Langmuir 1996, 12, 1947-1957
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Articles Self-Diffusion of Charged Polybutadiene Latex Particles in Water Measured by Pulsed Field Gradient NMR M. H. Blees,*,† J. M. Geurts,‡ and J. C. Leyte† Leiden Institute of Chemistry, Gorlaeus Laboratories, Leiden University, P.O. Box 9502, 2300 RA, Leiden, The Netherlands, and Laboratory of Polymer Chemistry, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Received April 13, 1995. In Final Form: December 14, 1995X The longtime self-diffusion of spherical polybutadiene latex particles in water was measured using the pulsed field gradient NMR technique. The particles were highly charged due to the adsorption of anionic surfactant. The volume fraction dependence of the self-diffusion was measured. Also the longtime selfdiffusion coefficient was measured as a function of the concentration of surfactant at a fixed latex volume fraction. Comparison of the results with theoretical predictions showed some qualitative agreement, but at a relatively low volume fraction, deviations from the theory were observed. The self-diffusion coefficient at infinite dilution obtained by extrapolating the experimentally observed values down to zero volume fraction was shown to be consistent with the self-diffusion coefficient obtained from dynamic light scattering.
Introduction In a colloidal dispersion the position of a colloidal particle is continuously changing. This “Brownian” motion of an isolated colloidal particle is caused by fluctuations in the force exerted on the particle by the molecules of the surrounding fluid. If the experimental time scale is much larger than the correlation time of the fluctuations in the mean force, the Brownian motion of a single colloidal particle can be described by the self-diffusion coefficient which (for spherical particles) can be obtained from the well-known Stokes-Einstein relation. At higher volume fractions the Brownian motion of a colloidal particle is influenced by direct and hydrodynamic interactions with other particles. Experimental results on the Brownian motion of colloidal particles are usually obtained by optical techniques. In this respect dynamic light scattering has been widely used.1 This technique is used to study the dynamics of the particles by examining the correlation function of either the intensity or the amplitude of the scattered light. The amplitude of the scattered light depends on the difference in refractive index of the solvent and the colloidal particle. In order to avoid contributions of multiple scattered light which would seriously complicate the analysis, for particles of typical colloidal size, the refractive index of the particle and supporting fluid must be quite closely matched. However, in order to obtain the * To whom correspondence should be addressed. Present address: Philips Research Laboratories, P.O. Box 80.000, 5600 JA Eindhoven, The Netherlands. † Leiden University. ‡ Eindhoven University of Technology. X Abstract published in Advance ACS Abstracts, February 15, 1996. (1) See, for instance: (a) Kops-Werkhoven, M. M.; Fijnaut, H. M. J. Chem. Phys. 1982, 77, 2242. (b) Kops-Werkhoven, M. M.; Pathmamanoharan, C.; Vrij, A.; Fijnaut, H. M. J. Chem. Phys. 1982, 77, 5913. (c) van Megen, W.; Underwood, S. M.; Ottewil, R. H.; Williams, N. St. J.; Pusey, P. N. Faraday Discuss. Chem. Soc. 1987, 83, 47. (d) Philipse, A. P.; Vrij, A. J. Chem. Phys. 1988, 88, 6459. (e) Ha¨rtl, W.; Versmold, H.; Zhang-Heider, X. Ber. Bunsen-Ges. Phys. Chem. 1991, 95, 1105. (f) van Megen, W.; Underwood, S. M. Phys. Rev. E 1993, 47, 248. (g) Na¨gele, G.; Zwick, T.; Krause, R.; Klein, R. J. Colloid Interface Sci. 1993, 161, 247.
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single particle (tracer) diffusion coefficient instead of the collective diffusion coefficient, a more complicated procedure must be used. In this procedure the dispersion must consist of a high concentration of (host) particles which are refractive index matched with the surrounding fluid. In this “matrix” a small number of particles are contained, of which the refractive index differs significantly from the solvent. However, in order to be able to compare experiments with theory, these (chemically different) particles should have a size and interaction potential identical to those of the host particles. Another optical technique which has been used to measure self-diffusion has been termed “fluorescence recovery after photobleaching” (FRAP).2 This technique requires labeling the colloidal particles with a fluorophore. An intense pulse of light is used to create a fringe pattern of irreversibly photobleached fluorophores across the sample. Because the fluorophores are associated with the colloidal particles, the diffusion of the particles causes the pattern to gradually disappear with increasing time. This technique is conceptually closely related with yet another optical technique which is known by the term “forced Rayleigh scattering”3 which is used to study diffusion.4,5 This technique also uses an intense pulse of light to create a fringe pattern. The light pulse changes the absorption characteristics of the molecules. The disappearance of the fringe pattern is monitored by a lowintensity reading beam of a wavelength which is close to the absorption frequency of the excited molecules. Nonoptical techniques that in principle allow selfdiffusion measurements in colloidal systems are radioactive tracer measurements, dynamic neutron scattering, and pulsed field gradient (PFG) NMR. In the PFG-NMR (2) See, for instance: (a) Gorti, S.; Plank, L.; Ware, B. R. J. Chem. Phys. 1984, 81, 909. (b) van Blaaderen, A.; Peetermans, J.; Maret, G.; Dhont, J. K. G. J. Chem. Phys. 1992, 96, 4591. (c) Imhof, A.; van Blaaderen, A.; Maret, G.; Mellema, J.; Dhont, J. K. G. J. Chem. Phys. 1994, 100, 2170. (3) Pohl, D. W.; Schwarz, S. E.; Irniger, V. Phys. Rev. Lett. 1973, 31, 32. (4) Qiu, X.; Ou-Yang, D.; Chaikin, P. M. J. Phys. France 1988, 49, 1043. (5) Dozier, W. D.; Lindsay, H. M.; Chaikin, P. M. J. Phys. (Paris) 1985, 3, c3.
© 1996 American Chemical Society
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technique a radio frequency pulse is applied to a sample (located in a magnetic field) which rotates the macroscopic magnetization of the nuclear spins into the transverse plane which is perpendicular to the main homogeneous magnetic field. Subsequently the positions of the nuclear spins are labeled by imposing a linear magnetic field gradient for a short time period. Then a second radio frequency pulse is applied which inverts the phase of the local magnetization in the sample. After the application of a second (identical) magnetic field gradient pulse, the spin echo is recorded. Diffusion of spins along the direction of the magnetic field gradient in the time span between the gradient pulses causes irreversible loss of phase coherence. The reduction of the amplitude of the spin echo caused by this loss of phase coherence is related to the self-diffusion coefficient of the molecules containing the nuclear spin of interest. It is also possible to obtain the diffusion coefficient of several components by separation according to chemical shift, which is achieved by Fourier transformation of the spin echo. Pulsed field gradient NMR has been used in the study of the dynamics of micellar systems6 and also in studying the diffusion of small molecules in colloidal latex systems.7 The macromolecular self-diffusion in polymer8 and polyelectrolyte9 systems has also been studied by this technique. However, to our knowledge, no PFG-NMR experiments on the diffusion of colloidal particles of polymeric or inorganic material have been reported (see, however, the experiments on spherical polystyrene microgels in solution10). Due to its high gyromagnetic ratio, the nucleus which is most suitable for PFG-NMR measurements is 1H. This would make latex systems in which the polymer chains, which together form the colloidal particle, usually have a high 1H content ideally suited for PFG-NMR experiments. However, it turns out that in most widely used latices, such as polystyrene and poly(methylmetacrylate), the local reorientational mobility of the polymer segments is very low. This low mobility causes a very fast decay of the NMR signal due to relaxation, which makes 1H PFGNMR measurements on these systems impossible. In this paper we report PFG-NMR measurements on polybutadiene latex particles in water. The polybutadiene polymer is sufficiently mobile to allow PFG-NMR measurements at room temperature. Here we study the selfdiffusion coefficient on a sufficiently long time scale that the particle has interacted with many other particles. In this regime the diffusion is described by the so-called longtime self-diffusion coefficient (DLS). The latex particles used in this report were highly charged due to the adsorption of anionic surfactant. The adsorption behavior of the added surfactant was studied by the method of conductometric titration. The volume fraction dependence of DLS was studied in different salt (surfactant) concentrations. Also DLS was measured as a function of the total amount of surfactant at a fixed latex volume fraction. (6) See, for instance: (a) Nilsson, P. G.; Wennerstro¨m, H.; Lindman, B. J. Phys. Chem. 1983, 87, 1377. (b) Nilsson, P. G.; Lindman, B. J. Phys. Chem. 1983, 87, 4756. (c) Jonstro¨mer, M.; Jo¨nsson, B.; Lindman, B. J. Phys. Chem. 1991, 95, 3293. (d) Wa¨rnheim, T. Colloid Polym. Sci. 1986, 264, 1051. (e) Cheever, E.; Blum, F. D.; Foster, K. R.; Mackay, R. A. J. Colloid Interface Sci. 1985, 104, 121. (7) (a) Jo¨nnson, B.; Wennerstro¨m, H.; Nilsson, P. G.; Linse, P. Colloid Polym. Sci. 1986, 264, 77. (b) Venema, P.; Struis, R. P. W. J.; Leyte, J. C.; Bedeaux, D. J. Colloid Interface Sci. 1991, 141, 360. (c) Blees, M. H.; Leyte, J. C. J. Colloid Interface Sci. 1993, 157, 355. (8) von Meerwall, E. D. Rubber Chem. Technol. 1985, 58, 527 and references therein. (9) Oostwal, M. G.; Blees, M. H.; de Bleijser, J.; Leyte, J. C. Macromolecules 1993, 26, 7300. (10) Fleischer, G.; Silescu, H.; Skirda, V. D. Polymer 1994, 35, 1936.
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Experimental Section Synthesis and Purification of the Latex. The polybutadiene latex used in this investigation was synthesized by emulsion polymerization. A stainless steel reactor was charged with 576.4 g of doubly distilled water which had been purged with nitrogen, 2.38 g of sodium carbonate (Janssen), 20.87 g of Aerosol MA 80 (Cynamide B. V., Rotterdam, The Netherlands), 0.50 g of sodium styrenesulfonate, and 1.36 g of sodium peroxydisulfate (both Fluka AG). Butadiene (DSM Chemicals, Geleen, The Netherlands) was distilled into a storage vessel and pumped into the reactor. The emulsion polymerization was performed at a temperature of 80 °C at an elevated pressure. The stirring speed was adjusted to 500 rpm. For experimental details concerning the emulsion polymerization see ref 11. The latex dispersion obtained was purified by dialysis against a 100-fold excess of daily refreshed deionized water. The water was stirred and constantly purged with nitrogen. The dispersion was contained in 30 mm seamless cellulose dialysis tubes (Visking). The dialysis membranes were cleaned before use by heating in dilute solutions of sodium hydrogencarbonate and EDTA (both Merck, p.a.) and were thoroughly rinsed with deionized water. After 4 days the conductivity of the water approximately equaled that of pure water, but the dialysis was continued up to a total period of 2 weeks. Subsequently, the latex was mixed with a Dowex 50-W4 sulfonic acid ion exchange resin and was stirred gently every 0.5 h by means of a glass rod. After a few hours the ion exchange resin was removed by filtering, and the procedure was repeated with a Dowex 2-X8 quaternary ammonium ion exchange resin. The ion exchange procedure as described above was repeated, after which the purified latex was stored at 4 °C under nitrogen atmosphere. Both ion exchange resins were thoroughly cleaned before use by the method described by Hul et al.12 The weight fraction of the purified latex was determined by drying a known quantity under reduced pressure at 50 °C in the presence of P2O5 (Sicapent, Merck). The latex was concentrated by exerting an external pressure on a dialysis bag filled with the latex resulting from the previous purification steps.13 An Anton Paar DMA 02C precision density meter was used to determine the density of a purified polybutadiene latex dispersion of known weight fraction (at 298.15 ( 0.03 K). The density of the latex particles was calculated by assuming the density of the latex dispersion to be a weighted average of the density of the colloidal particles and water. In this way we obtained F ) 895 ( 2 kg/m3 for the density of the latex particles, which is in excellent agreement with literature data for bulk mixed cis/trans-polybutadiene.14 Conductometric Titrations. The conductometric titrations were performed by coupling a Schott Gera¨te CG 855 conductometer to a Radiometer VIT90 Video titrator equipped with a 1 mL autoburet station. The conductometer operated at a frequency of 373 Hz at an electrical field of about 10 V/m. A modest flow of nitrogen, saturated with water vapor isothermal (25 °C) to the sample titrated, was administered to the space above the liquid/gas interface of the sample titrated. The duration of the titration was always about 1.5 h. Homogenization was maintained by means of a magnetic stirrer. The conductometer was calibrated by measuring dilute KCl solutions. Surfactants. The surfactants used were sodium dodecyl sulfate (SDS) (Serva, twice crystallized, analytical grade) and sodium decyl sulfate (SDeS) (Eastman Kodak, HPLC grade). Sodium dodecyl sulfate was Soxhlet extracted over a glass filter with n-heptane (Baker). After 2 days of drying under reduced pressure, the surfactant was recrystallized twice from methanol (p.a., Baker) and recrystallized three times from deionized water. The sodium decyl sulfate was of high purity and was used as received. Electron Microscopy. The particle size distribution of the polybutadiene latex was determined by calibrated transmission electron microscopy (TEM, JEOL 2000 FX). The particles were (11) Verdurmen, E. M.; Dohmen, E. H.; Verstegen, J. M.; Maxwell, I. A.; German, A. L. Macromolecules 1993, 26, 268. (12) Van den Hul, H. J.; Vanderhof, J. W. J. Colloid Interface Sci. 1968, 28, 336. (13) Hachisu, S.; Kobayashi, Y. J. Colloid Interface Sci. 1974, 46, 470. (14) Brandrup, J.; Immergut, E. Polymer Handbook, 2nd ed.; Wiley: New York, 1981.
Self-Diffusion of Polybutadiene Latex Particles
Langmuir, Vol. 12, No. 8, 1996 1949 chosen in such a way that G2 increased linearly. At the highest gradient value the echo attenuation was typically a factor e-4.
Results and Discussion
Figure 1. The time parameters of the stimulated echo experiment. stained by adding a solution of OsO4 to a dilute latex dispersion, which was then dried onto a TEM sample grid. The size of 1439 particles taken from four micrographs of different parts of the sample was determined. Static and Dynamic Light Scattering. Light scattering on highly dilute (volume fraction Φ < 10-4) latex dispersions was conducted to determine the size of the latex particles. The dispersions were filtered through a 0.22 µm filter (Millipore, GV) directly into a quartz cuvette. An argon ion laser (Spectra Physics, model 2000) operating at a wavelength of 514.5 nm was used. The intensity correlation function was computed by a correlator (ALV5000 Multiple Tau Digital Correlator ALV, Langen, Germany). A thorough discussion of the theory of dynamic light scattering (DLS) can be found in ref 15. Fourier Transform Pulsed Field Gradient NMR. Pulsed field gradient 1H self-diffusion measurements were performed using a Bruker AM 200 (4.7 T) wide bore magnet linked to an Aspect 3000 (Bruker) unit. The temperature was controlled by a gas thermostat using pressurized air to obtain a stability of 0.1 °C. All measurements were performed at 25 °C. The pulsed magnetic field gradient was generated by an actively shielded gradient coil capable of generating an amplitude (G) up to 7.2 T m-1 at a maximum current of 20 A. The axis of the magnetic field gradient was parallel to the main magnetic field. The gradient coil was designed and manufactured at Massey University, Palmerston North, New Zealand, by Prof. P. T. Callaghan and co-workers. Rectangular gradients, with carefully controlled rise and fall times (about 100 µS), were delivered by a Techron 7570 amplifier which was coupled to the spectrometer. The calibration of the gradient was performed by measuring the selfdiffusion of pure water (2.30 × 10-9 m2 s-1).16 We used 5 mm (outer diameter) NMR tubes, in which the filling height of the dispersion was always between 5 and 6 mm to ensure good gradient homogeneity over the sample volume (approximately 75 µL). Because of the large difference between the transversal (T2) and longitudinal (T1) relaxation rate of the 1H nuclei of the polybutadiene, the stimulated (pulsed field) echo sequence was used17 (Figure 1). The duration of the gradient pulses (δ) for the measurements described in this investigation was 2 ms. After the gradient pulses, a 1 ms delay was used before either applying the second RF pulse or acquiring the signal at τ1 + τ2. The time duration between the two gradient pulses (∆), for the measurements reported here, was always 253 ms. An exception is formed by measurements intended for the investigation of a possible time dependence of the self-diffusion coefficient, where ∆ was reduced up to a factor 5. The signal, starting from the top of the stimulated echo (τ1 + τ2), was Fourier transformed. No significant phase distortion indicative of pulse mismatch or residual gradients was observed. Therefore no (small) corrections on the duration of either of the gradient pulses were necessary. Because of the homogeneity of the main magnetic field, the constant background gradient (G0) can be neglected. The echo attenuation is now given by
AG ) A0 exp(-γ2G2δ2(∆ - δ/3)Dself)
(1)
where, in the experiment, the amplitude of the pulsed field gradient was varied. Typically, about 50 gradient values were used to measure the self-diffusion coefficient. These values were (15) Berne, B. J.; Pecora, R. Dynamic Light Scattering; Wiley and Sons: New York, 1976. (16) Mills, R. J. Phys. Chem. 1973, 77, 5. (17) Tanner, J. E. J. Chem. Phys. 1970, 52, 2523.
Hydrodynamic and Optical Particle Radius. The intensity correlation function of the dilute polybutadiene latex in deionized water was measured at angles from 40° up to 150° at intervals of 10° at a temperature of 24.8 ( 0.1 °C. The correlation functions were fitted to a secondorder cumulant expansion. The normalized second cumulant was typically 0.02, which indicates that the polydispersity of the colloidal particles was not very high. The resulting correlation time at a given angle was related to the self-diffusion coefficient. Within experimental error, the self-diffusion coefficient obtained was independent of the scattering angle. An analysis using a second-order cumulant expansion of the intensity correlation functions at different angles results in D ) 5.82 ((0.07) × 10-12 m2 s-1 (from a single exponential fit one obtains D ) 5.72 ((0.06) × 10-12 m2 s-1). Using the Stokes-Einstein relation, the radius (a) was found to be 41.9 ( 0.5 nm. In an analogous manner the hydrodynamic radius of the latex particles in a dilute dispersion containing 8.0 mM SDS was determined to be 42.0 ( 0.7 nm. For charged spheres the Stokes friction factor contains an additional term, which originates from the finite mobility of the ions in the electrical double layer surrounding the particle. The diffusion of the colloidal particle leads to an increase in the fluctuations of the local ion concentration, which gives rise to additional friction. This electrolyte dissipation term depends on the surface charge (Ψ0), the Debye screening length (κ-1), and the friction coefficient of the small ions.18 Theoretical predictions, confirmed by experiments, indicate that this term has a maximum near κa ) 1 and becomes very small for κa g 10.19,20,2a In the study of the volume fraction dependence of the self-diffusion as described below, the salt concentration is such (g8.2 mM) that the Debye screening length (κ-1) is smaller than 3.3 nm. Since we have κa > 12, we do not expect to observe effects originating from electrolyte dissipation. The optical particle radius was determined from the angular dependence of the scattering intensity. The scattering intensity was measured from 30° up to 150° at intervals of 5°. The scattering intensity of pure water measured at corresponding angles was subtracted from these intensities. Finally, the optical radius was obtained by fitting the resulting angular dependence to the Rayleigh-Gans-Debye form factor P(Θ).21 The variation of P(Θ) over the range of angles measured was about 30%. For the optical radius a value of 41 ( 1 nm was obtained. Therefore, within experimental error, the hydrodynamic and optical particle radius were equal. Adsorption of Surfactants on Polybutadiene Latex. The surface charge density of the purified latex dispersion was investigated by means of a conductometric titration using 0.01 M NaOH. A minimum of the conductivity with increasing titrant addition was observed. Assuming, at the minimum, the amount of added titrant to be equivalent with the number of sulfonic acid groups at the surface, the surface charge density was found to be 0.2 µC/cm2 (using the radius (42 nm) obtained from light (18) (a) Schurr, J. M. Chem. Phys. 1980, 45, 119. (b) van de Ven, T. G. M. Colloidal Hydrodynamics; Academic Press: London, 1989. (19) Ohshima, H.; Healy, T. W.; White, L. R.; O’Brien, R. W. J. Chem. Soc., Faraday Trans. 2 1984, 80, 1299. (20) Schumacher, G. A.; van de Ven, T. G. M. Faraday Discuss. Chem. Soc. 1987, 83, 75. (21) See, for instance: Kerker, M. Scattering of Light and Other Electromagnetic Radiation; Academic Press: New York, 1969; Chapter 8.
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Figure 2. Upper curve: conductivity (σ) of SDS in pure water as a function of the SDS concentration. Lower curve: conductivity of a polybutadiene latex dispersion (0.015 < Φ < 0.016) as a function of the SDS concentration.
scattering). This rather low surface charge is about 1 order of magnitude less than that anticipated on the basis of the recipe of the emulsion polymerization. Further evidence for the low surface charge was provided by the very low conductivity of the purified latex dispersion (∼15 µS/cm at Φ ) 0.02). It seems that only a minor part of the styrenesulfonate used in the emulsion polymerization was copolymerized with butadiene. Alternatively, most of the copolymerized styrenesulfonate may be located in the interior of the colloidal particles. Despite the low surface charge, the system was stable over a period of more than 6 months. This was probably due to the low ionic strength of the dispersion. With increasing salt concentration, the stability was poor. Since we intended to vary the ionic strength of the dispersion, surfactant was added to increase the surface charge. The adsorption of anionic surfactant at the particles was investigated by the method of conductometric titration. It is well known that the low-frequency electrical conductivity is a very sensitive tool for determining the critical micelle concentration (cmc) of an ionic surfactant in solution.22 At low concentrations the conductivity increases approximately linear with the surfactant concentration. As soon as the cmc is reached, the increase of conductivity with surfactant concentration changes markedly. This is due to the fact that the addition of micelles contributes less to the conductivity than the corresponding number of free (dissociated) surfactant molecules. If the concentration of surfactant in a latex dispersion increases, the conductivity also increases. In the latex system, part of the added surfactant is adsorbed at the surface of the colloidal particles. Therefore, if the intrinsic concentration of charges of the latex system is very small, the conductivity is significantly reduced with respect to the pure surfactant solution at the same surfactant concentration. If the concentration of surfactant in the water phase equals the cmc of the surfactant, micelles start to form and the increase of conductivity with surfactant concentration diminishes sharply.23 In Figure 2 the conductivity of SDS in pure water and in a polybutadiene latex dispersion is shown. The cmc of SDS in water (8.25 mM), obtained from the intersection of the two straight lines obtained by linear regression of parts of the curve above and below the cmc, is in good (22) See, for instance: (a) Mysels, K. J.; Otter, R. J. J. Colloid Sci. 1961, 16, 462. (b) Rehfeld, S. J. J. Phys. Chem. 1967, 71, 738. (23) Maron, S. H.; Elder, M. E.; Ulevitch, I. N. J. Colloid Interface Sci. 1954, 9, 89.
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Figure 3. Upper curve: conductivity of SDeS (σ) in pure water as a function of the SDeS concentration. Lower curve: conductivity of a polybutadiene latex dispersion (0.045 < Φ < 0.052) as a function of the SDeS concentration.
agreement with literature values.22 A marked change of slope can also be observed in the titration of the polybutadiene dispersion. Again, a numerical value can be obtained from the intersection of two straight lines. At this point, the concentration of SDS in the water phase of the dispersion equals the cmc. This assumption is accurate because of the very low intrinsic electrolyte concentration of the dispersion (,cmc), which therefore will not alter the cmc with respect to the value in pure water. The concentration difference of these two intersection points is therefore equivalent with the total amount of SDS adsorbed at the polybutadiene/water interface. Using the known particle volume fraction and size (a ) 42 nm), the total area of adsorption can be calculated. Finally the surface area per molecule at “saturation”24 can be obtained. This calculation yields 55 Å2 (surface charge density (σ) of -29 µC/cm2), which is typical of SDS adsorption at other nonpolar/water interfaces.25 In Figure 3 we examine the cmc and the adsorption behavior of SDeS on polybutadiene latex. The cmc of SDeS obtained from Figure 3 is much higher than the cmc of SDS. The value obtained (32.9 mM) is consistent with that reported in the literature.26 Because of the higher cmc, a more concentrated latex dispersion was used in the titration in order to increase the relative effect of the adsorption of the surfactant on the polybutadiene. Analogous to the titration of SDS, we obtain for SDeS a surface area per molecule of 62 Å2, which is equivalent to a surface charge density of -26 µC/cm2. Pulsed Field Gradient NMR Self-Diffusion Measurements. Often, for research purposes, commercially available latex such as polystyrene is used. Though industrially being an important polymer, not many experiments on polybutadiene latex have been reported. The reason for the use of polybutadiene latex in this investigation is its very low glass transition temperature (Tg ) -86 °C)11 compared to Tg ) 100 °C for polystyrene.14 Below Tg, the rotational motion of the polymer segments is largely inhibited. Therefore the spectral density J(ω) at zero frequency is very large. Since the expressions for (24) The term “saturation” should not be identified with a closely packed monolayer of surfactant molecules. It indicates that a maximum of the surface concentration of adsorbed surfactant is reached. This is due to the fact that the chemical potential of the surfactant in the water phase is essentially constant above the cmc. (25) Rosen, M. J. Surfactants and Interfacial Phenomena; John Wiley: New York, 1978. (26) Mysels, K. J.; Otter, R. J. J. Colloid Sci. 1954, 9, 89.
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Langmuir, Vol. 12, No. 8, 1996 1951
Figure 4. 1H spectrum obtained from Fourier transforming the stimulated echo in a PFG self-diffusion experiment.
the transverse relaxation rate (R2) contain a term proportional to J(0),27 R2 is too large to enable self-diffusion measurement. Above Tg, the motion of the polymer segments is relatively fast, which causes the spectral density to be spread over a much wider frequency domain. Since J(ω) is normalized, J(0) is lowered with increasing temperature, which leads to a decrease of R2. At room temperature, the R2 of the 1H nuclei in the polybutadiene phase is sufficiently reduced to enable pulsed field gradient self-diffusion measurement on the latex particles. In order to study the volume fraction dependence of the self-diffusion in charged latex dispersions, we used the following method. A concentrated polybutadiene dispersion was titrated with surfactant solution up to the characteristic point (Figures 2 and 3), where the particle surface is fully saturated with surfactant and the surfactant concentration in the water phase equals the cmc. Lower volume fractions were prepared by mixing part of this concentrated dispersion with a surfactant solution, of which the concentration equaled the cmc. In this way, both the surface charge and the salt (surfactant) concentration were independent of the latex volume fraction. Samples for studying the effect of the surfactant concentration on the self-diffusion (at constant Φ) were prepared in the following way. Two stock dispersions were prepared, one without surfactant and the other with a high surfactant concentration. The volume fraction of latex particles was equal for both dispersions. Dispersions at intermediate surfactant concentration were prepared by simply mixing known amounts of both stock solutions by weight. In Figure 4, the 1H spectrum obtained from Fourier transforming the stimulated echo in a self-diffusion measurement is shown.28 The system was polybutadiene latex saturated with SDS, where the surfactant concentration in the water phase was 8.2 mM. Two polybutadiene resonances were observed at 2.0 and 5.2 ppm, in agreement with the data on cis-polybutadiene.29 Estimates of the longitudinal relaxation rate of both resonances were obtained by determining the “zero” of the peaks following an inversion recovery sequence (T1,2.0 ppm ≈ 0.25 s, T1,5.2 ppm ≈ 0.3 s). The transverse relaxation rate was estimated from the reduction of the amplitude of the resonances caused by the increase of the time delay following the gradient pulses (both peaks T2 ≈ 2.5 ms). Already at low gradients the water signal was completely suppressed. This is due to the fact that the self-diffusion (27) Abragam, A. Principles of nuclear magnetism; Oxford University Press: Oxford, U.K., 1961. (28) In samples with a large filling height (about 30 mm), both peaks showed a much smaller line width (about 0.3 ppm at half peak height). In that case water suppression was incomplete due to the larger range of the RF coil with respect to the gradient coil. (29) English, A. D.; Dybowski, C. R. Macromolecules 1984, 17, 446.
Figure 5. Echo-attenuation plot obtained from the integration of the 2.0 ppm peak of a PFG-NMR experiment on polybutadiene latex (Φ ) 0.144). Full line: fit to eq 1.
coefficient of pure water is at least 2 orders of magnitude higher than the diffusion coefficient of the latex particles. At the highest latex volume fractions used here (∼0.15), the obstruction effect of the colloidal particles reduces the self-diffusion of the water by less than 10%.30 Though a significant amount of surfactant was contained in the system, resonances originating from the surfactant (mainly around 1.2 ppm) could only be observed at low gradient values. This probably indicates that the self-diffusion of the surfactants is much faster than the diffusion of the latex particles. Therefore, the value of the lowest gradient value of the echo attenuation curve was chosen such that no water or surfactant signal could be observed. The echo attenuation curves obtained from separate integration of the two polybutadiene peaks could always be well described by eq 1 (Figure 5). This indicates that the polydispersity of the latex particles was sufficiently low to enable a description of the self-diffusion with a single coefficient (see further). Though the standard error of the self-diffusion coefficient obtained from a nonlinear least squares analysis of the echo-attenuation curves using eq 1 was usually about 0.5%, the reproducibility after changing the sample was about 1%. As expected, within experimental error, the diffusion coefficients of both polybutadiene peaks were identical. Therefore always the mean value of both self-diffusion coefficients was used in further analysis. Another point which must be examined is the distance scale of the self-diffusion experiment. The echo-attenuation measured is directly related to the Brownian motion of the 1H nuclei of the polybutadiene polymer. Therefore there are three distinct diffusion processes which may, in principle, contribute to the echo-attenuation: (i) The diffusion of polymer chains or segments inside the latex particle. (ii) The rotational diffusion of the latex particle as a whole. (iii) The center of mass self-diffusion of the colloidal particle. If the conditional probability distribution for the b0, b r, t) is a Gaussian function, displacement of 1H nuclei P(r eq 1 is satisfied, and the diffusion coefficient obtained from this equation may be identified (for an isotropic system) with the mean square displacement,
〈(r b-b r0)2〉 ) 6Dt
(2)
where t is the diffusion time. In the self-diffusion (30) (a) Jo¨nsson, B.; Wennerstro¨m, H.; Nilsson, P. G.; Linse, P. Colloid Polym. Sci. 1986, 264, 77. (b) Venema, P.; Struis, R. P. W. J.; Leyte, J. C.; Bedeaux, D. J. Colloid Interface Sci. 1991, 141, 360. (c) Zwetsloot, J. P. H. Ph.D. Thesis Leiden University, The Netherlands, 1994.
1952 Langmuir, Vol. 12, No. 8, 1996
measurements the diffusion time (∆ - δ/3) was set to 0.25 s. As mentioned before, the fit of the echo-attenuation did not show any systematic deviations from eq 1. Therefore in the measurements described below, the rms displacement can be calculated from eq 2 and was between 1.5 and 3 µm. On the one hand, the typical time scale for self-diffusion of a polymer chain over a distance scale corresponding to the size of a latex particle is unknown. On the other hand, the correlation time for the rotational diffusion of an isolated sphere can be calculated from the Debye relation: τr ) 4πηa3/3kT, where η denotes the viscosity of the solvent. For spheres with a radius (a) of 42 nm in water (25 °C) this relation gives τr ) 67 µs. Hydrodynamic interactions will increase the correlation time at higher volume fractions,31 but τr will still be shorter than the time scale of the diffusion measurement. It is obvious that the combined action of rotational diffusion of the latex particle and possible internal diffusion of polymer chains or segments within the particles will result in a rms displacement of the 1H nuclei of about one particle radius. Since the measured rms displacement (between 1.5 and 3 µm) is much larger than the particle radius, rotational and internal motions were neglected with respect to the translational self-diffusion of the particle. Experimentally this was confirmed by the observation that the selfdiffusion coefficient was independent of the diffusion time (∆ - δ/3) between 0.05 and 0.25 s. If internal and rotational motions would play a significant role, the measured self-diffusion coefficient would generally be expected to depend on the diffusion time. Particle Size Distribution. A transmission electron micrograph of the stained polybutadiene latex particles is shown in Figure 6. The particle size distribution obtained from the micrographs is given in Figure 7. From the distribution a number average diameter of 61 nm and a relative standard deviation σ of 13% were obtained. The distribution can reasonably be described by a Gaussian function (Figure 7) yielding a mean diameter of 62 nm (σ ) 12%). The distribution given in Figure 7 was used to simulate the PFG-NMR echo attenuation curve for independent latex particles. Assuming the relaxation rates R1 and R2 of the 1H nuclei to be independent of particle size, the self-diffusion coefficient obtained by PFG-NMR is weighted by the number of 1H nuclei of the particle which in turn is proportional to the particle volume (∝r3). The echo attenuation curve was numerically simulated down to an attenuation factor of e-4 and showed an rms deviation from a fit to eq 1 of 0.11% (with respect to the amplitude at zero gradient). Since this rms deviation is less than one-third of the rms deviation of the measurement given in Figure 5, it follows that these deviations are small with respect to the effects of noise and slow temperature fluctuations on the experimental PFG-NMR results. The diameter obtained from a fit of the simulated data to eq 1 (64 nm) was 5% higher than that calculated from the number average diameter. The intensity correlation functions obtained from dynamic light scattering are weighted approximately (because the form factor P(Θ) is not rapidly varying for the distribution of radii (Figure 7)) by the squared particle volume (∝r6). A second-order cumulant fit to a simulated correlation function resulted in a diameter (67 nm) which was 9% higher than that calculated from the number average diameter. The diameter obtained from dynamic light scattering (84 nm) is significantly higher than that expected from (31) Jones, R. B. Physica A 1989, 157, 752.
Blees et al.
Figure 6. TEM micrograph of the polybutadiene latex particles. The bar corresponds to 200 nm.
Figure 7. Distribution of the particle diameter (d) of the latex obtained from electron microscopy. Full line: fit to Gaussian function.
the electron microscopy data (67 nm). It is now known whether the size of the particles is affected by the staining procedure. The staining and drying of the latex onto the sample grid does clearly lead to aggregation of the particles (Figure 7). It should be noted that even in the case of characterization of (nonstained) solid silica spheres, the size obtained from TEM is systematically smaller than the optical and hydrodynamic radius. This effect is often attributed to the intensity of the electron beam and/or the high vacuum.2b Despite the uncertainty in the absolute magnitude of the size, the TEM micrographs provide a clear picture of the polydispersity. In the calculation of the surface charge density of the latex particles as given above, the radius obtained from light scattering (which is weighted with r6) was used to
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Langmuir, Vol. 12, No. 8, 1996 1953
calculate the surface area. The relative particle size distribution can be used to obtain a better estimate of the surface area, which alters the calculated surface charge density (which is weighted with r2). This leads to surface charged densities of σSDS ) -25 µC/cm2 and σSDeS ) -22 µC/cm2 (instead of -29 µC/cm2 and -26 µC/cm2, respectively). However, the analysis given below proved to be rather unsensitive to such variations of the surface charge density. Particle Interactions. Because the surface charge density of the particles was obtained from the adsorption experiments, we can now give a quantitative discussion of the interaction between the latex particles. The interaction between the charged latex particles consists of both an attractive and a repulsive part. The repulsive part, which originates from electrostatic interactions, dominates at short distances. If the polybutadiene latex particles would be forced together, they would probably coalesce to form one larger spherical particle. However in the experiments discussed here, the ionic strength is rather low and the surface of the particles is highly charged. Therefore at very short range the electrostatic interaction energy is orders of magnitude higher than the thermal energy kT. The system is now stable, and the potential (V(r)) at contact can be effectiven taken as infinite.
V(r) ) ∞
(r < 2a)
(3)
The Debye length (κ-1) plays a central role in the description of the particle interactions. For simplicity, because of the nature of the added salt (surfactant), the equations below will be written in a form which is specific to 1-1 electrolytes. Because we are studying relatively high volume fractions of small particles with a high surface charge density, the counterions originating from the adsorbed anionic surfactant must be taken into account in calculating κ-1,32 which, for negatively charged particles, can be obtained from
κ2 )
[
]
e2 3σΦ 2NAC r0kT ae(1 - Φ)
(4)
where C denotes the added salt (surfactant) concentration (mol/m3) in the aqueous phase, NA is Avogadro’s constant, e is the elementary charge, 0 is the electrical permittivity of the vacuum, and r is the relative electrical permittivity of the continuous phase. Because of the high charge, the dimensionless surface potential (ψ ) ψ0e/kT) is higher than unity and usual approximation for the double-layer repulsion break down.33 Therefore, to describe the electrostatic repulsion, we use the approximation for the double-layer repulsion given by Honig and Mul34 (their eq 22) written as
VR )
(
32πr0(kT)2a
[
e
2
1 γ2e-κh 1 - e-κh + 2
γ4e-2κh (1 + γ2) -
(
)
])
γ2 - κh 2(1 + γ2)
(r > h) (5)
where h denotes the shortest distance between the spheres (r - 2a) and γ ) tanh(ψ). Equation 5 differs by less than 1% from a numerical solution (based on the Poisson(32) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, U.K., 1989. (33) See, for instance: Hunter, R. J. Foundation of Colloid Science; Oxford University Press: Oxford, U.K., 1986. (34) Honig, E. P.; Mul, P. M. J. Colloid Interface Sci. 1971, 36, 258.
Boltzmann equation, using the Derjaguin approximation) for the double-layer repulsion if κh g 3.34 It was pointed out35 that the calculations of the doublelayer interactions by Honig and Mul ignored the effects of the electric field within the interacting bodies induced by the overlapping of their double layers. These effects have been taken into account in ref 36. For the doublelayer interaction between two plates, the relative magnitude of this effect depends on the parameter ′r/rκd, where ′r is the relative electrical permittivity of the plates and d is the thickness of the plates, and is negligible if ′r/rκd e 0.01 (see Table 5 of ref 36). In view of the ratio of relative electrical permittivity of polybutadiene and water (∼0.03), and the large size of the latex particle with respect to the Debye length (κa > 12), this effect can be neglected. ψ can be obtained from the known surface charge density σ, using
σ)
r0kT 4 ψ ψ + κ 2 sinh tanh e 2 κa 4
[
( )]
()
(6)
which gives σ to within 5% for κa > 0.5 for any surface potential.32 In principle, an accurate description of the van der Waals attraction between the latex particles can be obtained from the Lifshitz theory.37 However, this requires knowledge of the dielectric absorption spectrum at all frequencies, which is difficult to obtain for mixed cis/trans-polybutadiene. Furthermore, at the intersphere distance where the electrostatic interaction falls significantly below kT and van der Waals attraction becomes important, the tractability of the problem is deteriorated further by retardation effects, which set in roughly above 5 nm.38 The nonretarded van der Waals interaction between two spheres is given by
VA(r) ) -
(
(
))
A 2a2 2a2 r2 - 4a2 + + ln 6 r2 - 4a2 r2 r2
(7)
where A is the Hamaker constant. The Hamaker constant of polybutadiene across water is not known, but for the chemically closely related polyisoprene (which also nicely matches the refractive index of polybutadiene) A ) 7.4 × 10-21 J was obtained.39 In water the (nonretarded) zero frequency contribution to A, which for organic substances in water is almost 3/4kT (≈2.8 × 10-21 J), is screened roughly according to e-2κh by the presence of salt.38 The distance at which the electrostatic inter action is sufficiently reduced for the van der Waals forces to become important is about 9 nm at Φ ) 0.11 in 32.5 mM SDeS (κ-1 ) 1.4 nm) and about 25 nm at Φ ) 0.005 in 8.2 mM SDS (κ-1 ) 3.3 nm). Therefore at interparticle distances where the van der Waals forces become important, the zero frequency contribution to A is essentially fully screened. Another important point which can be made now is that the van der Waals interaction at these typical distances is dominated by the bulk polybutadiene of the latex particles instead of the adsorbed layer, because these distances are much larger than the thickness of the adsorbed surfactant layer.38 Because retardation effects always lead to a reduction of (35) Ohshima, H.; Kondo, T. J. Colloid Interface Sci. 1989, 133, 523. (36) Ohshima, H. Colloid Polym. Sci. 1976, 254, 484. (37) See, for instance: Mahanty, J.; Ninham, B. W. Dispersion Forces; Academic Press: London, 1976. (38) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1991. (39) Hough, D. B.; White, L. R. Adv. Colloid Interface Sci. 1980, 14, 3.
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Blees et al.
Table 1. Volume Fraction Dependence of DSL of the Latex Particles in 8.2 mM SDS and 32.5 mM SDeS 8.2 mM SDS Φ 0.00521 0.0110 0.0197 0.0306 0.0403 0.0490 0.0598 0.0701 0.0820 0.0944 0.106 0.117 0.127 0.137 0.147
D
(10-12
32.5 mM SDeS m2
6.28 6.17 6.09 5.83 5.61 5.51 5.33 5.11 4.85 4.69 4.37 4.21 4.13 3.84 3.59
Φ
D (10-12 m2 s-1)
0.0112 0.0215 0.0365 0.0455 0.0570 0.0681 0.0793 0.0875 0.0977 0.107 0.115
6.17 5.99 5.65 5.46 5.33 5.04 4.85 4.71 4.65 4.53 4.28
s-1)
Figure 8. DLS of SDS-saturated polybutadiene latex as function of the volume fraction. Full line: fit to eq 9.
the van der Waals forces, it is reasonable to use eq 7 with A ) 4.6 × 10-21 J as an upper limit for the attractive forces. Experimental Time Scale and Diffusion Regimes. The translational self-diffusion of a labeled colloidal particle in an environment of other colloidal particles can be divided into two different regimes.40 The first regime is characterized by the short time self-diffusion coefficient (DSS), which describes the motion of the colloidal particles over distances much shorter than the interparticle distance. On the one hand the time scale is much larger than the viscous relaxation time which characterizes the decay of the velocity correlation function of the colloidal particle. On the other hand the time scale is still much shorter than the structural relaxation or interaction time, defined by
ξ2 τI ) D0
(8)
which characterizes the time scale on which a colloidal particle diffuses a typical distance ξ, where ξ is of the order of the location of the main peak in the pair correlation function. The second regime is characterized by the longtime selfdiffusion coefficient which describes self-diffusion at times larger than τI. For concentrated systems ξ is of the same order of magnitude as a. There are some indications41 that the longtime limit is already reached at times when the particle has typically diffused over a distance of only one particle radius. From the known volume fraction and the particle radius, the average interparticle distance can be estimated, whereas the rms displacement can be obtained from eq 2. In all our measurements the ratio of the rms displacement and the estimated interparticle distance was larger than 7. These results, together with the experimentally observed independence of the measured diffusion coefficient at time scales between 0.05 and 0.25 s, indicate that the self-diffusion measured can be identified with the long-time self-diffusion coefficient which is denoted by DLS. Volume Fraction Dependent of DLS . The experimental results for the volume fraction dependence of DLS for SDS-saturated polybutadiene latex particles containing 8.2 mM SDS are presented in Table 1, together with the corresponding results for SDeS at 32.5 mM. These (40) Pusey, P. N. J. Phys. A 1975, 8, 1433. (41) van Megen, W.; Underwood, S. M. J. Chem. Phys. 1989, 91, 552.
Figure 9. DLS of SDeS-saturated polybutadiene latex as function of the volume fraction. Drawn line: fit to eq 9.
results are plotted in Figures 8 and 9, respectively. Both figures show a decrease of DLS with increasing volume fraction. Phenomenologically, the decrease of DLS seems to be reasonably described by a linear dependence on the volume fraction of the form
DLS ) D0(1 - kΦ)
(9)
In the case of the latex system containing SDS (Figure 8), a fit to eq 9 results in D0 ) 6.41 ((0.02) × 10-12 m2 s-1 and k ) 2.91 ( 0.03, whereas for the latex system containing SDeS we obtain 6.34 ((0.04) × 10-12 m2 s-1 and 2.83 ( 0.07, respectively. From the results of the numerical simulations using the particle size distribution (see above), one would expect that D0, being the value of DLS obtained from PFG-NMR extrapolated to zero volume fraction, would be 4% higher than the diffusion coefficient obtained by dynamic light scattering DDLS ) 5.82 ((0.07) × 10-12 m2 s-1. The actual difference is 9%, but in view of the large differences in time scale, concentration range, and DLS and PFG-NMR technique, the remaining difference of 5% is quite small. The volume fraction dependence of DLS can be compared with Batchelor’s result for particles with hard sphere interactions which is first order in Φ. This result was
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Langmuir, Vol. 12, No. 8, 1996 1955
confirmed by calculations of Cichocki and Felderhof42 and is given by
DLS ) D0(1 + (λA + Rs)Φ)
(10)
The coefficient λA which equals -1.83 is the first-order correction in the short time self-diffusion coefficient which describes the hydrodynamic slowing down of a colloidal particle in an equilibrium configuration of other particles. The coefficient Rs is the long-time correction due to the modification of the pair distribution of the interacting particles and equals -0.27. Though Rs is relatively small with respect to λA, there is an important contribution from hydrodynamics to this value, since it is well known that without hydrodynamic interaction Rs is -2.43 The value of the first-order coefficient for hard spheres (λA + Rs) is -2.10. This theoretical value is supported by some experimental evidence,44 where the experiments were compared with eq 10 up to volume fractions of 0.5. The accuracy of the measured self-diffusion coefficients is often rather limited for low volume fractions (e0.1) which hampers comparison with theory. In this range the PFGNMR data for the latex systems reported here are still rather accurate. However, the interaction cannot simply be described by hard sphere repulsion because of the strong electrostatic repulsion at short range and rather weak attractive interactions at longer distances. Moreover, the strength of the interaction is varying with the volume fraction (see eqs 4 and 5). Therefore it is not surprising that the experimentally observed values for the first-order coefficients k differ significantly from the theoretical predictions for hard spheres. There are a few quantitative theoretical predictions for DLS in systems with interactions other than hard sphere interactions.45-48 Also some numerical results have been obtained, in which semiempirical two-particle mobility tensors were used to describe hydrodynamics.49 In some of the theoretical descriptions of DLS, the repulsive interaction was modeled by an infinitely high potential at an effective radius (b) larger than the hydrodynamic radius (a).45,47 Other potentials which have been used are the square step/well potential (at b g a),48 attractive short range attraction described by a power law combined with hard core repulsion (at b g 1.1a),45 and repulsive or attractive interaction described by a screened Yukawa potential.46 Rescaling of the Particle Radius. In view of the rather steep decrease of the interaction potential at distances where the potential is of the order of kT, it seems appropriate to use the most simple approximation for DLS beyond the first-order hard sphere result. In that case the hydrodynamic interactions are described using the particle radius (a), whereas the repulsion is described by an increased effective hard sphere interaction radius (b).45,47 We use a thermodynamic rescaling procedure which is similar to procedures which have been used to describe equilibrium phase behavior in colloidal systems.32 The effective hard sphere radius b is chosen in such a way that the second virial coefficient calculated from a hard (42) Cichocki, B.; Felderhof, B. U. J. Chem. Phys. 1988, 89, 3705 and references therein. (43) Lekkerkerker, H. N. W.; Dhont, J. K. G. J. Chem. Phys. 1984, 80, 5790 and references therein. (44) van Megen, W.; Underwood, S. M. Langmuir 1990, 6, 35. (45) Jones, R. B.; Burfield, G. S. Physica A 1985, 133, 152. (46) Venkatesan, M.; Hirtzel, C. S.; Rajagopalan, R. J. Chem. Phys. 1985, 82, 5685. (47) Cichocki, B.; Felderhof, B. U. J. Chem. Phys. 1991, 94, 556. (48) Cichocki, B.; Felderhof, B. U. J. Chem. Phys. 1991, 94, 563. (49) van Megen, W.; Snook, I. J. Chem. Soc., Faraday Trans. 2 1984, 80, 383.
Figure 10. DLS as a function of the rescaled volume fraction. Full line and 0 denote theoretical and experimental results for the latex saturated with SDS. Dashed line and O denote the corresponding results of the SDeS system.
sphere system (with radius b) equals the corresponding coefficient calculated from the potential of the system of interest. Therefore, the effective radius can be obtained from ref 2b
∫2a∞(1 - e-V(r)/kT)r2 dr
b3 - a3 ) (3/8)
(11)
The pair interaction potential V(r) contains both electrostatic repulsion eq 5 and van der Waals attraction eq 7. At short distances V(r) is dominated by electrostatic repulsion. At distances where the contributions obtained from eqs 5 and 7 are equal, both contributions are much lower than kT (between 0.08kT and 0.02kT). At these distances (h between 12 and 32 nm), retardation leads to a large reduction of VA(r). Therefore, in a first approximation, we neglect the van der Waals attraction in the calculation of b. The value of b can be obtained by numerically performing the integration of eq 11. The resulting scaling factor b/a varies from 1.26 (for Φ ) 0.005 in SDS) down to 1.11 (for Φ ) 0.115 in SDeS). Obviously there are alternatives to eq 11 for chosing an effective hard sphere diameter. For instance, a more sophisticated choice could be obtained from the perturbation treatment by Barker and Henderson50 or Monte Carlo simulation. Alternatively, in a rather crude approximation the effective radius may be identified with the distance at which the potential equals kT. In fact, the latter method provides results which are virtually identical to the rescaling procedure using eq 11. Therefore we do not expect the results to depend sensitively on the details of the rescaling procedure. In Figure 10 the experimental results for DLS for both the SDS and the SDeS system are given as a function of the effective volume fraction Φ′ ) (b/a)3Φ. According to the calculations of Cichocki and Felderhof,47 the firstorder coefficient is no longer given by the hard sphere value (-2.10). Instead, the absolute value of this coefficient decreases in the range 1.016 < b/a < 1.6 (see Table II of ref 47). In order to be able to compare their results with our experiments, we have interpolated their numerical results for the first-order coefficient as a function of b/a. We observed that their data in the range 1.00 e b/a e 1.3 could be accurately represented by a fourthdegree polynomial which was obtained by least squares analysis. In Figure 10, the numerical results for DLS obtained in this way are plotted as a function of Φ′. For (50) McQuarrie, D. A. Statistical Mechanics; Harper & Row: New York, 1976; Chapter 14.
1956 Langmuir, Vol. 12, No. 8, 1996
the SDeS system the slope predicted is around -1.6, whereas the slope for the SDS system changes with Φ′ from -1.3 to -1.5. The increase of the slope predicted for the SDS system is due to the volume fraction dependence of b/a (which originates from the variation of κ-1 with Φ). The rescaled experimental data fall significantly below the theoretical predictions, and the slope for both the SDS and SDeS data is close to -2. However, it also seems that the rescaling procedure has introduced some concavity in the SDS results. Some qualitative agreement is obtained by the observation that in both theory and experiment DLS for the latex with SDS is significantly higher than that in the SDS system. The concave behavior for DLS in the rescaled SDS results is also correctly predicted. Van Blaaderen et al.2b used a similar rescaling procedure to describe the self-diffusion of colloidal silica particles in DMF. DLS was measured up to Φ′ ≈ 0.5, but the amount of data below Φ′ ≈ 0.1 was rather limited. In that case the rescaled experimental data also showed a larger volume dependence than expected from the theoretical predictions. They concluded that for higher volume fractions the effect of the hydrodynamics on the contribution which describes the modification of the pair distribution function is (much) less important than for low volume fractions. Our measurements (Figure 10) indicate that deviations from the first-order calculation are already significant at rescaled volume fractions larger than about 0.05. In view of this result, it may be interesting to note that in the “exact” calculations of the first-order coefficient the low density approximation (g(r) ) exp[-V(r)/kT]) of the radial distribution was used.42,47,48 It was also remarked that the hydrodynamic interactions show a rapid variation in the near range which causes the results to depend sensitively on the details of the direct potential.48 At finite volume fractions, a more realistic approximation of g(r), obtained for instance from the solution of the PercusYevick equation, shows large deviations from the low density approximation at contact (and short range) already at low volume fractions (30% at Φ ) 0.1). In view of these facts the range of validity of the exact results for DLS is probably very limited. The experiments seem to indicate that (at least for b/a e 1.25) the inclusion of higher order coefficients in the volume fraction, together with more accurate approximations of the particle pair correlation function, should lead to a dependence on the rescaled volume fraction which can roughly be described by a slope of -2. Effect of Surfactant Concentration. The effect of the concentration of SDS at a constant latex volume fraction (0.144) is shown in Figure 11. The concentration at which the concentration of SDS in the aqueous phase is equal to the cmc is indicated. At low SDS concentrations, DLS increases strongly with the SDS concentration. In the neighborhood of the cmc, the increase of DLS is much less pronounced. No abrupt change of DLS around the cmc is observed. If the concentration of SDS is lowered below the cmc, both the surface concentration and the concentration of SDS in the aqueous phase are reduced. Because the surface charge is almost entirely determined by the charge provided by the adsorbed SDS ions, the surface potential is also lowered. The reduction of the SDS concentration in the aqueous phase causes κ-1 to increase. On the one hand, the repulsion decreases if the surface potential drops, but on the other hand, the range of interaction is increased with the increase of κ-1. Since Figure 11 shows
Blees et al.
Figure 11. DLS as a function of the concentration SDS (Φ ) 0.144). Arrow: cmc in water phase. O: measurement. Dashed curves: DLS obtained from the calculations for the effective hard sphere model of Cichocki and Felderhof (lower curve adsorption isotherm (i), upper curve isotherm (ii). Full curves: DLS calculated using the rescaled particle radius assuming a volume fraction independent first-order coefficient of -2.
a continuous increase of DLS with increasing SDS concentration, it can be concluded that the latter effect dominates. In order to enable a quantitative comparison with theory, we can again rescale the particle radius. Since the SDS concentration is varied, this would in principle require knowledge of the full adsorption isotherm of SDS on polybutadiene. However, it turns out that the rescaled particle radius is not very sensitive to the adsorption equilibrium. We shall illustrate this by using two rather extreme adsorption isotherms to calculate the effective particle radius as function of the surfactant concentration. (i) Essentially all the SDS is adsorbed at the polybutadiene surface; the SDS concentration in the aqueous phase is negligible up to the point were the surface is saturated. If the amount of SDS is further increased, the SDS concentration in the aqueous phase must of course also increase accordingly. (ii) The surface concentration increases linearly from zero with increasing SDS concentration in the aqueous phase. This increase is continued up to the point where the surface is saturated and the concentration of SDS in the aqueous phase is equal to the cmc. In many cases the adsorption is intermediate between case i and ii. For instance, the Langmuir isotherm shows concave behavior with the surfactant concentration. Even if the adsorption shows more complicated behavior (for instance the adsorption of SDS on polystyrene51), the adsorption is often still intermediate between i and ii. Figure 11 shows the curves for DLS calculated using the rescaled radius according to isotherm i and ii, together with the theoretical first-order coefficient (again interpolated from Table II of ref 47 by a least squares polynomial fit). The bounds for DLS obtained in this way are quite close to each other. The experimentally measured diffusion coefficients show an increase with surfactant concentration as expected from the calculations. However, the experimentally measured diffusion coefficients are much lower than theory predicts. Moreover, the theory for DLS predicts a much smaller variation with the (51) Zhu, B.-Y.; Gu, T. J. Chem. Soc., Faraday Trans. 1 1989, 85, 3813.
Self-Diffusion of Polybutadiene Latex Particles
surfactant concentration than that observed experimentally. In view of the results as a function of the volume fraction, we also indicate (in Figure 11) DLS calculated by assuming the first-order coefficient to be -2. Although there remains a significant discrepancy, there is better agreement between theory and experiment. These findings are consistent with the experimental results of the volume fraction dependence of DLS. Summary and Conclusions We have shown that PFG-NMR can be used to accurately measure self-diffusion of the colloidal particles in an aqueous latex system. This technique does not suffer from the limitations of most optical techniques at higher volume fractions, where usually the index of colloidal particle and solvent must be quite closely matched to avoid contributions from multiple scattering or even to allow the light to penetrate the sample. This usually precludes diffusion measurement of colloidal particles in water at high volume fractions because the refractive index of water in the optical regime is significantly lower than that of almost any organic or inorganic material. In this investigation the experimental time scale was sufficiently long to allow the measured diffusion coefficients to be identified with the long-time self-diffusion coefficient (DLS). The latex particles were highly charged due to the adsorption of anionic surfactants. The surfactants used were sodium dodecyl sulfate (SDS) and sodium decyl sulfate (SDeS). In the study of the volume fraction dependence, the dilutions were prepared in such a way that the concentration of the surfactant in the aqueous phase always equaled its critical micelle concentration. Therefore the volume fraction dependence of DLS was investigated under conditions of constant surface charge and constant added salt (surfactant) concentration in the aqueous phase. However, especially for the SDS system, the electrostatic interaction was still volume fraction dependent due to the non-negligible amount of counterions originating from the adsorbed surfactant.
Langmuir, Vol. 12, No. 8, 1996 1957
The strong electrostatic repulsion causes DLS to decrease more rapidly with increasing volume fraction than DLS for hard spheres. A quantitative comparison with a theoretical model was made. In this theoretical model, an effective hard core radius larger than the hydrodynamic one was used.47 The first-order coefficient in the rescaled volume fraction obtained from the experimental results is about -2, whereas the theoretical model predicts a significantly less pronounced volume fraction dependence. It may seem tempting to suggest that hydrodynamics can be neglected in these charged systems, since a number of calculations have shown that without hydrodynamics the first-order coefficient is exactly -2.43 However theoretical results47 indicate that especially the long-time correction to the first-order coefficient in the volume fraction due to the modification of the pair distribution of the interacting particles differs significantly from the corresponding value without hydrodynamics even if the repulsion is very long ranged. So far, available theoretical results for DLS are only first order in the volume fraction. Therefore, it is possible that compensating effects of higher order coefficients lead to a volume fraction dependence which can roughly be described by the first-order coefficient without hydrodynamics (-2) over a large volume fraction range. At constant volume fraction, DLS shows a continuous increase with the amount of SDS, even up to concentrations beyond the cmc of the surfactant. This shows that with regard to the electrostatic interaction between the particles the reduction of the Debye length with increasing SDS concentration dominates the concomitant increase of the surface potential of the particles. Irrespective of the precise shape of the adsorption isotherm, the theory using the rescaled particle radius does not adequately describe the experiments. Again the agreement is significantly improved by simply using a first-order coefficient of -2 irrespective of the rescaled particle radius. LA9502960
