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Langmuir 2009, 25, 2696-2702
Role of Counterion Association in Colloidal Stability Lyonel Ehrl,† Zichen Jia,‡ Hua Wu,† Marco Lattuada,† Miroslav Soos,† and Massimo Morbidelli*,† Institute for Chemical and Bioengineering, Department of Chemistry and Applied Biosciences, ETH Zurich, 8093 Zurich, Switzerland, and NoVartis Pharma AG, TRD/PHAD/PDU Topical and Other, WSJ-145.8.51, NoVartis Campus, Forum 1, CH-4056 Basel, Switzerland ReceiVed October 16, 2008. ReVised Manuscript ReceiVed December 3, 2008 A generalized model for colloidal stability has been validated against experimentally measured values of Fuchs stability ratio and critical coagulation concentration (ccc) for electrolytes with mono- or divalent cation, i.e., potassium chloride and magnesium chloride, respectively. Besides the classical DLVO theory, the generalized model accounts for the interplay between colloidal interactions and the association of cations with the particles surface charge groups. The model parameters are either obtained or estimated purely on the basis of independent information available in the literature. For the monovalent salt, the predictions agree well with literature experimental data, forecasting both the ccc values and stability ratios quantitatively. For the divalent salt the predictions for large values of the stability ratio tend to deviate from the experimental data produced in this work, but it is noted that the onset of stability, i.e., the ccc, and small stability ratios are correctly predicted. Moreover, a comparison of the above results with those neglecting the effect of counterion association with the particles surface charge groups indicates that the latter substantially overestimates stability ratios in the presence of high salt concentration in the case of the monovalent salt, and leads to unrealistic large values of the ccc for the divalent salt. Including the association of cations with the particles surface charge groups can explain the relatively low values of experimental ccc for divalent salts compared to the theoretical predictions by the classical DLVO theory neglecting ion association, which is a point of interest in industrial coagulation processes.
1. Introduction We encounter colloidal dispersions, on a day-to-day basis, in consumer goods, such as dairy products, cosmetics, and paints.1 In addition, many final products are based on intermediates of colloidal nature, may it be plastics2 or simply clean water.3 The stability of colloidal dispersions with respect to aggregation is relevant for their processing, as, on one side, stability is required during transport and storage, and, on the other side, aggregation is desired in the downstreaming (e.g., separating polymer from latex). Commonly, the stability of such systems is guaranteed by electrical charges on the particle surface, leading to an electrostatic double layer that impedes aggregation.4,5 The surface charge groups can result from polymer end groups as a result of the initiator and, therefore, are covalently bound to the surface (fixed), or they may rise from adsorbed ionic surfactant molecules and, thus, are physically bound (mobile).6,7 Colloidal stability is a result of the extent of surface charge and the condition of the electrostatic double layer, and thus depends on the parameters *Corresponding author. E-mail:
[email protected]; phone: +41-44-63-23033; fax: +41-44-63-21082. † Institute for Chemical and Bioengineering. ‡ Novartis Pharma AG. (1) Holmberg, K. Handbook of Applied Surface and Colloid Chemistry; John Wiley & Sons: Chichester, U.K., 2002; Vol. 1-2. (2) Odian, G. Principles of Polymerization, 4th ed.; John Wiley & Sons, Inc.: Hoboken, NJ, 2004. (3) Letterman, R. D.; Amirtharajah, A.; O’Melia, C. R. Coagulation and Flocculation. In Water Quality and Treatment - A Handbook of Community Water Supplies, 5th ed.; Letterman, R. D., Ed.; McGraw-Hill: New York, 1999; pp 6.1-6.66. (4) Hunter, R. J. Foundations of Colloid Science, 2nd ed.; Oxford University Press: Oxford, 2001; p 816. (5) Isrealachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1991. (6) Lyklema, J. Fundamentals; Academic Press: London, U.K., 1991; Vol. 1. (7) Lyklema, J. Solid-Liquid Interfaces; Academic Press: London, U.K., 1995; Vol. 2.
affecting those quantities, i.e., the type and concentration of the electrolyte and the solution pH. Since the 1950s, the standard approach in modeling colloidal interactions has been based on the Deryaguin-Landau-VerweyOverbeek (DLVO) theory, which accounts for the competing effects between van der Waals attraction and electrostatic double layer repulsion. Under certain conditions, additional interaction forces such as long-range dispersion forces, short-range hydration forces, capillary condensation, and specific ion adsorption are known to be important and need to be considered specifically.4-7 Due to the difficulties in assessing the additional parameters by either experimental measurement or theoretical treatment of such non-DLVO interactions, their utilization is still infeasible for complex colloidal systems such as those usually encountered in industrial practice. Whereas for several model systems the classical DLVO theory has proven to provide correct predictions of colloidal stability in quantitative terms,8,9 for some systems its predictions are erroneous.10-13 It is often argued that this is due to ion specificity, which is proposed to be accounted for in a computational effortful manner.14,15 However, the validation of models including ion specificity is challenging, as an experimental proof or test of such theories is difficult to be adduced in the first place. Another fact is that although standard text books clearly illustrate the (8) Hanus, L. H.; Hartzler, R. U.; Wagner, N. J. Langmuir 2001, 17, 3136– 3147. (9) Kobayashi, M.; Skarba, M.; Galletto, P.; Cakara, D.; Borkovec, M. J. Colloid Interface Sci. 2005, 292, 139–147. (10) Hiemstra, T.; Van Riemsdijk, W. H. Langmuir 1999, 15, 8045–8051. (11) Swanton, S. W. AdV. Colloid Interface Sci. 1995, 54, 129–208. (12) Kihira, H.; Ryde, N.; Matijevic, E. J. Chem. Soc., Faraday Trans. 1992, 88, 2379–2386. (13) Elimelech, M.; O’Melia, C. R. Langmuir 1990, 6, 1153–1163. (14) Ruckenstein, E.; Manciu, M. AdV. Colloid Interface Sci. 2003, 105, 177– 200. (15) Bostrom, M.; Deniz, V.; Frank, G. V.; Ninham, B. W. AdV. Colloid Interface Sci. 2006, 123, 5–15.
10.1021/la803445y CCC: $40.75 2009 American Chemical Society Published on Web 01/28/2009
Role of Counterion Stability
effects of physically well established phenomena such as surfactant adsorption and counterion association on colloidal stability,4-7 they are often not implemented in quantitative models mainly because their physical parameters are unknown functions of ionic strength, surfactant concentration, solution pH, or temperature, where changes of the latter affect all the equilibrium dissociation constants. Moreover, the basic information needed for calculating the particle interaction potential, i.e., the effective surface charge of the particles, is not easily accessible.16 For the standard measurement techniques, i.e., conductometric or potentiometric titration and electrophoretic mobility measurements, certain limitations are inherent, and often they provide only an indirect measure for the effective surface charge of the particles.6,7,16 The effects of counterion association have been indeed accounted for in a few cases. The experimental results for electrostatic properties of membranes17 as well as on Langmuir monolayers18,19 can be well explained by accounting for association of monovalent and divalent cations with the surface charge groups. Manning20 introduced a description of counterion condensation in polyelectrolytes, which, e.g., plays a role in the binding of proteins to desoxyribonucleic acid (DNA), and interest is growing in the effect of counterion association on the bending and flexibility of DNA.21,22 To our knowledge, however, there exists little literature that considers the counterion association in the evaluation of the colloidal stability, i.e., Poisson-Boltzmann theory combined with a Langmuir adsorption theory describing ion association on the surface.23 In our previous work,24 a generalized model for the colloidal stability was proposed, which accounts simultaneously for the interplay among three physicochemical processes: colloidal interactions, surfactant adsorption equilibria, and association equilibria of surface charge groups with counterions. In this way, the estimated surface charge and potential are a result of this interplay and the ion distribution in the diffusive layer. In other words, a change in any parameter that affects one of the above processes may lead to variations in the estimated surface charge and potential. Note that, for the surfactant adsorption and counterion association, their equilibria are considered to occur at the particle-liquid interface. For the colloidal interactions, both DLVO and non-DLVO interactions can be included in the model. The applicability of the generalized model in describing the colloidal stability has been verified for several industrial latexes.24,25 The objective of this work is to further validate the generalized model, using experimental stability data of model colloids in the presence of monovalent or divalent salt. The colloidal systems are all surfactant-free with only fixed surface charges in order to minimize the number of involved parameters. In the case of monovalent salt, we apply the model to the experimental data reported by Behrens et al.26 It was shown by the authors26 that at low electrolyte concentrations the DLVO theory describes correctly their experimental results, but at high electrolyte concentrations it fails. In the present work, we demonstrate that (16) Lyklema, J. Colloids Surf., A 2006, 291, 3–12. (17) McLaughlin, S. Annu. ReV. Biophys. Biophys. Chem. 1989, 18, 113–136. (18) Bloch, J. M.; Yun, W. B. Phys. ReV. A 1990, 41, 844–862. (19) Sano, M.; Kamino, A.; Shinkai, S. J. Phys. Chem. B 2000, 104, 10339– 10347. (20) Manning, G. S. Acc. Chem. Res. 1979, 12, 443–449. (21) Lu, Y. J.; Stellwagen, N. C. Biophys. J. 2008, 94, 1719–1725. (22) Nordmeier, E. Macromol. Chem. Phys. 1995, 196, 1321–1374. (23) Grahame, D. C. Chem. ReV. 1947, 41, 441–501. (24) Jia, Z. C.; Gauer, C.; Wu, H.; Morbidelli, M.; Chittofrati, A.; Apostolo, M. J. Colloid Interface Sci. 2006, 302, 187–202. (25) Jia, Z. C.; Wu, H.; Morbidelli, M. Ind. Eng. Chem. Res. 2007, 46, 5357– 5364. (26) Behrens, S. H.; Christl, D. I.; Emmerzael, R.; Schurtenberger, P.; Borkovec, M. Langmuir 2000, 16, 2566–2575.
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when the generalized model is applied, accounting for the counterion association, the DLVO theory can well describe the experimental results even at high electrolyte concentrations. In the case of divalent salt, the considerable differences between neglect and consideration of counterion association are compared with relevant experimental data produced in this work, and the higher accuracy of using the generalized model in the prediction of critical coagulation concentrations is pointed out. It is emphasized that in both cases any fitting of parameters and contributions due to non-DLVO forces are avoided. All the physical parameters involved in the model necessary to predict the experimental results stem from independent literature data, and if at all, they are only allowed to change within the tight intervals reported in critical literature sources.
2. Monovalent Counterion Behrens et al.26 performed titration and aggregation experiments under various potassium chloride (KCl) concentrations and pH, for two surfactant-free polystyrene latexes with carboxylic surface groups, which allow one to vary the surface charge density, thus to compare the corresponding colloidal stability measured experimentally with that predicted by the DLVO theory. The particle radii of the latexes were 52 and 155 nm, respectively. The charge density and the association constant of the carboxylic surface groups with protons were determined by fitting potentiometric titration curves with a model combining the association of the surface groups (neglecting the association with the potassium cations) and the ion activity due to the diffusive double layer, resulting in maximum surface charge densities at full deprotonation of 78 and 98 mC/m2 for 52 and 155 nm latexes, respectively. The obtained pKa was 4.9, which is larger than the pKa ()4.76) of ethanoic acid27 as well as those measured by other authors for carboxylic latexes (4.6428). Behrens et al. explained this relatively large pKa by the relatively low dielectric constant inside the particles. This information was then used in predicting the colloidal stability by solving the nonlinear Poisson-Boltzmann equation allowing for charge regulation and applying the classical DLVO theory, with the Hamaker constant as a fitting parameter. Deviations of the simulation results to experimental data were observed, especially for high electrolyte concentrations. 2.1. Surface Density of Chemical Species. Let us introduce, at the particle-liquid interface, the association/dissociation equilibria between the anionic carboxylic surface groups and the cationic species in the electrolyte solution, i.e., protons and potassium ions: Ka
-COOH {\} -COO- + H+ 1
(1)
KK+
-COO- + K+ {\} -COOK
(2)
where Ka1 is the dissociation constant of the carboxylic headgroup and KK+ is the association constant of the carboxylic headgroup with potassium ions, which are defined via the respective species activities (27) Speight, J. G. Lange’s Handbook of Chemistry, 16th ed.; McGraw-Hill: New York, 2005. (28) Ottewill, R. H.; Shaw, J. N. Kolloid-Zeitschrift and Zeitschrift Fur Polymere 1967, 218, 34.
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Ehrl et al.
Ka1 )
a-COO-aH+ a-COOH
(3)
KK+ )
a-COOK a-COO-aK+
(4)
Consequently, the surface charge density σ is given by
σ)
FFtot s 1 + aH+ ⁄ Ka1 + aKs +KK+
(5)
where F ) NAe is the Faraday constant (being the product of elementary charge, e, and Avogadro number, NA), Ftot is the total density of chargeable sites, and aHs + and aKs + are the surface activities of the protons and potassium ions. The surface activities, asj , are related to those of the bulk, aj, following the Boltzmann equation:
ajs ) ajexp(-zjβeψ0)
(6) -1
where zj is the valence of the j-th ionic species, β ) kBT, the thermal energy, and ψ0, the surface potential. Furthermore, the distributions of all ionic species in the system are of Boltzmann type, and the electrostatic potential profile is described by the Poisson-Boltzmann equation. In the following, we will make the usual assumption that activities, aj, can be replaced by the corresponding concentrations, cj. For convenience, in the framework of the classical Gouy-Chapman theory we apply the following expression to relate the surface charge density to the surface potential24
√ ∑ c [exp(-z βeψ ) - 1]
σ ) - R0
j
j
0
(7)
where R0 ) 2Fε0εr/βe with ε0 being the vacuum permittivity, and εr the relative dielectric constant of the medium. Behrens et al.26 applied an additional correction factor in eq 7 accounting for the surface curvature of the particles. In the present work this term is neglected as for particle diameters larger than 100 nm where no significant effect was observed, especially for large electrolyte concentrations. Concomitant solution of the association equilibria defined in eq 5, using surface activities as defined in eq 6, and the boundary condition due to the Poisson-Boltzmann distribution of ions in the diffusive layer in eq 7, provides the surface charge and surface potential at each given pH and electrolyte concentration. In order to compare predictions of the generalized model with the results from the potentiometric titrations of Behrens et al.,26 we have considered that potentiometric titration does not provide a direct measure of the surface charge density, but rather provides a measure of the amount of protons exchanged by counterions. When the K+ association with the surface groups is accounted for, potentiometric titration provides a measure of the sum of the densities from both the deprotonated surface groups and the surface groups associated with potassium ions, Fexp ) F-COO- + F-COOK. Since Fexp also includes noncharged species, in the following, it is referred to as molar surface density (mol/m2), instead of surface charge density (mC/m2). Then, in the following, the simulated surface density of deprotonated surface charge groups neglecting or accounting for the K+ association will be denoted as F′sim ) F-COO- or Fsim ) F-COO- + F-COOK, respectively. A comparison of the experimental potentiometric titration data (in molar surface densities) of Behrens et al.26 with the model predictions is presented in Figure 1 for both primary particle sizes at various ionic strengths, corresponding to those used in later aggregation experiments. The values of the parameters involved in the model predictions and their sources are listed in
Figure 1. Absolute molar surface densities of the various surface species as a function of the pH for a primary particle size of (a, b) Rp ) 52 nm and (c, d) Rp ) 155 nm and ionic strengths equal to (a, c) 10 mM, (b) 100 mM, and (d) 300 mM. Legend: (0) Fexp, experimental data obtained from Behrens et al.;26 (- - -) Fsim ′ , prediction of experimental data without association; (s) Fsim, prediction of experimental data with association; (- -) F-COO-, with association; and (- · -) F-COOK, with association. For the model prediction the parameters listed in Table 1 were used.
Table 1. Note that all model parameters have been estimated from independent literature sources, i.e., with no fitting of the experimental data. It can be noted that for most of the range of pH values investigated the predictions with (solid line) and without (short dashed line) K+ association are very similar and in good agreement with the experimental data (open squares), especially for the initial slope (at low pH values) and the total number of deprotonated carboxylic surface groups (at high pH values). Some differences appear in the region of intermediate pH in the range from 5 to 7, where the results neglecting the K+ association provide a better fit to the experimental data. It should be noted that this is the region close to the pKa of the surface groups, where small changes in pH would lead to large changes in the surface charge. Moreover, the driving force for establishing a stable potential, i.e., the proton concentration, becomes substantially small in this case, resulting in a substantially long time (several hours) for the system to reach equilibrium.7 In fact, Behrens et al. reported that in this region the signal of the potentiometer was still shifting while recording the value, which could be a reason for the observed deviation. In Figure 1 are also plotted the molar surface densities of the resulting free deprotonated carboxyl groups, F-COO- (--), and those associated with K+ ions, F-COOK (- · -). It can be seen that the K+ association reduces the amount of the free deprotonated carboxyl groups, which are the source of the surface charge and determine the surface potential. Thus, it is expected that increasing the electrolyte concentration (ionic strength) promotes the K+ association, depleting the surface charges. Figure 2 shows two examples in the cases of pH ) 4 and 8 for the particle size, Rp ) 155 nm, where the values of the relative surface charge, σ/σ′, defined as the ratio of the surface charge accounting for the K+ association to that neglecting the K+ association, are plotted as a function of the ionic strength, I. The values of the corresponding relative surface potential, ψ0/F0′, are also reported in the same figure. It is seen that, for pH ) 8, values of both σ/σ′ and ψ0/ψ0′ decrease monotonically as the ionic strength increases. At I ) 300 mM, for example, the effective surface charge (surface potential) considering the K+ association is only 54% (61%) of the surface charge (potential) neglecting the K+ association. In the case of pH ) 4, however, both the σ/σ′ and ψ0′ values are practically equal to unity, and significant deviation from unity occurs only at I > 100 mM. This arises because, under acidic conditions, there are protons (H+) competing with K+ for association with the carboxyl charge groups. Since the association constant is much larger for H+ than for K+, see Table 1, the H+
Role of Counterion Stability
Langmuir, Vol. 25, No. 5, 2009 2699 Table 1. Table of Model Parameters as Used in the Model Prediction
parameter
description
value used in computations
Hamaker constant for polystyrene
pKa,1
dissociation constant of the latexes used by Behrens et al.26 dissociation constant of the 4.312 latexes used in this work association constant of K+ -0.41 with -COO2+ association constant of Mg 0.55 (0.05 with -COO-
pKa,2 log KK+ log KMg2+
source
remarks
1.65 ×10-20J Hough and White40 (0.92 × 10-20 J or theoretical values depend on the 0.95 × 10-20 J); Prieve and Russel41 spectroscopic data they are -20 (1.37 × 10 J); Parsegian based on and the numerical method 42 5 -20 and Weiss (1.3 × 10 J); Isrealachvili used and are in the range from -20 -20 (0.9 × 10 J to 1.3 × 10 J) 0.9 × 10-20 to 1.4 × 10-20 J, a slightly broader range is obtained by fitting experimental data reaching from 0.7 × 10-20 to 1.9 × 10-20 J26 4.9 Behrens et al.26 measured by potentiometric titration
AH
Speight27 Martell and Smith43
correspond to the pKa of phenylacetic acid as expected for carboxyl-ended polystyrene, cf. Ottewill and Shaw28 25 °C, 0.1 M
Martell and Smith43
25 °C, 0.1 M
association is dominant, while the K+ association manifests significant effect only at very high ionic strength. 2.2. Predictions of Stability Ratios and Critical Coagulation Concentrations. Behrens et al.26 experimentally measured the values of the Fuchs stability ratio W for the two latexes at each
given ionic strength, as a function of pH. The results are reproduced in Figure 3a,b (symbols) for Rp ) 52 and 155 nm, respectively. Now let us apply the generalized stability model to predict the W values shown in Figure 3. The Fuchs stability ratio is defined as the ratio of the Smoluchowski rate constant over the measured one, which can be expressed as
W)2
Figure 2. Relative surface charge density (σ/σ′) and relative surface potential (ψ0/ψ′0), defined as the ratio of the values accounting for the counterion association (no prime) to those neglecting the counterion association (with prime) for Rp ) 155 nm. At pH ) 4 both curves for σ/σ′ and ψ′0 overlap and practically equal one.
Figure 3. Values of the Fuchs stability ratios (W) as a function of pH at different fixed ionic strengths for the latex with the primary particle radius, (a) Rp ) 52 nm and (b) Rp ) 155 nm. The symbols represent experimental data reproduced from Behrens et al.,26 the solid curves correspond to the predictions of the generalized model, with parameters as listed in Table 1, and the dashed curve is the model prediction neglecting the counterion (K+) association.
dl ∫2∞ exp(βU) Gl2
(8)
where l ) r/Rp, and r is the center-to-center distance between two particles. U is the total interparticle interaction energy, which is described here using the classical DLVO model accounting for the van der Waals attractive (UA) and the electrostatic repulsive (UR) potentials, i.e., U ) UA + UR. The applied expression for UA is the typical Hamaker relation5 (UA ) -(AH/6){[2/(l2 - j4)] + (2/l2) + ln[1 - (4/l2)]}, where AH is the Hamaker constant) and for UR is the modified Hogg-Healy-Fuersteneau expression29 (UR ) (4πε0εrRpψ02/l)ln{1 + exp[-κRp(l - 2)]}, where the Debye-Hu¨ckel parameter is defined as κ ) [(e2NA∑j z2j aj)/ (ε0εrkBT)]1/2.24 The quantity G in eq 8 is a hydrodynamic function accounting for the additional resistance caused by the squeezing of the fluid during the particle approach30 (G ) (6l2 - 20l + 16)/(6l2 - 11l)). Therefore, in the predictions of the Fuchs stability ratio, we first determine the surface potential, as described in the previous section, which is used in the UR expression in order to compute the total interparticle interaction energy U. Then, with U, the W value is calculated using eq 8. The predicted W values (continuous curves) are shown and compared with the experimental results of Behrens et al.26 in Figure 3a,b for Rp ) 52 and 155 nm, respectively. Note that no fitting parameter has been used in both cases. The only parameter that was allowed to be changed inside a range provided by theoretical considerations and experimental evidence (see Table 1) in order to get a better fit was the Hamaker constant, where a value of 1.65 × 10-20 J was used. It is seen that the model predictions are in good agreement with the experimental data for both particle sizes. We have also computed the W value neglecting the K+ association, and found that in the case of Rp ) 52 nm the results are very similar to those accounting for the K+ association. For this reason, the results neglecting the K+ association are not reported in Figure 3a. This can be explained by considering two (29) Sader, J. E.; Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1995, 171, 46–54. (30) Honig, E. P.; Roeberse, Gj.; Wiersema, P. H. J. Colloid Interface Sci. 1971, 36, 97.
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Figure 4. Values of the critical coagulation concentration (ccc) as a function of pH for the latex with the primary particle radius, (a) Rp ) 52 nm and (b) Rp ) 155 nm. The symbols represent experimental data obtained from Behrens et al.,26 and the solid and dashed curves correspond to the model predictions considering and neglecting the counterion (K+) association, respectively.
Figure 5. Values of the Fuchs stability ratio (W) as a function of the MgCl2 concentration at various fixed pH values for the latex with the primary particle radius, Rp ) 150 nm. The symbols are experimental data obtained in this work, and the solid and dashed curves correspond to the predictions of the generalized model and the model neglecting the counterion (Mg2+) association, respectively. For the model prediction the parameters listed in Table 1 were used.
Figure 6. Values of the critical coagulation concentration (ccc) as a function of pH for the latex with the primary particle radius, Rp ) 150 nm. The solid and dashed curves correspond to the predictions of the generalized model and the model neglecting the counterion (Mg2+) association, respectively.
main factors. First, at relatively low ionic strengths (1-100 mM), since the diffusive layer is relatively thick, also due to the low K+ association constant with the carboxyl group, see Table 1, the K+ association under such conditions is substantially reduced. Second, all the W values for the ionic strength in the range 1-100
Ehrl et al.
mM are located in the range pH < 6, where H+ competes with K+ to associate with the carboxyl charge groups. Obviously, because of the much larger H+ association constant, the association of the carboxyl charge groups with H+ is dominant with respect to that with K+. Thus, the difference in the predictions between considering and neglecting the K+ association becomes negligible in these cases. An insignificant effect was also observed for the other extreme of very high ionic strength (1 M) in Figure 3a. This is because the salt concentration in this case is above the critical coagulation concentration, all the systems are fully destabilized, and the W value is determined by the Smoluchowski rate constant. In the case of the larger primary particles, Rp ) 155 nm, for the same reasons, the predictions with and without considering K+ association also provide very similar results for the extreme cases of low electrolyte concentrations (2-100 mM) and very high electrolyte concentrations (1 M). However, for the intermediate ionic strength, large differences in the predictions between neglecting and considering the K+ association have been observed. This is clearly shown in the case of I ) 300 mM in Figure 3b, where the modeling results accounting for the K+ association are in excellent agreement with the experimental results, i.e., having predicted correctly the position of the critical coagulation concentration and the occurrence of the plateau as well as the corresponding W values. Instead, the computed results neglecting the K+ association (the dashed curve in Figure 3b) do not capture any features of the experimental data. This arises because in the case of I ) 300 mM in Figure 3b, most of the W values are located in the range pH > 6, where the H+ association with the carboxyl charge groups becomes rather weak and the K+ association becomes dominant. Considering also the increased K+ concentration, the K+ association leads to substantial changes in the surface charge, and consequently in the surface potentials, which result in large differences in the calculated stability ratios between neglecting and considering the K+ association. In practical applications, it is often required to know the critical coagulation concentration (ccc) of a colloidal system, which represents the border between the fast DLCA and the slow RLCA regimes. The experimental data of Behrens et al. in Figure 3 clearly indicate the presence of a ccc for each set of data. We have estimated the experimental ccc value for each case, by extrapolating the slope of the W curve to the W value of 1.5, as is expected for DLCA experiments.31 Such obtained ccc values from the experimental data are shown in Figure 4a,b (symbols) as a function of pH for Rp ) 52 and 155 nm, respectively. The same approach has been used to estimate the ccc values from the predicted W curves, and the results are also reported in Figure 4a,b, where the solid and dashed curves correspond to the predictions accounting for and neglecting the K+ association, respectively. It is seen that the agreement between the predictions considering the K+ association and the experimental results is very good. The predictions neglecting the K+ association are also very good in the low pH region, again due to the dominant H+ and negligible K+ associations. For pH > 5, however, the model neglecting the K+ association overestimates the ccc value, and such overestimation increases as pH increases. This reflects that, at high pH, also due to the ccc being located at high ionic strength, the K+ association becomes substantial. Therefore, in this case, the K+ association has to be accounted for in modeling colloidal stability. We conclude that, for the two colloidal systems studied by Behrens et al., the generalized stability model, which in the present case is simply based on the classical DLVO theory and accounts for the interplay between colloidal interactions and counterion association, can well describe their colloidal stability behavior (31) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, U.K., 1999.
Role of Counterion Stability
for the monovalent salt (KCl) and in the entire experimental ranges of ionic strength and pH. We note that there is a difference in the treatment of the Poisson-Boltzmann equation by the two models. Behrens et al. solved the Poisson-Boltzmann equation applying the Derjaguin approximation allowing for full charge regulation. In this work, to facilitate the computations in the generalized model, the Debye-Hu¨ckel approximation is used, strictly valid only for low surface potentials (ψ0 < 25 mV), together with the assumption of a constant surface potential when calculating the Fuchs stability ratio. While it is known that the constant surface potential model tends to underestimate the height of the repulsive energy barrier between two particles compared to the full charge regulation model, the Hogg-Healy-Fuersteneau expression tends to overestimate it in the case of highly charged surfaces. These two errors compensate for each other, and the values of the Fuchs stability ratio are not too sensitive to the type of model used, especially when taking into consideration the uncertainties in the Hamaker constant value. However, our results clearly indicate the importance of accounting for the counterion-binding effect in predicting the correct evolution.
Langmuir, Vol. 25, No. 5, 2009 2701
(carboxyl) groups with the protons and the divalent magnesium ions (Mg2+):
Ka2 ) KMg+2 )
a-COO-aH+ a-COOH a-COOMg+ a-COO-aMg+2
(9) (10)
Consequently, the surface density of free charge groups F-COOis given by
F-COO- )
Ftot s s 1 + aH+ ⁄ Ka2 + aMg +2KMg+2
(11)
Since the association of one Mg2+ ion leads to the charge reversal of a surface charge site from a negative to a positive one, the net surface charge density must be given by
σ ) F(F-COO- - F-COOMg+)
(12)
2+
3. Divalent Counterion 3.1. Colloidal System and Aggregation Experiments. In order to study the applicability of the generalized model to divalent counterion electrolytes, we have produced suitable experimental data using surfactant-free carboxylic polystyrene latex with a primary particle radius of 150 nm, whose aggregation was induced by magnesium chloride (MgCl2). The latex was provided by Interfacial Dynamics Corporation (Portland, OR) (product 7-300, coefficient of variation ) 3.2%, batch 2440, solids 4.3 g/100 mL, surface charge density ) 99 mC/m2 corresponding to a molar surface density of carboxyl groups ) 1.0 × 10-7 mol/m2). This stock solution was prediluted using Millipore water; then, an equal volume of the MgCl2 solution (proper amount of hydrochloric acid was also added to adjust pH) was mixed with the prediluted latex to start aggregation. The stock MgCl2 solution was filtered with a 200 nm filter (Whatman, Clifton, NJ) prior to use. One half of this final solution was covered by a lid and used to monitor the aggregation extent by dynamic light scattering, and the second half was used to verify if the pH was stable over time during the experiment. The time evolution of the average hydrodynamic radius of aggregates was monitored, up to a maximum of 20% conversion of the primary particles, and from it the stability ratio W was evaluated. Details for the evaluation of the W value from the dynamic light scattering data can be found in the literature.32-34 In this way, for each fixed pH value, the stability (W) curve as a function of the MgCl2 concentration was obtained. Note the difference with respect to the W curves of Behrens et al.,26 which were evaluated for constant ionic strengths with varying pH. The dynamic light scattering measurements have been performed using a BI-200SM goniometer (Brookhaven, Holtsville, NY) and a solid-state laser, Ventus 532-500 (Laser Quantum, Stockport, U.K.), with a wavelength of 532 nm. All the aggregation experiments and measurements were carried out at a temperature of 25 °C. 3.2. Comparison of the Model Predictions with Experimental Data. In Figure 5 are shown the obtained W values (symbols) as a function of the Mg2+ concentration for various fixed pH values. To predict the experimental results using the generalized stability model, similar to the monovalent case, see eqs 3 and 4, we consider the association of the surface charge (32) Lattuada, M. Aggregation Kinetics and Structure of Gels and Aggregates in Colloidal Systems; ETH Zurich: Zurich, 2003. (33) Lattuada, M.; Sandku¨hler, P.; Wu, H.; Sefcik, J.; Morbidelli, M. AdV. Colloid Interface Sci. 2003, 103, 33–56. (34) Vaccaro, A.; Sefcik, J.; Morbidelli, M. Polymer 2005, 46, 1157–1167.
It is clear that, at high degree of Mg association, the surface charge density can be neutralized and even charge reversal can occur. For the considered 150 nm latex, the same Hamaker constant was used as for the previous two latexes. For the present latex, we did not measure the pKa value, and instead we used a value reported in the literature, which was pKa2 ) 4.312, equal to that of phenyl acetic acid.27 This value was chosen as according to Ottewill and Shaw28 and Shaw35 the surface charge of carboxylated latexes originates from both carboxylic and phenyl acetic groups, where it is noted that the used value represents a lower limit for the pKa. Also for the decadic logarithm of the association constant of Mg2+ with the surface charge groups, log KMg2+, a literature value has been used, which is 0.55 (see Table 1). Figure 5compares the model predictions considering (solid curves) and neglecting (dashed curves) the Mg2+ association with the experimental data. In the case of accounting for the Mg2+ association, when one considers that no fitting parameter has been used, the agreement between the model predictions and experimental data is rather satisfactory, particularly in the regions of small W values close to the ccc. In fact, as shown in Figure 6, the ccc values (solid curve) predicted by the model accounting for the Mg2+ association are in good agreement with the experimental ccc values (symbols). For larger W values the predictions significantly overestimate the experimental data. Such deviations of predicted W values, obtained by classical DLVO, with measured ones is common for larger W values.12,36-38 Possible explanations of these deviations are distribution of the particle surface potential,12 nonhomogenoues surface potential,39 and surface roughness.12 For a detailed discussion the reader is referred to the review by Swanton.11 Although the particle system is assumed to be quite monodisperse and the mentioned single properties are expected to be close to ideal, their concomitant effect could easily cause the differences between predicted and measured W values. (35) Shaw, J. N.; Marshall, M. C. J. Polym. Sci., Part A: Polym. Chem. 1968, 6, 449. (36) Reerink, H.; Overbeek, J. T. G. Discuss. Faraday Soc. 1954, 74–84. (37) Ottewill, R. H.; Shaw, J. N. Discuss. Faraday Soc. 1966, 154. (38) Lips, A.; Willis, E. J. Chem. Soc., Faraday Trans. 1 1973, 69, 1226– 1236. (39) Vreeker, R.; Kuin, A. J.; Denboer, D. C.; Hoekstra, L. L.; Agterof, W. G. M. J. Colloid Interface Sci. 1992, 154, 138–145. (40) Hough, D. B.; White, L. R. AdV. Colloid Interface Sci. 1980, 14, 3–41. (41) Prieve, D. C.; Russel, W. B. J. Colloid Interface Sci. 1988, 125, 1–13. (42) Parsegian, V. A.; Weiss, G. H. J. Colloid Interface Sci. 1981, 81, 285– 289. (43) Martell, A. E.; Smith, R. M. Critical Stability Constants. Plenum Press: New York, 1989; Vol. 6, 2nd supplement.
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In the case of neglecting the Mg2+ association, the predicted results shown in Figure 5 indicate that when pH is rather low (e.g., the cases with pH ) 2.2 and 3.4), considering or neglecting the Mg2+ association leads to very similar results; i.e., the effect of counterion association is insignificant in these cases. This is due to the fact that at very low pH the H+ association (due to the larger association constant) is prevailing against the Mg2+ association, and also because the W values in the two cases are located in the region of very low MgCl2 concentrations. However, for the other two cases with pH ) 4.0 and 6.6, the deviations of the predictions neglecting the Mg2+ association from those considering the Mg2+ association increase dramatically. It should be noted that, unlike the monovalent cases in Figure 3, for this divalent case, substantial deviations occur in the acidic region and low MgCl2 concentrations. This involves two main factors: First, the association constant of Mg2+ is much larger than that of K+, and second, association of one Mg2+ ion leads to the charge reversal of a surface charge site from a negative to positive one. Also for the ccc values, when Figure 4 is compared to Figure 6, in the former the deviation starts around pH ) 5, while for the latter it starts around pH ) 3.5. Therefore, it is concluded that accounting for counterion association for divalent ions becomes even more important than for monovalent ions.
4. Concluding Remarks In this work, we have applied a generalized model for colloidal stability, developed previously, to describe the stability behavior of three model colloids: surfactant-free polystyrene latexes stabilized by fixed carboxyl surface charge groups, with particle radii equal to 52, 155, and 150 nm, respectively. The main objectives are first to further validate the generalized model and second to emphasize the importance of accounting for the interplay among different physicochemical processes when modeling colloidal stability. In the present cases, the interplay occurs between colloidal interactions and counterion association. For the first two latexes with particle radii equal to 52 and 155 nm, the values of the Fuchs stability ratio (W) were measured by Behrens et al.26 as functions of pH and ionic strength. A substantial attempt has been made by the same authors to model the obtained experimental results, using a sophisticated approach including the appropriate choice of a Hamaker constant, the effect of nonlinearity in the Poisson-Boltzmann equation on stability predictions, as well as the role of charge regulation and the error introduced by the Derjaguin approximation. However, they ignored the counterion association effect. It turns out that such a complex approach was unable to predict the colloidal stability behavior in the range of high ionic strength. Now, for the same sets of experimental results, when the generalized model is applied, where the colloidal interaction model is the classical DLVO model but the counterion association is accounted for, with the values of all the model parameters coming from independent literature sources, we are able to quantitatively predict the W values in the entire ranges of the experimental conditions. Further, for the third latex with a particle radius of 150 nm, we have experimentally determined the W values at different pH, using a divalent salt, magnesium chloride. The obtained results have been simulated using the generalized model, again with the DLVO interaction model and considering counterion association. With all the model parameters taken from independent literature sources, the generalized model predicts quantitatively the values of the critical coagulation concentration in the entire pH range. However, the model fails if one neglects the interplay induced by the association of magnesium cations.
Ehrl et al.
Acknowledgment. This work was financially supported by the Swiss National Science Foundation (Grant 200020-113805/1).
Nomenclature aj,asj (mol/L) AH (J) cj,csj (mol/L) e (C) F (C/mol) G
I (mol/L) kB (kg2 m s-2 K-1) Ka, Ka1, Ka2 (mol/L)
KK+ (L/mol) KMg2+ (L/mol) l NA (1/mol) pKaj r (nm) R0 (F mol-1 m-1) Rp (nm) T (K) U,UA,UR (J) W zj (mol/L)
Roman Letters activity of the j-th ionic species in the bulk and at the surface, respectively Hamaker constant activity of the j-th ionic species in the bulk and at the surface, respectively elementary charge Faraday constant, F ) NAe hydrodynamic function accounting for the additional resistance caused by the squeezing of the fluid during the particle approach ionic strength Boltzmann constant (1.3807 × 10-23 kg2 m s-2 K-1) dissociation constant of the acidic surface charge groups, 1 for Behrens latexes, 2 for the latex of this work, see Table 1 association constant of the monovalent potassium ions with the surface charge groups association constant of the divalent magnesium ions with the surface charge groups normalized center-to-center distance, l ) r/Rp Avogadro constant (NA ) 6.022 × 1023 mol-1) negative common logarithm of Kaj, pKaj ) -log [Ka/(mol/L)] center-to-center distance between two particles R0 ) 2Fε0εr/βe particle radius absolute temperature (298 K) total, attractive, and repulsive interparticle interaction energy, respectively Fuchs stability ratio, defined in eq 8 valency of the j-th ionic species
Greek Letters inverse of the thermal energy, β-1 ) kBT vacuum permittivity, relative dielectric constant of the medium κ (1/m) Debye-Hu¨ckel parameter, κ ) ([e2NA∑jz2j aj]/[ε0εrkBT])1/2 2 Fj (mol/m ) molar surface densities of j-th species Fexp, Fsim, Fsim ′ (mol/m2) Fexp ) F-COO- + F-COOK, Fsim ) F-COO+ F-COOK, Fsim ′ ) F-COOFtot (mol/m2) total density of chargeable surface groups surface charge density considering and σ, σ′ (mC/m2) neglecting (prime) counterion association surface potential considering and neψ0, ψ0′ (mV) glecting (prime) counterion association β (1/J) ε0 (F/m), εr
LA803445Y
