Soil Science Society of America Journal 66:1207-1217 (2002)
© 2002 Soil Science Society of America
DIVISION S-2—"PARTICLE INTERACTIONS IN COLLOIDAL SYSTEMS"
Diffuse Double-Layer Models, Long-Range Forces, and Ordering in Clay Colloids
M. B. McBride* and
P. Baveye
Dept. of Crop and Soil Sciences, Cornell University, Ithaca, NY 14853
* Corresponding author (mbm7{at}cornell.edu)
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ABSTRACT
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The modern view of colloidal particle behavior in water is based on the DLVO (Derjaguin-Landau-Verwey-Overbeek) theory, which hypothesizes strictly repulsive long-range interactions among charged particles. However, numerous phenomena of colloidal behavior, some recently discovered, cannot be explained quantitatively or qualitatively by this theory. An alternative description of the fundamental forces involved in the formation of dispersions and gels was outlined in 1938 by Langmuir, and further developed theoretically by Sogami, Ise, and Smalley. This model, unlike DLVO, hypothesizes a long-range Coulombic attractive force countering osmotic repulsion, which may explain experimentally confirmed multiparticle phenomena such as long-range attraction and transitions among ordered (liquid crystalline) and disordered phases of colloidal suspensions. The key concepts of Sogami–Ise theory and its experimental support are reviewed in the present article. This review begins with an analysis of the extent to which dynamic (Brownian) motion and interparticle steric hindrance influence the macroscopic properties of clay dispersions and gels. It is shown that the covolume theory of Onsager in some cases could explain the existence of apparent long-range interparticle interactions, and phase transitions between ordered and disordered states in colloidal dispersions, without invoking long-range forces. Nevertheless, in situations where the composition of the electrolyte solution influences the observed interparticle interactions, the experimental evidence for long-range attraction among charged colloidal spheres points to a force that is fundamentally electrostatic in nature, of the type envisioned by Langmuir. After consideration of all experimental and theoretical results to date, it is concluded that the DLVO theory adequately describes the repulsive interaction between isolated like-charged particles, whereas a long-range attractive force is needed to explain multiparticle interactions in suspensions under conditions of low electrolyte concentration and high particle charge.
Abbreviations: CAT, Coulombic attraction theory • DLVO, Derjaguin-Landau-Verwey-Overbeek theory • PB, Poisson-Boltzman
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INTRODUCTION
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EVIDENCE FOR LONG-RANGE interaction among colloidal clay particles in suspension has been based on the observation of several intriguing physical phenomena. These include the fact that fully dispersed Na+-smectites possess viscosities higher than that of water, even when the clays comprise only a fraction of 1% (by weight) of the suspension (van Olphen, 1977). Suspensions of 1 to 2% Na+-smectite form thixotropic gels on standing. Furthermore, as certain suspensions of rod or plate-shaped particles (i.e., particles with high aspect ratios) are allowed to settle for long periods of time, the particles become concentrated in the bottom of the container, and interference colors are seen. This phenomenon is attributed to the separation from suspension into two phases: a dilute isotropic (disordered) phase and a concentrated (ordered) one that is often birefringent (Langmuir, 1938; Marshall, 1949; Fitzsimmons et al., 1970). In the case of Fe hydroxide suspensions, the interference colors have been used to calculate a separation distance between layers of particles, termed Schiller layers, in the range of 200 to 400 nm (Overbeek, 1952). The remarkably uniform spacing between particles or assemblies of particles, as indicated by light interference effects such as iridescence (Maeda and Hachisu, 1983), suggests that a balance has been achieved between sedimentation pressure and electrical repulsion. This is the explanation, based on DLVO theory, that is generally given for Schiller layer formation in ß-FeOOH sols (Maeda and Hachisu, 1983). Supporting this explanation is the fact that increasing the electrolyte concentration decreases the spacing, while diluting the electrolyte with water dissolves the ordered phase, forming an isotropic suspension. It is perhaps instructive to note, however, that ß-FeOOH sols quickly sediment and spontaneously form Schiller layers only at extremely low pH (<2), whereas at higher pH they form only the isotropic phase unless the particles are physically forced together by centrifugation. This appears to contradict DLVO theory, because surface charge on FeOOH at low pH is positive, and will increase in magnitude if the pH is further lowered. Thus, enhanced Schiller layer formation at lower pH, which in itself implies stronger interparticle attraction, seems to indicate that the attractive force in these systems is electrostatic. Certainly there is no reason to believe that the van der Waals forces, which are the attractive forces assumed in DLVO theory, would increase at lower pH. As these ß-FeOOH systems are diluted and the pH is raised, the particle-particle separation increases and disorder sets in, indicating that repulsive (osmotic) forces predominate once the magnitude of surface charge is diminished. The coexistence (at equilibrium) of a condensed ordered phase with a disordered phase, observed in numerous monodisperse colloidal systems in water, is generally viewed as evidence for a long-range attractive force, in contradiction to DLVO.
Studies in which Na+-smectites of different charge density are compared (Slade et al., 1991) or where the magnitude of negative charge is modified (Stucki and Tessier, 1991), provide evidence that interlayer spacing and charge density have a tendency to be negatively correlated, in contradiction with the DLVO theory. Again, it seems that there must be an important electrostatic component in the attractive interparticle force, at least for charged clay plates separated by less than a few nanometers. This is not a new concept, as Norrish (1954) contended that for separations between smectite plate surfaces of 2 nm or less, the electrostatic attractive potential was much greater than that attributable to van der Waals forces. Conversely, the swelling pressure of clay gels (at interlayer separations >2 nm) has long been known to be too large to be balanced by van der Waals forces (Norrish and Rausell-Colom, 1963).
On the basis of observations of Schiller layers and other accumulated inferential evidence for long-range forces, scientific consensus was reached among most colloid scientists in the mid-20th century that the London–van der Waals attraction was the only possible attractive force of sufficient generality to account for the observed phenomena in a wide variety of colloidal materials (Verwey and Overbeek, 1946; Overbeek, 1952). Prior to this consensus, however, Langmuir (1938) had argued for very different perspectives on long-range interactions of colloids. First, he suggested that the high aspect ratio (anisodimensional character) of colloidal particles can in itself explain some of the observed long-range interactions. That is, phenomena related to thixotropy, phase separation, and mutual orientation of particles can be explained physically without necessarily invoking long-range attractive forces. This viewpoint was further developed with remarkable success by Onsager (1949) who showed that long-range interactions could arise from the thermal motion of anisodimensional colloidal particles and the consequent occupation of solution volume far in excess of the particle volume. In situations where electrolyte concentrations influenced colloidal behavior, and therefore where some additional process needed to be invoked, Langmuir (1938) further argued that van der Waals attraction could not possibly operate in aqueous media over the distances necessary to explain the observed long-range interactions between colloids. He believed that the attractive force among charged colloidal particles was fundamentally electrostatic, arising from the arrangement of counterions between the particles. Langmuir's (1938) model of attractive and repulsive forces in colloids was summarily rejected soon after it was proposed (Levine, 1946; Verwey and Overbeek, 1946). Later, however, Adamson (1976) reevaluated the case for long-range forces and concluded that, unless one believes that "weak evidence if piled high enough becomes strong evidence", there is little indication that the observed long-range interactions arise from the action of long-range forces as hypothesized by the DLVO theory. Except for the Coulombic (electrostatic) force, Adamson argued that the known fundamental types of forces would not be important at distances greater than a few molecular diameters. With evidence mounting recently that London–van der Waals types of interactions do not provide a viable explanation for observed attractive forces among like-charged colloidal particles (Grier, 1998), and with the theoretical developments of Sogami, Ise, and Smalley in the last two decades, the old arguments of Langmuir (1938) have regained considerable credibility.
The foregoing leads naturally to the conclusion that a detailed reanalysis of the fundamental nature of attractive forces in suspensions of soil colloids is needed. Such a revisitation is the main purpose of the present article. Initially, we explore to what extent the available experimental evidence of thermal motion of anisodimensional soil colloids in suspension supports an Onsager-like explanation for their long-range interactions. This provides a suitable background when, in subsequent sections, we turn to situations where other processes than just thermal motion are at play. After a short review of the Coulombic attraction theory (CAT) developed by Sogami and Ise (1984) and Smalley (1994), we assess in what measure the principles embodied in this theory are compatible with experimental observations.
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Theoretical Background on Particle Motion in Water
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To appreciate the nature of dynamic motion of suspended clay particles, it is useful to review the process which generates this motion. Brownian movement arises from collisions of water molecules with suspended particles. If numerous successive unbalanced collisions occur, the particles will move rapidly. The process if repeated produces random thermal translation and rotation of clay particles. The smaller the particles, the smaller is the number of collisions with water that occur in a given time interval. The probability that the collisions are not balanced on all sides is greater for small than for large particles. Hence, small particles demonstrate more violent and rapid Brownian motion than larger particles. This inverse relation between Brownian motion and particle size is in agreement with the Waterston–Maxwell equipartition theorem of the kinetic theory of fluids (see e.g., Brush, 1968) according to which the mean square translational velocity, 
2c, of suspended particles is determined by the equation:
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where mc represents the mass of an individual particle (kg), kB = 1.38066 x 10-23 J K-1 is the Boltzmann constant and T (K) is the absolute temperature.
Equation [1] predicts that a 1 nm-thick layer silicate particle with a diameter of 0.3 µm, containing about 65 unit cells and weighing 
6.5 x 10-23 kg, would have a root mean square translational velocity equal to
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Even though instantaneous velocities of this magnitude may be achieved in clay suspensions, they are however not accessible to direct measurements, as pointed out by Einstein (1956). The only measurable velocity of particles in a suspension is that, much smaller than 2c, calculated on the basis of the mean square displacement of a particle, (
x)2, given by Einstein (1956) as
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where t is the time that has elapsed since the particle moved from its initial location and D is the translational diffusion coefficient. Only if the time interval during which the particle is observed is made exceedingly short would the observed velocity approach the velocity expected from the kinetic theory of fluids. However, even after extremely short time intervals, the original trajectory and velocity of the particle will have been altered many times and in a random manner (Einstein, 1956).
For spherical particles, the translational diffusion coefficient, D, in Eq. [3] is given by the Stokes–Einstein equation
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where 
is the viscosity of the solvent and r is the radius of the suspended particles. This equation results from Stokes' equation for the hydrodynamic drag, which greatly dampens particle velocity in liquids, balanced against the driving force for motion, quantified by the van't Hoff osmotic equation.
Applying the Stokes–Einstein equation to a spherical particle with a 1-µm diameter suspended in water at room temperature, the value of D is found to be D = 4.40 x 10-12 m2 s-1, which produces a root-mean-square displacement in 1 s of (2Dt)-1/2 = 0.94 µm. This means that a 1 µm-diam. particle should be translated about the distance of its own diameter along any particular direction in 1 s. However, after a minute, it will only have moved 7 µm along this direction (on average) because of its random motion.
The rotational diffusion of particles can be estimated from an equation based on Stokes' calculation that a sphere of radius r, rotating in a liquid of viscosity, , has a fractional torque of 8
r3 times the angular velocity of the sphere (Debye, 1929). Consequently, the rotational diffusion coefficient, DR, of spherical particles of 1-µm diameter in water at room temperature is given by
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Since the rotational relaxation time, 
R, is given by R = (6DR)-1, then R in this case equals 0.13 s. For disc-shaped particles, the translational and rotational motions interfere with each other. However, for time intervals long compared with the rotational relaxation time, R, the two motions are mutually independent as a first approximation (Perrin, 1934), and the rotational diffusion coefficient is given by (Hunter, 1989):
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so that DR for a disc would be about 2.3 times greater than for a sphere of the same maximum dimension. Thus, for approximately disc-shaped layer silicate clay particles of 1-µm diameter, DR is 3.0 s-1, and R = 0.06 s.
Since Brownian rotational motion is by nature erratic, these values of DR and R cannot be unambiguously related to a number of revolution per seconds. However, if clay particles could be constrained to rotate in only one direction along a given axis, this number of revolutions would be equal to 1/R = 17 s-1. By comparison, somewhat smaller clay particles with a diameter of 0.3 µm would rotate several hundred times per second if unhindered by particle-particle interactions in dilute suspensions.
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Experimental Measurements of Clay Particle Motion in Suspensions
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It is now informative to compare these calculated diffusion coefficients with those measured experimentally. Clay platelets, in suspension, when subjected to an electrical field, orient relative to the field because of their permanent and induced electrical dipoles (Kahn and Lewis, 1954; Shah et al., 1963a,b). Optical birefringence develops in the suspension once the randomly oriented particles begin to align relative to the applied field. By measuring decay of this birefringence as a function of time after the applied field is turned off, it is possible to measure the rotational diffusion coefficient (DR) of the clay particles, according to the following equation (Benoit, 1951):
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where n0 is the initial birefringence and n is the birefringence at time t. When this was done with a very dilute Na+-montmorillonite suspension with an average particle diameter of 0.3 µm, estimates of DR ranged from 30 s-1 to over 100 s-1 depending on the electrolyte concentration in the suspension (Schepers and Miller, 1974; Schepers et al., 1976). From Eq. [6], this particle size would be expected to have a rotational diffusion coefficient of DR = 113 s-1. Only the clay suspensions with lowest electrolyte concentration produced measured DR values approaching this value, suggesting that particle-particle interactions were induced and effective rotational diffusion coefficients decreased at higher electrolyte concentrations. Interparticle associations even in dilute Na+-smectite suspensions were later detected by different methods (Lubetkin et al., 1984; Cebula et al., 1980).
Studies on different size fractions of Na+-montmorillonite confirmed that the measured DR increased with decreasing particle radius (Schepers and Miller, 1974; Schepers et al., 1976), essentially in accord with Perrin's equation (Eq. [6]). We conclude, therefore, that dilute, low-electrolyte, highly dispersed suspensions of smectite particles are characterized by rotational motions that essentially obey classical hydrodynamic equations.
Much earlier studies of the decay of birefringence by Langmuir (1938) also produced estimates of rotational diffusion times of smectite particles in water. In these experiments, birefringence was induced by flow processes in the smectites which created shear forces and the resultant temporary ordering of suspended particles. The time, , necessary for the disappearance of birefringence was then measured. The relationship of to smectite concentration in water is plotted in Fig. 1
, showing a high sensitivity of the rate of particle disordering to the concentration of particles. Langmuir (1938) further noticed that became shorter at higher temperature, reducing by a factor of 5 from 0 to 40°C. This indicated an energy barrier to randomization of particle orientation of 9.6 kJ (2.3 kilocalories). (The temperature dependence of viscosity of water would account for only about half of this effect.)
At the higher concentrations of smectite suspensions studied (1.2–2.0%), Langmuir (1938) noted that an ordered phase (permanently birefringent) separated out from the isotropic (nonbirefringent) phase. It appears, then, from these early studies, that particle-particle interactions are appreciable at smectite concentrations below 1%, and phase separation may be a consequence of this interaction once the concentration exceeds 1%. Shah et al. (1963a,b) noted anomalous behavior in birefringence measurements of dispersed smectites, suggesting long-range particle-particle interactions. Since they had fractionated the clay into sizes and had obtained good estimates of the platelet diameters using electron microscopy, the average interparticle separations could be calculated from their data. These data will be used below to support Onsager's covolume theory.
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Onsager's Covolume Theory and Phase Transitions
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Now that a picture has emerged from experimental results that clay particle rotation (tumbling) occurs on a rapid time scale, the covolume theory attributable to Onsager (1949) can be shown to explain the apparently anomalous behavior of highly anisometric colloidal particles. Covolume (excluded volume) can be defined as the "volume around a given particle ... from which the center of a second one is excluded" (Adamson, 1976). It is easily shown, that for spherical particles, the covolume is four times the particle volume (Fig. 2)
. When anisometric particles are considered, Onsager (1949) realized that the volume swept out by the freely rotating particle would be greatly in excess of the particle volume itself. In fact, for plate-shaped particles, the covolume, b, divided by the particle volume, Vp, is approximated by (Onsager, 1949):
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where d is the plate diameter and T is the plate thickness. That is, the ratio of covolume/particle volume is a large number, of magnitude roughly equal to the ratio of the long/short dimension of the particle.
Shah et al. (1963a,b) noted apparently large particle-particle interaction effects with 0.6% suspensions of smectites having d 
300 nm. Since these suspensions were highly dispersed, most particles could be assumed to consist of single platelets with d/T = 300 nm/1 nm = 300. That is, the ratio of covolume/particle volume is predicted to be close to 300. The clay particles themselves (not including diffuse double layer volume) would occupy about 0.23% of the suspension volume, so that their covolume would be 300 x 0.23% = 69% of the total suspension volume. Clearly, most of the volume of the suspension is occupied by covolume space, leaving little room for unhindered rotational thermal motion of the individual particles. This being the case, it is not surprising that Shah et al. (1963a,b) saw evidence of particle interaction effects in the birefringence behavior of a 0.6% suspension.
These interactions are predicted because of the rapid rotational and translational motion of the particles (described in the previous section), and no long-range forces need be invoked to explain this long-range interaction. Particle alignment necessarily ensues once covolume overlap occurs because this orientation lowers the free energy of the system.
Onsager (1949) went further to explain the tendency of anisometric particles in suspension to form coexisting isotropic and anisotropic phases. A qualitative explanation of Onsager's equations, adapted from Adamson (1976), is instructive here. Using the analogy to the kinetic theory of gases, we find that the van der Waals term, B, in the ideal gas equation
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is in fact a correction term to account for finite molecular volume. Suppose that the gas molecules are anisodimensional; then as the gas is compressed, molecular interaction decreases B because the molecules are no longer free to sweep out a spherical volume of diameter equal to the long dimension of the molecule. A precipitous decrease in the effective volume of B is then expected at some threshold pressure. This means that (V - B) may actually increase as V decreases, the pressure could diminish and a nonmonotonic function of the type shown in Fig. 3
could result. In reality, the pressure of the system would not go through a maximum and then a minimum with increasing compression; instead a phase transition would occur at pressure P1.
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Fig. 3. The pressure-volume isotherm for a gas composed of anisometric molecules, with P1 representing the pressure at which a phase transition is predicted (adapted from Adamson, 1976).
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In an analogous manner to nonsymmetrical gas molecules, individual clay particles are freely rotating until the clay concentration is increased, and a point will be reached where the covolumes begin to overlap significantly. The particles will be forced into alignment, and rotation about the platelet's c-axis will become more probable than rotation about the a or b axis. The result will be a reduction in rotational degrees of freedom of the particles (lower entropy), and a dramatic decrease in the effective covolume to a value approaching the volume of the particle itself. As a consequence, the osmotic pressure of the clay might actually be expected to diminish as clay concentration reaches the threshold value, then increase again. Instead, however, a phase transition would occur, associated with the abrupt entropy change, and a dilute isotropic phase could be in equilibrium with a more concentrated anisotropic phase.
Forsyth et al. (1978) used the Onsager equations for anisometric particles to calculate the osmotic pressure of disc-shaped particles as a function of their concentration in suspension. This relation is shown in Fig. 4
, revealing the discontinuity in osmotic pressure (plateau) identified with the phase change. The initial stage represents the dilute isotropic clay phase, whereas both the isotropic and ordered phases coexist at the plateau. At even higher concentrations, only the ordered (anisotropic) phase should exist. None of these phenomena requires an attractive long-range force to be involved. While it could be argued that the electrical double layer of clay particles provides a repulsive force that increases the effective particle thickness (t) in the Onsager model, the qualitative relationship between osmotic pressure and particle concentration remains the same without the introduction of double layer forces (Forsyth et al., 1978). The explanation of long-range interactions proposed in this section is expected to be most relevant for dilute clay suspensions, in which attractive forces are weak. However, the importance of the thermal motion of particles relative to the forces invoked by DLVO theory to explain the stability and behavior of colloidal suspensions has not been assessed.
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Fig. 4. Calculated osmotic pressure of idealized thin cylindrical clay plates as a function of their concentration. The plates are assumed to have negligible thickness relative to their diameter of 300 nm (adapted from Forsyth et al., 1978).
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Self-assembly of Anisodimensional Particles by Covolume Effects
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As discussed in the section above, Onsager's 1949 paper on covolume effects explained for the first time that reversible phase transitions from disordered to ordered states in colloidal suspensions, depending upon particle shape, could be driven by entropy alone (Frenkel, 2000). This would at first seem to be counterintuitive because the disordered colloidal phase (isotropic) appears to have higher disorder (greater entropy) than the ordered (nematic) phase, and that ordering could only be driven by a lowering of the energy (enthalpy). However, as the particles align, their excluded volumes decrease, and they are then free to explore a larger fraction of the volume without interference (covolume overlap) from other particles—consequently, translational entropy actually increases with alignment into the more dense ordered phases.
It is now known that entropy-driven self assembly of anisodimensional particles can produce liquid crystals with varying degrees of order. Thus, plate-like particles can form ordered phases with a preferred orientation but no long-range positional order (nematic), one-dimensional periodic arrays of layers (smectic), or two-dimensional lattices of columns (columnar) (van der Kooij et al., 2000; Brown et al., 1998). These liquid crystalline structures in suspension are shown schematically in Fig. 5
. To define the conditions favorable to these various phases, a phase diagram can be constructed for particular colloidal suspensions, as is shown in Fig. 6
for a smectite, based upon the experimental results of Gabriel et al. (1996). This diagram illustrates the fact that the clay particle concentration as well as the electrolyte concentration determine particle ordering. The role of clay concentration is easily understood in light of the foregoing discussion in which covolume overlap was shown to lead to ordering and phase transition. However, the effect of electrolyte concentration on phase transitions (Fig. 6), favoring the isotropic phase at low concentration, and the nematic phase or flocculated state at high concentration, implies that the electrostatic interaction between the negatively charged clay particles must also influence particle-particle interaction forces. It is this phenomenon, the effect of electrolyte concentration (and valence) on clay interparticle ordering and dispersion, that has provided much of the evidence for the existence of diffuse double layers and long-range attractive or repulsive forces among colloidal particles in suspension. A brief review of the present state of knowledge about these forces is given in the next section.
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Fig. 5. Diagrammatic view of the disordered (isotropic) and several ordered (liquid crystalline) structures in suspensions of plate-like particles.
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Theoretical Limitations of DLVO and Evidence for Long-range Non-DLVO Forces
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The DLVO model of charged colloidal interaction has been generally accepted since the 1940's (McBride, 1997), having had success in predicting the interlayer spacings of free-swelling clays such as Na and Li-montmorillonite in simple electrolyte solutions (Quirk and Marcelja, 1997; Quirk, 1994). The model assumes that the coulombic interaction between like-charged particles in water or electrolyte solutions is pure repulsion, a result of the imagined overlap of diffuse double layers associated with neighboring particles. The extent of this overlap, and the magnitude of the resulting repulsion force, is calculated by the Poisson-Boltzmann (PB) approximation. Under the assumptions of the DLVO model, attraction is caused by van der Waals forces, meaning that clay flocculation is only possible when these (short-range) van der Waals forces overcome electrostatic repulsion as particles approach closely.
It is now generally accepted that van der Waals attraction forces can only exceed thermal energies for colloidal particles in solution that are almost in contact, separated by less than a few nanometers (Grier, 1998; Chu and Wasan, 1996). At separations of charged particles less than 3 nm, repulsive hydration forces dominate (Quirk, 1994), however, so that DLVO and other continuum models cannot predict behavior of colloids in aggregated states. Furthermore, the DLVO model cannot correctly explain phenomena arising from long-range interparticle attraction because the van der Waals forces are much too weak over the separation distances of relevance.
A further problem with the DLVO model is that the PB mean-field approximation is a serious oversimplification of the electrical forces between charged surfaces or particles. Ion-ion correlations, attributable to the ordering effects of an ion on other ions in its environment, are not accounted for in DLVO. In practice, as two highly charged particles approach, the counterion clouds around these particles reposition in response to the altered local electrical field. When such correlated fluctuations in the counterion clouds of two surfaces or particles are accounted for, an attractive rather than repulsive interaction is found, particularly when the counterions are divalent or polyvalent (Oosawa, 1968; Guldbrand et al.,1984). Ion-ion correlations have been shown to provide an explanation for the attractive force that prevents Ca2+ -montmorillonite clay platelets from separating in water beyond 1 nm (Kjellander et al., 1988). However, the predicted particle separation distances are much <5 nm in these cases, and it does not appear that ion-correlation forces alone could explain long-range attractions over 10 to 100 nm or more. At short separation distances between charged layer silicate surfaces, a strong repulsive force because of hydration of interlayer ions is measured (Pashley, 1981), and for this reason the DLVO and other mean field models are unable to make reasonable predictions of short-range behavior in colloids.
Sogami and Ise (1984) point out a further weakness of the DLVO model, namely that it is based on the forces interacting between only two particles, and any conclusion from the model is therefore valid only at infinite dilution. Dilution causes the counterions to diffuse away from the charged particles, so that interparticle repulsion can be expected under these conditions of low particle density. Attractive forces not considered by DLVO could be anticipated at higher particle densities, when counterions are simultaneously attracted by multiple nearby particles. Thus, the interaction between an isolated pair of like-charged spherical particles is believed to be purely repulsive, whereas many body interactions in colloid suspensions should be expected to result in an attractive force (Larsen and Grier,1997).
Hansen and Lowen (2000), in their review of interactions between electric double layers, state that "the effective force between two uniformly plates may turn attractive at short distances if correlations between microions are properly accounted for." They further stated that a similar attractive force can be expected for two isolated spheres, and if the separation distance, d, between their surfaces is much less than their radii, a, then the problem reduces to that of two charged parallel plates, where the force is clearly attractive. However, these correlation-induced attractions are relatively short-range in nature. Belloni (2000) also discussed the more recent evidence for "effective electrostatic attraction of pure electrostatic origin between like-charged particles via their counterions," but believed that such an attraction could only be predicted outside the PB-DLVO mean field approximation by including ion-ion correlation forces. Belloni stated that attraction between like-charged plates is well-accepted, arguing that the case for attraction between spherical colloids is less clear, despite the work of Patey (1980). Patey calculated the interaction of colloidal spheres with high charge to be strongly attractive at separation distances on the order of the particle diameter, with the counterions between the particles appearing to overcome interparticle repulsion. Patey (1980) noted that other researchers had not found an attractive interaction, and suggested that the choice to hold surface charge density rather than surface potential constant may have led to the very different result.
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Experimental Evidence for Long-range Electrostatic Attraction among Like-charged Particles
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As discussed in the introduction, Langmuir (1938) may have been the first to hypothesize a long-range attractive electrostatic force in colloidal suspensions, stating that "the Coulomb attraction between the micelles (clay particles) and the oppositely charged ions gives an excess of attractive force which must be balanced by the dispersive action of thermal agitation and another repulsive force. Thus, there is no reason to assume long range van der Waals forces." In effect, Langmuir argued that there is a long-range attractive force among charged particles that is electrostatic in nature, that the opposing repulsive force evident at short interparticle separations is thermal agitation, and that the van der Waals force is over negligible significance because of the fact that the attractive forces are operating in suspension of distances of hundreds of nanometers. Langmuir likened the electrostatic attractive force to that in an ionic crystal, where "the distance between any ion in this lattice and its oppositely charged neighbor is less than the distance between the ion and its nearest neighbor of like sign." The attractive forces therefore exceed the repulsive forces, and the lattice has a tendency to contract. A similar argument could be applied to charged plates (layer silicate sheets) with layers of counterions between; contraction of the interlayer spacing would be predicted from electrostatic forces alone.
Much later, research in Japan on suspensions of synthetic latex particles, which are clay-sized spherical negatively charged colloids, provided experimental evidence of a long-range attractive force among the like-charged colloids, as predicted by Langmuir (Ise et al., 1990; Ise, 1986). Briefly, the colloidal particles were seen to undergo reversible phase transitions between liquid crystalline arrays and the disordered isotropic state in very dilute electrolyte solutions, depending upon particle concentration, electrolyte concentration, and temperature. Voids were formed as transition to the ordered phase occurred, and measured particle-particle distances were shorter than would be the case if the particles occupied all of the solution volume. The crystalline arrays of charged spheres compress as the temperature is raised, the extent of which is better predicted by the Sogami potential, with its attractive tail at large interparticle separations, than the DLVO potential (Ise and Smalley, 1994). The main cause of liquid crystalline compression is the temperature-dependence of the Debye screening length, K-1. Under equilibrium conditions, the DLVO theory cannot explain any thermal contraction, as particles must repel at all separation distances over which van der Waals forces are negligible (<100–>1000 nm).
These phenomena are all inconsistent with DLVO, which would predict that no voids could form if long-range particle-particle interaction were strictly repulsive, since voids would always become occupied by particles maximizing their interparticle separation distances. As the stable interparticle separation distances in these liquid crystals are on the order of microns (1000 nm), only electrostatic forces, and certainly not van der Waals forces, are sufficiently long-range to satisfactorily explain the apparent particle-particle attraction. Later studies of these model colloids further confirmed the evidence for a long-range attractive interaction in aqueous suspensions (Larsen and Grier, 1997). The nature of the ordering forces are essentially electrostatic, because of the sensitivity of the ordering and interparticle distances to added electrolyte, and this reversible phase transition is therefore not driven by Onsager's excluded volume (entropy-driven) effect.
Crocker and Grier (1996) point to "mounting evidence that the effective pair interaction in dense suspensions sometimes has a long ranged attractive component," which is at odds with the purely repulsive screened Coulomb interaction of the DLVO theory. They cite several experimental studies showing that colloids can in fact attract over large distances. In their own work, the experimentally observed interparticle attraction of charged colloids operating over distances of 100 to 1000 nm in water (Larsen and Grier, 1997; Crocker and Grier, 1996) are not explained by the ion-ion correlation forces described in the above section; those forces are too short-ranged. Crocker and Grier (1996) have demonstrated experimentally that the interaction between an isolated pair of similarly charged colloidal spheres in water is repulsive, as predicted by DLVO. They have also found, most interestingly, that confining these spheres in a volume <5000 nm deep appears to create an attractive interaction, with a potential minimum at 2000 to 3000 nm. The authors suggested that the attraction could arise from a multiparticle effect, with the confining walls having a similar effect in favoring attraction as would an increase in particle density. Because the CAT considers that thermal motion (entropy) is the dominant repulsive force, it is easy to imagine that the imposition of confining walls on a colloidal dispersion would decrease the particles' degrees of freedom of translational motion, and shift the balance in favor of the attractive multiparticle electrostatic force.
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Coulombic Attraction Theory (CAT) Versus DLVO
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A theoretical basis for a long-range Coulombic attractive force between like-charged particles was proposed by Sogami and Ise (1984), and its development continued in a series of subsequent publications (Ise and Matsuoka, 1994; Smalley, 1990; Smalley and Sogami, 1995), in part in response to early criticisms and misunderstandings (e.g., Belloni, 2000). Schmitz (1993) provides a discussion of the Sogami–Ise interaction potential, the criticism leveled against it, and responses to these criticisms.
The Sogami–Ise theory, like the DLVO model, begins with the assumption of a Boltzmann distribution of counterions (exchange ions) at the colloidal spherical particle surfaces. Via the PB equation, the total electrostatic energy of the electrolyte solution is obtained, and the Gibbs free energy is calculated. It is thereby shown that there is an interparticle repulsion at small separation distances, and an attraction at larger distances. A secondary free energy minimum (deepest at about Ka = 1, where a is the particle radius) results at intermediate interparticle separations, where the attractive electrostatic potential balances the purely repulsive contribution because of entropic contributions of the counterions (Ise, 1986). Because the Gibbs free energy shows a minimum with a potential well that is deeper than that of thermal energy, it allows particles to order into stable but fragile lattice arrays at room temperature at separation distances comparable in magnitude to several particle radii, which for some colloids is in the range of 1000 nm or more. The theory appears to adequately explain the interparticle spacing of latex spheres that spontaneously form liquid crystalline lattices in water, and the tendency of the ordered phase to melt with small temperature increases, attributable to the shallow potential well.
Smalley (1994) has explained the essential features of the departure of the Sogami–Ise CAT from the DLVO theory for a pair of parallel charged plates. The former theory has three terms that contribute to the Helmholz pair potential, V(x):
 | [10] |
where x is the plate separation. The second term, Voi
, is purely repulsive, and represents the osmotic contribution of the small counterions trapped between plates. The third term, Voo, is purely attractive, arising from the osmotic pressure of the small ions, exerted against the plate surfaces from the outside. The first term, Vel(x), results from the electrical interaction at equilibrium between the counterions and the charged plates. The DLVO potential is given as:
 | [11] |
where K is the Debye screening length. The DLVO potential corresponds essentially to the osmotic component, the last 2 terms in Eq. 10, and therefore fails to account for the electrical interaction, predicting pure (osmotic) repulsion. This electrical term in Eq. 10 leads to a long-range attraction between the plates. The two opposing effects, the electrical and the osmotic, are balanced at equilibrium, and the pair potential has a minimum that leads to a stable plate separation distance. Thus, the repulsive force in the DLVO model originates in the electrostatic repulsion inherent to the Debye–Huckel theory of small ions (in this regard, it is noteworthy that Debye–Huckel requires that ionic interaction energies be less than thermal energy, an assumption that does not hold for colloidal particles with high charge or for nondilute solutions of ions). Langmuir (1938), and much later Sogami and Ise (1984), Smalley (1994) and others (e.g., Kjellander et al., 1988) considered a purely repulsive potential (entropic), arising from the thermal motion (pressure) of the excess concentration of counterions trapped between particles, separately from an electrical potential, which is generally attractive. Under certain circumstances, the electrical term is sufficiently large to completely balance the repulsive term, causing particles to attain a stable separation distance at the potential minimum.
The fundamental reason for the difference of the Sogami–Ise theory from the DLVO theory is that the former theory allows for the clouds of counterions in colloidal suspensions to compress around and between highly charged particles, particularly when particle density is high, generating interparticle attraction via the counterions, and lowering the free energy (Sogami and Ise, 1984). In the DLVO model, the assumption is that of two particles in infinite volume, so that the diffuse-double layer overlap is too negligible to effect a redistribution of counterions to accomodate multiple particles in a finite volume. Thus, the conclusion from DLVO that the particles repel, while valid for dilute suspensions of low-charge particles, may be incorrect for finite concentrations of particles with high charge (Sogami and Ise, 1984). Under these more realistic conditions for clay suspensions, the counterions tend to be held more closely to the particles and are at the same time attracted to more than one particle. Thus, as depicted diagrammatically in Fig. 7
, whereas particle-particle repulsion of DLVO arises from double-layer overlap in the case of low-charge particles, particle-particle attraction is attributed to the rearrangement and confinement of counterions in the volume around and between high-charge particles, particularly when the average particle-particle separation distance is reduced by a high particle concentration in suspension.
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Fig. 7. Depiction of particle-particle Coulombic interaction according to Sogami–Ise model at low and high particle charge, leading to repulsive (a) and attractive (b) interaction.
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To summarize, according to the Sogami–Ise theory as well as experimental observation, the conditions favorable to interparticle attraction in colloidal suspensions are (Ise, 1986):- Low temperature, as higher temperature increases kinetic energy and favors counterions escaping the influence of the charged particles, that is, increasing the diffuse-double layer thickness and weakening the attractive force.
- High particle concentration, decreasing the tendency of counterions to migrate away from the particles.
- High particle charge, which redistributes the counterion clouds around the particles, favoring attraction.
- Presence of multivalent rather than monovalent counterions, which intensifies counterion-particle attraction and consequently, particle-particle attraction.
- Low ionic strength, as high electrolyte concentrations effectively screen the long-range electrostatic force.
The calculations of Chu and Wasan (1996), although questioned by Belloni (2000), predict an attractive interaction between two similarly charged colloidal particles, in contradiction of the Debye–Huckel and DLVO models. Their calculated pair potentials for different conditions predicted that like-charged colloidal particles will attract each other at low ionic strength (<10-3 M), at high particle concentration (>> 0.01% by volume) and at high particle charge (>100 charges per particle). These predictions are at least qualitatively consistent with experimental observations that attractive behavior should be expected in model charged colloid and natural clay systems with high particle charge density (e.g., smectites), and under conditions of low ionic strength. By increasing the electrolyte concentration, the long-range electrostatic attractive force is weakened, ultimately to the point of shielding the Coulombic field at a very short distance from the particle surface. At high ionic strength (approaching 10-1 M), therefore, particle interaction becomes hard sphere-like and particle flocculation can be expected because of the ease of close approach of particles.
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Experimental Evidence from Colloidal Systems
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The Sogami–Ise model predicts the interparticle distance to become smaller with increasing Ka, where K-1 is the Debye radius and a is the radius of the particle, and the potential well to become deepest at Ka = 1. Because K-1 can be viewed as the effective diffuse-double layer thickness, which is inversely proportional to the square root of the electrolyte concentration, the Sogami–Ise model predicts an effect of electrolyte strength on interparticle separation similar to DLVO; higher electrolyte concentration compresses the Debye radius (screening distance) and reduces interparticle separation distance. Additionally, the model predicts that this interparticle distance decreases with increasing particle concentration and with increasing temperature. In fact, for plate-like particles, such as layer silicate clays, the Sogami–Ise theory predicts that Kd = 4.0, where d is the interparticle spacing (Williams et al., 1994). Because K is proportional to the square root of the electrolyte concentration, 
, a plot of smectite interlayer spacing versus 1/ should be a straight line. Although the Sogami–Ise theory provides a good fit of measured to predicted d-spacings in Na-smectites within the osmotic swelling range of 2 to 200 nm (McBride, 1997), this good fit is not in itself a reason to favor the Sogami–Ise theory over DLVO. Both theories predict the relationship of clay d-spacing to because of compression of the Debye radius by electrolyte.
One aspect of clay behavior that DLVO has had success in explaining is the osmotic swelling of smectites (see e.g., Quirk and Marcelja, 1997 and Quirk, 1994). However, numerous experimental observations said to support the DLVO theory can be shown to be explained as well or better by the Sogami–Ise theory (Sogami and Ise, 1984; Ise and Smalley, 1994). Also, any experimental test of DLVO versus Sogami–Ise theories requires that conditions be used that are predicted by the Sogami–Ise model to be favorable to electrostatic attraction. This is frequently not done, as may have been the case with the study by Crocker and Grier (1996), which Tata and Ise (2000) criticized for not adequately characterizing the effective charge on the polystyrene microspheres. The latter authors presented evidence that the colloidal spheres used by Grier and Crocker (1996) had low charge, and were therefore more appropriate for testing screened Coulombic repulsion (DLVO behavior) than long-range attraction predicted by the Sogami–Ise theory.
As this short discussion illustrates, the CAT of Sogami and Ise remains controversial at this time, as studies claiming to support (e.g., Wang, 2000) or refute (e.g., Trizac, 2001; Wu et al., 1998) the theory continue to be published. Experimental evidence is still accumulating that charged colloidal particles of various materials in homogeneous dispersions actually form ordered domains with higher densities of particles and voids within the disordered bulk suspension (Ise et al., 1999; Ise, 1999; Grohn and Antonietti, 2000; Harada et al., 1999), clearly in contradiction of the purely repulsive DLVO theory. The experimental evidence for a long-range electrostatic force is now compelling, but the fundamental nature of this force is uncertain. Grohn and Antonietti (2000) have proposed two alternative models, the "fluctuation model" and the "colloidal orbital approach." The former allows the formation of temporary dipoles in the spherically symmetrical counterion clouds, analogous to the London dispersion force, but much stronger. The latter generates attraction by redistributing counterion density between colloidal particles, analogous to electrons forming interatomic bonds. It should be noted that the "colloidal orbital approach" is exactly the explanation given by Ise (1986) for the Sogami–Ise attractive force.
As increasingly detailed evidence of interparticle ordering in colloidal dispersions is gathered in years ahead, uncertainties that may persist about the physical nature of long-range interactions in these suspensions will hopefully evanesce.
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CONCLUSIONS
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It is clear from both experiment and theory that idealized monodisperse colloids have well-defined long-range ordering and reversible phase transitions that are dictated by both their shape (anisodimensionality) and charge. In many situations, a long-range attractive interparticle force exists, which appears to be electrostatic in nature. The van der Waals attractive forces are negligible when the separation distance between colloidal particles is more than a few nanometers, although ion-ion correlation forces can generate an attraction between like-charged particles at short separation distances, particularly when the counterions are divalent or polyvalent.
Real clay and organic colloids in soils are not likely to display the ideal behavior demonstrated by model colloids, partly because they are polydisperse (variable in size and shape), and also because they probably range widely in particle charge density. Contributing further to their nonideal behavior is the mixed electrolyte solution that they are dispersed in, and the likelihood of molecules and ions adsorbed on the surface that alter their charge and distribution of ions in the diffuse-double layer. Nevertheless, the origins of the fundamental attractive and repulsive forces in these charged colloidal systems are the same as for model colloids. We can therefore apply the theoretical and experimental concepts learned from these model systems to more complex colloidal systems, recognizing that many interesting physical phenomena observed in monodisperse colloids may not occur, or may be difficult to detect, in real colloids.
Received for publication June 20, 2001.
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REFERENCES
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- Hansen
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ABSTRACT
INTRODUCTION
