, 100, 4630. (55) Tsuge, H.; Yonese, M.; Kishimoto, H. J. Phys. Chem. 1987, 91, 1971. (56) Magdalenat, H.; Turq, P.; Trivant, P.; Chemla, M.; Menez, R.; Drifford, M. Biopolymers 1979, 18, 187.

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system

a (nm)

Z

Z*

Z/a

Z*/Z

ref

latexa latexa latexb latexb latexb SDSc SDSc SDSc latexb latexb latexb latexb latexb latexb latexb latexb latexb latexb latexb latexb latexb latexd latexb latexe latexf latexg latexe

54.5 51 45 51 43 2.2 2.5 2.8 70 94.2 43 58.5 43 43 60 51 45 45 101 65 94.5 44 180 326 51 51 150

600 580 570 950 1 010 64 94 110 3 410 7 190 3 660 6 200 4 570 4 800 8 472 7 880 6 970 7 200 19 100 15 000 22 416 10 750 58 700 133 550 22 000 22 000 70 686

305 395 360 385 558 19 37 37 1 712 2 466 673 1 228 658 706 847 1 040 871 929 2 655 1 950 2 242 260 5 694 7 300 490 370 1 000

11.0 11.4 12.7 18.6 23.5 29.1 37.2 39.4 48.7 76.3 85.1 106.0 106.3 111.6 141.2 154.5 154.9 160.0 189.1 230.8 237.2 244.3 326.1 409.7 431.4 431.4 471.2

0.508 0.681 0.632 0.405 0.552 0.300 0.394 0.336 0.502 0.343 0.184 0.198 0.144 0.147 0.100 0.132 0.125 0.129 0.139 0.130 0.100 0.024 0.097 0.055 0.022 0.017 0.014

25 26 22 26 23 30 29 29 23 23 23 23 23 23 21 23 23 23 23 9 21 this work 23 13 27 27 24

a Shear modulus. b Conductivity in the deionized sample. c Sodium dodecyl sulfate micellar solutions, light- or neutron-scattering. d Voltammetry. e Video microscopy f Torsional resonance. g Electrophoresis.

may be used to predict the effective charge number on the basis of the measured values of the intrinsic charge and particle radius. Second, as summarized in Table 3, these data were taken from several different kinds of experiments carried out on 25 different latex suspensions and three different micellar solutions, none of which contained deliberately added electrolyte. These data may be classified into two categories: those obtained from experiments which probe directly the electrostatic interactions between particles (methods a, c, f, and g), and those which infer an effective charge on the particle from measurement of a transport property of either the particles or the counterions (methods b, c, e, and h). The latter approach was used in this study. The agreement of both sets of data with the theory suggests that either type of experiment is useful in determining effective charge values on spherical macroions. Third, the utility of latex suspensions as model colloids is evident, as they can be made to span the entire range of ion binding. This may be contrasted with solutions of micelles, for which the effective charge (or the extent of ion binding) is localized to the region in which the effective charge is very sensitive to small changes in the ratio of the intrinsic charge to micelle radius. A wide range of effective charge values has been reported for micellar solutions.7,29,30 It has been suggested that this may be due to the different methods used to infer an effective charge. The present treatment, however, suggests that the strong dependence of Z*/Z on Z/a for micelles is responsible for the uncertainty in this value. Finally, from a practical perspective, a well-characterized, charge-stabilized colloidal system is best achieved by producing

a dispersion of particles with a large value of Z/a. In this region, the effective charge value is rather insensitive to uncertainties in the measured values of Z and a. The physical interpretation of the relationship between Z*/Z and Z/a is the following. At equilibrium, the distribution of counterions around a colloidal particle results from two opposing forces: electrostatic attraction to the surface of the oppositely charged particle and electrostatic repulsion between neighboring counterions. The accumulation of counterions around the charged particle (i.e., counterion binding) is the mechanism by which the system minimizes its electrostatic energy. At constant ionic strength, the electrostatic potential (and, therefore, the energy) at the surface of a charged sphere is directly proportional to the ratio of its intrinsic charge number to radius, Z/a (see section 7.4 of ref 57). As a sphere of a fixed radius is charged up (i.e., as Z increases), the response of the system is to bind a larger fraction of its counterions, lowering the charge to some effective value, (57) Feynman, R. P.; Leighton, R. B.; Sands, M. The Feynman Lectures on Physics, Volume II; Addison-Wesley Publishing Co.: Reading, MA, 1964.

Z*, and, therefore, lowering the electrostatic energy at the surface of the sphere. Within the two-state model proposed here, replacing Z with Z* is equivalent to removing counterions from the system and incorporating them into the internal energy of the particle. As a result, the (electrical) free energy with which the colloidal particle may interact with counterions, electrolyte ions, and other particles is diminished by 1 - Z*/Z and is proportional to Z*/a. ACKNOWLEDGMENT We thank F. Dennis and Lord Corp. for use of the Malvern ZetaMaster IV instrument. J.M.R. thanks M. Ciszkowska, A. Jaworski, and K. Schmitz for helpful and stimulating discussions. This work was supported by the National Science Foundation under Grant No. DMR 9711205. Received for review December 18, 1997. Accepted July 6, 1998. AC971364L

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