On the gel layer interpretation of anomalous viscosity of colloidal silica dispersions

The anomalous viscosity of colloidal dispersions of silica has been known and studied for a long time. Several different mechanisms have been proposed to explain it, related to the electrical double layer, hydration layer, and porous/gel layer at the silica-water interface. However, it is still not entirely clear which mechanism is actually operative. Here, based on measurements of viscosity dependence on the concentration of indifferent electrolyte in highly diluted colloidal silica systems and its interpretation using the well-known Einstein equation, the concept of a swellable polyelectrolyte gel layer is corroborated.


Introduction
It is widely recognized [1,2] that aqueous colloidal systems of various hydrous metal oxides, particularly amorphous silicon dioxide (silica), exhibit extraordinary behavior: a high (i) titration-determined surface electric charge as compared to the low electrokinetic charge or zeta potential derived from traditional models of electric double layer (EDL), (ii) concentration of electrolyte required to induce coagulation by neutralizing the EDL, and (iii) viscosity.
Unique properties of silica in water have persistently been noticed and associated with its spontaneous gelling propensity, from silicic acid solutions in 1864 [3] to model silica colloids about 100 years later [4] and even after another half a century (this and other cited papers).However, in a further development of colloid science, silica's unusual properties used to be analyzed primarily by applying the EDL models in conjunction with the ionization and complexation of surface functional groups, and/or with the hydration layer on an otherwise ideal (i.e., smooth, bare) surface.This approach evidently did not lead to an understanding of what happens on the hydrophilic surface of silica or other metal oxides in contact with water.
This prompted an introduction of the heuristic model of rigid porous surface layer on colloidal metal oxide particles in the late 1960s, which assumed the distribution of a significant portion of fixed electric charge and neutralizing ions within the layer [5][6][7][8].However, even these models, aimed at reconciling the density of titratable surface charge and zeta potential, were evidently not successful enough.The factors influencing the thickness (a fittable constant), homogeneity, and elasticity of such a layer were not given closer attention.This was largely due to the significantly limited knowledge regarding its structure, which is an essential aspect to consider.
On the other hand, the concept of the porous layer with uniformly distributed electric charges seems to have suggested a consideration of a possible effect of the wellknown, precisely defined, and easily explainable phenomenon of Donnan equilibrium of ions in such a layer [9][10][11].This phenomenon was derived as early as the beginning of the twentieth century for the ion equilibrium in a solution on two sides of a membrane that is permeable only to some of the ions.But, it generally applies to any water molecule-and ion-permeable structures carrying a charge, including free charged polymers and their crosslinked gel analogues.Initially, this phenomenon was associated with biological membranes, but surprisingly, it was also considered to exist on the surface of inorganic colloidal particles as early as 1912 [12].
In the further development of colloid science, as of 1963, the Donnan effect was considered a mathematical fiction for surface layers of unknown thickness and structure, known as the macroscopic Donnan model where the Donnan potential, due to local electroneutrality in the layer, is assumed to be constant across the layer and abruptly becomes zero at its outer boundary [13].But, the so-called "two-phase" concept, based on the similarity of the protolytic behavior of hydroxylated oxides and weakly acidic or basic gels to the excess neutral salt concentration, relying on Gibbs-Donnan logic for otherwise free ions, emerged in the 1980s as a simpler and thus better (more realistic) alternative to the significantly more elaborated and widespread but simultaneously more complex concept of surface charge varied through chemical complexation of ions in combination with the distribution of neutralizing ions in EDL, as already mentioned [14][15][16][17].
Nevertheless, although the complexity and variability of different modern complexation models with their numerous constants were considered confusing as late as in 1990 [18] and even in 2006 [19], there has evidently been no deeper incorporation of the Donnan equilibrium into traditional colloid science.A significant advancement was however the introduction of the concept of "soft" colloids-ionimpenetrable core particles coated with an ion-penetrable surface charge shell that mimics a polyelectrolyte layer [20][21][22].This simple Donnanian model assumed the distribution of completely dissociated functional groups (fixed charges), polymer segments (resistance centers), and ions in the shell to be uniform and thus brought very useful approximate solutions for the electrostatic potential within the particles themselves and in their vicinity, their interaction energy, electrophoresis, etc. (in the so-called microscopic Donnan model, the Poisson-Boltzmann distribution of electrolyte ions is taken into account).Still, this approach did not consider the structurally dependent elasticity of the shells inherent for polymers and was primarily directed towards the field of biological colloids.
A new endeavor involved a more detailed theoretical and experimental analysis of two fundamental factors influencing the anomalous viscosity of colloidal silica: pH and concentration of monovalent electrolytes in its aqueous environment [23][24][25][26][27].The main finding of these studies was that the observed viscosity can not be attributed to the so-called primary electroviscous effect (PEE) caused by shear-induced deformation of EDL and its consequent resistance.Although qualitatively consistent with the assumed impact of both the factors, the PEE significantly underestimated the viscosity.Therefore, it was speculated that the presence of a filamentous layer on the surface of silica particles could explain it.This outer layer could have been reversibly changing its volume due to variations in the aforementioned factors-it swells/shrinks at high/low pH and/or high/low electrolyte concentrations.It was suggested that this layer is an elastic polyelectrolyte gel, having a fuzzy or crosslinked structure.
Although more concrete evidence confirming these ideas was still lacking, an important contribution of these works was their anticipation of the role of elasticity and electrostatic repulsion between the fixed charges carried by flexible backbones of polyelectrolyte chains in the layer.They saw the gel-like layer as an inseparable and natural part of silica particles' surface, once immersed in water.Therefore, they did not consider it merely as a potential artifact that hinders a better understanding of the true ("uncontaminated") nature of the silica surface but as an important phenomenon that may not strongly depend on the type of silica (as it is often thought) and thus deserves further investigation, presumably from the standpoint of polymer physics, i.e., polyelectrolytes and their gels.Additionally, they emphasized the possibility of utilizing the well-known Einstein equation for the viscosity of dilute colloidal systems to determine the structure of silica colloidal particles.
It was evident that the structure of silica surface changes upon contact with water due to its reactivity, with the porosity of the newly forming swellable (elastic) polymer gellike surface layer going beyond the traditional notions of macro-, meso-, or even nanoporosity of rigid structures.In 2013 [28], basing on a theoretical analysis of the kinetics of anomalous coagulation of monodisperse colloidal silica spheres induced by the addition of electrolytes at pH values rendering these particles (declared as nonporous by the manufacturer, Bangs Lab.) highly charged, it was concluded that these particles do not behave as hard spherical particles according to the DLVO theory, which operates with the classical EDL (the fitted surface potentials changed inversely with experimental zeta potentials), and possibly not even according to its extended versions.Instead, it was proposed to consider them as composed or "soft" spheres, with their solid cores covered by a shell that satisfactorily reflects a structurally and electrically homogeneous, ion-and water molecule-permeable, loose, and elastic (swellable) Donnanian polyelectrolyte gel layer.
The main argument for our conclusion regarding the swellable character of the gel-like layer was that the volume or degree of swelling of the model shells of silica spheres deduced from experimental results decreases (in parallel with the thickness of the EDL) with the concentration or, more precisely, the activity of coagulation-inducing chaotropic ions (weakly hydrated and supposedly easily penetrable into and weakly adsorbable onto the silica surface) of a univalent electrolyte (KCl) in solution.This decrease follows a well-known and simple asymptotic mathematical (power-law) relationship with an exponent of −3/5 derived for the effect of electrolytes on swellable polyelectrolyte macro-and meso-gels, implicitly encompassing fundamental phenomena expressed in polymer physics and subsequently confirmed experimentally for colloidal gels (microgels and nanogels) as well [29][30][31][32].These phenomena include the Donnan equilibrium and the (rubber-like) elasticity of a three-dimensional, crosslinked network formed by charged and flexible polymer chains.
Later, in our 2019 work [33], we also found out a similar analysis leading to a very reasonable prediction of the electrophoretic mobility or the intrinsic dissociation constant of silanol functional groups of silica colloids.Interestingly, the scaling power law with the −0.6 exponent was found to hold good for gel layers supposedly deswelling under the effect of not only chaotropic potassium chloride but of all alkali metal chlorides, except that all the dependencies were mutually shifted, in agreement with the Hofmeister series.These shifts could still be well rationalized within the framework of the Donnan equilibrium by considering the limited penetration of cations into the gel phase.Its validity was also confirmed by analyzing coagulation kinetics of other oxides [34] and also of AFM images [35] or the distribution of ions on the silica surface [36].A similar paper dealing with the character of the gel layer under the influence of various ionic strengths and pHs has been published simultaneously [37] where the net potential (well depth) between Brownian silica colloids and a glass plate was analyzed, as obtained from nonintrusive TIMR (total internal reflection microscopy) measurements on the kT energy scale and fN force scale by comparing physically meaningful models of the gel layer.
In order to further support the idea of existence of the swellable polyelectrolyte gel layer on the surface of colloidal silica particles, the aim of this work is to verify whether the mentioned scaling law also captures the influence of the concentration of the indifferent electrolyte (KCl) on the viscosity of their diluted dispersions.The verification will mainly rely on the results of our own experiments, as well as of those from other researches.

Materials
The LUDOX ® HS 40 (Sigma-Aldrich) used in this study is a stable 40 wt% aqueous dispersion of colloidal silica with pH of ca.9.8.The dispersion was initially diluted without any purification by the factor of 8.In the first set of experiments, the desired pH of the diluted dispersion was adjusted by adding NaOH or HCl.The volume fraction of silica was then systematically reduced by replacing 1.4 ml of the dispersion in the 6 ml viscometer with water of the same pH.In the second set of experiments, the concentration of KCl in the initially diluted dispersion was systematically reduced by replacing the 1.4 ml aliquot with a dispersion of the same pH and silica content.To convert the mass fraction (initially 5 wt%) into the volume fraction, the density of the dry silica particles was assumed to be 2300 g cm −3 , a value determined with a high accuracy by other researchers for Ludox HS silica based on mass measurements before and after drying a known volume of the dispersion.As in our previous coagulation experiments, KCl was used at relatively high concentrations ranging from 0.0078 to 0.1 M, so the activities and not the concentrations of the solutions were considered and calculated using activity coefficients listed in the reference [38].

Methods
The dynamic viscosities of the silica dispersions were determined very precisely using a glass micro Ubbelohde capillary viscometer for transparent liquids (capillary size I, the second thinniest in the series), which was placed and sealed in a glass jacket.The temperature of the surrounding water in the jacket was maintained at 25 °C using a circulation thermostat (Alpha, Lauda, with the temperature stability of ± 0.05 K at 35 °C, and the display and setting resolution of 0.1 °C).The temperature was parallelly monitored with a more precise external thermometer Termio 1 with the resolution of 0.01 °C and the accuracy of 0.07 °C.The calibrated capillary constant of the viscometer was 0.0085411.The efflux time of the dispersion, being approximately 2 min, was manually measured using a stopwatch.Three to five consecutive measurements were performed, providing averages with a standard deviation of approximately 0.2 s under given conditions (silica or KCl concentration).No extra time intervals were allowed for the dispersions to equilibrate after changing the conditions, as the equilibrium was assumed to reach immediately.Throughout the experiments, no signs of coagulation or sedimentation were observed.

Results
The viscosity of highly diluted colloidal systems can be described by the well-known Einstein equation, as derived by Einstein [39,40,41]: In Eq. 1, η s and η 0 represent the viscosity of the colloidal system and its liquid medium, respectively.ϕ is the volume fraction of colloidal particles, and K is a geometric constant, which is equal to 2.5 for solid spheres.From several viscosity measurements, it has been observed that the linear relationship given by Eq. (1) holds true very accurately for ϕ < 0.05.However, for aqueous dispersions of model-synthesized silica colloidal particles, as well as for many other hydrophilic colloids, the value of K is usually higher than 2.5, as also in our case.It is obviously attributed to the PEE coefficient p in the (1) modified version of Eq. 1: Figure 1 shows dependencies of the so-called specific viscosity of dilute aqueous dispersions η s /η 0 − 1 on the volume fraction of dry particles ϕ (as determined from their weight fraction) of HS Ludox colloid silica at three pH values.As Eq. 1 predicts, the specific viscosity increases in a linear fashion with the increasing volume fraction and approaches to zero for ϕ = 0 of the particles, irrespective of pH of the solution.It shows all the sols Newtonian and the so-called secondary electroviscous effect due to the electrostatic interaction between particles absent.Moreover, not only is the slope of all these dependences, representing K in Eq. 1, higher than 2.5 but it also increases with pH, i.e., as it is well known, with the surface charge density of silica.Fitting the dependences by straight lines (R 2 ≥ 0.997), K values are obtained to be 2.67, 3.48, and 7.68 for pH 2.3, 7.0, and 9.8, respectively.These values are in line with the values obtained for the same kind of silica in the presence of the lowest KCl concentration tested (3 × 10 −4 M) by Laven and Stein [23]: 3.12, 4.87, and 8.12 for pH 2.7, 5.7, and 8.7, respectively.Note that although the fitted intercept of the lines in Fig. 1 is not exactly zero as theoretically predicted by Eq. 1, the deviation is negligible.
Figure 2 shows the dependence of the specific viscosity of HS Ludox colloid silica dispersions on the activity of the KCl solution at ϕ = 2.5 wt% and ϕ = 5 wt% (pH = 9.8).It can be seen that the viscosity (and so K) decreases with increasing KCl activity in both cases, indicating a neutralizing effect of KCl to the surface charge of silica.For a comparison, Adamczyk et al. [24,26] obtained K = 4.4 for their S1 sample at pH 10 and 20 mM KCl which value is very close to ours K = 4.2 at the comparable pH and KCl concentration.
Based on the measurements shown in Figs. 1 and 2, the overestimated values of K sensitively reflect the pH and ionic strength of the aqueous medium.From the perspective of colloid science, these observations could be explained by phenomena associated with the formation-and-neutralization of electric charge at the interface between silica particles and the surrounding water solution, particularly with the EDL, its structure, and/or deformation (PEE, as mentioned) as well as with the hydration layer.From the perspective of polymer science, the gel layer may be operative.In fact, all these layers have been considered in the literature to elucidate the viscosity (as well as coagulation or colloid stability) of hydrophilic colloidal silica dispersions.None of them however were able to be brought in the quantitative agreement with the experiments.It seems that the simplest way of identifying a relevant layer is to define and verify the dependence of its thickness on the ion activity in the solution.In fact, the increasing ionic concentration leads theoretically to a thinning of all these layers.
Assuming that none of the assumptions on which Einstein originally derived Eq. 1 for a given colloidal system is violated, it is likely that the higher values of K, not reflecting other than the spherical particle shape factor, can be simply attributed to the fact that the viscosity is not influenced by "dry" spherical silica particles but by their "kinetic units" with larger "wet" dimensions.In other words, Eq. 1 actually involves not ϕ but the so-called effective or hydrodynamic volume fraction (ϕ eff ), where it holds that ϕ eff > ϕ and simultaneously: (2) K × = 2.5 × ef f .Furthermore, if we assume that the increase in particle size is due to an as yet unspecified layer on the surface of silica particles in contact with the aqueous solution, and we consider the silica particles as core-shell spheres as in our previous and other works, then ϕ eff can be simply expressed as the sum of two contributions: the (invariable) solid core that can be identified with ϕ and a (changeable) permeable shell, i.e., ϕ eff = ϕ + ϕ shell .Then we can express ϕ shell to depend on the difference between the K and the Einstein viscosity coefficient of 2.5: It should be noted that Eq. 3 also holds in cases where the shell is assumed to be not hydrodynamically permeable enough when the degree of correction, c, defined as Finally, if we assume a direct proportionality between ϕ shell and the volume of the layer and it is the swellable polyelectrolyte gel layer in question, a very simple relation between K − 2.5 and the activity of the electrolyte solution (see the "Introduction" section, [28]) can be obtained: It follows from the mean-field Flory-Huggins thermodynamic theory, which expresses the total osmotic pressure in the neutral, homogeneous, and unstrained crosslinked polymer gel as a combination of several phenomena in polymer physics.These include (i) the mixing term with the Flory polymer-solvent interaction parameter χ (playing the same role as temperature in the equation of state for a van der Waals fluid) and (ii) the rubber-like elastic term.When the polymer chains of the gel are partially charged via ionized functional groups, two additional terms must be added: (iii) The Donnan effect of ions and (iv) the electrostatic repulsions between these charges themselves.While it is not possible to find an explicit general solution that compensates for all the osmotic pressure contributions, special cases can be considered.So, for certain experimental conditions, the total thermodynamic equilibrium, i.e., zero osmotic pressure, can be determined by balancing only the elastic and Donnanionic contributions, while still accounting for the essential physics of the system.This provides a simple asymptotic behavior for the scaling effect of salt concentration on the swelling volume of the polyelectrolyte gel V ~ a −3/5 , as reflected by Eq. 4.
The experimentally observed inverse scaling of K -2.5 with the activity of KCl solutions is shown in Fig. 3 in loglog coordinates for highly charged HS Ludox silica samples at 2.5 wt% and 5 wt% (gray and black circles, respectively, data from Fig. 2) and at pH 9.8.In these coordinates, each power function appears as a straight line with the position and slope depending on the prefactor and exponent of that function, respectively.The figure clearly shows that our data, when plotted in this manner for both weight fractions of silica particles, merge into a single strictly linear dependence.This demonstrates that the assumed degree of swelling, reflected by the value of K, under the given conditions, does not depend on ϕ but solely on the activity of the KCl solution.However, the most important fact is this experimental dependence is indeed very accurately fitted (R 2 = 0.99903) by Eq. 4, i.e., a power law function with an exponent of −3/5 = −0.6 (−0.5948 ± 0.00819).

Discussion
The observed close agreement between our experimental data and the theoretically based scaling power law is truly surprising considering on one hand the simplicity of the given law and of the core-shell model of silica particles and, on the other hand, the straightforward character of the experimental evidence of anomalous viscosity in aqueous colloidal silica dispersions.In our opinion, this strongly supports the long-standing notion proposed by a few proponents of colloid science regarding the inherent existence of a swellable polyelectrolyte gel layer on the surface of silica particles brought into contact with aqueous solutions.
Figure 3 includes three datasets of K calculated from p's which all can be considered supportive of the gel layer Fig. 3 A log-log dependence of K − 2.5 for Ludox HS silica dispersion on the activity of electrolyte solution.Our measurements using KCl: ϕ = 2.5 wt% (gray circles) and ϕ = 5.0 wt% (black circles), pH = 9.8.The solid line is the best power law fit with the function parameters denoted.Other experimental data included Laven and Stein [23], Ludox HS silica, pH = 8.7, KCl, ϕ = 1 vol% (orange triangles); Honig et al. [42], Ludox HS silica, pH ~ 9.5, NaCl, ϕ = 2.2 vol% (blue diamonds); and Rasmusson et al. [25,27], prolate silica particles, SOL2 (Eka Chemicals AB), pH = 9.5, NaCl (green squares) concept by their authors, although without a thorough quantitative analysis.The position of the (K − 2.5)-to-a dependence, following a power law that becomes linear in log-log coordinates, should depend on pH, salt type, and the shape of silica particles in a colloidal silica dispersion.None of these factors are strictly identical in the measurements by others.However, because only the slope of the dependence matters here, all these measurements captured in Fig. 3 can be seen to indicate a good agreement with our measurements, at least in the asymptotic limit for the highest salt activities presented.It should be noted that, in principle, although the dependence reflecting supposedly gel layer deswelling manifests for all the datasets over the given range of electrolyte activities, it does not mean that it continues to have the same trend up to highest possible activities, where other mechanisms may apply.
According to Laven and Stein [23] (orange triangles in Fig. 3), there must be another mechanism apart from PEE that significantly affects the linear dependence of viscosity on ϕ.They consider this mechanism to be manifested through a "combined electroviscous" function p′ = p + p other .Indeed, the values of p′ were found to be much higher by a factor of 10 than those expected for p (although there is a similar trend between the two), even under conditions where the zeta potential and hence p should be close to zero, such as at low pH and/or high salt levels.Consequently, Laven and Stein propose that an osmotically driven and reversibly swellable gel-like oxide surface layer, formed by a crosslinked polyelectrolytic network, is the only possible cause of p other and, simultaneously, of anomalous behavior observed in colloidal silica dispersions in general.
They also argue that the mechanical strength of such a layer might not be very high and refer to a paper by Churaev et al.where swollen silica gel layers were proposed to rationalize anomalous streaming potentials in vitreous silica capillaries, and the gel layer on their surface was assumed to be easily torn off by hydrodynamic forces.The interpretation of their and our results is consistent with the assessment by Dukhin and Derjaguin, who attributed the apparent viscosity of water in fine glass capillary systems, significantly higher than the calculated values, to "boundary layers" of immobile water on the capillary surface or to a slight swelling of surface layers due to hydration effects.
Be that as it may, Laven and Stein still did not see the existence of such a layer fully proved or interpretable in then-known theoretical framework.They speculated that the layer with its swelling mechanism can be associated with the third electroviscous effect.Such an interpretation would be correct only formally since it is the silica surface that would provide the material (polysilicic acid) for the layer rather than any other external source.(The tertiary electroviscous effect has just been evolved to represent the distortion of polymers due to their electrostatics sensitive to pH and salts, playing no role for supposedly rigid colloidal particles such as silica.)Also, it is fair to mention that they had speculated that the underestimation of PEE can be caused by an additional conductance (the phenomenon not being incorporated in the theoretical derivation of the effect) occuring in the gel layer and reducing the electrophoretic mobility and subsequently underestimating the zeta potential.
Rasmusson et al. [25,27] (green squares in Fig. 3) measured viscosity and mobility of polydisperse sols of charged oblate (ellipsoidal) silica particles at ionic strength varying from 2 to 50 mM NaCl.They attributed the deviations from the Einstein equation to a combination of shape factor and again PEE, "other" effects such as the expanded gel layer being difficult to separate out."In many respects," the gel layer model was seen as a generalization of the dynamic Stern layer model of otherwise hard spheres, which is essentially in agreement with Laven and Stein's view.However, specifically in this work, it is necessary to emphasize the apparent validity of the power law scaling also for particles that have a significantly different shape than a spherical one.
Honig et al. [42] (blue diamonds in Fig. 3) stated PEE in Ludox HS sols conforms to the theory (Booth equation).However, to fit the experimental viscosity data, parameters in this equation (charge of the particles and the residual electrolyte concentration in the solution) were adapted.Yet, the fits were not very good in the whole range of NaCl electrolyte concentrations, and no fits could be obtained when adapting zeta potentials instead of the surface charge.These facts might again indicate the presence of the charged gel layer.

Conclusions
Our work hopefully advances the understanding of the structure of model colloidal silica particles in aqueous environments as soft colloids-solid cores encircled with an elastic gel layer.The responsible mechanisms identified, the Donnan equilibrium and the rubber-like elasticity, are well known in the field of polymer science but not yet in colloid science, at least for the rationalization of viscosity in dilute dispersions of silica or some other non-polymeric colloids.
To our best knowledge, no paper has been published that would present the relationship between the viscosity and the electrolyte concentration in a systematic way as we do.Also, although the experiments presented in this work, being consistent with the experiments of others and also with our previous measurements, are modest, we believe that they can have a more far-reaching impact to the future research-not only in terms of our view of the structure of the silica particles themselves and its effect on anomalous viscosity (silica particles but also other hydrophilic particles in the dilute to dense regime of their dispersions), but also on, of course, other phenomena.However, this is already well beyond the scope of this short report.
Still, although scaling laws in general mean more than correlations between variables, they do not prove a causality completely.Therefore, a further study is needed to explore a broader range of factors that influence other seemingly anomalous properties of colloidal systems of silica and other types of hydrophilic colloidal and nanometric particles (e.g., hydrous metal oxides as a group) from the perspective of polyelectrolyte solution and gel theory, which after all also continue to experience a significant development.

Fig. 1 Fig. 2
Fig. 1 Specific viscosity η s /η 0 − 1 versus dry volume fraction ϕ of HS Ludox colloid silica dispersions at various pHs (no KCl added).Straight lines with the analytical functions are best fits.Error bars represent standard deviations from the average of at least three measurements