Application of the Charge Regulation Model to the Colloidal

May 11, 2001 - The model predicts a weakly attractive but nontouching particle network when sufficient salt is added, in agreement with experiments. T...
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Langmuir 2001, 17, 3557-3562

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Application of the Charge Regulation Model to the Colloidal Processing of Ceramics P. Maarten Biesheuvel† and Fred F. Lange* Materials Department, College of Engineering, University of California, Santa Barbara, California 93106 Received October 2, 2000. In Final Form: February 20, 2001 We implement competitive adsorption of ions in the charge regulation (CR) model to describe several aspects of the interaction and compaction behavior of ceramic particles. This model shows that there is an electrostatic origin to the experimental observation that the compaction of particles to high volume fractions can still result in a body of nontouching particles that exhibits fluidlike rheological behavior. Analytical expressions based on the model describe the influence of pH and ionic strength on the pressure needed to force particles into touching contact. The model predicts a weakly attractive but nontouching particle network when sufficient salt is added, in agreement with experiments. The CR model further predicts that when salt is added to a dispersed particle network, surface dipoles of the indifferent co-ion will make up much of the surface sites; the fraction of dipoles increases slightly when opposing surfaces approach one another and eventually discharge when they touch.

Introduction The stability and compaction behavior of concentrated particle networks influenced by electrostatic forces in aqueous environments are of importance to many applications such as the behavior of paints, the processing of minerals,1 and the manufacturing of ceramic materials. The purpose of this article is to describe how the charge regulation (CR) model can explain several phenomena that have been combined and used to form ceramic powders into complex, engineering shapes by a method called colloidal isopressing.2 The first of these phenomena is the observation that a weakly attractive but nontouching particle network can be formed by adding excess salt to an aqueous slurry formulated far from the isoelectric point (IEP) of the powder. This observation indicates that the addition of sufficient salt induces a strong repulsion at small separations.3 However, DLVO theory,4,5 based on the concept of constant surface potential, predicts that on increasing the ionic strength, the primary maximum decreases and at a critical salt concentration the system agglomerates spontaneously to form a touching particle network.3,5-11 Instead, experiments show that at high * Corresponding author. E-mail: [email protected]. † Current affiliation: Shell Global Solutions International B.V., Badhuisweg 3, 1031 CM Amsterdam, The Netherlands. E-mail: [email protected]. (1) Johnson, S. B.; Franks, G. V.; Scales, P. J.; Boger, D. V.; Healy, T. W. Int. J. Miner. Process. 2000, 58, 267. (2) Yu, B. C.; Lange, F. F. Adv. Mater. 2001, 13, 276. (3) Velamakanni, B. V.; Chang, J. C.; Lange, F. F.; Pearson, D. S. Langmuir 1990, 6, 1323. (4) Derjaguin, B. V.; Landau, L. Acta Physicochim. URSS 1941, 10, 25. (5) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the stability of lyophobic colloids; Elsevier: New York, 1948; p 161. (6) Chang, J. C.; Lange, F. F.; Pearson, D. S. J. Am. Ceram. Soc. 1994, 77, 19. (7) Colic, M.; Franks, G. V.; Fischer, M. L.; Lange, F. F. Langmuir 1997, 13, 3129. (8) Israelachvili, J. N. Intermolecular and surface forces; Academic Press: London, 1992; p 248. (9) Sigmund, W. M.; Bell, N. S.; Bergstro¨m, L. J. Am. Ceram. Soc. 2000, 83, 1557. (10) Tadros, Th. F. Solid/Liquid Dispersions; Academic Press: London, 1987; p 99. (11) Velamakanni, B. V.; Chang, J. C.; Lange, F. F.; Pearson, D. S. Langmuir 1990, 6, 1323.

concentrations of salt (e.g., 0.2-2.0 M), attractive particles experience a very high repulsive force at short separations (∼1 nm). The second phenomenon concerns the relation between the short-range repulsion and the pressure required to force particles into contact during consolidation, e.g., by pressure filtration. Franks et al.12 showed that during consolidation, particles formulated to be highly repulsive (far for the IEP) could be easily forced into contact to produce bodies that were elastic and brittle, i.e., when stressed, they failed by crack extension. On the other hand, particles formulated with a short-range repulsion (by adding salt to a dispersed slurry) could be consolidated to a high particle packing density and produce plastic bodies; i.e., the bodies exhibited a yield stress smaller than the fracture stress. The yield stress increased with the concentration of added salt. In addition, Franks et al.12 showed that a transition pressure exists that separates the plastic and brittle regime without a significant change in the particle packing density. They concluded that a critical (transition) pressure was required to force the particles into contact to produce the elastic, brittle body. Colic et al.7 and Franks et al.12 showed that the transition pressure was dependent on the type of counterion used to form the short-range repulsion. More recently, Yu and Lange2 reported how the two phenomena could be combined to produce a new method to form engineering shapes. They add a small amount of salt to a dispersed slurry and consolidate the particles below the transition pressure. Although the consolidated body, still saturated with water, had a high relative density, the small amount of salt produced a very low yield stress; i.e., the body appeared fluidlike after consolidation. After injecting the fluidlike body into a rubber cavity, they applied an isostatic pressure to the particle network that was higher than the transition pressure. The high isostatic pressure forced the particles into contact to transform the fluidlike body in the cavity to an elastic body that could be removed from the cavity without shape distortion. As shown below, a recent analysis13 suggests that both the short-range repulsion and the transition pressure (12) Franks, G. V.; Colic, M.; Fisher, M. L.; Lange, F. F. J. Colloid Interface Sci. 1997, 193, 96.

10.1021/la001388e CCC: $20.00 © 2001 American Chemical Society Published on Web 05/11/2001

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required to force particles into contact can be explained by the CR model that was developed by Ninham and Parsegian14 and applied to amphoteric surfaces by Chan et al.15 The CR model does not require an arbitrary, fixed surface potential (as in the constant potential (CP) approach of the DLVO theory) but formulates the electrostatic repulsion in terms of a small set of well-defined surface properties such as the isoelectric point, the number of available surface sites, and the equilibrium constants of the mass-action equations that describe the surface reaction with species in the surrounding liquid. The CR model predicts that the surface potential increases when surfaces approach and correctly predicts that the repulsive force (as a function of separation) is highly dependent on surface properties, pH, and ionic strength. In this work we develop several analytical expressions based on the CR model and use these to consider some aspects of the particle behavior briefly discussed above, the maximum electrostatic repulsion, and the composition of approaching oxidic surfaces in water with added salt. Theory The CR model is essentially similar to the DLVO theory but replaces the assumption of a constant, fixed surface potential by an equality that relates the field strength at the surface with the adsorption behavior of the different ions in solution. Electrostatic models are based on the electrochemical potential of species i in solution, µi (J/mol)

µi ) µi0 + RT ln γixi + ziFφ + ViP

(1)

Here, R is the gas constant (8.3144 J/(mol‚K)), T temperature (K), γi the activity coefficient, xi the fraction of component i, zi the charge number, F Faraday’s number (96485 C/mol), φ the electrostatic potential (V), Vi the specific volume (m3/mol), and P the pressure (Pa). At chemical equilibrium (∇µi ) 0), for γi ) 1, for a constant specific volume (as for liquids) and for a constant pressure (∇P ) 0), eq 1 results in the Boltzmann distribution of the ionic species

ci ) ci,∞ exp(-ziφF/RT)

dx

)-

d2φ dx2

σ)F

∑i zici

) (r0)-1 F

(3)

dE

∫m0 F∑zici dx ) -r0∫m0 dx dx ) i

r0(Em - E0) ) -r0E0 ) r0

|

dφ dx

(6)

When competitive adsorption of monovalent indifferent ions is implemented, e.g., by assuming cation (C) adsorption to the [O-] sites forming a [O-C+]s dipole and anion (A) adsorption to the [OH2+]s site forming [OH2+A-]s,19,20 the total amount of surface sites (adsorbed hydroxylgroups, mol/m2), cs,tot, is given by16

cs,tot ) [O-C+]s + [O-]s + [OH]s + [OH2+]s + [OH2+A-]s (7) On the basis of equilibrium thermodynamics,15 mass action equations can be set up that relate the concentrations of surface sites with equilibrium constants K. The four relevant equations for the five surface groups in eq 7 are for monovalent cation and anions

KC ) [O-]sc0,C[O-C+]s-1 K- ) [O-]sc0,H[OH]s-1 K+ ) [OH]sc0,H[OH2+]s-1 KA ) [OH2+]sc0,A[OH2+A-]s-1

(8)

Combination of the mass action equations with eq 6 and eq 7 results in

(

K

K-c0,C

+

σ)F

(

[OH]s ) cs,tot (4)

(5)

σ ) F([OH2+]s - [O-]s)

with the condition of charge neutrality, which results in5,8

σ)-

∑i zics,i

with cs,i surface concentrations (mol/m2). For an amphoteric surface consisting of adsorbed hydroxyl groups [OH]s that may either dissociate to form [O-]s sites and oxonium ions (H3O+, symbol H) or react with oxonium ions to form [OH2+]s sites, eq 5 results in

(2)

In both the constant potential approach (CP, as in the DLVO theory) and in the CR model, surface charge σ (C/ m2) is determined by combining the Poisson equation (planparallel opposing plate geometry; valid for field strengths below ∼20 MV/m16)

dE

10-12 C/(V‚m)), and x the coordinate perpendicular to the surface; 0 refers to the solution phase next to the Nernst plane and m to the midplane. The Nernst plane is the plane parallel to the surface where charging occurs. This plane does not coincide with the particle surface when an adsorbed layer of hydroxyl groups is present.5,8,13,17 Interestingly, eq 4 directly implies that when the Nernst planes touch, they must have discharged completely (σ f 0). Though this condition is the direct consequence of assuming charge neutrality,15,18 it is not always used to its fullest potential. The CR model uses the fact that σ given by eq 4 also equals σ given by

c0,H

KCc0,H

+

-

)

K[OH]s c0,H

(9)

)

c0,H c0,Hc0,A K+1+ + + + A c0,H K K K

-1

(10)

0

with E the electric field (V/m), r the relative permittivity (water, r ) 78), 0 the permittivity of vacuum (8.854 × (13) Biesheuvel, P. M. Langmuir 2001, 17, 3553. (14) Ninham, B. W.; Parsegian, V. A. J. Theor. Biol. 1971, 31, 405. (15) Chan, D. Y. C.; Perram, J. W.; White, L. R.; Healy, T. W. J. Chem. Soc., Faraday Trans. 1 1975, 71, 1046. (16) Basu, S.; Sharma, M. M. J. Colloid Interface Sci. 1994, 165, 355.

Neglecting competitive adsorption (infinitely high values of KC and KA), combination of eqs 6 and 7 results in15,19 (17) Overbeek, J. Th. G. Adv. Colloid Interface Sci. 1982, 16, 17. (18) Prieve, D. C.; Ruckenstein, E. J. Theor. Biol. 1976, 56, 205. (19) Healy, T. W.; White, L. R. Adv. Colloid Interface Sci. 1978, 9, 303. (20) Van Riemsdijk, W. H.; Bolt, G. H.; Koopal, L. K.; Blaakmeer, J. J. Colloid Interface Sci. 1986, 109, 219.

Colloidal Processing of Ceramics

σ ) Fcs,tot

c0,H2 - K-K+ K-K+ + K+c0,H + c0,H2

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(11)

With competitive adsorption of monovalent indifferent cations and anions, the result is13,16

σ ) Fcs,tot ×

imply that the surface is positively charged, thus the gap between the surfaces must contain an excess of negative ions and the concentration of all positive species is lower than the bulk values, pHm > pHIEP; this contradicts the premise of pHm < pHIEP. For pH∞ < pHIEP a similar argument can be set up concerning an excess of positive ions. As a result

|pHm - pHIEP| e |pHm - pH∞|

c0,H2 - K-K+ K-K+ + K-K+c0,C/KC + K+c0,H + c0,H2 + c0,H2c0,A/KA (12) All four mass action equations, eq 8, are based on ideal Langmuir adsorption behavior. For more detailed studies one might consider non-Langmuir isotherms.20,21 However, eqs 11 and 12 suffice for the purpose of this work. At the isoelectric point, concentrations in the liquid far away from the surface (∞) are identical to surface concentrations (0) and specifically, c0,H ) c∞,H ) cIEP, with cIEP the oxonium (proton) concentration of the isoelectric point (cIEP ) 103-pHIEP; concentrations in mol/m3). Thus, eq 12 results in (K-K+)1/2 ) cIEP. From this point on, we will focus on touching flat surfaces to simplify the analysis. However, the analysis can be extended to describe the full interaction of curved surfaces by solving the CR model at several separations, implementing van der Waals forces and using Derjaguin’s approximation.13 We make calculations for the case that the Nernst planes touch, thus at a separation of two times the Nernst layer thickness, NL (NL being of the order of the size of a water molecule).13 For touching Nernst planes, σ f 0, and thus, according to eq 12, c0,H ) (K-K+)1/2 ) cIEP. In terms of pH, this results in

(16)

Equation 15 indicates that the maximum midplane potential φm,max is found when pHm is as far away as possible from pH∞. Because of (16), φm,max requires pHm ) pHIEP, which occurs when the Nernst planes touch (see eqs 13 and 14), resulting in

φm,max ) φ0,max )

RT ln(10)(pHIEP - pH∞) F

(17)

At equilibrium, the electrostatic repulsion between flat parallel plates is (for all γi ) 1) given by8

∑i (cm,i - c∞,i)

PESR ) RT

(18)

Combining eqs 2, 17, and 18 results for the electrostatic repulsive pressure for touching Nernst planes in

( (

ziF

) )

∑i c∞,i exp - RT φm,max

PESR,max ) RT RT

-1 )

∑i c∞,i(exp(zi ln(10)(pH∞ - pHIEP)) - 1)

(19)

For a 1:1 salt, eq 19 can be simplified to22

(13)

PESR,max ) 2RTc∞(cosh(ln(10)(pH∞ - pHIEP)) - 1) (20)

Further, as σ f 0, and Nernst planes touch, the field strength E0 has gone to zero (see eq 4), resulting in the fact that

where c∞ is the ionic strength ()1/2∑izi2c∞,i). Prieve and Ruckenstein18 derived

pH0 ) pHm

PESR,max ) RTc∞ ln2(10)(pH∞ - pHIEP)2

pH0 ) pHIEP

(14)

where m and 0 refer to the midplane and to the solution phase next to the Nernst plane, respectively. Equation 2 gives the midplane potential

φm )

RT ln(10)(pHm - pH∞) F

which is correct to within 10% for a pH∞ that is less than 0.5 points from pHIEP. The van der Waals force (Pa) between flat surfaces is given by8

PVDW ) A/(6πD3)

(15)

As we will explain next, pHm is always intermediate between pH∞ and pHIEP. First we must realize that the field E is zero at the midplane due to symmetry and the gradient in E (dE/dx) and the electrostatic potential φ are nonzero at the midplane for overlapping double layers. Further, the charge (∑zici) and the magnitudes of dE/dx, E, and φ increase monotonically from the midplane to the Nernst plane (“0”). Now, assume pH∞ > pHIEP for which the surface is negatively charged and the electrostatic potential everywhere in the gap between the surfaces is negative. For this case, the concentration of the positive species is larger at the midplane than in the bulk solution: cm,i > c∞,i, as shown in eq 2, thus pHm < pH∞. Further, for pH∞ > pHIEP, pHm can never be smaller than pHIEP because it would (21) Koopal, L. K.; Van Riemsdijk, W. H.; De Wit, J. C. M.; Benedetti, M. F. J. Colloid Interface Sci. 1994, 166, 51.

(21)

(22)

with A the Hamaker constant (J) and D the separation between the oxidic surfaces (distance between Nernst planes + 2‚NL). The van der Waals force will be combined with the electrostatic/osmotic repulsion, eq 18, to describe the total repulsion between two approaching surfaces. The composition of the surface in terms of the five possible groups can be calculated for each separation by eqs 8 and 10 as is done in Figure 2. However, if we focus solely on the three neutral groups, [OH]s, [O-C+]s, and [OH2+A-]s, and define the fraction Xi as relative to the total amount of neutral groups, e.g., for the [OH]s sites, X[OH] is given by

X[OH] )

[OH]s [OH]s + [OH2+A-]s + [O-C+]s

(23)

(22) Prieve, D. C.; Ruckenstein, E. J. Colloid Interface Sci. 1978, 63, 317, fourth entry in Table 2.

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Table 1. Data Used in Simulations + 2(pK + pK ) ) 9.25 pK- - pK+ ) 3 a 8.3 × 10-6 mol/m2 a 2.8 Å c 0.7 mol/m3 a 5.3 × 10-20 J d

IEPalumina ) pHIEP,alumina ∆pK cs,tot b thickness of Nernst layer NL KA ) K C Hamaker constant Aalumina

1/

a Chan et al.15 b Total number of available surface sites. c Biesheuvel.13 d Israelachvili.8

then it turns out that the three fractions Xi are independent of separation, and given by

A ) K+KAKCcH,∞, B ) KCcA,∞cH,∞2, C ) K+K-KAcC,∞ X[OH]s )

A , A+B+C

B , A+B+C C (24) X[O-C+]s ) A+B+C

X[OH2+A-]s )

Equation 24 provides a simple means to assess the composition of the surface in terms of the dipoles and the hydroxyl groups [OH]s as a function of salt concentration and pH in the solution far away from the surface.

Figure 1. Surface charge for alumina as function of separation. Solid line is pH ) 7 without salt, dashed line pH ) 7 with 0.1 M monovalent salt. Data are in Table 1. Surface charge becomes zero when the Nernst layers touch.

Results Surface Charge and Composition. The CR model is based on overall charge neutrality which implies that surfaces discharge (σ f 0) when they are forced to touch one another. To show that this part of the model is correct, we calculate σ as a function of separation by numerically solving eq 2 together with eqs 3, 4, and 12 for alumina (see Table 1), the material used for many colloidal experiments, such as discussed in the Introduction. Figure 1 shows that σ indeed goes to zero when surfaces approach each other. In addition, Figure 1 shows that the surface charge is much higher for an aqueous system containing 0.1 M of a monovalent salt relative to the same system without added salt. A second condition of the model is that as the surfaces approach, the concentration of the two charged surface groups, [O-]s and [OH2+]s become the same, which is necessary to comply with the condition of charge neutrality. Figure 2 reports the calculations of the surface concentrations normalized to the total concentration cs,tot for alumina. In this case, when surfaces approach, the concentration of negatively charged sites increases, while the concentration of the positive surface sites decreases. (At a large separation the surface concentration of negative sites is much lower than the concentration of the positive sites.) At contact, the concentration of positive sites has become the same as the concentration of the negative sites. Additionally, the total amount of charged sites decreases when surfaces approach, which implies that the concentrations of each of the three neutral groups increases (because their relative concentrations, Xi, are fixed) to compensate for the diminishing (total) concentration of charged sites. The increase in concentration of the neutral groups is minute, namely, ∼0.5%. The prediction that the concentration of cations and anions is not decreasing but even slightly increasing when surfaces approach, is contrary to the initial hypothesis to understand the shortrange repulsive force, which suggested that the indifferent ions were removed from the oxidic surface.12 Instead, we see that both the counterions and the co-ions remain at the surface and pair with charged surface sites in the form of dipoles. The influence of pH and salt concentration on the surface composition can most simply be analyzed by considering

Figure 2. Surface composition for alumina surfaces in a solution of 0.1 M of monovalent salt (pH ) 7) as function of separation between Nernst planes. Simulations are based on data in Table 1.

the concentration of each of the three neutral groups normalized by the sum of their concentration, Xi, defined by eq 23; these fractions are independent of surface separation. Fractions Xi are plotted in Figure 3 for a pH range of 4-12 without salt addition and with 0.1 M of monovalent salt. As shown, for a pH between the IEP for alumina and pH ) 7, most surface sites are in the [OH]s form. However, outside this pH region, the fraction of [OH]s decreases and these sites are replaced by either [OH2+A-]s at lower pH or by [O-C+]s at higher pH. For the addition of 0.1 M of monovalent salt, the fraction of [OH]s sites is less than 10% for all pH values. Interaction Force. The total interaction between two charged surfaces, Ptot, can be calculated as the summation of the van der Waals attraction, PVDW, eq 22, and the electrostatic repulsion, PESR, eq 18. With the PoissonBoltzmann equation solved (eqs 2 and 3), the midplane

Colloidal Processing of Ceramics

Figure 3. Composition of alumina surfaces in terms of the three neutral groups, [O-]s, [OH]s, and [OH2+]s, without added indifferent salt (solid lines) and with 0.1 M monovalent salt (dashed lines) (data in Table 1).

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Figure 5. Electrostatic repulsion PESR and van der Waals attraction PVDW (horizontal line) for alumina for the condition of touching Nernst planes (separation 2‚NL), calculated from eq 20 and eq 22, for 0 M, 0.01 mM, 0.1 mM, 1 mM, 0.01 M, 0.1 M, and 1.0 M monovalent salt. Salt concentration increasing in direction of arrow.

a strong repulsive force can develop at separations smaller than the potential minimum described above. One can determine the salt concentration needed to produce this short-range repulsion by combining eqs 19 and 22 for the electrostatic repulsion for touching Nernst planes and the van der Waals force at a separation of 2‚NL, with NL the thickness of the Nernst layer. When this total force is positive, a potential minimum and strong repulsion can be expected. In the presence of a 1:1 indifferent salt, combining eqs 20 and 22 gives an explicit criterion for the minimum ionic strength c∞* necessary for the presence of a potential minimum

c∞* ) A(24RTπ NL3(cosh(ln(10)(pH∞ - pHIEP)) - 1))-1 (25) Figure 4. Total interaction Ptot (Pa) as sum of electrostatic repulsion, PESR, and van der Waals attraction, PVDW, for alumina based on solution of PB equations. Solid line is pH ) 7 without salt (c∞ ) 1 × 10-4 mol/m3), dashed line is pH ) 7 with 0.1 M monovalent salt (c∞ ) 100 mol/m3). Data are in Table 1.

concentrations cm,i are found that are necessary in eq 18. For alumina at pH ) 7 the total interaction force between surfaces is given in Figure 4 with and without the addition of 0.1 M monovalent salt. Without salt addition, the interaction is predicted to be purely attractive at pH ) 7 and we can expect a touching network, thus agglomerated particles. This is in agreement with experiments. With the addition of salt, a short-range repulsion develops resulting in an interaction profile characterized by an attraction down to a small separation, a deep potential minimum (located at the separation at which Ptot ) 0), and a steep repulsion for even smaller separations. Thus we surmise that the properties of weakly attractive particle networks, which make them so useful for colloidal isopressing,2 depend in essence on the interaction of van der Waals forces, eq 22, with the osmotic repulsion, eq 18, with midplane concentrations cm,i determined by the CR model. Criterion for the Existence of an Attractive Nontouching Network. With the addition of sufficient salt,

For c∞ > c∞*, a potential well precedes repulsive interaction at smaller separations such as depicted in Figure 4 for alumina. For alumina in water (pH ) 7), eq 25 predicts a value of c∞* ) 37.8 mol/m3, which is between the two entries in Figure 4 and indeed is a potential well only found for the higher c∞. Equation 25 shows that with a zero Nernst layer thickness, c∞* is +∞, which suggests that weakly attractive networks require the presence of a Nernst layer. Furthermore, eq 25 correctly predicts that the critical salt concentration necessary for the development of a short-range repulsion increases asymptotically (c∞* f +∞) for a pH approaching on the isoelectric point (pH f pHIEP). In the limit of pH ) pHIEP, c∞* ) +∞. Forces for Touching Nernst Planes. Figure 5 illustrates the pressure required to force two Nernst planes into contact (where particles are separated by twice the Nernst layer thickness). This pressure follows from subtracting the van der Waals attractive force from the electrostatic repulsive force which is a function of pH∞ and concentration of indifferent, monovalent salt. A shortrange repulsion will develop only when the electrostatic repulsion exceeds the van der Waals force. The pH∞ range for which this occurs increases with increasing salt content. In other words, given a certain bulk pH, pH∞, a shortrange repulsion will develop above a certain critical salt

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concentration c∞* as discussed before. Figure 5 shows that at the isoelectric point, where the surface potential is zero at all separations, the van der Waals force always exceeds the electrostatic repulsion, independent of salt concentration. Thus particles are predicted to agglomerate spontaneously, which is indeed observed. At zero salt addition and moving away from the IEP the electrostatic repulsion increases and exceeds the van der Waals attraction sufficiently far from the IEP, which suggests that nontouching networks can be prepared at sufficiently low and high pH values. This prediction agrees well with the fact that dispersed alumina suspensions can be prepared without salt addition at pH values of 5 and lower and at pH values of 10 and higher. A very low electrostatic repulsion is found between the IEP and pH ) 7 without salt addition, which suggests that the CR model explains the observed asymmetry of colloidal stability around the IEP (page 443 in ref 23; Figure 1B in ref 24, and page 517 in ref 25). The CR model predicts that for touching Nernst planes the electrostatic repulsion increases with salt concentration and that a critical salt concentration exists above which the electrostatic repulsion exceeds the van der Waals attraction. Above this critical concentration an external pressure must be applied to push the particles into touching contact. The magnitude of the external pressure needed to push particles into touching contact must increase with salt concentration. These predictions are opposite to DLVO theory which teaches that spontaneous agglomeration can be induced either by changing the pH to the IEP or by the addition of a sufficient amount of salts (see references in Introduction). Experiments show that adjusting the pH to the IEP does produce a touching network, while adding excess salt does not produce a touching network.6 Both observations are in agreement with predictions based on the CR model.26 Furthermore, the prediction based on the CR model that the necessary external pressure that needs to be applied to bring surfaces in touching contact increases with increasing salt concentration is in agreement with experiments that show that the transition pressure necessary to form a touching, brittle, and elastic network from a viscous saturated slurry increases with salt content.12 Discussion The electrostatic repulsive forces between hydrophilic surfaces in polar liquids, such as alumina in water, may be described by the Poisson-Boltzmann (PB) equation. Within the framework of this equation, we replaced the boundary condition of a constant surface potential, as used in the DLVO theory, by a relation that describes the charging reactions that take place at the surface. That is, we use the charge regulation model, which dates back to 1971. The charging reaction we use implements competition between oxonium ions on one hand, and the indifferent cations and anions on the other. We further implement the presence of a layer of hydroxyl groups on both of the approaching surfaces. With these conditions, the CR model is able to describe typical experiments with alumina. Without salt addition, the monotonic attraction into the primary minimum at pH ) 7 is predicted in agreement with observations (see Figure 4), while with the addition (23) Ducker, W. A.; Xu, Z.; Clarke, D. C.; Israelachvili, J. N. J. Am. Ceram. Soc. 1994, 77, 437. (24) Velamakanni, B. V.; Lange, F. F. J. Am. Ceram. Soc. 1991, 74, 166. (25) Chapel, J. P. J. Colloid Interface Sci. 1994, 162, 517. (26) Buscall, R.; Ettelaie, R.; Healy, T. W. J. Chem. Soc., Faraday Trans. 1997, 93, 4009.

Biesheuvel and Lange

of salt a potential minimum at a small separation is predicted, together with a strong but finite repulsion in agreement with experiments with concentrated slurries containing excess salt.2 The development of this shortrange repulsion when the salt concentration is increased is not predicted by the DLVO theory. The short-range repulsion can be overcome by exerting a high external force on the particle network12 to bring particles into touching contact. The force to push particles into touching contact increases with salt concentration. These observations are correctly predicted by the CR model; see Figure 5, which shows that for a given bulk pH, pH∞, the electrostatic repulsion at the distance of touching Nernst planes increases with increasing salt concentration, trespasses the van der Waals attraction at a critical salt concentration, and continues to increase with salt concentration, thus necessitating an increasingly high externally applied pressure to bring particles into touching contact. The model presented here only considers thermodynamically ideal systems. However, activity coefficients at the concentrations of importance to colloidal isopressing (order of 0.1 M) may deviate significantly from unity.27,28 Implication of these effects in the Boltzmann equation, eq 2, and the expression for the electrostatic (osmotic) repulsion, eq 18, may increase the predictive power of the CR approach, especially when the influence of different salts is concerned. This effect, investigated in detail,7,12 comes to the fore in the CR model not only by considering the different activity coefficients for the bulk activities27,28 but also via the different adsorption equilibrium constants KA,C for different ions. Conclusions We have analyzed several aspects of the charge regulation (CR) model with regards to the colloidal processing of ceramic powders. One of its key premises results in the fact that surface charge goes to zero as the surfaces approach one another. When the CR model incorporates a Nernst layer with the thickness of a water molecule, it predicts that the monotonic attraction for low ionic strengths, resulting in agglomeration, is truncated with a steep short-range repulsion of electrostatic origin when the ionic strength is above a critical value. This prediction is in agreement with experiments on the behavior of consolidated slurries of weakly attractive but nontouching particle networks and cannot be explained by the DLVO theory. The CR model explains that when the surface charging occurs via adsorbed hydroxyl groups, such as is the case for most oxides, including silica and alumina, the electrostatic repulsion for touching Nernst planes is very low between the IEP and pH ) 7 but increases rapidly on both sides of this region. The electrostatic repulsion increases rapidly when indifferent salt is added, even near the IEP. The adsorption equation predicts that the concentration of adsorbed indifferent ions increases slightly when surfaces approach, with a maximum concentration at contact. The percentage of surface sites that are in the dipole form (thus containing an indifferent ion) increases rapidly outside the pH region 7-IEP and is >90% for all pH values when sufficient indifferent salt is added. LA001388E (27) Eigen, M.; Wicke, E. Z. Elektrochem. 1954, 58, 702. (28) Bromley, L. A. AIChE J. 1973, 19, 313.