Quantitative Interpretation of Anomalous Coagulation Behavior of

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Quantitative Interpretation of Anomalous Coagulation Behavior of Colloidal Silica Using a Swellable Polyelectrolyte Gel Model of Electrical Double Layer Jiří Škvarla* Institute of Montaneous Sciences and Environmental Protection, Technical University in Košice, Park Komenského 19, 04200 Košice, Slovak Republic ABSTRACT: Electrolyte-induced coagulation of colloidal dispersions of silica has remained a puzzle for many decades, and it is widely considered anomalous from the viewpoint of traditional Gouy−Chapman theory of diffuse double layer and ζ-potential at ideal surfaces and of their electrostatic interaction (Derjaguin−Landau−Verwey− Overbeek, DLVO theory). It is suggested that this anomaly is caused by the fact that silica particles are covered with swellable gel layers. Theoretical stability ratios are calculated combining the attractive van der Waals and repulsive electrosteric interactions between core−shell (soft) model spheres with homogeneously distributed fixed charges in the shells and matched with the experimental ones measured for nonporous silica microspheres of different diameters (50, 150, and 320 nm) in an univalent electrolyte (KCl) of increasing concentration and pH (2.6, 4, 6, and 8). The variation in the shell thickness with the KCl concentration (mimicking the charged gel layer swelling) as the only adjustable parameter, deduced in such a way from data at pH 6 and 8, not only can explain the parallel experimental electrophoretic mobilities but also conforms itself to a scaling law derived from the thermodynamic theory of polyelectrolyte hydrogels. A resulting inapplicability of the DLVO theory and the ζ-potential concept for a quantitative predicting the coagulation kinetics of gel layer-covered colloids is discussed. In order to limit the number of fitting parameters, the zeta (ζ) potential converted from independently determined electrophoretic mobilities of colloids has been tried to replace the surface potential. However, a bad or no correlation between the two potentials could be found. For example, in the paper3 on coagulation of monodisperse polymer latexes by KBr, the values of ζ-potential varied from ca. −150 to −50 mV when the KBr concentration was increased while the values of surface potential fitted from the coagulation kinetics were on the order of only −10 mV. Apparently, should the ζ-potential replace the surface potential in the calculation of EDL repulsion, the coagulation rate would be underestimated grossly. This seems to be true not only for polymer colloids. For the same reason, when coagulation experiments on essentially monodisperse particles of synthetic titania and silica4 were analyzed by promoting the ζ-potential to the surface potential, the fitted value of Hamaker constant exceeded that expected theoretically by one order, and thus the theoretical radius of particles (becoming a next fitting parameter) had to be minimized drastically to the level of a few nanometers as an alternative. Later on, more sophisticated light scattering procedures had allowed researchers to precisely determine absolute rate constants of aggregation,5 and a (hopefully) more complete

1. INTRODUCTION In colloid science, two regimes of the coagulation process are discriminated phenomenologically, depending on the degree of screening of the surface charge by electrolyte ions in the diffuse and Stern part of the electric double layer (EDL) around colloid particles. In the so-called slow or reaction-limited coagulation, a partial screening is achieved by an electrolyte at low concentrations, letting the repulsive EDL interaction between the particles (colliding due to their thermal motion) be strong enough in comparison with the attractive van der Waals (vdW) interaction. When the electrolyte concentration attains a value (critical coagulation concentration) at which the surface charge and thus the EDL repulsion are suppressed to such an extent that the vdW attraction becomes important, the fast or diffusion-limited coagulation comes into operation. A rigorous theoretical description of the perikinetic coagulation of thermally encountering colloids stimulated by electrolytes in the slow regime was worked out in the past by defining the coagulation rate constant with the account of Derjaguin−Landau−Verwey−Overbeek or DLVO (i.e., EDL + vdW)1,2 and hydrodynamic interparticle interactions. Since then, numerous studies have appeared on model colloidal dispersions, all with the pair of key parameters of the EDL and vdW interactions, i.e., the surface potential and the Hamaker constant, respectively, adjusted to fit coagulation experiments. However, values of these parameters have been found to change in an unwanted way and/or to exceed reasonable physical ranges. © 2013 American Chemical Society

Received: April 23, 2013 Revised: June 17, 2013 Published: June 18, 2013 8809

dx.doi.org/10.1021/la401502f | Langmuir 2013, 29, 8809−8824

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surface charge segregation (the average surface potential and its standard deviation) were adjusted without verifiying them independently. Finding ourselves in this inconclusive situation, an equivalent alternative would be to let the charge distribute across a porous layer on the surface rather than over the essentially ion- and water-impenetrable surface. In fact, many hints of the presence of layers with a radially spread fixed ionic charges inside the surface of oxides and specially silica have been given. They had resulted in the elaboration of a semiquantitative porous double layer (PDL) concept15 that was refined.16,17 In the last paper,17 it was concluded that upon considering some uncertainties (e.g., the location of the electrokinetic shear plane) the site-binding-PDL model of the surface layer with a few parameters optimized, including its dielectric and penetration depth constants, can account simultaneously for experimental surface charge-pH and ζ-pH data for all analyzed oxide surfaces. (The values of ζ-potential were compared with the electrostatic double layer potential placed arbitrarily at 2 nm below the surface.) This however can be made possible by enhancing the model arbritrariness, too. In more recent studies,18 a gel layer or the “soft” particle model has been developed to match not only the surface charge density and transport properties (electrophoretic mobility and the primary electroviscous effect) but also those of colloidal metal oxides, preferentially silica. Unfortunately, although benefiting from the coupled analysis of titration and transport experimental data, the complexity of this model, allowing for the charge regulation (nonlinear Poisson−Boltzmann equation), partial flow of an incompressible fluid (Brinkman equation), and binding of ions (MUSIC-like complexation model) within the layer, required a numeric solution with many parameters. An important result of the latter model however is that a gel layer should exist at the surface of 9 nm colloidal Ludox silica, whose thickness is ca. 4 nm and shrinks down to almost zero when the concentration of KCl increases from 0.3 to 80 mM at pH 8.7. Just for silica, a lot of evidence about a sparse or fussy gel layer of the size spanning from that of nanometric silica particles themselves to values of several tens of nanometers on submicrometer particles has been gathered by experimental techniques applied to silica dispersions, such as static and dynamic light scattering (SLS, DLS), small-angle X-ray scattering (SAXS), field flow fractionation (FFF), and viscosity.19 The problem is that the presence of the gel layer has been perceived as a factor complicating the studied phenomena so that its character and variation with respect to parameters of external solution was not in the center of interest in most of these experiments. In fact, there is a dilemma as to the background electrolyte concentration which cannot be too high nor too low to prevent the aggregation and swelling, respectively. Altogether, one can expect that the progress in the interpretation of the surface charging on oxides should depend not so much on the development of ever-increasingly sophisticated models (of the double layer) but rather on the availability of more precise varied experimental information and that coagulation studies could also serve as a test for basic ideas. On one hand, the premise of the swellable gel layer on the silica surface suggests itself as a prima facie choice, but on the other hand, one is inclined to consider gel-layer models too complex for explaining the anomalous coagulation of charged colloids,

picture of surface charging behavior could be achieved by combining experimental titration (surface charge) and electrophoretic mobility data using the basic Stern model with the ζpotential in place of the Stern potential. A sweeping achievement, i.e., to bring the DLVO theory in a quantitative agreement with aggregation experiments based solely on explicitly evaluated parameters, seemed to be accomplished.6 Unfortunately, the latter approach turned out to fail when applied to colloid polymer latices and oxides (such as hematite and silica), both with the surface charge exceeding ca. 3 mC/ m2.7−9 For these colloids, the absolute aggregation rate was theoretically underestimated enormously in the slow regime, reflecting again the overestimation of the surface (Stern) potential parameter used in the calculation of the DLVO interaction free energy. It should be mentioned that, already in the earlier article10 studying the slow coagulation kinetics of PS latexes and silica, this inverse paradox (consisting of the idea that the Stern potential obtained from coagulation rates would be lower than the ζ-potential, which fact contradicts the traditional picture of EDL) was chanced upon. We should point out here that for colloidal metal oxides in solutions of low ionic strengths the traditional Gouy− Chapman−Stern−Grahame model (GCSG) of the double layer was found unable to equalize a high electrostatic potential (derived from the very high titrable charge) with the ζ-potential positioned at the Outer Helmholtz Plane (OHP) or at some distance from it.11 The same conclusion has been made in more recent studies on colloidal oxides12 as well as polymer colloids.13 As we know, the above disability of the GCSG model has widely been resolved by introducing a capacitance of the Stern layer. Nevertheless, high values of the Stern layer capacitance obtained for nonporous silica (3−4 F/m2) exceed the acceptable upper limit for nonporous metal (hydr)oxides (1.7 F/m2) and so appear physically unreasonable. This shortcoming has been advocated by a less condensed surface structure and the presence of reactive surface groups protruding from the surface but is still viewed from the perspective of the so-called MUSIC model.14 Silica is electrostatically and chemically homogeneous since it has no distinct crystal planes and only one type (singly coordinated SiOH) of reactive surface group is present on its surface that can be either negatively charged or noncharged. It means that the modern MUSIC (multisite complexation) model, counting upon the difference in the proton affinity of various types of surface oxygen groups at different crystal planes, is not a resource. One may acknowledge, therefore, that the introduction of another unmeasurable parameter (capacitance) allows researchers to align the surface charge and ζ-potential data just at the expense of enhancing a model’s arbitrariness. Be that as it may, the apparent underestimation of experimental slow aggregation of colloids by the DLVO theory, i.e., the overestimation of the surface potential by the ζpotential, cannot be reasoned out adhering to the concept of the mean-field electrostatic repulsion between ideal surfaces with homogeneously spread or “smeared” charges giving the average surface potential (identified with the ζ-potential). Indeed, to attenuate the inadequacy, following the former study,10 a lateral charge distribution over the plane surface was considered7−9 providing an extra electrostatic attraction between alternating stronger and weaker repulsion sites on interacting surfaces. However, not to mention the partial successfulness of this modification, parameters representing the 8810


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some (macroscopically based) approximations are preferred. So, the van der Waals interaction energy can be expressed by combining the Hamaker (eq 3) and the Lifshitz theory23

not excluding yet the possibility of their false self-consistency because of the introduction of more parameters. However, it will be shown in this study that both the seemingly indecipherable and diverging coagulation and electrophoretic behavior of nonporous monodisperse and spherical silica colloids in a presence of uniunivalent electrolyte can be rationalized and unified by using a simple core−shell model with the fixed charge distributed uniformly in the loose shell of which thickness (as the only adjustable parameter) varies according to a scaling law derived from a thermodynamic theory of osmotically driven swelling of polyelectrolyte hydrogels.

V vdW(u) = −

+ ln

0

8kBT 3η V (u) β(u) exp k T (u + 2)2 B

du

κ=

(6)

A(κ , h) = 0.75kBT (1 + 2κh)e‐2κh + {1 + (h/λ)q }−1/ q

2 3ℏω (B1 − B3) 16 2 (B1 + B3)3/2

(7)

Here B1 and B3 are the square of the refractive index of particles and aqueous solution medium (1.77), ℏ is the Planck constant divided by 2π (1.054 × 10−34 J s), c is the speed of light, ω is the characteristic frequency, and q is a best fit parameter (=1.185). The characteristic wavelength of the retardation effect, λ, can be described as λ=

c π 2ω

2 B3(B1 + B3)

(8)

Several approximative analytical expressions have been provided for the electrostatic double layer repulsion under the assumption of a surface potential lower or comparable to the thermal potential kBT/ze so that the nonlinear Poisson− Boltzmann equation can be linearized by applying the Debye− Hückel limit. The linearization approximations (LA) have been derived for large τ and small τu (τ = κa and τu = κh being the radius of spheres and their surface separation, respectively, scaled to the Debye screening length) by applying the wellknown Derjaguin integration method. When the surface potential φ0 is maintained to be constant during the particle’s approach

(2)

A⎛ 2 2 u 2 + 4u ⎞ V vdW(u) = − ⎜ 2 + + ln ⎟ 2 6 ⎝ u + 4u (u + 2) (u + 2)2 ⎠ (3)

where A is the Hamaker constant. Equation 3 simplifies if u ≪ 1 to the form A 12u

2z 2e 2c ∞ ε0εrkBT

(1)

When the electromagnetic retardation of pairwisely summed dispersion forces is neglected (nonretardation regime), the attractive van der Waals interaction energy reads as21

V vdW(u) = −

(5)

In eq 6, z is the valency of ions (symmetric electrolyte), e is the elementary electron charge, c∞ is the number of ions per unit volume, ε0 is the dielectric constant of free space, and εr is the relative permittivity of the solution. (τu is identical to κh, expressing the surface separation scaled to the Debye length). It is seen that the “effective Hamaker” constant A(κ,h) is considered to be a function depending on the distance between the spheres (reflecting the retardation effect) as well as the Debye screening length of the solution due to the electrolyte ions. In fact, this screened retarded function comprises the zero-frequency term (due to the orientation and induction vdW interactions) and the nonzero frequency term (due to the dispersion interactions):

where kB is the Boltzmann constant, T is the temperature (kBT being the thermal energy), and η is the dynamic viscosity of water. The stability ratio W incorporates the interaction energy V(u) term and the so-called hydrodynamic retardation β(u) term, accounting, respectively, for the surface and viscous forces manifested between two spheres during their encounters. Both terms are functions of the dimensionless intersphere surface separation u = h/a (a is the radius of the spheres and h is the closest distance between their surfaces). Surface Forces. The potential energy V(u) of interaction between spheres is obviously considered to be a sum of the DLVO components, namely the van der Waals VvdW(u) and the electrostatic double-layer VEl(u) one:1,2 V (u) = V vdW(u) + V El(u)

u 2 + 4u ⎞ ⎟ (u + 2)2 ⎠

where κ is the (inverse) Debye screening length of diffuse layer around the spherical particles in the electrolyte solution:

2. THEORETICAL SECTION Aggregation Kinetics. The kinetics of aggregation of monodisperse colloids of a spherical form has been described by von Smoluchowski who considered the process as a steady diffusion of the colloidal particles toward each other with no interaction. The rate constant of the dimer formation in this “fast” or diffusion-limited aggregation is kSm ≈ 1.2 × 10−17 m3 s−1. The “slow” or reaction-limited aggregation can be considered as the diffusion process of colloids under the influence of their mutual interactions, having the rate constant k of the dimer formation given as kSm reduced by the so-called stability ratio (modified Fuchs expression)20 k k = Sm = ∞ W 2∫

A(κ , h) ⎛ 2 2 + ⎜ 6 ⎝ u 2 + 4u (u + 2)2

V El(u) = C2 ln[1 + exp( −τu)] (4)

C2 = 2πε0εraφ02

To exactly calculate the van der Waals interaction in an electrolyte solution, the complicated Lifshitz approach has to be used, treating the interacting spheres as well as the solution as continua with certain macroscopic electrodynamic properties (dielectric constant and/or refractive index).22 That is why

(9) (10)

When the surface charge is kept constant, the Derjaguin result is (τ > 30, τu > 1)24 V El(h) = −C2 ln[1 − exp( −τu)] 8811

(11)


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provide the final silica−water−KCl system of required composition. The Spectronic 401 spectrophotometer (Milton Roy) was operated at the wavelength of 380, 540, and 800 nm for the 50, 150, and 320 nm spheres, respectively, by using a computer program (Quant software). The experimentally recorded absorbance of the silica dispersion as a function of time, reflecting the evolution of aggregates provoked by the addition of KCl, was fitted by using a Fortran program provided by Dr. Antonio Puertas (Department of Physics, University of Granada, Spain). This iterative program repeatedly generates absorbance-vstime curves according to a theoretical model of homoaggregation in the Rayleigh−Gans−Debye regime and matches them with the experimental ones until an acceptable fit is obtained. In the model (see ref 27 for the model formulation and/or ref 28 for its complete description and basic verification), an unlimited number of aggregates can be included. The parameters needed to launch the program are the wavelength of light λ, and the initial concentration, radius, and refraction index of colloidal spheres. The kinetic constant kkin of aggregation and the initial time (at which the aggregation begins) are fitting parameters. The absorbance records were collected in the time period from 2 to 100 s, disregarding the very initial period since there is a high dispersion of data (leading even to a transient decrease in absorbance) due to the turbulence stimulated by the initial mixing of the colloidal dispersion with the KCl solution. Because the software used to fit our absorbance data points is not able to analyze the goodness of the fit, we have checked it independently and found it high enough and found it to increase slightly with the KCl concentration. For example, for 150 and 320 nm silica spheres at pH 6, the coefficient of determination R2 was as high as 0.98 or higher for the concentrations of KCl above 0.05 M. There is a direct relation between the rate constant of the dimer formation k introduced in the theoretical part and the kinetic constant of the whole aggregation process determined experimentally kkin, i.e., k = 2kkin.

and φ0 in eq 10 must be replaced with the unperturbed surface potential φ∞ defined at the infinite distance between surfaces. Because in general there is a relation between the surface charge and the surface potential as the surfaces with ionizable groups (maintaining the chemical equilibrium) approach each other, a modification of eq 9 was proposed25 in the framework of the Debye−Hückel theory V El(u) =

C2 ln[1 + η exp( −τu)] η

(12)

where η = 1 − 2p is the regulation parameter characterizing the location of the interaction energy between the constant potential (p = 0, η = 1, eq 9) and the constant surface charge (p = 1, η = −1, eq 11). (The regulation parameter η is defined by the competition of the capacity associated with the diffuse part and the compact inner part of the electric double layer; a high diffuse layer capacity leads to the constant charge-like behavior, and a predominant capacity of the compact layer indicates a constant potential-like behavior.) Viscous Forces. The essentially repulsive hydrodynamic retardation β(u) can be expressed by the approximation26 β(u) =

6u 2 + 13u + 2 6u 2 + 4u

(13)

which is in good agreement with the exact solutions of the motion of two spheres in a viscous fluid.

3. EXPERIMENTAL SECTION Colloids. Uniform nonporous silica microspheres of three different diameters covered with natural hydroxyl SiOH groups were obtained in aqueous stock suspensions from Bangs Laboratories Inc. (surfactant-free, 10% solids content, corresponding to the number of particles 3.92 × 1014 cm−3, 2.978 × 1013 cm−3, and 3.156 × 1012 cm−3 for the 50, 150, and 320 nm SEM diameters, respectively). The microspheres were made from pure tetraethyl orthosilicate Si(OC2H5)4 reacted with water and ammonia, Si(OC2H5)4 + H2O → SiO2 + 4 C2H5OH; being pure SiO2, they should have no surfaceactive impurities, and therefore no cleanup was necessary before use. The density of the silica particles, provided by the producer, is 2.00 g cm−3. For experiments, the supplied stock suspensions were diluted with doubly distilled water so as to have dispersions of 1.21 ± 0.32 × 1012 particles/cm3, 1.48 ± 0.47 × 1011 particles/cm3, and 3.16 ± × 1010 particles/cm3, respectively. Electrokinetic Measurements. The electrophoretic mobility μe of silica spheres was measured at pH 2.6, 4, 6, and 8 for increasing concentration of KCl in the fully automated Zeta Plus instrument (Brookhaven Instruments Corporation) utilizing electrophoretic light scattering. The instrument was operated in the heterodyne mode with a wavelength of the laser source of 670 nm, the sampling time of 256 μs, modulation frequency of 250 Hz, and scattering angle 15°. For κa > 6, the conversion of μe to zeta (ζ) potentials was made by using the Zeta program (originated by M. Kosmulski) accounting for both the retardation and the relaxation effect. In this program, the setting parameters are the molar conductivity of ions (0.007 35 and 0.007 64 m2 Ω−1 mol−1 for K+ and Cl−, respectively), dielectric constant, ionic strength, temperature, and viscosity of aqueous solutions, and the radius of particles. The modified Booth equation (MBE) for nonconducting spheres was used to check the conversion independently. At least three consecutive measurements were recorded for each KCl concentration, providing averaged values of μe. The standard error of the conversed ζ-values is typically |φ0′|. It is also tacitly assumed that d is greater than the Debye length 1/κ to keep the potential deep inside the shell as the Donnan potential. If the condition is not satisfied the potential does not reach φDON; it follows from Table 1 that d is at least two times higher than 1/κ for pH 6 and 8 (but not for pH 4). When φDON is low, the potential deep inside the layer approaches the linearized Donnan potential in the form

φDON = eN/εrε0κ 2

(17)

Table 2. Parameters Characterizing the Donnan Equilibrium in the Model Shell

and φ0′ = φDON /2 = eN /2εrε0κ 2

(18)

In the limit of large φDON we have φ0′ = φDON − kT /e ≈ φDON − 0.025

(20)

The values of Donnan and surface potentials, as calculated from eqs 15 and 16, respectively, for the adjusted d and ρfix are also included in Table 1 for varying KCl concentration [cKCl = c∞/ (NA × 103)] and drawn in Figure 4. We see in the latter figure that φDON values at pH 6 and 8 tend to be constant at lower cKCl, and then, falling between the descending ζ-potentials and the ascending surface potentials φ0 and φ∞, decrease slightly as cKCl increases. The small decrease in φDON is remarkable when realizing the approximately 20-fold and 40-fold decrease of d at pH 6 and 8, respectively, in response to the two-order increase of cKCl and, correspondingly, the one-order decrease of κ−1 (κ ∝ √c∞, see eq 6). Also, φ0′ is always around (a little bit higher than) one-half of φDON, as predicted by eq 18 for low potentials. At pH 8 both potentials are higher by ca. 2 mV than these at pH 6. Let us step over to the problem of semipermeable membrane that separates two solutions containing electrolytes, the ions which are able to permeate this membrane (small ions of the supporting electrolyte) and the other ions (macromolecular anions) which are not. In order to attain a Donnan equilibrium in the system, the small ions must redistribute between the macromolecular and the external solution, so that51 c −′ = [c −(c − + zc)]0.5

c+ (mM)

1 2.5 5 10 25 50 70 100 250

1.753 4.27 8.45 16.68 39.61 78.31 105.6 140.75 307.56

2.21 5.42 10.77 21.38 51.7 102.74 140.5 192.75 447.56

5 10 25 50 70 100 250 500

9.58 18.49 45.73 90.865 104.74 150.50 380.55 781.64

11.72 22.92 56.75 113.07 139.84 200.5 504.55 1025.64

c− (mM) pH 6 0.453 1.15 2.32 4.70 12.09 24.43 34.90 52.00 140.0 pH 8 2.14 4.43 11.02 22.2 35.1 50.0 124 244

c+/c+′ (=c−′/ c−)

φDON (eq 22) (mV)

φDON (eq 15) (mV)

2.20 2.17 2.154 2.133 2.068 2.05 2.00 1.925 1.788

19.96 19.54 19.34 19.10 18.33 18.11 17.56 16.52 14.65

19.94 19.70 19.34 19.13 18.31 18.13 17.56 15.46 14.66

2.34 2.27 2.269 2.257 1.996 2.02 2.017 2.05

21.43 20.72 20.66 20.53 17.72 17.51 18.69 18.10

21.45 20.84 20.66 20.55 17.43 17.51 17.99 18.10

follows from this table that the values of the Donnan potential obtained from eq 15 by fitting the coagulation data are in an excellent agreement with these calculated for the Donnan equilibrium from eq 22. It is noticeable that the redistribution of indifferent electrolyte (K+ and Cl−) ions documented in Table 2 by the ratio of Cl− anions in the solution to these in the gel layer c−′/c− ≈ 2.0 at cKCl from 10 and 70 mM for pH 6 and pH 8, respectively, until the pseudocritical concentrations. The same ratios are obtainable for c+/c+′. Surprisingly, φDON values for pH 6 and 8 tend to merge with ζ-potentials at cKCl above ca. 10 mM. Figure 1 also shows that φDON approaches φ∞ at highest cKCl. It can be easily shown that when the product of ρfix and d (i.e., the amount of the charge contained in the layer per unit area of the spheres) is kept constant, in the limit of d → 0 it becomes the surface charge density σ0 and eq 14 reduces to eq 11 derived for two hard spheres interacting in the constant surface charge regime25 (assuming a ≫ κ−1, the surface charge of a sphere σ0 ≈ φ∞ε0εrκ).

(21)

In eq 21, c is the (invariable) concentration of the anionic macroions P z− , and c − and c − ′ are the equilibrium concentrations of the small anions in the macromolecular and the external solution, respectively. Because of the unequal diffusion redistribution, an electric potential develops across the membrane at the equilibrium Δφ = (kT /e) ln(c −′/c −)

N (mM)

(19)

It follows from the above that φDON /2 < φ0′ < φDON

cKCl (= c+′ = c−′) (mM)

(22)

We recall that the solutions are regarded as ideal-diluted so that the ion concentrations are given instead of their activities (moreover, the activity coefficients in eq 22 cancel out). 8818


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Figure 6. Averaged electrophoretic mobility (lower part) and the corresponding standard deviations (upper part) as a function of KCl concentration.

Figure 7. Comparison of the electrophoretic softness parameter of the gel layer estimated in two ways. (See the text for details.)

5.3. Electrophoretic Mobility. Electrophoretic mobilities, from which the ζ-potentials (presented partly in Figure 1) were calculated, can be reinterpreted by using the Ohshima expression derived for core−shell spheres52 under assumptions that are in agreement or not in conflict with ours. Figure 6 summarizes averaged negative values of the electrophoretic mobility (lower part) together with the corresponding positive standard deviations (upper part) obtained for the silica spheres at pH 6 and 8. It is apparent that there are a few steps on the μ−cKCl dependences, which in fact harmonizes with what has been said about the W−cKCl dependences. The main assumptions in the Ohshima analysis are as follows: (i) the liquid flowing (at low Reynolds numbers) through the polyelectrolyte shell is incompressible, (ii) the applied electric field is relatively weak, (iii) the slipping plane is

located on the spherical particle core, (iv) ions of electrolyte cannot penetrate the particle core but the shell is permeable to mobile ions, (v) the relative permittivity takes the same value both inside and outside the shell. This approximative expression depends on a weighed average of the Donnan potential and the surface potential of the shell layer (eqs 15 and 16) provided that λa ≫ 1, κa ≫ 1, λd ≫ 1, κd ≫ 1: μe =

⎤ ρ 2ε0εr φ0 /κ m + φDON /λ ⎡ 1 + fix2 ⎢1 + 3⎥ 3η 1/κ m + 1/λ ⎣ 2(1 + d /a) ⎦ ηλ (23)

The effective Debye−Hückel parameter of the shell, κm, is defined as 8819


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κm = κ 4 1 +

⎛ ρfix ⎞2 ⎜ ⎟ ⎝ 2c ∞ ⎠

gel layer, as expected. (At closer inspection of Figure 7 it can be found that both 1/λ values increase slightly with the silica diameter.) The fitted ap magnitude is around 0.75 nm for which value ϕ ranges from ca. 0.002 to 0.15 (i.e., 0.2−15%) for the 150 nm silica at pH 6. Incidentally, with eq 23 improved by taking into account the relaxation effect,54 the adjusted values of the electrophoretic softness have been found unchanged. It is also worth mentioning that almost identical 1/λ values were obtained when the Stokes equation was used instead of the Brinkman equation (eq 26) in method 2. Nevertheless, the degree of dissociation of silanol groups on the silica surface α is far from 100% at pH 6 and even at pH 8 and also depends on the KCl concentration. We can estimate α by simply dividing the experimentally evaluated surface charge density σ0 presented in Table 1 with the charge density σmax determined for the maximum number of ionizable OH groups per square nanometer (say, 2.43 nm−2, providing σmax = 0.389 C/m2). For example, at pH 6, only 1.68% of silanols are expected to ionize in the presence of 1 mM KCl, respectively. This estimate seems to be rough but acceptable for our purpose as the value of 2.56% can be obtained for pH 6 from an empirical Henderson−Hasselbach-like equation for weak acids with the internal dissociation constant of polysilicic acid pKa,int = 6.81:

(24)

Finally, 1/λ is the “softness” parameter of the shell and, having the dimension of length, represents a resistance to the liquid flow penetration (γ is the frictional coefficient): ⎛ η ⎞1/2 1/λ = ⎜ ⎟ ⎝γ ⎠

(25)

Even if assuming a ≫ d so that the d-containing term in the square brackets in eq 23 becomes 3/2, both shell parameters in eq 23, i.e., the fixed charge density and the liquid flow resistance (softness), are obviously estimated via a curve fitting procedure of the whole experimental mobility-vs-electrolyte concentration plot. It means however that the shell thickness and structure are taken as constant while the electrolyte concentration increases. (We must precede that this procedure did not allow us to fit the experimental mobilities even if a radial distribution of the charge was allowed.) Apparently, this is not in agreement with the concept of swellable gels, and even a more complicated problem is encountered than that with eq 14 for the electrosteric interaction. The problem is that not only ρfix but also 1/λ must be considered to change if (no matter whether non-negligible with respect to a or not) d varies. Moreover, the hydrodynamic friction in the polyelectrolyte shell is taken in the Ohshima model simply as a sum of independent Stokes frictions of “resistive elements” of radius ap representing polymer segments distributed uniformly at the density N, i.e., γ = 6πηapN (ρfix = Ne for completely ionized silanol groups). This is argued acceptable for higher electrolyte concentrations, but hydrodynamic interactions should be taken into account at lower electrolyte concentrations. Now, we can evaluate the softness parameter by matching experimental mobilities presented in Figure 6 with theoretical mobilities calculated according to eq 23 in which the values of d and ρfix from coagulation experiments (Table 1) and the corresponding Donnan and surface potential (given by eqs 15 and 16) are input parameters (method 1). Then, as method 2, we will determine the same parameter independently by using the Brinkman formula53 relating 1/λ directly with the mean volume fraction of polymer segments ϕ in a homogeneous group of beads [assuming all segments charged, ϕ = (4/ 3)πap3N]:

⎛ α ⎞ 1.9(α + α 2) ⎟ + pH = pK a,int + log⎜ ⎝1 − α ⎠ 0.039 + α

Restricting our consideration to σ0 data from Table 1, α is found to rise in 0.1 M KCl solution to 6.8% and 24.5% at pH 6 and pH 8, respectively. On the other hand, pH well above 12 would be necessary for α to reach values close to 1 (100%) using eq 27. Consequently, the total density of uncharged plus charged silanols is much higher than N used in calculating 1/λ, and thus Ohshima’s eq 23, which is derived for completely dissociated ionic sites within the strong polyelectrolyte layer, should be not applicable. Unfortunately, a simple correction of the second term of eq 23 proposed by Dukhin et al.55 to account for the incomplete dissociation led only to a minor amendment to 1/λ by method 1 while that obtained by method 2 is supposed to decrease enormously. However, a perfect agreement can be achieved again (see the lower parts of Figure 7) by inserting the total instead of fractional density of silanol groups in the second term of eq 23 with the parameter ap brought down to 0.33 ± 0.03 nm, which not only becomes commensurable wih the monomer unit length of the polysilicic acid (the double of physical bond length of the Si−OH group is ca. 0.165 nm56) but also reverts the volume fraction of the polymer network to a low enough range. So, the value of 1/λ will decrease from 1.23 nm (ϕ = 1.34%) to 0.14 nm (ϕ = 26.5%) at cKCl 1 mM and 100 mM KCl, respectively, for 150 nm silica now. 5.4. Physicochemistry of Swelling Polyelectrolyte Gels. Up to now, we have analyzed how some parameters vary in response to the thinning of the Donnan gel layer deduced from the experimental electrolyte-induced coagulation rates by applying the core−shell model (eq 14) and the standard surface charge data. However, in order to validate these parameters and thus the model with its assumptions in general, one has to justify primarily and appropriately (from first principles) the deduced variation in the very thickness of the shell. For this reason, a complex analysis must be made including not only the Donnan effect due to the ion

1/2 ⎧ ⎡ ⎛8 ⎞1/2 ⎤⎫ ⎪ ap ⎪ 4 1/λ = ⎨ ⎢3 + − 3⎜ − 3⎟ ⎥⎬ ⎪ 18 ⎢ ⎪ ϕ ⎝ϕ ⎠ ⎥⎦⎭ ⎣ ⎩

=

ap

1/2

{ 18 } f (ϕ)

(27)

(26)

The upper doublets of data points in Figure 7 visualize the values of 1/λ (×) obtained by manipulating the ap parameter (method 2) so as the best agreement is reached with 1/λ values (−) resulting from the application of method 1. It can be seen that both values 1/λ correspond to each other very well for the silica spheres of selected diameters at pH 6 and 8, both diminishing together linearly as cKCl increases from 10 to 100 mM (in the log−log form of Figure 7). For example, for 150 nm silica, 1/λ works out at 9 nm at 1 mM KCl to ca. 0.2 nm at 100 mM KCl. This clearly documents that the friction between the polymer network and water strengthens within the thinning 8820


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Figure 8. Deduced shell thickness (left) and the degree of swelling (right).

redistribution, which becomes necessarily “passive” when considered in a surface charge layer of a constant thickness, but all pertinent effects participating actively in an establishment of actual swelling equilibrium of the polyelectrolyte gel layer. (We mean that the gel layer should be responsive enough in the sense of its capability to attain an equilibrium with the actual external electrolyte solution before realizing coagulation tests but also to re-equilibrate and reswell in a period much shorter than the time scale of the electrolyte-stimulated coagulation process.) Rubber-like materials are characterized by a combination of two features: a capability to sustain large deformations without rupture when subjected to an external stress and to recover spontaneously nearly to its initial dimensions after removal of the stress. This combination of properties, denoted as a longrange elasticity, is by no means singularly characteristic of hydrocarbon polymers, and even inorganic polymers may display it under appropriate conditions. This is when a network consists of cross-linked long chain polymers with a high number of mobile segments (able to accommodate the stressinduced deformation through reversible rearrangements of the chains configurations) between the permanent cross-linkage points. The problem is that the structure of nonionic and especially ionic polymer networks reflects in a complicated way not only interactions with solvent molecules and ions but also the character of the cross-linking of the polymer segments. The influence of basic parameters of the polymer and solution on the equilibrium swelling degree of macroscopic three-dimensional anionic polymer network-hydrogels was predicted theoretically in 1991.57 In the mathematical model used, the Flory−Huggins thermodynamic theory and the rubber elasticity theory58 were combined with ionic interactions. A sharp sigmoidal increase in swelling over a narrow pH range of external solution has been predicted, having an inflection point at the dissociation constant of ionizable groups on the polyelectrolyte, pKa, and reaching a constant volume at a highest pH. Given the same pKa, an increase in the ionic strength leads to a decrease of swelling but, at the same time, with the inflection preserved at pH ≈ pKa. The model also indicates that a finite degree of swelling does not vanish even if the polyelectrolyte is neutral, i.e., at pH ≪ pKa, provided that the Flory−Huggins polymer−solvent interaction parameter of the system χ1 is very low (good solvent), i.e., χ1 → 0.1. Fernández-Nieves et al.59 proved that the Flory−Huggins thermodynamic theory of macroscopic gels combined with the Donnan relations contains the essential physics for describing the osmotic pressure-driven swelling−deswelling process of

mesoscopic pH-responsible gels in the presence of salts. Their numerical analysis showed and experiments confirmed that the osmotic pressure and consequently the size D of (no matter whether completely or partially) ionized spherical microgel particles decreases as the salt concentration c in the external solution will exceed a limit, following an asymptotic power law: D ∼ c −1/5

(28) 58

Now, let us follow original Flory’s reasoning. The swelling equilibrium state of a homogeneous and unstrained (isotropic) macroscopic ionic polymer network of low degree cross-linking is attained when three osmotic pressure contributions compensate in that gel: (i) the mixing contribution due to the mixing of water molecules with the polymer segments (entropy of dilution) which is augmented or diminished by the heat of dilution, depending on whether the Flory−Huggins parameter χ1 is negative or positive, respectively; (ii) the elastic contribution due to the retractive forces of the polymer network; (iii) ionic contribution due to the osmotic pressure of mobile counterions whose concentration in the gel is inevitably higher than outside (Donnan effect). Because an explicit general solution of the equilibrium of the osmotic pressure contributions is impossible, Flory, like many others after him, resorted to the consideration of special cases. When the differences in the concentration of counterions in the external solution and in the gel are comparable in magnitude, the ionic contribution is greatly reduced. Then, it has been found that, at the equilibrium, the swelling ratio qm (the ratio of the volumes of the swollen and unswollen gel V/V0) to the fivethirds power is proportional to the square of the fixed charge of the network and to a reciprocal of the ion concentration in the external solution. The same power of the ion concentration (c−1) is obtainable from the above asymptotic solution, eq 28, omitting the mixing osmotic pressure and assuming isotropic swelling of microgel spheres,59 since V/V0 = (D/D0)3. Analogously, it can be predicted that the ratio of the volumes of the isotropically swollen and unswollen shell around a rigid core sphere (core−shell model) is qm =

(D* + 2d)3 − D*3 V = = c −3/5 V0 (D* + 2d0)3 − D*3

(29)

where D* (≡2a) is the core diameter given as the diameter of “plain” silica spheres in absence of the gel layer. Figure 8 (right) shows the magnitude of qm calculated by eq 29 as a function of KCl concentration with d0 being the expected gel-layer thickness at the pseudocritical coagulation concentration of 8821


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cKCl (i.e., when W attains a minimum), D* taken as the diameter of silica microspheres with a collapsed gel layer determined by electron microscopy (i.e., 50, 150, and 320 nm), and, naturally, with d obtained from coagulation experiments by applying the core−shell interaction model (Table 1). It can be seen in Figure 8 that the function is indeed linear and even identical, when expressed in the log−log form, for all examined samples of silica (diameters) at pH 6 and 8. Moreover, the linear slope is −3/5. For example, for 150 nm silica at pH 6, the following power law function is the best fit: qm = 0.2295cKCl−0.6051 with R2 = 0.989, see Figure 8. This can be considered as a strong argument supporting the validity of eq 29 and, simultaneously, the gel-layer approach represented by the core−shell model with its numerous assumptions, namely that the gel layer may be viewed as loose (permeable to small ions and water molecules), homogeneous, pH- and saltresponsive, isotropically swelling polyelectrolyte network that is, in the course of the presented experiments, always at equilibrium. It should be added that the above conformity of experimental data with the asymptotic scaling is observable only for a restricted (roughly one-order) KCl concentration scale from ca. 10 mM to the pseudocritical 100 and 300 mM for pH 6 and 8, respectively. For cKCl lower than 10 mM, a graduating deviation from the linearity would be observed in Figure 8. (For pH 8 the negative exponent of the best fit power law is slightly higher than 3/5, i.e. 0.6134.)

soft spherical particles (core spheres covered with ion- and water-permeable shells with homogeneously distributed fixed charges), the shell thickness is obtainable as the only adjustable parameter. Its magnitude is deduced to decrease with the increasing KCl concentration in a perfect conformity with the scaling law derived for the degree of unrestrained isotropic swelling of macroscopic polyelectrolyte networks at equilibrium. We believe that this conformity proves a core−shell structure of the silica/water interface and, at the same time, the deswelling character of the shell (gel layer) as to its looseness, homogeneity, responsiveness, and reversibility with respect to external stimuli (pH and ionic strength), elasticity, isotropy, and resiliency. The total pairwise interaction free energy curve obtained as a sum of the repulsive electrosteric and the attractive vdW contributions between the core−shell spheres for a given pH is characterized by a barrier (preventing the spheres from their adhesive contact) resulting from the repulsive electrosteric contribution. However, preceded by this barrier is a pseudosecondary minimum whose depth increases with the electrolyte concentration, a result of the combined effect of the attractive vdW contribution and the shell shrinkage. This is in accordance with the phenomenological Healy’s model of reversible electrolytic coagulation of gel-covered silica.49 Theoretical electrophoretic mobilities calculated by using the core−shell electrokinetic theory with the shell thicknesses (together with the Donnan and surface potential at the shell boundaries) being extracted from the coagulation experiments match very well the experimental ones. This underlines the correctness of the swellable gel-layer approach. It also implies that electrophoresis with the soft particle electrokinetic theory may well be an alternative way to asssess how the thickness or other parameters of adsorption layers of polyelectrolytes covering nanoparticles or colloids can be determined with a better confidence. Hence, the alleged anomalous behavior (in view of the inconsistency between the ζ-potential and coagulation trends) of colloidal silica dispersions can be reconciled from the position of one and the same three-dimensional, yet simple swelling gel-layer model of EDL, while efforts of using EDL models on two-dimensional surfaces with whatever complicated modifications by other researchers have been found unsuccessful so far. Incidentally, aside from the aforementioned papers,7−9 the MSPM (microsurface potential measurement) method has been declared as inapplicable to evaluate the stability ratio of sulfonated latexes if coagulated by an indifferent monovalent electrolyte.38 It was argued that the DLVO theory cannot describe the stability ratio quantitatively due to specific ion effects, surface roughness, and hydration forces manifesting at separations below 1 nm, where the DLVO barrier occurs. This argument is, naturally, groundless in the presented gel layer model. We are aware of the fact that the homogeneous distribution of fixed charges across the gel layer is an approximation and that ρfix in our concept represents rather an average quantity. Indeed, the swelling degree of a gel layer integrated with the hard core may be to a certain degree inhomogeneously deformed even when it is in the equilibrium state since it is subjected to the constraint of the core, producing a field of stress at the core/shell interface but not at the shell/solution interface.60 Nonetheless, our results indicate that these effects are subordinated to a high degree to the swelling/deswelling phenomenon, at least for the KCl concentration below the

6. CONCLUSIONS Stability ratios calculated from the DLVO theory with ζpotentials in place of electrostatic surface potentials (as obtained by converting experimental electrophoretic mobilities according to theories for smooth spheres) exceed markedly experimental stability ratios of model dispersions of uniform and nonporous silica spheres in the presence of KCl electrolyte at higher pHs (6 and 8) whereas the situation is opposite at lower pHs (2.6 and 4). In other words, surface potentials determined by matching the experimental stability ratios with the theoretical ones at higher and lower pH values, respectively, are lower and higher in absolute value than experimentally based ζ-potentials. At the higher pH’s, the deviation increases with decreasing the electrolyte concentration so that it cannot be associated with the adsorption of ions into the Stern layer. It seems that something is wrong with the idea of smooth silica surface, at least at high pH values, when it is in a charged state, since a ζ-potential must always be lower than the Stern or diffuse layer potential (in absence of Stern layer) in the realm of traditional electric double layer (EDL) theory. We suggest that for higher pH values the above problem is explicable by envisioning a swellable gel layer of hydrolyzed charged material on the silica surface which is loose to such a degree that the reference plane of van dar Waals interaction is close to its inner boundary. The slipping plane would also be placeable there, but the ζ-potential, if one persists using it, must be considered only as an apparent parameter determined from the electrophoretic mobility by applying a model taking into account the presence of the gel layer. At the same time, encircling solid silica particles, the charged gel layers should be stiff enough to withstand the particle’s thermal encounters so that the reference plane of their mutual EDL interaction can be located close to their outer boundary. Matching repeatedly the experimental to theoretical stability ratios, the latter being calculated with the account of Donnan potential-governed electrosteric repulsion between the so-called 8822


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(9) Kobayashi, M.; Skarba, M.; Galletto, P.; Cakara, D.; Borkovec, M. Effects of Heat Treatment on the Aggregation and Charging of StöberType Silica. J. Colloid Interface Sci. 2005, 292, 139−147. (10) Kihira, H.; Ryde, N.; Matijevic, E. Kinetics of Heterocoagulation. 2. The Effect of the Discreteness of Surface Charge. J. Chem. Soc., Faraday Trans. 1992, 88, 2379−2386. (11) Hunter, R. H.; Wright, H. J. L. The Dependence of Electrokinetic Potential on Concentration of Electrolyte. J. Colloid Interface Sci. 1971, 37, 564−580. (12) Attard, P.; Antelmi, D.; Larson, I. Comparison of the Zeta Potential with the Diffuse Layer Potential from Charge Titration. Langmuir 2000, 16, 1542−1552. (13) Marlow, B. J.; Rowell, R. L. Electrophoretic Fingerprinting of a Single Acid Site Polymer Colloid Latex. Langmuir 1991, 7, 2970− 2980. (14) Hiemstra, T.; van Riemsdijk, W. H. Physical Chemical Interpretation of Primary Charging Behaviour of Metal (Hydr)oxides. Colloids Surf. 1991, 59, 7−25. (15) Lyklema, J. The Structure of the Electrical Double Layer on Porous Surfaces. J. Electroanal. Chem. Interfacial Electrochem. 1968, 16, 341−348. (16) Perram, J. W. The Oxide-Solution Interface. Aust. J. Chem. 1974, 27, 461−475. (17) Kleijn, J. M. The Electrical Double Layer on Oxides: SiteBinding in the Porous Double Layer Model. Colloids Surf. 1990, 51, 371−388. (18) Allison, S. Analysis of the Electrophoretic Mobility and Viscosity of Dilute Ludox Solutions in Terms of a Spherical Gel Layer Model. J. Colloid Interface Sci. 2004, 277, 248−254. (19) Laven, J.; Stein, H. N. The Electroviscous Behavior of Aqueous Dispersions of Amorphous Silica (Ludox) in KCl Solutions. J. Colloid Interface Sci. 2001, 238, 8−15. (20) Fuchs, N. Zur Theorie der Koagulation. Z. Phys. Chem., Abt. A 1934, 171, 199−208. (21) Hamaker, H. C. London-van der Waals Attraction between Spherical Particles. Physica 1937, 4, 1058−1072. (22) Dzjaloshinskii, I. E.; Lifshitz, E. M.; Pitaevskii, L. P. The General Theory of van der Waals Forces. Adv. Phys. 1961, 10, 165−209. (23) Mahanty, J. H.; Ninham, B. W. Dispersion Forces; Academic Press: London, 1977. (24) Wiese, G. R.; Healy, T. W. Effect of Particle Size on Colloid Stability. Trans. Faraday Soc. 1970, 66, 490−499. (25) Carnie, S. L.; Chan, D. Y. C. Interaction Free-Energy between Identical Spherical Colloidal ParticlesThe Linearized PoissonBoltzmann Theory. J. Colloid Interface Sci. 1993, 155, 297−312. (26) Honig, E. P.; Roeberson, G. J.; Wiersema, P. H. Effect of Hydrodynamic Interaction on the Coagulation Rate of Hydrophobic Colloids. J. Colloid Interface Sci. 1971, 36, 97−109. (27) Škvarla, J. Does the Hydrophobic Attraction Contribute to the Interaction Between Colloidal Silica Spheres Coagulated by an Adsorbing Cationic Surfactant? Colloids Surf., A 2012, 397, 33−41. (28) Puertas, A. M.; de las Nieves, F. J. A New Method for Calculating Kinetic Constants within the Rayleigh-Gans-Debye Approximation from Turbidity Measurements. J. Phys.: Condens. Matter 1997, 9, 3313−3320. (29) Reerink, H.; Overbeek, J. Th. G. The Rate of Coagulation As a Measure of the Stability of Silver Iodide Sols. Discuss. Faraday Soc. 1954, 18, 74−84. (30) Kihira, H.; Matijevič, E. Kinetics of Heterocoagulation. 3. Analysis of Effects Causing the Discrepancy between the Theory and Experiment. Langmuir 1992, 8, 2855−2862. (31) Chang, S. Y.; Ring, T. A.; Trujillo, E. M. Coagulation Kinetics of Amorphous Colloidal Silica Suspensions with and without Hydroxypropyl Cellulose Polymer. Colloid Polym. Sci. 1991, 269, 843−849. (32) Killmann, E.; Adolph, H. Coagulation and Flocculation Measurements by Photon/correlation SpectroscopyColloidal SiO2 Bare and Covered by Polyethylene Oxide. Colloid Polym. Sci. 1995, 273, 1071−1079.

pseudocritical value of 100 and 300 mM for pH 6 and pH 8, respectively. We believe that the presented results are original and encouraging and may also contribute to the elucidation of the dilemma as to the origin of the additional, non-DLVO repulsion observed by direct surface force measurements for silica and some other surfaces. In fact, we subscribe to the (electro)steric instead of the solvation origin of the repulsion. The argument is that the repulsion between structural layers of water on silica particles would not resolve the problem of overestimating their stability by the DLVO theory in conjunction with the electrophoretic mobility (at high pH’s). We have to stress that the DLVO theory is not disapproved nor should be extended. This is because the apparent coagulation anomaly is supposedly a result of invalidity of the basic precondition (ideality of surfaces independent of time, adjacent liquid, adsorption layers, etc.) rather than of the theory itself. It should be kept in mind that the DLVO theory has been intended for lyophobic colloidal systems.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This contribution/publication is the result of the project implementation “Research Excellence Center on Earth Sources, Extraction and Treatment −2nd phase supported by the Research & Development Operational Programme funded by the ERDF” (ITMS: 26220120038). The author also acknowledge the financial support of the Slovak Agency for Research and Development (grant no. APVV-0423-11) and of the Slovak Scientific Grant Agency (grant VEGA No. 1/1222/12).

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REFERENCES

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