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Maxwellian Double Layer Forces: from Infinity to Contact Jan J. Spitzer IPM Emulsion Polymer Research, 6643 Lyndonville Drive, Charlotte, North Carolina 28277 Received January 7, 2003. In Final Form: May 3, 2003 A linear Maxwellian model of dissociative electrical double layers (DEDL) is formalized and extended to cover separation distances from infinity to contact in planar geometry. The entire interaction range is covered by three electrostatic models: the low potential (LP) model, the co-ion exclusion (CX) model, and the high potential (HP) model. The three models are contiguous variations of the linear Poisson-Boltzmann equation, which is consistent under all physicochemical conditions, including high potentials. Infinitely separated (single) double layers are classified into the LP double layers and the CX double layers. The LP double layers consist of the Debye-Hu¨ckel (DH) ionic cloud only. The CX double layers consist of the DH ionic cloud, which is separated from the counterion ionic cloud by a co-ion exclusion surface. The single LP and CX double layers become identical under a singular physicochemical condition. The transitions between the LP, CX, and HP double layers are discussed in detail as a function of separation. Examples of repulsive pressures are calculated from infinity to contact. The DH exponential decay of repulsive forces is recovered at large separations; at closer separations, the CX model predicts positive deviations from the DH limiting slope, which are similar to the predictions of the nonlinear Poisson-Boltzmann equation. In the contact limit, the HP interaction model predicts forces that can be much larger than those predicted by the nonlinear Poisson-Boltzmann equation. These limiting contact forces decrease according to the inverse square of the separation. The previously discovered Lubetkin-Middleton-Ottewill law is incorporated into the DEDL theory to predict new ionic strength effects similar to the classical DH effects on bulk ionic equilibria.
Introduction The Poisson-Boltzmann equation (PBE) has provided essential insights into interactions in electrolyte solutions1-4 and electrical double layers5-8 for over 90 years. It explains ionic strength effects in many natural and industrial systems.9-24 In its linear form, the LPBE is the (1) Debye, P.; Hu¨ckel, E. Phys. Z. 1923, 24, 185. (2) Debye, P.; Pauling, L. J. Am. Chem. Soc. 1925, 47, 2129-2134. (3) Pauling, L. Personal communication, January 20, 1981. (4) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworth & Co. Ltd.: London, 1959; Chapters 4 and 9. (5) Gouy, G. J. Phys. 1910, 9, 457. (6) Chapman, D. L. Philos. Mag. 1913, 7, 129. (7) Verwey, E. J.; Overbeek, W. J. Th. The Theory of Stability of Lyophobic Colloids; Elsevier: Amsterdam, The Netherlands, 1948; p 23. (8) Overbeek, J. Th. G. In Colloid Science; Kruyt, H. R., Ed.; Elsevier: Amsterdam, The Netherlands, 1952; Chapter 4, p 129. (9) van Olphen, H. Clay Colloid Chemistry; Interscience Publishers: New York, 1963. (10) Israelachvili, J. N. Intermolecular and Surfaces Forces; Academic Press: London, 1985. (11) Stumm, W.; Morgan, J. J. Aquatic Chemistry; John Wiley & Sons: New York, 1981; Chapter 10. (12) Honig, B.; Nicholls, A. Science 1996, 268, 1144. (13) Davis, M.; McCammon, J. A. Chem. Rev. 1990, 90, 509. (14) Ottewill, R. H. In Emulsion Polymerization and Emulsion Polymers; Lovell, P. A., El-Aasser, M. S., Eds.; Wiley & Sons: Chichester, U.K., 1997; Chapter 3. (15) Leckband, D.; Israelachvili, J. Q. Rev. Biophys. 2001, 34, 105. (16) Fitch, R. M. Polymer ColloidssA Comprehensive Introduction; Academic Press: San Diego, 1997. (17) Hsu, J.-P. Interfacial Forces and Fields. Theory and Applications; Marcel Dekker: New York, 1999. (18) Russel, W. B. The Dynamics of Colloidal Systems; The University of Wisconsin Press: Madison, WI, 1987. (19) Lucassen-Reynders, E. H. Anionic Surfactants; Marcel Dekker: New York, 1981. (20) Evans, D. F.; Wennerstro¨m, H. The Colloidal Domain; WileyVCH: New York, 1999. (21) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: Oxford, 2001; Chapters 7 and 12. (22) Hiemenz, P. C. Principles of Colloid and Surface Chemistry; Marcel Dekker: New York, 1977, Chapter 9. (23) Bockris, J. O.; Reddy, A. K. N. Modern Electrochemistry, 2nd ed.; Plenum Press: New York, 1998; Vol. 1, Chapter 3.
basis of the Debye-Hu¨ckel (DH) theory of ionic activities4 and of other physicochemical properties. In its nonlinear form, the NLPBE is the principal component of the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory7,8 of colloidal stability and is used in other colloidal applications. The literature on both the LPBE and NLPBE continues to grow in many diverse applications, including the complex modeling of biological surfaces.12,13,15 The LPBE is regarded as a simplifying approximation of a more “accurate” NLPBE, which has been traditionally used in colloidal applications.7 However, such “accuracy” is questionable because the proportionality between charges and potentials is not preserved, as is required by Maxwellian electrostatics. This unresolved electrostatic issue has been acknowledged in some expositions of the double layer theory, including the original promulgation by Verwey and Overbeek.7,13,22,24 In electrolyte theory,4,23 the NLPBE has not been used on account of its inconsistency.4 The emerging model of dissociative electrical double layers (DEDL)25-31 is based on linear ionic distributions that guarantee electrostatically consistent results. The DEDL model was introduced25 more than 10 years ago with a focus on a new limiting law27,28,30 for close-range Coulombic repulsions. The purpose of this article is to formalize the DEDL model as a consistent basis for a new theory of ionic atmospheres. Because the DEDL theory requires strict adherence to Maxwellian electrostatics, the assumptions in the PBE are revisited first. (24) Bockris, J. O.; Reddy, A. K. N. Modern Electrochemistry; Plenum Press: New York, 1970; Vol. 2, p 725 and the footnote. (25) Spitzer, J. J. Nature 1984, 310, 396. (26) Spitzer, J. J. Langmuir 1989, 5, 199. (27) Spitzer, J. J. Colloids Surf. 1991, 60, 71. (28) Spitzer, J. J. Langmuir 1992, 8, 1659. (29) Spitzer, J. J. Langmuir 1992, 8, 1663-1665. (30) Spitzer, J. J. Colloid Polym. Sci. 1992, 270, 1147-58. (31) Spitzer, J. J. Langmuir 2002, 18, 7906-7925.
10.1021/la034028a CCC: $25.00 © 2003 American Chemical Society Published on Web 07/03/2003
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Poisson-Boltzmann Equation The development of the ionic atmosphere theory, based on the PBE, has progressed concurrently in colloidal science and electrolyte theory. These two historical strands are reviewed first. The following discussion reiterates the non-Maxwellian nature of the NLPBE and the need for a more consistent theory. HistoricalsDouble Layer Theory. In 1910 and 1913, Gouy5 and Chapman6 (GC) used the model of a 100% diffuse ionic atmosphere as an alternative to the 100% associated Helmholtz double layer model to explain electrocapillary phenomena. They combined the Poisson equation
∇2ψ ) -F/0
(1)
with the Boltzmann theorem
[
]
z+eψ + kT z-eψ (2) z-eν-n0 exp kT
F ) z+eν+n+ + z-eν-n- ) z+eν+n0 exp -
[
]
to obtain the NLPBE
[
-0∇2ψ ) z+eν+n0 exp -
]
z+eψ + kT
[
z-eν-n0 exp -
]
z-eψ (3) kT
The Poisson equation, eq 1, relates the second derivative (Laplacian in general) of the average electrostatic potential, ψ, to F, the volume charge density, in a linear way (0 is the permittivity of vacuum, is dielectric constant). The volume charge density F is determined additively from local ionic concentrations around a charged surface z+ zBν- of bulk concen(immersed in strong electrolyte Aν+ tration n0). The local ionic concentrations are, in turn, determined exponentially from the value of the average electrostatic potential used in the Boltzmann theorem, eq 2. The GC theory contributed initially to the rationalization of double layer capacities at mercury/solution interfaces. However, the unphysically large local concentrations of counterions led to the concept of a Stern layer32 of “adsorbed” ions, which defines the “inner double layer”. The inner (Helmholtz) double layer turned out to be very important, and, hence, the NLPBE cannot be critically tested24,33 by electrocapillary experiments. The question of “to linearize or not to linearize” has remained alive. The following is a quote from a well-known electrochemistry textbook (1970):24 “Attempts to achieve so-called “rigorous” solutions by not linearizing the PoissonBoltzmann equation led to certain inconsistencies. Nevertheless, it has been customary, in diffuse-double layer treatments, based on PBE, to proceed with the solution of the unlinearized differential equation...” And in the footnote, it is added: “What is customary need not necessarily be right. A more consistent theory of the double layer is clearly required.” In 1948, the GC NLPBE was used in the DLVO theory of colloidal stability7 to calculate repulsions between charged surfaces. Verwey and Overbeek were also fully aware of the Maxwellian inconsistency of the GC theory: (32) Stern, O. Z. Elektrochem. 1924, 30, 508. (33) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989; p 121.
“This method of determining n_(x,y,z) and n+(x,y,z) from ψ(x,y,z) is correct only for such small values of the electric potential ψ(x,y,z) that it is permissible to develop the exponential form according to MacLaurin series and to break off after the linear term. For in that case there is a linear relation between ψ and F. For larger values of the electric potential the latter is no longer true, and the determination of the mean value of n(x,y,z) from ψ(x,y,z), as indicated is then, strictly speaking, not allowed.” Verwey and Overbeek then explained their arcane reason to proceed with the NLPBE: to make the effect of counterion valency unsymmetrical by retaining the Boltzmann exponentials (to explain ion-valency effects on coagulation). Four years later,8 the warning was dropped, and since then a notion has been growing that the NLPBE is more exact than the LPBE.20,21 Since 1978, new measurements on charged clay surfaces (mica and montmorillonites)34-37 revealed unexpectedly large forces at close separations. (These results were determined in three independent laboratories using different techniques). The unexpected forces cannot be accounted for by the DLVO theory, which predicts universal, van der Waals adhesion at close separations.10 Such catastrophic failure of the DLVO theory is remarkable. Current explanations38-40 suggest that the Maxwellian physical model may be inadequate at close separations and that new effects need to be invoked (for example, polarization effects). While such effects may well be operative, they are obscured by the spurious (nonMaxwellian) electrostatics of the NLPBE. HistoricalsElectrolyte Theory. In 1923, Debye and Hu¨ckel1 (DH) explained the nonidealities of “strong” electrolytes as electrostatic attractions between ions and their oppositely charged ionic atmospheres. They used the same Maxwellian model (ions as elemental charges and the solvent as a linear dielectric) as Gouy and Chapman. The main difference is the boundary surface definition: in the GC (DLVO) model the boundary is a “stationary” macroscopic surface; in the DH theory the boundary is the “moving” surface4 of the central ion. To obtain analytical results, Debye and Hu¨ckel linearized Boltzmann’s exponentials as
F ) z+eν+n0 + z-eν-n0 -
n0e2 2 (z ν + z-2ν-)ψ kT + +
(4)
where the first two terms add up to 0 (electrolyte electroneutrality). The linear PBE equation is then
∇2ψ ) -
F 0
)
n0e2 (z 2ν + z-2ν-)ψ ) κ2ψ 0kT + +
(5)
Equation 5 defines the famous DH constant κ. The success of the linear DH theory was immediate and profound. For example, the empirically known dependence of the properties of dilute electrolyte solutions on the square root of concentration (ionic strength) became understood as the overall attraction between an ion and its ionic atmosphere. (34) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc. Faraday Trans. 1 1978, 74, 975-1001. (35) Lubetkin, S. D.; Middleton, S. R.; Ottewill, R. H. Philos. Trans. R. Soc. London, Ser. A 1984, 311, 353. (36) Viani, B. E.; Low, P. F.; Roth, C. B. J. Colloid Interface Sci. 1983, 96, 229. (37) Zhang, F.; Low, P. F.; Roth, C. B. J. Colloid Interface Sci. 1995, 173, 34. (38) Israelachvili, J.; Wennerstro¨m, H. Nature 1996, 379, 219. (39) Bostro¨m, M.; Williams, D. R. M.; Ninham, B. W. Phys. Rev. Lett. 2001, 87, 168103. (40) Manciu, M.; Ruckenstein, E. Langmuir 2001, 17, 7061-7070.
Maxwellian Double Layer Forces
The Debye screening length 1/κ became important in colloidal science, leading to the concepts of double layer compression and to the definition of the screened Coulomb potential. We note that the NLPBE does not lead to these fundamental concepts unless it is linearized. The DH theory was subjected to many theoretical analyses to understand whether the linearization is responsible for the deviations from the DH limiting laws. The early analyses were reviewed by Onsager41 in 1933, with the following conclusions. (a) The NLPBE is inconsistent because different but equivalent charging processes do not give the same thermodynamic result (inconsistent electrostatics). (b) This inconsistency arises from the replacement of the statistical potential of the mean force with ionic electrostatic energies, the so-called central approximation. (c) The linearized theory does not suffer from this inconsistency. In 1959, Robinson and Stokes4 reanalyzed the effects of linearization in their classical book on electrolytes. They judiciously observed that the unphysical (negative) coion concentrations grow smaller with increasing bulk concentration, while the deviations from the limiting laws grow larger. The linearization clearly is not responsible for experimental deviations from the limiting laws. Given that, they concluded that the linear distribution laws are to be preferred to ensure consistency with the Maxwellian model (proportionality of charges and potentials). The failure of the linearization in the DH theory can be quantified as negative concentrations of co-ions close to a charged surface42 and remains an unresolved issue (cf. Bjerrum’s association theory).4 We conclude that the PBE theory of ionic atmospheres is intricate, and the issue of linearization remains unresolved. The LPBE led to the consistent DH limiting laws, which provide the essential insight into the nature of ionic strength both in electrolyte theory and in colloidal science. The NLPBE was found to be inconsistent in the electrolyte theory. In modern colloidal science, the DH limiting laws were experimentally confirmed by Israelachvili and his school10 for repulsive forces between mica cylinders at large separations. At closer separations, the NLPBE appears to account for the initial deviations from the DH repulsions but at still closer separations, the NLPBE fails. The preference for the NLPBE in the DLVO theory7 is rooted in the desire to treat counterions and co-ions unsymmetrically. Verwey and Overbeek, however, recognized that the resulting nonlinearity of potentials and charges is “strictly speaking, not allowed”. Their subsequent silence on this issue is enigmatic. Electrostatics of a Maxwellian Model. As a preamble to a new consistent model of ionic atmospheres, the non-Maxwellian nature of the NLPBE is demonstrated in the following. In setting up the ionic atmosphere model, two assumptions are made to exploit the Poisson equation. (a) Charged surfaces attract counterions and repel co-ions according to Coulomb’s law. (b) The thermal motion (energy) allows the average ionic distributions43 to be represented by continuous volume charge densities in a linear dielectric. These charge densities give rise to continuous electrostatic potentials that obey Poisson’s equation. The volume charge density is given by the average charge dq in a volume element dV and represents the excess charge arising (41) Onsager, L. Chem. Rev. 1933, 13, 73. (42) Spitzer, J. J. J. Chem. Soc., Faraday Trans. 1 1978, 74, 24182421. (43) Hunter, R. J. Introduction to Modern Colloidal Science; Oxford University Press: Oxford, 1993; p 205.
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Figure 1. Electrostatic model of a charged surface and volume. The contribution dψ arising from the volume and surface charges dq to the total potential ψ at any point is given additively by Coulomb’s law, eq 7. If the charge dq doubles, then its contribution to the total potential at r also doubles (proportionality of charges and potentials).
additively from counterions and co-ions as
dq ) dq+ + dq- ) (F+ + F-) dV ) F dV
(6)
The generalized electrostatic model is shown in Figure 1 (the charged surface need not be specified in detail). The main point is this: for fixed physicochemical conditions (e.g., ionic charges, their concentrations, dielectric constant, surface charge densities, temperature, etc.), the average electrostatic properties are also fixed. The mathematical solution, therefore, gives the “equilibrated” or time-averaged,43 quasi-static continuous distributions, which ought to obey Maxwellian electrostatics. One important and general Maxwellian result gives the desired mathematical solution by the integration of Coulomb’s law, compare Figure 1.
ψ(r) )
F
∫∫∫10 rv dV + ∫∫10 rσs dS
(7)
The first integral is a solution of Poisson’s equation, eq 1, and the second integral is a solution of Laplace’s equation (no ionic atmosphere), eq 8.
∇2ψ ) 0
(8)
The linearity of the differential Poisson equation, eq 1, and of the integral eq 7 guarantees that the chargepotential linearity is preserved in the algebraic solution also. This linearity is not satisfied in the NLPBE. Further, the NLPBE solution cannot be constructed by the superposition of solutions of Poisson’s and Laplace’s equations. Therefore, the NLPBE solution does not satisfy the superposition principle. Solutions of the LPBE have long been known to satisfy the electrostatic proportionality and additivity principles, which, in turn, guarantee the consistent calculations of electrostatic energies by different charging paths.4,41 Thus, the LBPE has an appealing and consistent character, which transcends the easy interpretation that the LPBE is a low-level approximation of the more “accurate” NLPBE. A simple, physically consistent theory based on the LPBE is, therefore, of interest. The new DEDL theory postulates diffuse co-ion exclusion boundaries and related transitions, which accurately preserve the Maxwellian consistency without predicting unphysical ionic distributions. These new boundaries define three contiguous variations of the LPBE with precisely defined transitions. The three variations of the LPBE cover low potentials (the LP double layers), medium potentials with progressive co-ion exclusion (the CX double layers) and high potentials with complete co-ion exclusion (the HP double layers). Such linear models do not
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ψLP(a) e ψt+ )
kT z+e
(12)
to maintain physically meaningful (positive) local concentrations of co-ions. The potential ψt+ is termed the thermal electrostatic potential of co-ions. This boundary condition differs from that of the DH approximation, which fails to properly distinguish between the repulsive energies of co-ions and the attractive energies of counterions.3,4 The general solution of the DH PBE is Figure 2. LP double layer. The Stern potential ψ(a) is less than the thermal potential of co-ions ψt+ ) kT/(z+e).
ψ3(x) ) A3 exp(-κx) + B3 exp(κx)
necessarily contradict statistical mechanics. The setting up of the LPBE (or of the NLPBE, for that matter) represents only a shortcut to fundamental statistical mechanical results. The proper statistical mechanics44 of the unscreened Coulomb’s potential does not rely on the exploitation of Poisson’s equation, and the issues of linearity and nonlinearity do not arise.
where κ is the Debye constant. 1.3. Boundary Conditions. To determine the integration constants A3 and B3, we recognize the symmetry of the electrostatic problem by the minimum value of the potential at the midplane x ) d:
( ) d2ψ3 dx2
Theory 1. LP Double Layers. The LP model is shown in Figure 2 in terms of the electrostatic charge and potential distributions. The model consists of two electrostatic regions; the Laplace region in the interval 0 e x e a (the Stern layer) and the Poisson region (the DH ionic atmosphere) in the interval a e x e d. 1.1. Laplace Region. The surface charge is represented by σ0, the (analytical) surface charge density. The solvent in the Laplace region is characterized by the dielectric constant 1 (which may be lower than the solvent , hence, the subscript 1). The electrostatic solution is
ψ(x) ) ψ0 -
σ0 x 01
(9)
σ0 a 01
(10)
The Stern potential ψ(a) is independent of the inner dielectric constant 1 and the Stern thickness a. The Laplace region plays a role in the electrostatic modeling12,13 of biological surfaces. The surface potential ψ0 is connected to the aqueous environment through the value of the Stern potential according to eq 10. Thereby, physicochemical changes in the diffuse double layer are monitored inside the double layer. 1.2. Poisson Region. The LP Poisson equation is identical to the DH linearized PBE; however, the boundary conditions are different. First, there is the Stern layer charge distribution σa, which defines the degree of association R of the double layer as
R)-
σa σa ) σ0 σT
(11)
where σ0 is the surface charge and σT is the total compensating charge (the sum of Stern charge and DH diffuse charge). The Stern layer has a finite thickness a, which represents the radius of Stern ions. Second, the value of the Stern potential is limited by the condition (44) Mayer, J. E. J. Chem. Phys. 1950, 18, 1426.
(14)
x)d
The Stern charge density at the Stern plane x ) a is given by Maxwellian electrostatics as
( )
σa ) -0
dψ3 dx
x)a
( )
+ 01
dψ1 dx
x)a
(15)
The amount of the Stern charge is related to the primary surface charge σ0 by the degree of double layer association R, given by eq 11. The Stern layer is an integral part of the overall electrostatic solution and does not imply any ad hoc adsorption phenomena. The degree of association R is to be determined from experimental data. The integration constants are then
Therefore,
ψ0 ) ψLP(a) +
)0
(13)
exp[κ(d - a)] A3 ) Qκ 2 sinh[κ(d - a)]
(16)
exp[-κ(d - a)] B3 ) Qκ 2 sinh[κ(d - a)]
(17)
and
with Qκ defined as
Qκ )
(1 - R)σ0 ) ψ(a)df∞ 0κ
(18)
which is equivalent to the Stern Potential when the LP double layers are infinitely separated. 1.4. Stern and Midplane Potentials. The Stern potential is now given by
1 ψLP(a) ) Qκ tanh[κ(d - a)]
(19)
The midplane potential is given by
1 ψLP(d) ) Qκ sinh[κ(d - a)]
(20)
1.5. Repulsive Pressure. The repulsive pressure is given by the Maxwellian electrostatic expression
∫0ψ(d)F dψ
P(d) ) -
(21)
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This transition distance depends on ionic strength through both the Debye length 1/κ and the degree of double layer association R. At plate separations of d < dLP, the CX model in Figure 3 becomes applicable. 1.8. Classification of Double Layers. We now consider the LP double layer at infinite separation. In this case, the middle potential is 0, and the Stern potential is given by eq 18. We now define singular double layers (infinite separation) by specifying that the Stern potential equals the thermal electrostatic potential of co-ions. Figure 3. Structured CX model. The Stern potential is limited by ψ(a) g ψt+ ) kT/(z+e), the thermal potential of co-ions. The co-ion exclusion boundary b is a constant potential boundary at ψ(b) ) kT/(z+e). The midpoint potential is limited by ψ(d) e kT/(z+e). The DH ionic atmosphere is given by the excess volume charge density F3 and the counterion-only ionic atmosphere by F2.
Because the double layers overlap in the DH region, where the charge density F is related linearly to the potential ψ by the usual DH expression, we obtain
PLP(d) ) 1/20κ2ψ2(d)
(22)
The potential ψ(d) is given by eq 20. 1.6. DH Limiting Law. We now check that in the limit of large separations the DH limiting law is obtained. In eq 20, the sinh function is then dominated by the positive exponential, and considering that d . a, then on substitution into eq 22 and rearrangement, we get
[(1 - R)σ0]2 exp(-2κd) lim P(d) ) df∞ 0
(23)
This is the limiting behavior of the LP Maxwellian double layers, which shows the DH exponential decay of repulsive forces at large separations. The preexponential factor contains only the dielectric constant of the solvent and the “effective surface charge” (the sum of surface charge and Stern charge):
σeff ) (σa + σ0) ) (1 - R)σ0
(24)
The concept of effective charge is useful in the sense that at large separations the details (e.g., solvation) of the Stern layer are unimportant. The incorporation of the electrostatic Stern layer into the model allows relatively large analytical surface charge densities to give “low” Stern potentials (zeta potentials) when the degree of association is large. The LP double layer model is not the DH approximation of the NLPBE equation. 1.7. Onset of Co-Ion Exclusion (LP/CX Transition). Let us now push the LP double layers in Figure 2 closer until the value of the Stern potential is increased to the equal sign of eq 12, that is, ψ(a) ) ψt+. At this condition, the double layers are separated by the LP/CX transition distance dLP, and the co-ion contribution to the volume charge density at the Stern boundary drops to 0 according to the Maxwellian distribution laws. Therefore, at closer separations, a co-ion exclusion boundary b emerges from the Stern layer, compare Figure 3, that describes the coion exclusion (CX) model. The LP/CX transition distance dLP is calculated as
dLP ) a +
[
]
(1 - R)σ0 1 tanh-1 κ 0κψt+
(25)
ψS(a) )
[
]
(1 - R)σ0 0κ
S
) ψt+ )
kT z+e
(26)
Equation 26 is a special case of the boundary condition given by eq 12 and defines a singular physicochemical condition, according to which single (noninteracting) double layers are classified into either LP or co-ion exclusion (CX) double layers. Specifically, the LP double layers have “lower” Stern potentials, limited by
ψLP(a) e
[
]
(1 - R)σ0 0κ
S
) ψt+
(27)
and the CX double layers have “higher” Stern potentials, limited by
ψCX(a) g
[
]
(1 - R)σ0 0κ
S
) ψt+
(28)
The singular double layers represent the upper bound of the LP double layers and the lower bound of the CX double layers. Equation 26 provides the relationship of physicochemical values that define singular double layers to which any experimental double layer is to be compared. Such a comparison determines which electrostatic model (LP or CX) is applicable. The singular condition, eq 26, can also be obtained from eq 25 by realizing that the LP/CX transition distance dLP is located at infinity for singular double layers. The equality of the numerator and denominator in eq 25 gives the singular condition, eq 26, when dLP ) ∞. In other words, singular double layers have an incipient co-ion exclusion boundary in the Stern layer. When such double layers are brought closer from infinity, there is an immediate appearance of the co-ion exclusion boundary close to the Stern layer. 2. Co-Ion Exclusion (CX) Double Layers. 2.1. Definition of the CX Electrostatic Model. The distinguishing feature of the CX model of the Maxwellian double layers is the presence of explicit co-ion exclusion boundaries, designated by the symbol b. The CX double layers have diffuse ion structure, which comprises the diffuse DH region farthest away from the surface, and the counterion-only diffuse region, closer to the surface, as is shown in Figure 3. In sections 1.7 and 1.8, two occasions are identified that lead to the explicit co-ion exclusion boundaries b (the CX model). First, when two LP double layers approach each other, the LP/CX transition distance dLP is eventually reached, given by eq 25. At this transition distance, the Stern potential reaches the value of the thermal electrostatic potential of co-ions, ψt+, and the incipient exclusion boundary forms in the Stern layer. At closer separations than dLP, the co-ion exclusion boundary moves away from the Stern layer toward the midplane separation distance, as is shown in Figure 3. Its location is determined by the constant value of ψt+. It is a free boundary (location
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unspecified) with a constant potential and continuous electric field across the boundary (no dielectric discontinuity). Second, the CX double layers, defined by the condition of eq 28, have the co-ion exclusion boundary even at infinite separation because their Stern potential is higher than ψt+. This boundary is designated b∞. Therefore, the interaction of two CX double layers pushes the co-ion exclusion boundary from b∞ toward the middle of the plates, rather than from the Stern layer (cf. interacting LP double layers). At infinite separation, the CX double layers have a diffuse ion structure, whereas the LP double layers do not. The CX and LP double layers merge under the singular physicochemical condition given by eq 26 at infinite separation. The CX interaction model is defined in Figure 3. It represents both the interacting LP double layers after they passed the LP/CX transition and the single and interacting CX double layers. The electrostatic solution for single CX double layers is obtained as the limit of d f ∞. 2.2. Laplace’s Region. The solution is given in section 1.1. However, the Stern potential becomes ψCX(a) and always has a higher value than ψLP(a). 2.3. Poisson’s Equation in the CX Region. The CX region a e x e b (subscripted 2) is characterized by the “average” absence of co-ions. It contains only counterions. Freely moving co-ions do not contribute to the volume charge densities in the regions where, on average, their repulsive electrostatic energy would be higher than their average thermal energy. The value of the potential ψ(b) ) ψt+ defines the location of the co-ion exclusion plane at x ) b. The counterions contribute to the net charge density according to their attractive electrostatic energies as
dq dV
[
) z-en-(x) ) F2(x) ) z-en-0
]
kT - z-eψ2(x) kT
(29)
When eq 29 is substituted into Poisson’s equation, we obtain 2
d ψ2 2
dx
) 0λ2[ψ2(x) - ψt-] ) -
F2 0
(30)
where λ is defined analogously to the DH constant κ as
λ2 )
(z-e)2n00kT
kT z-e
(33)
with integration constants A2 and B2 to be determined. Equation 29 shows that the effect of counterions is electrostatically enhanced by the absence of co-ions and that the short-range, specific effects of co-ions are eliminated in the CX model. These effects emphasize the role of counterions in an unsymmetrical way, as was originally desired by Verwey and Overbeek.7 (They emphasized the role of counterions by retaining the “full” exponentials in
(34)
The boundary conditions are quite different, however, and are discussed next. 2.5. Boundary Conditions. We shall omit the CX superscript. Only standard electrostatic conditions are used. The five unknowns are as follows: the integration constants A3 and B3 and A2 and B2 and the co-ion exclusion plane at x ) b. The five boundary conditions are as follows. The first one is the midplane symmetry condition, giving the minimum value of the potential at the midplane x ) d
( ) d2ψ3 dx2
)0
(35)
x)d
The next two conditions are the continuity of potentials and the value of the potential at the exclusion boundary x ) b.
A2 exp(-λb) + B2 exp(λb) + ψt- ) A3 exp(-κb) + B3 exp(κb) (36) and
A2 exp(-λb) + B2 exp(λb) + ψt- ) ψt+
(37)
and it follows that
A3 exp(-κb) + B3 exp(κb) ) ψt+
(38)
Equation 38 is useful for algebraic manipulations (not an independent condition). The next condition also refers to the co-ion exclusion boundary, where the electric field is continuous, because we assume the same value of the dielectric constant on both sides at the co-ion exclusion boundary.
( ) ( ) dψ3 dx
)
x)b
dψ2 dx
(39)
x)b
The last and fifth condition gives the amount of Stern charge at the Stern plane x ) a by standard electrostatics.
( )
σa ) -0
(32)
The general solution of eq 30 is
ψ2(x) ) A2 exp(-λx) + B2 exp(λx) + ψt-
CX CX ψCX 3 (x) ) A3 exp(-κx) + B3 exp(κx)
(31)
and the thermal electrostatic potential of the counterions is defined as
ψt- )
the PBE.) The CX model is in accord with VerweyOverbeek’s original sentiments. 2.4. Poisson’s Equation in the DH Region. The DH region is in the interval b e x e d and subscripted as 3. The general solution is the same as that in the case of the LP model in section 1.2. Adding the superscript CX, we get
dψ2 dx
x)a
( )
+ 01
dψ1 dx
x)a
(40)
2.6. Outline of the Algebraic Solution. The basic strategy is to derive an expression for the location of the exclusion plane b. When eqs 35 and 38 are used, the integration constants A3 and B3 are
exp(κd) A3 ) ψt+ 2 cosh[κ(d - b)]
(41)
exp(-κd) 2 cosh[κ(d - b)]
(42)
and
B3 ) ψt+
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When eqs 37 and 39 are used with the above two results, eqs 41 and 42, the integration constants A2 and B2, can be expressed as follows: +
-
κ + ψt+ tanh[κ(d - b)] (43) λ
+
-
κ - ψt+ tanh[κ(d - b)] (44) λ
2A2 exp(-λb) ) ψt - ψt 2B2 exp(λb) ) ψt - ψt
Now, when eq 40 is used, eqs 33 and 9 give the relation between the integration constants A2 and B2 in terms of the surface charge as
Qλ ) A2 exp(-λa) - B2 exp(λa)
(45)
where the symbol Qλ is shorthand for
Qλ )
(1 - R)σ0 0λ
(46)
which is analogous to Qκ, compare eq 18. And finally, eliminating the constants A2 and B2 from eqs 44-46 yields the desired expression for the location of the co-ion exclusion boundary b as
2Qλ ) T1 - T2
(47)
where the terms T1 and T2 are given by
{
}
κ T1 ) exp[λ(b - a)] ψt+ - ψt- + ψt+ tanh[κ(d - b)] λ (48)
{
T2 ) exp[-λ(b - a)] ψt+ - ψt- -
}
κ + ψ tanh[κ(d - b)] (49) λ t We define the function F(b) ) 0 as
F(b) ) 2Qλ - T1 + T2 ) 0
(50)
which can be differentiated in respect to b to give
F′(b) ) dF/db ) -λT1 - λT2 + κU1 + κU2 (51) where the terms U1 and U2 are defined by
exp[λ(b - a)] κ U1 ) ψt+ λ cosh2[κ(d - b)]
(52)
exp[-λ(b - a)] κ U2 ) ψt+ λ cosh2[κ(d - b)]
(53)
and
The exclusion boundary b can now be calculated by Newton’s iterations as
b 1 ) b0 -
F(b0) F′(b0)
(54)
2.7. Electrostatic Potentials and Forces. The Stern potential is derived from eqs 33, 43 and 44, and 48 and 49 as
ψCX(a) ) 1/2(T1 + T2) + ψt-
(55)
The important midplane potential is derived from eqs 34, 41, and 42 as
1 ψCX(d) ) ψt+ cosh[κ(d - b)]
(56)
To calculate the repulsive pressure between the double layers, we need to integrate the standard electrostatic expression, eq 16. Because the double layers overlap in the DH region, where the volume charge density is related to the potential by the linear DH relation, we obtain
PCX(d) ) 1/20κ2ψ2(d)
(57)
The potential in midplane ψ(d) is given by eq 56 above. Thus, the repulsive pressures are calculated by the same expression for the LP and CX double layer models, each having its own expression for the midplane potential ψ(d). 2.8. CX DH Limiting Law. We can now check that the DH limiting behavior of the repulsive forces is obeyed at large separations. In eq 56, as we take the limit to infinity, the co-ion exclusion distance b moves toward the Stern plane and becomes b∞ when d f ∞. The cosh function is then dominated by the positive exponential. Substitution into eq 53 and tidying up gives
PCX(d)df∞ ) 20κ2ψt+2 exp(2κb∞) exp(-2κd) (58) Both the LP double layers and the CX double layers exhibit the DH exponential limiting behavior at large separations. However, the preexponential factor is different for the LP and the CX double layers. This difference reflects their distinct diffuse structures. From eq 58, it is possible to experimentally determine the co-ion exclusion distance at infinity as a function of the ionic strength. If we had a singular double layer, then b ) b∞ ) a exactly, when d f ∞ (singular double layers have incipient co-ion exclusion in the Stern layer). If we had a LP double layer, then we would encounter the LP/CX transition first, when b ) a at d ) dLP, before reaching the infinite separation limit. Then the results of the LP model are used to obtain the DH limiting behavior, as was discussed in section 1 for LP double layers. 2.9. Continuity of the Stern Potential across the LP/CX Transition. We check that the lower bound of the CX double layers coincides with the upper bound of the LP double layers. Thus, we need to recover eq 26 with the equal sign, that is, ψCX(a) ) ψt+ ) ψLP(a) when b f a. This can be done mentally (starting from eq 51) by letting b f a in eqs 48 and 49, which makes the exponential terms equal to 1 in the T1 and T2 terms, and substituting the result into eq 55. 2.10. Co-Ion Exclusion Boundary at Infinite Separation. The CX double layers defined by eq 23 possess a co-ion exclusion boundary at infinite separation, which can be calculated by letting the midplane separation distance d f ∞ in eq 47. Then the tanh terms in eqs 48 and 49 become equal to 1, and the unknown b∞ occurs only in the exponential terms. Equation 47 then can be regarded as a quadratic polynomial when a substitute variable is defined for the exponential terms. The result is
{
1 b∞ ) a + ln λ
[
Qλ + Qλ2 + (ψt+ - ψt-)2 κ ψt+ - ψt- + ψt+ λ
+
t
}
2 1/2
(λκψ ) ]
(59)
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Thus, single CX double layers are characterized by ψ(a) g ψt+ with a finite co-ion exclusion distance b∞ given by the previous expression. The exclusion boundary b∞ defines the structure of the diffuse ion distributions of the CX double layers. 2.11. Physicochemical Continuity of CX/Singular/LP Double Layers. We can now check that singular double layers have Stern potentials equal to ψt+ when we approach this singular condition from the CX model at infinite separation. Therefore, when b∞ f a, the equal sign in eq 26 should be recovered from eq 59. To get b∞ ) a in eq 59, the denominator must equal the numerator in the ln function. The rest is an algebraic exercise; in working through it, the following identity is useful: + κ2 z- - z+ ψt - ψt ) ) zλ2 ψt+
(60)
Thus, all expressions are consistent and continuous on going from the LP double layers to the CX double layers under all physicochemical conditions and interaction distances. 2.12. Complete Co-Ion Exclusion (CX/HP Transition). When the CX double layers are pushed together until the co-ion exclusion boundary b coincides with the midplane separation distance d (Figure 3), we reach the condition of complete co-ion exclusion from between the double layers. A further increase of pressure will compress the diffuse counterions into a smaller volume, which defines the HP interaction model. Because the boundary b “carries” the thermal electrostatic potential of the co-ions, ψt+, the midpoint potential will become equal to ψt+, as is shown by eq 56 when b ) d. This condition represents a switch from the CX mode of interactions to the HP mode of interactions, or the CX/HP transition. To derive the necessary condition for the CX/HP transition distance, we set dHP ) b ) d in eqs 47-49, which can be algebraically worked out as
dHP ) a +
[
(1 - R)σ0 1 sinh-1 λ 0λ(ψt+ - ψt-)
]
(61)
For comparison, the LP/CX transition distance is given by eq 25. We notice that in both expressions the “effective surface charge” plays a decisive role. The CX/HP transition distance dHP is also obtained as the lower-bound solution of the HP model in the next section. 3. HP Interacting Double Layers. 3.1. Definition of the HP Model. In section 2.12, the CX/HP transition distance dHP is established, at which all co-ions are expelled from between the plates. The co-ion exclusion distance b coincides with the midplane separation distance d. At this separation distance, the midplane potential ψ(d) has reached the value of the thermal electrostatic potential of co-ions ψt+. The midplane potential is now increasing as the charged plates are brought together in the HP mode of interactions. There are no HP double layers at infinite separation, as the “very” high potentials are brought about by the additivity of the potentials arising from both charged plates. The electrostatic model is shown in Figure 4. 3.2. Definition of Contact Separation. From Figure 4 we define the limiting contact separation when d ) a, where the Stern thickness represents hard-sphere interactions (Born electron repulsions) given by the geometry and solvation of the surface ions and counterions. However, in electrostatic terms we have a model of a sheet of
Figure 4. HP interacting double layers. The Stern potential is limited by ψ(a) g ψt+ ) kT/(z+e), and the midpoint potential is ψ(d) g kT/(z+e). Only counterions are present between the plates (total exclusion of co-ions, i.e., “added” electrolyte).
counterions sandwiched between two sheets of surface charges of half the charge density. It is conceivable that the Stern ions and any diffuse ions become immobilized, when d f a, when a discrete ion model would be more appropriate. Here, we assume continuous electrostatic distributions in the limit of d f a, while the discrete ion model will be dealt with in a future communication. 3.3. Laplace Region. As in sections 1.1 and 2.2, the electrostatic solution here is the same. The value of the Stern potential ψ(a) is different for all three contiguous models of the LP, CX, and HP double layers and is derived from the properties of their respective diffuse regions. 3.4. Poisson Region. This region lies in the interval a e x e d (subscripted 2), and is analogous to the co-ion exclusion region treated in the previous section, except for the boundary conditions. Importantly, there is no adjacent DH region of low potentials, where both co-ions and counterions contribute to the volume charge density. The HP Poisson region is defined by the condition of the midplane potential being higher than the thermal electrostatic potential of the co-ions ψt+ as
ψ(d) g ψt+ )
kT z+e
(62)
Therefore, this model is applicable only to interacting double layers, when the co-ions (electrolyte) have been completely expelled from between the plates. The contribution of the counterions to the volume charge density is on average
dq dV
[
) z-en-(x) ) F2(x) ) z-en-0
]
kT - z-eψ2(x) kT
(63)
When eq 63 is substituted into Poisson’s equation, we obtain
d2ψ2 dx2
) 0λ2[ψ2(x) - ψt-]
(64)
where the symbols have been defined previously and have their usual meanings. The general solution of eq 64 is
ψ2(x) ) C2 exp(-λx) + D2 exp(λx) + ψt-
(65)
with the integration constants C2 and D2 to be determined. 3.5. Boundary Conditions and Integration Constants. The solution is analogous to the LP case. There are only two integration constants to determine. All that is needed are the usual Maxwellian electrostatic boundary conditions plus the dissociation of the Stern layer. The midplane
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symmetry condition at the midplane x ) d is
( ) d2ψ2 dx2
)0
repulsion arising from the HP interaction mode. The relation between the volume charge density and the potential in the DH region is given by the usual relation
(66) F3 ) -0κ2ψ3
x)d
The second condition determines the Stern charge by the usual Maxwellian relation:
( )
σa ) -0
dψ2 dx
x)a
( )
+ 01
dψ1 dx
x)a
(67)
From these conditions, the integration constants are determined as
(68)
exp(-λd) D2 ) Q λ 2 sinh[λ(d - a)]
(69)
and
These expressions can be compared with the results of the LP model, eqs 16 and 17. The LP and the HP models have analogous mathematical structures. 3.6. Stern and Midplane Potentials. These important colloidal properties are now given by the following expressions. The Stern potential, from eqs 65, 68, and 69 is worked out as
1 + ψtψ(a) ) Qλ tanh[λ(d - a)]
(70)
The midplane potential is worked out similarly as
1 + ψtψ(d) ) Qλ sinh[λ(d - a)]
(71)
dfa
dfa
2
0λ
1 d-a
(72)
The contact limiting electrostatic potential tends to infinity according to the inverse of the separation distance. 3.7. Calculation of the Repulsive Pressure. The usual Maxwellian electrostatic expression, eq 21, must be now split into two terms. When the midplane potential reaches the value of the thermal electrostatic potential ψt+ at the CX/HP transition, the relationship between the volume charge density changes from the DH relation to the HP relation, eqs 74 and 75 that follow. The Maxwellian force integral is then composed of two parts:
∫0ψ
PHP(d) ) -
+
t
F3 dψ3 -
∫ψψ(d)F2 dψ2 +
t
F2 ) -0λ2[ψ2 - ψt-]
(75)
The result of the integration of eq 73 is
PHP(d) ) 1/20λ2[ψ22(d) - ψt-ψ2(d) + ψt-ψt+] + /20ψt+2(κ2 - λ2) (76)
The contact-limiting expression of eq 76 for close separations, when d f a, is given in Results and Discussion. 3.8. Co-Ion Exclusion and Electroneutrality. It can be shown explicitly that all interactions take place under electroneutral conditions for all three electrostatic models of the LP, CX, and HP double layers. The electroneutrality is satisfied directly by the linear distribution laws, compare the derivation of the DH constant κ in the LPBE. The “co-ion exclusion” is to be understood then as the exclusion of electroneutral combinations of ions (e.g., salt), the coions being merely the “active agents” for this phenomenon to take place. In other words, the changes of structure and transitions between these three models do not lead to any imbalances of charge distributions. Hence, the thermodynamics of electrolytes, being meaningful only for electroneutral combinations of ions, is not violated. In other geometries, co-ions may be expelled to the regions farthest away from charged surfaces, possibly leading to new “Madelung-like” structures of charged colloids (e.g., colloidal crystals).29,30 Results and Discussion
We have already derived the CX/HP transition distance dHP as eq 61 from the CX model. Here, we can obtain the same relation as the lower bound of the HP electrostatic solution. When eqs 62 and 65 and eq 68 with eq 69 are used, eq 61 is easily recovered. The limiting behavior of the potentials, as we take the contact limit d f a, is convergent (both tanh and sinh at small values of the argument give the value of the argument) as
(1 - R)σ0
For the HP mode of interactions, the relation is somewhat analogous:
1
exp(λd) C2 ) Qλ 2 sinh[λ(d - a)]
lim ψ(a) ) lim ψ(d) )
(74)
(73)
The first integral represents the contribution from the LP and CX double layers that overlap in the DH region up the CX/HP transition. The second integral represents the
The mathematics of the linear electrostatic models is much simpler than the generally intractable mathematics of the NLPBE. In planar geometry, many detailed calculations can be done conveniently in a spreadsheet program from the analytical results derived above. In this article, only the following, most important results are discussed: the discovery of the Lubetkin-MiddletonOttewill (LMO) law, the physicochemical classification of double layers, the contact repulsive limiting law, and the total interaction regime from infinity to contact. Some results obtained in spherical geometry were published previously.29-31 LMO Law. The previous application of the DEDL theory in the HP mode of interactions gave a good quantitative account of the pressures in montmorillonite gels.25,26,30 The unique, extensive set of data35 of Lubetkin, Middleton, and Ottewill over a large range of ionic strengths with different counterions was used to derive an approximate empirical law26 (the LMO law), which predicts the dependence of the degree of double layer association on ionic strength as
σa 1 R)- ) σ0 1 + pκ
(77)
The LMO parameter p describes the “physical chemistry” of the double layer. Different montmorillonite counterions have LMO parameters in the range of 2-4 Å that indicate double layers that dissociate slightly (∼1-20%) with increasing ionic strength. The DEDL theory accounts
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Spitzer Table 1. Singular (Critical) Surface Charge Densities σ0S (µC cm-2) as a Function of the Molarity of 1:1 Electrolyte (Water at 25 °C)a LMO parameter p (Å) molarity
1/κ (Å)
∞
1000
100
10
3
1
10-7
9619.9 3042.1 962.0 304.2 96.2 30.4 9.6 6.8 4.3 3.0
0.00 0.01 0.02 0.06 0.19 0.59 1.86 2.63 4.15 5.87
0.02 0.02 0.04 0.08 0.20 0.60 1.87 2.64 4.17 5.89
0.18 0.18 0.20 0.24 0.36 0.77 2.03 2.80 4.33 6.05
1.79 1.79 1.80 1.84 1.97 2.37 3.64 4.41 5.94 7.66
5.95 5.96 5.97 6.01 6.14 6.54 7.81 8.58 10.10 11.82
17.86 17.86 17.88 17.92 18.04 18.44 19.71 20.48 22.01 23.73
1× 1 × 10-6 1 × 10-5 1 × 10-4 1 × 10-3 1 × 10-2 1 × 10-1 2 × 10-1 5 × 10-1 1 × 100
a The values of the LMO parameter p (Å) in the top row vary from ∞ to 1 Å.
Figure 5. Graphical representation of the LMO law. The degree of association R from eq 77 is plotted over a wide range of ionic strengths on the x axis (10-7-100 M 1:1 electrolyte). The LMO parameters p cover a wide range from 1, 3, 10, 100, 1000, to 10 000 Å for curves from top to bottom (Helmholtz to GC double layers).
consistently both for the strong, close-range repulsions (the HP interactions) and for the Stern potentials at infinity (CX noninteracting model) with the same LMO parameter. The LMO law is very useful when it is incorporated into the DEDL theory because the ionic-strength dependence of various calculated results, such as the Stern potential, can be predicted in different geometries. For example, on going from plates to spheres, the DEDL-LMO formulation predicts maxima of Stern potentials for spheres but not for plates. The general predictions of the LMO law, eq 77, are given in Figure 5 over wide ranges of ionic strengths and values of the LMO parameter p. The LMO p values below 1.0 Å lead to (highly associated) Helmholtz double layers above the uppermost curve in Figure 5. The LMO p values above 105 Å lead to (slightly associated) GC double layers below the bottommost curve in Figure 4. We conclude that the classical Helmholtz double layer reappears as the limit of the LMO parameter p f 0, when R f 1. Similarly, the classical GC double layer becomes the limiting case of the LMO parameter p f ∞, when R f 0. The provisional interpretation of this law is that the addition of electrolyte (increased ionic strength by free ions) lowers the activity of diffuse ions, which is compensated by the release of ions from the Stern layer (increased dissociation). This effect is similar to the classical DH effect that predicts the increased solubility of ionic salts in the presence of electrolyte (or increased dissociation of weak electrolytes). In other words, the surface charges and the Stern charges can be thought of as a two-dimensional, ionic salt, the solubility of which increases with increasing ionic strength. Classification of Double Layers. Equation 26 defines the singular double layer as a common case of the LP and CX double layers: the Stern potential equals the thermal electrostatic potential of co-ions (infinite separation). When the LMO law is incorporated in eq 26 and we focus on the analytical surface charge density as a physicochemical variable, the singular condition can be written as
[σ0]S ) [0ψt+(κ + 1/p)]S
(78)
The singular charge densities allow the classification of double layers into two kinds: the LP double layers and the CX double layers, according to eqs 27 and 28. Double layers with lower charge densities than those given in Table 1 are the LP double layers. Double layers with higher charge densities are the CX double layers. The singular double layers are defined by the charge densities given in Table 1. They represent the physicochemical continuity between the LP and the CX double layers. The data in Table 1 show that LP double layers can sustain appreciable (analytical) surface charge densities, depending on the value of the LMO parameter p. For example, with p ) 3.0 Å, the LP surface charge densities can range up to over 5.95 µC/cm2 at the lowest ionic strength of 10-7 M and increase up to 6.54 µC/cm2 at 0.01 M (the penultimate column in Table 1). The calculated charge densities at higher ionic strengths, where the DH interionic theory becomes progressively less reliable, are given only as a guide. The LP double layers are expected to behave qualitatively differently from the CX double layers in their property versus volume fraction dependencies because the CX double layers exclude co-ions (electrolyte) explicitly already in the limit of 0 volume fraction. This observation will be quantified only when solutions of the DEDL model are obtained for interacting, many-body assemblies of double layers. Maxwellian Limiting Laws. The full expression, eq 76, for the repulsive pressure can be simplified in the limit of a close approach of the surfaces (the HP model). The limiting repulsive pressure is obtained by incorporating the LMO law for the degree of dissociation and taking the limit d f a as
lim PHP(d) ) dfa
( )(
p 1 2 1 + pκ
2
)
ψt+ - ψt- σ02 1 (79) 2 + ψ 0 (d - a) t
Further simplification is achieved by taking a DH limit to low electrolyte concentrations:
lim P
dfa;κf0
(
)
+ 2 1 2 ψt - ψt σ0 1 (d) ) p (80) 2 + 2 ψt 0 (d - a)
HP
The first essential insight is that these short-range Coulombic repulsions decay according to the inverse square of the separation. The second essential insight is that these forces become independent of ionic strength in the limit of 0 ionic strength. These predictions can be checked with new data for montmorillonites.
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Table 2. Colloidal Properties of a CX Double Layer with Surface Charge Density σ0 ) 11.8 µC cm-2, Stern Thickness of 3.0 Å, and LMO Parameter 2.0 Å molarity (mol dm-3) 10-6
1× 1 × 10-5 1 × 10-4 1 × 10-3 1 × 10-2 1 × 10-1
R
ψ∞(a) (mV)
b∞ (Å)
dHP (Å)
ψ(dHP) (mV)
P(dHP) (kPa)
d0.50mV (Å)
P1.0nm (kPa)
0.9993 0.9979 0.9935 0.9796 0.9383 0.8279
34.5 34.4 34.3 33.7 32.2 28.2
904.9 286.7 91.2 29.4 9.9 3.9
3590 1136 360 115 37 13
25.69 25.69 25.69 25.69 25.69 25.69
0.00 0.02 0.25 2.48 24.79 247.89
14 997 4743 1500 475 151 48
1644 1632 1600 1524 1294 670
Figure 6. Experimental data37 replotted to check for the limiting repulsive law of Maxwellian electrical double layers. The thick line represents an overlap of data at 10-4 and 10-3 M. The bottom line is the 10-1 M set of data.
The swelling of montmorillonite clays has been experimentally studied,35-37 and the DEDL model was shown to account25,26 for the previous data. The latest available data37 for lithium montmorillonite, taken directly from the published article’s Figure 5, are plotted against the inverse of the square of distance in Figure 6. Importantly, the experimental straight lines are in agreement with the fundamental limiting law at close separations. The topmost curve corresponds to two superposed data sets at 10-4 and 10-3 M that are very close together, in agreement with the double limiting law given by eq 80. The 0.01 M data, the middle straight line, also obey the inverse square law. The 0.1 M data (the bottom curve) are not entirely in the limiting repulsive range; this implies the presence of free electrolyte between the plates. The traditional DLVO theory cannot predict these large repulsive forces because the NLPBE appears to underestimate the correct Maxwellian repulsions. Electrostatic Repulsions from Infinity to Contact. The electrostatic solutions obtained for the contiguous models of the LP, CX, and HP interactions make it possible to present new calculations of repulsive forces from infinity to contact separations. Two illustrative examples are presented. In Figure 7, we have chosen a low electrolyte concentration (for graphical purposes, because the CX model then extends over an appreciable distance range). The particulars of the double layer are: 1:1 electrolyte at 10-6 M (water at 25 °C), σ0 ) 5.90 µC/cm2, the LMO parameter p ) 3.0 Å, and the Stern layer thickness a ) 3.0 Å. The total repulsion profile reflects the three models used for calculations, as is shown by the arrows. The LP/CX and CX/HP transition points are highlighted, which shows that the potentials and pressures remain continuous at these transitions, without any apparent breaks. In the second example, we use a CX double layer with colloidal properties described in Table 2. It is a slightly dissociating double layer, as is shown in the first column of Table 2. In the second column, this double layer has a
Figure 7. Long-range dependence of repulsive pressures at 10-6 M ionic strength, showing the LP/CX and CX/HP transition separation distances (bigger circles). The CX and HP models show positive deviations from the DH limiting slope, qualitatively similar to the results of the NLPBE.
Stern potential of about 34 mV at infinite separation. In the next column, the CX diffuse structure is characterized by the exclusion distances b∞ at infinite separation. In the fourth column, the important CX/HP transition distance (completeness of co-ion expulsion) is shown. Next, two columns give the midplane potential (constant) and the repulsive pressure at this transition distance. The last two columns contain ad hoc colloidal properties. First, we define the “practical infinity” of interacting double layers as the midplane at 0.50 mV (neglecting the 0.00-0.50 mV regime). This distance is about 4 times the Debye length. Second, the last column gives electrostatic repulsive pressures at a constant 1.0-nm midplane separation. These pressures are large with a weak ionic-strength dependence. Overall, the data in Table 2 show that the LP/ CX/HP models can be used to characterize the double layer in detail. Such data are not easily obtained from the intractable NLPBE. The repulsive electrostatic pressures for this CX double layer are plotted in Figure 8 over a range of ionic strengths. The theoretical repulsions in Figures 7 and 8 bear a striking resemblance to the published data on the repulsions between crossed mica cylinders in electrolyte solutions.10,34 The exquisite mica data appear to suggest a single, underlying force law by themselves. Here, we make an important observation that the predicted forces deviate positively from the DH large separation limiting law. These positive deviations (the CX model) appear to be similar to the positive deviations obtained from the NLPBE predictions. Clearly, the experimental data qualitatively support the DEDL theory. More importantly, the HP model accounts for the very large forces in the close range, where the NLPBE predictions fail catastrophically. In this closerange region, the limiting repulsive laws, eqs 79 and 80, are confirmed by the montmorillonite data. When the DEDL theory is worked out in detail for the crossed cylinder geometry, detailed quantitative analyses will become possible over the total interaction regime. In
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Table 3. Summary of Transition Phenomena in Maxwellian Double Layer Interactions Based on Precise Physicochemical Criteria separation
Stern boundary
midplane boundary
double layer type
Single Double Layers ψ(d) f 0 LP double layers ψ(d) f 0 singular double layer ψ(d) f 0 CX double layers
phenomena
df∞ d f ∞ (b ) a) d f ∞ (b ) b∞)
ψ(a) e ψt+ ψ(a) ) ψt+ ψ(a) g ψt+
∞ e d e dLP d ) dLP (b ) a) dLP e d e dHP (a e b e d) d ) dHP (b ) d) dHP e d e a
ψ(a) e ψt+ ψ(a) ) ψt+ ψ(a) g ψt+ ψ(a) > ψt+ ψ(a) > ψt+
Interacting Double Layers: LP Type ψ(d) < ψt+ LP mode of interaction ψ(d) < ψt+ LP/CX transition ψ(d) e ψt+ CX mode of interaction ψ(d) ) ψt+ CX/HP transition + ψ(d) g ψt HP mode of interaction
no co-ion exclusion incipient co-ion exclusion partial co-ion exclusion end of co-ion exclusion no co-ions between plates
∞ e d e dHP (b∞ e b e d) d ) dHP (b ) d) dHP e d e a
ψ(a) g ψt+ ψ(a) g ψt+ ψ(a) g ψt+
Interacting Double Layers: CX Type ψ(d) e ψt+ CX mode of interaction ψ(d) ) ψt+ CX/HP transition + ψ(d) g ψt HP mode of interaction
partial co-ion exclusion end of co-ion exclusion no co-ions between plates
Figure 8. Example of a CX Maxwellian double layer covering a broad range of separation distances and ionic strengths (1:1 electrolyte). The ionic strength is 10-4-10-1 for the curves from right to left. The colloidal properties are described in Table 2. The lowest pressure for each concentration corresponds to the midplane potential of 0.50 mV.
general, the Maxwellian electrostatic double layers do not predict a universal adhesive minimum, though adhesive minima can be predicted at low charge densities or with less dissociating double layers. Ion Distributions and Repulsions. The three Maxwellian models of the electrical double layer give a clear and simple picture of what happens to the “added” electrolyte and repulsive pressures when charged interfaces interact. In the LP region, the weak DH overlapping atmospheres give low repulsive pressures, and no electrolyte exclusion. In the CX region, the “added” electrolyte is being expelled from between the plates, when the coion exclusion boundary moves toward the midpoint separation. The pressure then becomes higher compared to the DH limiting-law prediction. In the HP region, there is no electrolyte left between the plates, and the very large repulsions arise from the “squeezing” of the diffuse counterions into a smaller volume under electroneutral conditions. Surface Conditions during Interactions. In the colloidal theory of the NLPBE, three conditions of surface “regulation” are recognized: the constant potential conditions (conducting surface), the constant charge conditions (nonconducting surface), and the charge regulation model,10,45 which gives intermediate results between the conducting and the nonconducting surfaces. The charge regulation involves ad hoc surface ionization equilibria (45) Ninham, B. W.; Parsegian, V. A. J. Theor. Biol. 1971, 31, 405.
no co-ion exclusion incipient co-ion exclusion co-ion exclusion
incorporated into the NLPBE. The DEDL theory presents a somewhat different model, partly because of the adherence to linear electrostatics. The theory is developed for a single physicochemical condition, which involves the analytical surface charge and a degree of association of counterions into the Stern layer. The calculated examples make the simplest assumption that the degree of association does not change when ionic atmospheres (surfaces) interact and the repulsive pressures increase. This assumption is in accord with a generally weak pressure dependence of chemical equilibria in condensed phases. The traditional ionization model for the analytical surface charge can be easily incorporated into the DEDL theory to determine how the degree of association depends on the surface charge density (at constant ionic strength). The related application of the theory to clay surfaces showed that the double layer association depends on the ionic strength according to the LMO law (at a constant surface charge). A more complex dependence of the degree of association is likely to emerge when the ionic strength and analytical surface charge are varied. The DEDL model also clarifies the meaning of the “effective” charge, eq 24 (the sum of surface and “Stern” charge), while retaining the classical meaning of the Debye length (ionic strength). This classical concept has been recently confirmed experimentally within the context of colloidal interactions.46,47 Summary of Maxwellian Double Layers and Their Interactions. The development of the model of Maxwellian electrical double layers provides a new way of thinking about interactions in colloidal systems. Out of necessity, a number of new concepts and new terminology were developed. Table 3 summarizes the classifications and interactions of Maxwellian double layers and related co-ion exclusion phenomena. In this article, only planar interactions between double layers of the same kind are considered. The DEDL theory can be developed for interactions of unequal double layers in a multitude of curved geometries and physicochemical conditions. Such an approach could be applicable to interactions between biological surfaces;12,13,15 the conditions in Table 3 may represent “Maxwellian (electrical) switches” that control the assembly and breakup of biochemical structures inside living cells. The chaos-prone, non-Maxwellian NLPBE does not predict any such transitions. (46) Kohonen, M. M.; Karaman, M. E.; Pashley, R. M. Langmuir 2000, 16, 5749-5753. (47) Tulpar, A.; Subramanian, V.; Ducker, W. Langmuir 2001, 17, 8451-8454.
Maxwellian Double Layer Forces
Conclusions A systematic mathematical derivation of the DEDL theory is given in planar geometry. The DEDL theory comprises three Maxwellian variations of the linear PBE with proper boundary conditions. The theory provides a physically consistent description of ionic distributions at charged planar surfaces and a general mechanism to define more complicated Maxwellian models. The theory covers interactions from infinity to contact in terms of three contiguous and continuous electrostatic models: the LP model, the CX model, and the HP model. The theory is clear and straightforward. A new contact limiting law predicts that repulsive forces decay inversely with the square of the separation at high potentials and at close separations. This prediction is observed experimentally for repulsive pressures of montmorillonite gels. The DH LP (large separation) limit is also recovered. The theory gives a qualitative account of repulsions between crossed mica cylinders over the entire interaction regime. The CX model gives positive deviations from the DH limit, which is qualitatively similar to the predictions of the NLPBE. The HP model, which is applicable to the short-range and high-potential interactions, predicts very large repulsions that are described by
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the Maxwellian limiting contact laws. The NLPBE underestimates the Maxwellian repulsions in the closerange regime. This underestimation leads to the unrealistic prediction of universal adhesion in the DLVO theory and to catastrophic discrepancies between experiment and theory. The LMO law was derived previously from the application of the DEDL theory to the experimental data. The law gives an empirical prediction of the dependence of double layer association on ionic strength. The LMO law is useful in the DEDL formulation to predict the ionicstrength dependence of various colloidal variables. The combination of the Maxwellian electrostatics with the LMO law predicts a decreasing sensitivity of the electrostatic repulsions to decreasing ionic strength in the high-repulsion regime. Such decreasing sensitivity is experimentally observed in montmorillonite gels. Acknowledgment. Extensive and valuable suggestions of two anonymous referees significantly improved the original manuscript. Prof. Ottewill is thanked for his continued interest in this work. LA034028A
