8554
J. Phys. Chem. B 2007, 111, 8554-8559
Counterion Condensation on Charged Spheres, Cylinders, and Planes† Gerald S. Manning* Department of Chemistry and Chemical Biology, Rutgers UniVersity, 610 Taylor Road, Piscataway, New Jersey 08854-8087 ReceiVed: October 28, 2006; In Final Form: December 28, 2006
We use the framework of counterion condensation theory, in which deviations from linear electrostatics are ascribed to charge renormalization caused by collapse of counterions from the ion atmosphere, to explore the possibility of condensation on charged spheres, cylinders, and planes immersed in dilute solutions of simple salt. In the limit of zero concentration of salt, we obtain Zimm-Le Bret behavior: a sphere condenses none of its counterions regardless of surface charge density, a cylinder with charge density above a threshold value condenses a fraction of its counterions, and a plane of any charge density condenses all of its counterions. The response in dilute but nonzero salt concentrations is different. Spheres, cylinders, and planes all exhibit critical surface charge densities separating a regime of counterion condensation from states with no condensed counterions. The critical charge densities depend on salt concentration, except for the case of a thin cylinder, which exhibits the invariant criticality familiar from polyelectrolyte theory.
I. Introduction Counterion condensation on linear polyelectrolytes is a welldocumented laboratory phenomenon.1,2 It is reflected, for example, in an electrophoretic mobility that is independent,3-6 or nearly so,7 of the amount of charge placed on a polymer of fixed length once a critical charge load has been reached. The condensation of counterions on the high side of the threshold charge density maintains the net value of the polymer charge invariant. Theoretical analyses and computer simulations are consistent with this interpretation.8-26 In the theories, the polymer is usually represented as a linear or helical array of discrete charges or as a cylinder with uniform surface charge. The interaction of charged spheres and planes with counterions differs from the cylinder. Zimm and Le Bret examined the behavior of counterions as they are diluted away from these three geometical shapes.11 For the sphere, all counterions dilute away to infinity; for the plane, all of the counterions remain at a finite distance from the plane; and for the cylinder, a fraction of counterions sufficient to balance the cylinder charge in excess of a threshold value remains at finite distances, and the other counterions dilute away to infinity. Experimental behavior for the electrophoretic mobility of charged spheres does not indicate the qualitative differences in condensation properties anticipated by the Zimm-Le Bret dilution criterion. Instead, the mobility of a charged spherical macroion looks very much like that of a charged linear polymer.27-31 A plot of mobility vs surface charge density exhibits a plateau, as though a threshold value of the charge density initiates counterion collapse on the surface, maintaining a constant net value of macroion charge and hence mobility. Actually, the idea that counterions can condense on spherical surfaces as well as on cylinders was advanced quite awhile ago,32-35 when it was demonstrated from Poisson-Boltzmann † Part of the special issue “International Symposium on Polyelectrolytes (2006)”. * E-mail:
[email protected].
theory that counterion condensation on a sphere occurs at a threshold surface charge density, but in such a way that ZimmLe Bret behavior (no condensation) is recovered in the limit of vanishing concentration. A recent paper by Netz and Orland22 describes how a variational formulation of the theory of fluctuating fields may be used to demonstrate the existence of effective charges on planes, cylinders, and spheres that are less than their bare surface charges. To lend tractability to the calculations, Netz and Orland worked their way down to the mean field level, and the results they obtained for the sphere are related to those previously derived from numerical and asymptotic analyses of the Poisson-Boltzmann equation. Lau et al.36 have studied a charged plane in the context of a two-fluid model for the counterions (condensed and uncondensed), emphasizing strong spatial correlations among counterions collapsed on the plane. They find a first-order transition for multivalent counterions. Above a critical surface charge density, nearly all the counterions collapse onto the plane, so that the effective charge on the plane nearly vanishes. This behavior occurs because an approaching counterion clears away from its vicinity those counterions already condensed, and therefore, it encounters only the bare surface charge of the plane. Univalent counterions are only weakly correlated, however, and there is presumably a close relationship of the results of Lau et al. for the more gradual behavior of univalent counterions to the mean-field calculations. The purpose of the present paper is to explore condensation on spheres, cylinders, and planes from the point of view of standard counterion condensation theory.12,21 In this theory, an undetermined fraction of condensed counterions is assumed at the outset and then optimized by a free energy minimization that points to the competition between electrostatic binding of the counterions and their dissociation entropy. Although our results have various degrees of similarity to the mean-field formulas, the derivations are comparatively transparent, both mathematically and physically. They may be regarded as a refinement and generalization of the simple introductory argument of Alexander et al.33
10.1021/jp0670844 CCC: $37.00 © 2007 American Chemical Society Published on Web 03/08/2007
Counterion Condensation on Spheres/Cylinders/Planes We will consider spheres, both small and large (appropriately defined); cylinders, both thin and thick; and planes. We find that threshold condensation for counterions of arbitrary valence happens for all of these systems, in such a way that Zimm-Le Bret limiting behavior is preserved. Our result for the small sphere is the same as Ramanathan’s from Poisson-Boltzmann theory,34 although as indicated in the section on numerical results we believe it may be literally applicable to small ions (and therefore to the venerable idea of ion pairing) as well as to spherical micelles and other macroions. For the large sphere, we continue to find a clear distinction between condensed and uncondensed counterions, which are blurred in some previous results for this case. The result for the thin cylinder is the familiar canonical one;1,21 the thick cylinder looks more like the plane and large sphere. We hope that the formula for the threshold charge density on a thick cylinder will be useful in understanding why the counterions of thick biological filaments behave in a manner reminiscent of thin polyelectrolytes.37,38 Our results are also related to those of Netz and Orland22 but differ in being restricted to dilute bathing electrolyte. In each of our formulas, the Debye screening length is greater than some characteristic constant length, be it a spherical or cylindrical radius, a linear charge spacing, or the Bjerrum length, depending on the case. The formalism of Netz and Orland is able to handle screening lengths down to zero. In their calculation for cylinders, ion condensation diminishes with increasing salt, ultimately disappearing at sufficiently high salt. For linear polyelectrolytes, this behavior has not been observed in experiments.5,39,40 (A likely reason is that the condensed counterions in real systems are an integral part of the solvated structure of the polymer, held to it by collective short-range forces, and are thus effectively buffered against external salt.) II. Analysis We begin our analysis by listing electrostatic free energies Gel for an impenetrable sphere of radius a, an impenetrable cylinder of radius a, and a planar wall. The sphere and cylinder are immersed in an infinite solution of bulk simple electrolyte characterized by the inverse Debye screening length κ. The wall is bathed on only one side by the electrolyte solution. The free energies represent the work required to charge these structures up to their net surface charge densities after possible counterion condensation. The bare surface charge density before counterion condensation is designated by σ. The counterions have unsigned valence z and are the same as one of the ion species in the simple electrolyte. The free energies ascribe any departure from linear Debye-Hu¨ckel electrostatics to collapse of counterions from the diffuse ion atmosphere onto the surface of the macroion, effectively reducing, or “renormalizing”, the bare surface charge density. This eventuality is represented by inclusion of a factor (1 - zθ)2, where θ is the number of counterions per bare unit surface charge collapsed, or condensed, on the macroion. In fact, the Mayer virial expansion indicates that, for sufficiently dilute electrolytes, the Debye-Hu¨ckel free energy should become a good approximation unless the expansion diverges because of physical ion collapse.41 The product zθ could be zero if it turns out on free energy minimization that no counterions are condensed. In any event, it cannot exceed unity, because our formalism is too simple for consideration of charge reversal. The total number of condensed counterions is Nθ if the covalent structure of the macroion includes N unit charges e on its surface. Also appearing in the expressions for the free energies is the Bjerrum length lB ) e2/DkBT (esu/cgs units) where the denominator is the product
J. Phys. Chem. B, Vol. 111, No. 29, 2007 8555 of the dimensionless dielectric constant D of the pure solvent, Boltzmann constant, and absolute temperature. In eq 3 below for the cylinder, the functions K0(x) and K1(x) are modified Bessel functions of the second kind. Finally, the cylinder and plane are infinite; the finite number N of unit charges on each of them cancels in the subsequent calculation (or, alternatively, it may be regarded as counting the charge in central sections large enough to overwhelm edge effects). With this discussion, the electrostatic surface free energies of the sphere, cylinder, and planar wall are obtained by applying a standard charging procedure to the respective solutions of the Debye-Hu¨ckel linearization of the Poisson-Boltzmann equation in the appropriate coordinate systems,42
Gel ) 2πNkBT(1 - zθ)2(lBaσ/e)f(κa)
(1)
where for the sphere
f(κa) )
1 1 + κa
(2)
and for the cylinder
f(κa) )
K0(κa) κaK1(κa)
(3)
and for the planar wall
f(κa) )
1 κa
(4)
Notice that for the planar wall there is no radius a, and indeed, this quantity cancels from eqs 1 and 4. Further, for a charged plane with electrolyte bathing both sides, f(κa) is half the value given in eq 4. There is an additional entropic component of the overall free energy common to each macroion geometry. The Nθ condensed counterions come from the ion atmosphere in solution, where, to leading order in dilute solution, their translational entropy was -NθkB ln c, where c is the bulk counterion concentration. Except for a multiplicative constant, irrelevant in the present context, c is also the concentration of simple salt, which, importantly for the following development, scales like the square of the inverse Debye length. The lost entropy must be added as a positive contribution to the overall free energy,
Gtr ) -NθkBT ln c
(5)
Note that a more complete expression may be given and is useful, indeed essential, in various contexts,12,21 but it will become clear that our present exclusive focus on the number of condensed counterions does not require anything further. The aim is to determine the conditions, if any, for which the number of condensed counterions is greater than zero, that is, 1 - zθ e 1, the equality marking the threshold condition for condensation. We seek an equilibrium state determined by the vanishing of the derivative of total free energy Gel + Gtr with respect to θ, that is,
∂Gel/∂θ ) NkBT ln c
(6)
We begin with the case of the small sphere, meaning that the radius of the sphere is small compared to the Debye screening length, κa > 1, the ratio of Bessel functions K1(κa)/K0(κa) tends to unity, and σcrit for the very thick cylinder approaches the value common to it, the plane wall, and the very large sphere. The difference between thick and thin (polyelectrolyte) cylinders emerges more clearly if we work in the highly charged polyelectrolyte environment of small κb and use the relation σ ) eξ/(2πalB) between cylinder surface charge density and the polyelectrolye parameter ξ ) lB/b, where b is the axial length per unit charge. Substitution into eq 27 produces ξcrit,
1 ξcrit ) - ln(κb)[κaK1(κa)/K0(κa)] z
(28)
or, for large κa but small κb, The equilibrium condition, eq 6, is
1 4πz(1 - zθ)(lBaσ/e) ) -ln c 1 + κa
(22)
We seek counterion condensation when the left-hand side is asymptotically equal to -2 ln(κlB). In other words, we choose our small dimensionless quantity to be κlB, noticing that κa will not do, because it is O(1), and so -ln(κa) is not an asymptotically large quantity for dilute salt solutions. With this choice,
e(1 + κa) ln(κlB) 1 - zθ ) 2πzlBaσ
e(1 + κa) ln(κlB) 2πzlBa
(23)
(24)
Notice that when κa greatly exceeds unity (the ion atmosphere is a thin layer surrounding the sphere) the critical charge density for the large sphere approaches that for the charged wall, eq 20. It must also be remembered in practical applications that, although the numerical value of κa may exceed unity, the solution must still be dilute in electrolyte concentration, such that the argument of the logarithm, κlB, must be numerically small. The final case to be considered is the thick cylinder. As for the large sphere, “thick” means that the cylinder radius a is comparable to the width 1/κ of the diffuse ion atmosphere surrounding the cylinder, κa ) O(1). The full eqs 1 and 3 must be used for the electrostatic free energy of the charged cylinder, and the equilibrium condition eq 6 then reads
K0(κa) ∼ -ln c 4πz(1 - zθ)(lBaσ/e) κaK1(κa)
(25)
Because κa is not a small quantity, we work instead in an asymptotic environment of small κlB, and proceeding exactly as for the large sphere, we then find that
1 - zθ ) -
e ln(κlB) [κaK1(κa)/K0(κa)] 2πzlBaσ
(26)
which implies the occurrence of counterion condensation for surface charge densities σ greater than the critical value,
σcrit ) -
e ln(κlB) [κaK1(κa)/K0(κa)] 2πzlBa
(29)
In these conditions ((b/a)
