Effective charges of colloidal particles obtained from collective diffusion experiments

Transcription

1 Journal of Colloid and Interface Science 263 (2003) Effective charges of colloidal particles obtained from collective diffusion experiments M. Tirado-Miranda, a C. Haro-Pérez, b M. Quesada-Pérez, c J. Callejas-Fernández, b and R. Hidalgo-Álvarez b, a Departamento de Física, Universidad de Extremadura, Escuela Universitaria Politécnica, Cáceres, Spain b Grupo de Física de Fluidos y Biocoloides, Departamento de Física Aplicada, Facultad de Ciencias, Universidad de Granada, Granada, Spain c Departamento de Física, Universidad de Jaén, Escuela Universitaria Politécnica de Linares, Linares, Jaén, Spain Received 29 July 2002; accepted 19 March 2003 Abstract In this work, the collective diffusion coefficient of highly charged colloidal particles in dilute dispersions has been measured by means of dynamic light scattering. The possibility of obtaining valuable information about the particle charge from these data is looked into with the help of electrophoresis experiments. Our results suggest that this is possible in the case of slight or moderately interacting particles as long as experimental data are properly treated. For highly interacting colloids, however, such information could not be so reliable, presumably due to certain shortcomings of the experimental technique at low angle. The role of charge renormalization is also discussed in this work Elsevier Science (USA). All rights reserved. Keywords: Collective diffusion coefficient; Charge-stabilized colloidal suspensions; Dynamic light scattering; Effective charge 1. Introduction Diffusion processes in colloidal dispersions have attracted considerable attention for decades. The Stokes Einstein equation for the Brownian motion of a single sphere is well established, which reads D SE = k BT (1) 6πηa, where D SE is the Stokes Einstein diffusion coefficient, k B T is the thermal energy, η is the fluid viscosity, and a is the particle radius. When considering an ensemble of particles, however, the situation becomes more complex since the diffusive motion of one particle is influenced by the presence of its neighbors. Then Eq. (1) is not valid any longer and, furthermore, one should distinguish between two diffusion coefficients. On the one hand, the self-diffusion coefficient D S describes the fluctuating trajectory of a tracer particle among others. On the other hand, the collective diffusion coefficient D C describes the relaxation of a concentration gradient and appears in Fick s law of diffusion. D S and D C * Corresponding author. address: rhidalgo@ugr.es (R. Hidalgo-Álvarez). coincide with D SE at infinite dilution but differ for concentrated dispersions. The knowledge of D C is a problem of great practical interest because it is a key parameter in a large number of processes involving colloids, such as membrane filtration, sedimentation, or biochemical reactions [1 3]. From an experimental viewpoint, dynamic light scattering (DLS) is said to be a feasible technique to determine its value (providing that certain optical requirements are fulfilled) [4,5]. Theoretically, it is possible to achieve relatively simple expressions for the collective diffusion coefficient at low volume fractions. If this transport property depends on direct interactions between colloidal particles, such expressions could be applied to obtain information about parameters characterizing them. For example, Petsev and Denkov used analytical formulae (which they had derived themselves) to estimate the surface charge of latex particles [4]. However, these authors did not compare their charge values with others obtained from different techniques. This comparison would have allowed them to check the reliability of their charge estimations. Muschol and Rosenberger tried to determine the charge of lysozyme molecules from the collective diffusion coefficient and a similar theoretical background [5]. However, they also performed static light scattering (SLS) mea /03/$ see front matter 2003 Elsevier Science (USA). All rights reserved. doi: /s (03)

2 M. Tirado-Miranda et al. / Journal of Colloid and Interface Science 263 (2003) surements and consider the dispersion forces (which have been ignored by Petsev and Denkov) in the interaction potential. At any rate, it should be stressed that the analysis of D C -data to estimate the particle charge involves the use of certain models or assumptions (e.g., a Debye Huckel interaction potential). Accordingly, the charges so obtained should be considered phenomenological parameters whose value could differ from the actual one. In this sense, they are usually termed effective charges. There exist differentexperimental techniques providing effective charges as well as a large number of theoretical works predicting their values and discussing their dependence on the particle charge, the particle concentration, and the ionic strength. These experimental and theoretical studies have been reviewed in recent papers [6 8]. However, many controversial issues remain to be clarified. This work also examines the use of collective diffusion coefficient data to obtain information about the interaction potential of colloidal particles. In order to test the reliability of such information, a well-characterized latex has been employed. In particular, electrophoretic mobility measurements have been performed to determine the so-called electrokinetic charge. This parameter has been proved to be intimately related to the effective charges that account for the liquid-like structures observed in colloidal fluids and, therefore, it can be considered a parameter featuring the longranged electrostatic interaction [9]. In this way, we look into to what extent D C can provide reliable information about the electrostatic forces. Dealing with highly charged colloidal particles, a renormalization approach has been considered in the discussion. The remainder of the paper is organized as follows. First, the theoretical framework for the analysis of D C -data is briefly outlined. Then, the experimental techniques and sample characterization are described. Finally, the results are presented and discussed. 2. Theoretical background Generally speaking, a short-time cumulant expansion of the dynamic light scattering (DLS) autocorrelation functions results in an effective diffusion coefficient given by H(q; φ) D eff (q; φ) = D 0 (2) S(q; φ), where q is the modulus of the scattering vector, φ is the volume fraction, S(q; φ) is the structure factor, H(q; φ) is the so-called hydrodynamic function, and D 0 is the diffusion coefficient at infinite dilution [10]. D C is obtained in the limit q 0. Hence, the collective diffusion coefficient is intimately related to the behavior of S(q; φ) at low q and, more specifically, to S(0; φ). The reader will have noted that the dependence on the volume fraction has been indicated explicitly in the magnitudes appearing in Eq. (2). Hereafter, however, such dependence will not be pointed out (for simplicity). With regard to the structure factor, this function includes the same information about the spatial ordering as the radial distribution function, g(r). Given a suspension of particles interacting via a known potential, g(r) can be calculated (from a theoretical viewpoint) through different ways. For instance, it can be obtained solving the Ornstein Zernike equation together with a suitable (approximate) closure. In this work, the structure factor was calculated using the rescaled mean spherical approximation (RMSA) [11]. H(q) contains the configuration-averaged effect of hydrodynamic interactions on the dynamics of particles. The diffusion of highly charged particles as well as some matters concerning H(q)were reviewed extensively by Nägele [12]. Particularly, for a colloidal dispersion in which the separation between colloidal particles is higher than the particle radius (a), Nägele and Baur showed that [13] H(z)= 1 15φ J 1(z) + 18φ y ( (g(y) 1) ) z 1 ( J 0 (zy) J 1(zy) + J ) 2(zy) (3) zy 6y 2 dy. In this equation, z = 2qa, y = r/2a,andj n (z) is the spherical Bessel function of order n. Whichever the case, the interaction potential must be specified in calculating these functions. Dealing with suspensions of low ionic strength, dispersion forces were assumed negligible. In relation to the electrostatic interaction, the widely known Debye Hückel (DH) potential was used [1], u(r) = Z2 e 2 4πε 0 ε r ( exp(κa) 1 + κa ) 2 exp( κr), r where e is the elemental charge, ε 0 and ε r are the dielectric constant of vacuum and the relative constant of the solvent, and κ is the reciprocal Debye screening length. It should be stressed, nevertheless, that Eq. (4) was derived using the DH approximation for the distribution of small ions around the particles. Accordingly, this expression could be applied successfully to slight or moderately charged colloidal particles (such as proteins or small micelles) but the linearization assumption could not be fulfilled in the case of highly charged colloids. However, a large number of theoretical works claim that Eq. (4) could be valid if Z is considered a renormalized charge rather than the actual one (see references cited in [6 8]). We applied Eq. (4) in this context, using Z as an effective parameter in the attempt to fit experimental data. 3. Experimental The latex used in this work was prepared from butyl acrylate and styrene by emulsion polymerization with sodium (4)

3 76 M. Tirado-Miranda et al. / Journal of Colloid and Interface Science 263 (2003) dodecyl sulfate as surfactant, potassium peroxidisulfate as initiator, and NaHCO 3 as buffer. The mean diameter was measured by DLS and was found to be 40 ± 3nm.We assumed that a reliable value for D 0 would be obtained extrapolating D C in the limit of vanishing volume fractions and for sufficiently high ionic strengths. The latex was cleaned by dialysis. The titration experiments to determine the number of ionizable groups on the particle surface were carried out with a Crison Instruments ph-meter and conductimeter. It was estimated that there are about 1200 sulfate groups per particle. However, the titration also revealed the presence of weak negatively charged acid groups (2800 per particle in round numbers). They are likely to be carboxylate. Electrophoresis experiments were performed using a ZetaSizer 4 (Malvern). Average values (and the corresponding error bars) were obtained from 9 measurements (3 samples 3 measurements/sample). The light scattering setup was the 4700C System from Malvern Instruments, with an Argon laser operating at 488 nm wavelength and about 15 mw. The particle concentration in scattering experiments was chosen carefully. It must be low enough to avoid multiple light scattering. The effective diffusion coefficients (D eff ) were determined from the initial decay of the normalized field autocorrelation functions following a widely known procedure (cumulant analysis [14]). All the experiments were performed at 25 C and use potassium bromide as the electrolyte. increases with decreasing q, reaching certain value at low scattering angles, which is just the collective diffusion coefficient for the corresponding volume fraction. In this way, we determined D C for six φ-values. Similarly, this procedure was repeated for five different electrokinetic radii. The results are summarized in Fig. 2. One clearly concludes that the set of points of each κa can be fitted according to a linear function, whose independent coefficient is just D 0.This quantity is shown as a function of the electrokinetic radius in Fig. 3 (normalized by D SE ). A remarkable feature is that D 0 seems to depend slightly on κa. More precisely, D 0 passes through a minimum at κa 1. This behavior, which was also reported experimentally by other authors [4,15], has been theoretically predicted as a result of an electrokinetic effect [16 18]. Schumacher et al. showed that D 0 is given by the Stokes Einstein equation for κa 3. Consequently, our result for the highest salt concentration would lead to the best estimation for the hydrodynamic radius. In fact, we used this value for D SE. Whichever the case, our major interest is discussing the results shown in Fig. 2 in terms of charge. In fact, such data were fitted using Eq. (2) (4), with Z as fitting parameter. The so-obtained fits are plotted with solid lines. It should be stressed that the linear behavior of D C found experimentally is perfectly captured by Eq. (2) (4) using only a Z-value for each electrokinetic radius. In other words, the effective charge does not depend on φ (in the range of volume fractions studied in this work). A similar conclusion has been reported through another experimental technique [19]. 4. Results and discussion In Fig. 1, we show the effective diffusion coefficient as a function of the modulus of the scattering vector q for several volume fractions (ranging from to ) and for κa = 0.6. It should be stressed that this parameter (to which we will refer as electrokinetic radius hereafter) includes the contributions of all ionic molecular species, not only the ions arising from added salt but also the counterions dissociated from colloidal particles. As can be seen, D eff Fig. 2. Collective diffusion coefficient as a function of the volume fraction for several κa = 0.6, 0.8, 1.2, 1.8, and 4, which are denoted by squares, circles, triangles, upside-down triangles, and diamonds, respectively. The solid lines stand for the fits obtained using Eq. (2). Fig. 1. Effective diffusion coefficient as a function of the modulus of the scattering vector for κa = 0.6 and several volume fractions, φ = ,5 10 5,1 10 4,2 10 4,6 10 4,and , which are denoted by solid squares, circles, triangles, upside-down triangles, diamonds, and open squares, respectively. Fig. 3. Normalized diffusion coefficient (D 0 /D SE ) as a function of κa.

4 M. Tirado-Miranda et al. / Journal of Colloid and Interface Science 263 (2003) ζ through the following relation (for a 1:1 electrolyte) Z ζ = 4πε 0ε r k B Tκa 2 e 2 [ ( eζ 2sinh 2k B T ) + 4 ( ) ] eζ κa tanh, 4k B T (5) Fig. 4. Electrokinetic charge (open squares), renormalized charge (solid squares), and the Z-values calculated from the experimental D C -data using Eq. (2) (circles) as well as the analytical expression given in Ref. [4] (triangles). All of them are plotted as a function of κa. Fig. 5. Electrophoretic mobility (squares) and ζ potential (circles) as a function of the logarithm of the KBr molar concentration. However, the effective charge does depend on κa,whichis shown in Fig. 4 (circles). As can be seen, the former parameter increases with the latter. Such dependence will be discussed later. At any rate, the most remarkable fact is that the charges determined from D C are significantly smaller than the number of ionizable groups (1200 electrons) determined by titration, which is only due to the dissociation of sulfate groups under our experimental conditions. In relation to this, it would be instructive to include in this analysis the charge estimated from electrophoresis experiments. The so-called electrokinetic charge (which is the net charge resulting from the colloidal particle and the electric double layer limited by the shear surface) has been proved to be closely related to the effective charges determined from liquid-like structures observed in dilute colloidal dispersions (as a result of the long-ranged electrostatic interaction) [9,20]. This means that this electrokinetic property could provide us a fairly reliable estimate of the charge that particles feel interacting electrostatically at large distances. In order to determine the electrokinetic charge, the electrophoretic mobility was first measured as a function of the electrolyte concentration. These data, which are plotted in Fig. 5, were converted into ζ potential applying the widely known O Brien White theory [21], which is also shown in Fig. 5. As can be observed, this parameter is almost constant. Finally, the electrokinetic charge was calculated from which gives the electrokinetic charge to within 5% for κa >0.5 andanyζ potential [1]. The results (corresponding to the electrokinetic radii studied in the previous diffusion experiments) were estimated (from ζ) and plotted also in Fig. 4 (open squares) to be compared with the effective charges obtained from D C. As can be seen, the electrokinetic charge is smaller than that determined from titration, particularly for low electrokinetic radii. In this way, the smallness of the effective charges is not so remarkable (as in the case of being compared to the titration result). However, the discrepancies have not disappeared completely. As stated above, Eq. (4) was derived under the DH approximation, which involves weakly interacting systems. It is interesting to find out if charge renormalization can account for the mentioned differences since the ζ potential and some of the electrokinetic charges derived previously are not small. A renormalization procedure (bearing certain resemblance to that proposed by Alexander et al. [22]) was applied. If one solves the nonlinear Poisson Boltzmann (PB) equation for an isolated charged sphere, the electrostatic potential (ψ) at large enough distances is known to have the form ψ(r) A e κr. (6) r Comparing this expression with the solution of the linear PB equation (see, for instance, [1]) ψ(r) Ze e κa e κr, (7) 4πε 0 ε r 1 + κa r it is easily concluded that the prefactor A can be interpreted in terms of a renormalized charge. Thus we obtained A by numerical integration of Eq. (6), matching at r = a the boundary condition (Gauss law) [ ] dψ = Ze (8) dr 4πε 0 ε r a 2. r=a It should be emphasized, however, that we preferred to use the electrokinetic charge (rather than the value obtained by titration) as the particle charge appearing in Eq. (8). The rationale for this can be found in previous studies, which suggest that the number of surface groups obtained by conductimetric and potentiometric titrations could not be a suitable input parameter for the renormalization theories [9,23 25]. Instead of such a number, a reduced charge is more appropriate. For instance, Palberg et al. presented the shear modulus titration as a direct experimental access to the actual surface charge [24], whereas Quesada-Pérez et al. concluded that electrophoresis experiments could provide an

5 78 M. Tirado-Miranda et al. / Journal of Colloid and Interface Science 263 (2003) acceptable estimate of this quantity [9]. The renormalized charges obtained in this way for the κa-values studied in this work are shown in Fig. 4 as well. As can be seen, there are not large differences between these renormalized values and the electrokinetic charges. Both are of the same order because the electrokinetic charges are not large (especially at low ionic strengths). Comparing the renormalized charges with those obtained fitting D C with Eq. (2), one concludes that the disagreement between experimental and predicted data has disappeared for the largest electrokinetic radius but persists for the other cases. Being more precise, the discrepancies between both quantities grow with decreasing κa. As mentioned previously, the collective diffusion coefficient is straightforwardly related to S(0). In the case of intense repulsive forces (i.e., low electrokinetic radii), this quantity should be extremely small on paper. In practice, however, several experimental studies on liquid-like order in colloid dispersions reveal S(q) is larger than expected for q<q 0 (q 0 being the modulus of the scattering vector at which S(q) reaches its first peak) [9,25]. What is more, this discrepancy was observed for the latex of this study in a previous work [26]. This effect has been attributed to polydispersity and the presence of aggregates, which scatter preferentially into the forward direction. Irrespective of its cause and according to Eq. (2), the anomalous behavior of the structure factor in the limit q 0 may involve obtaining experimental collective diffusion coefficients smaller than predicted and, therefore, phenomenological charges smaller than those theoretically expected. This could be a feasible cause for the disagreement discussed earlier. In relation to this topic, it should also be pointed out that other authors have compared values of S(0) obtained from an optical technique (frequency domain photo migration) and the corresponding predictions [27]. Both agreed for high ionic strengths, but the agreement deteriorated with decreasing this parameter. Obviously, this also suggests that the analysis of highly interacting colloidal particles through optical methods involving S(0) must be carried out carefully. As mentioned above, the effective charge depends on the electrokinetic radius. From the viewpoint of the renormalization approaches, analyzing this behavior could become a vexed question since different models lead to different predictions. For instance, according to Alexander s renormalization theory, the effective charge should increase slightly with the electrolyte concentration [22] whereas Belloni reported a nonmonotonic relationship between both quantities [28]. In our opinion, however, the increase of the effective charge with κa is rather related to the increase of the electrokinetic charge with the same parameter (see Fig. 4). In fact, the coincidence in the qualitative behavior of these two charges would also suggest that Z ζ is better than the surface charge obtained by titration as input parameter of renormalization approaches. Finally, the results obtained through the analytical expression derived by Petsev and Denkov [4] will be also discussed briefly. Such results are shown in Fig. 4 (triangles). As can be seen, these effective charges are even smaller than those obtained from D C and Eq. (2) (4) (particularly at high ionic strengths). This is somewhat logical since the achievement of analytical formulae involved the use of quite severe approximations, such as g(r) exp( u(r)/k B T) (where T and k B are the absolute temperature and Boltzmann s constant, respectively). 5. Conclusion This work confirms the widely reported fact that electrostatic forces can have a considerable effect on the collective diffusion processes in dilute colloidal dispersions. For slightly or moderately interacting particles, an appropriate treatment of experimental data could yield precise charge values. In the case of highly interacting systems, however, our results show that one must be careful when D C -data are analyzed in order to obtain information about the interaction potential. Certain experimental limitations of the light scattering technique at low angle could lead to misleading results. Acknowledgments Financial support from the Ministerio de Ciencia y Tecnología, Plan Nacional de Investigación, Desarrollo e Innovación Tecnológica (I + D + I) is gratefully acknowledged. Particularly, C.H.P., J.C.F., and R.H.A. are indebted to project MAT C03-01, whereas M.T.M. and M.Q.P. express their gratitude to project MAT C We also thank Dr. Jacqueline Forcada for supplying the latex used in this work. References [1] W.B. Russel, D.A. Saville, W.R. Schowalter, Colloidal Dispersions, Cambridge Univ. Press, Cambridge, [2] W.R. Bowen, A. Mongruel, Colloids Surf. 138 (1998) 161. [3] B.M. Fine, A. Lomakin, O.O. Ogun, G.B. Benedek, J. Chem. Phys. 104 (1996) 326. [4] D.N. Petsev, N.D. Denkov, J. Colloid Interface Sci. 149 (1992) 329. [5] M. Muschol, F. Rosenberg, J. Chem. Phys. 103 (1995) [6] L. Belloni, J. Phys. Condens. Matter 12 (2000) R549. [7] J.P. Hansen, H. Löwen, Ann. Rev. Phys. Chem. 51 (2000) 209. [8] M. Quesada-Pérez, J. Callejas-Fernández, R. Hidalgo-Álvarez, Adv. Colloid Interface Sci. 95 (2002) 295. [9] M. Quesada-Pérez, J. Callejas-Fernández, R. Hidalgo-Álvarez, Colloids Surf. A 159 (1999) 239. [10] J.P. Hansen, D. Levesque, J. Zinn-Justin, Liquids, Freezing and Glass Transition, Part 2, Course 10, North-Holland, Amsterdam, [11] J.P. Hansen, J.B. Hayter, Mol. Phys. 46 (1982) 651. [12] G. Nägele, Phys. Rep. 272 (1996) 216. [13] G. Nägele, P. Baur, Physica A 245 (1997) 297. [14] R. Pecora, Dynamic Light Scattering, Plenum, New York, 1985, Chap. 4. [15] J.M. Delgado-Calvo-Flores, J.M. Peula-García, R. Martínez-García, J. Callejas-Fernández, J. Colloid Interface Sci. 189 (1997) 58. [16] J.M. Schurr, Chem. Phys. 45 (1980) 119.

6 M. Tirado-Miranda et al. / Journal of Colloid and Interface Science 263 (2003) [17] M. Medina-Noyola, A. Vizcarra-Rendon, Phys. Rev. A 32 (1985) [18] G.A. Schumacher, T.G.M. Van de Ven, Faraday Discuss. Chem. Soc. 83 (1987) 75. [19] P. Wette, H.J. Schöpe, T. Palberg, J. Chem. Phys. 116 (2002) [20] C. Haro-Pérez, M. Quesada-Pérez, J. Callejas-Fernández, E. Casals, J. Estelrich, H. Hidalgo-Álvarez, J. Chem. Phys. 118 (2003) [21] R.W. O Brien, L.R. White, J. Chem. Soc. Faraday Trans (1978) [22] S. Alexander, P.M. Grant, G.J. Morales, P. Pincus, D. Horne, J. Chem. Phys. 80 (1984) [23] M. Quesada-Pérez, J. Callejas-Fernández, R. Hidalgo-Álvarez, J. Chem. Phys. 110 (1999) [24] T. Palberg, J. Kottal, F. Bitzer, R. Simon, M. Würth, P. Leiderer, J. Colloid Interface Sci. 169 (1995) 85. [25] T. Gisler, S.F. Schulz, M. Borkovec, H. Sticher, P. Schurtenberger, B. D Aguanno, R. Klein, J. Chem. Phys. 101 (1994) [26] M. Quesada-Pérez, J. Callejas-Fernández, R. Hidalgo-Álvarez, J. Colloid Interface Sci. 206 (1998) 354. [27] Y. Huang, E.M. Sevick-Muraca, J. Colloid Interface Sci. 251 (2002) 434. [28] L. Belloni, Colloids Surf. A 140 (1998) 227.