Voltammetric Studies of Counterion Diffusion in the Monodisperse

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Voltammetric Studies of Counterion Diffusion in the Monodisperse Sulfonated Polystyrene Latex James M. Roberts,† Per Linse,‡ and Janet G. Osteryoung*,† Department of Chemistry, North Carolina State University, Box 8204, Raleigh, North Carolina 27695-8204, and Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University, P.O. Box 124, S-221 00 Lund, Sweden Received June 2, 1997. In Final Form: August 19, 1997 Steady-state voltammetry for the reduction of hydrogen ion at a Pt microelectrode was carried out in an aqueous suspension of a monodisperse sulfonated polystyrene latex with hydrogen ion as counterion to the latex particles. The measured transport-limited current for the reduction of the hydrogen ion was related to the gradient-diffusion coefficient of the hydrogen ion over a wide range of concentration of inert, 1:1 electrolyte. The gradient-diffusion coefficient for hydrogen ion in the deionized, ordered latex crystal was less than 1/40th of the value found without the latex. Addition of a 100-fold excess of electrolyte (with respect to hydrogen ion concentration) resulted in a value for the gradient-diffusion coefficient close to that without the latex. A neutral molecule, 4-hydroxy-TEMPO, oxidized anodically in the suspension under these same conditions yielded transport-limited currents that were little affected by changes in salt concentration. This suggests that most of the hydrogen ions are electrostatically accumulated near the particles. This observation can be described in terms of an effective charge on the latex particles. The gradient-diffusion coefficient of hydrogen ion was found to be insensitive to the transition from an ordered crystalline phase of the latex to disorder upon the addition of electrolyte. The dependence of the gradientdiffusion coefficient on the concentration of 1:1 electrolyte agrees with the self-diffusion coefficient values predicted from the cell model using the nonlinear Poisson-Boltzmann equation and with that predicted by a simple, semiempirical model.

Introduction The interactions between small ions in dilute solution is well understood in terms of the Debye-Hu¨ckel (DH) theory of electrolytes.1,2 For example, nonspecific Coulombic interactions in aqueous solution slow the diffusion of small ions. The DH theory predicts correctly that the diffusion coefficient of a small ion, D°, in a solution with an ionic strength I is suppressed relative to its ideal value expected at infinite dilution, D∞, by an amount described by

D°/D∞ ) 1 - AI1/2/2

(1)

where A is a constant derived from the DH theory and has the value 0.512 M-1/2 in water at 25 °C.2 Diffusion of Ions in Macroionic Systems. The diffusion coefficient of counterions in the presence of charge-stabilized colloidal particles in suspension or polyelectrolytes in solution is less than D° due to electrostatic attraction of the counterion to the more slowly diffusing macroion.3-14 The magnitude of this effect depends strongly on the amount of added electrolyte in * To whom correspondence may be addressed. E-mail: chejgo@ chemdept.chem.ncsu.edu. † North Carolina State University. ‡ Lund University. X Abstract published in Advance ACS Abstracts, December 15, 1997. (1) Debye, P. The Collected Papers of P. J. W. Debye; Interscience Publishers: New York, 1954. (2) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry; Plenum Press: New York, 1970; Vol. 1. (3) Bell, G. M.; Dunning, A. J. Trans. Faraday Soc. 1970, 66, 500. (4) Jo¨nsson, B.; Wennerstro¨m, H.; Nilsson, P. G.; Linse, P. Colloid Polym. Sci. 1986, 264, 77. (5) Nilsson, L. G.; Nordenskio¨ld, L.; Stilbs, P.; Braunlin, W. H. J. Phys. Chem. 1985, 89, 3385. (6) Manning, G. S. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 909.

the solution or suspension. The most dramatic suppression of the counterion diffusion coefficient value occurs in deionized systems, in which the only ions in solution or suspension are the macroion and its accompanying counterion. We have found that in deionized solutions of linear polystyrenesulfonic acid (PSSA) the gradient-diffusion coefficient of hydrogen ion is only 35% of the value found in polyelectrolyte-free solution.9 Addition of electrolyte to this deionized solution increases the gradient-diffusion coefficient, and at approximately 100-fold concentration excess of univalent electrolyte the gradient-diffusion coefficient approaches the value found in polyelectrolytefree solution. This can be understood as a screening of the electrostatic attraction between hydrogen ion and PSS- or as replacement of hydrogen ion in the Stern layer of the PSS- anion by the added cation.15 The general dependence of the hydrogen counterion diffusion coefficient value in solutions of PSSA on the molar concentration of added 1:1 electrolyte is described accurately by a simple, semiempirical equation.9 This result for PSSA, (7) Morris, S. E.; Osteryoung, J. G. In Electrochemistry in Colloids and Dispersions; Mackay, R. A., Texter, J., Eds.; VCH Publishers, Inc.: New York, 1992; p 245 (8) Morris, S. E.; Ciszkowska, M.; Osteryoung, J. G. J. Phys. Chem. 1993, 97, 10453. (9) Ciszkowska, M.; Osteryoung, J. G. J. Phys. Chem. 1994, 98, 3194. (10) Ciszkowska, M.; Osteryoung, J. G. J. Phys. Chem. 1994, 98, 11791. (11) Ciszkowska, M.; Zeng, L.; Stejskal, E. O.; Osteryoung, J. G. J. Phys. Chem. 1995, 99, 11764. (12) Scordilis-Kelley, C.; Osteryoung, J. G. J. Phys. Chem. 1996, 100, 797. (13) Ciszkowska, M.; Osteryoung, J. G. J. Phys. Chem. 1996, 100, 4630. (14) Ciszkowska, M.; Osteryoung, J. G. In Proceedings of the International Symposium on “New Directions in Electroanalytical Chemistry”; Leddy, J., Wightman, R. M., Eds.; The Electrochemical Society, Inc.: Pennington, NJ, 1996; p 263. (15) Rouzina, I.; Bloomfield, V. A. J. Phys. Chem. 1996, 100, 4292.

S0743-7463(97)00572-6 CCC: $15.00 © 1998 American Chemical Society Published on Web 01/06/1998

Monodisperse Sulfonated Polystyrene Latex

a model for highly charged linear macroions, has practical application to understanding and control of ion binding, ion-exchange processes, and tertiary structure in macroionic systems such as DNA,16 ionic-exchange membranes used in water purification systems,17 ion-exchange chromatography,18,19 and the uptake of nutrients by plants via cations bound to colloidal clay particles.20 We have now extended our voltammetric studies of diffusion to include the transport of hydrogen ion in a colloidal suspension of spherical macroionic particles. Many experimental and theoretical studies of counterion diffusion in spherical association colloids (micellar solutions) have shown that counterion diffusion is suppressed in these systems.3,21,22 However, to our knowledge, the diffusion behavior of counterions in a spherical, lyophobic colloidal suspension has not been addressed. The model system chosen is a monodisperse sulfonated polystyrene latex in the acid form. The latex is a spherical analog to the solutions of linear PSSA discussed above, with the important difference that the macroion forms a separate phase from the suspending medium containing the counterion.23 It is well accepted that the electrostatic energy of interaction between the particles in the deionized latex dominates thermal energies, which otherwise tend to randomize the particles in suspension.24 The result is an ordering of the particles into a crystalline state; typical lattice spacings are on the order of a few particle diameters, depending on the volume fraction of particles in suspension. Because visible light is diffracted from crystal planes in typical particle size ranges (100-1000 nm), the latex exhibits a brilliant iridescence in addition to the milkywhite color produced from scattered light.25 This ordering phenomenon, combined with the rather large size of the particles (most are visible using a conventional optical microscope) and the long time scale of processes, has made this system a useful model for phase transitions in atomic systems.26,27 Although ordering is believed to be caused by electrostatic interactions between particles, the mechanism of this interaction is not fully understood. For this reason, the monodisperse polystyrene latex has been used extensively to test various theories of interactions in colloidal suspensions.28-31 The major disagreements surrounding the issue of structure in latex suspensions revolve around the role of small ions.31 In our experiments we focus not on particleparticle interactions but rather on the effect of the highly charged latex particles on the diffusion of the small (16) Duguid, J. G.; Bloomfield, V. A. Biophys. J. 1996, 70, 2838. (17) Evans, D. F.; Wennersto¨m, H. The Colloidal Domain; VCH Publishers, Inc.: New York, 1994. (18) Sta˚hlberg, J. Anal. Chem. 1994, 66, 440. (19) Sta˚hlberg, J.; Appelgren, U.; Jo¨nsson, B. J. Colloid Interface Sci. 1995, 176, 1. (20) Hedin, L. O.; Likens, G. E. Sci. Am. 1996, 275 (6), 88. (21) Almgren, M.; Stilbs, P.; Alsins, J.; Linse, P.; Kamenka, N. J. Phys. Chem. 1985, 89, 2666. (22) Jansson, M.; Linse, P.; Rymden, R. J. Phys. Chem. 1988, 92, 6689. (23) Fitch, R. M. In Polyelectrolytes and Their Applications; Rembaum, A., Selegny, E., Eds.; D. Reidel Publishing Company: Dordrecht, 1975; p 51. (24) Arora, A. K.; Rajagopalan, R. In Ordering and Phase Transitions in Charged Colloids; Arora, A. K., Tata, B. V. R., Eds.; VCH Publishers, Inc.: New York, 1996; p 1. (25) Hiltner, P. A.; Krieger, I. M. J. Phys. Chem. 1969, 73, 2386. (26) Murray, C. A.; Grier, D. G. Am. Sci. 1995, 83, 238. (27) Monovoukas, Y.; Gast, A. P. J. Colloid Interface Sci. 1989, 128, 533. (28) Larsen, A. E.; Grier, D. G. Nature 1997, 385, 230. (29) Murray, C. A. Nature 1997, 385, 203. (30) Arora, A. K.; Tata, B. V. R. Ordering and Phase Transitions in Charged Colloids; VCH Publishers, Inc.: New York, 1996. (31) Schmitz, K. S. Macroions in Solution and Colloidal Suspension; VCH Publishers, Inc.: New York, 1993.

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counterion. As with the PSSA system, we investigate the influence of electrolyte on the diffusion behavior of the counterion in the macroionic suspension. Theoretical treatments based on fundamental principles have been successful in describing the diffusion behavior of counterions in a variety of macroionic systems. For example, the cell model has described correctly ionic diffusion in several model polyelectrolyte systems.3-5,32 Since this colloidal system can be well characterized, we compare our experimental results for hydrogen counterion diffusion with predictions of the cell model. We also wanted to know if the semiempirical treatment previously used for solutions containing linear polyelectrolytes is applicable to other macroionic systems as well.9,10 We therefore compare counterion diffusion in the latex with this semiempirical theory. Finally, we show that the voltammetric measurement of the gradient diffusion of the counterion in the deionized latex gives a reasonable estimate of an effective charge on the particle. Experimental Methods Used To Investigate Diffusion. The method used most commonly to measure the self-diffusion coefficient of small ions and molecules in colloidal systems is pulsed-field-gradient, spin-echo NMR (PFG-SE NMR).33,34 One of the goals of this work is to show that steady-state voltammetry at a microelectrode yields quantitative data on the gradient-diffusion coefficient of small ions and molecules in colloidal systems. Typically, a microelectrode has at least one dimension that lies in the range 100 nm to 50 µm.35 Various geometries of microelectrodes have found applications in analytical chemistry, electrosynthesis, biology, battery technology, corrosion, and metal deposition.36 Important to this study is the ability of the microelectrode to measure mass-transport-limited, steady-state currents in relatively short times over a wide range of concentration of electrolyte, particularly in solutions of very low ionic strength. This makes the microelectrode a useful tool for studying changes in the gradient diffusion of ions in complex media, because typically a range of 106 in electrolyte concentration must be used to reveal the dependence of that diffusion on electrolyte concentration. We have shown that this technique yields gradient diffusion coefficients that are, within experimental uncertainty, equal to the self-diffusion coefficients determined by NMR for 205Tl+ in aqueous solution.11 In addition, the presence of the polyelectrolyte has been shown to reduce the diffusion of Tl+ and H3O+ to the same extent, indicating that the mechanism of diffusion is irrelevant.9,10,37-39 By exploiting both the nuclear spin and electroactive properties of the analyte, we hope to increase the number of ions and molecules available for this kind of investigation. The following ions and molecules are examples of small species that have favorable electrochemical behavior in aqueous solutions: H3O+, Tl+, Pb2+, Cd2+, Cu2+, and TEMPO. Atoms that have favorable nuclear spin properties for NMR studies include 1H, 2H, 7 Li, 14N, 17O, 19F, 23Na, 31P, 195Pt, and 205Tl. (32) Yoshida, N. J. Chem. Phys. 1978, 69, 4867. (33) Stejskal, E. O.; Tanner, J. E. J. Chem. Phys. 1965, 42, 288. (34) Stilbs, P. Prog. Nucl. Magn. Reson. Spectrosc. 1987, 19, 1. (35) Pletcher, D. In Microelectrodes: Theory and Applications; Montenegro, M. I., Queiros, M. A., Daschbach, J. L., Eds.; Kluwer Academic Publishers: Dordrecht, 1991; p 3. (36) Montenegro, M. I., Quieros, J. L., Daschbach, J. L., Eds. Microelectrodes: Theory and Applications; Kluwer Academic Publishers: Dordrecht, 1991. (37) Lobaugh, J.; Voth, G. A. J. Chem. Phys. 1996, 104, 2056. (38) Halle, B.; Karlstro¨m, G. J. Chem. Soc., Faraday Trans. 2 1983, 79, 1031. (39) Halle, B.; Karlstro¨m, G. J. Chem. Soc., Faraday Trans. 2 1983, 79, 1047.

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Roberts et al. determined by measuring the concentration, ci, of each monovalent counterion of type i in a deionized latex sample43

∑i ci(1 - φ)/3φ

σ ) Fa

Figure 1. SEM micrograph of monodisperse polystyrene latex particles. The latex was sputtered with chromium to reduce charging of the polymer. Particle radius, a ) 44 nm.

In the following, we discuss the response of the voltammetric technique in simple solution and in the latex and the effect of changing the concentration of inert electrolyte on the measured signal. The influence of electrolyte on the gradient-diffusion coefficient value of the hydrogen counterion is measured and compared with predictions of existing theories. Experimental Section Latex Synthesis. Water had a resistance of 18 MΩ (Millipore, Bedford, MA). An emulsifier-free synthesis of styrene with sodium styrenesulfonate as an ionic comonomer was employed. This method, described by Juang and Krieger,40 produces a model lyophobic colloid; the particles are monodisperse, highly charged from incorporation of the sulfonate comonomer, and free of adsorbed charged surfactant, as there is no conventional surfactant used in the synthesis. After cleaning and characterization of the particles, the measured surface charge density is constant with time. The following were added to a 3-L kettle, purged with nitrogen, and equilibrated to 70 °C with an internal thermometer for 1 h: 0.3726 g of KOH, 0.0117 g of KH2PO4, 0.7721 g of NaCl, 315 mL of styrene, 1685 mL of water. Then 3.3786 g of NaSS and 0.9540 g of K2S2O8 were added to the vessel and the entire mixture was stirred at 70 °C and bubbled with nitrogen for 23 h.27 All the chemicals were used as received (Aldrich). The latex was then filtered through polypropylene wool (Aldrich) and glass fiber filters (Millipore APFF) to remove the coagulum and placed in an oven at 75 °C for 72 h to hydrolyze residual sulfate groups.41 Latex Characterization. The particle radius was determined from the SEM micrograph in Figure 1. Diameters of 125 particles were measured and averaged to give a particle radius of 44 nm with a uniformity ratio of 1.01.40 Random sampling of the dried latex at wider fields of view showed that the particles are highly monodisperse. Particle volume fractions were measured gravimetrically in triplicate assuming the polystyrene density of 1.05 g/cm3.42 For each experiment, the particle volume fraction was measured at a particle density large enough to obtain an accurate measurement of the particle volume fraction (typically a few percent). When a dilute suspension of particles was required, the particle volume fraction was calculated from the measured volume fraction of a more concentrated sample that was diluted volumetrically. The surface charge density, σ of the latex particle is difficult to measure directly.41 Instead, the charge on the particle can be (40) Juang, M. S. D.; Krieger, I. M. J. Polym. Sci. 1976, 14, 2089. (41) Hearn, J.; Wilkinson, M. C.; Goodall, A. R. Adv. Colloid Interface Sci. 1981, 14, 173. (42) Stecher, P. G., Ed. The Merck Index; Merck & Co., Inc.: Rahway, NJ, 1968.

(2)

where F is Faraday’s constant, a is the particle radius, and φ is the measured particle volume fraction. Equation 2 follows from the spherical geometry of the macroion and the electroneutrality of the suspension. After synthesis, the particle suspension contains excess electrolyte from the polymerization. To prepare a deionized latex in the acid form, excess ions were removed from the suspension and then the remaining alkali ions were ion-exchanged for hydrogen ion. The concentration of each ionic species was measured quantitatively during the course of the cleaning to ensure the completion of the ion-exchange process. As the procedures for cleaning latex suspensions of this type have been the subject of considerable discussion, we provide a detailed description. First the latex was centrifuged in six 85-mL polycarbonate tubes (Nalgene) at 13 500 rpm (20 980 maximum g) in a Marathon 22KBR centrifuge (Fisher) at 15 °C for approximately 4 h. The supernatant was decanted, and the pellet of latex at the bottom of the centrifuge tube was resuspended with ultrapure water overnight. This process was repeated three times, after which the conductivity of the slightly turbid supernatant was typically 2 µS/cm (Fisher Traceable digital conductivity meter with a dipcell configuration operating at 1000 Hz). The resulting latex suspension was brilliantly iridescent, indicating a deionized suspension of ordered and monodisperse latex particles. In addition to removing excess electrolyte, centrifuging the particles and resuspending them with different amounts of ultrapure water is a simple way to control the particle volume fraction. To quantify the concentrations of ions in this deionized latex, we assume that the only simple ions in the medium are Na+, K+, and H3O+ (since these alkali metal ions were the only source of ions used in the synthesis and H3O+ arises from the dissociation of the acid groups). The amount of hydrogen ion was measured by conductometric titration of the latex with NaOH solution (Fisher), standardized in triplicate with dried KHP.44 The concentrations of sodium and potassium ions in the deionized latex sample were measured using atomic emission spectroscopy (AES, Perkin-Elmer 2380 atomic absorption spectrophotometer operated in emission mode).45 The surface charge density of the particles in this water-washed sample, calculated from eq 2, had the relatively high value of -7.34 µC/cm2 (11 100 e/particle). However, only 4% of this charge corresponded to hydrogen ion. Therefore ion-exchange was necessary to replace the alkali metal counterions with hydrogen ion. There are in principle several ways to achieve this. Dialysis is time-consuming and ineffective in ion-exchanging hydrogen ion for alkali metal ions in the latex.41 We attempted serum replacement of the water with HClO4, but this proved ineffective because the small particle size caused a prohibitively slow flow rate through the pore membrane.46 The least unattractive method was treatment of the latex with a mixed bed of ion-exchange resins (Bio-Rad AG 501-X8).41 Even with analytical grade resin beads, there is a possibility of contaminating the latex with polyelectrolytes that leach from the ion-exchange resins. We took the following steps to ensure the purity of the ion-exchange resin beads. Following the procedure of Van den Hul and Vanderhoff,47 we stirred the ion-exchange beads in (43) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. (44) Kolthoff, I. M.; Sandell, E. B.; Meehan, E. J.; Bruckenstein, S. Quantitative Chemical Analysis; The Macmillan Company: London, 1969. (45) Ingle, J. D.; Crouch, S. R. Spectrochemical Analysis; Prentice Hall: Englewood Cliffs, NJ, 1988. (46) Ahmed, S. M.; El-Aasser, M. S.; Pauli, G. H.; Poehlein, G. W.; Vanderhoff, J. W. J. Colloid Interface Sci. 1980, 73, 388. (47) Van den Hul, H. J.; Vanderhoff, J. W. J. Electroanal. Chem. 1972, 37, 161. (48) Aoki, K.; Akimoto, K.; Tokuda, K.; Matsuda, H.; Osteryoung, J. G. J. Electroanal. Chem. 1984, 171, 219.


Langmuir, Vol. 14, No. 1, 1998 207 Table 1. Latex Particle Surface Charge Characterization (Particle Radius, a ) 44 nm) σ Z (charges/ charge (µC/cm2) particle) spacing (nm) titration (H3O+) atomic emission (Na+, K+) total

Figure 2. Conductometric titration of the latex showing a strong acid end point. [NaOH] ) 0.0921 M; volume end point ) 141 µL; initial latex volume ) 10.00 mL; φ ) 2.37%; the resulting charge density per particle attributable to hydrogen ion is -7.73 µC/cm2. The total particle charge density of -8.03 µC/cm2 was found by quantifying the concentrations of all the ions in the deionized latex: H3O+, K+, Na+. ultrapure water for 24 h and measured the transmittance of ultraviolet light through this treated water (Hewlett Packard 8452A diode array spectrophotometer, 1-cm path length). There was significant absorbance from 180 to 300 nm (relative to ultrapure water), indicating the presence of low-molecular-weight organics. However, these are probably neutral compounds rather than polyelectrolytes, as this same water had an extremely low conductivity value (e0.1 µS/cm) and contained negligible amounts of sodium, potassium, and hydrogen ion, as measured by AES and titration, respectively. After stirring an aliquot of resin beads in water for 24 h, we used the “batch” ion-exchange method by mixing directly these water-rinsed beads with the deionized, iridescent latex at a volume fraction of about 4% and stirring the suspension for 24 h. The latex was then filtered to remove debris from the resin and titrated to determine the amount of hydrogen ion in the sample. This treatment was repeated two times, and the titrated amount of hydrogen ion was found to be similar for consecutive exposures to the resins. Figure 2 shows a representative titration curve, illustrating the strong acid end point. We found no evidence for weak acid moieties on the particle surface. Two titrations were averaged to give a surface charge density value of -7.73 µC/cm2 (11 700 e/particle). Measurements of sodium and potassium concentrations by AES in this deionized latex suspension after the resin treatment showed that about 3% of the ions were sodium and potassium. Determination by AES of alkali ions from the supernatant of this treated latex showed negligibly small concentrations of sodium and potassium (2 µM 1:1 electrolyte in the supernatant from a sample spun at an initial particle volume fraction of 2.37%). This indicates that the ionexchange process was 97% effective in removing the alkali ions and that there is only a low concentration of background electrolyte arising from these alkali ions. These measurements of the concentrations of hydrogen ion and sodium and potassium ions account for all cations present in the latex. Using eq 2 and accounting for the measured concentrations of all ionic species, we conclude that the particle surface charge density is -8.03 µC/cm2. The concentration of hydrogen ion in a given sample is calculated from the particle volume fraction measured gravimetrically and the surface charge density value of -7.73 µC/cm2, corresponding to the titrated amount of hydrogen ion. Note that there is relatively good agreement between the total charge measured before and after the ion-exchange procedure (-7.34 and -8.03 µC/cm2, respectively). After completion of the electrochemical experiments, the latex was titrated again to determine the concentration of hydrogen ion. This differed by only 1.5% from its initial value, an amount within the experimental error. These results are summarized in Table 1. Electrochemical Experiments. Platinum disk microelectrodes as shown in Figure 3 were used as working electrodes. The electrode radii were measured using an inverted metallurgical microscope (Leitz, Diavert) equipped with a micrometer.

-7.73 -0.30 -8.03

11 700 450 12150

0.4

A two-electrode cell arrangement was used, with a platinum wire serving as both quasi-reference and counter electrodes. A true reference electrode (e.g., SCE) might contaminate the deionized latex with adventitious electrolyte from the liquid junction.9 This procedure is permissible, because the potential of the working electrode does not need to be well specified in this experiment. The electrochemical cell was enclosed in an aluminum box (Faraday cage) to prevent distortion of the small currents from external electric fields. A diamond suspension (0.25 µm, Buehler) was used to polish the electrodes on a soft, wet pad (Microcloth, Buehler) between scans. After they were polished, the electrodes were rinsed vigorously with water and inspected under the 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), and injections of electrolyte into the latex were made with either micropipets (Eppendorf) or microsyringes (Hamilton). Voltammetry was carried out by means of a PAR 273 potentiostat controlled with in-house software on a PC computer. The potential was changed according to a staircase waveform with a 5-mV step height and a frequency of 1 Hz. This frequency ensures that the measured limiting current is within 3% of the predicted steady-state value.48 The resulting current was amplified by means of a Keithley 427 current amplifier connected to the potentiostat. One voltammetric scan requires about 3 min. Limiting currents were measured with respect to the baseline current. Typical latex sample volumes range from 5 to 10 mL. 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.

Results and Discussion Voltammetric Measurement of Diffusion in Solutions of Strong Acid. The reference system is HClO4 in an aqueous solution of LiClO4. The effects of viscosity and activity with change in ionic strength on the transport properties of hydrogen ion in these solutions are well understood. The transport properties of hydrogen ion are obtained by scanning the potential at the Pt microelectrode through the region where hydrogen ion is reduced according to the reaction

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

(3)

At sufficiently negative potentials, the current due to reaction 3 is limited by the rate at which hydrogen ion is transported to the electrode surface.49 For a disk microelectrode of radius r0 this steady-state transport-limited current for the reduction of hydrogen ion is given by35

il ) 4nFTcr0

(4)

where n is the stoichiometric number of electrons transferred (unity for the hydrogen evolution reaction), F is Faraday’s constant (96 485 C/(mol e)), c is the bulk concentration of hydrogen ion, and T is the transport coefficient of hydrogen ion. (49) Bard, A. J.; Faulkner, L. R. Electrochemical Methods; John Wiley & Sons: New York, 1980.

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Figure 3. Schematic of a two-electrode electrochemical cell. The working electrode was a Pt disk inlaid into a glass insulating sheath. A Pt-wire was used as the quasi-reference. Argon was bubbled through Peek tubing to remove dissolved oxygen. Electrodes were immersed into the sample through a friction fit at the top of the polycarbonate tube.

The transport of hydrogen ion arises both from migration (i.e., flow of ions under the influence of an electric field) and from diffusion (i.e., transport due to a concentration gradient). Since the goal of this work is to obtain diffusion coefficients, the migrational contribution has to be subtracted from the total transport. The dependence of the transport-limited current on electrolyte concentration is well-known both experimentally and theoretically.50 For one-electron reduction of a singly charged cation in a solution containing a 1:1 inert electrolyte, the transportlimited current, il, is enhanced with respect to the diffusionlimited current, id, by an amount

il/id ) 2 + 2γ - 2[γ(γ + 1)]1/2

(5)

where γ is the ratio of concentrations of added electrolyte to the species reduced, in this case [Li+]/[H3O+]. It is also clear from eq 5 that, in the absence of electrolyte, migrational transport and diffusional transport contribute equally to the measured limiting current, whereas in a solution with 100-fold excess electrolyte the current is diffusion-limited (i.e., T ) D in eq 4). Steady-state voltammetry at the microelectrode brings selectivity to the investigation of ion transport because, in contrast with conductivity measurements, the signal is specific to the particular electroactive ion being studied. It also brings sensitivity, because small changes in the transport behavior of the electroactive species are measured as reproducible changes in the height of the limiting current plateau.9 These aspects are evident in Figure 4, which shows voltammograms for reduction of hydrogen ion in perchloric acid solution without added electrolyte (Figure 4a) and with a 100-fold excess of lithium perchlorate (Figure 4b). The increase in the magnitude of the limiting current at the lower electrolyte concentration is due almost entirely to changes in the transport of hydrogen ion to the electrode surface, as expressed through the value of T in eq 4 (about 3% of the increase arises from the effect of ion-ion interactions expressed in eq 1). In the voltammogram shown in Figure 4b, the transport is controlled by diffusion only. In the solution containing no supporting electrolyte, the steady-state limiting current (Figure 4a) contains equal contributions from migration and diffusion, as predicted by eq 5. Curve b occurs at a (50) Myland, K. J.; Oldham, K. B. J. Electroanal. Chem. 1993, 347, 49.

Figure 4. Steady-state voltammograms for the reduction of hydrogen ion in perchloric acid solution at a Pt microelectrode, r0 ) 11 µm: (a) [H3O+] ) 0.210 mM, no added electrolyte, T(H3O+) ) 1.70 × 10-4 cm2/s; (b) [H3O+] ) 0.210 mM, [Li+] ) 21 mM, T(H3O+) ) D(H3O+) ) 9.14 × 10-5 cm2/s.

Figure 5. Dependence of the measured limiting current on the amount of supporting electrolyte in simple perchloric acid solution. A Pt electrode with a radius of 11 µm was used. 0.598 mM < [H3O+] < 0.626 mM; 1 µM < [Li+] < 5 mM.

more negative potential because of the larger iR drop in the solution without electrolyte. This gradient diffusion may be contrasted with the self-diffusion which is determined in NMR measurements. Recall that in the case of polyelectrolyte solutions the gradient-diffusion and self-diffusion coefficients are the same.11 Figure 5 shows good agreement between eq 5 (solid line) and experimental points (open circles) for the reduction


Langmuir, Vol. 14, No. 1, 1998 209

Figure 6. Optical photograph of a polished Pt microelectrode with a diameter of 21 µm immersed in the deionized latex acid crystal. The green and blue iridescence results from Bragg diffraction of visible light from the ordered particle structure.

of hydrogen ion in perchloric acid containing LiClO4 as supporting electrolyte. The background concentration of adventitious electrolyte, corresponding to the lowest value of γ, was assumed to be 1 µM. In the following discussion of D-values in colloidal suspensions, the ratios D/D° are calculated from the ratio of measured limiting current in the suspension to the current, il, that would exist in simple strong acid solution for the same concentrations of hydrogen ion and salt, calculated from eqs 1, 4, and 5. This current ratio is taken as equal to the diffusion coefficient ratio. In eq 1, the value of the hydrogen ion diffusion coefficient at infinite dilution, D∞ ) 9.30 × 10-5 cm2/s, is calculated from its ionic conductivity at infinite dilution.49 In some cases (e.g., Figure 7) the value of il for strong acid without colloid is measured directly in a parallel experiment. Implicit in this calculation of D/D° is the assumption that migration contributes equally to the transport of ions in the perchloric and macroionic latex acids. This is not intuitively obvious. Our experience with PSSA, however, suggests this approach is valid, since the NMR and voltammetric measurements give the same results for D/D°.11 Voltammetric Measurement of Hydrogen Counterion Diffusion in the Latex Acid Suspension. Figure 6 is a photograph of a platinum microelectrode with a radius of 10.5 µm immersed in the deionized latex suspension. The photograph was taken by placing a drop of the deionized latex on the electrode surface and mounting it on the stage of an inverted microscope equipped with a camera. The diameter of the electrode is more than 200 times larger than the diameter of the particles in suspension. The particles are too small to resolve optically, but the green and blue iridescence indicates an ordered structure. We first discuss the voltammetric results obtained from these deionized, iridescent latex suspensions at various particle volume fractions.

Figure 7. Steady-state, transport-limited currents for the reduction of hydrogen ion in perchloric acid and in the latex acid. The hydrogen ion concentration range in the latex corresponds to a particle volume fraction of 0 < φ < 2%. Perchloric acid slope ) 39.23 nA/mM; r2 ) 0.999. Latex acid slope ) 0.9596 nA/mM; r2 ) 0.986. The ratio of the slopes, R, is 0.024.

At a given volume fraction of particles, the hydrogen ion concentration in the sample (c of eq 4) is known from the strong acid end point of the conductometric titration (Figure 2). Figure 7 shows a concentration calibration plot for the reduction of hydrogen ion in the deionized latex and in a solution of perchloric acid without added electrolyte. The highest concentration of hydrogen ion corresponds to a particle volume fraction of almost 2%. In these systems without electrolyte the currents are enhanced by migration. The slope of the transport-limited current against concentration is proportional to the transport coefficient of the hydrogen ion. Assuming that the currents measured in each system contain equal contributions from migration, the ratio of the slopes is D/D° for hydrogen ion in the deionized latex, which has the value 0.024. Thus, gradient diffusion of hydrogen ion

210


Figure 8. Steady-state voltammograms taken in the latex at a Pt disk microelectrode with a radius of 10.5 µm. Oxidation of 0.500 mM TEMPO in the latex at a volume fraction of 2.22%: (a) deionized (no added electrolyte), D(TEMPO) ) 5.16 × 10-6 cm2/s; (b) [LiClO4] ) 124 mM, (γ ) 100), D(TEMPO) ) 5.82 × 10-6 cm2/s. Reduction of 0.260 mM H3O+ in the latex at a volume fraction of 0.474%: (c) deionized (no added electrolyte, γ ≈ 0.03), T(H3O+) ) 6.70 × 10-6 cm2/s, D(H3O+) ) 3.35 × 10-6 cm2/s; (d) [LiClO4] ) 26.46 mM (γ ) 100), T(H3O+) ) D(H3O+) ) 8.18 × 10-5 cm2/s.

in the macroionic latex acid is only 1/40th as fast as that in simple strong acid solution. This is a much stronger reduction than typically observed in most polyelectrolyte systems, for which the D-value of the counterion rarely is less than 25-35% of that observed in simple solution.34 In the simple acid solution the perchlorate anions diffuse freely; in contrast, the sulfonate anions in this latex suspension are bound covalently to the particle, which does not move on the time scale of the experiment. A Stokes-Einstein argument shows that the diffusion coefficient of the macroion is at most 0.05% of the hydrogen ion diffusion coefficient in simple solution.51,52 Therefore, the suppressed transport of counterion in the latex suspension suggests that the counterion is electrostatically accumulated by the latex particle. (In the present range of particle volume fractions, the diffusion coefficient should not be reduced due to excluded volume effects.4) Hence the observed reduction of the limiting current should be of purely electrostatic origin. To support this hypothesis, we use the molecule 2,2,6,6tetramethyl-1-piperidinyloxy (TEMPO) as a probe of all effects other than electrostatic that might affect the transport of species to the electrode (e.g., viscosity). Since this is a neutral molecule, electrostatic interactions with the particles and migration are precluded. Figure 8a and b shows currents for the oxidation of TEMPO in the ionexchanged latex at a volume fraction of 2.22% before and after the addition of electrolyte. At this particle volume fraction, the iridescent deionized latex is visibly more viscous than it is at volume fractions below 1%; adding electrolyte drastically reduces the bulk viscosity. The limiting currents for the oxidation of TEMPO, however, are little affected by the addition of electrolyte. This result may be contrasted with the currents shown in Figure 8c and d for the reduction of hydrogen counterion in the deionized latex and in the same latex with a 100-fold excess of supporting electrolyte. Currents in the deionized latex are suppressed relative to currents in the suspension containing excess electrolyte. Thus, changes in the current for the reduction of hydrogen ion in the latex are due only to changes in the electrostatic interaction of the counterion with the particles. (51) Ross, S.; Morrison, I. D. Colloidal Systems and Interfaces; John Wiley & Sons: New York, 1988. (52) van de Ven, Th. G. M. Langmuir 1996, 12, 5254.

Roberts et al.

Figure 9. Transport-limited currents for the reduction of hydrogen counterion in the latex at a volume fraction of 0.474%; [H3O+] ) 0.260 mM: (a) deionized (1 µM background); (b-e) [Li+]/mM ) (b) 0.101; (c) 0.261; (d) 0.861; (e) 8.461.

Figure 10. Influence of LiClO4 on the diffusion coefficient of the hydrogen counterion in the latex at a volume fraction of 0.237%. The observed diffusion coefficient of hydrogen ion, D, is calculated from transport-limited currents (Figure 9) and eqs 4 and 5. The diffusion coefficient of hydrogen ion in HClO4 solution, D°, was calculated according to eq 1.

Figure 8 shows the extremes of the range of electrolyte concentration used in these experiments (no added electrolyte and 100-fold concentration ratio of electrolyte to hydrogen counterion). The influence of smaller changes in salt concentration on the limiting current is shown in Figure 9. Small volumes of electrolyte solutions were added to the latex, keeping the particle volume fraction constant. Each limiting current shown in Figure 9 contains a different contribution from migration, which depends on the concentration of LiClO4. Normalized diffusion coefficients, D/D°, arising from currents such as the ones shown in Figure 9, are shown in Figure 10. As the arrow in the figure indicates, the iridescence of the latex was lost after the first addition of electrolyte. The fact that the gradient diffusion of the counterion apparently is insensitive to this first-order phase transition suggests that the diffusion is not affected by the degree of correlation between the particles under these conditions.53 However, this point must be investigated much more thoroughly before this conclusion is justified. As electrolyte is added, the diffusion increases, and at sufficiently large concentrations of added electrolyte, log γ ∼ 2, hydrogen ion diffuses freely and D/D° approaches unity. Deviations of D/D° from unity at higher electrolyte concentrations can be ascribed to uncertainty in the titrated charge and artifacts such as flocculation (53) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: San Diego, CA, 1992.


Langmuir, Vol. 14, No. 1, 1998 211

On the basis of the cell model, a theory based on irreversible thermodynamics has been developed and applied to calculate the self-diffusion coefficient of small charged species in solutions containing macroions.3,4 This model assumes a linear gradient in the concentration of labeled ions across the cell in the z-direction (Figure 11) along with a compensating gradient of unlabeled ions such that the ions are in equilibrium with the particle. Since there is no gradient in the electrochemical potential of the ions, the resulting diffusion coefficient of labeled ions calculated from the theory is a self-diffusion coefficient. This model is analogous to the physical situation of spinlabeled ions present in the PFG-SE NMR experiment. Following Jo¨nsson et al.,4 we have calculated the selfdiffusion coefficient of monovalent counterions according to

Ds ) D0,sc(b) U(b)/〈c〉

Figure 11. Cell model. The latex particle is modeled as a charged sphere of radius a centered at the origin of the coordinate system. The particle is enclosed with a spherical cell of radius b with a volume given by the total volume of the suspension divided by the number of particles in the suspension. The cell is electrically neutral, containing the particle and all of its accompanying counterions. This figure is drawn to scale for a suspension with a particle volume fraction of 0.237%. Counterion diffusion is calculated in the volume bounded by the particle and the cell boundary, a < r < b.

of the latex. These points will be the subject of future work. The results of Figure 10 are next examined in light of existing theories. Cell Model: Solution of the Nonlinear PoissonBoltzmann Equation. An accurate description of counterion diffusion in the latex over the entire range of electrolyte concentrations used in these experiments is provided by solving the (nonlinear) Poisson-Boltzmann equation for the local electrostatic potential in the region bounded by the cell shown in Figure 11. The physical model is that of a stationary particle at the center of a spherical cell with a volume equal to the total volume of the suspension divided by the total number of particles. Each cell is electrically neutral, containing one particle and all of its accompanying counterions. Once the electrostatic potential (and therefore the ion concentration profile) is determined in the region bounded by the cell, the self-diffusion coefficient of the ion in the presence of the known electric field can be calculated from a model that employs irreversible thermodynamics. The Poisson-Boltzmann (PB) equation predicts reasonably the electrostatic field and the distribution of small ions outside a macroion, provided that only univalent small ions are present in the aqueous solution. The cell model as described above is frequently used together with the PB equation for the description of systems with finite concentrations of macroion. Within the PB cell model, the electrostatic potential φ(r) is given by

∑i zici(b) exp[-zieφ(r)/kT]

0r∇2φ(r) ) -e

(6)

with the boundary conditions φ(b) ) φ′(b) ) 0 and φ′(a) ) -σ/0r, where b and a are the radii of the cell and the macroion, respectively, zi is the valency of species i, ci(r) is the concentration of species i at r, σ is the surface charge density of the macroion, and the other symbols have their usual meaning.

(7)

where D0,s is the self-diffusion coefficient at zero electrostatic potential (i.e., 9.30 × 10-5 cm2/s for hydrogen ion), c(b) is the counterion concentration at the cell boundary, and 〈c〉 is the number of ions divided by the total volume of the cell. In our application with a spatially independent diffusion coefficient, U(r) satisfies the differential equation

rU′(r) ) 2 - U(r) - U2(r) - rU(r)(ln(c(r)))′

(8)

where U(a) ) 0. Obviously, we only need to know the (unnormalized) radial density profile of the counterions to calculate U(b) needed in eq 7. Equation 8 was obtained by combining the continuity equation in spherical coordinates under steady-state conditions with Fick’s first law for the diffusion of labeled ions and by minimizing entropy production under the constraints of nonzero concentration and flow of ions at the cell boundary, r ) b. The quantity Ds was calculated from eq 7 as a function of the amount of added 1:1 electrolyte by solving eq 8. The counterion distribution profile c(r) was calculated from the PB equation using the parameters for the present experimental system at a volume fraction of 0.237%: σ ) -8.0 µC/cm2, a ) 44 nm, b ) 330 nm, T ) 298 K, and r ) 78.3. For low concentration of added electrolyte, γ < 0.23, the PB equation was solved traditionally by providing initial ion concentrations at r ) b and integrating the PB equation to r ) a. For the high-salt-concentration regime, γ > 0.23, the concentrations of the cations and anions are essentially equal at r ) b and the potential was assumed to behave as exp(-κr)/r at the cell boundary. For the purposes of the calculation, the Debye screening constant, κ is given by

κ ≡ ((e2/0rkT)

∑i ni0zi2)1/2

(9)

where ni0 is the bulk number density of added inert electrolyte ions of type i and zi is the corresponding valency. Figure 12 displays the calculated self-diffusion coefficients (solid line) as a function of the ratio of the concentration of added 1:1 electrolyte to the counterion concentration (γ). The small feature barely visible at log γ ) -0.64 arises from the switch to the limiting highsalt-concentration form. Experimental gradient-diffusion coefficients measured at several volume fractions are also included in Figure 12. Clearly the agreement is excellent; the calculated self-diffusion coefficients are very closely related to the experimentally determined gradient-diffusion coefficients.

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Roberts et al.

Figure 12. Calculated self-diffusion coefficients (see text) and experimental gradient-diffusion coefficients as a function of γ. Latex volume fraction/%: (*) 0.237; (4) 0.356; (O) 0.474; (0) 0.592; (×) 0.710.

Figure 13. Empirical theoretical treatment of hydrogen counterion diffusion in the latex using eq 10: (-) R ) 0.024. Measured D/D° for the hydrogen counterion in the latex at the following volume fractions (*) 0.237; (4) 0.356; (O) 0.474; (0) 0.592; (×) 0.710.

The calculated self-diffusion coefficient value without added electrolyte can be related to the effective charge on the particle by assuming a two-state model of “bound” and “free” ions. We define the fraction of “free” counterions, f, as the number fraction of counterions with an attractive electrostatic energy less than kT at 25 °C. The value of f predicted from the PB equation is 0.038. Assuming that the “bound” counterions neutralize the anionic sites on the particle, then the effective charge on these particles becomes Z* ) fZ ) 462. Semiempirical Model. In our previous work with solutions of the model linear polyelectrolyte, PSSA, we found that the diffusion behavior of the small counterion is determined only by the amount of added electrolyte and the extent to which counterion diffusion is suppressed in the deionized polyelectrolyte solution.9,10 The diffusion behavior of counterion in solutions of PSSA was described by the following simple, semiempirical equation

D/D° ) (R + γ)/(γ + 1)

(10)

where γ is the ratio of concentrations of 1:1 electrolyte to counterion. In eq 10, R is clearly the limiting value of D/D° as γ f 0. Since R can be measured directly in completely deionized solutions, as shown in Figure 7, eq 10 has no adjustable parameters. Equation 10 was developed from considering experimental data from solutions of PSSA plotted in the form of D/D° vs log γ, as shown in Figures 10, 12, and 13 for the latex. It does, however, have a physical basis in its

form. This is evident if eq 10 is written as D/D° ) Rmc + ms, where mc and ms are the mole fractions (with respect to the total number of counterions) of native counterion and added salt, respectively. Thus, the physical meaning of eq 10 is that the observed diffusion is attributable to the sum of diffusion of counterions which are free (equivalent to ms) and those bound, mc. The quantity R then expresses the extent to which the diffusion of counterions, in the absence of added salt, is hindered by interaction with the macroions. A value of R ) 0 describes the situation in which all the counterions must diffuse with the macroion (which has a negligibly small diffusion coefficient compared to that of the free counterion). A value of R ) 1 arises when diffusion of the counterion is unaffected by the macroion. This is similar to Lindman’s first law in the case of a macroion that is large enough to have negligible diffusion, relative to that of the free counterion.34,54 The advantage of casting the model in the form of eq 10 is that it encompasses the entire range of D/D° in one semilogarithmic plot. We wanted to know if this simple expression describes diffusion in the latex suspension as well as in solutions of PSSA. Figure 13 shows the same experimental data from Figure 12 plotted with the semiempirical theory from eq 10 (solid line) using a value of R ) 0.024 obtained from the data of Figure 7. Obviously, the semiempirical model also describes well the diffusion of counterions in the latex suspension. Since the R-value from the voltammetric experiments is analogous to the fraction of “free” counterions, f, it can be used to determine the effective charge on the particle. Using the same argument as with the PB results, the effective charge on the particle predicted from the voltammetric data in Figure 7 is Z* ) RZ ) 291. This agrees qualitatively with values obtained in previous studies on latexes.55,56 Conclusions We propose the use of voltammetry at a microelectrode as a general experimental method for investigating the diffusion behavior of small ions in solutions or suspensions containing highly charged macroions. In principle, at a fixed bulk concentration of electroactive ions in a macroionic system, any change in the magnitude of the limiting current plateau from that found in simple solution is a result of changes in the transport of electroactive ions to the electrode surface. This diminished transport-limited current is then related to the gradient-diffusion coefficient of the small ion by removing migrational transport via eqs 4 and 5. Once the gradient-diffusion coefficient has been determined for a given set of conditions, it is desirable to relate the suppressed diffusion to interactions between the counterion and the macroion. We have studied the diffusion behavior of hydrogen ion as counterion to the monodisperse sulfonated polystyrene latex over a wide range of concentration of inert electrolyte. Hydrogen ion gradient diffusion in the deionized latex was only 2.4% as fast as that in simple solution, under similar conditions of total concentration, suggesting that most of the counterion is strongly associated with the particles. This measurement allows an estimate of the effective particle charge, assuming the counterions that are “bound” to the particle do not contribute to the measured plateau current. Using these assumptions, the (54) Stilbs, P.; Lindman, B. J. Magn. Reson. 1982, 48, 132. (55) Alexander, S.; Chaikin, P. M.; Grant, P.; Morales, G. J.; Pincus, P.; Hone, D. J. Chem. Phys. 1984, 80, 5776. (56) Ito, K.; Ise, N.; Okubo, T. J. Chem. Phys. 1985, 82, 5732.


effective charge inferred from the experiment agrees qualitatively with those determined in previous studies. The addition of 1:1 electrolyte to the deionized latex increases the gradient-diffusion coefficient of the counterion; at a 100-fold excess of electrolyte to native counterion, the diffusion of the counterion is close to that found in simple solution. Deviations of D/D°-values from unity at high salt concentration are attributed to flocculation of the latex and uncertainty in the titrated charge. These results agree with two existing theories of ion diffusion in macroionic systems: the Poisson-Boltzmann cell model and a semiempirical model. The agreement of the semiempirical model with the diffusional behavior of hydrogen ion in the latex suspension and in solutions of linear polystyrenesulfonic acid suggests that the model does not depend on the geometry of the charged aggregate. We emphasize that the shape of the curve fit by eq 10 is only well defined if the diffusion coefficients of the ion in the deionized system and with large excess of electrolyte are measured accurately. The microelectrode makes this semiempirical treatment possible, because it yields reproducible, well defined limiting currents in highly resistive media. The ubiquity of electrochemical equipment and the relative ease with

Langmuir, Vol. 14, No. 1, 1998 213

which the experiment can be carried out provide a voltammetric complement to NMR studies of diffusion of small ions in macroionic systems. This study of gradient diffusion is being supplemented by NMR studies of self-diffusion of 205Tl+, which are now in progress. Voltammetric studies of gradient diffusion of Tl+ in the same system will test the identity of these two quantities for the latex. The good agreement between theory and experiment presented here for the latex, and for previous work on linear polyelectrolytes,9-13 suggests that measurement of the diffusion coefficient of the counterion in deionized solution is sufficient to predict its behavior over the entire range of electrolyte concentration. Acknowledgment. This work was supported by the National Science Foundation under grant number CHE 9208987. J.M.R. thanks M. Ciszkowska, J. O’Dea, and C. Keefer for stimulating discussions concerning this work. P.L. acknowledges the Swedish National Science Research Council (NFR) for financial support. LA9705726

Voltammetric Studies of Counterion Diffusion in the Monodisperse

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