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Dissociation of Colloidal Spheres According to LMO Law Jan J. Spitzer* IPM Emulsion Polymer Research, 6643 Lyndonville Drive, Charlotte, North Carolina 28277 Received May 12, 2002. In Final Form: July 30, 2002 New detailed results for ionic distributions around charged spherical surfaces in electrolyte solutions are presented using the formulation of dissociative electrical double layers (DEDL) with the Lubetkin, Middleton, and Ottewill dissociative law. Wide ranges of spherical radii and ionic strengths are considered. From these theoretical results the extent of ionic atmosphere and the number of ions contained therein are calculated to give an estimate of how many ions interact with a colloidal sphere. A new measure of the range of electrostatic repulsions is defined as the sum of the Debye length and the thickness of the co-ion exclusion shell. New results for extremely low charge densities indicate a possibility of significant electrostatic effects in such systems. These results, representing a synthesis of the classical theory of Debye-Hu¨ckel interionic interactions with the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, provide new insights into the co-ion exclusion phenomena in colloidal dispersions and related consequences. The DEDL theoretical approach, with emphasis on electrostatic consistency, has shown a greater explanatory power than the classical DLVO theory. The latter contradicts Maxwellian electrostatics unnecessarily.
Introduction In this paper the theoretical drafting of the Dissociative Electrical Double Layer (DEDL) model, strengthened with the Lubetkin, Middleton, and Ottewill (LMO) law,1-7 is continued in spherical symmetry. For the sake of completeness and clarity, we take a broader, more historical view of ionic atmospheres and re-approach the topic by comparing the Debye-Hu¨ckel (DH) interionic theory with the Derjaguin-Landau-Verwey-Overbeek (DLVO) colloidal theory. Both theories employ essentially the same physical model of electrostatic charges in a linear dielectric. And yet both theories contain inconsistencies that persist. These long-standing inconsistencies are reviewed first. In 1923 Debye and Hu¨ckel explained large nonidealities of dilute ionic solutions in terms of long-range Coulombic interactions between dissociated ions.8-10 These interactions were treated in terms of ionic atmosphere: the thermally smoothed-out volume charge density of moving ions around each other in solution. The resulting limiting laws of the DH theory of activities of strong electrolytes are a classic achievement of physical chemistry of solutions. Early on, however, the experimentally observed deviations from the limiting laws became a topic of intense scrutiny, and critiques and corrections to the DH treatment were advanced. These extensive investigations were critically summarized about 40 years later in the classical book on electrolyte solutions by Robinson and Stokes.11 It became clear that the deviations from the limiting laws are due to the specificity of ionic solvation and to the structure of ionic atmospheres at higher concentrations.11,12 Germane to the present paper, Robinson and Stokes discussed the issue of negative co-ion distributions, * Corresponding author e-mail:
[email protected]. (1) Spitzer, J. J. Nature 1984, 310, 396-397. (2) Spitzer, J. J. Colloids Surf. 1984, 12, 189-193. (3) Spitzer, J. J. Langmuir 1989, 5, 199-205. (4) Spitzer, J. J. Colloids Surf. 1991, 60, 71-77. (5) Spitzer, J. J. Langmuir 1992, 8, 1659-1662. (6) Spitzer, J. J. Langmuir 1992, 8, 1663-1665. (7) Spitzer, J. J. Colloid Polym. Sci. 1992, 270, 1147-1158. (8) Debye, P.; Hu¨ckel E. Phys. Z. 1923, 24, 185. (9) Debye, P.; Pauling, L. J. Am. Chem. Soc. 1925, 47, 2129-2134. (10) Pauling, L. Personal communication, January 20, 1981. (11) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworth & Co., Ltd.: London, 1959; Chapters 4 and 9.
when the exponential Boltzmann distributions laws are linearized. They commented on the counterintuitive trend of the co-ion concentrations: as experimental deviations from the limiting laws grow larger (with increasing bulk concentration), the negative co-ion concentrations grow smaller, and in fact disappear. (These trends can be calculated explicitly.13) This observation implies that the deviations from the limiting laws are not due to the linearization of the Boltzmann exponential. Furthermore, they argued that to try to avoid these negative concentrations by using the full Boltzmann exponential is tantamount to violation of Coulomb’s law (Maxwellian electrostatics). For example, one consequence is that free energy functions, such as activity coefficients (i.e., electrostatic interaction energies), cannot be calculated consistently. Meaningful thermodynamic tests of the nonlinear theory cannot be done. They concluded that the nonlinear Poisson-Boltzmann (PB) equation does not offer a more rigorous theoretical alternative to the linear PB equation. Rather, the fundamental issue is the replacement of the potential of mean force Wi of statistical mechanics with Maxwellian electrostatic energy zieΨ.11,14 In other words, the linearization is not a numerical approximation to the full nonlinear PB equation but an expediency to remain within the Maxwellian electrostatics. Although the linear DH theory predicts unphysical co-ion concentrations, it has proved useful in diverse ways. For example, ionic transport phenomena could be theoretically explained, or reliable extrapolations could be devised to obtain standard thermodynamic properties at infinite dilution.15,16 Importantly, the ionic strength effect on ionic equilibria (solubility products or dissociation constants) and on ionic reaction kinetics could be understood as the lowering of activity coefficients of free diffuse ions. This (12) Bennetto, H. P.; Spitzer, J. J. J. Chem. Soc., Faraday Trans. 1 1976, 72, 2108-2124. (13) Spitzer, J. J. J. Chem. Soc., Faraday Trans. 1 1978, 74, 24182421. (14) McQuarrie, D. A. Statistical Mechanics; Harper & Row: New York, 1976; Chapter 15. (15) Spitzer, J. J. J. Chem. Soc., Faraday Trans. 1 1973, 69, 14911497. (16) Spitzer, J. J.; Bennetto, H. P. In Thermodynamic Behavior of Electrolytes in Mixed Solvents; Furter, W. F., Ed.; Advances in Chemistry Series 155; American Chemical Society: Washington, DC, 1976; Chapter 12, pp 197-212.
10.1021/la025937n CCC: $22.00 © 2002 American Chemical Society Published on Web 09/12/2002
Dissociation of Colloidal Spheres
classical DH interionic effect has not been considered in double layer theories until the discovery of the LMO law.1-7 Before the advent of the DH theory, Gouy17 and Chapman18 (GC) used the concept of ionic atmospheres, arising from complete dissociation of electrical double layers, in connection with electrocapillary phenomena. The resulting exponential PB theory provides a semiqualitative rationalization of double layer capacities at mercury-solution interfaces. The GC PB equation was used 40 years later as an essential component of the DLVO theory of colloidal stability19 to represent electrostatic repulsions between any charged surfaces. Thus, the resulting GC-DLVO nonlinear theory is in direct opposition to the Robinson and Stokes analysis.11 It suggests that the nonlinear PB equation is to be preferred to the linearized equation but does not address the issue of implied violation of Coulomb’s law. Because of this serious disrespect for Coulomb’s law, the current claim of general validity20 [sic] of the DLVO theory cannot be sustained. Perhaps not surprisingly then, some experimental observations disagree with the GC-DLVO predictions. One example is the prediction of the universal adhesive minimum that is not always observed: rather, infinite repulsions are sometimes found. In the last 20 years these observations of unexpected large repulsions have led to a growing number of ad hoc hypotheses such as hydration forces and other non-DLVO forces, assuming fundamental validity of the GC theory. Though some of these non-DLVO effects may indeed be of importance in specific instances, the proposition of the general DLVO validity is questionable. Interestingly, some claims of agreement with DLVO are in fact agreements with the DH linearized expressions, which are obtained ad hoc because of the intractability of the nonlinear PB equation. These linear approximations are sometimes claimed to be valid up to high Stern potentials of 50 mV, despite the implied negative co-ion distributions, as analyzed by Robinson and Stokes.11 The second issue in the DLVO approach is the GC assumption of complete double layer dissociation. As a rule, only partial dissociation is experimentally observed. Early on, it was recognized that complete double layer dissociation is rather unrealistic, and Stern21 postulated a layer of adsorbed counterions. The current concept of the effective charge22 can be viewed as the summation of the primary charge and the Stern charge. The frequently observed discrepancies between the analytical surface charge and the GC-DLVO derived charge density (usually much lower than the analytical charge) reflect the unrealistic 100% dissociation assumption in the GC theory. Such discrepancies imply the existence of a Stern layer of charge. The DEDL formulation is a synthesis of the GC-DLVO and DH approaches, drafted with Occam’s Razor to keep things as simple as possible in a model that is sufficiently realistic. Occam’s simplification is to do away with the nonlinear PB equation by retaining the linear ionic distributions, as argued convincingly by Robinson and Stokes.11 The resulting issue of the co-ion negative concentrations is solved elegantly by the introduction of a free co-ion exclusion boundary b located away from the charged surface. Thereby, the theory becomes applicable to any (high) surface charge densities or potentials. The (17) Gouy, G. J. Phys. 1910, 9, 457. (18) Chapman, D. L. Philos. Mag. 1913, 7, 129. (19) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of Stability of Lyophobic Colloids; Elsevier: Amsterdam, The Netherlands, 1948. (20) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: Oxford, 2001; Chapter 12. (21) Stern, O. Z. Elektrochem. 1924, 30, 508. (22) Alexander, S.; Chaikin, P. M.; Grant, P.; Morales, G. J.; Pincus, P. J. Chem. Phys. 1984, 80, 5776-5781.
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linearity of ionic distributions guarantees electrostatic consistency (free energy calculations); it also greatly improves mathematical tractability. The GC point-charge model is made more realistic by the explicit recognition of incomplete double layer dissociation, as required by experimental findings. Some counterions (of finite size) are assigned to form a Stern layer against the primary charge on the surface, and the ratio of the Stern charge to the surface charge is the degree of association R. No assumption is made about the dependence of the degree of association on ionic strength or other parameters; rather, experimental data are used to discover such dependencies. (Strong dependence on ionic strength and on primary surface charge density is expected.) This basic theoretical draft is applicable to any particle (or surface) shape and to any concentration of charged colloidal particles. The initial results for interactions between infinite flat surfaces1,3,5 using the DLVO approach of adding electrostatic repulsions and van der Waals attractions met with unequivocal success. The large repulsive pressures in montmorillonite gels could be accounted for quantitatively without invoking any non-DLVO forces, such as hydration forces. Further work led to a new fundamental limiting law for electrostatic repulsions of flat surfaces in the limit of surface contact: the electrostatic repulsions decay according to the inverse square of the separation distance. Subsequent application of the theory3,5 to LMO23 data on swelling pressures of homo-ionic montmorillonites proved unexpectedly insightful. The analysis of the LMO data (covering broad ranges of ionic strengths, repulsive pressures, and specific counterions) revealed a new relationship for the dependence of the degree of double layer association R on ionic strength (Debye’s κ), as
R)
1 1 + pκ
(1)
where p is an unknown parameter describing the chemistry of the double layer (the susceptibility of the double layer to dissociation). This equation was designated as the LMO law,3 and it reflects the classical DH interionic effect when bulk electrolyte increases the dissociation of the double layer by lowering the activity of diffuse ions. (With the GC-DLVO theory this effect could not be discovered or utilized because of the 100% dissociation assumption.) When the LMO law is incorporated into the contact limiting law, it is found that the high repulsive forces become only weakly dependent on ionic strength. In the limit of zero ionic strength and zero separation distance, a new double limiting law is obtained in which the dependence of the repulsive forces on ionic strength disappears entirely. Significantly then, the DEDL-LMO theoretical draft provides a natural transition from the ionic strength dependent, DH exponential law at infinite separations (which has been well-established experimentally by Israelachvili and his school,24-26) to the new ionic strength independent limiting law at contact separations. To present new derivations and calculations for spherical symmetry, this paper is organized as follows. First, a general solution is obtained in detail for a single elec(23) Lubetkin, S. D.; Middleton, S. R.; Ottewill, R. H. Philos. Trans. R. Soc. London 1984, A311, 133. (24) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975-1001. (25) Kohonen, M. M.; Karaman, M. E.; Pashley, R. M. Langmuir 2000, 16, 5749-5753. (26) Tulpar, A.; Subramanian, V.; Ducker, W. Langmuir 2001, 17, 8451-8454.
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troneutral sphere of arbitrary size immersed in a strong electrolyte solution. Some new theoretical justifications are also provided. Second, general predictions of the theory are presented, including examples of the range of electrostatic forces, the calculation of the extent of the ionic atmospheres, and the actual number of ions contained therein. A similar approach was used before to analyze the workings of the DH formalism,13 according to Robinson and Stokes’ analysis. Theory of a Single Colloidal Sphere Preliminary Considerations. We consider a twophase colloidal system with electrolyte solution being one phase and a colloidal sphere being the second phase. Both phases are independently electroneutral. In the reference state in the solvent the sphere has an undissociated, solvated, Helmholtz double layer. On introduction of a strong electrolyte, the dissociation of this sphere can be represented as z+e z-e Snz+e Cz-e + ν-Bz-e] ) Snz+e Cn-x + xCz-e + 1 n + [ν+A 1
[ν+ Az+e + v-Bz-e] (2) The increasing concentration of strong electrolyte AB (in square brackets) lowers the activity of the C anions, thereby increasing the dissociation of the colloidal sphere SC. Here we take the view (subject to future refinement) that in a two-phase colloidal system (lyophobic colloids), a colloidal particle with a Helmholtz double layer does not spontaneously dissociate in the absence of added ions (cf. LMO law, eq 1). If it did, the result would be a spontaneous charging of two macroscopic phasessthe colloidal sphere and the solventsimplying a positive free energy change of such a process. Thus, by this criterion, the LMO law (though empirical) is thermodynamically sensible. Self-ionizing solvents, such as water, provide added ions as a matter of course in the limit of infinite dilution of the electrolyte, though ion exchange will typically complicate matters. The present theoretical results are valid for an infinitely dilute colloidal dispersion. For simplicity, we assume that this sphere is positively charged and that it possesses counterions of the same type (B) as those in the electrolyte solution. Thus, any specific ion-exchange phenomena are not considered and neither is the operation of Le Chatelier’s equilibrium principle (which may need to be invoked in special cases such as silver iodide sols in silver nitrate or sodium iodide solutions). Spherical DEDL Model. The DEDL electrostatic problem is shown in Figure 1. The physical model is that of DH (or GC-DLVO) with familiar approximations to reality: the solvent is assumed to be a linear dielectric with dielectric constant , and both co-ions and counterions are charged spheres of the same solvated size, given as the difference (a - R) in Figure 1. The model defines three regions around the charged sphere. Region 1 is the counterion association region, characterized by the primary surface charge density σ0+ and by the Stern layer surface charge density σa interacting through the dielectric 1. Region 2 is the co-ion exclusion region where only diffuse counterions are present, interacting through the solvent dielectric . Under certain critical conditions this region disappears, when b ) a, and only the DH region exists. Region 3 is the conventional DH region for low electrostatic potentials. The dielectric constant of the solvent will be used in all three regions. Region 1 (Counterion Association Region) R e r e a. In this region the model is a relatively poor representation
Figure 1. Colloidal sphere has radius R and surface charge density σ0+. The negative counterions partly remain associated with the charge on the sphere at distance a giving rise to negative surface charge density σa. Other symbols have their usual meanings and are described in text.
of the short-range interactions between (solvated) charges on the central sphere and (solvated) counterions. All the specific effects of such interactions are represented by the degree of association R defined as
R)-
Qa 4πa2 σa )Q0+ 4πR2 σ0+
(3)
A lower dielectric constant 1 than that of the bulk solvent could be used to reflect the electrostriction (nonlinear dielectric response) of solvent molecules close to the bare surface charges and close to the bare counterions. But this is a contributing factor to the assumed counterion association in the first place, and such an approach would overparameterize the model. Hence only the dielectric constant of the solvent is used in all three regions, although the symbol 1 will be retainedsits value determines the potential drop from the surface charge to the Stern charge only and hence determines the value of the surface potential. The Stern potential is independent of this inner dielectric constant because this region is inside the Stern sphere. The potential ψ1(r) in region 1 is given by
ψ1(r) ) ψ0 -
R2σ0+ 1 1 01 R r
(
)
(4)
The potential ψ0, the surface potential at r ) R, is the total potential arising from all charge distributions assumed in the system. They are as follows: the central sphere charge σ0+ giving rise to potential ψs(R), the associated charge σa giving rise to potential ψa(R), the counterion volume charge F2 between a < r < b giving rise to potential ψab(R), and the DH counterion and co-ion volume charge F3 giving rise to potential ψDH(R). These potentials are additive, satisfying
ψ0 ) ψ1(R) ) ψs(R) + ψa(R) + ψab(R) + ψDH(R) (5) not only on the surface of the central sphere but also at any point outside it. In fact, the potentials at any point can be calculated by standard electrostatics by integrating Coulomb’s law over respective volume or surface charge densities. This additivity condition is not satisfied in the GC-DLVO theories, which is the consequence of replacing the potential of mean force with Maxwellian electrostatic energy without linearization. Indeed, in applying the DH approach to the larger size colloidal particles (compared to smaller ions and molecules of the DH theory), the requirement of consistent macroscopic electrostatics ap-
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pears even more compelling. The corresponding charges that give rise to the potentials in eq 5 are also additive, satisfying the electroneutrality condition:
Q0+ + Qa + Qab + QDH ) 0
(6)
to DH κ but involves only bulk number density of counterions.1-7 Region 3 (Debye-Hu¨ ckel Region) b e r e ∞. The potential satisfies the Poisson equation
∇2ψ3 )
where the charges are given by
Q0+ ) 4πR2σ0+
(7)
Qa ) 4πa2σa
(8)
∫abF2(r)r2 dr
(9)
∫b∞F3(r)r2 dr
(10)
Qab ) 4π
QDH ) 4π
The electroneutrality condition (eq 6) reflects the fact that the two-phase colloidal dispersion has no excess charge (is electrostatically grounded), and it is a self-consistency check on the electrostatic solution of the charged system (i.e., it is not an independent boundary condition, although it can be used in lieu of one). Region 2 (Co-Ion Exclusion Region) a e r e b. Under certain conditions (to be determined later) the co-ions are on average excluded away from the positive central sphere up to the exclusion boundary b, where their local concentration ν+n is just zero:
ν+nz+e ) ν+n0z+e[1 - z+eψ2(b)/(kT)] ) 0
ψ2(b) ) kT/(z+e) ) ψt+
(12)
where z+e is the charge on the co-ions, and kT is their average thermal (and electrostatic) energy. In this way the electrostatic consistency to calculate electrostatic energies (free energies) is maintained, and no unphysical co-ion distributions are predicted (as they are in the DH theory). Thus, at r ) b the potential is known but the location of the boundary surface (spherical surface of radius b) is not. This represents, mathematically speaking, a free boundary value problem, where the value of the potential ψ2(b) is known and where the electric field is continuous across the boundary in the absence of polarization charges (the dielectric constant being the same on both sides of the boundary). In this region only counterions are present as volume charge density F2. The general physical-chemical condition for co-ion exclusion to become operational (i.e., when b ) a) will be derived from the general electrostatic solution defined in Figure 1. The electrostatic potential satisfies Poisson’s equation, assuming electric field independent dielectric constant as
∇2ψ2 )
( )
F2 1 d 2dψ2 r )2 dr dr r 0
(13)
ψ2(r) ) A2
exp(-λr) exp(λr) + B2 + ψtr r
ψ3(r) ) A3
(14)
where A2 and B2 are integration constants to be determined from the boundary conditions. The constant λ is analogous
exp(-κr) r
(16)
where we assumed that the potential decays to zero at infinity. Boundary Condition Results. We have five unknowns, three integration constants (A2, B2, and A3), the free boundary distance b, and the value of the potential on the central sphere ψ0. The boundary value conditions are, first and second, the continuity of potentials at r ) a and at r ) b
ψ1(a) ) ψ2(a)
(17)
ψ2(b) ) ψ3(b)
(18)
and at r ) b
Third, the value of the potential at r ) b, as given by eq 12 is
ψ2(b) ) ψ3(b) ) ψt+
(19)
Fourth, the absence of surface charge accumulation at r ) b and assuming no change in the dielectric constant across this boundary (which would lead to a dielectric polarization charge), we have the continuity of the electric field as
(dψ2/dr)b ) (dψ3/dr)b
(20)
And fifth, the accumulation of the associated surface charge at r ) a that is given by
( )
σa ) -0
dψ2 dr
a
( )
+ 01
dψ1 dr
(21)
In this way the Stern counterion layer becomes an inherent part of the electrostatic model. Moreover, we have the charge association assumption expressly written to relate the associated charge density σa to the primary surface charge density σ0 according to eq 3. Thus, the Stern concept of adsorbed ions is treated in an entirely new way (i.e., as an electrostatic phenomenon). The working out of these boundary conditions gives the following main results. The constant A3 is given by
A3 ) ψt+b exp(κb)
(22)
The constants A2 and B2 are given by
A2 ) -
the solution of which is
(15)
with the familiar solution
(11)
Thus the potential ψ2(r ) b) is defined by the average thermal energy of the co-ions as
( )
F3 1 d 2dψ3 r ) dr 0 r2 dr
1 exp(+λb)[(ψt+ - ψt-)(1 - λb) 2λ ψt+(1 + κb)] (23)
1 B2 ) + exp(-λb)[(ψt+ - ψt-)(1 + λb) 2λ ψt+(1 + κb)] (24)
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The exclusion distance b can then be calculated by Newton’s method from the following transcendental equation:
R2(1 - R)σ0+ - A2(1 + λa) exp(-λa) F(b) ) 0 ) 0 B2(1 - λa) exp(λa) (25) by repeating
bi+1 ) bi -
F(bi) F ′(bi)
(26)
to any desired accuracy. It is advantageous to work out eq 25 with eqs 24 and 23 analytically (as well as its first derivative) to avoid overflows in the exponential functions in practical calculations in SI units. Calculation of Charge Distributions. To check the self-consistency of the electrostatic solutions (e.g., electroneutrality), it is necessary to calculate the charge distribution integrals given by eqs 9 and 10. The total charge in the DH region is given by
QDH ) -ψt+b(1 + κb) 4π0
(27)
The counterion diffuse charge (or the Stern diffuse charge) in the co-ion exclusion region (b - a) is given by the following expression:
Qab ) T 1 + T2 2λ 4π0
+
T2 ) {(ψt - ψt )(1 + λb) - ψt (1 + κb)}{+(1 - λb) (1 - λa) exp[+λ(b - a)]} (30) reveal a pleasingly muted symmetry. The Stern charge in eq 8 can be calculated through the basic definition of the degree of association given by eq 3, and the electroneutrality condition (eq 6) then can be evaluated. Range of Electrostatic Forces. Conventionally, the Debye length 1/κ is taken as the measure of the range of the electrostatic forces in the DH theory of electrolyte solutions, and the same interpretation has been adopted in the colloidal theory. This is a distance from a charged spherical surface at which the net (excess) local charge (sum of charges arising from counterions and co-ions) has a maximum. Being derived from the simple DH theory, this interpretation is correct only if the electrostatic potential at the Stern boundary is less than the thermal electrostatic potential of the co-ions. This is not always the case.13 The present model (Figure 1) shows that the exclusion boundary distance (b - a) needs to be added to the Debye length. Thus, we define the extent of the range of electrostatic forces between the colloidal sphere and all the ions around it as the distance l defined by
l ) (b - a) + 1/κ
(32)
When fDH ) 1, then x ) ∞ (as shown in eq 10), which implies that mathematically the ionic atmosphere extends to infinity. The extent of the ionic atmosphere x can be calculated from the transcendental equation
G(x) ) 0 ) (1 + κx) exp[-κ(x - b)] (1 + κb)(1 - fDH) (33) by the usual methods. Number of Ions in Ionic Atmosphere. This calculation gives a real and practical estimate of the many-body Coulombic interactions between the colloidal sphere and all the surrounding ions for any desired accuracy of electroneutrality. For instance, we may choose fDH ) 0.995 and then calculate the distance x and then the number of ions between (x - b). The number of ions of type i is in general
∫bx(1 -
Ni ) ni0
)
zieψ3 dV kT
(34)
which can be worked out to give
T1 ) {(ψt+ - ψt-)(1 - λb) - ψt+(1 + κb)}{-(1 + λb) + (1 + λa) exp[+λ(b - a)]} (29) -
∫bxr2F3(r) dr
fDHQDH ) 4π
(28)
where the terms on the right, shown below
+
than the DH theory. Interestingly, the nonlinear terms of the GC theory tend to shorten the range of electrostatic forces, possibly the chief failure of the DLVO theory at close separations. While this expression is a measure of the range of the forces, the actual extent of the ionic atmosphere is best described by a distance x defined by a fraction of the DH charge fDH, chosen close to unity for this purpose.13 In this way the real extent of the ionic atmosphere can be calculated as well as the actual numbers of ions in the ionic atmosphere. The distance x is defined then by
[]
z+ b 3N+ ) (x3 - b3) + 3 {(1 + κx) 4000πν+NAc z+ κ2 exp[-κ(x - b)] - (1 + κb)} (35) for co-ions and
[]
z- b 3N) (x3 - b3) + 3 {(1 + κx) 4000πν- NAc z + κ2 exp[-κ(x - b)] - (1 + κb)} (36) for counterions, with symbols having their usual meanings. The valence terms in square brackets would be more complex if a mixture of electrolytes were considered. In this model, the ratio of the valencies becomes 1 for positive co-ions, as spelled out in eq 35 for comparison with eq 36. Condition for Co-Ion Exclusion. In this paper the solution of the model in the DH regime (Stern potential lower than the termal electrostatic potential of the coions) will not be explicitly given as it is a simple variation of the DH theory. It is important to know, though, the general conditions under which the explicit co-ion exclusion becomes operational. By setting b ) a in the solution of the electrostatic model in Figure 1, a general condition for the critical (highest) surface charge density can be derived above in which the co-ions begin to be spatially excluded to b > a:
(31)
We reach an important conclusion that this model predicts a longer range of electrostatic forces at higher potentialss calculated by self-consistent macroscopic electrostaticss
(
/ ) 0ψt+ κ + σ0+
1 a 1 + κa p R κR
)( )(
)
(37)
This critical charge density depends on ionic strength,
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Langmuir, Vol. 18, No. 21, 2002 7911
charge on the co-ions, LMO parameter p, and radius of the colloidal sphere. In the limit of infinite radius, a simpler expression valid for plane surfaces is obtained as
Table 1. Critical Surface Charge Densities for 1:1 Electrolyte of Molarity C and for Sphere Radii R in the Range of 150-5000 Åa -log(C)/R
(
/ σ0+ ) 0ψt+ κ +
1 p
)
Thus, the onset of co-ion exclusion is favored by decreasing ionic strength and with increasing the value of p (increasing susceptibility of the double layer to dissociate). In the case of spheres, the co-ion exclusion also becomes favored by increasing radius. Another way to calculate the critical surface charge density is to solve the model in Figure 1 for the DH case (b ) a) and then set the potential at r ) a equal to the thermal electrostatic potential of the coions. This approach is represented by
ψ3(a) ) ψt+ )
/ pκ σ0+ R R 1 + pκ 0 a (1 + κa)
(
) ()
7 6 5 4 3 2 1
(38)
(39)
In the above descriptions, the term “critical” was assigned to the surface charge density, although it can be assigned to any of the variables given in eq 39 (for instance, to the ionic strength). Results and Discussion Consideration of Colloidal Variables. In charged colloidal systems there are four independent variables of importance. It is useful to briefly discuss these variables and associated issues before considering specific, calculated results. (1) Radius of the Sphere and Concentration of the Spheres. These are interrelated variables but conceptually clear-cut; there maybe some discrepancies in the particle sizes determined by different experimental techniques. Polydispersity is another effect that may complicate interpretation of experimental results. This model deals only with a single sphere; concentrated interacting double layers will be considered later. (2) The ionic strength of the solution (or of self-ionizing solvent) in which the colloidal spheres are dispersed is characterized by the Debye length 1/κ. This seems to be a precise variable that can be relatively easily controlled. Recently, however, some confounding theoretical suggestions have been made that the colloidal sphere itself should be regarded as contributing to the Debye length27 or that a colloidal sphere behaves as a strong electrolyte.27,28 Experimental data (e.g., as summarized in the LMO law that describes the dissociation of a charged surface as the classical DH interionic effect) do not support the 100% dissociation (strong electrolyte) assumption. It is implicit in the DH concept of the ionic atmopshere that it is the number of free ions (charges) of comparable size that determine the Debye length. The radii of colloidal spheres are typically more than an order of magnitude larger than simple ions and hence are unlikely to contribute to the Debye length by the same thermal diffusional processes in two-phase colloidal systems. Recent experimental data25,26 confirm the correctness of the original (1923) DH concept of the ionic atmosphere from a colloidal point of view. (3) Primary Surface Charge Density. This is a troublesome quantity from both experimental and theoretical points of view; it reflects the chemistry of the charged surface, and various model colloids appear to have their (27) Chan, D. Y. C.; Linse, P.; Petris, S. N. Langmuir 2001, 17, 42024210. (28) Schmitz, K. S. Langmuir 2000, 16, 2115-2123.
150
350
29.79 12.88 9.83 4.43 3.53 1.762 1.571 0.957 1.071 0.818 1.289 1.142 2.546 2.406
750
1000
2500
5000 Rf∞
6.23 2.298 1.068 0.715 0.719 1.084 2.351
4.78 1.835 0.916 0.662 0.697 1.071 2.339
2.176 1.004 0.645 0.568 0.658 1.049 2.317
1.312 0.728 0.555 0.537 0.645 1.041 2.310
0.448 0.452 0.465 0.505 0.632 1.033 2.303
R 0.996 0.987 0.960 0.884 0.706 0.432 0.194
a The surface charge densities are in µC/cm2. The case of highly dissociating spheres with large p ) 40.0 Å.
own peculiarities arising from their chemical compositions and reactivities. However, it is observed experimentally that charged surfaces release ions according to the LMO law in response to increased ionic strength (i.e., the adsorbed ions do not obey Langmuirian or similar adsorption mechanisms). The most reasonable interpretation of this non-Arrhenius type dissociation is the DH lowering of ionic activity coefficients as discussed in the theoretical section above and previously. This interpretation does not involve the postulation of new phenomena ad hoc and is supported by the known effects of ionic atmospheres on classical thermodynamic quantities, such as dissociation constants and solubility products. Often experimental results are discussed in terms of effective charge, sometimes used as an adjustable parameter rather than a measured quantity. According to the present model (Figure 1), we can see that the concept of effective charge may be somewhat intricate. We have a choice to define an effective charge as the sum of the surface charge and the Stern charge as
Qeff ) Q0+ + Qa
(40)
Qeff ) Q0+ + Qa + Qab
(41)
or
where we have added the diffuse Stern counterion charge in the co-ion exclusion region. The most pragmatic way to define the effective charge will depend on the chemistry and characterization techniques used for its determination in each specific case; in this paper eq 40 will be used. Clearly, there remains a pressing need to search for model spherical colloids and for associated techniques to characterize accurately and extensively their surface charge densities and sizes. The calculated results below are presented as functions of broad ranges of radii and ionic strengths of a 1:1 electrolyte. First, we consider the critical boundaries between the applicability of the simple DH model and the DEDL model, when the co-ion exclusions need to be considered explicitly. Then a specific behavior of various quantities derived theoretically will be considered for two typical cases of colloidal spheres. The first case is a sphere with a high surface charge density of 12.0 µC/cm2 and a low LMO parameter p ) 4.00 Å, values consistent with montmorillonite swelling clays. The second case represents a charged sphere of lower charged density of 3.0 µC/cm2 with a high LMO parameter p ) 40.0 Å, representing a slightly charged sphere with more dissociable double layers. Such a sphere may be more typical of polymer latexes. Also the extreme condition of a very low surface charge density will be considered. Quantification of Critical Surface Charge Density. In Tables 1 and 2, the critical charge densities are calculated as a function of ionic strength of a 1:1 electrolyte
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Table 2. Critical Surface Charge Densities for 1:1 Electrolyte of Molarity C and for Sphere Radii R in the Range of 150-5000 Åa -log (C)/R 7 6 5 4 3 2 1
150
350
750
1000
2500
5000
296.8 128.4 62.02 47.58 21.68 13.07 97.12 43.73 22.71 18.14 9.927 7.197 33.99 16.99 10.29 8.835 6.221 5.351 14.06 8.566 6.401 5.930 5.085 4.804 7.880 6.012 5.286 5.127 4.840 4.745 6.300 5.581 5.298 5.236 5.125 5.088 6.990 6.605 6.453 6.420 6.360 6.341
Rf∞
R
4.446 4.470 4.483 4.523 4.650 5.051 6.321
1.000 0.999 0.996 0.987 0.960 0.884 0.706
a The surface charge densities are in µC/cm2. The case of highly dissociating spheres with large p ) 4.0 Å.
at different spherical radii. First, looking at the ionic strength dependence at an infinitely large sphere (going up in column R f ∞), we see that the critical surface charge decreases with the decreasing electrolyte concentration according to eq 38. This effect is stronger for the highly dissociating sphere in Table 1 (cf. the data in Table 2 for a slightly dissociating sphere). This trend is due to the double layer decompression effect at a planar surface that is modulated by the double layer dissociation. Second, looking at the same trend at smaller radii, we see that initially the critical surface charge decreases in the same way (double layer decompression) but when the spherical radii become comparable to the Debye length, then the critical surface charge densities begin to increase again according to eq 39. Thus, when the ions in ionic atmosphere see the colloidal particle as a sphere rather than as a plane, the onset of co-ion exclusion is shifted to higher charge densities; this is an electrostatic effect as the DH potential decays faster from a spherical surface than from a planar surface and can thereby accommodate higher surface charge. In general, when the surface charge increases (or the LMO parameter p), the adequacy of the DH approximation becomes more and more limited, and the co-ion exclusion case (shown in Figure 1) becomes applicable over a wider range of electrolyte concentrations and spherical radii. However, the transition from the DH case to the modified DH case with the co-ion exclusion is a smooth one, and the choice of “critical” may be too strong a term to describe this phenomenon. In Tables 1 and 2, the DH region (no co-ion exclusion) is located toward low electrolyte concentrations and small radiisa direction toward dilute, one-phase electrolyte solutions (as if going left from R ≈ 150 Å in Tables 1 and 2). Thus, on going from a two-phase to a single-phase case, we cross a delicate conceptual boundary of the size of the charged sphere when the LMO law ceases to be applicable in a single-phase system (e.g., micellar and ionic solutions). The LMO law is then colloidal by its very nature. The issue of when a colloidal system may be described as a two-phase or one-phase system needs more theoretical and experimental work; it is evidently related to various nucleation and phase separation phenomena. However, once a stable two-phase colloidal system is born, the LMO law appears to be applicable. As mentioned before, the degree of association is likely to depend on the surface charge density, although there appears no experimental data from which such dependence might be determined. Calculations of the Range of Electrostatic Forces. This important quantity is defined as the distance l by eq 31. In Tables 3 and 4, the co-ion exclusion shell thickness (b - a) is calculated at different radii and ionic strengths, also showing the Debye length, which together determine the range of electrostatic forces.
Table 3. Case of Highly Dissociating Sphere with Charge Density of 3.0 µC/cm2 and p ) 40.0 Åa -log(C)/R
150
350
750
1000
2500
5000
1/κ
R
7 6 5 4 3 2 1
0 0 0 77 71 24 1
0 0 159 223 116 32 2
0 176 528 372 147 35 2
0 427 705 424 156 36 2
706 1829 1290 553 173 38 3
3289 3245 1662 614 180 38 3
9620 3042 962 304 96 30 10
0.996 0.987 0.960 0.884 0.706 0.432 0.194
a Thickness of the co-ion exclusion shell for 1:1 electrolyte of molarity C and for sphere radii R in 150-5000 Å range. The length units are in Å (10-10 m).
Table 4. Case of Slightly Dissociating Sphere with Charge Density of 12.0 µC/cm2 and p ) 4.0 Åa -log(C)/R
150
350
750
1000
2500
5000
1/κ
R
7 6 5 4 3 2 1
0 0 0 0 27 18 5
0 0 0 58 58 24 6
0 0 67 150 80 28 7
0 0 159 183 86 28 7
0 270 503 267 98 30 7
0 1047 736 306 103 30 7
9620 3042 962 304 96 30 10
1.000 0.999 0.996 0.987 0.960 0.884 0.706
a Thickness of the co-ion exclusion shell for 1:1 electrolyte of molarity C and for sphere radii R in 150-5000 Å range. The length units are in Å (10-10 m).
In the region of very small spheres and extremely dilute solutions the simple DH model applies and the co-ion exclusion region is zero, and the range of electrostatic forces is given by the Debye length only. It is interesting to look at the trend of the co-ion exclusion shell with increasing radii at constant ionic strength. At all ionic strengths this co-ion exclusion region becomes larger, and in fact it is the largest at planar surfaces, which accounts for the strong repulsions in planar montmorillonite clays. The effect arises from the slower decay of electrostatic potential at planar surfaces as compared to curved surfaces. Looking at the trend of the co-ion exclusion with increasing ionic strength, we see that the co-ion exclusion has a maximum at some intermediate ionic strength; the maximum being dependent on spherical radii. In the case of a planar surface, there is no maximum, and the double layer compression effect is dominant. In Table 4 we see that for the less dissociating sphere, the co-ion exclusion regions are smaller, as expected. In general, the model predicts a longer range of electrostatic forces because of the appearance of the co-ion exclusion region under certain conditions of the chemistry of the surface (the LMO parameter p), ionic strengths, and curvature of radii. In fact, if we chose a larger value of p than that in Tables 3 and 4 to reflect even higher dissociation of the colloidal sphere, then the co-ion exclusion region will become operational in the whole experimental space of ionic strengths and sphere curvatures. By adding the Debye length 1/κ to the co-ion exclusion distance, the range of electrostatic forces is significantly extended. Two additional comments can be made regarding the co-ion exclusion shell. First, we notice that the co-ion exclusion is present even at a 0.1 M electrolyte solution (concentration comparable to the physiological ionic strength) and that it is of the order of a few Ångstroms (i.e., of molecular dimensions). It seems plausible to suggest that the co-ion exclusion region may play a significant part in enzymatic catalysis of biochemical reactions. For example, the charges on a protein surface may have evolved in a way to keep away charged molecules that would interfere with a particular biochemical reaction
Dissociation of Colloidal Spheres
Langmuir, Vol. 18, No. 21, 2002 7913
Table 5. Case of Highly Dissociating Sphere with Charge Density of 3.0 µC/cm2and p ) 40.0 Åa -log(C)/R
150
350
750
1000
2500
5000
7 6 5 4 3 2 1
3 8 22 50 76 64 30
6 17 44 85 106 75 32
12 34 75 120 126 80 33
16 42 88 133 131 82 33
35 80 135 163 142 84 34
60 116 165 177 146 85 34
a Stern potentials for 1:1 electrolyte of molarity C and for sphere radii R in 150-5000 Å range. The potentials are in mV.
Table 6. Case of Slightly Dissociating Sphere with Charge Density of 12.0 µC/cm2 and p ) 4.0 Åa -log(C)/R
150
350
750
1000
2500
5000
7 6 5 4 3 2 1
1 3 9 22 40 51 46
2 7 18 36 54 60 50
5 14 30 50 63 64 51
6 17 35 55 66 65 51
14 31 52 66 71 67 52
24 44 62 71 73 67 52
a Stern potentials for 1:1 electrolyte of molarity C and for sphere radii R in 150-5000 Å range. The potentials are in mV.
within the co-ion exclusion region. Another role of the co-ion exclusion might be the provision of space with a large electrostatic potential gradient (protein surface charge to the co-ion exclusion boundary), which, being an electrostatic force, acts on the chemical bonds of molecules within the co-ion exclusion region. Clearly (as indicated by the data in Tables 3 and 4) the size or curvature of the protein surface would be one factor to control the co-ion exclusion boundary distance but still be within molecular dimensions. In general, it appears that the co-ion exclusion region is a factor in spatial separation of various biochemical reactions in the packed cytoplasm. Second, when the co-ion exclusion becomes operative, it probably plays a part in the stability of colloidal crystals. A Madelung-like structure of colloidal crystals was suggested before6 but more detailed solutions need to be worked out, perhaps using the lattice theory based on the Wigner-Seitz cell approach.22 It seems possible that the co-ions are in fact expelled to localized regions between the colloidal spheres that are of the same net charge sign as that of the colloidal spheres, thereby imparting a Madelung-like structure to colloidal crystals. In this case the Wigner-Seitz cell would not be electroneutral and spherical. In any case such a structure would impart additional electrostatic attractive forces needed to understand the coexistence of colloidal crystals with their disordered colloidal phase.29,30 Preliminary results suggest that the Stern potentials under these conditions can reach values of hundreds of milivolts, indicating not only an extended range of electrostatic forces but also a significant increase of the strength of these forces. However, the model for an array of spheres needs to be solved by more sophisticated mathematical methods. Calculation of the Stern Potential. The Stern potentials are calculated in Tables 5 and 6 for the case of a highly dissociating sphere and a less strongly dissociating colloidal sphere. As expected, a more dissociating sphere generally gives higher Stern potentials in Table 5 than a colloidal sphere with less tendency to dissociate in Table 6. The Stern potentials always increase with an (29) Okubo, T.; Yoshimi, H.; Shimizu, T.; Ottewill, R. H. Colloid Polym. Sci. 2000, 278, 469-474. (30) Ise, N. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 841-848.
Figure 2. Ionic strength dependence (1:1 electrolyte) of a dissociating negatively charged sphere according to the DEDLLMO formulation as described in the text. Increasing ionic strength increases the effective charge on the sphere Qeff, as the Stern charge Qa decreases and the DH charge increases. The counterion Qab charge in the co-ion exclusion region (b a) has a maximum.
increasing radius of the colloidal sphere, again an expected electrostatic effect when the potential decays faster from a small sphere than from a very large sphere. At higher ionic strengths the Stern potentials are predicted to be essentially independent of the curvature of the radii, again showing the expected effect of double layer compression. The Stern potentials show generally a maximum in their ionic strength dependence. Maxima in the ζ potential are often observed in the case of polymer latexes.31 While the identification of the ζ potential with the Stern potential is not necessarily straightforward (and the use of the simple Smoluchowsky proportionality between electrophoretic mobility and ζ potential may be questionable), it is nevertheless satisfying that a theory, originally devised for planar interactions, does in fact predict these maxima without any additional assumptions. Remarkably, for planar geometry the DEDL theory does not predict maxima in the Stern potential, in agreement with electrophoretic mobilities of flat montmorillonites. Electrophoretic mobility data for a broad range of spherical radii, ionic strengths, and charge densities involving different counterions do not appear to be available to test the theoretical predictions that are given in Tables 5 and 6 as illustrative examples. Illustrative Calculations of Net Charge Distributions. In Figure 2 the net amounts of charge in proton units in different regions of the double layer are calculated for a sphere with R ) 750 Å, a ) 753 Å, p ) 40 Å, and σ0 ) -3.0 µC/cm2 as a function of ionic strength of a 1:1 electrolyte with the distance of closest approach of 3.0 Å. These total charges were calculated from eqs 8-10. In addition, the effective charge on the sphere is calculated from eq 40. With increasing ionic strength, the charge on the sphere dissociates according to the LMO law, with the counterion being released first into the co-ion exclusion region and then into the DH region as the co-ion exclusion region begins to disappear at high ionic strength. To get some sense of the amounts of charges, we may first note that the total charge on the sphere is -13 235 protons. At 0.00001 M there are -12 707 electrons in the Stern layer, -110 electrons in the co-ion exclusion region of thickness (b - a) ) 528 Å, and -419 electrons in the DH region (31) Elimelich, M.; O’Melia, C. R. Colloids Surf. 1990, 44, 165.
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Table 7. Compensatory Effect at 0.001 M Electrolyte for a Sphere with R ) 5000 Åa -log(C) 7 ψ(a) ψ(a) Qeff Qeff (b - a) (b - a)
6
5
4
3
2
1
10 28 61 98 101* 63 26 7 20 46 78 101* 105 89 46 143 438 1271 3320* 6110 8596 33 104 329 1034 3320* 9505 25170 0 50 419 317 125* 27 0.3 0 0 264 259 125* 44 13
p (LMO)
σ0
41.5 3.0 41.5 3.0 3.0 41.5
2.40 24.00 2.40 24.00 24.00 2.40
a Considering the compensation of 10-fold increase of charge density (from 2.4 to 24 µC/cm2) by decreasing the p parameter from 41.5 to 3.0 Å. The compensation is shown for the Stern potential at 101 mV, effective charge density at 3200 electrons, and co-ion exclusion distance of 125 Å.
(being the net sum of counterions and co-ions). The molarity of the counterions in the exclusion region is about 2.6× higher than in the bulk. The effective charge is then only 528 electrons. At 0.01 M at a greater degree of dissociation there are -5718 electrons in the Stern layer, -4547 electrons in the co-ion exclusion region (diffuse Stern counterions), and -2971 electrons in the DH region. The molarity of the counterions in the co-ion exclusion region is about 2.9× higher than in the bulk. The effective charge has now increased to 7517 electrons. Similar calculations for other cases have shown that these computed charges and concentrations are always very reasonable. The previous calculation can be extended to determine how many ions comprise the DH region and how far this region reasonably extends. We shall consider the example in the previous paragraph at 0.00001 M as an illustrative example. As above (except for the units), there are 12 707 Stern adsorbed ions and 110 Stern diffuse counterions in the co-ion exclusion region. We now fix the extent of the DH region by considering a 0.995 fraction of the total DH charge according eq 32, which gives the distance x (the extent of ionic atmosphere to 99.5% accuracy) as 7675 Å. This can be compared with the range of the ionic atmosphere given by the co-ion exclusion distance of 1281 Å plus the Debye length of 962 Å giving the total of 2243 Å. Thus the ionic atmosphere, considered as essentially an electroneutral lattice cell, extends considerably farther when nearly all ions are taken into account. The 0.5% accuracy represents about 2-3 ions that are unaccounted for. The number of co-ions in the DH region (x - b) is 11 144, and the number of counterions is 11 560, calculated according to eqs 35 and 36. The net charge is then 416, which can be compared to the 419 net charge in the DH region when it extends to infinity (the 2-3 ions being unaccounted for by 0.5% charge cutoff). Many similar calculations have been performed for a large number of cases, and in all instances reasonable and physically meaningful quantities have been calculated. These results are necessary checks of reasonableness of the model predictions, preparatory to calculations of distorted ionic distributions of two approaching spheres. Effective Charge of the Surface. Though the surface charge densities in the present model are well-defined, it is intuitively felt that there will be a compensatory effect between the increasing surface charge density and decreasing value of the LMO parameter p, at least from an electrostatic point of view. The Maxwellian electrostatic laws suggest that when the effective charge density is the same, the ionic atmosphere will be the same also, regardless of the chemistry up to the Stern layer. In Table 7 we compare a relatively low primary spherical charge that dissociates easily (p ) 41.5 Å) with a highly charged sphere that tends to dissociate far less (p ) 3.0 Å). It is seen that the Stern potential, the effective charge density,
and the co-ion exclusion distance are the same for these two different double layers at 0.001 M of electrolyte (data marked by asterisks). In fact, all electrostatic properties must be the same outside the Stern layer, while obviously the chemistry of the Stern layer is different at this ionic strength. It is satisfying to see that at other ionic strengths there is no compensation that is just limited to the singular condition or point. In other words, the specific chemistry at the surface will be propagated far away into the ionic atmosphere in general, although under a singular condition (or small range of variables) different surface chemistries may look the same. Extreme Condition of Very Low Surface Charge Density. It is sometimes assumed that in certain noncharged systems there are strong repulsive forces that are not of coulombic origin. However, it is not always clear what precautions have been taken to ascertain that no ionic impurities are present in the surface; additionally, when deionized water is used as the solvent, it is not clear how well the ionic strength has been controlled. In this example it is shown that 1% ionic impurity may be a cause of significant electrostatic effects, particularly at low ionic strengths. For this purpose we have chosen a very low surface charge density of 0.10 µC/cm2 (1% of a reasonably large 10 µC/cm2) and a very large LMO paremeter p of 1000 Å for a sphere with a 5000 Å radius. For ionic strengths of 10-7, 10-6, 10-5, and 10-4 M of a 1:1 electrolyte, the Stern potential is calculated as 45, 71, 64, and 32 mV, respectively, always with an extensive co-ion exclusion region of 2060, 2090, 761, and 62 Å. At higher ionic strengths the Stern potential drops quickly toward zero, being about 1 mV at 0.10 M. This theory suggests that detailed characterization of surface charge densities is necessary when dealing with nominally uncharged surfaces, as ionic impurities (sometimes surface active) need to be controlled. The large LMO parameter chosen for this example implies as yet an unknown dependence on the primary charge density. Conclusions The DEDL theoretical draft of electrostatic interactions in two-phase colloidal systems (lyophobic colloids) does away with the known inconsistencies of the DH and GCDLVO theories. This theoretical approach provides a consistent framework to formulate colloidal problems within the limited scope of the DH and GC-DLVO physical model. Four important results have emerged so far: (a) the derivation of fundamental limiting laws of planar repulsions in the limit of surface contact and their transition to the DH limiting laws at infinite separation; (b) the prediction of stronger and longer-range electrostatic forces, quantitatively and consistently accounting for repulsions in colloidal montmorillonites; (c) the discovery of the classical DH interionic effect being operative in increasing the double layer dissociation by lowering the activities of diffuse ions via the LMO law; and (d) the prediction of maxima of Stern potentials in spherical symmetry without any further assumptions and the prediction of no maxima for platelike montmorillonites. In light of these new insights, the concepts of hydration forces and other non-DLVO forces may need to be revised as well as the origin of maxima of electrophoretic mobilities (Stern potentials) for spherical colloids. The DEDL theoretical draft satisfies macroscopic Maxwellian electrostatics under all conditions and suggests that distinct co-ion exclusion boundaries exist as a physical fact. The calculated numbers of diffuse ions and their
Dissociation of Colloidal Spheres
distributions are always sensible and reasonable. The coion exclusion boundaries are bound to play a significant role in many colloidal phenomena, including those at biological surfaces (spatial separation of biochemical reactions in packed cytoplasm) and in colloidal crystallization phenomena (Madelung-like structure of colloidal crystalline phases). It is suggested that the search for well-behaved model colloids be intensified in order to test the DEDL theoretical draft over broad ranges of spherical radii, surface charge densities, and ionic strengths. Future
Langmuir, Vol. 18, No. 21, 2002 7915
work will follow in the footsteps of Overbeek and his school19 to provide new expressions for classical DLVOlike interactions of two spheres and of a sphere and an infinite plate as well as new results for cylindrical symmetries. Acknowledgment. An anonymous reviewer is thanked for his/her comments. LA025937N
