Langmuir 1994,10, 2450-2461
2450
A n Attractive Force between Two Rodlike Polyions Mediated by the Sharing of Condensed Counterions Jolly Ray and Gerald S. Manning* Department of Chemistry, Rutgers, The State University of New Jersey, New Brunswick, New Jersey 08903 Received January 3, 1994@ We calculate the potential of mean force of a pair of identical rodlike polyions, oriented in parallel, in 1:1electrolytesolution. The polyions are modeled as infinitelinear arrays of uniformly spaced monovalent
charge sites. The analysis is accomplished within the framework of counterion condensation theory. The mean force between the polyions is repulsive for distances much less than the Debye length and for distances on the order of a Debye length. There is an intermediate range of distances, however, not small but less than the Debye length, where the polyions attract each other. The predominant cause of the effect is expansion of the condensation region intothe spacebetween the polyions, allowing condensed counterions to be shared with greatly enhanced translational entropy.
Introduction Coulomb's law in a vacuum or constant dielectric medium provides a repulsive r-l potential between two like charges. The potential is generated from Laplace's linear equation and is subject to the superposition principle. The interaction potential between two assemblies of like charges, all in fixed positions, is therefore also purely repulsive; it is the sum of rg-l potentials, i a charge in one assembly,j a charge in the other, over all pairs of charges. If the charges are continuously (and nondeformably) distributed on surfaces or in volumes, the s u m is replaced by an integral, leaving the repulsive character of the interaction potential unchanged. In real solution conditions, two polyions or charged colloid particles interact not only with each other but also with small ions and solvent molecules. In the Mayer diagram theory of ionic solutions,' the lowest-order approximation to the potential of mean force for two charges is the screened Coulomb potential, with distance dependence r-le-". The effect of the small ions is incorporated into the screening factor e-M,and K - ~is the familiar Debye screening length. The solvent is represented by a constant dielectric medium. The screened potential is also subject to pairwise superposition. Thus, in this approximation,the polyions, modeled as assemblies of fured charges, continue to repel. The only change has been replacement of a Coulomb potential with a screened Coulomb potential. A variant of this approximation, also leading to pure repulsion, is use of a screened potential between two effective charges. The effective charge has the same sign as the actual charge but is less in absolute value. It represents a "complex"between the actual charge and counterions which interact with the charge in a manner more intimate than simple screening. This variant, of course, also leads to pure repulsion. Parenthetically, we note that counterion condensation theory does not provide an example of this type of model. Indeed, we obtain an attractive force in this paper, which would be impossible if the theory did not contain elements that go qualitatively beyond the screened potential. The nonlinear Poisson-Boltzmann equation provides a level of approximationhigher than the screened Coulomb potential. It is believed that Poisson-Boltzmann theory, used correctly, predicts nothing other than pure repulsion between like-charged objects. It should be remembered, ~~
* Abstract published in Advance ACS Abstracts, May 15,1994. W M a y e r , J. E.J. Chem. Phys. 1960,18,1426.
however, that the Poisson-Boltzmann mean field is not exact, even for pointlike electrolyte ions. SanchezSanchez and Lozada-Cassou2 provide a recent paper on nonlinear Poisson-Boltzmann theorywith references that lead back to the classical literature. Despite the expected behavior resulting from these familiar methods of analysis, there is no general requirement that like-charged particles repel each other in manybody thermalized systems. Indeed, attractive forces are found in Monte Carlo simulations and on levels of approximation higher than the mean field. It is difficult to rank counterion condensation theory in the statistical mechanical hierarchy from mean field to computer simulation, because the theory does not provide pair potentials either for two condensed counterions or for a condensed counterion and a polyion. Instead, the polyion, or, in the present case, assembly of polyions,is considered to be a single molecular complexconsisting of fixed charge sites and condensed counterions. The noncontinuum, dynamic, molecular nature of the solvent renders the assumption of a continuous twice-differentiable electrostatic potential inexact, even obscure,at the close distances important in macroionic solutions. Counterion condensation theory circumvents such problems, at the cost, to be sure, of more than a rudimentary description of the condensed layer. At any rate, the theory has proved its reliability in many applications. In the present case, it produces an interaction potential between polyions qualitatively similar in shape to potentials reported by other workers using different methods. We proceed to cite examples. In all cases, the solvent is represented by a constant dielectric. Employing a hypernetted chain analysis, Pate? has analyzed the potential of mean force for two identically charged large spheres immersed in 1:1electrolyte,modeled as much smaller spherical ions. He concludes that if the surface charge density of the macroionic spheres is sufficiently high, there is a stable position (minimum in the interaction potential) at small separations. The potential, set to zero a t infinite reference distance between the spheres, rises through positive values with decreasing distance, as expected, but then hits a maximum and falls to negative values before rising again as the spheres approach to contact (see Figure 6 below for a picture of (2)Sanchez-Sanchez,J. E.;Lozada-Cassou, M. Chem. Phys. Lett. 1992,190,202. (3) Patey, G. N. J . Chem. Phys. 1980,72, 6763.
0743-7463/94/2410-2450$04.50/00 1994 American Chemical Society
Attractive Force between Two Rodlike Polyions a potential with this shape). Teubnelff has demonstrated, within the framework of the same approximation, that this shape persists even when the electrolyteions are taken as pointlike (we mention this result, because our own theory contains no size effects). Belloni5has sought for phase transitions in a mixture of spherical polyions, counterions, and co-ions, also using hypernetted chain theory. He finds a transition analogous to a liquid-gas transition, implying the existence of an effective attractive force between the polyions of purely ionic origin. The shape of the interaction potential as a function of distance between spheres6 is similar to Patey’s result. Hypernetted chain theory7and Monte Carlo simulations of the electrostatic interaction between two identically charged, parallel planes are in close agreement. For divalent counterions they converge on a potential with characteristics similar to the Patey-Belloni potential for spheres;Table 1of the paper by Kjellander and Marcelja’ indicates that the planes repel each other at large distances, attract in an intermediate range, then repel again at small separations (again, see Figure 6 below for a qualitatively similar picture). We may mention at this point a report of experimental confirmation of the attractive electrostatic force for planes, realized by mica surface^.^ Monte Carlo simulation of the electrostatic interaction between a sphere and a planelo generates the same type of potential if the counterions are divalent. If the value of the dielectric constant is set lower than 80, an attractive potential appears with univalent counterions as well. A solution of the nonlinear Poisson-Boltzmann equation for two (or indeed any number of) parallel zero-radius lines of continuously distributed charge leads to a purely repulsive potential.llJ2 In this case the primary nonlinear effect is entrapment of the (point)counterions on the lines. The net line charges then interact with a repulsive potential similar to that given by integrating a screened Coulomb potential along the lengths of the lines. In the context of the Poisson-Boltzmann mean field, the counterions trapped on the line charges are uniformly distributed along the lines. 00sawa’~J~ considered the effect of fluctuations away from uniform distribution. The fluctuations on one line are correlated with the fluctuations on the other. Correlated counterion fluctuations provide an intrinsically attractive component to the force, for the same reason provided by the classical description of van der Waal’s forces. A transient excess of counterions in a segment along the length of one line is associated with a counterion deficit in the oppositely located segment on the other line, so that the two segments are effectively oppositely charged and attract each other. Oosawa found that the attractive fluctuation force dominates the repulsive mean field force at small separation distances, whereas the latter determines the behavior at larger distances. The overall electrostaticpotential of mean force therefore rises through positive values as the distance between the line charges decreases from infinity, but then hits a maximum and decreases with further approach of the lines as the attractive fluctuation force takes hold. (4)Teubner, M. J. Chem. Phys. 1981,75, 1907. (5)Belloni, L. Phys. Rev. Lett. 1986,57, 2026. ( 6 ) Belloni, L. Statique et dynamique duns les solutions depolyelectrolytes spheriques; These d’Etat, Pans, 1987. (7)Kjellander, S.;Marcelja, R. Chem. Phys. Let.. 1984,112,49. (8) Guldbrand, L.; Jonsson, B.; Wennerstrom, H.; Linse, P. J.Chem. Phys. 1984,80,2221. (9)Kjellander, R.;Marcelja, S.; Pashley, R. M.; Quirk, J. P. J.Chem. Phys. l f M , 92,4399. (10)Svensson, B.; Jonsson, B. Chem. Phys. Lett. 1984,108,580. (11) Imai,N.; Ohnishi, T. J. Chem. Phys. 1969,30,1115. (12)Oosawa, F.PolyelectroZytes;Marcel Dekker: New York, 1971. (13)Oosawa, F.Biopolymers 1968,6,1633.
Langmuir, Vol. 10,No. 7, 1994 2451 The Oosawa potential therefore has essential features of the potentials discussed above €or spheres and planes. The same qualitative behavior is found for univalent as well as divalent counterions, although for the former, the attractive region is at separation distances between the lines small enough possibly to be masked by a realistic distance of closest approach. Monte Carlo simulation of an approximate model for a parallel assembly of charged rods indicates attraction if the counterions are di~a1ent.l~ In this paper we represent a rodlike polyion by a linear array of discrete charge sites. It will become clear that counterions are not trapped on the line in this model. We fmd that counterion condensation the0ryl6-l~provides a potential of mean force for a parallel pair of these arrays immersed in 1:l electrolyte similar in shape to those that we have discussed above. As the lines of charge sites approachfrom infinity, the interaction potential increases as expected. But when the Debye screening atmospheres interpenetrate, the potential attains a maximum and decreases to negative values before stabilizing at a minimum and rising again. The attractive region is not generated by Oosawa “van der Waal‘s” fluctuations, which are not present in our model. Instead, we find a quantum mechanical analogy in the covalent bond. In their minimum free energy configurationthe two lines of charge sites are surrounded by condensed counterions that are delocalized and belong to both, like the valence electrons in a covalent bond between two atoms. We have found that the Oosawa stabilizing energy is small in comparison with the effect of sharing of condensed counterions, just as in molecular theory van der Waal’s attractions are small in comparison with a covalent bond. The current experimental situation, as it impinges on the possibility of effectively attractive ionic forces between like-charged linear polyions, may not be conclusive but is certainly intriguing. We have already mentioned the measurements of Kjellander et aL9 on mica surfaces. For linear polyions, the papers by Sedlak and Amis19120 report results of dynamic light scattering experiments and contain many references. In a paper that may also be used as a resource for references, Wang and BloomfieldZ1 raise the question of “apparent attraction”between DNA rods and other polyions in their interpretation of neutron scattering data. We would like to keep the considerations in our presentation as simple as possible. The two polyions are identical, their individual charge densities are above the critical value for counterion condensation, and the salt is uni-univalent. Generalizations to polyions of differing charge densities, to a pair of polyions with charges of opposite sign, and t o salts of general valence type will be reported separately. The discussion urges caution in extrapolating our results for an idealized model to real polyelectrolyte solutions.
The Free Energy of Two Interacting Lines of Like Charge We formulate the various contributions that, in counterion condensation theory,17J8define the free energy associated with two infinitely long parallel lines of (14)Guldbrand, L.;Nilsson, L. G.; Nordenskiold, L. J.Chem. Phys. 1986,85,6686. (15)Manning, G. S.J. Chem. Phys. 1969,51,924. (16)Manning, G. S.J. Chem. Phys. 1969,51,3249. (17)Manning, G. 5. Q. Rev. Bwphys. 1978,11,179. (18)Manning, G. S.J.Phys. Chem. 1984,88,6654. (19)Sedlak, M.; Amis, E. J. J. Chem. Phys. 1992,96,817. (20)Sedlak, M.; A m i s , E. J. J. Chem. Phys. 1992,96,826. (21)Wang, L.;Bloomfield, V. A. Macromolecules 1991,24,5791.
Ray and Manning
2452 Langmuir, Vol. 10, No. 7, 1994
'1-
9
0-
9
2 reflects the presence of two lines of charge sites. We will always work with screening lengths K - ~much larger than the charge spacing b. The term e-Kbmay therefore be linearized, and eq 4 takes the form Gt' = -2PkT&1 -
t
h(Kb)
Now let G2e1be the total free energy associated with the ionic repulsion between the two lines. Pairwise summation of potentials of the type of eq 2, with rc given by eq 3, gives US the formula
b 1
G:' = 2PkTg(1 - 8)2&(K@)
-q
i
Figure 1. Two model polyions in parallel orientation with lattice spacing b, separation distance 4,and charge q (or -4) on each lattice site.
univalent charge sites (see Figure 1). The spacing of sites on each line is b; the distance separating the lines is e. The unit of distance is cm. With q the protonic charge (esu), we assume that all sites bear charge q (both lines are polycations), or all sites bear charge -q (both lines are polyanions). We recall the dimensionless charge density 5' of each line
5' = q2fDkTb,
(1)
where D is the dielectric constant of the solvent, k is Boltzmann's constant, and T i s the Kelvin temperature. The following analysis is presented in a self-contained manner, but some of the detail in refs 17 and 18, which consider counterion condensation theory for a single line of charges, has been omitted. Condensed counterions effectively reduce the charge on each site by the fractional amount 1 - 8, where 8 is the number of condensed counterions per site and is to be determined. For simplicity in the presentation of the theory, we take the counterions as univalent. The free energy associated with the ionic repulsion between two sites i a n d j is taken as a screened Coulomb potential
where K is the reciprocal of the Debye screening length. Sites i a n d j may be on the same line or on different lines. If they are on the same line, then r, = nb, where n is a positive integer. If they are on different lines, we have
r y = (e2-t(nb)2)1/2
(3)
where the integer n may be zero. Let Glelrepresent the total free energy associated with ionic repulsion among sites on the same line. Pairwise summation of potentials of the type in eq 2, with ry = nb, yields
G:' = -2PkT&1 -
ln(1 - e-Kb)
(5)
(4)
whereP is the number of charge sites on each line; strictly, the equality is an asymptotic relation as P- 00. The factor
(6)
where KOis the zeroth-order modified Bessel function of the second kind. In obtaining this result,lsJ6 we replaced the sum by an integral, an accurate procedure when the separation distance between the lines is much greater than the site spacing b within a line. In addition to the two electrostatic free energies of eqs 5 and 6, there is an ideal free energy of transfer of 0 counterions (per site) from the relatively dilute bulk solution to the relatively concentrated condensationregion
ioooe/u Gided= 2PkT6 In cs
(7)
In this formula, u is the volume (cm9 of the condensation region per mole charge site (the total condensation volume is m),and 6, as above, is the number of moles of condensed counterions per mole charge site (total moles condensed counterions equals We), SO the numerator in the argument of the logarithm is the local concentration c l d (molarity) of condensed conditions,
The condensation volume u is to be determined along with 8 as part of the calculation. The quantity cs is the concentration (molarity) of free counterions. We are consideringa systemlike NaPolymer immersed in aqueous NaC1, with the salt (NaC1) concentration much greater than the sodium ion concentration contributed by the polyelectrolyte. Then the free counterion concentration is indeed accurately represented by the salt concentration cs. Similarly, the screening parameter is based on the salt concentration, K c . ~ . In terminating the list of free energy contributions with eqs 5-7, written as they are, we have taken some inconsequential short-cuts. We have indicated neither the contribution from the free energy of the solvent nor the contribution from a higher-order term in Gided that cancels it. Other terms that cancel in free energy differences or derivatives are also omitted. We may restate the physical origins of each of the three contributions to the polyelectrolyte free energy. Positive work against electrostatic repulsion is required to assemble P effective charges (1- 6)q, initially dispersed at infinity, onto a line. This amount of work, times 2, since we have two lines, is denoted by GI"'. Given by eq 5, it is evidently positive, since Kb 1. To bring the assembled linear arrays of charge sites from infinite separation distance to separation distance requires that work be done against the electrostatic repulsion between the two lines. This additional work is given by Gzelin eq 6. The process of decreasing the separation distance from infinity to e changes the value of 8, as we shall see. Thus, eq 6 indicates an indirect dependence on e, separate from the explicit dependence of the Bessel function. AB indicated
-
Attractive Force between Two Rodlike Polyions
Langmuir, Vol. 10,No. 7,1994 2453
by eq 5, Glel also changes, indirectly through 8, in the process of approach of the two lines. The third contribution to the free energy, Gidedin eq 7, proves in the sequel to be of great importance. The effective charges (1 - 0)q are net charges that include the charge of the condensed counterions. Condensation of counterions, however, is more than a structureless merging of the counterions into the charge sites on the line. The assembly of two lines at separation distance 8 is associated with a spatial region within which the condensed counterionsrandomly translate. We shall see that the volume (and shape) of this region depends on 0. In other words, there is a local concentration of condensed counterions, given by eq 8, which depends on the separation distance g between the two lines. The entropically derived work required to assemble the local condensed counterion concentration from the more dilute bulk solution is the positive free energy term Gidealof eq 7 and depends on the distance e. Define a total “polyelectrolyte”free energy G(e)
(9) where, as discussed, all three components depend on the separation distance e. The free energy G(e) is the total work of assembling 2P effective charges (1 - 8)q onto two parallel lines separated by distance e, with uniform spacing b between the P charge sites on each line. The corresponding work per site is g(g) g(e)= G(eY2P
(10)
Let us now introduce a free energy difference per charge site A&)
&(e)
= g(g) - g(-)
(11)
Thus defined, &(e) is the total work (per charge site) required to bring two parallel linear arrays of sites, P sites on each array, each array with spacing b, each site with charge q , from infinite separation distance to separation distance e. In other w %rds, &(e) is the potential of mean force of the two lines in parallel orientation. It is the fundamental quantity to be calculated in this paper. It is instructive to begin with a pure Debye-Huckel calculation of &(e). Counterion condensation does not occur in the Debye-Huckel approximation. Therefore 6 is set to zero in our formulas. In consequence, Gideal vanishes, whether at a finite value of e or at = Further, Glel(e)= Gle’(=),so &lel(e)= 0. We are left with the result that in the Debye-Huckel approximation,&(e) = &ze’(e), or, from eq 6 00.
&(e) = kTE&(K@)
(12)
In Figure 2 we portray the Debye-Huckel result for &(e), which essentially amounts to a plot of the Bessel function KO. The work required to bring the two charged lines from infinity in parallel orientation to distance e increases smoothly as the two lines approach each other. The force between the lines (equal to -aAg/ae) is positive, or repulsive, at all distances. We shall see that counterion condensation changes this picture utterly.
The Reference State at Infinite Separation We consider first the reference state at infinite separation distance, g = m. In this paper, when we refer to e as infinite, large, small, etc., we mean a relation to the Debye screening length K - ~ . Further, our theory is a “limiting
50
150
100
200
p (angstrom)
Figure 2. Potential of mean force, Ag(e ,in the Debye-Huckel approximation,eq 12. 6 = 4.2 ( b = 1.7 ,as for DNA, D = 78), K - ~= 96 A (salt concentration 0.001 M).
A
law”, in the sense that it is valid asymptotically as the salt concentration cs tends to zero (together with the polymer concentration, such that the ratio of salt to polymer site concentrations remains large). Since K scales like cSm,the screening length in this limit becomes infinite. By the statement g = =, we mean something precise, K
-1
-m,
(13)
KQ-m
In other words, the screening length K - ~becomes infinite, but e remains much larger than the screening length, such that the ratio of g to the screening length (equal to K g ) also becomes infinite. Loosely,we say that the distance between the two lines of charge sites is much greater than the screening length. In their reference state, the two lines are then completely screened from each other. Since KO(=) = 0, the contribution to the reference free energyg(=) fromgzel,the electrostatic repulsion between the two lines, vanishes, and we are left with the expression,
where 8, and u, are values of the condensed counterion fraction and condensation volume, respectively, at infinite separation distance. This equation for the reference free energy is, of course, exactly the same as the formula for the free energy in single-polyioncounterion condensation theory,17which does not consider polyion-polyion interactions. Analysis of the free energy g(-) proceeds as in singlepolyion counterion condensation theory.17Js The equilibrium condition is
There are two terms in eq 15 proportional to In cs, one from the In K term in eq 14, the other from the explicit In c3 term of this equation. The requirement that an equilibrium state exists in the limit cs 0 means that a solution of eq 15 must exist in this limit. In turn, a solution of eq 15 when csvanishes is possible only if the coefficient of the combined In cs term in eq 15vanishes. Setting the coefficient of the combined In cs term equal to zero yields the equilibrium value of 8,
-
(16)
When this value of 8, is substituted back into the equilibrium condition, eq 15, the result is an equation with a single unknown, the equilibrium value of v,.
2454 Lungmuir, Vol. 10, No. 7, 1994
Ray and Manning
Solving the equation gives us a formula for this value u, = 8neLAv({- 1)b3
(17)
where e is the base of natural logarithms and LA,.is Avogadro’s number. Equations 16 and 17 are the same as the corresponding ones in single-polyion counterion condensation t h e ~ r y . ’ ~For J ~ obviousreasons, neither the condensed counterion fraction nor the condensation volume in the reference state reflects the presence of a second polyion. Both depend only on the structure of a single polyion, namely, the charge spacing b (see eq 1). Now that we know the free energyg(m1of the reference state at equilibrium, we may write a general formula for the pair potential &(e), defined in eq 11
Ag(e)/kT= -150 - el2-
e-’] h(Kb) +
0.15 ’
140
160
180
200
p (angstrom)
Flgure 3. Potential of mean force, &(@I, for large separation distances. Parameters as in Figure 2.
in the range of large separation distances is (18)
In this equation the free energy of the reference (initial) state at infinity has been equilibrated with respect to counterion condensation, and the equilibrium values of 8, and u, from eqs 16 and 17 have been used. The equilibrium condition for counterion condensation has, however, not yet been applied to the final state at finite separation distance e. This condition, ag(e)/atl= 0, is the same as
where u, is the expression in eq 17. Substitution of eqs 21 and 22 into eq 18 gives us the pair potential for large e. To understand the general character of &de) in this range of separation distances, we calculate its derivative with respect to
(19)
where&(Ke) is the first-order member of the same family of Bessel functions as KO(Ke). The right-hand side is negative for all values of e, so &(e) is a monotone decreasing function of e in the range of large distances under consideration, and the force between the two polyions is repulsive. We plot a numerical example of &(e) in Figure 3,where the range ofvalues ofq are restricted to Yarge”separations between the polyions. Comparison with Figure 2 shows qualitatively similar behavior of the pair potential (Figure 3) and the Debye-Huckel approximationto it (Figure 2) for large polyion separations. These distances are of the order of the Debye length. The polyions do not penetrate across each other’s Debye atmospheres. Qualitative similarity with a simple screened repulsion is therefore the expected result. A quantitative comparison of Figures 2 and 3 for overlapping distances indicates that counterion condensation causes polyion-polyion repulsion to be weaker than in the Debye-Huckel approximation. Since condensation lowers the net charge on each polyion, this result, too, is expected. Closer inspection, however, discloses a quantitative feature that, at first glance, seems hard to explain. If weakening of the repulsion is a simple consequence of a lower net charge, then the repulsive energies in Figure 3 should be lower than in Figure 2 by a factor of E-2, or about 18 for the value 6 = 4.2 used in the numerical example. The reason is that each of the polyion charges q is lowered by a factor by condensation, and there is a q2 dependence of the free energy. But the repulsive free energies of Figure 3 are lower than those in Figure 2 by only a factor of about 2. We see here the importance of the entropic effect of changes in the condensationvolume. According to eq 21, the number of condensed counterions does not change in this region as the polyions approach. But the interaction is more complicated than simple screening of net charges, because the net charges have structure, and the entropy of the net charges (the translational entropy of the condensed counterions)
We apply eq 19 under distinct sets of limiting conditions, each leading to a formula for the pair potential in a distinct range of separation distances e. In this way, we shall obtain the behavior of the pair potential for large, small, and intermediate distances.
Large Separation Distances In this section we shall analyze &(e) under limiting conditions differing from those of eq 13, specifically K
-1
--,
e--,
Ke=constant
(20)
Here, the screening length becomes infinite, and so does the distance between the lines, but e remains on the same order as K - ~ . We say loosely that the distance between the polyions is about the same as the Debye screening length. Even more loosely, we call separation distances in this range “large”. We find the values of 8 and u by requiring that the equilibrium condition for counterion condensation, eq 19, have a solution in the limit cs 0. In the derivative of Ag with respect to 8, there are the same two In c. terms as in the reference state equilibrium equation, eq 15. The reason is that the polyion-polyion interaction term corresponding to the term with the Bessel functionKo(Ke) in eq 18 does not depend on c. in the limiting conditions of eq 20. Thus, in the range of large e, as well as at infinite separation
-
e = 1 - -1 5
(21)
The equilibrium value of u differs from u,, however, becausewhen eq 21 is substituted into eq 19,the resulting equation in the unknown v now contains a polyion-polyion interaction term proportional to Ko(Ke). “he result for u
(23)
Attractive Force between Two Rodlike Polyions
Langmuir, Vol. 10,No. 7,1994 2455
decreases as the polyions approach in accord with the contraction of u indicated by eq 22. 0.25 O I
Small Separation Distances
\
p (angstrom)
To obtain small separation distances, we need different limiting conditions K
-1
-=,
=constant
(24)
The separation distance e remains fixed while the screening length K - ~becomes infinite. Certainly, then, the distance between the polyions is small relative to the screening length, e1K-l = K e 0. In the limiting conditions of eq 24 the interaction between the polyions is strong, for the Bessel function KO in Gzel, eq 6, has a singularity at the zero value of its argument q , the ratio of separation distance to screening length. Equation 24 implies that K e 0. The asymptotic relation
-
-
where y is Euler’s constant, y = 0.5772 ...,is thus available for our use. It provides a third In cs term that was not present in the analysis of large polyion-polyion distances. There is another limiting condition implicit in the appearance of the Bessel function, and hence in the use of eq 25, in our analysis. As explained in connection with eq 6, the Bessel function KO(K@) emerges when a certain sum of screened Coulomb potentials is replaced by an integral. The replacement is accurate ifthe linear charge spacing b is small relative to the distance e between the linear arrays (Figure 1). In other words, we need ble 0. For the reference state, and for the range of large e, we were automatically working in this limit, since in both cases, e =. But in the present case of small e, we have chosen the condition e = constant, which means that b must tend to zero. We would like to impose this limit under the restriction 5 = constant, so that familiar formulas of counterion condensation theory like eq 21 remain valid without qualification. For this purpose, noting eq 1, we also let the charge q on each site tend to zero in such a way that q21bstays constant. In turn, this restriction implies that the linear charge density on the polyion becomes infinite, since qlb is then O ( b - 9 We may hope that our analysis is applicable to highly charged, but real polyions, in the same way that single-polyion counterion condensationtheory, valid strictly in the limit of infinite dilution, has turned out to be an accurate description of realistically dilute (and even moderately concentrated) solutions. We proceed to apply the equilibrium condition eq 19 to the formula for &(e), eq 18, using eq 25 for KO(w). There are three terms proportional to In cs that are to be combined. Setting the combined coefficient of these terms equal to zero, we get the value ofthe condensed counterion fraction
-
-
15
10
8
-
-0.75
~
,
(27)
25
30
\ ~
~
-1.25 -It
Figure 4. Potential of mean force, &(e), for small separation distances. Parameters as in Figure 2. where y is defined above in connection with eq 25. Recalling that a requirement for the appearance of the Bessel function KOin our general equations is e >> b, we note that u in the range of “small” is substantially larger than the reference value at infinity. We cannot take e 0 in eq 27. Substitution of eqs 26 and 27 into eq 18 gives us the pair potential &(e) for small e. We record the derivative of &(e) with respect to e in this range
-
(28)
We see that,just as for large e,&(e) increases in monotone fashion as the two polyions approach;the force is repulsive. A suggestion of interesting behavior emerges only in the numerical example in Figure 4. Over most of the range shown, Ag is negative. The assembly of two polyions at small separation distance has lower free energy than the isolated polyions at infinite separation. Since A,&) is zero at infinity, there must be a stable position for a value of e larger than those shown in Figure 4. The curve in Figure 4, if extended to larger e, must pass through a free energy minimum and rise.
Intermediate Separation Distances The evidence for a stable position in Figure 4 is numerical, hence suggestive only. In this section we exhibit a range of separation distances where the function &(e) can be written down as an analytical formula and shown to be monotone decreasing as the polyions approach. The force between the two polyions of like charge is attractive in this range. We shall see that the attractive range of distances is intermediate between small and large distances. There must therefore be a free energy minimum somewhere between small and intermediate distances, although we shall not be able further to specifyits location. Consider the following limiting conditions K
-1 -03,
e-=,
Ke-0
(29)
The screening length becomes infinite, and so does the distance between the polyions, but the polyions are always located at a mutual distance well within the screening length (ratio of e to K - ~= KQ 0). These separation distances are smaller than the “large”values, eq 20, which are on the order of the screening length. The condition e -=guarantees also that they are larger than the “small” values of eq 24. Thus the values of e under the limiting conditions of eq 29 are intermediate between small and large. We cannot implement these intermediate limiting conditions in the same straightforward way as eqs 20 and 24. Instead, we have to exhibit an explicit formula for e
-
It seems reasonable that the value of 8 for small distances between the two polyions should turn out to be the same as for a single polyion of twice the charge density. Substitute eq 26 back into the equilibrium condition and solve for the condensation volume. The result is
20
Ray and Manning
2456 Langmuir, Vol. 10, No. 7, 1994 which is consistent with eq 29. To this end, letx be a fixed number greater than 0 but less than 1, and let eo be a fmed length. It is easily checked that the formula
If we substitute eq 32 into the equilibrium condition and solve for the condensation volume, we get a formula with a complicated dependence on the polyion-polyion distance (through the parameter x )
-(-)1
1 242-x)
u = 2-x 2
is consistent with eq 29; indeed, as K tends to zero, Q becomes infinite while K e vanishes like ~ l - In ~ .the sequel it will also prove useful to havex as a function
[email protected] eq 30 for x , we get
(31) That eq 30 bridges the gap between small and large values of 4 becomes especially clear if we include the end points 0 and 1for the parameter x . When x = 0, eq 30 states that e = eo, a constant. Thus, the value x = 0 corresponds to a ”small” value of e (see eq 24). On the other extreme, if x is set to unity in eq 30, the statement becomes q = 1,which, according to eq 20, means that e is “large”. Further, if 0 < x1 < x2 < 1,eq 30 implies that e(x2) is larger than e(x1) (in the sense that e(xdle(x1) as K 0). In summary, as x varies from 0 to 1,e begins as a small value, runs through an increasing continuum of intermediate values, and winds up as a large value. A disadvantage of our procedure of working in limiting conditiop.sis its inabilitytoprovide smoothnessconditions at the transitions from small to intermediate, and intermediate to large, values of e. The value of eo is arbitrary, as is the value unity for the constant value of K e a t x = 1. We will be able to demonstrate the existence of a minimum in the pair potential in the transition from small t o intermediate values, and a maximum in the transition from intermediate to large values, but we will not be able to locate these extrema other than qualitatively (at their stable position the two polyions are separated by a distance well within a Debye length). To proceed with the analysis, notice first that the asymptotic form eq 25 may be used for the Bessel function in eq 18, since, with intermediate as well as small values of e, we are working in limiting conditions such that KQ 0. Then, replace K@ by the expression in agreement with eq 30. Apply the equilibrium condition eq 19to the resulting formula for &(e). In the equilibrium equation there appear three In c8 terms. The combined coefficient of these terms contains the unknown 8and the parameter x . Setting the combined coefficient equal to zero, we find
-
-
-
-
(32) Since the value of x corresponding to the lower bound of the intermediate range of polyion separations is 0, we recover eq 26, the result for 8 in the range of small e, for the smallest separation distance of the intermediate range. For x = 1,correspondingto the largest of the intermediate distances, eq 32 is the same as eq 21, the value of 8 for large separations. As the polyions approach from large to small distances, the number of condensed counterions progressively increases. The distance dependence becomes explicit when we use eq 31 for x in eq 32 (33)
e2Y/(2-x)
6-1
u,
(34)
This expression is continuous at e = 8 0 , that is, at x = 0,
with u as given by eq 27 for small e. It would be continuous (but is not) a t K e = 1,x = 1,with eq 22 for large e if we could use the asymptotic form eq 25 in eq 22 (but we cannot). This type of comparison does not reflect inconsistencyin the analysis, but it does indicate its limitations. We can consistently define small, intermediate,and large ranges of e, and derive consistent expressions for the pair potential in these regions; but we cannot handle the transitions from one region to the next. Following our standard procedure, we substitute eqs 33 and 34 into eq 18,thus obtaining a formula for the pair potential in the intermediate range of polyion-polyion separations. We inspect the distance dependence
We seek conditions for which dAg/de is positive, implying an attractive force between the polyions. The factor to the left of the bracketed terms is negative, since K e o is small. We therefore want the entire bracketed factor to be negative. We are studying the case 5 > 1(existence of condensed counterions in the reference single-polyion state at infinity). Since 0 < x < 1,the factor 4 - (2 - x1-l is alwayspositive. The term y - In 2 is a negative number, -0.1159.... The term ln(b/eo) is negative, since b/@ois a small quantity. Thus the sign ofthe bracketed factor rests on the interplay between the negative ln(b/eo)term and the positive term -ln(Keo). Recall that we are working in the dual, and independent, limiting conditions K e O 0 (or, equivalently, since eo is a constant length, K 0)and b/eo 0 (Le., b 0). We can have negative values of the bracketed factor in eq 35, over nearly the entire range of intermediate polyion-polyion distances, in the following way: choose a fixed value x1 of x as close to 1as we wish, and, for all x , 0 < x < xl, let the term 2[5 - (2 - x)-lI ln(b/eo) tend to more strongly than ln(Ke0). In these conditions, the force between the polyions is attractive throughout essentially the entire range of intermediate separation distances. Although the conditions for an attractive force may at first glance seem arcane, they are not, as the numerical illustration in Figure 5 shows.
-
-
--
--
Thermodynamic Description of the Physical Origin of the Attractive Force We have looked in Figures 3-5 at fragments of the pair potential in different regions of the polyion-polyion separation distance. The fragments are combined in Figure 6 to get an overview of the entire function. Although we are unable to draw a continuous curve with smooth connections between the fragment, we have from Figure 6 an impression of a single minimum located somewhere in the transition from small t o intermediate distances, and a single maximum in the transition from intermediate to large distances. When the polyions approach from infinity, the interaction force is increasingly repulsive as long as the separation distance is on the order
Attractive Force between Two Rodlike Polyions
Langmuir, VoE. 10,No. 7, 1994 2457
p (angstrom)
-0.2
I
60
70
0.5
I\ ' .
\\. . . , . . ,
,
,
p (angstrom) , . , , , , ,
,
,
.
,
,
.
,
.
,
P
8 -0.5
-1
.
.- -
'
Age1 + Agideal
Figure 5. Potential of mean force, &(e), for intermediate separation distances. Parameters as in Figure 2.
0.25 0.5
A
+
-
p (angstrom)
-0.2 -0.4
-'1
0
0
-1.2
Figure 6. Potential of mean force, &(e), across the entire range of separationdistances. The plot combinesFigures 3-5. Parameters as in Figure 2.
of a Debye length. The force becomes attractive only when the polyionshave mutually penetrated each other's Debye atmospheres. Finally, when the polyions approach to within small distances, the force again becomes repulsive. We ask whether we can trace from the theory the physical origin of the interesting behavior a t distances less than K-1.
From eq 9 we see that it is possible to analyze &(e) as the sum of two contributions
The electrostatic contribution is associated with the direct ionic interactions of eq 2, summed over both intra- and interpolymer pairs of charge sites, while the ideal term is the work of transferring 0 counterions per charge site from bulk solution to the condensation volume u. In Figure 7 we have plotted &(e), &,I(@), and &ideal(@) for all three ranges of polyion separation distances. Both &,I(@) and &ideal(@) show the same trends as the total interaction free energy &(e). In particular, both contribute to the attractive force for intermediate distances. It is evident, however, that the ideal transfer free energy &ideal(@) is the dominant influence. It is easy to interpret the trends in &el(@), which is a painvise sum of ionic repulsions between charge sites. As the two lines of charge sites approach in the region of large distances, the net charge on each site (including condensed counterions) remains constant, eq 21, and &de) increases simply because the two polyions of like charge repel each other. On further approach into the range of intermediate distances, there is an increase in the number of condensed counterions, eq 32,and the net charge on each site decreases. The decreased repulsion among charge sites due to the decreased charge dominates the increased repulsion due to the closer approach of the sites, and the electrostatic free energy of interaction
I
I
140
160
180
200
p (angstrom)
Figure 7. Analysis of the potential of mean force into its electrostatic and ideal components: (a, top) small distances; (b,middle)intermediatedistances;(c,bottom) large distances. Parameters as in Figure 2.
decreases. Described in terms of physical forces, the matter is simply that the opposite charge ofthe additional condensed counterions pulls the polyions together. The approach of like-charged polyions in the range of intermediate distances is a spontaneous process. One of the reasons for this interesting behavior is that additional counterions condense on the assembly of two polyions, and the direct attractive force between the commonly held condensed counterions and the polyions pulls the assembly together. According to Figure 7, however, direct electrostatic forces are not the major reason for spontaneity of approach of like-charged polyions. The dominant effect is the trend with polyion-polyion separation distance of the entropic transfer of counterions from bulk solution to the smaller condensation volume. We proceed to make transparent the physical processes involved. We may write down the expression for &ideal(@) from eqs 7 and 8. In units of kT, it is &ideal(@)
= 0 ln(clocal~c.J - 0, ln(cwl,alc,)
(37)
where the dependence on separation distance e is carried by 8 and c l d , the number (per site) and local concentration,
Ray and Manning
2458 Langmuir, Vol. 10, No. 7, 1994
>
~-1
\ \ \ \
2000
\ \
1000
I
50
- - 100
c
150
-
-
200
50
100
150
200
p (angstrom)
p (angstrom)
Figure 8. Condensation volume as a function of separation distance. Parameters as in Figure 2.
Figure 9. Local concentration of condensed counterions as a function of separation distance. Parameters as in Figure 2.
respectively, of condensed counterions, the analogous quantities in the reference state of infinite separationbeing denoted by subscript or superscript 00. Equation 8 indicates that the local condensed counterion concentration is a ratio of 8 to u , the condensation volume. There are thus two factors to consider, the variation of 8 with e, and the variation of u with e. Now the difference between 8 and 8, is not a large number. It equals at most 1/(2() (eqs 16 and 26) or 0.12for the numerical example we have been using, 5 = 4.2 (the smallest value of 5 consistent with counterion condensation is unity). Besides, the transfer of additional condensed counterions from bulk solution to the relatively small condensation volume is a process with negative entropy; by itself, it does not lower the free energy but instead raises it. It is a factor contributing to increased repulsion between polyions, not attraction. Increased counterion condensation on approach of the polyions is not the reason for the interesting behavior of the pair potential between the polyions, and we turn our attention to the condensation volume u. The condensation volume is given by eqs 22,27,and 34 as a function of e over all three ranges of separation distances, long, short, and intermediate,respectively. We plot u ( e ) in Figure 8. ComparingFigure 8 with Figure 6, we note a striking inverse correlation of u(Q) and the polyion-polyion interaction free energy &(e); decreases of &(e) are accompaniedby increases of the condensation volume, indeed, very large increases, as indicated by the numerical scale in Figure 8. Since u ( e ) is the dominant influence on the behavior of &(e), we have found the beginning of our physical interpretation. As the polyions approach from infinity into the range of large separation distances, the condensation volume decreases slightly (Figure 8);the translational entropy of the condensed counterions decreases,thus increasing the interaction free energy, and the polyions repel each other. The repulsion is not primarily caused by direct electrostatic repulsion between the like-charged polyions. The predominant reason for repulsion (spontaneity of polyion drift away from each other) at long distances is the increased translational entropy of the condensed counterions as the polyions move apart. When the polyions are moved into the range of intermediate distances, the condensation volume increases dramatically on further approach, the translational entropy of the condensed counterions increases, and approach of the polyions is a spontaneous process. The polyions attract each other. On still further approach into the range of short distances, the condensation volume contracts sharply, and the polyions repel each other, because closer approach involves a decreased translational entropy of the condensed counterions. To interpret the polyion-polyion interaction correctly, it is crucial to realize that the increase in the number of
condensed counterions entailed by the approach of the two polyions is not an important factor. It is the spatial rearrangement of all of the condensed counterions that predominantlydetermines the behavior of the interaction free energy. In the region where the polyions attract each other, all ofthe condensed counterions,both those already condensed on the single polyions at infinite separation and the additional condensed counterions, occupy a common volume u, a volume that expands greatly as the polyions approach. To reinforce this picture, we have constructed Figure 9, a plot of the local concentration of condensed counterions, according to eq 8. In the region of attraction, the local condensation of condensed counterions diminishes by an order of magnitude as the polyions approach, even though the number of condensed counterions increases somewhat (from 8 = 0.76to 8 =
0.88). Geometric Description of the Physical Origin of the Attractive Force In the range of separation distances where the polyions attract each other, the condensed counterions occupy a certain region of space. All we know of this region at this point of our analysis is the value of its volume u. To gain further insight, we must ask ourselves where the condensed counterions are located. In other words, what is the shape ofthe condensation region, and how is it situated relative to the polyions? We begin with the single-polyion case, which has served as our reference state of infinite separation of the two polyions. If we neglect the small perturbing influence of the discrete spacing of charge sites, symmetry demands that the condensationregion be a cylindricalvolume with the polyion (a line charge) at the axis. The surface separatingthe condensation region from the bulk solution is therefore a cylinder. The radius r of the cylinder may be numerically determined from the formula u = dbLA,, where u is given by the right-hand side ofeq 17,and lengths cm, r works out are in cm. When 5' = 4.2,b = 1.7 x to equal 14A. (Note that in previous work on the singlepolyion case,17we modeled the DNA polymer as occupying a cylindrical region of radius 10 A. In that model, the condensation region is the space between two coaxial cylinders with inner radius 10 A and outer radius 17 A. In this paper we would like to maintain strict consistency with the line-charge model, perhaps at the cost of some realism). Figure 10 shows a cross-section through the parallel assembly of two polyions separated by "infinite" distance (i.e., a distance well outside the Debye screening length). In this perspective the line-charge polyions appear as points at the respective centers of circles of radius 14A. The circles are cross sectionsofthe cylindrical
Attractive Force between Two Rodlike Polyions
Langmuir, Vol.10,No. 7,1994 2459 by the Cassinian oval. There are two cases. If a < /3
-100
-50
50
100
Figure 10. A cross section through an assembly of two arallel The lines of charge sites at “infinite”separation (e = 300 circlesbound the condensation regions. Parametersas in Figure
1).
2.
surfaces separating the condensation regions from bulk solution. The condensed counterions (not shown) are uniformly distributed within the circles. We would now like to arrive a t the results for the singlepolyion case in a different way, which can be generalized to the two-polyion parallel assembly. Since counterion condensation theory takes the condensed counterions as uniformly distributed in the condensation volume, the condensed counterions are, in particular, uniformly distributed on the surface (or, in cross section,closed curve) bounding the condensation volume. Assuming a Boltzmann distribution, we find that the bounding curve (in cross section) is an isopotential. The potential of a line charge is proportional to the Bessel functionKO(Kr),where r is distance from the line. Distances r on the scale of the condensation region are less than the Debye length, so, in counterion condensationtheory, K r is small. The Bessel function is then approximated by -ln(Kr) = -In K - In r. The isopotential curves at fixed ionic strength are thus given by r = constant, or, by circles. Now consider a cross section through the two parallel lines of charge a t finite distance of separation. The potential at a point a t distance rl from one ofthe lines and distance r2 from the other is proportional to KO(Kr1) Ko(Kr-2) = -1n(Krl) - h ( u - 2 ) = -2 In K - ln(rlr2). The isopotential curves are those with rlrz = constant. The loci of points which have constant product of distances from two fixed points are closed curves known as the ovals of Cassini (they are fourth-order generalizations of ordinary ellipses).22The curve bounding the condensation region is thus a Cassinian oval. The family of Cassinian ovals is generated by two parameters, the distance betweeen the two fixed points and the value of the constant product r1r2. In our problem the distance between the two fxed points is the polyionpolyion separation distance e, and for a given value of e, we want to find the Cassinian oval that bounds a condensation region of known area A (twice the condensation volume u per site is equal toAb, soA is known from the known values of u and site spacing b). We will show that a unique Cassinian oval is determined by given values of g and A, and thus we will have the shape of the condensation region as it changes with polyion-polyion separation distance. As Cassinian ovals are easily graphed, we will be able to display our results visually. We begin with a more formal presentation of the Cassinian ovals. Consider two fxed points (the foci) a distance 2 a apart and the locus (a Cassinian oval) of a point whose bipolar coordinates rl and r2 with respect to the foci satisfy the relation
+
rlr2 = p2 where p2 is a constant. The equation of the oval in polar coordinates (r,6)with respect to an origin at the midpoint of the line connecting the foci is r4 - 2a2r2cos 26 = p4 - a4
(39)
It is a textbook exercise to calculate the area A enclosed (22) Lawrence, J. D. A Catalog of Special Plane Curues; Dover: New York, 1972.
A = 2p2E(m)
(40)
where E(m) is the complete elliptic integral of the second kind with parameter m = a4/b4. If a > 8,
U
where K(m) is the complete elliptic integral of the first kind with parameter m; in this case m = p4/a4. Note that the two formulas for A coincide when a = p. In our application of Cassinian ovals, we want to find the boundary of the condensation region for a given separation distancee between two polyions. The boundary is a Cassinian oval described by eq 39 with a = e/2. We need the value of p corresponding to g in order to draw the specific Cassinian oval bounding the condensation region. Corresponding to g, we know the condensation volume u from eq 22, eq 27, or eq 34, depending on whether 8 is large, small, or intermediate, respectively. The area A of the cross section of the volume is known from the formula A = 2vlL~&(v is defined on a ‘per mole site” basis, whereas the cross section A involves two sites, one on each polyion). Hence, for given g we know A. Either eq 40 or eq 41 is then numerically invertible to yield the corresponding value of p. With both a and /3 known, eq 39 provides a specific Cassinian oval, which may be graphed, for the given value of g. Figure 11 shows the results for the numerical case employed throughout this paper. Starting with large distances between the polyions, we see that counterions condense separately around each polyion. As the polyions approach, the condensation regions interpenetrate, forming a greatly expanded common volume filling the space between the polyions. The two polyions share a common population of condensed counterions. Merging of the two condensation regions into a single one occurs in a narrow range of distances, from about 82 A to 78 A in Figure 11. Finally, on very close approach, the common region contracts down on the polyions toward a small cylindrical condensation volume characteristic of a single polyion with twice the charge density. Figure 6 indicates that the polyions attract each other in the 50-90 A range, which in Figure 11 is where, with decreasing distances, the separate condensation regions distort, interpenetrate, merge into a single common region, and greatly expand to fill the space between the polyions. There is a distance a t which the polyions form a dimer that is stable relative to parallel displacement(somewhere between the first two distance ranges in Figure 6). The counterions condensed on the dimer are shared by both polyions and stabilize the dimer through their enhanced translational entropy. Discussion We have modeled a rodlike polyion as a linear array of uniformly spaced charge sites. We have calculated the pair potential between two such lines of charge sites, oriented in parallel, as a function of their separation distance. As the lines approach, they repel each other, and the potential increases. This behavior persists, however, only while the distance of approach remains on the order of a Debye length. When the lines of charge come inside a Debye length, the potential attains a maximum, then decreases; the lines of like charge attract each other. Finally, on still closer approach, the potential turns up again, creating a minimum, and the lines then
Ray and Manning
2460 Langmuir, Vol. 10, No. 7, 1994
25
75
p =looA
p =%A
p =a2 A
p =20A
p =80A
0
75
p =lOA
p= 7 a ~
p =70A
p =5 A
p =2 A -75
-50
-25
23
50
75
p =SOA Figure 11. Evolution of the condensation region as the two polyions approach each other. Parameters as in Figure 2.
repel each other as they move inside this position of stability. The shape of the potential as a function of distance is qualitatively similar to the potentials found by other workers (see the Introduction) for a pair of spheres, a pair of parallel planes, and a sphere and a plane.
Our analysis is rooted in the limiting procedures of counterion condensation theory. An advantage of the method is that the reason for the attractive force is laid bare, at least within the framework of the model. As the lines of charge approach within a Debye length, their layers ofcondensed counterions intersect and merge into a greatly
Attractive Force between T w o Rodlike Polyions expanded shared region of space between the lines. Stabilization of the polyion dimer is achieved by the increased translational entropy of the commonly held condensed counterions. A disadvantage of the method is its inability to specifyother than qualitativelythe location of the minimum and the maximum in the interaction potential. We reemphasize, however, that the interesting non-monotone behavior of the potential occurs at distances within a Debye length. It is our impression from the experimental literature (see references in the Introduction) that if the attractive part of the potential that we have predicted for two lines of charge is present in solutions of real polyelectrolytes, it is most likely to be at low ionic strength. All of the numerical examples given in this paper have been at M salt. We have, however, looked at a wide range of ionic strengths and find only a slight dependence on salt concentration. Since the predicted attraction is an effect of the condensed counterions, insensitivity to bulk salt concentration is the expected behavior. How then can we
Langmuir, Vol. 10, No. 7, 1994 2461 reconcile theoretical lack of dependence on electrolyte concentration with the impression that an attractive force in real solutions, if itexists at all, is significant only a t low ionic strengths? A possible answer lies in the model of the polyion as a line of pointlike charge sites. The minimum in the pair potential for two such lines is located at a separation distance well within a Debye length. For any ionic strength, two infinitely thin lines can approach within a Debye length. But two real rodlike polyions, possessing nonzero radial extent in addition to axial length, can approach within a Debye length only if the ionic strength is relatively low.
Acknowledgment. The research reported herein has been partially supported by the U.S.Public Health Service under Grant GM36284. We are grateful to Luc Belloni, Victor Bloomfield, and G. N. Patey for informative correspondence and discussion.
