Furuday Discuss. Chem. SOC., 1987, 83, 75-85 Brownian Motion of Charged Colloidal Particles surrounded by Electric Double Layers Gerhard A. Schumacher and Theodorus G. M. van de Ven" Pulp and Paper Research Institute of Canada and Department of Chemistry, McGill University, Montreal, Quebec, Canada H3A 2A7 Using photon correlation spectroscopy, diffusion coefficients of charged colloidal particles surrounded by electrical double layers have been deter- mined. The diffusion constant equals the value of a neutral sphere at high and low electrolyte concentrations, but is reduced by several per cent when the electrical double layer is comparable to the radius of the particle. The reduction in diffusion constant depends on the zeta potential of the particle and the sizes of the ions in the double layer.The diffusion of charged particles can be explained by the theory of Ohshima ef al. (J. Chem. Soc., Furaday Trans. 2, 1984, 80, 1299) for the friction coefficient of charged spheres, assuming that the friction coefficient of a charged sphere in Brownian motion equals the equilibrium friction coefficient of a sedimenting sphere. Many experiments have been described in the l i t e r a t ~ r e l - ~ in which the diffusion coefficients of small particles, such as enzymes, polyelectrolytes or colloidal particles with adsorbed polymer, have been measured as a function of pH or salt concentration. Variations in the effective particle radius obtained from diffusion constants were ascribed to changes in particle dimensions (swelling, shrinking) or to changes in configurations of the adsorbed polymers.In those studies what has been overlooked is that the very presence of a diffuse ionic double layer affects the diffusion of small particles. Although not at all obvious, there are reasons to believe that the Stokes-Einstein relation continues to be applicable to charged particles, and hence the diffusion coefficient or equivalent sphere radius can be related to the friction coefficient of a charged sphere, expressions for which exist in the l i t e r a t ~ r e . ~ To test these theories we performed photon correlation spectroscopy (P.c.s.) measurements on classical gold sol particles of ca. 20nm and polystyrene latex spheres of ca. 40 nm, as a function of salt concentrations for various salts. Salt concentrations were varied in the range 0.1 d K a d 3 for the gold sol and 0.3 s K a s 7 for the latex, a being the sphere radius and K-' the Debye length or double-layer 'thickness'. We used three different salts: sodium chloride, sodium benzoate and tetraethylammonium chloride.These salts were chosen because of the widely different sizes of their ions and because they demonstrate little or no surface activity, thus avoiding specific adsorption. As predicted by theory, the diffusion was found to be maximum at low and high Ka values and minimum around Ka = 1. The depth of the minimum depended mainly on the positive ion size. At the minimum, the diffusion constants were 3 4 % lower than the values at high or low electrolyte concentration, in which case the diffusion constant equals that for a neutral sphere.According to the theory, in the limit of low 6 potentials (eJ/kT<< 1, e being the unit electric charge and kT the thermal energy), the depth of the minimum at K a = 1 should depend equally on the size of the positive and negative ions, for a given zeta potential. However, as the potential increases, the depth of the minimum becomes primarily dependent on the counterion (the ion opposite in charge to the colloidal particles). Since both the gold and the polystyrene are negatively charged, one would expect the minimum depth to 7576 Brownian Motion of Charged Colloidal Particles be deeper when tetraethylammonium rather than sodium is the cation. Sodium benzoate would be expected to have a slightly deeper minimum than sodium chloride, since there is still a small contribution from the coion.Experimental observations were in excellent agreement with these predictions, proving conclusively that the decrease in diffusion constants is due to the presence of electrical double layers around the particles. Theory One can consider Brownian motion to be a random walk in which the particle executes jumps of a characteristic length. Charged colloidal particles will execute this random walk, however, in the presence of an ionic double layer. Since the particle is moving, the ionic double layer will be distorted from its spherical symmetry and this will give rise to a microscopic electric dipole moment, which will increase the drag on the particle and thus reduce its velocity. Estimating the time of a Brownian jump of a colloidal particle of radius a, mass m, friction coefficient f and diffusion constant D as r,=rn/f and the relaxation time of an ion (of diffusion constant Di) in the double layer after a jump of length 1 by rr = 12/ Di it follows that the double layer can be considered to keep its equilbrium shape during a jump when r,<< rj.( 3 ) Estimating 1 from 1 == U ~ T ~ and the particle jump velocity uo from mu: = 3kT, using the Stokes-Einstein relation Df = kT and Stokes law (for a spherical particle) f = 6 ~ 7 a (4) which, with some modification, also holds for ions (7 being the viscosity of the medium), condition (3) can be restated as For most Brownian particles this condition is satisfied. The effect of ionic double layers on the sedimentation velocity of charged spherical colloidal particles in a dilute suspension has been determined theoretically by Ohshima et aL4 The ionic double layer loses its spherical symmetry because of the particle motion (sedimentation), setting up a microscopic electric dipole, which decreases the sedimenta- tion velocity.Brownian motion of charged colloidal particles can be thought of as being analogous to the effect of ionic double layers on sedimentation, if the equilibrium distorted double layer can follow the Brownian particle throughout its random walk. We can then replace the relative sedimentation velocity with the relative diffusion constant, to obtain theoreti- cal expressions on which to base our experiments. The diffusion constant is predicted to be maximum at low and high K a values and minimum at KU =: 1 (see fig.1). Besides general numerical calculations, Ohshima et uL4 also derived an analytical expression for the relative sedimentation velocity u / u, ( oo being the Stokes sedimentation velocity) as a function of KU and zeta potential, valid at low zeta potentials ( e f / k T < < 1).G. A. Schumucher and T. G. M. van de Ven 77 1.00 0.98 6 0.96 1 0.94 0.92 0.90 I I 1 I I J - 2 -1 0 1 2 1% (.a) Fig. 1. Relative diffusion constant as a function of ~a for three potentials: e l / k T = ( a ) 3, ( b ) 5 and ( c ) 8. The points are numerical results taken from Ohshima et ai.4 The solid lines are smooth fits. The dashed lines indicate the prediction of eqn (10) valid for low potentials.Calculations are for KCI. Replacing v / v o with D/Do, where Do is the Stokes diffusion constant, we obtain N I = 1 where with and where is the scaled drag coefficient of the ith ion; J; is the drag coefficient of the ith ion; zi is the valency of ion i and n: the bulk concentration; E , is the dielectric constant of the medium and E~ the permittivity of free space. For 1 : 1 electrolytes eqn (6) can be written as D/ Do = 1 - f i a* f ( KU ) (10) where78 - 90 - 80 > -70 E \ h - 60 - 50 -40 Brownian Motion of Charged Colloidal Particles I I I log (NaCl/mol dm-3) Fig. 2. Zeta potential us. sodium chloride concentration for colloidal gold particles. The inset shows a typical doppler spectrum from laser doppler electrophoretic experiments (relative intensity, I, us.frequency shift, Av; E = 509 V m-', [NaCl] == lop4 moI dmP3). The indicated spread is estimated from the width of the laser doppler spectrum. is the average scaled ion drag coefficient, or [ c j eqn (4) and (9)] the average scaled ion size; <D is the reduced y potential Predictions of eqn (10) are included in fig. 1 (dashed lines) for an aqueous KC1 solution. Experimental Gold sols were prepared according to the procedure used by Enustun and Turkevich.' The extraneous ions were removed by allowing the sol to flow through ion-exchange resin (Amberlite MB-1) until the conductivity of the sol dropped to 8 x lo-' K1 m-', which corresponds to a salt concentration of ca. 6 x lop6 mol dmP3 1 : 1 electrolyte. Electrophoretic mobilities u were determined as a function of NaCl concentration (see fig.2) using laser doppler electrophoresis from the relation6 A Av 2nE sin (8/2) U = where A is the wavelength of light, hv the frequency shift, n the refractive index of the medium, E the electric field strength and 0 the scattering angle. The zeta potentials, 5, were calculated from u, from the relation u ( & Ka) given by Oshshima et al.7 As can be seen from fig. 2, the measured [ potentials are in the range - 40 to - 90 mV, corresponding to @ in the range - 1.5 to -3.5. Electron micrographs of the gold sol (plate 1) were taken using a Philips EM400T transmission electron microscope. The micrographs were analysed to obtain the particle size distribution (see fig. 3). The histogram was obtained from measurement of 198 particles. From this distribution the mean diameter of the gold particles was found toFaraday I>iscuss.Chem. Soc., 1987, Vol. 83 Plate 1. Electron micrograph of gold particles. G. A. Schumacher and T. G . M. van de Ven Plate 1 (Facing p . 78)G. A. Schumacher and T G. M. van de Ven a/nm IS 20 25 I I I 1 30 20 10 0 - ._ 5 a cr 0 30 20 10 2 .o 2.5 D/ lo-" m2 s-l 1 3.0 79 Fig. 3. ( a ) Particle size distribution of gold sol and ( b ) particle diffusion constant distribution where the diffusion constant has been calculated from the size by means of the Stokes-Einstein relationship. The dashed curve is a Gaussian fit. be 20.2 nm (the standard deviation was 10%). The distribution is, however, skewed and therefore it is not completely justified to approximate the distribution as Gaussian.The distribution of diffusion constants, which were calculated from the size distribution using the Stokes-Einstein relationship, is quite nicely approximated by a Gaussian fit [see fig. 3 ( 6 ) ] . The standard deviation of this distribution was found to be 9%. Polystyrene latex was provided to us by Prof. R. H. Ottewill at the University of Bristol. The latex was prepared using the procedure of Ottewill and Richardson' and was subsequently extensively dialysed against distilled water. It was then stored over mixed-bed ion-exchange resin. The surface charge density u was measured conduc- tometrically and found to be -45.3 mC m-'. Using the Gouy-Chapman theory for flat plates, this surface charge density corresponds to a surface potential, $o, of - 166 mV at K = 5 x lo7 m-' and T = 298 K.Measured zeta potentials have been shown' to be 2 to 3 times smaller than the surface potential calculated from Couy-Chapman theory at salt concentrations of mol dm-3, the difference decreasing as the salt concentration increases. This suggests a zeta potential for the latex of ca. -60 to -80 mV. The reason for the difference between measured [-potentials and potentials calculated from the Gouy-Chapman theory is the following. For low potentials, the Debye-Huckel theory predicts that for flat plates,80 Brownian Motion of Charged Colloidal Particles Here E = E , E ~ . For spherical particles the equivalent equation is 1+Ka a (T=E:- *O. At K a = 1, for a given surface charge density, t,hO from eqn (15) will be half the value for t,ho calculated from eqn (14).This indicates that the difference between the calculated surface potentials and the measured zeta potentials is largely due to the use of the flat-plate approximation, which is not valid for spherical particles: Electron microscopy data for the latex particles gave an average diameter of 32 nm with a standard deviation of 18%. Diffusion constants of both the gold sol and the latex particles were measured by dynamic light scattering l o using a Brookhaven Instruments Corporation photon correla- tion spectrometer with a BI-2030 digital correlator and a Spectra Physics 120 helium-neon laser. All measurements were carried out at 25 f O . l "C and with a scattering angle, 8, ranging from 45 to 150". The gold sol was not diluted for the light scattering measure- ments, whereas it was necessary to dilute the latex. The volume fractions for both the gold sol and the latex were estimated as 2 x Samples were filtered through a 0.2 p m millipore filter before measurement of the diffusion constant.The normalized time correlation function of the electric field of scattered light, g(q, T), was assumed to be in the form of a cumulant expansion where only the first (r) and the second (p2) moments were calculated. The diffusion constant is related to r through the following expression where q is the absolute value of the scattering vector q : 4nn h q = - sin (:) . Results and Discussion Photon Correlation Spectroscopy (P.C.S.) Measurements The ratio, Q, of the second moment, p2, to the square of r was found to be < 0.1 in all cases for the polystyrene latex and between 0.1 and 0.2 for the gold sol.The second moment can be related to the standard deviation of the particle size distribution through the following: p2= ( r - T ) 2 G ( r ) d r (19) I where G(T) is the distribution of r values, which we have assumed to be Gaussian [cJ: fig. 3 ( b ) ] . Using the average sizes of the latex and the gold sol which were obtained from electron microscopy (see Experimental section), we found Q=0.03 for the latex and Q = 0.01 for the gold sol, using eqn (19). These values are low enough'"' to analyse g(q, 7) as a single exponential: d q , 7) = exp (- r4 (20) which assumes that the scattering of light is due to a monodisperse solution of scatterers.The calculated value of Q for the latex corresponds to the experimentally observed0.: 0.: 3 m v 0. - O.! -1 .I h t- ef; W -1.' E: - -2. G. A. Schumacher and T. G. M. van de Ven 81 1 1 I I 1 . 25 50 75 100 125 TIPS Fig. 4. ( a ) Experimental time correlation function for gold sol (not normalized). The inset is an expansion of the data at low sampling time. ( b ) Plot of the natural logarithm of the time correlation function against T. values, whereas for the gold sol the observed Q value is higher than the calculated one. This can be attributed to the presence of 1-4% doublets5 found in the gold sol, which will broaden the size distribution and hence increase Q. Although in the presence of doublets the gold sol diffusion constant distribution [fig. 3( b ) ] will no longer be Gaussian, an estimate of the broadening of the distribution can be made by calculating the standard deviation of this new distribution (with doublets), which was found to be 20%. This result is in agreement with the P.C.S.data. A typical experimentally observed time correlation function for the gold sol is shown in fig. 4(a). If these data are to be analysed according to eqn (20), then a plot of In [g(q, T)] us. T should yield a straight line. A typical example is shown in fig. 4(b). As Q increases, eqn (20) is no longer valid and must be replaced by eqn (16). However, from fig. 4(6) we can see that the P.C.S. data from the gold sol are easily approximated by a single exponential, which means that despite having 1-4% doublets in the gold sol, effects of polydispersity can be ignored.The first 2 ps of fig. 4 ( a ) are expanded (inset) to show that there is a decay of much smaller order than that for translational Brownian diffusion of the gold sol. The timescale of this decay leads us to believe that this is due to rotational Brownian diffusion, which for particles of this size should be of the order of a microsecond. As can be seen from plate 1 the gold particles are not perfectly spherical. The small percentage of doublets82 Brownian Motion of Charged Colloidal Particles 60 90 120 scattering angle/" 150 Fig. 5. Angular dependence of the measured diffusion constant for gold sol (a) and for polystyrene (0) latex. The error bars for the gold sol data indicate reproducibility, whereas for the latex they refer to the error in calculating z'.(Data for distilled water as medium.) could also be responsible for this effect. Including the datum point at 2 ps in the analysis did not significantly alter the results and therefore the complete time correlation function was used when calculating I'. For a dilute monodisperse suspension of scatterers, the diffusion constant measured should be independent of the scattering angle. Realistically, however, there is always some degree of polydispersity and therefore one expects some curvature in a plot of. diffusion constant us. scattering angle. Measured diffusion constants as a function of scattering angle are shown for both the latex and the gold sol in fig. 5 . The diffusion constant for the latex remains roughly constant with angle, while for the gold sol it begins to deviate at 90".The measured diffusion constant of the gold sol is constant over a substantial range, bringing us again to the conclusion that effects of polydispersity can be ignored. This reasoning applies even more to the latex. For the analysis of the variation of diffusion constant with salt concentration, a scattering angle of 135" was chosen. All values of K a are determined using the radius calculated from P.C.S. measurements. In the case of the gold sol the average diameter obtained from P.C.S. measurements, 22 nm, is slightly higher than that obtained from electron microscopy, 20 nm, and in the case of latex, P.C.S. measurements give a diameter of 40nm, whereas electron microscopy give a diameter of 32 nm.The difference in diameters found from P.C.S. and electron microscopy for the gold sol is small. The difference in diameters for the latex is larger, but not uncommon,'"l and can be attributed mainly to the difference in size when the latex is solvated (P.c.s.) or unsolvated (electron microscopy), and also due to the inherent differences in the types of measurement being made. The Effect of Ionic Double Layers on Brownian Diffusion Fig. 6 and 7 show the change in diffusion constant as a function of Ka for the gold sol and the latex, respectively. In both cases the depth of the minimum is smallest for sodium chloride, largest for tetraethylammonium chloride and somewhere between these two for sodium benzoate. Eqn (10) predicts that the depth of the minimum should beG.A. Schumacher and T. G. M. van de Ven 83 concentration/ mol dm-3 2.3 d IV1 2 . 2 "E + .- I \ 4: 9 2 . 1 0 0.5 1 .o 1.5 Ka Fig. 6. Plot of the experimentally determined diffusion constants us. KU for gold sol for three different salts: NaCl (e), NaC,H5C02 (0) and (CH3CH2)4NC1 (0). For the last two salts no data were collected for KU b 1.5 because of the occurrence of coagulation. The triangle corresponds to the measurements in distilled water (ca. lo-' mol dm-3). A typical error bar is shown to indicate the reproducibility of the results. proportional to the average of the scaled drag coefficients, m,, of the constituent ions for 1 : 1 strong electrolytes. Table 1 gives the scaled drag coefficients, m,, for various ions, from their limiting molar conductances. I 2 * l 3 Also included are the effective radii of the ions, from which it can be seen that condition (5) holds even for the largest ion and smallest particle.From the data in table 1, the reduction in diffusion constant should be in the ratio of 0.213 : 0.287 : 0.328 for NaCl : (CH3CH2)4NCl : NaC6H5C02. This is, however, not what is observed. Eqn (lo), however, is only applicable in the limit of low potentials (a<< 1). Fig. 1 is calculated using KCl as an electrolyte, in which case the scaled drag coefficients of K+ and C1- are essentially equal. Eqn (10) provides a good fit for e l / kT = 3 with KCl as electrolyte. However, for an electrolyte with ions of unequal drag coefficients, eqn (10) may no longer provide a good fit. This hypothesis is based on another approximate analytic expression from Ohshima et aL,4 in the limit of large Ka, where the ions can have different mobilities and all values of y are applicable.In this case it is no longer the average of the m, for the constituent ions which is important, but predominantly rn, of the counterion, typically of the order of 6 : 1 for the potentials of the gold sol and the latex. If we then use this result to interpret fig. 6 and 7, we can see that the minimum depth is correctly predicted within experimental error. The gold sol and the latex are negatively charged and therefore tetraethylammonium chloride should have the largest effect, due to its large positive ion. Sodium benzoate will have a larger effect than sodium chloride since there is still a small contribution from the anion.We can then re-express eqn ( 1 1 ) for when the condition e l / kT << 1 is not met, as rn=WTZ,+(l-a)m-. (21) From our data we estimate that for our experimental conditions LY eO.80.84 Brownian Motion of Charged Colloidal Particles 1.25 - 'm E : 9.20 0, 4 N l --. 1.15 concentration/mol dm-3 5. 1 I I I I I I I I 1 2 3 4 5 6 K a Fig. 7. Plot of the in distilled water ( (W, NaC,H5C02 diffusion constant us. KU for polystyrene latex for three different salts: NaCl (a), and (CH3CH2)4NCI (0). The triangle corresponds to the measurements ca. lop5 mol dm-3). A typical error bar is shown to indicate reproducibility of the results. Table 1. Limiting molar conductances, scaled drag coefficients and sizes for selected ions at 25 "C limiting molar scaled drag effective conductance coefficient, ion radiusa ion / lo2 K' m-' equiv-' m, /nm 50.11b 73.52b 32.3' 32.3' 73.34h 0.257 0.28 0.175 0.19 0.399 0.43 -0.169 0.18 0.399 0.43 a Calculated from eqn (9), assuming the friction coefficient of an ion is given by f; = 47rr)uetf (corresponding to the slip boundary condition). Ref.(12). Ref. (13). Using eqn (10) with this modified fi, we can solve for the reduced zeta potentials of the gold sol and the latex. The reduced zeta potentials, @, (zeta potentials, 5 ) for the gold sol are -3.2*0.1 (- 82*3 mV) at lop4 mol dmP3 salt concentration and -3.1 f 0.1 (- 80* 3 mV) at mol dm-', where the potentials given were averaged from the potentials calculated for each individual salt. The error given is the standard deviation.Similarly, the reduced zeta potential, a, (zeta-potential, 5) for the latex is -2.7f0.3 (- 69 k 8 mV) at Ka = 1.0. These results are in good agreement with the zeta potentials measured for the gold sol and with the zeta potential for the latex, estimated above. From the diffusion experiments one can even deduce the sign of from the dominant contribution of the counterion to fi.G. A. Schumacher and T. G. M. van de Ven 85 Conclusions We have shown conclusively that the double layer around a colloidal particle contributes to the diffusion constant of the particle, especially when the double layer thickness is comparable to the radius of the particle. When K a = 1, the double layer can reduce the diffusion coefficient of a colloidal particle by several per cent, depending on the zeta potential and the reduced friction coefficients of the ions in solution. The experimental results are in general agreement with the Stokes-Einstein relation Of = kT, with f given by the equilibrium friction coefficient of a sedimenting charged sphere, expressions and numerical results for which are given by Ohshima et al." For gold sol and latex particles of 20 and 40 nm, respectively, of reduced potentials Q> of ca.3, it was found that the data can be explained by the asymptotic theory valid for @<< 1, provided the mean drag coefficient given by eqn ( 1 1 ) is replaced by the modified eqn (21), i.e. the contribution of the counterions is much larger than the contribution of the coions. The agreement between the zeta potentials of the gold sol and latex particles with the values obtained from the diffusion coefficients [using eqn (lo)] indicates that zeta potentials of small particles can be obtained from measurements of diffusion constants (without the need for an external electric field).Alternatively, the method can be used to measure the friction coefficients (and hence the sizes) of ions in solution. In this respect it is interesting to observe that diffusion constant measurements have gone full circle. Early measurements by P e r ~ i n ' ~ were intended to measure the Boltzmann constant k or, equivalently, Avogadro's number, from which the sizes of atoms can be calculated (thus proving their existence). With the present method atomic dimensions (of ions) can be measured directly. Special thanks are in order to Dr K. Takamura for measuring zeta potentials of the gold sol, to Prof. R. H. Ottewill and associates for giving us the polystyrene latex, to Dr R. M. Fitch for providing an initial gold sol and to J-F. Revol and Dr P. Pradkre for taking electron micrographs. References 1 S . Sasaki, Colloid Polym. Sci., 1984, 262, 406. 2 T. Raj and W. H. Flygare, Biochemistry, 1974, 13, 3336. 3 J. W. S. Goossens and A. Zembrod, Colloid Polym. Sci., 1979, 257, 437. 4 H. Ohshima, T. W. Healy, L. R. White and R. W. O'Brien, J. Chem. Soc., Faraday Trans. 2, 1984, 80, 5 B. V. Enustun and J. Turkevich, J. Am. Chem. SOC., 1963, 85, 3317. 6 K. Takamura, R. S. Chow and D. L. Tse, in Proc. Inf. S.ymp. on Flocculation in Biotechnology and 7 H. Ohshima, T. W. Healy and L. R. White, J. Chem. Soc., Faraday Trans. 2, 1983, 79, 1613. 8 R. H. Ottewill and R. A. Richardson, Colloid Pol-ym. Sci., 1983, 260, 708. 9 M. E. Labib, Ph.D. Thesis (McGill University, Montreal, 1979), p. 175. 1299. Separation Systems (to be published). 10 B. J. Berne and R. Pecora, in Dynamic Light Scattering (Wiley-Interscience, New York, 1976), chap. 1-5. 11 M. M. Kops-Werkhoven and H. M. Fijnaut, J. Chem. Phys., 1981, 74, 1618. 12 D. A. MacInnes, in The Principles of Electrochemistry (Dover, New York, 1939, 1961), chap. 18, p. 342. 13 M. Tissier and C. Douhkret, Solution Chem., 1978, 7, 87. 14 J. Perrin, Les Atomes (Fklix Alcan, Paris, 1913), chap. 4. Received 12rh December, 1986