688
Langmuir 2009, 25, 688-697
A Novel Approach to Determine the Critical Coagulation Concentration of a Colloidal Dispersion with Plate-like Particles Longcheng Liu,* Luis Moreno, and Ivars Neretnieks Department of Chemical Engineering and Technology, Royal Institute of Technology, S-100 44 Stockholm, Sweden ReceiVed August 14, 2008. ReVised Manuscript ReceiVed NoVember 3, 2008 The critical coagulation concentration (ccc) of counterions is commonly described by the classical Derjaguin-Landau-Verwey-Overbeek (DLVO) theory on the basis of a static force balance. It can, however, also be estimated from a kinetic point of view by studying the process of colloidal coagulation, or from a dynamic point of view by considering colloidal transport in nonequilibrium systems where other processes such as diffusion and the influence of gravity come into play. In particular, in a test tube where colloidal expansion takes place, the ccc can be interpreted as the electrolyte concentration below which expansion of colloids would always lead to full access to the entire volume of the test tube and above which a sharp boundary is established between a colloidal gel and pure water. On the basis of this perception and the dynamic force balance model that we developed to describe colloidal expansion in a test tube, accounting for the effects of particle diffusion and gravity in contrast to the DLVO theory, we propose an alternative way to assess the ccc of counterions. We also derive an approximate expression for the case of homointeraction at constant charge for montmorillonite. The estimated ccc values agree quite well with those observed experimentally for both Na+ and Ca2+ counterions for montmorillonite dispersions, at pH ∼ 6.5. This is in contrast to the DLVO theory, which overpredicts the ccc by about 2 orders of magnitude. In addition, the detailed analyses suggest that the ccc of counterions decreases with increasing surface area and with the thickness of the particles. For montmorillonite, the ccc is nearly independent of the surface charge density of the particles for the range of typical charge densities.
1. Introduction The critical coagulation concentration (ccc), defined as the minimum concentration of counterions required to induce coagulation, is one of the most important characteristics of a colloidal dispersion and is often applied to assess its status.1 For some dispersed entities, the ccc of counterions was found on the basis of experimental observations to be inversely proportional to the sixth power of its valence, the so-called Schulze-Hardy rule.2 For montmorillonite dispersions, however, a significant deviation from this rule has been reported, which suggests that the relationship between the ccc values can be roughly expressed as3 +
ccc(Na ) ≈ 12ccc(Ca ) ≈ 63ccc(Al ) 2+
3+
(1)
In particular, as tabulated in Tables 1 and 2, a compilation of experimental results from different methods indicates that at pH ∼ 6.5 the ccc values would span a wide range between 5 and 250 mM in Na+ solutions, whereas they might vary between 0.36 and 3 mM in Ca2+ solutions. These observations cannot be explained successfully by the classical DLVO theory,10,11 but have been attributed to the * Corresponding author. E-mail:
[email protected]. Tel: +46 8 790 6414. Fax: +46 8 10 52 28. (1) Hsu, J. P.; Yu, H. Y. J. Colloid Interface Sci. 2005, 285, 719. (2) Hiemenz, P. C. Principles of Colloid and Surface Chemistry, 2nd ed.; Marcel Dekker, Inc.: New York and Basel, 1986. (3) Lagaly, G.; Ziesmer, S. AdV. Colloid Interface Sci. 2003, 100-102, 105. (4) Hetzel, F.; Doner, H. E. Clays Clay Miner. 1993, 41, 453. (5) Frey, E.; Lagaly, G. J. Colloid Interface Sci. 1979, 70, 46. (6) Chheda, P.; Grasso, D.; van Oss, C. J. J. Colloid Interface Sci. 1992, 153, 226. (7) Tomba´cz, E.; Szekeres, M. Appl. Clay Sci. 2004, 27, 75. (8) SKB Report R-04-35; Swedish Nuclear Fuel and Waste Management Co.: Stockholm, Sweden, 2004. (9) Tomba´cz, E.; Bala´zs, J.; Lakatos, J.; Sza´nto´, F. Colloid Polym. Sci. 1989, 267, 1016. (10) Derjaguin, B.; Landau, L. Acta Physiochem. URSS 1941, 14, 633.
Table 1. The ccc of Sodium Chloride for Montmorillonite at pH ∼ 6.5 authors
methoda
clay/water (w/w %)
ccc (mM)
Lagaly et al.3 Hetzel et al.4 Frey et al.5 Lagaly et al.3 Chheda et al.6 Lagaly et al.3 Lagaly et al.3 Tomba´cz et al.7 SKB8 Tomba´cz et al.9
RM AM TTS RM TOM RM RM CKM unspecified VI
0.025 0.1 0.025 0.5 0.015 1 2 4 unspecified 0.2
5 10 10-15 15 18 20 30 52 100 250
a The experimental methods include rheological measurement (RM), adsorption method (AM), turbidity and optical density measurement (TOM), coagulation kinetic method (CKM), test tube series (TTS), and visual inspection (VI).
Table 2. The ccc of Calcium Chloride for Montmorillonite at pH ∼ 6.5 authors
method
clay/water (w/w %)
ccc (mM)
Hetzel et al.4 Lagaly et al.3 Chheda et al.6 SKB8 Tomba´cz et al.9 Lagaly et al.3 Lagaly et al.3
AM RM TOM unspecified VI RM RM
0.1 0.025 0.015 unspecified 0.2 0.5 1
0.355 0.4 0.45 1 2 2 3
pronounced properties of montmorillonite and the strong effect of the Stern-layer adsorption.12 (11) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids: The Interaction of Sol Particles HaVing an Electric Double Layer; Elsevier: New York, 1948. (12) Bergaya, F.; Theng, B. K. G.; Lagaly, G. Handbook of Clay Science, DeVelopments in Clay Science; Elsevier: Amsterdam, Netherlands, 2006; Vol. 1, Chapter 5.
10.1021/la802658g CCC: $40.75 2009 American Chemical Society Published on Web 12/24/2008
NoVel Approach to Determine the ccc of Counterions
Montmorillonites are the most abundant minerals within the smectite group of 2:1 layered clays,13 which are distinguished from other colloidal materials by the highly anisometric and often irregular particle shape, the broad particle size distribution, the different types of charges (permanent charges on the faces, pH-dependent charges at the edges), the large cation exchange capacity and the different modes of aggregation.3,12 Montmorillonite particles are never true crystals but are more like assemblages of aluminosilicate sheets with very large surface area. In some dispersions, the sheets may indeed be very nearly completely separated, but this is by no means the general case. More often than not, especially in montmorillonite suspensions of low electrolyte concentration, the sheets tend to approach each other in a face-to-face orientation and aggregate to form particles. Such particles may have up to tens of sheets, separated by about 1 nm, for the case of Ca dominated counterions on montmorillonite, but only one or two sheets for the case of Nadominated counterions on montmorillonite.12,14 These multisheet particles can be compact stacks but, mostly, they are foliated and look like paper sheets torn into irregular pieces.3 The sheet has on its faces a permanent negative charge that results from isomorphous substitutions in the crystal lattice. However, the degree of substitution changes from sheet to sheet within certain limits but also on the individual sheet itself.3 On average, the surface charge of montmorillonites varies between 0.2 and 0.4 equiv/formula unit (Si, Al)4O10 but most montmorillonites have surface charges around 0.3 equiv/formula unit,3,12 which corresponds to a surface charge density of 0.10 C m-2. At the edges, the sheet has a charge that arises from adsorption or desorption of protons, attributed to the presence of aluminol and silanol groups, and hence it is pH dependent:7 positive at acid pH, and negative at basic pH. Also cations can compete with protons for the edge groups influencing the charge of the edges. As a result, montmorillonite particles show different modes of aggregation, depending on both pH and solid content.3 It was observed that, at pH between 4 and 6, the positively charged edges can interact with the negatively charged surfaces to generate T-type contacts and card-house type aggregation.7,12 At pH above 8, face-to-face contacts would dominate in the case of low solid content, because the area between two faces is much larger than that between an edge and a face, leading to band-type aggregation.7,12 In an intermediate pH range, in particular in the region near the point of zero charge of edge sites (∼6.5) where positive edge charges are no longer present or their number is very small, the probability of negatively charged edge-to-face random collisions is larger than that of oriented face-to-face collisions because of the spillover of the negative electrostatic field emanating from the surface of montmorillonite particles.7,12 However, such T-type contacts can only be formed at low solid content. In other cases, especially for delaminated montmorillonite, the very strong repulsion between the faces disrupts the edge-to-face contacts more easily, and the van der Waals attraction must be enhanced to reach the face-to-face coagulation condition.3 All these special properties (the high degree of structure- and charge-heterogeneity) of montmorillonite particles make the ccc of counterions dependent on pH, the solid content, the type of anion,12 and the finite size of counterions.15 As a result, it renders the full interpretation of experimental observations rather difficult, due mainly to the fact that the point of zero charge of edge sites (13) van Olphen, H. An Introduction to Clay Colloid Chemistry: For Clay Technologists, Geologists, and Soil Scientists; John Wiley and Sons: New York, 1977. (14) Schramm, L. L.; Kwak, J. C. T. Colloids Surf. 1982, 3, 43. (15) van de Ven, T. G. M. J. Colloid Interface Sci. 1988, 124, 138.
Langmuir, Vol. 25, No. 2, 2009 689
is still not well established, and the effect of pH on the modes of aggregation still remains under discussion.7,12,16 To date, the classical Derjaguin-Landau-Verwey-Overbeek (DLVO) theory is the only one that can give approximate solutions to the ccc values, and hence it is commonly employed to explain the coagulation of montmorillonite dispersions.7,12 However, in the modeling, it is mostly assumed that the particles are spherical, and the interaction takes place under the condition that the surfaces are at a constant potential.17,18 In addition, the classical DLVO theory cannot describe the pH-dependent stability of montmorillonite dispersions,7 because it is inherently not capable of accounting for the ionizable surface groups. Consequently, the prediction of the stability of montmorillonite colloids with the classical DLVO theory has been observed to be far from their actual behavior.19 The extended DLVO theory16,20 can, in principle, take into account the effects of pH, permanent charges, and electrolytes on the swelling of clays.7,21 However, the inclusion of Sternlayer adsorption needs more information, and is not generally applicable to systems other than the specific system. Also, it no longer leads to transparent analytical expressions, and hence it cannot be easily applied over a wide range of conditions. In this study, we do not account for the pH effect but restrict ourselves to the region close to the point of zero charge of edge sites, at pH ∼ 6.5, where montmorillonite particles can be considered to be plate-like with a constant surface charge density.12 We acknowledge that the Poisson-Boltzmann approximation underlying the diffuse layer model is theoretically not applicable to high surface charge densities and divalent counterions such as calcium.22 Such systems should collapse into a very dense gel caused by ion-ion correlations. It is not understood why this does not happen. However, this may be a reason why calcium montmorillonites form dense stacks of up to tens of sheets. It is outside the scope of this paper to discuss this problem. We treat the stacks as particles with surface electrical properties that can be described by the Poisson-Boltzmann equation in the same way as sodium montmorillonites. We find that this approach gives model predictions that agree reasonably well with experiments. We then approach the problem by setting up a dynamic force balance model to give an approximate expression to the ccc, without invoking the Gouy-Stern double layers. This approach gives reasonably good agreement with the experimental results tabulated in Tables 1 and 2. The paper is organized as follows. In section 2, for later reference and comparison with our model, we first present the approximate solution of the ccc of counterions for the case of homointeraction between parallel plates at constant charge, from the classical DLVO point of view. Then we shortly discuss the extended DLVO theory. In section 3, we describe the forces acting on a particle in a gradient of particle concentrations, and then introduce the concept of the thermal deViation factor. For illustrative purposes, we exemplify the system with colloidal expansion in a test tube, exploring its significance in determining the ccc, and then derive an approximate expression for the ccc of counterions. The test (16) Dura´n, J. D. G.; Ramos-Tejada, M. M.; Arroyo, F. J.; Gonza´les-Caballero, F. J. Colloid Interface Sci. 2000, 229, 107. (17) Hsu, J. P.; Tseng, M. T. AIChE J. 1996, 42, 3567. (18) Hsu, J. P.; Liu, B. T. J. Colloid Interface Sci. 1998, 198, 186. (19) Missana, T.; Adell, A. J. Colloid Interface Sci. 2000, 230, 150. (20) Lyklema, J. Fundamentals of Interface and Colloid Science: Particulate Colloids; Elsevier: New York, 2005; Vol. IV. (21) Mohan, K. K.; Fogler, H. S. Langmuir 1997, 13, 2863. (22) Kjellander, R.; Marcelja, S.; Quirk, J. P. J. Colloid Interface Sci. 1988, 126, 194.
690 Langmuir, Vol. 25, No. 2, 2009
Liu et al.
tube example is relevant because most ccc experiments are performed in vertical tubes or vessels with an upward or downward movement of the colloids. In this or similar systems, the particles will move as long as equilibrium has not been reached so as to attain an equilibrium situation for the whole system. It can be likened to particle diffusion, although other forces than just thermal forces are active. We include the effects of van der Waals force, double layer force, particle diffusion, and gravity. This is in contrast to the classical DLVO theory, which considers only the interaction between two isolated particles. In section 3.4, we present a comparison of the predicted ccc values and the experimental results for montmorillonite in both Na+ and Ca2+ solutions. Following this, we draw conclusions in section 4.
2. The Classical DLVO View of the ccc To start with, we briefly present the approximate solution of the ccc of counterions for the case of homointeraction between semi-infinite parallel plates, and then work on the numerical solution of the ccc by accounting for the thickness of the particles. We need to go over this in some detail because we will show that the DLVO model overpredicts the ccc by orders of magnitude for montmorillonites. 2.1. The Approximate Solution of the ccc for Two SemiInfinite Parallel Plates. Within the framework of the classical DLVO theory,10,11 it is assumed that the total free energy of interaction between two particles, GTOL, consists only of the energy of van der Waals interaction, GvdW, and the energy of electrostatic interaction, GDDL. Other non-DLVO interactions, such as those due to structural or hydration forces, are all neglected. This leads in general to interaction curves with a maximum Gmax and a minimum Gmin if the energy is plotted against the distance. The van der Waals attractive term always dominates over the double-layer repulsive term at both large and small separations. In between, however, the behavior depends critically on the ionic strength, and hence the concentration and the charge number of the counterions, of the dispersion. If the surface is sufficiently charged and if the electrolyte ions do not screen too much, the double-layer force gives rise to a substantial energy barrier, and the colloidal system is typically stable. Only when Gmax is small compared to the unit of thermal energy, kBT, can the system coagulate and become unstable.20,23 Thus, from the DLVO point of view, both the total free energy and its derivative with respect to the separation distance h should vanish when the concentration c reaches the ccc of counterions.20,23 This gives, if we describe the electrical double layers as pure Gouy-Chapman type,24 for the interaction of two semi-infinite parallel plates in a symmetrical electrolyte solution of valance z,
ccc )
Cψ AH2z6
()
tanh4
y∞0 4
(2)
with the constant given by
Cψ ) 9 exp(-4)
217π2(ε0εr)3(RT)5 F6
(3)
where ε0 is the permittivity of the vacuum, εr is the relative permittivity of the dispersion medium, R is the gas constant, T is the absolute temperature, F is the Faraday constant, and at room temperature Cψ ) 8.3e-36 J2 mol m-3. (23) Evans, D. F.; Wennerstrom, H. The Colloidal Domain Where Physics, Chemistry and Biology Meet, 2nd ed.; John Wiley: New York, 1999. (24) Overbeek, J. T. G. Pure Appl. Chem. 1980, 52, 1151.
In eq 2, AH is the Hamaker constant, depending on the properties of the particles and of the dispersion medium,23 and y is a scaled potential for which the subscript ‘∞’ refers to an isolated particle and the superscript ‘0’ refers to the surface of the particle, and it is defined as
y)
zFψ RT
(4)
For the case of homointeraction at constant potential, it is commonly considered that the surface potential ψ0 is a known quantity, and thus eq 2 can be used directly to give the ccc of counterions if ψ0 is independent or weakly dependent on the ionic strength. For the case of homointeraction at constant charge, eq 2 still holds, but now the surface charge density σ0 is the known quantity, and hence the dependence of ψ0 on κ and σ0. In this case, the term of the hyperbolic tangent in eq 2 can be written as25
()
tanh
y∞0 √4κ2 + s2 - 2κ ) 4 s
(5)
with the reciprocal Debye length κ given by
κ)
( ) 2F2cz2 ε0εrRT
1⁄2
(6)
and the scaled surface charge density defined as
s)
zFσ0 ε0εrRT
(7)
Thus, only when s . 2κ may we approximate the hyperbolic tangent term as unity and use eq 2 directly. For not too large σ0 cases, or if we wish to know the explicit dependence of the ccc on σ0, substitution of eq 5 into eq 2 gives, after some mathematical rearrangements with the help of eqs 3 and 6,
κ ) 3 exp(-2)
29πε0εr RT 2 8κ2 - 4κ√4κ2 + s2 + s2 (8) AH Fz s2
( )
The solution to this equation can be written as, by using the cubic formula and eq 6,
ccc )
(σ0)2 5 1 u+ 8ε0εrRT 72u 3
(
)
(9)
with
q q u3 ) + 2 2
1 - ( 725 ) ( q2 ) 3
2
(10)
and
q)
ε0εrRTCψ 2AH2z6(σ0)2
-
7 216
(11)
Equation 9 explicitly gives the analytical solution of the ccc for the case of homointeraction at constant charge, upon which approximate expressions can be derived in several ways. In particular, it has been found that a reasonably good approximation is
ccc ) Cσ
(σ0)4⁄3 (σ0)2 2⁄3 2 40ε0εrRT AH z
(12)
with the constant given by (25) Liu, L.; Neretnieks, I. Colloids Surf. A: Physicochem. Eng. Aspects 2008, 317, 636.
NoVel Approach to Determine the ccc of Counterions
(
Cψ1⁄3 2 Cσ ) 16 ε0εrRT
)
Langmuir, Vol. 25, No. 2, 2009 691
2⁄3
(13)
This expression agrees quite well with the analytical solution of the ccc, i.e., eq 9, with an accuracy always better than 15%. 2.2. The Numerical Solution of the ccc for Two Parallel Plate-like Particles. Now we consider the interaction between two parallel plate-like particles that are very thin compared to their other dimensions. In this case, the energy GvdW of van der Waals interaction per unit area can be well approximated,2,23 if considering only nonretarded forces, by
Ga,vdW ) -
[
AH 1 1 2 + 12π h2 (h + 2δ )2 (h + δ )2 p p
]
(14)
where the subscript “a” denotes per unit area of the particles, and δp is the particle thickness. The energy GDDL of electrostatic interaction falls off, however, as an exponential function of the separation and has a range on the order of the thickness of the electrical double layer.20,23 In particular, for the case where κh is so large that the linear superposition approximation holds, Verwey and Overbeek11 found that GDDL per unit area can be simply written as
Ga,DDL )
()
y∞0 64cRT exp(-κh) tanh2 κ 4
Figure 1. The ccc value as a function of particle thickness for a monovalent electrolyte, according to the classical DLVO theory, given AH ) 2.5 kBT.
(15)
Thus, for the case under consideration, the total interaction energy per unit area becomes
Ga,TOL ) -
[
] ()
AH 1 1 2 + + 12π h2 (h + 2δ )2 (h + δ )2 p p y0 2 ∞
64cRT tanh κ
4
exp(-κh) (16)
This leads to, by applying the conditions that both the total free energy and its derivative with respect to h should vanish when c ) ccc,
[
]
AH 1 2 1 + 12π h 2 (h + δ )2 (h + 2δ )2 ccc ccc p ccc p
()
y0 2 ∞
64cRT tanh κ
4
exp(-κhccc) ) 0
(17)
and
[
]
AH 1 2 1 + 6π h 3 (h + δ )3 (h + 2δ )3 ccc ccc p ccc p
()
y0 2 ∞
64cRT tanh
4
exp(-κhccc) ) 0
(18)
where the subscript “ccc” has been added to h as a reminder that it is the separation h at the ccc of counterions. In this manner, combination of the above two equations with the aid of eq 6 would give κ at the ccc of counterions. This, however, leads to a rather tedious and complicated expression. Consequently, it is preferable to use a numerical method instead to solve these equations simultaneously, by applying eq 9 or 12 to give an initial guess of the ccc for the case of interaction at constant charge. Once κ is known, the ccc can be given immediately with the help of eq 6:
ccc )
ε0εrRT 2F2z2
κ2
(19)
Following this procedure, we calculated the ccc values for both mono- and divalent counterions for montmorillonite particles,
Figure 2. The ccc value as a function of particle thickness for a divalent electrolyte, according to the classical DLVO theory, given AH ) 2.5 kBT.
which are commonly considered to be plate-like with a constant surface charge density varying between 0.069 and 0.137 C m-2. Most montmorillonite particles have, however, a surface charge density12 around 0.1 C m-2. Given AH ) 2.5 kBT, a more or less representative value for montmorillonite particles across water, in Figure 1 the numerical results of the ccc are plotted as a function of particle thickness for a monovalent electrolyte for three surface charge densities. Similar plots are given in Figure 2, but they are for a divalent electrolyte. It is seen that, in both figures, the predicted ccc is approximately proportional to the surface charge density σ0, and it decreases with increasing particle thickness when δp < ∼4 nm but levels off for thicker particles. Since Na-montmorillonite particles consist typically of one or two sheets with a thickness ranging from 0.6 to 2.4 nm,14,26 we may conclude from Figure 1 that the predicted ccc would vary between 0.8 and 1.95 M. For Camontmorillonite particles, which consist typically of seven sheets with a thickness of 13 nm or more,6,14 the predicted ccc would be in the range between 0.13 and 0.26 M. Thus, provided that both types of montmorillonite have the same surface charge densities, the predicted ccc ratio would be about 7. These findings are certainly at odds with experimental results, as tabulated in Tables 1 and 2, which suggest that, for montmorillonite, the ccc values are mostly in the range between 5 and 50 mM in Na+ solutions, whereas they vary between 0.36 and 3 mM in Ca2+ solutions, at pH ∼ 6.5. This discrepancy (26) Cadene, A.; Durand-Vidal, S.; Turq, P.; Brendle, J. J. Colloid Interface Sci. 2005, 285, 719.
692 Langmuir, Vol. 25, No. 2, 2009
Liu et al.
between theory and experiment indicates that the classical DLVO theory, under the assumption of pure Gouy-Chapman double layers, is not applicable to the case of interaction at constant charge. This has led to a significant development of extended DLVO theory, DLVOE.20 In this paper, we shall not go into this area in more detail but emphasize that DLVOE covers a number of phenomena that are quite common in practice and it goes far beyond the Poisson-Boltzmann level, as compared to the classical DLVO theory on which it is based. Essentially, the extended DLVO theory combines different kinds of spatial or surface charge regulation models12,20 with the classical DLVO theory in such a way that the Stern-layer adsorption could be appropriately taken into account.20,24 The consequence of describing the electrical double layers as the Gouy-Stern type is that only a small but very relevant fraction of the counterions resides in the diffuse part of the double layers. As a result, the diffuse potential could be decreased to whatever degree, by allowing ion fluxes from one part of the electrical double layer to the other, in order to fulfill the requirement of sufficiently explaining experimental results. More importantly, these models enable us to subsume the effects of multivalent counterions and ion correlations in the Stern layer,20 to consider the complex formation of counterions in the solution,12 and to explain the pH-dependent colloidal stability of aqueous montmorillonite suspensions.27 However, such an implementation of spatial or surface charge regulations on the Gouy-Stern level becomes increasingly complex and no longer facilitates the development of transparent analytical expressions. Instead we must rely on numerical methods. Furthermore, additional efforts have to be made to determine some key parameters such as the intrinsic ionization constants, the density of adsorption sites, etc.20,27 Thus, for the time being, the significance of the DLVOE theory for practical use remains somewhat academic as it does not lead to formulas that can be easily applied and are valid over a wide range of conditions.20 A question arises, then, as to whether we can develop an approximate solution to the ccc, without invoking the Gouy-Stern double layers but still giving reasonably good estimates.
Such a kinetic interpretation is, however, difficult to give an analytical expression to the ccc of counterions.23 Nevertheless, it encourages us to rethink the concept of the ccc in nonequilibrium situations where other processes come into play. Consider, for example, colloidal expansion in a test tube, it can be intuitively imagined that the ccc also plays a critical role in determining the expansion mode, i.e., only when the concentration of counterions is lower than the ccc could the colloidal particles always reach the top of the test tube. Thus, study of this system would facilitate understanding of the ccc of counterions. Recently, we developed a force balance model that describes the dynamic expansion of a colloidal gel.30 We have exemplified it by applying it to an initially compacted tablet of bentonite in an electrolyte in a test tube of the same diameter. The colloidal particles are assumed to consist of one or more thin sheets with the other dimensions much larger than their thickness. The forces considered include van der Waals force, diffuse double layer force, thermal force giving rise to Brownian motion, gravity, as well as friction force. The model results in an expression resembling the instationary diffusion equation but with an immensely variable diffusivity. This diffusivity is strongly influenced by the concentration of the electrolyte as well as by the particle concentration in the colloidal dispersion. It was found, in particular, that the normalized diffusivity, which we call the thermal deviation factor, is very important in determining the nature of colloidal transport, and it can be used to find an approximate expression for the ccc of counterions. In the following, we shall describe the thermal deviation factor in detail, discussing its significance in determining the basic behavior and the properties of colloidal expansion in a test tube, and then derivation of an approximate expression to the ccc of counterions will be presented. 3.1. The Thermal Deviation Factor. In developing the dynamic force balance model for colloidal expansion,30 the rate of change of the volume fraction φ of the particles with time is obtained from the mass balance of flux, J, in and out of a control volume. It results in the well-known equation of continuity,
3. The Dynamic DLVO View of the ccc
where Fs is a combination of the gravitational force and buoyant force, and DF is the diffusivity function. The derivation of this expression is detailed in our companion paper.30 The first term shows the local rate of change of colloid concentration. The second term shows the influence of gravity. The third term is central to our discussion that follows. It accounts for the diffusion-like movement of the colloidal particles in a concentration gradient. When the system has reached equilibrium, there is no more change in φ anywhere in the system, and the sum of second and third terms is zero. With this dynamic force balance model, we can study the colloid concentration profile at equilibrium when changing the cation concentration and the charge of the cations. The aim is to find the concentration at which the system turns from a sol to a gel, i.e., the ccc of counterions. Before proceeding, we need to say something about the influence of gravity on the upward expansion of a colloidal system. The gravity will balance the dispersion due to diffusion and the other forces in DF. If DF is very small, a pure sedimentation system will result where there is a region with all particles in the bottom of the test tube and a clear liquid above without any particles. In common cases, however, it was found30 that gravity has a marginal influence on the dynamics of the bentonite gel
The classical or the extended DLVO theory predicts the ccc of counterions from a static point of view, because it focuses on studying the equilibrium properties of colloids. Alternatively, we can approach the ccc from a kinetic point of view, because coagulation is a rate phenomenon. During the early stages of the coagulation process, where monomers and dimers dominate, we can distinguish between two regimes of coagulation of colloids; one is the rapid coagulation regime and the other is the slow coagulation regime.20 In the rapid regime, the rate constant of coagulation is typically close to the Smoluchowski rate23,28 for purely diffusional coagulation and the stability ratio W is close to unity (W is experimentally accessible). Upon further tuning of the electrolyte concentration, one usually enters the slow regime, where the rate constant starts to decrease and depends strongly on the electrolyte concentration, resulting in W > 1. Between these two regimes, there is typically a rather sharp transition point, which is the ccc and it corresponds to the intersection of the bilinear branches of the log W-log c plot.20,29 ´ braha´m, I.; Gilde, M.; Sza´nto´, F. Colloids Surf. 1990, 49, (27) Tomba´cz, E.; A 71. (28) Overbeek, J. T. G. J. Colloid Interface Sci. 1977, 58, 408. (29) Guaregua, M. J. A.; Squitieri, E.; Mujica, V. J. Mol. Struct.: THEOCHEM 2006, 769, 165.
∂φ ∂ φ ∂φ ∂ + DF ) Fs ∂t ∂x f ∂x ∂x
()
(
)
(30) Liu, L.; Moreno, L.; Neretnieks, I. Langmuir 2009, 25, 679.
(20)
NoVel Approach to Determine the ccc of Counterions
Langmuir, Vol. 25, No. 2, 2009 693
expansion. Thus, in the following, we disregard gravity and concentrate the analysis on the third term with the diffusivity function DF in eq 20. The diffusivity function DF, accounting for the effects of van der Waals and double-layer forces as well as particle diffusion due to thermal movement, can be written as30
DF )
(
kBT (h + δp)2 ∂FA ∂FR + f f ∂h ∂h
)
(21)
where f is the friction coefficient, FA is the van der Waals force, and FR is the diffuse double layer force. In contrast to the classical DLVO theory we also account for the effects of particle diffusion caused by the thermal movement of the particles, i.e., the second term in eq 21. For very dilute systems, eq 21 reduces to the diffusion coefficient D of the colloidal particles,2
D)
kBT f
(22)
We may then introduce a dimensionless quantity, i.e., the thermal deviation factor ξ, as a normalized diffusivity function,30
ξ)
DF D
(23)
In this manner, ξ not only measures the extent to which colloidal expansion deviates from pure diffusion but also it determines the nature of colloidal transport. As long as ξ is positive, expansion of colloids would always continue until at least some particles have reached the top. If there are no external body forces such as, e.g., gravity or forces exerted by an electrical field, then only the particle concentration will become equal everywhere. When ξ or the particle concentration gradient becomes zero, the expansion can stop part way. If and when ξ should become negative, the colloidal suspension will contract. As we are only discussing steady-state systems in this paper, we need not discuss the friction factor f. This is done in the companion paper30 where the expansion over time and space is explored. To better understand ξ, we may combine eqs 21 and 22 to give
ξ)1+
(
(h + δp)2 ∂FA ∂FR kBT ∂h ∂h
)
(24)
This expression states that ξ consists essentially of two parts: one is h-dependent, resulting from the van der Waals contribution, and the other is h- and c-dependent, resulting from the electrostatic contribution. Because the former contribution is always negative, whereas the latter one is positive, it can be envisaged that the ξ-h or ξ-φ curves would exhibit very different features, depending on the electrolyte concentration. To demonstrate this, in Figure 3 ξ is given as a function of volume fractions for different electrolyte concentrations with z ) 1, on the basis of the simple relationship between h and φ that we have developed previously and on the approximate expressions for FA and FR.25,30 The parameters used for these calculations are tabulated in Table 3, where we set φmax ) 1 for simplicity by noting that the maximum volume fraction is not a fraction that could be attained in reality in compacted bentonite, and it has a marginal influence on the results.30 It is seen in Figure 3 that ξ increases rapidly with increasing volume fraction and with decreasing electrolyte concentration. ξ spans many orders of magnitude for the cases studied. It changes, however, more dramatically for larger c cases than otherwise, and, in particular, the ξ-φ curves smooth out somehow when c becomes very small such as c e 0.01 mM.
Figure 3. The thermal deviation factor ξ (base 10 logarithm) as a function of particle volume fractions for different concentrations of monovalent electrolytes for the case of σ0 ) -0.131 C m-2 and φmax ) 1. Table 3. Input Data Used for Particles and Hamaker Constant property
notation
value
maximum volume fraction surface charge density particle thickness particle surface area Hamaker constant
φmax σ0 δp Sp AH
1.0 -0.131 C m-2 1.0 nm 9.0e-14 m2 2.5 kBT
On the other hand, it is noticed that, over the entire range of φ, ξ remains positive only for small c′s. When c is larger than ∼20 mM, a minimum is clearly seen on the ξ-φ curves. This minimum goes nearly to zero when c is about 50 mM and then it becomes negative for larger c′s. This could be thought of as a negative diffusivity caused by the attractive force domination over the repulsive and thermal forces. To emphasize the important features of the ξ-φ curves around ξ ) 1, a sketch is given in Figure 4, through which we shall discuss in detail the basic behavior and the properties of colloidal expansion in a test tube. 3.2. The General Features of Colloidal Expansion. As a general rule, the ξ-φ curve always has a volume fraction φb at which ξ reaches its minimum ξb ( 45%, as a result of the limited volume of the test tube, and then the φ profile would become a straight line. At the other extreme, the gel layer would vanish if φ0 < 4%, and a diffusion-like profile of φ would be observed at steady state with a volume fraction at the bottom of the test tube somewhat larger than φ0. This could be understood if we think about similar cases of sedimentation, and it results mainly from the fact that the point S in Figure 4 is the only stable point, to the left of which negative diffusion would take place. It should be noted that, if no gravity was present, the volume fraction would be constant in the entire test tube, similar to how a gas would fill a volume. When c is larger than the ccc, the vertical expansion of the colloidal gel cannot reach the top of the test tube for all the cases shown in Figure 6, even if φ0 is as large as 40%. More importantly, it is found that the dilute layer totally disappears for all φ0 values, even if φ0 is as small as 2%, and the volume fraction at the edge of the gel drops immediately from ∼4.5% to zero. This is expected, however, because this volume fraction corresponds to the stable point S in Figure 4, at which ξ has become zero. The point U also has a zero value of ξ, but it is unstable. The reason for this is that, to the left of point S, the van der Waals force would always pull back any particles that are trying to escape from the colloidal gel, whereas to the right of point S the diffuse doublelayer force would always push the particles to disperse further into the electrolyte solution. By comparison, the point U does not have such a property, because around this point the van der Waals force always dominates over the diffuse double-layer and thermal forces. As a result, when c is larger than the ccc, colloidal expansion would always be stabilized at the point S, even if in the beginning φ0 is smaller than the volume fraction of the point S, and the point U would never be reached along the tube. If no gravity was present, the volume fraction would be constant up to a certain height and then zero, similar to sedimentation of large particles. Having discussed the general features of the ξ-φ curves and their importance in determining the steady-state behavior of colloidal expansion in a test tube, analytical analyses can be made to find the approximate expression for the ccc of counterions. 3.3. Approximate Solution to the ccc of Counterions. As we have discussed above, the ccc can be interpreted from a dynamic point of view as the electrolyte concentration below which the expansion of colloids would always lead to full access to the entire volume of the test tube and above which it would stabilize at a certain height delineating a particle free region from one with all the particles. Thus, when c reaches the ccc, the thermal deviation factor ξ and its derivative with respect to φ or h should both vanish at the minimum, i.e.,
ξb ) 0
(25)
∂ξb )0 ∂h
(26)
and
The difference between our dynamic view and the classical DLVO view of the ccc comes, then, into sight. In the DLVO theory, the assumption is that of two particles in an infinite volume, so that only the pairwise interactions operate. In our view, there are multiple particles in a finite volume, where thermal energy leading to Brownian motion is essential for the very existence of stable colloidal dispersions.
NoVel Approach to Determine the ccc of Counterions
Langmuir, Vol. 25, No. 2, 2009 695
Thus we obtain, by using eq 24 to make ξb explicit,
-
∂FA ∂FR kBT + ) ∂h ∂h (hb + δp)2
The van der Waals term on the left-hand side of eq 33 can further be approximated (always better than 2% when h/δp > 5) by
[
(27)
and
∂2FA ∂h2
-
∂2FR ∂h2
This gives, by combination of eqs 33 and 34,
2kBT
)
(28)
(hb + δp)3
hb ) -δp +
where the subscript “b” has been added to h as a reminder that it corresponds to ξb. Now if we still use the Hamaker-De Boer theory to approximate the van der Waals force for the interaction between two parallel plate-like particles,2 we have
FA )
[
AHSp 1 2 1 + 6π h3 (h + δ )3 (h + 2δ )3 p p
]
(29)
where Sp is the surface area of the particles. For the diffuse double-layer force FR, the linear superposition approximation11 remains valid, because at ξb the separation between particles has become so large that κhb . 2. Thus, we can write
()
FR ) 64SpcRT tanh2
y∞0 exp(-κh) 4
(30)
As a result, substitution of eqs 29 and 30 into eq 27 yields,
[
]
AH 1 2 1 + 2π h4 (h + δ )4 (h + 2δ )4 b b p b p
()
64κcRT tanh2
]
2AH 1 2 1 + π h5 (h + δ )5 (h + 2δ )5 b b p b p
()
y0 2 ∞
64κ2cRT tanh
4
exp(-κhb) )
kBT 2 (32) Sp (h + δ )3 b p
Equations 31 and 32 show that the ccc depends not only on z, AH, and δp, but also on the surface area Sp of the particles. This is not surprising, however, because the thermal energy causing Brownian motion is independent of particle size or mass, whereas the van der Waals and double-layer forces are proportional to the particle size. With the help of eq 6, eqs 31 and 32 can be solved numerically to give κ at the ccc, which can then be determined in turn by eq 19. For practical purposes, however, it is preferable to have an approximate solution to the ccc of counterions. To that end, we notice that when c reaches the ccc, the electrical contribution in eq 31 is small in comparison with the van der Waals contribution, and hence we can write approximately
[
]
kBT AH 1 R 2 1 + ) (33) 4 4 4 2 2π h S (hb + δp) (hb + 2δp) p (hb + δp) b where R is a correction factor expected to be somewhat larger than 1, introduced to compensate for the loss of neglecting the electrical contribution, and for simplicity we may set R ) 1.
(
10Spδp2 AH πR kBT
)
1⁄4
(35)
For square particles with sides between 50 and 300 δp, hb/δp ranges from 11 to 28 for R ) 1 and AH ) 2.5 kBT. This is equivalent to φ between 0.033 and 0.084, and hence it agrees well with Figure 3. As a result, we expect that the gel phase at a c just larger than the ccc will have a particle volume fraction in this range. This agrees with Figure 6, where, however, gravity is also accounted for. With eq 35 at hand, we can now work on eq 32 to find the ccc of counterions. First, it is noted that the van der Waals contribution in eq 32 can be approximated fairly well (always better than 2% when h/δp > 5) by
[
]
60AHδp2 2AH 1 2 1 + ≈ (36) π h5 (h + δ )5 (h + 2δ )5 π(h + δ )7 b b p b p b p Second, for the cases that we are mostly interested in, i.e., for the interactions of highly charged montmorillonite particles, the term of hyperbolic tangent in eq 32 goes nearly to unity so that we can write approximately, by combination with the above equation,
kBT y∞0 1 exp(-κhb) ) (31) 4 Sp (h + δ )2 b p
Where and henceforth c should be understood as ccc, and correspondingly κ should be related to ccc.Likewise, eq 28 leads to
[
]
10AHδp2 AH 1 2 1 + ≈ (34) 2π h4 (h + δ )4 (h + 2δ )4 π(h + δ )6 b b p b p b p
64κ2cRT exp(-κhb) )
60AHδp2 π(hb + δp)
7
-
kBT 2 (37) Sp (h + δ )3 b p
Thus, substitution of eq 19 for c gives
1 Fz κ exp(-κhb) ) 16ε0εr RT
[
2
( )
4
]
30AHδp2
kBT π(hb + δp)7 Sp(hb + δp)3 (38)
This can readily be solved numerically for κ and hence the ccc of counterions. Alternatively, an analytical solution can also be obtained. This is done by introducing two dimensionless variables,
1 W ) - κhb 4 and
X)-
{
hb 1 Fz 8 ε0εr RT
[
2
( )
30AHδp2
kBT 7 π(hb + δp) Sp(hb + δp)3
(39)
]}
1⁄4
(40)
With these two entities, eq 38 can be rewritten as
W exp(W) ) X
(41)
This is the Lambert W function,31 the value of which in different branches has been well implemented in both Mathematica and MATLAB. For the present case, X is negative, and therefore the solution is the value of the Lambert W function in the branch satisfying W(X) e -1, commonly denoted as W-1(X), for which an asymptotic series expansion is available.31 Thus, from eq 39, we have
696 Langmuir, Vol. 25, No. 2, 2009
κ)-
Liu et al.
4 W (X) hb -1
This gives, with the help of eq 19,
[
ccc ) 8ε0εrRT
W-1(X) hbFz
(42)
]
2
(43)
This equation, in combination with eq 35 for hb and eq 40 for X, shows that the ccc depends not only on z, AH, and δp, but also on Sp, as already suggested by eqs 31 and 32. Note, however, that this approximate expression of the ccc applies only to highly charged colloidal particles with |σ0| g 0.05 C m-2. In these cases, the ccc is insensitive to σ0. 3.4. Comparison with Experimental Results. Given R ) 1.0, AH ) 2.5 kBT, and σ0 ) -0.131 C m-2, the numerical results and the approximate solutions of the ccc are plotted in Figure 7 as a function of surface area of the particles for two particle thicknesses for a monovalent electrolyte. It is seen that the approximate solutions agree excellently with the numerical results over the entire range of Sp that Namontmorillonite particles typically span, i.e., from 1.1 × 103 to 1.0 × 105 nm2, for both the average particle thickness of 1.2 nm and the maximum particle thickness of 2.4 nm.14,26 Although the ccc values shown in Figure 7 are only for the case of σ0 ) -0.131 C m-2, they do not differ much if σ0 varies in the typical range that montmorillonite particles spread,12 i.e., from 0.069 to 0.137 C m-2. The predicted ccc of counterions decreases with increasing surface area but also the thickness of the particles. More importantly, it is found that, for both δp cases, the predicted ccc values fall well into the range between 5 and 250 mM, indicating that they are in good agreement with experimental results tabulated in Table 1. Thus, it is the particle thickness δp and the surface area Sp that should take main responsibilities for the fact that the ccc of Na+ solutions often spans a wide range. On the other hand, although montmorillonite particles typically have a broad size distribution, they mostly have a surface area14 close to 1.0 × 105 nm2. Hence, if these large particles dominate the ccc of counterions, we may well understand from Figure 7 that the ccc should often be in the range between 20 and 50 mM.12 By comparison, Ca-montmorillonite particles are much thicker than Na-montmorillonite particles. They may consist of 5-15 parallel sheets6 but, mostly, they have seven sheets or so, separated by about 1 nm.14 Thus, the thickness of Ca-montmorillonite particles is about 15 nm, on average. For this reason, in Figure 8 the numerical results and the approximate solutions of the ccc
Figure 8. The ccc value as a function of the surface area of the particles for a divalent electrolyte, given R ) 0.8, where the numerical results are represented by lines, whereas the approximate results are shown by solid markers.
are plotted as a function of surface area of the particles for δp ) 15 and 30 nm, respectively, for a divalent electrolyte, given R ) 0.8. Although the agreement between the approximate solutions and the numerical results in Figure 8 is not as good as in Figure 7 for a monovalent electrolyte, especially in the region of small surface areas of the particles in the case of δp ) 30 nm, they are still acceptable. In practice, it has been found that R ) 1.0 works much better than R ) 0.8 in the case where the surface areas of the particles are larger than 1.0 × 104 nm2 for all reasonable values of δp for a divalent electrolyte. A matter of great interest in Figure 8 is, however, that the predicted ccc shows the same dependences on δp and Sp as in Figure 7, and, for both δp cases, it is in the range between 0.4 and 5 mM over the entire range of Sp from 1.1 × 103 to 1.0 × 105 nm2. These results agree quite well with experimental observations of the ccc of Ca2+ solutions tabulated in Table 2. In addition, with Figure 8 at hand, it may now become understandable why the ccc of Ca2+ solutions often varies between 0.4 and 2 mM, but mostly is about 1 mM, if accounting for the fact14 that montmorillonite particles mostly have a surface area close to 1.0 × 105 nm2. Thus, the detailed analyses of Figures 7 and 8 suggest that the approximate expression, as given in eq 43, gives reasonably good estimates to the ccc of counterions for the case of homointeraction at constant charge. Moreover, it seems that mainly the thick particles with large surface areas dominate the ccc of counterions for montmorillonite dispersions with a broad size distribution, whereas the surface charge density plays a less important role as long as it is larger than 0.05 C m-2.
4. Discussion and Conclusions
Figure 7. The ccc value as a function of the surface area of the particles for a monovalent electrolyte, given R ) 1.0, where the numerical results are represented by lines, whereas the approximate results are shown by solid markers.
In developing a force balance model for colloidal expansion,30 it was found that the thermal deviation factor ξ, as given in eq 24, plays a critical role in determining the behavior and the properties of colloidal expansion. ξ is a normalized diffusivity function, and it measures the collective effect on the driving force of colloidal expansion by thermal motion and by interaction between particles in a particle concentration gradient, including the diffuse double layer force and the van der Waals force. As long as ξ is positive, expansion of colloids would always continue until the entire volume of the test tube has been reached. This is analogous to ordinary diffusion caused by the random movement due to the thermal energy of the particles, tending to spread out the particles evenly in a volume if external body forces can be neglected. In this case, however, we also account
NoVel Approach to Determine the ccc of Counterions
for van der Waals and double layer forces. In high ionic strength solutions ξ may become zero or negative, which would be analogues to a zero or negative diffusivity. Only in cases where ξ becomes zero can the expansion stop part way, resulting in the formation of a sharp front with all particles on one side of the front and no particles on the other side. We do not conceive this situation as the presence of two phases in equilibrium, although it could be likened to a case where atoms in a liquid cannot be vaporized and enter the gas phase because the thermal motion of the atoms is not sufficiently large to overcome the attractive forces between the atoms. The study of this closed system indicates that the ccc can be interpreted from a dynamic point of view as the electrolyte concentration below which expansion of colloids would always lead to full access to the entire volume of the test tube and above which it cannot. Hence, when c reaches the ccc, the thermal deviation factor ξ and its derivative with respect to h should both vanish at the minimum of the ξ-φ curves. On the basis of this analysis, an approximate expression of the ccc was derived in this paper for the case of homointeraction at constant charge for montmorillonite dispersions. This expression, as given in eq 43, is somewhat complex in form and is very different from the one obtained with the classical DLVO theory. However, its estimates agree quite well with experimental observations of the ccc of counterions for montmorillonite at pH ∼ 6.5, as tabulated in Tables 1 and 2. In particular, it is found that the predicted ccc of counterions decreases with increasing the surface area but also the thickness of the particles, and, as long as σ0 is larger than about 0.05 C m-2, it is nearly independent of the surface charge density. In addition, the predictions suggest that the ccc of Na+ solutions would be in the range between 20 and 50 mM if most of the montmorillonite particles have surface area larger than 1.0 × 104 nm2 with a thickness of about 2.4 nm, whereas the ccc of Ca2+ solutions would vary between 0.4 and 2 mM if the montmorillonite particles are about 15 nm thick or more. Hence, it is the particle thickness δp and the surface area Sp that should take main responsibilities for the fact that the ccc of counterions often spans a wide range for montmorillonite dispersions. The classical DLVO theory tends, however, to overestimate the ccc by at least 2 orders of magnitude, and therefore it is not applicable to the case of interaction between plate-like particles
Langmuir, Vol. 25, No. 2, 2009 697
at constant charge. By comparison, it is actually found that the separation hb between particles at the estimated ccc, as given in eq 35, is very close to the secondary minimum of the interaction curves of the classical DLVO theory, rather than to the primary maximum. In addition, it should be stressed that the dynamic force balance model30 we have developed is based on the notion that the smectite sheets are arranged parallel in a columnar pattern.32 This seems to be a reasonable assumption when the volume fraction of the particles is so large that the particles, if rigid, cannot rotate freely without hitting their neighbors.30 The model is, however, not valid for low φ′s when c > ccc. This can be understood from Figure 4 when φ lies to the left of the point U. Then ξ > 0, and Brownian motion dominates, seemingly causing the particles to disperse. This view, however, neglects the sticking together of particles when they randomly collide and come near enough for the van der Waals force to let them “stick”. This will happen when c > ccc because then the van der Waals force dominates over the double-layer force. A double sheet particle will form. This in turn will stick to other particles, and the particles will grow. In a finite volume, finally a sole large particle consisting of all the individual sheets would form. If gravity is also acting, the agglomeration process will speed up because the larger the particles have grown, the faster they will sediment and collect near the bottom of the vessel. These mechanisms are not contained in our present model, and the model thus is not valid in the region just described. It may be noted that we discuss the ccc as if it were not influenced by external body forces, caused, for example, by gravity, which will tend to decrease the ccc somewhat. The effects of external body forces are not discussed further in the present paper, however. Acknowledgment. The authors gratefully acknowledge the encouragement and financial support of the Swedish Nuclear Fuel and Waste Management Company (SKB). LA802658G (31) Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; Knuth, D. E. AdV. Comput. Math. 1996, 5, 329. (32) van der Kooij, F. M.; Kassapidou, K.; Lekkerkerker, H. N. W. Nature 2000, 406, 868.
