Langmuir 1994,10, 962-966
962
Fluctuations of Counterions Condensed on Charged Polymers Gerald S. Manning* and Jolly Ray Department of Chemistry, Rutgers, The State University of New Jersey, New Brunswick, New Jersey 08903 Received September 28,1993. In Final Form: November 29, 199.9
We derive a formula for the mean-square number fluctuation of counterionscondensed on a polyion. The local average concentration of condensed counterions tends toward a finite value greater than zero as the charge density of the polyion tends from above toward ita critical value, but the concentration fluctuations increase without bound. As an exercise in the applicatiQnof the formula, we calculate the mean-square fluctuating dipole moment and, thereby, recover a previously derived formula for the static low-field polarizability of the condensed counterions. Introduction
nonconducting solution, the fluctuation-dissipation theorem5 states that
The dimensionless charge density 5 of a linear polyion1P2 is defined in electrostatic (cgs) units by
C; = q2/DkTb
p = (m2)/kT
(1)
where q is the unit charge, D the dielectric constant of the solvent, k T the product of Boltzmann constant and Kelvin temperature, and b the average spacing of charge sites along the contour of the polyion (for DNA, along the double-helical axis). If C; exceeds the critical value 14-1, where Z is the valence of the counterion, then there is a layer of counterions condensed on the polyion. The number of condensed counterions per charge site may be denoted by 0, and its equilibrium average value by 6'0. The equilibriumvalue is uniform along the length of the polyion (except near its ends3v4)and is given by the formula 6'0
=
&-m) 1
Equation 2 for 6'0 is derived as a "limiting law," that is, from the properties of the free energy in the limit of vanishing ionic concentrations. In this limit the minimum at 6'0 is infinitely deep; the sides of the free energy curve as one proceeds away from 6'0 in either direction have infinite slope. In these circumstances, we do not expect fluctuations of the number of condensed counterions. However, eq 2 may be applied at nonzero ionic concentrations as well, up to quite high values.' At nonzero concentrations,the free energy curvehas a typical parabolic shape in the neighborhood of its minimum, and 6'0 takes on the significance of the thermally averaged value of a fluctuating quantity 0. In the next section we will derive a formula for the mean-square fluctuation of 6' and examine ita behavior as 5 14-1. The low-field static polarizability p of a solution is defined as the ratio mlE, where m is the average dipole moment induced by a small static electric field E. For a
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* Abstract published in Advance ACSAbstracts, February 1,1994. G.S. Q. Rev. Biophys. 1978,11,179. (2) Manning, G.S. J. Phys. Chem. 1984,88,6654. ( 3 ) Odijk, T.Biophys. Chem. 1991,41,23. (4)Odijk, T.Physica 1991,A176,201. (1) Manning,
0743-7463194124lO-Q962%04.5OI 0 ,
(3)
where (m2) is the mean-square zero-fieldfluctuating dipole moment. Having at hand a formula for the fluctuating number of condensed counterions, we can derive from it a formula for the contribution of condensed counterions to ( m2) and thus top. In this way, we recover a result for the condensed counterion polarizabilitypreviouslyderived directly from its definition.6~~ The polarizability result is of proven interest,819so it is nice to see that is is based on a theory that consistently handles both zero-field fluctuations and linear response to a small field. Both derivations are based on the assumption that the condensed counterions polarize but do not conduct. Discussion of this assumption, as well as of the fluctuations of 6' at the critical point, appears in the discussion section. We are pleased to acknowledge Oosawa's treatment of fluctuations in polyelectrolyte solutions10 as an important motivating factor in our work. With the benefit of years of hindsight, we may have been able to introduce some improvements of his model and results. Number Fluctuations of the Condensed Counterions Let 6' be the number of condensed counterionsper charge site on the polyion, and let g(6') be the free energy per charge site. The expression for g in counterion condensation theory1v2is g(6') = -(1- (q8)'C;kT ln(Kb)
+ 6'kT l n ( VCm )
(4)
where the quantities K , u, and c, not defined in the Introduction, are, respectively, the Debye screening constant (cm-9, the volume (cm3per mole charge site) within which counterions are considered to be condensed, and ~~
(5)Landau, L. D.;Lifshitz, E. M. Statistical Physica, 3rd ed., Part 1; Pergamon: Oxford, 1980. (6)Manning, G.S.Biophys. Chem. 1978,9,65. (7)Manning, G.S.J. Chem. Phys. 1993,99,477. (8)Saif, B.; Mohr, R. K.; Montrose, C. J.; Litovitz, T. A. Biopolymers 1991,31, 1171. (9)Bowers, J. S.;Prud'homme, R. K. J. Chem. Phys. 1992,96,7135. (IO) Oosawa, F. Polyelectrolytes; Marcel Dekker: New York, 1971.
0 1994 American Chemical Society
Langmuir, Vol. 10, No. 3, 1994 963
Fluctuation of Counterions Condensed on a Polyion
the concentr@ion(molarity)of counterionsfrom the buffer salt. The counterion from the salt is the same species as that of the polyion. Furthermore, c may be small but at any rate is much larger than the concentration of charge sites on the polyion. Finally, under these conditions, c may be thought of simply as the bulk counterion concentration, so that the argument of the logarithm in the second term of the right-hand side of eq 4 is the ratio of the local concentration of condensed counterions to the bulk counterion concentration far from the polyion. Now let 80 be the equilibrium (average) value of 8, and let the differences 8-00 and g(8)-g(80) be denoted, respectively, by A8 and Ag. Expand Ag in powers of A8 up to second order. The first term of the expansion is dgl d81+eoA8. This term vanishes because the derivative is zero,.g being a minimum at equilibrium. Indeed, if the derivative is computed from eq 4 and set to zero, we get eq 11 of ref 1, from which eq 2 for the equilibrium value 80 was derived. The remaining term is (1/2)a2g/d821e,~(A8)2. With the second derivative as calculated from eq 4, we have,
- -
ations of the number of condensed counterions to exist in the limit of vanishing concentration. In accord with our expectation, we seerthat 0 as K 0. The polyelectrolyte interaction free energy becomes infinite in this limit and totally dominates random thermal energy. It is of interest to ask what happens when the polyion charge density t approaches ita critical value lql. This value is that which marks the onset of counterion condensation. According to eq 2,Bo 0 as 5 lql.The root-mean-squarerelative fluctuation is ( 1/2/80, and, from eq 6, tends to infinity like 1/801/2as 80 0. Thus, although the average number of condensed counterions is small near the critical point, the fluctuations away from the average are large (except in the theoretical limit of vanishing concentration). This behavior calls to mind the density fluctuations of an ordinary fluid near ita critical point.6 If we want to deepen the analogy, we should examine the local concentration of condensed counterions,not simplytheir number. We have defined the condensation volume u in eq 4 as a volume in cm3per mole charge site on the polyion. With this definition, the local concentration of condensed counterions in molarity units, which we shall call p, is
- -
(5)
We recall that Ag is the deviation of the free energy from its minimum value per charge site. Let S designate the number of charge sites in the segment of the polyion, the fluctuationsof which we wish to consider. The segment may be the entire polyion (or, more precisely, an interior segment extending out to a distance of about a Debye length from the ends3p4),or it may consist of only one interior charge site, or it may be an interior segment of any intermediate size. Let AG be the free energy difference for the segment, so that AG = S Ag. But (AG) = (1/2)kT, where the brackets imply a thermal average. Therefore, from eq 5
our desired result. The structure of this formula is not hard to interpret. If the bracketed factor in the denominator is at first ignored,and the equation is multiplied through by S2,the left-hand side becomes the mean-square of the excess number of condensed counterions in the segment, while the right-hand side becomes 8oS, the average number of counterions condensed on the segment. The formula is then nothing but the ideal gas formula for themean-square fluctuation of the number of gas particles in agiven volume elements (which may be obtained from the Poisson distribution without passage to the Gaussian limit and, hence, is not restricted to a large number of particles). The bracketed factor must then arise from ionic interactions among the condensed counterions and between the condensed counterions and the polyion. This factor is always greater than unity (Kb being less than unity in the approximation of eq 4), and so makes the fluctuations smaller than they would be in the absence of interactions. The net interaction is an effective attraction between the polyion and the average equilibrium layer of condensed counterions(effective,because the net interaction includes free energy gradients of entropic origin). A greater number of counterions than the equilibrium number costa free energy becausethe counterions effectivelyrepel each other; a smaller number costa free energybecause the counterions are effectively attracted by the polyion. We indicated in the Introduction that we did not expect thermal fluctu-
= 1o3wU
(7)
A general formula for u is given by eq 13 of ref 1. For the usual case of buffer salt of type 2 1 or 1:14 (examples: NaC1,Z = 1, Na+the counterion; NazSO4,Z = 2, Sodz the counterion)
where A is a universal dimensionlessconstant, 80 is given by eq 2, and b, as before, is the spacing of the charge sites. In counterion condensation theory, u is not a fluctuating quantity. At the critical point, t = 14-l,and, from eq 1, bmit = const X 1 4, whereupon
where B is a constant. We can now see from eqs 7 and 9 that phto, the local equilibriumconcentrationof condensed counterions at the critical point, has a nonzero value, even though 80 itself, a measure of the number of condensed counterions, vanishes at the critical point. For univalent counterionsin water at 25 "C, p & ' = 0.068M, asubstantial concentration. In summary, the equilibrium number of condensed counterions vanishes at the critical point, but so does the condensation volume, in such a way that the local equilibrium concentration of condensed counterions is high even at the critical point. Turning to the fluctuations of the local concentration of condensed counterions,and letting po be the equilibrium value of the local concentration, not necessarily at the critical point, we see from eq 7 that
where the asymptotic relation pertains to the approach to the critical point and follows from eq 6. The relative rms local condensed counterion concentration, like the relative rms number of condensedcounterions,divergesto infinity as 80 0 at the critical point.
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964 Langmuir, Vol. 10, No. 3, 1994
Manning and Ray
Application to the Fluctuating Dipole Moment Suppose that a static electric fielil E is applied to the polyelectrolyte solution. The condensed counterions polarize in the field and produce an average dipole moment m on each polyion. If the field is small, the induced dipole is linear =pE
(11)
We refer to the proportionalityconstantp as the condensed counterion polarizability. Even when there is no field applied, there is still an instantaneous dipole moment on each polyion, one that fluctuates with time as the number of condensed counterions on different segments of the polyion fluctuate. Under the assumption that the condensed counterions polarize but do not conduct, the fluctuation-dissipation theorem, stated in eq 3, provides a relation between the zero-field mean-squarefluctuating dipole (m2)and the polarizabilityp . In a previous paper: dimensionless measures p, a,and e were introduced for, respectively, m , p , and E
of the value of Am, which can then just as well as be the midpoint of the polyion, Am = ‘/z. With this value of Am, the term proportional to Bo, which is the zero-field average dipole, vanishes, and we may write the average induced dipole
(17)
where r( is now simply the average condensed counterion dipole in the field. The result for the polarizability a = ;/e is
Our goal now is to find a formula for the mean-square zero-field fluctuation ( p 2 ) and to see whether or not ( p 2 ) and a are related by eq 14,as the fluctuation-dissipation theorem demands. The first step is to write an expression for the zero-field dipole moment p of an arbitrary instantaneous condensed counterion distribution B(X). In analogy with eq 17, we set
where the only quantity thus far undefied is L, the length of the polyion. In these dimensionless quantities, the polarization equation is
(19)
and the fluctuation-dissipation theorem takes the form
Next, square both sides of eq 19, write the square of the integral as a double integral, and take a time average. The result of these operations is
Note that Llb is the totalnumber of charges on the polyion. In previous work,s7 we derived a formula for a directly from ita definition in eq 13 by adding a term -B(s)ZqE(s -50) to the free energyg. This term represents the energy of the condensed counterions in the field and embodies the assumption that the condensed counterions polarize but do not conduct. It recognizes that the number of condensed counterions in the field is a function of position s along the length of the polyion. The reference location SO is the point on the polyion where the time-averaged value of 0 is the field-free value Bo; for small fields, 80 coincides with the midpoint of the polyion. The average condensed counterion dipole moment A i induced by the field is defined as the difference between the average condensed counterion dipole in the field and the average condensed counterion dipole when the field is zero. In dimensionless quantities
where the term of the integral proportional to Bo gives the average zero-field dipole, X is a measure of length along the polyion
x = SIL
(16)
and A, is the position on the polyion with-respect to which the moment is taken. Thus defined, A p is independent
where the integration region R is a unit square in the AX’ plane, R = {OSXSl;OSX’Sl}. At this stage, we must say something about the structure of the correlation function ( B ( X ) B ( X ’ ) ) . The product B ( X ) dX measures the number of counterions condensed on a segment of length dX, and B(X’)dX’ is the same measure for segment dh’. We assume that these segments are small enough to allow use of differential and integral calculus, as in eq 20, but also long enough to ensure that B(X) and B ( X ) are uncorrelated if h # A’. For h = A’, we need only assume that end effects may be neglected, that is, (B(h)2) = ( 0 2 ) does not depend on location X along the polyion. Thus, we take
We return now to the integral in eq 20. The region of integration is a unit square in the AX’ plane with lower left vertex at (0,O)and upper right vertex at (1,l). Subdivide this region into two subregions R1 and & such that R i= R1+ R2. For R1 take a narrow strip running along the diagonal from (0,O)to (1,l); in R1, X = A’. The subregion RZconsists of the points of R not in R1, where X # A’. Let the integral in eq 20 be designated by I, and the corresponding integrals over R1 and RZ by 11 and I 2 ,
Fluctuation of Counterions Condensed on a Polyion
Langmuir, Vol. 10, No. 3, 1994 965
respectively, so that I = I1 +I,. For I1 we have the following calculation
= (e2) dX
E( - f,” A’
dX’
For IZ
= - -2 ; 8; dX
(23)
Combining eqs 20, 22, and 23,we find ($)
1 =E Z2((A@’) dX
(24)
where, as before, A8 = 8 - BO. In eq 6 we have a formula for ( ( A W ) . The quantity S in this formula is the number of charge sites in the segment for which we are measuring the statistical fluctuations of 8. The number of charge sites in a segment of dimensionlesslength dX is the number of charge sites in a segment of corresponding physical length ds, which is dsj b. So S = dsjb, and, by definition, dX = dslL, or, dh/S = b/L. With this result, substitution of eq 6 into eq 24 gives
where we get the second equality on comparison with eq 18. But this result is the one we wanted to prove, namely, that independent calculations of (L/ b) (p 2 ) and CY satisfy the fluctuation-dissipation theorem, eq 14.
Discussion A large variety of measurable properties of polyelectrolyte solutions show anomalous behavior at, or near, the critical charge density [ = 1Zl-1.10The -14 behavior is consistent with the onset of counterion condensation at the critical point but does not always yield to easy interpretation in all of its details. Following up on a suggestion by Ander,12 we have pointed out that the singular properties of the equilibrium free energy at the critical point may be consistent with local conformational change of the polymer triggered by the onset of counterion condensation.16 Although local folding of the polymer when the charge density drops below the critical value is clearly suggested by some of the measurements, direct detection of conformationalchange has yet to be reported. (11)Zana, R.; Tondre, C.; Riaudo, M.; Milas, M.J. Chim. Phys. Physicochim. Biol. 1971,68,1258. (12) Ander, P.;Knrdan, M.Macromolecules 1984,17, 2431. (13) Klein, J. W.; Ware,B.R.J. Chem. Phys. 1984,80, 1334. (14) Penafial, L.M.;LitowitZ, T. A. J. Chem. Phys. 1992,93, 3033. (16) Manning, G. S. J. Chem. Phys. 1988,89,3772.
Our present finding that critical number and density fluctuations of condensed counterions are large adds another element to the interpretation of critical polyelectrolyte behavior. The equilibrium number 80 of condensed counterions falls continuously to zero as the criticalcharge density is approachedfrom above; it is equal to zero for subcriticalchargedensities. For a locally rodlike polymer configuration below the critical point, we have no reason to modify previous notions that there are no, or very few, condensed counterions. When the critical point is crossed, however, we now understand the effective number of condensed counterions to jump abruptly (or nearly so), to a value greater than zero. The reason is that, although the static equilibrium value of 8 is very small for charge densities close to the critical value, the root-mean-square value of 8 is substantial. The behavior of the local concentration of condensed counterions near the critical point can only reinforce the discussion. The condensation volume falls to zero along with the equilibrium number of condensed counterions as the critical charge density tends to the critical value from above. The result is that the equilibrium value of the local counterion concentration is not zero at the critical point. The local concentration jumps abruptly from zero to about 0.07 M as the critical point is crossed from below. Counterionscondensed near the critical point may be few in number but are nevertheless intimately associated with the polyion, and their effect on local conformation cannot be discounted. These considerations apply to the static equilibrium value of the local concentration of condensed counterions. Fluctuations of the local concentration at the critical point are large in comparison with the equilibrium value. At any instant, many polymer segments at the critical point will be in contact with a local concentration of condensed counterions much greater than the already substantial equilibrium value of 0.07 M. As an illustration of the possible applications of the fluctuation theory to other problems, we have given an alternate derivation, via the fluctuation-dissipation theorem, of a known formula for the static low-field polarizability of a polyion, or, more precisely, of the contribution of condensed counterions to the polarizability. Although this result has become well-established, having been used with a measure of success by other it has been criticizedl6and unfavorably compared with other theories developedwith greater rigor.”J8 The critique is based on the observation that a polyelectrolyte solution is a conducting system in which field-induced fluxes of uncondensed small ions couple with the flow of condensed counterions. In contrast, the original derivation of our polarizability formula assumes equilibration of condensed counterions with an applied electric field (in much the same way as electronic polarizability is calculated for a isolated molecule), and the present alternate derivation relies on the fluctuation-dissipation theorem, which, in the form of eq 3, is not valid for a conducting system. Our position can be stated briefly enough. We believe the assumption that condensed counterions polarize in a small field but do not conduct to be a justifiable approximation in the conditions of high dilution (low ionic strength)required for validity of counterion condensation theory and used in polarizability measurements. The uncondensed counterions are far from the polyion. The bulk solution more or less approximates the “vacuum” assumed in calculations of electronic polarizability of isolated molecules. The field-induced flow of ions into (16) Fixman, M.Macromolecules 1980, 13, 711. (17) Mandel, M.; Odijk, T. Annu. Rev. Phys. Chem. 1984,35,75. (18)Fixman, M.; Jagannathan, S. J. Chem. Phys. 1981, 75, 4048.
966 Langmuir, Vol. 10,No. 3, 1994
and out of the condensed counterion subsystem is expected to be minimal. The degree of success of the result based on this approximation, as assessed by agreement with measurement:$ supports the approximation as at least reasonable, even if not necessarily brilliant. Unrealistic aspecta of the models used by other theories (the counterions of DNA,for example, are located primarily within structural grooves, not outside a mythically smooth
Manning and Ray
cylindrical surface; the Poisson-Boltzmann equation overestimates the extent of diffuseness of the condensed layer; etc.) negatively compensate, at least partially, for the rigor of the calculations once the model is in place.
Acknowledgment. The research reported herein has been partially supported by the US. Public Health Service under Grant GM36284.
