Langmuir 2007, 23, 5305-5314
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Hard versus Soft Particle Electrokinetics of Silica Colloids Jirˇ´ı Sˇ kvarla* Department of Mineralurgy and EnVironmental Technologies, Technical UniVersity in Kosˇice, Park Komenske´ ho 19, 04384 Kosˇice, SloVak Republic ReceiVed December 7, 2006. In Final Form: February 26, 2007 To verify the existence of a gel layer at the surface of silica, dependences of the electrophoretic mobility of fresh and aged colloidal silica particles on the KCl concentration are measured. These dependences, corrected for the relaxation/polarization effect, are fitted by analytical expressions based on the model of hard, soft, and brush surfaces. A bad fit is obtained for both silicas when its surface is considered ideal (hard). Much better fits are achieved with the invariable soft layer model for the fresh silica but especially for the aged silica whose surface is less charged probably as a result of an extension and/or loosening of the layer. A perfect fit is found for aged silica when applying a trivial model of the soft polyelectrolyte layer combined with the scaling model of polyelectrolyte brushes.
Introduction Electrophoresis is a traditional experimental tool for determining electric characteristics such as the charge and potential at the putatively ideal (hard, bare, or smooth) particles with no features at their surfaces, immersed in an electrolyte solution.1-5 The soft or “hairy” particles are hypothetized to consist of a rigid core covered by an adsorbed or inherent polyelectrolyte layer with a spatial distribution of ionogenic groups. The electric potential distribution with the associated penetration of hydrodynamic flow inside of such a 3D surface must logically differ from that at the 2D hard surface. That is why the interpretation of electrophoretic mobility of soft particles by theories developed for hard particles may be inappropriate on principle and thus pointless because such theories may be rather complicated and inconsistent; as a rule, the polarization and relaxation phenomena must be reckoned with, especially for highly charged particles.6-14 For soft particles, the distribution of completely dissociated functional groups (fixed charges) and ions in the polyelectrolyte layer is, for the sake of simplicity of expressing the Donnan potential, the commonly assumed uniform (i.e., isotropic). Consequently, the uniformly distributed polymer segments (resistance centers) allow relatively simple theories to tackle the hydrodynamics inside the layer, and analytical solutions of the governing transport and electrostatic equations have been * E-mail:
[email protected]. (1) Wiersema, P. H. On the Theory of Electrophoresis; Rijkuniversiteit: Utrecht, The Netherlands, 1964. (2) Duchin, S. S.; Derjaguin, B. V. Electrophoresis; Nauka: Moscow, 1976 (in Russian). (3) Hunter, R. J. Zeta Potential in Colloid Science: Principles and Applications; Academic Press: London, 1981. (4) Williams, R. A., Ed. Colloid and Surface Engineering. Applications in the Process Industries; Butterworth Heinemann Ltd.: Oxford, England, 1992. (5) Ohshima, H., Furusawa, K., Eds. Electrical Phenomena at Interfaces; Marcel Dekker: New York, 1998. (6) Wiersema, P. H.; Loeb, A. L.; Overbeek, J. Th. G. J. Colloid Interface Sci. 1966, 22, 78. (7) Overbeek, J. Th. G. AdV. Colloid Sci. 1950, 3, 97. (8) Booth, F. Nature 1948, 161, 83. (9) Booth, F. Proc. R. Soc. London, Ser. A 1950, 203, 514. (10) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 1978, 77, 1607. (11) O’Brien, R. W.; Hunter, R. J. Can. J. Chem. 1981, 59, 1878. (12) O’Brien, R. W. J. Colloid Interface Sci. 1983, 92, 204. (13) Dukhin, S. S. In Research in Surface Forces; Consultants Bureau: New York, 1963, p 313. (14) Dukhin, S. S.; Shilov, V. N. AdV. Colloid Interface Sci. 1980, 13, 153.
provided to examine the electrophoretic behavior of soft particles under the condition that polarization and relaxation are absent.15-18 An approximate analytical mobility expression is also possible for the exponential distribution of segments of adsorbed polyelectrolytes,19 but more cumbersome derivations are obviously necessary when a (perpendicular) distribution of charges and/or of the dielectric constant in the surface layer is assumed.20,21 Inhomogeneities manifested by continuous spatial profiles of the volume fraction of polymer segments, the density of fixed charges, and the friction coefficient across the interphasial region (the so-called diffuse interface) have recently been taken into account recently numerically.25-28 (In the latter study, the possibility of partial dissociation of the ionogenic groups as well as specific interactions between these sites and electrolyte ions are also considered.) The relaxation phenomenon is also important in layers with a high charge density and makes us more reliant on numerical evaluations of the electrophoretic mobility.27,29-32 Last, the polarization caused by the so-called anomalous (additional) surface conductivity is also manifested in soft layers.33 An advanced method has been established for the electrokinetic characterizationofisotropicpolyelectrolytelayersonflatsurfaces34-36 (15) Ohshima, H. AdV. Colloid Interface Sci. 1995, 62, 189. (16) Ohshima, H. J. Colloid Interface Sci. 1994, 163, 474. (17) Ohshima, H. Electrophoresis 1995, 16, 1360. (18) Ohshima, H.; Nakamura, M.; Kondo, T. Colloid Polymer Sci. 1997, 270, 873. (19) Ohshima, H. J. Colloid Interface Sci. 1997, 185, 269. (20) Hsu, J-P.; Fan, Y-P. J. Colloid Interface Sci. 1995, 172, 230. (21) A similar situation is encountered when evaluating the electrokinetics of bare spheroidal colloidal particles with a charge distribution along their surface22-24 deviating from that of otherwise equivalent but uniformly charged particles. (22) Anderson, J. L. J. Colloid Interface Sci. 1985, 105, 45. (23) Fair, M. C.; Anderson, J. L. J. Colloid Interface Sci. 1989, 127, 388. (24) Pysher, M. D.; Hayes, M. A. Langmuir 2005, 21, 3572. (25) Duval, J. F. L.; van Leeuwen, H. P. Langmuir 2004, 20, 10324. (26) Duval, J. F. L. Langmuir 2005, 21, 3247. (27) Hill, R. J.; Saville, D. A.; Russel, W. B. J. Colloid Interface Sci. 2003, 258, 56. (28) Duval, J. F. L; Ohshima, H. Langmuir 2006, 22, 3533. (29) Hill, R. J.; Saville, D. A.; Russel, W. B. J. Colloid Interface Sci. 2003, 263, 478. (30) Hill, R. J.; Saville, D. A. Colloids Surf., A 2005, 268, 31. (31) Hill, R. J. Phys. ReV. E 2004, 70, 051406. (32) Saville, D. A. J. Colloid Interface Sci. 2000, 222, 137. (33) Yezek, L. P. Langmuir 2005, 21, 10054. (34) Dukhin, S. S.; Zimmermann, R.; Werner, C. AdV. Colloid Interface Sci. 2006, 122, 93.
10.1021/la0635451 CCC: $37.00 © 2007 American Chemical Society Published on Web 04/07/2007
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by taking into account their surface conductance as well as the incomplete ionization of functional groups. However, this integrated approach is limited when polyelectrolyte layers are on spherical particles. To overcome this limitation, two models have been unified, which disregard alternatively the concentration polarization (induced dipole moment) due to the surface current or the hydrodynamic permeability of the layers.34 Nevertheless, the parameter reflecting the polarization effect has to be evaluated independently. We see that there are analytical and numerical solutions available for a theoretical description of the electrokinetics of simple and more complex soft particles, respectively. Notwithstandingly, the thickness of the polyelectrolyte layer has been a priori set to a constant value in all theoretical approaches, irrespective of the method and complexity of treating the layer. However, the sine qua non fact, namely, that polyelectrolyte molecules in a solution shrink and swell (with a direct impact on the polyelectrolyte layer structure and thickness), depending among other things on the solution ionic strength and pH, is omitted.37 In fact, as revealed by diffusivity or photon correlation spectroscopy (PCS) measurements,38,39 the hydrodynamic radius of colloidal spheres of polymer latices decreases with increasing ionic strength of the electrolyte solution. Apparently, it is a result of the charge shielding and accompanying shrinkage of flexible polyelectrolyte chains protruding from the latex surface into the solution. The hydrodynamic radius was also found to be reduced for nanometric silica particles at lower pH values and higher concentrations of salts, although this time it was considered to be due to an electrostatically mediated weakening of the surface hydration layer.40 In both cases, the lower limit of the hydrodynamic radius approached the radius determined by TEM.41 In line with the “wet” diameter determinations by dynamic light scattering, anomalous viscosities of dilute silica dispersions have been explained by the surface conductance effect within a gel-like network at the particles’ surface.42 This cross-linked polyelectrolytic network would have such a porosity that the silica particles can irreversibly swell or shrink by osmotic forces, depending on the solution pH and electrolyte concentration, up to even four times the dry volume. Moreover, aside from the above data, special analyses of the electrophoresis data per se, acquired on polymer (electrophoretic fingerprinting)43 as well as metal oxide colloid particles (hypernetted chain integral function, including the excluded volume and electrostatic correlation effects, both neglected in the mean field approach to the electric double layer - Poisson-Boltzmann approximation),44 are indicative of the presence and variation of (35) Dukhin, S. S.; Zimmermann, R.; Werner, C. J. Colloid Interface Sci. 2004, 274, 309. (36) Dukhin, S. S.; Zimmermann, R.; Werner, C. J. Colloid Interface Sci. 2005, 286, 761. (37) The authors of the study34 hold to the constancy of the polyelectrolyte layer thickness while meaning “swelling” surfaces. (38) Seebergh, J. E.; Berg, J. C. Colloids Surf., A 1995, 100, 139. (39) Gittings, M. R.; Saville, D. A. Colloids Surf., A 1998, 141, 111. (40) Sasaki, S.; Maeda, H. J. Colloid Interface Sci. 1994, 167, 146. (41) It has however been argued39 that in calculating the hydrodynamic particle size the effects of the particle charge and diffuse layer together with the associated polarization and relaxation effects (in the absence of an electric field) are usually omitted so that the colloids might not be considered to have a surface layer at all. Indeed, the hydrodynamic particle radius with increasing ionic strength was found to decline less gradually by taking into account the above effects (e.g., when increasing the electrolyte concentration above 0.1 mM, the hydrodynamic radius of the latices was only 4% larger than the TEM size and became almost constant). (42) Laven, J.; Stein, H. N. J. Colloid Interface Sci. 2001, 238, 8. (43) Marlow, B. J.; Rowell, R. L. Langmuir 1991, 7, 2970. (44) Attard, P.; Antelmi, D.; Larson, I. Langmuir 2000, 16, 1542.
Sˇ kVarla
a hairy, gel, or porous layer on the surface of colloids. We should also not forget independent experiments, such as the surface force measurements45-52 and aggregation tests,53-58 giving us another hint of the softness of the surface of silica (with its curious polywater character59) but also of alumina particles,60-64 let alone the spheres of polymer latices. It seems, therefore, that the accepted notion of the ideal surface of synthetized colloidal particles, including even model colloids, is illusory. On one hand, it should be mentioned that simple analytical/approximative expressions of the electrophoretic mobility have already been applied successively for some specific types of soft particles (e.g., hydrogels or biocolloids) assuming their surface layer is isotropic and of a consistent thickness. On the other hand, the mere fact that there is no direct (implicit or explicit) theoretical linkage of solution parameters such as the pH and ionic strength as well as the solubility parameters of the polymer backbones with the structure of polyelectrolyte layers hinders the derivation and verification of more proper expressions of the electrophoretic mobility (irrespective of whether analytical or numerical) not only for the “true” soft particles but also for any colloids whose surfacial part would have been expected to swell. In the pioneering publication of Pincus,65 simple analytical scaling laws (power asymptotics) for the mean thickness of polyelectrolyte layers (i.e., brushes in which the polyelectrolyte chains are tethered on the surface by one end) were obtained in the limits of high and low densities of fixed charge immobilized (quenched) on the brush chains (i.e., strong homopolyacids) approximating the numerical solutions of the electrostatic meanfield equations (full Poisson-Boltzmann equation) coupled to the Gaussian elasticity of the polyelectrolyte chains, as published by Miklavic and Marcˇelja66 and Misra et al.67 For the sake of simplicity, a uniform distribution of polymer monomers throughout the dense brush (i.e., a step function concentration profile or a simple box model) was considered, ignoring the detailed variation in the concentration of the monomers and counterions and replacing it with constants over certain regions. (The excluded-volume interactions were also omitted.) To go beyond the boxlike model and to describe the internal structure of polyelectrolyte brushes, a self-consistent field theory (45) Peschel, G.; Belouschek, P.; Mu¨ller, M. M.; Mu¨ller, M. R.; Ko¨nig, R. Colloid Polymer Sci. 1982, 260, 444. (46) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991, 353, 239. (47) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Langmuir 1992, 8, 1831. (48) Horn, R. G.; Smith, D. T.; Haller, W. Chem. Phys. Lett. 1989, 162, 404. (49) Horn, R. G.; Smith, D. T. J. Non-Crystal. Solids 1990, 120, 72. (50) Grabbe, A.; Horn, R. G. J. Colloid Interface Sci. 1993, 157, 375. (51) Chapel, J. P. Langmuir 1994, 10, 4237. (52) Vigil, G.; Xu, Z.; Steinberg, S.; Israelachvili, J. J. Colloid Interface Sci. 1994, 165, 367. (53) Allen, L. H.; Matijevicˇ, E. J. Colloid Interface Sci. 1969, 31, 287. (54) Allen, L. H.; Matijevicˇ, E. J. Colloid Interface Sci. 1970, 33, 420. (55) Depasse, J.; Watillon, A. J. Colloid Interface Sci. 1970, 33, 430. (56) Harding, R. D. J. Colloid Interface Sci. 1971, 35, 172. (57) Kobayashi, M.; Juillerat, F.; Galletto, P.; Bowen, P; Borkovec, M. Langmuir 2005, 21, 5761. (58) Kobayashi, M; Skarba, M; Galletto, P.; Cakara, D.; Borkovec, M. J. Colloid Interface Sci. 2005, 292, 139. (59) Yaminsky, V. V.; Ninham, B. W.; Pashley, R. M. Langmuir 1998, 14, 3223. (60) Beattie, J. K.; Cleaver, J. K.; Waite, T. D. Colloids Surf., A 1996, 111, 131. (61) Karaman, M. E.; Pashley, R. M.; Waite, T. D.; Hatch, S. J.; Bustamante, H. Colloids Surf., A 1997, 130, 239. (62) Ducker, W. A.; Xu, Z.; Clarke, D. R.; Israelachvili, J. N. J. Am. Ceram. Soc. 1994, 77, 437. (63) Runkana, V.; Somasundaran, P.; Kapur, P. C. AIChE J. 2005, 51, 1233. (64) Waite, T. D.; Cleaver, J. K.; Beattie, J. K. J. Colloid Interface Sci. 2001, 241, 333. (65) Pincus, P. Macromolecules 1991, 24, 2912. (66) Miklavic, S. J.; Marcˇelja, S. J. Phys. Chem. 1988, 92, 6718. (67) Misra, S.; Varanasi, S.; Varanasi, P. P. Macromolecules 1989, 22, 4173.
Electrokinetics of Silica Colloids
(SCF) was evolved by Zhulina et al.68 Later on, an analytical SCF approach was developed without the assumption of the distribution of counterions given by the local electroneutrality approximation (LEA, valid for strongly charged or densely grafted osmotic brushes), allowing an interpolation between the asymptotic regimes.69-71 Simultaneously, specific structural characteristics (nonmonotonic dependence of the brush thickness on the grafting density and the ionic strength of the solution) of the so-called annealed brushes of weak polyacid chains with the degree of ionization being dependent on the local concentration of protons inside the brush has been revealed by scaling-type conformation analysis72 as well as SCF theory.73 Interestingly, both of the above delineated theoretical descriptions of polyelectrolyte layers (accentuating the electrokinetics and the conformation, respectively) have been developing independently during the last almost 20 years. They have, nevertheless, one thing in common, namely, the apparent endeavor at progressing the complexity of viewing a polyelectrolyte layer and consequently the demands of an adeqate mathematical treatment. When evaluating this situation, an integration of most complex approaches seems to be possible and desirable; we could consider an actual but nontrivial concentration profile of the polyelectrolyte layer (derived by the SCF theory for polyelectrolyte brushes) and apply it in complete electrokinetic formulations. However, it is apparent that such an integration, although providing us with more or less exact results, would fall under the necessity of a numerical solution. Such a solution will be restricted only for preselected scales of parameters, which might not be fulfilled in experiments. However, because software solving the governing electrokinetic equations exactly (such as the MPEK computer program, see the Experimental Section) have become available to researchers, electrokinetics may give us more detailed information about the structure of surfacial parts of colloids. Nevertheless, despite of primacy and availability of more exact theories and associated (numerical) solutions, a question is whether or to what degree the analytical electrokinetic theories that are simplified under specific conditions may benefit from the conjunction with corresponding polyelectrolyte brush theories to explain experimental electrophoretic data taken on swelling collodal particles under these conditions. It has been proven that the electrophoretic mobility of micrometer-sized polystyrene particles coated with an ionic surfactant74 or microgel94 particles within the Smoluchowski regime (where all of the electric fieldinduced relaxation/polarization phenomena are marginal) scales with the electrolyte concentration according to the approximative electrophoretic theory of soft particles combined with the scaling theory of quenched brushes, with respect to the homogeneity axiom. However, for submicronal colloids, the above phenomena complicating the analysis must be taken into account. To resolve the possible existence of the polyelectrolytic network inherent in the surface of colloids, as manifested exclusively through their electrokinetics, the electrophoretic mobility of model nanometric silica spheres is measured in aqueous KCl solutions and analyzed by applying the theory of hard particle electro(68) Zhulina, E. B.; Borisov, O. V.; Birshtein, T. M. J. Phys. II 1992, 2, 63. (69) Zhulina, E. B.; Borisov, O. V. J. Chem. Phys. 1997, 107, 5952. (70) Zhulina, E. B.; Borisov, O. V. Macromolecules 1998, 31, 7413. (71) Zhulina, E. B.; Wolterink, J. K.; Borisov, O. V. Macromolecules 2000, 33, 4945. (72) Zhulina, E. B.; Birstein, T. M.; Borisov, O. V. Macromolecules 1995, 28, 1491. (73) Israels, R.; Leermakers, F. A.; Fleer, G. J. Macromolecules 1994, 27, 3087. (74) Lie´tor-Santos, J. J.; Ferna´ndez-Nieves, A.; Ma´rquez, M. Phys. ReV. E 2005, 71, 042401.
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phoresis along with the simple theory of soft particle electrophoresis, combined further with a scaling theory of polyelectrolyte brushes.
Theory Hard Particle Electrophoresis. The electrophoretic mobility (i.e., the velocity per unit applied electric field) of a rigid nonconducting colloidal particle with no surface structure is usually expressed by the simple Smoluchowski equation75
µe )
0rζ η
(1)
where 0 is the permittivity of vacuum and r and η are, respectively, the relative permittivity and the viscosity of the electrolyte solution. The zeta potential ζ of the particle is defined as a potential at the so-called slipping or shear plane where the liquid velocity relative to the electrophoresing particle is zero. If the shear plane is located at the particle surface, then ζ becomes equal to the surface potential φ0. For a spherical particle of radius a, the average and low φ0 or ζ can be related to the density of charge σ “smeared out” on the surface by using the linearized solution of the Poisson-Boltzmann equation
φ0 )
σ 0rκ(1 + 1/κa)
(2)
where κ-1 stands for the Debye-Hu¨ckel screening length of the 1:1 electrolyte solution
κ)
( ) 2e2c 0rkT
1/2
(3)
with c being the bulk concentration of the electrolyte ion and e being the electron charge. On one hand, the Smoluchowski equation expresses the limiting mobility for large particles whose dimension is much larger than κ-1 (κa . 1) so that the particle surface can be considered to be locally planar. On the other hand, the mobility of relatively small spheres (κa , 1) is expressed by the Hu¨ckel equation, which differs from eq 1 only by a factor of 2/3:76
µe )
20rζ 3η
(4)
For low ζ potentials, Henry derived a general mobility equation77
µe )
0 r ζ f(κa) η
(5)
where the function
f(κa) ) 1 - eκa{5E7(κa) - 2E5(κa)}
(6)
accounts for the so-called retardation effect due to the movement of the ionic atmosphere and the associated fluid surrounding the charged particle in a direction opposite to the particle’s movement. Hence, eq 5 is valid for any value of κa and bridges the above limits because it reduces to eq 1 for κa .1 and to eq 4 for κa ,1. The Henry function that embraces exponental integrals of the nth order En(κa) can be simplified by the Ohshima approximation78 (75) von Smoluchowski, M. Z. Phys. Chem. 1918, 92, 129. (76) Hu¨ckel, E. Phys. Z. 1924, 25, 204. (77) Henry, D. C. Proc. R. Soc. London, Ser. A 1931, 133, 106. (78) Ohshima, H. J. Colloid Interface Sci. 1994, 168, 269.
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5308 Langmuir, Vol. 23, No. 10, 2007
f(κa) )
{
}
2 1 1+ 3 2[1 + (δ/κa)]3
(7)
with the δ parameter being a constant (2 to 3) or by a polynomial function of order 3.79 Along with the retardation effect, the allowance for the relaxation or polarization effect (i.e., an asymmetry or distortion of the ionic astmosphere around the particle due to its movement) causes the electrophoretic mobility of particles with a high potential to be further reduced. The O’Brien and White10 numerical solution of the standard model for bare particles is obviously adopted. However, because the “normal” conduction associated with the charge transfer above the shear plane through the mobile portion of the edl is taken into account only in the O’Brien and White approach, lower values of the mobility (or higher zeta potentials) will be expected after considering the additional surface conductance inside the shear plane, at least for electrolyte concentrations below 10-3 M (i.e., at moderate κa values). Therefore, after making allowance for the additional surface conduction, as done by Semenikhin and Dukhin.,80,81 a simple analytic equation can be applied, which is referred to as the Lyklema, Minor, and van der Wal (LMW) equation, for large potentials (e|ζ|/kT > 2):34,82
|µe| )
0r|ζ| 1 + Du 0rkT 4Du e|ζ| ln cosh η 1 + 2Du eη 1 + 2Du 4kT
(
)
(8)
which can also be rewritten in another form:
|µe| )
0r|ζ| 0rkT 2Du + η eη 1 + 2Du
(9) {2 ln 2 - 2 ln[1 + exp(e|ζ| 2kT)]}
The dimensionless Dukhin number Du accounts for the degree of the edl polarization for curved interfaces. The difference between the O’Brien and White and the Semenikhin and Dukhin results generally increases with the surface charge density. For Du , 1 the effect itself is negligible, and eqs 8 and 9 becomes eq 1, whereas for Du . 1 any dependence of µe on ζ may disappear. (For highly charged polystyrene latices, eq 8 yields larger ζ potentials as compared with these obtaine from the O’Brien and White numerical solution).83 Soft Particle Electrophoresis. Ohshima84 derived the following approximate analytic expression of electrophoretic mobility for a hard spherical core coated with a weakly (in order to neglect the relaxation effect) charged polyelectrolyte layer of thickness d (so that the outer radius of the whole soft sphere would be b ) a + d), assuming that (i) the fixed charge density Ffix and the so-called softness parameter 1/λ of the polyelectrolyte layer are constant, (ii) the slipping plane is located on the (uncharged) particle core, (iii) electrolyte ions can penetrate into the polyelectrolyte layer but not the particle core, (iv) the relative permittivity is the same both inside and outside the polyelectrolyte layer, (v) functional groups of the polyelectrolyte chains are completely dissociated, and (iv) the density of the polymer segments is low (not influencing the applied electric field): (79) Deshikan, S. R.; Papadopoulos, K. D. Colloid Polymer Sci. 1998, 276, 117. (80) Dukhin, S. S.; Semenikhin, N. M. Kolloid. Zh. 1970, 32, 366. (81) Dukhin, S. S. AdV. Colloid Interface Sci. 1995, 61, 17. (82) Lyklema, J. In Fundamentals of Colloid and Interface Science; Academic Press: London, 1995; Vol. 11, Chapter 4. (83) Moncho, A.; Martı´nez, F.; Hidalgo-A Ä lvarez, R. Colloids Surf., A 2001, 192, 215. (84) Ohshima, H. Electrophoresis 2006, 27, 526.
[
(
)
Ffixb2f0 a3 κb 2L2 2L3 a3 e E (κb) 1+ 2 2 eκbE5(κb) 3 3 2η 6b3 3L 2b λ b L1 1 L3 2L2 2 1 a3 1 1 + 1+ 3+ + + 1+ 2 3κb λ2b2L κb 3κbL κb 2b (κb) 1
µ)
( )] L +L F a (1 + κb) 1 e 1 - )f (κ, λ) + ( L κa [ 4ηλ b (1 + κa) L -L (1 + κa1 )f (-κ, λ)} - ( L ){(1 - κa1 )f (κ, -λ) + F (1 + κb) (1 + κa1 )f (-κ, -λ)}] - 3ηλ (1 + κa)e ([ 1 - LL ){(1 - κa1 )f (κ) + (1 + κa1 )f (-κ)} L 1 1 1 - )f (κ) + (1 + )f (-κ)} (10) L {( κa κa ]
(
1
2
-κ(b - a)
fix 3 3
3
) ){(
4
1
3
1
4
1
1
1
-κ(b - a)
fix 2
1
3
2
2
1
3
3
3
1
The definitions of terms f0 to f3 and L1 to L4 can be found in Ohshima’s article. The rather lengthy general equation (eq 10, written in terms of the exponential integrals) reduces to simpler expressions in various limiting situations. For the so-called large, soft particles (i.e., when a .1/λ, a . 1/κ, d . 1/λ, and a . 1/κ), the following expression holds, which considers the averaged Donnan potential φDON in the polyelectrolyte layer, the potential between the layer and the surrounding solution φ0, and the effective Debye-Hu¨ckel parameter of the layer κm:15 µe )
[
]
20r (φ0/κm) + (φDON/λ) Ffix 1 1+ + 2 (11) 3 3η (1/κm) + (1/λ) 2(1 + (d/a)) ηλ
φDON )
kT ln[crel + xcrel2 + 1] e
φ0 ) φDON +
(12)
kT [1 - xcrel2 + 1] ecrel
(13)
κm ) κx{1 + crel2} 4
(14)
where crel ) Ffix/(2ec). When assuming a thin polyelectrolyte layer such as a >> d, eq 11 can be recast as
µe )
0r (φ0/κm) + (φDON/λ) Ffix + 2 η (1/κm) + (1/λ) ηλ
(15)
For low potentials, eq 15 takes the form
µe )
Ffix ηλ
2
[
1+
(λκ) (11++(λ/2κ) (λ/κ) )] 2
(16)
Apparently, eqs 15 and 16 do not depend on the polyelectrolyte layer thickness d because of the initial restriction κd . 1 and thus is insensitive to the position of the shear plane and the zeta potential at this plane; therefore, the zeta potential concept loses its physical meaning. However, when a charge is considered on the particle core together with Ffix, eq 15 is extended:
µe )
0r (φ0/κm) + (φDON/λ) Ffix + 2+ η (1/κm) + (1/λ) ηλ -λd -κmd 80rkT /κm) eζ (e /λ) - (e (17) tanh 2 2 ηλe 4kT (1/λ ) - (1/κ ) m
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It is evident that the third term of eq 17 is due to the zeta potential of the core. This term becomes negligible for sufficiently large d with respect to κ-1 unless the core charge is very high. It should be pointed out that polyelectrolyte layers were considered up to now, for which the product λd . 1. However, the general equation (eq 10) merges into the following approximate analytic expression for the particle electrophoretic mobility in the limits a f ∞ and κ f ∞ (a “plate-like” soft particle84)
µe )
Ffix
[1 - sech(λd)] ηλ2
(18)
which hold for 1/λ . 1/κ but λd may be arbitrary. It should also be noticed that the Henry function does not appear in eq 18 although it has been taken into account when formulating the problem. (It does appear in some of Ohshima’s other approximations of eq 10.) If we assume that Ffix ) Q/{4π/3[(a + d)3 - a3]} with [(a + d)3 - a3] ≈ 3a3d for a . d and that 1/λ . d, then the hyperbolic secant can be expanded in a Taylor series around λd ) 0, and eq 18 simplifies to the form74
µe ≈
Qd 8πa2η
(19)
To account for the surface current in the layer or to confer the hard surface with hydrodynamic permeability, eqs 8 and 15 can be combined as follows:34
0r (φ0/κm) + (φDON/λ) 1 + Du | | η (1/κm) + (1/λ) 1 + 2Du eφ0 0rkT 4Du κ 1 + (ee|φ0|/2kT - 1) (20) ln cosh eη 1 + 2Du 4kT λ
|µe| )
[
]
The last term of eq 20 in squared brackets represents the additional effect of capillary osmosis due to the hydrodynamic permeability of the grafted polyelectrolyte layer. When Du is sufficiently small and 1/λ is large, Ohshima’s model can be used. However, when 1/λ is sufficiently small and Du is not small, the second term in eq 15 may be disregarded, and the LMW model applies. Furthermore, when 1/λ , 1/κ, the term [(φ0/κm) + (φDON/λ)]/ [(1/κm) + (1/λ)] in eq 20 may be replaced by φ0. Finally, when considering any degree of dissociation (deprotonation reaction of anionic sites) in the polyelectrolyte layer, the following equations should be used to express φDON, φ0, and κm as function sof pH and the concentration of the electrolyte solution:34,36
(x
10pH - pK 2
e-(eφDON/kT) )
1+4
|Ffix| pK- pH 10 -1 ec
)
(21)
φ0 - φDON ) kT expR (pH)|tanh{eφDON/2kT}| + R (pH) - 1 (22) ln e R-(pH)
κm ) κ
x
1-
R- -eφDON/2kT e 2
(23)
where the polyelectrolyte fractional charge R j is defined as
[ (x
1 R ) 1+ 2 -
|Ffix| pK - pH 1+4 -1 10 ec
)]
-1
(24)
Scaling of Polyelectrolyte Brushes. Considering a homogeneous brush layer of polyelectrolyte chains, with one ends grafted on a surface and the other ends located at the outer boundary of the brush, the constant monomer concentration cp is given as σ*N/L. (σ* is the mean grafting density, N is the number of monomers per chain, and L is the layer thickness.) Similarly, the charge density due to the polyelectrolyte chains with fixed (i.e., quenched) charge is σ*fN/L. (f is the fraction of elementary charged monomers.) Next, when assuming counterions of the polyelectrolyte brush with the mean concentration cci and the (added) salt counterions at the finite concentration of monovalent and completely ionized salt molecules cs (with the corresponding Debye-Hu¨ckel screening lengths κci-1 ) (4πlBσ*f1/2/a*)-1/2 and κs-1 ) (8πcslB)-1/2, with lB being the Bjerrum length and a* being the chain monomer unit length), four limits can be distinguished, depending on the counterions’ distributions.65,85-87 If cci < cs, then the salt ions screen the brush charge, and when L . κs-1 or cci , cs (the strong screening limit, SSL), the ions are confined inside the brush and a simple scaling law for the equilibrium brush layer thickness can be derived by balancing the swelling effect of the counterion entropy and the chain elasticity:65
LSSL = (fa*)2/3N
( ) σ* cs
1/3
(25)
Notice the shrinkage of the brush induced by the added salt ions according to the relatively weak power law L ∝ cs-1/3. Importantly, cs does not appear in scalings predicted for the remaining three limits of quenched brushes. For example, in the so-called strong charging or osmotic limit (SCL), when cci > cs, the external salt plays a marginal role in the behavior of the polyelectrolyte brush. If L . κci-1 (or cci . cs), then the counterions are again trapped inside the brush where cci is equal to the immobilized charge density σ*fN/L (i.e., fcp). The osmotic brush thickness was obtained in the simplest form:
LSCL = Na*f1/2
(26)
It should be pointed out that, for annealed brushes of weak polyacids in the SSL regime, eq 25 still holds, unlike the situation in the SCL regime where the brushes are even predicted to expand with increasing cs.72,73 Integration of the Theory of Soft Particle Electrophoresis and Polyelectrolyte Brushes. Lie´tor-Santos et al.74 have shown that the dependence of log(electrophoretic mobility of large polystyrene spheres) on log(1:1 electrolyte concentration) is linear, with a slope of -0.33. This is in conformity with the prediction provided by combining a limiting solution of the general electrophoretic theory of soft particles (eq 19) and a very simple scaling dependence of the brush layer thickness on the salt concentration (in the strong screening regime, eq 25). Therefore, µe is predicted to vary linearly with d and, when identifying d and LSSL, with cs as µ ≈ cs-1/3. Meanwhile, eqs 25 and 26 were proved experimentally by direct measurements of parameters N, f, L, and σ* (for polystyrene sulfonate chains covalently grafted on a silicon wafer by terminal trichlorosilane end-groups);86 in both the SCL and SSL regimes, the only adjustable parameter, namely, the monomer size a*, was found to attain a reasonable magnitude (0.22 and 0.29 nm, respectively), indicating the scaling prefactors in both equations (85) Prinz, C; Muller, P.; Maaloum, M. Macromolecules 2000, 33, 4896. (86) Tran, Y.; Auroy, P.; Lee, L-T. Macromolecules 1999, 32, 8952. (87) Wittmer, J.; Joanny, J. F. Macromolecules 1993, 26, 2691.
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5310 Langmuir, Vol. 23, No. 10, 2007
to be close to 1. Thus, inserting Q ) 4πa2fσ*Ne, one finally arrives at
µe ) AcS-1/3 A)
(2ηe )f
5/3
σ*4/3N2a*2/3
(27) (28)
Note that the combination with eq 25 eliminates the dependence of µ on d, as is also the case with eqs 15 and 16. Experimental Section Uniform silica microspheres were obtained in an aqueous stock dispersion (surfactant-free, 5% solids content corresponding to 1.48 × 1016 particles L-1) from Bangs Laboratories Inc. As stated by the producer, the microspheres were made from pure Si(OC2H5) reacted with water and ammonia, and because of the pure SiO2, it should have no surface-active impurities so that no cleanup is necessary before use. The mean diameter, as determined by SEM, was 150 nm with a standard deviation of 100). Therefore, to convert the electrophoretic mobilities (as input data) to zeta (ζ) potentials of silica particles, the modified Booth equation (MBE)3 for nonconducting spheres (accounting for both the retardation and relaxation correction terms to Smoluchowski’s equation, eq 1) was used. To simplify this conversion, the Henry function and the Booth relaxation correction functions were substituted by third-order polynomial functions, as proposed by Deshiikan and Papadopoulos.79 For κa > 6, the conversion was made by using the simple Zeta computer program developed by Kosmulski on the basis of the approximative analytic expression for the electrophoretic mobility (derived by Ohshima et al.98,99). In this program, the setting parameters are the molar conductivity of ions (0.00735 and 0.00764 m2 Ω-1 mol-1 for K+ and Cl-, respectively), dielectric constant, ionic strength, temperature and viscosity of aqueous solutions, and radius of particles under consideration. The reason that the widely accepted Mobility and Winmobil computer programs based on the O’Brien and White10 numerical solution providing exact conversions for bare colloids were not adopted here is simply the fact that both of these programs calculate mobilities from zeta potentials as input data and not vice versa. To verify the correctness of all conversions, the MPEK-0.01 computer program was utilized (kindly provided by R. J. Hill, Department of Chemical Engineering, McGill University, Montreal, and described in detail by Hill et al.27); it gives numerically exact solutions of the standard electrokinetic model of bare spherical colloids, but it is extended to include uncharged or charged polymer coatings characterized by Brinkman rheology. To carry out a curve fitting of the experimentally obtained electrophoretic mobilities, the TableCurve 2D automated curve fitting and equation discovery computer software from SPSS Inc. was used.
Figure 1. Experimental dependence of µe on (a) pH (in the absence of KCl) and (b) cKCl at pH ∼6.0 for fresh (b) and aged (O) silica colloids. The circles are arithmetic means, and vertical error bars are for (a) the whole range of data variation and (b) the standard deviation of the arithmentic means.
Results The experimental dependence of the electrophoretic mobility µe of fresh and aged silica dispersion on pH (in the absence of KCl) and on the KCl concentration cKCl is shown in Figure 1 (left and right, respectively). We can see a difference between both dependences in that the fresh silica particles carry a markedly higher charge than that of aged particles at pH below 7. Apparently, the mobility and its scattering decrease with an initial increase in the KCl concentration in both dispersions. However, the trend is suddenly reversed at cKCl above ca. 30 mM (probably due to the effect of electrolysis, as indicated by the enormous data scattering) for the aged silica dispersion. Nevertheless, minimal individual values of µe detected at KCl concentrations above 90 mM (not shown here) disclose a continuation of the generally descending trend; a linear part at lower KCl concentrations and a shoulder at cKCl above 0.7 mM are identifiable in the µe versus cKCl dependence. No such trend is observed with the fresh silica. To fit the experimental dependence of µe on cKCl by analytical functions presented in the theoretical part designed for a weakly charged surface with the relaxation effect being omitted, contributions due to this effect to the µe values should be numerically calculated and eliminated. Therefore, we first calculated ζ potentials from the average values of experimentally determined mobilities depicted in Figure 1 by using the modified Booth equation (MBE),79 which supplies the O’Brien and White numerical procedure (Experimental Section), taking into account both the retardation and relaxation effects for bare colloids. For κa > 6, the Zeta computer program was employed as well. Subsequently, the mobilities were recalculated in turn by disregarding the relaxation effect (Booth relaxation functions).
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Langmuir, Vol. 23, No. 10, 2007 5311
Table 1. Original (µe) and Corrected (µe*) Electrophoretic Mobilities and ζ Potentials of Silica cKCl (M)
κa
-µe (× 10-8 m2 V-1 s-1)
4.0000 × 10-5 8.0000 × 10-5 1.0000 × 10-4 2.8000 × 10-4 4.6000 × 10-4 6.4000 × 10-4 1.0000 × 10-3 1.2000 × 10-3 1.8000 × 10-3 2.1000 × 10-3 2.4000 × 10-3 2.7000 × 10-3 3.0000 × 10-3 6.0000 × 10-3 8.0000 × 10-3 9.0000 × 10-3 1.0000 × 10-2
1.5 2.2 2.5 4.1 5.3 6.2 7.8 8.6 10.5 11.3 12.1 12.8 13.5 19.1 22.1 23.4 24.7
3.63 3.17 2.97 2.36 2.26 1.82 1.85 1.86 1.79 1.82 1.62 1.57 1.29 1.45 1.73 1.72 1.79
Aged Silica 0.504 0.575 0.318 0.195 0.209 0.137 0.140 0.204 0.077 0.168 0.124 0.146 0.384 0.453 0.576 0.438 0.561
5.0000 × 10-4 1.0000 × 10-3 2.0000 × 10-3 1.0000 × 10-2 2.0000 × 10-2
5.5 7.8 11.1 24.7 34.7
2.99 2.83 2.46 1.99 1.67
Fresh Silica 0.296 0.140 0.166 0.09 0.05
sN - 1 (× 10-8 m2 V-1 s-1)
It can be done directly in the above way at least when bare surfaces are considered, for which the MBE has originally been evolved. The values of the ζ potential as well as the original and corrected mobility values, µe and µe* respectively, are summarized in Table 1, together with their percentage differences. (For aged silica, the data for cKCl above 3 mM were excluded from this analysis because of the possible effect of electrolysis.) We can see that the corrected values are all higher than the uncorrected ones, as expected, especially at lower KCl concentrations where the surface charge of particles is high. The lower the cKCl (and κa), the greater the difference, apparently reflecting the hindering effect of the relaxation effect of electrophoresing particles. The difference, even being above 20% for the lowest cKCl, indicates that the relaxation effect is not negligible for silica and hence the correction is necessary. However, the difference (the surface charge of silica) is still believed to be low enough to ensure that the corrections by MBE are comparable to those determined by concurrent numerical procedures. Actually, for all zeta potentials in Table 1, the MPEK computer program with the polymer layer thickness set to zero returned the same mobilities (with the error within ca. 0.5%) as those in Table 1. A possible role of the additional surface conductance and the surface layer for the whole edl polarization of silica colloids can be estimated as follows. First, assuming bare surfaces, for Du ≈ 0.2 (the value obtained for silica colloidal dispersions at pH 5.5 and 10-3 M NaCl from conductivity measurements by Sonnefeld et al.),88 the ζ potential calculated from eq 8 for the experimental value of µe of fresh silica at 10-3 M KCl (-2.83 × 10-8 m2 V-1 s-1) is -48.4 mV. This value, although possibly not relevant to our experiments, is a little bit lower than that calculated by the MBE (-49.6 mV, Table 1). Therefore, in the absence of any other data we can only expect that the surface conductance does not enhance the edl polarization of silica. Second and more importantly, it follows from calculations according to eqs 20-24 that the decrease in µe caused by the polarization effect is not modified considerably in the presence of a strongly charged polyelectrolyte layer (Ffix/ec ) 40, 1/λ ) (88) Sonnefeld, J.; Lo¨bbus, M.; Vogelsberger, W. Colloids Surf., A 2001, 195, 215.
-ζ (mV)
-µe* (× 10-8 m2 V-1 s-1)
difference (%)
84.2 67.7 62.0 43.5 40.0 30.8 30.6 30.4 28.5 28.6 25.3 24.2
4.59 3.74 3.49 2.56 2.42 1.90 1.93 1.94 1.86 1.88 1.67 1.61
26.6 18.3 16.3 8.3 7.1 4.4 4.2 4.1 3.6 3.5 2.9 2.6
55.7 49.6 40.0 29.1 23.4
3.37 3.12 2.61 2.10 1.70
12.9 10.2 6.2 5.5 1.8
0.6 nm), especially when the capillary osmosis due to its hydrodynamic permeability is counted.34 This is in line with results of the numerical study by Saville.32 Adapting the O’Brien analytical asymptotic methodology (set out for smooth particles), he showed semianalytically that the distribution of immobile charge throughout the homogeneous and hydrodynamically highly permeable and charged fuzzy polymer layer (with the thickness of half the Debye screening length) provides an electrophoretic mobility that is lower than the Smoluchowski mobility limit (which ignores the polarization and relaxation effects) but higher than that when the same amount of charge is on the surface of the smooth particle; when the hydrodynamic permeability decreased, the mobility approached that for the smooth particle. More exactly, the MPEK program has provided the mobility of (bare) aged silica at the lowest KCl concentration of 5 × 10-4 M (with the recalculated surface charge density of 2.538 × 10-3 C m-2) to be -3.65 × 10-8 m2 V-1 s-1 (Table 1). With the same amount of charge being hypothetically spread in a homogeneous layer over the otherwise neutral silica surface, the MPEK results are -3.68 × 10-8 and -3.88 × 10-8 m2 V-1 s-1 for layer thicknesses of 1 and 10 nm, respectively. The corresponding segment (and charge) densities and the Brinkman screening lengths (as input data) are 0.026 M and 5.97 nm and 0.0026 M and 18.9 nm, respectively. The latter (absolute) values of mobility due to the presence of uniform polyelectrolyte layers are higher than those of the equivalently charged bare surface. However, the difference is at most a few percent for a 10-nm-thick layer. Therefore, it can be concluded that the above corrections of experimental values of electrophoretic mobility for polarization are justifiable, irrespective of whether the surfaces of the particles are hard or soft, especially when realizing a rather low surface charge of silica. To check the above correction procedure independently, the ζ potentials of fresh silica calculated from the measured µe data (Table 1) by the MBE polynomials are compared with surface potentials determined by measuring the surface forces between a silica microsphere and a silica substrate by AFM in NaCl
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5312 Langmuir, Vol. 23, No. 10, 2007
Figure 2. ζ potential of fresh silica calculated from the experimentally determined electrophoretic mobilities by using the MBE (b), the diffuse layer potential calculated by using the 1 - pK basic Stern model with parameters from ref 57 (-), and the surface potential determined (in the plane of charge situated at the plane of closest approach, see ref 89) by fitting the AFM surface force curve between silica surfaces (O) and by the simple mass-action model (shaded area) (adapted from ref 46) as a function of the 1:1 electrolyte concentration.
solutions at pH ∼5.7.46,47,89 As can be seen in Figure 2, there is fair agreement between the ζ potentials (full circles) and surface potentials (open circles) as a function of electrolyte concentration. Surface potentials of silica are also shown in Figure 2 (shaded area) for comparison, as calculated by Pashley46 himself according to a simple mass-action model (assuming that the ion sizes are smaller than an individual silanol site and there is no binding of alkalic ions to the silanol sites). Despite the model’s simplicity, the calculated values of the surface potential vary within 10 to 15 mV because of the uncertainty in the H+ binding constant pKa ) 6.8 ( 0.2. However, even better congruence is found between the zeta potentials and diffuse layer potentials (solid line in Figure 2), as calculated by the simple 1 - pK basic Stern model (in which the surface charge of silica in KCl solutions is assumed to originate only from the deprotonation of silanols) with approved input parameters, namely, the Stern layer capacitance Cs ) 2.9 F m-2, the total number density of surface sites Γ0 ) 2 × 1018 m-2, and the intrinsic equilibration constant Ks ) 2 × 10-7 (pKs ) 6.7).57 Fresh Silica. The experimental dependence of the averaged electrophoretic mobility (corrected for the relaxation effect, see Table 1) of fresh silica on the KCl concentration is shown in Figure 3. It was fitted by available analytical functions for µe. Assuming the hard surface of silica particles, a very bad fit indicated by the coefficient of determination r2 ) 0.016 is obtained with the equation obtained by combining eqs 2, 5, and 7 (equating φ0 with ζ)
µe )
{
}
2σ 1 1+ 3ηκ[1 + (1/κa)] 2[1 + (δ/κa)]3
(29)
(89) A two-stage force versus distance (force curve) was detected: an electrostatic repulsion at longer distances and a strong repulsion at short distances. It was expected that the short-range repulsion is at least partially resolvable because the electrostatic force by simply positioning the origin of charge at a negative surface separation is equal to half the maximum roughness (∼1.5 nm) of the silica substrate. Moreover, because the extra force in the last few nanometers before contact was much smaller at pH 2 (zero charge) than at pH 5.6 (in 10-2 M NaCl), the effect of the swelling of a gel layer promoted at the higher surface charge density could also participate in the force. (This means further moving the origin of charge to negative values). The surface potential was evaluated by fitting the surface force curves at infinite separation, assuming both the plane of charge situated at the plane of closest approach and that of charge shifted -1.5 nm relative to the position of constant compliance.
Figure 3. Experimental dependence of µe* on cKCl for fresh silica colloids with the best fit (-) by using (a) eq 29 and (b) eq 15. The two outer (--) and two inner (- -) lines denote the 95% prediction and confidence intervals, respectively.
as can be seen from Figure 3a (the solid line). A much better fit (Figure 3b, r2 ) 0.901) is observed when the function derived for soft particles (eq 15) is used. Fitting parameters Ffix/e and 1/λ are equal to 1.679 × 10-3 M and 1.09 × 10-8 m, respectively. It should be recalled that the position of the shear plane has been set to be invariable, being identified with the core surface of spherical colloidal particles, or indefinable up to now. According to the Gouy-Chapman theory, there is a simple relationship between the ζ potential at a distance ∆ from a flat surface (at the shear plane) and the potential φ0 at the surface:
ln tanh
( )
( )
zeφ0 zeζ ) ln tanh - κ∆ 4kT 4kT
(30)
By plotting ln tanh(zeζ/4kT) as a linear function of the electrolyte concentration represented by κ, the shear layer thickness ∆ (a constant by definition) can be determined from the slope of the straight line. Of course, we can adopt eq 30 only if the potential distribution is assumed not to be influenced by a layer at the surface. Moreover, it is valid for κa . 1, a condition that is not fulfilled entirely in the present experiments. However, the error that comes with the curvature is only ∼10% at κa ) 5.4. Whether the zeta potentials are taken as calculated from the present experiments or by using the basic 1 - pK Stern model, accounting for data for KCl concentrations up to 0.1 M (Figure 2), a nonlinear dependence is observed (Figure 4, triangles). This fact indicates an inward shift of the shear plane with increasing ionic strength. If so, the slope of the tangent (to the concave curve) drawn at a point corresponding to a value κ (∝ c1/2) or the derivative of the curve at this point will provide the position of the shear plane. By doing so for the given KCl concentrations, a series of shear plane positions are obtained, which decrease from ca. 4 nm at 5 × 10-4 M KCl to 0.55 nm
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Langmuir, Vol. 23, No. 10, 2007 5313
Figure 4. Experimental dependence of ln tanh(zeζ/4kT) on κ (4) and ∆ on cKCl (0, on a log-log scale) for fresh silica colloids.
at 0.1 M KCl. Interestingly, a straight line with a slope of -0.35 is obtained after plotting the ∆ versus cKCl dependence on the logarithmically scaled axes (Figure 4, squares). This means that the shear plane position scales almost as predicted by eq 25 (i.e., c-0.33). Nevertheless, the goodness of fit is worse (r2 ) 0.73). A decrease in the shear plane position with increasing electrolyte concentration has also been expected when analyzing surface electrical properties of a polystyrene latex.90 The decrease was explained by introducing a pH- and electrolyte-dependent hairy layer, although the Stern layer condunctance might also take effect. Also, the shear plane has been shown to shift inwardly with increasing c but only if the O’Brien and White calculus was not applied in the analysis.91 However, the dependence was not detailed in these articles. Aged Silica. The seemingly two-stage course of the experimental dependence in Figure 1b suggests a structural collapse on the aged silica surface at higher KCl concentrations, explained by a weakening (shielding) of the electrostatic repulsion between charged silanols accompanied by a strengthening of the polymerpolymer contacts. However, for the moment we will assume that any changes at the silica surface with cKCl are continuous. Figure 5 shows the whole experimental dependence of µe* versus cKCl for aged silica. In Figure 5a, it can be seen that eq 29 fits this dependence well for lower KCl concentrations but deviates markedly for higher KCl concentrations. The coefficient of determination r2 is equal to 0.941, and the only fitting parameter σ is found to be 2.39 × 10-3 C m-2. When analyzing the experimental data by using approximate analytical expressions of the electrophoretic mobility for soft particles, numerous assumptions are needed. In fact, the only assumption that could be made for certain is that the particle size dominates the layer thickness (a . d). As follows from Figure 5b, the goodness of fit of the data is better (r2 ) 0.987) when eq 15 is used instead of eq 29. The fitting parameters, Ffix/e and 1/ , are equal to 3.99 × 10-4 M and 2.07 × 10-8 m, respectively. λ We have to recall that eq 15 is valid under the assumption that a . d . 1/λ, 1/κ. These inequalities do not seem to be entirely valid because the fitted value of 1/λ is as high as ca. one part in four of the hard core radius of the silica spheres in question (taken to be 7.5 × 10-8 m). Finally, let the fitting expression be eq 27, which holds for a f ∞ and 1/λ . d . 1/κ and which on a log-log scale becomes a straight line with a slope equal to the exponent -1/3. As shown in Figure 5c, the fit is again quite good, with r2 being 0.976, but (90) Rasmusson, M.; Wall, S. J. Colloid Interface Sci. 1999, 209, 312. (91) Folkersma, R.; van Diemen, A. J. G.; Stein, H. N. Langmuir 1998, 14, 5973.
Figure 5. Experimental dependence of µe* on cKCl for aged silica colloids with the best fit (-) by using (a) eq 29, (b) eq 15, and (c) eq 27. The two outer ()-- and two inner (- -) lines denote the 95% prediction and confidence intervals, respectively.
the slope is lower (-0.25) than expected (-0.33). However, it reaches a value of -0.3 (r2 ) 0.989) if points in the linear part of the experimental dependence (cKCl e 1 mM) are used in the fitting procedure only. It should be mentioned that unlike eq 29, eq 27 does not account for the retardation effect. In fact, the fit of eq 29 would be much worse if µ ∝ c-1/2, as predicted by the Smoluchowski concept, disregarding the bracketed term in eq 2. When considering comparable goodness of fits by eqs 27 and 15 and the absence of corresponding PCS data for the hydrodynamic size, it is difficult to prove that the polyelectrolyte layer at the silica surface really shrinks when the KCl concentration increases. However, such information on silica is provided independently (e.g., in ref 40). Moreover, that the soft layer shrinks as predicted by Pincus and that the electrophoretic mobility mimics this behavior in agreement with Ohshima are well demonstrated with model soft colloidal (microgel) particles.94 Recalling the above idea of the collapse of the surface layer of aged silica at higher KCl concentrations, it is noteworthy to mention again the study by Garcia-Salinas et al.,92 who revealed a two-step decrease (shrinkage) in the PCS diameter of colloidal microgels evoked by increasing NaCl concentration (ca. 420 nm for up to 2 × 10-4 M and 300 nm for up to 10-1 M and a minimum of 200 nm at 1 M). The same trend was reflected in a decrease in the electrophoretic mobility. To describe the more complex decline in µe, the Ohshima approach was applied but with the incorporation of the (variable) total charge density (ratio (92) Garcia-Salinas, M. J.; Romero-Cano, M. S.; de las Nieves, F. J. J. Colloid Interface Sci. 2001, 241, 280.
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5314 Langmuir, Vol. 23, No. 10, 2007
of the total charge and the shell volume, determined by the conductometric titration and PCS, respectively) and the (also variable) friction coefficient into the calculation as input (not free) parameters.
Conclusions The electrokinetic theory based on the simplified approach of a homogeneous polyelectrolyte layer with invariable thickness around a bare rigid particle (soft particle electrophoresis) was successively used to examine the surface properties of soft colloidal particles such as latex particles coated with a stabilizing layer of long-chain fatty acid soaps, proteins, polypeptides, or a hydrogel.17,96,97 Obviously, the soft particle electrophoresis analysis is applied to fit the dependence of electrophoretic mobility on the electrolyte concentration, which is characterized by a “tail” (i.e., a finite value of mobility at the highest electrolyte concentrations). In this analysis, the fitting parameters are the fixed charge density and the softness of the layer, Ffix and 1/λ, respectively. It is not necessary to know the layer thickness. Electrokinetic studies of more complex microgels have also been published92 in which the layer thickness, varying discontinuously with the ionic strength, is quantified independently (as are Ffix and 1/λ) and inserted into the calculation as an input parameter. Recently, the electrophoretic mobility of latex particles covered with an ionic surfactant74 as well as of microgel particles94,95 was found to scale with the electrolyte concentration as µ ∝ c-1/3 (mimicking the same scaling law for the particle diameter) as predicted by the theory of soft particle electrophoresis combined with the scaling theory of polyelectrolyte brushes, underlining the value of the homogeneous layer formalism. It has been shown in the present study that, adhering to the simple idea of the homogeneous soft layer/brush, colloidal particles with a structure of polyelectrolytes inherent in their surface may be identified and characterized by electrokinetic experiments. Having a prejudice in favor of the ideal (hard) character of colloids, the electrokinetic (ζ) potential at the surface of fresh silica particles under study, as determined from the electrophoretic mobility, is commensurate with the surface potential evaluated by independent procedures for other silicas. Nevertheless, the experimental dependence of the electrophoretic mobility on the 1:1 electrolyte (KCl) concentration, corrected for the relaxation/polarization effect, is not resolvable within this comprehension (eq 29). When allowing the shear plane to move with increasing cKCl inwardly, however, its position is found to shift continuously from 4 to 0.5 nm, as a homogeneous polyelectrolyte brush shrinks according to the scaling theory in the strong screening regime (i.e., as c-1/3). Apparently, this means that the shear plane would be at the outer boundary of the (shrinking) hydrodynamically impermeable layer of linear (93) Sˇ kvarla, J. In Role of Interfaces in EnVironmental Protection. Barany, S. Ed.; Nato Science Series IV; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2003, Vol. 24, p 201. (94) Sierra-Martı´n, B.; Romero-Cano, M. S.; Ferna´ndez-Nieves, A.; Ferna´ndezBarbero, A. Langmuir 2006, 22, 3586. (95) Borget, P.; Lafuma, F.; Bonnet-Gonnet, C. J. Colloid Interface Sci. 2005, 285, 136. (96) Hoare, T.; Pelton, R. Polymer, 2005, 46, 1139. (97) Ho, C. C.; Kondo, T.; Muramatsu, N.; Ohshima, H. J. Colloid Interface Sci. 1996, 178, 442. (98) Ohshima, H.; Healy, T. W.; White, L. R. J. Colloid Interface Sci. 1982, 90, 17. (99) Ohshima, H.; Healy, T. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1983, 79, 1613.
polyelectrolyte molecules tethered to the silica surface. Good resolution can also be found if a rather dense but still homogeneous soft layer of constant thickness is considered to be present on the silica particles (eq 15), irrespective of the shear plane positioning. However, the expected layer thickness must be higher than the fitted parameter of hydrodynamic permeability by definition (i.e., 1/λ, ca. 10 nm). The surface charge density of aged silica particles is lowered by the aging process, probably as a result of the layer extension and/or loosening. Therefore, the hard particle approach with the shear plane localized invariably at the particle (core) surface is paradoxically more successful for aged than for fresh silica, especially at lower KCl concentrations. Holding to the idea of a soft layer (eq 15), even better resolution of the µe versus cKCl dependence is possible, although the layer thickness should exceed ca. 20 nm. Moreover, the combination of the approach of a rather loose isotropic soft layer (eq 19 in which 1/λ dominates other parameters) with that of a swelling polyelectrolyte brush (eq 25) provides a perfect fit of the experimental dependence of µe on cKCl (eq 27). The above findings prove that the silica surface is not hard by nature, but because various limiting approximations (assuming the soft character of the surface) describe experiments at a comparable level of reliability, which is quite high, the surface structure of silica appears to be at the transition between the structures ascribed to the limits. The combination of the soft layer and brush approach, suggested earlier by me93 and others,28 has already been found to apply to special colloidal microgels92,94 and latexes.74,95 However, the electrokinetic experiments with the latter (rather coarse) soft colloidal particles benefit their interpretation by virtue of the fact that they are included in the Smoluchowski regime where κa . 1. In the case of colloids with κa g 1, as in this study, a correction for the relaxation/polarization effect is necessary. In this respect, a successful fit of the mobilities for aged silica by the trivial eq 27 can be considered to be rather surprising and original. The argument is that there are a number of simplifications implemented analogously in both the soft and brush polyelectrolyte layer theories employed. In fact, along with a number of parameters such as the structure, permittivity, hydrodynamic permeability, and (complete) ionization of the polyelectrolyte layer, which are all considered to be constant, there are other factors that have not yet been taken into account. It can especially be presumed that nonelectrostatic interactions between polymer segments and water molecules and between the segments themselves are also influental, apparently differing for oxides and polymer latices. In this respect, the influence of solvent quality on the polyelectrolyte brush conformation through the Flory-Huggins terms (excluded volume and second virial coefficient), being embraced in the scaling theory of polyelectrolyte brushes, seems to be more important to consider than the role of the polyelectrolyte layer structure. Also, scaling theories are available for considering branched instead of linear polyelectrolyte molecules and so forth. Acknowledgment. This article is based upon work supported by the Slovak Scientific Grant Agency under grant no. 1/3347/ 06. I thank Dr. R. J. Hill for providing the MPEK computer program. LA0635451
