J. Phys. Chem. 1996, 100, 9977-9989
9977
Macroion Attraction Due to Electrostatic Correlation between Screening Counterions. 1. Mobile Surface-Adsorbed Ions and Diffuse Ion Cloud Ioulia Rouzina and Victor A. Bloomfield* Department of Biochemistry, UniVersity of Minnesota, 1479 Gortner AVenue, St. Paul, Minnesota 55108 ReceiVed: February 15, 1996X
Highly charged macroion surfaces in solutions of multivalent electrolytes attract each other electrostatically through correlations in their counterion environment. We show that significant correlation occurs when the counterion distribution has a pseudo-two-dimensional character. This allows us to treat the electrostatic correlation attraction semianalytically by reducing the problem to interaction between layers of adsorbed but mobile counterions neutralizing surfaces of similar charge density. Both when the counterion distribution is in the two-dimensional limit, and when it has a more realistic three-dimensional character, Coulomb repulsion between counterions produces an alternation of positive and negative charges at the surface. Two such apposing patterns adjust complementarily to each other, resulting in electrostatic attraction of the surfaces. The magnitude of this attraction depends solely on the surface charge density and the solution dielectric constant, while its range is defined by the size of the planar correlation hole around each ion. The attraction is stable with respect to the disruptive influence of planar thermal motion of the ions. The theory enables construction of a universal function which, after being scaled with the appropriate parameters of the system, yields the attractive electrostatic correlation pressure.
1. Introduction 1.1. Electrostatic Nature of DNA Attraction. Doublestranded DNA molecules in dilute electrolyte solutions develop strong lateral attraction, and form ordered condensates, when small amounts of multivalent counterions are added.1-3 In this paper, we develop an electrostatic theory for DNA attraction, based on correlated fluctuations of counterions and emphasizing the essentially two-dimensional nature of the counterion distribution near highly charged surfaces. Some salient features of DNA attraction and condensation have been reviewed.4-7 The equilibrium spacing between condensed DNA helices is too large for cross-linking by condensing ligands to be a general mechanism.8 Condensation is relatively insensitive to the chemical nature and size of the counterion, but strongly dependent on its valency.2,3 The B-DNA structure remains largely unaltered upon condensation.9 Counterions stay mobile within the condensed DNA particle.10 Condensation is readily reversible by dilution of the solution with respect to multivalent counterions.3 DNA condensation occurs when about 90% of its charge is neutralized by the combination of multivalent and salt cations,2,3,11 where neutralization is calculated using a variant2 of Manning’s counterion condensation theory. Lowering of the dielectric constant of the solution by adding alcohol facilitates DNA condensation2,8,12 while increasing the dielectric constant with zwitterionic osmolytes inhibits it.13,14 These features indicate that attraction between DNA helices has an electrostatic basis and suggest possible similarities between DNA condensation and already familiar phenomena associated with ionic double layers including limited clay swelling, membrane fusion, mica surfaces attraction, and colloid instability. From this perspective, DNA condensation reduces to the traditional problem of interaction between double layers, i.e., interaction between two highly charged surfaces screened * Author for correspondence. Phone (612) 625-2268. Fax: (612) 6256775. E-mail:
[email protected]. X Abstract published in AdVance ACS Abstracts, May 1, 1996.
S0022-3654(96)00458-3 CCC: $12.00
by small ions. However, the traditional way of estimating the balance of forces in double-layer theory does not explain the net attraction between DNA segments. A mean field description of the role of the ions (the Poisson-Boltzmann approach) always yields a repulsion between the surfaces,15 while the shortranged van der Waals force16,17 is insufficient to produce an attraction that outweighs repulsion.8,18 Therefore, the importance of correlations in the macroion screening environment, first suggested as a significant source of attraction by Oosawa,19 has been investigated with increasing interest. Calculations using Oosawa’s theory in its original form have shown that it can predict attraction under conditions observed experimentally.5 A recent theory with somewhat similar approach20 describes solutions of highly charged polyelectrolyte chains by a model that introduces ion condensation as a random charge along the polymer. The extent of condensation is calculated from the cylindrical Poisson-Boltzmann (PB) equation. Short-range electrostatic attractions between the monomers through condensed multivalent counterions lead to reversible chain precipitation when applied to poly(styrenesulfonate); application to DNA has not yet been reported. Since Oosawa’s theory was derived heuristically rather than rigorously, there have been numerous attempts to treat correlations in a more fundamental manner. Macroion attraction has indeed been shown to exist within the framework of the primitive model of polyelectrolytes by Monte Carlo (MC) simulations21-26 and by hypernetted chain theory (HNC) (and other density functional theories).27-40 This model considers electrostatic interaction only within the system of small charged hard spheres which are between two uniformly highly charged macroscopic surfaces of low curvature. Attraction arises due to interaction of the charge inhomogeneities within the two halves of the system. Though conceptually simple, this model is very hard to treat analytically and has required extensive numerical treatment for particular sets of parameters: surface charge; ion charges, sizes, and bulk concentrations; dielectric constants of solution and macroions, and temperature. Calculations for the case of interest, trivalent counterions near B-DNA, © 1996 American Chemical Society
9978 J. Phys. Chem., Vol. 100, No. 23, 1996
Rouzina and Bloomfield
have not yet been performed since numerical methods for trivalent ions are not very efficient. 1.2. Outline of the Paper. The major goal of this paper, and those that will follow, is therefore to develop a simple but well-founded semianalytical theory to estimate electrostatic correlation interaction between arbitrary macroions, with particular application to DNA helices. In section 2 we develop the two-dimensional character of electrostatic correlation attraction. First, we show that strong correlation in the diffuse double layer system is always connected with an essentially two-dimensional (2D) distribution (section 2.1). Knowledge of the predominantly 2D nature of the correlation allows us to predict the scaling behavior of the electrostatic correlation attraction (section 2.2). This prediction is supported by all available numerical data for both diffuse (section 2.3) and fully adsorbed, strictly 2D (section 2.4) double layers. The numerical data also indicate that the attraction is very stable with respect to thermal disruption. To explain this result we analytically calculate the electrostatic attraction for the adsorbed counterion case. In section 3 we treat the limit of maximum correlation, corresponding to the ground state at zero temperature. This is done with two different models: lattice (section 3.1) and disk (section 3.2). The disruptive influence of planar ion thermal motion is then discussed in section 4. The distinction between thermal motion in and perpendicular to the plane is made in section 4.1, and section 4.2 shows that inplane thermal motion has little effect on the strength of correlation attraction. In section 4.3 we show that the thermal motions can be treated in the harmonic oscillator approximation. The details of these calculation are provided in the Appendixes, where repulsive free energies of surface interaction corresponding to phonons on the lattice (Appendix A) and oscillations on the disk (Appendix B) are derived. These two complementary approaches provide rather similar results, with the magnitude of thermal disruption of correlation attraction comparable to that obtained numerically (section 4.4). Features of the ionic planar pair correlation function are discussed in section 4.5. The difference between electrostatic correlation attraction in the diffuse and adsorbed double layer systems is analyzed in section 5. Finally, conclusions are presented in section 6. This first paper deals only with correlation attraction. The second paper in this series will introduce the other (repulsive) components of the full interaction: the kinetic and collisional pressures of the small ions. The second paper will also consider effects of higher bulk solution ionic strength and dielectric boundaries on correlation attraction. The third paper will apply these general results to a complete theory of the effect of ions on the interaction of DNA double helices. 2. Two-Dimensional Character of Electrostatic Correlation Attraction 2.1. Strongly Correlated Diffuse Counterion Double Layer. Electrostatic correlation in the ionic environment near a macroion is significant when the macroion surface is highly charged. That is, screening must be in the nonlinear regime in which electrostatic potential energy is much greater than thermal energy.15,41 This implies that the great majority of neutralizing counterions reside within a thin surface layer.41-43 In the PB approximation the thickness of this layer is a few λz (see Figure 1):
λz ) 1/4π(σ/qe)lbz
(2.1)
Here z is the counterion valence, qe the electron charge, the dielectric constant of the solution, σ the macroion surface charge
Figure 1. Primitive electrolyte model of surface-surface interaction between planes of charge density σ separated by distance h, with counterions of valence z. See text for description of other variables.
TABLE 1: Parameters of Some Diffuse Double Layer Systems Studied Numerically σ, C/m2
qe/σ, Å2
z
az, Å
ns, M
λz, Å
Γ2 ) Wc/kT
0.341 0.272
47.0 58.9 60.0
0.224
71.4
0.178 0.150
90.0 106.8
0.119 0.091
135 176.1
9.70 7.67 10.8 7.75 11.0 8.45 12.0 13.4 10.3 14.6 17.9 16.4 13.3 18.8
34.3 21.9
0.267
2 1 2 1 2 1 2 2 1 2 3 2 1 2
0.25 0.64 0.32 0.65 0.32 0.77 0.39 0.49 1.16 0.58 0.39 0.73 1.90 0.95
2.97 0.94 2.65 0.93 2.63 0.85 2.41 2.15 0.68 1.92 3.53 1.75 0.54 1.53
21.1 14.9 9.37 6.65 4.16 2.45
ref a b a, c b a d a b
a Reference 35 (HNC+MC) and ref 34 (HNC): z ) 2, δ ) 4.25 Å, bulk salt concentrations 0, 0.15, 0.4, 1, and 2 M. Total pressure only. b Reference 24 (MC) for z ) 2 (2:2 salt), σ ) 0.091 C/m2, n bulk ) 0.165 M, δ ) 4.2 Å; σ ) 0.224 C/m2, nbulk ) 0 M, δ ) 6 Å; σ ) 0.272 C/m2, nbulk ) 0.971 M, δ ) 4.2 Å. All pressure components. c Reference 36 (HNC) for z ) 1 and 2, δ ) 4.25 Å, n bulk ) 0 and 2 M. All pressure components. d Standard model of B-DNA with R ) 10 Å helix radius, and b ) 1.7 Å length per unit charge.
density, and
lb ) qe2/kT
(2.2)
is the Bjerrum length. λz is the distance at which the ionic potential energy zqeE0‚λz in the field of the unscreened surface E0 ) 4πσ becomes equal to the thermal energy kT. It is typically on the order of 0.5 Å (Table 1). Distances are measured from the closest approach of the counterion to the surface, so that for finite size ions the PB distribution is that for the ionic centers. The concentration in this layer near the surface is ns, which is typically several molar
ns ) 2π(σ/qe)2lb
(2.3)
Note that neither λz nor ns depends on the bulk ion concentrations. This type of short-range screening independent of the
Electrostatic Correlation Attraction
J. Phys. Chem., Vol. 100, No. 23, 1996 9979
bulk solution holds so long as the ionic strength
I ) 1/2∑nbizi2
(2.4)
i
is significantly less than ns:
I < ns
(2.5)
Here nbi is the bulk concentration of the ith species and zi its valence. The counterion profile between two highly charged surfaces is insensitive to the bulk solution so long as the surface spacing h is less than two Debye lengths:
h e 2rd
(2.6)
rd2 ) 1/(8πIlb)
(2.7)
where
In 0.1 M ionic strength, rd is about 10 Å. The average distance between z-valent ions on the surface is
az ) (zqe/σ)
1/2
(2.8)
Since attraction between macroion surfaces typically occurs at spacings h e az which are in the range 10-15 Å or closer (Table 1), condition 2.6 is not a more severe limitation on I than (2.5). These considerations allow us to consider a universal correlation attraction between the diffuse double layers, determined by properties of the surface and counterion valence rather than of the bulk solution, provided that condition 2.5 holds. Table 1 presents values of the parameters for several systems reported in the literature. We see that for B-DNA, condition 2.5 will be satisfied for I e 0.5 M. The 2D charge density in a counterion surface layer, zqensλz, neutralizes a significant portion of the total surface charge density σ. For example, about 80% of neutralizing charge resides within a layer of 5λz thickness, which for a trivalent ion near B-DNA is about 2 Å (see Table 1 and Figure 1a,b in ref 41). Correlation effects make the charge distribution even more compact.44,45 The layer can be considered essentially 2D if its characteristic thickness λz is small with respect to the average distance az between ions on the surface:
λz kT 1 ) ) ,1 az 4πWc 4πΓ2
(2.9)
Wc is the average Coulomb energy between neighboring mobile counterions at the surface
Wc ) (zqe)2/az
(2.10)
and Γ2 is the 2D coupling strength, the ratio of Wc to the thermal energy
Γ2 ) Wc/kT
(2.11)
Γ2 will be an important parameter in our subsequent considerations. It defines the degree of planar ordering in the counterion system and thereby the correlation attraction. The mutual repulsion of the counterions within the surface layer results in a planar pattern with a positive ionic charge surrounded by the negative charge of the background in a correlation hole. Since apposing macroion surfaces are similar, and the ions on them are mobile, two such patterns can adjust to each other, positive against negative, resulting in attraction. Equation 2.9 shows that the condition for strong 2D coupling, Γ2 > 1,
parametrically coincides with the condition for an essentially 2D screening layer. This means that strong electrostatic correlation in the system is always coupled to its predominantly 2D nature. We will see in paper 2 of this series that, at the point of spontaneous association of macroions in electrolyte, Γ2,cr ≈ 2. Under these conditions λz/az ≈ 0.04, so that the 2D character of the correlation is still well-defined. Therefore, our 2D description of correlation attraction between diffuse double layers is self-consistent everywhere, including the collapse transition region. 2.2. Scaling Laws for Electrostatic Correlation Attraction between 2D Double Layers. Two main features of the correlation attraction follow directly from its 2D nature and can be checked by comparison with numerical data from HNC and MC studies. The first is that the maximum of the attraction should be reached at zero separation, when the two complementary surface patterns are superimposed. This means that the actual distance between the macroion surfaces
H ) h + 2δ
(2.12)
equals the ion diameter 2δ (Figure 1). Each ion in that case, being located in the center of a planar correlation hole, feels the unscreened attractive field E0 of the opposite surface. The resulting electrostatic pressure (i.e., the force zqeE0 per ion over the surface per ion -zqe/σ) is then
Pel(h)0) ) Pel0 ) E0σ
(2.13)
The field E0 at the unscreened surface of a macroscopic, uniformly charged body is zero inside and 4πσ/ outside. This value was used as a boundary condition for the planar and cylindrical PB equations41 to obtain expressions for ns and λz. A twofold smaller field exists on each side of a charged plane in a uniform dielectric. This was the case in the HNC and MC studies which did not consider the image charge effect. The expected maximum value of the pressure in that case is
Pel0 ) 2πσ2/
(2.14)
which can also be expressed as
Pel0 ) nskT ) 2πWc/az3
(2.15)
This value of the maximum attractive pressure at h ) 0 should pertain to both diffuse and 2D adsorbed mobile counterion double layers. We thus predict the maximum attractive pressure for any particular system to be independent of the counterion valence, temperature, or bulk solution ionic strength, but to vary strongly with the surface charge density, and inversely with the solution dielectric constant. The second basic feature implied by the predominantly 2D nature of counterion correlations is that the decay length of the attractive interaction is of the order of the size of the correlation hole, az. This becomes evident if we note that az is the length scale of the electroneutrality violation for the otherwise completely screened surface. 2.3. Numerical Data on Electrostatic Correlation Interaction of Two Diffuse Double Layers. In Figure 2 we summarize numerical data from HNC and MC calculations21,25,28,45 on Pel(h) for systems with a diffuse monovalent or divalent counterion layer near various highly charged surfaces. Parameters for these systems are summarized in Table 1. Unfortunately, in many cases only the longer range of intersurface separations, h > 10 Å, was studied. At these separations the correlation interaction is greatly reduced with respect to its full strength. Some studies
9980 J. Phys. Chem., Vol. 100, No. 23, 1996
Rouzina and Bloomfield
Figure 3. Calculated electrostatic attractive pressure in scaled units between 2D adsorbed but mobile ionic layers for the systems described in Table 2. Γ2 increases from the top to the bottom line. The solid line is the maximum pressure at Γ2 f ∞ according to the disk model described in the text.
TABLE 2: Parameters of Systems with Fully Adsorbed Mobile Counterions Studied in the HNC Approximation
Figure 2. Calculated electrostatic attractive pressure between diffuse double layers in unscaled (a) and scaled (b) units for the systems in Table 1 with the specified surface area per unit charge, counterion valence, and coupling strength: (- - -)qe/σ ) 176.1 Å2, z ) 2, Γ2 ) 1.53; (- - -) qe/σ ) 71.4 Å2, z ) 2, Γ2 ) 2.41; (- -) qe/σ ) 58.9 Å2, z ) 2, Γ2 ) 2.65; (s s) qe/σ ) 60 Å2, z ) 1, Γ2 ) 0.93; (s) qe/σ ) 60 Å2, z ) 2, Γ2 ) 2.63. See Table 1 for source of each curve.
did not present the individual pressure components but only the total pressure which includes not only Pel but also the kinetic pressure Pkin and the collisional pressure Pcol. In those situations where complete Pel(h) curves are available, they appear to be different for each system (Figure 2a); but the curves coincide to a notable extent (Figure 2b) when pressure and distance are expressed in reduced units Pel/kTns and h/az, as predicted from our 2D model of correlation. Agreement worsens, as expected, at larger spacings where the pressure reaches its bulk 3D value determined by the solution ionic strength. The similarity between the reduced pressure curves is especially striking, considering that values of ns varied about 10-fold (Table 1). The attractive ranges for the monovalent and divalent ions near the same surface appear to differ solely by the factor of x2, as predicted by eq 2.8. The universal curve in Figure 2b can be fitted by the polynomial
()
Pel(h/az) h 2 h ) -1.0 + 0.63259 + 4.5003 kTns az az h 3 h + 2.9228 7.0911 az az
()
()
4
(2.16)
qe/σ, Å2
z
a z, Å
ns, M
λz, Å
Γ2
ref
71.4 250 40 200 1000
2 2 1 1 1
11.95 22.36 6.35 14.14 31.62
14.88 1.21 47.41 1.90 0.08
0.39 1.36 0.43 2.17 10.86
2.41 1.29 1.11 0.50 0.22
29 37 29 29 29
This approximation is good only for h/az e 1. At greater separations the pressure is less than 2% of its maximum strength at h ) 0 and cannot be reliably obtained from the numerical data. Equation 2.16 can be used to estimate electrostatic correlation attraction for an arbitrary system of two diffuse double layers with 2D coupling strength Γ2 in the range 0.9-3 explored numerically. Higher values of Γ2 should result in Pel closer to that for adsorbed ion layers, eq B.2. 2.4. Numerical Data for Adsorbed (Strictly 2D) Mobile Counterion Layers. In Figure 3 we display, again in reduced coordinates, available numerical data on Pel(h) between fully adsorbed, strictly 2D, mobile ion layers. The parameters of these systems are given in Table 2. We see that while the maximum value of the attractive pressure varied among these cases by about 2 orders of magnitude (as ns from 0.08 to 47 M), and az from 6.4 to 31.6 Å, all reduced Pel(h) curves are similar and only moderately weaker than the maximum possible reduced attractive pressure. Damping of the reduced Pel(h) relative to the maximum correlation case depends solely on the value of the 2D coupling strength Γ2. This is illustrated by the close similarity of the reduced Pel(h) curves for two systems as different as those in lines 2 and 3 of Table 2. Despite very large differences in the scaling parameters az and ns, the coupling strength Γ2 is comparable, and it is on that the reduced Pel(h/ az) depends. It is also worth noting that the system in the third line of Table 2 does not have fully adsorbed counterions, but rather a diffuse ion layer with an additional contact absorption energy of variable strength. The curve reproduced in Figure 3 corresponds to the maximum absorption energy Ea ) 8 kT per ion and is apparently equivalent to the fully adsorbed case. Lowering of the absorption energy gradually transforms a tightly adsorbed double layer into a diffuse one. This leaves the correlation attraction almost intact, while considerably increasing
Electrostatic Correlation Attraction
J. Phys. Chem., Vol. 100, No. 23, 1996 9981
Figure 4. Ground states of the 2D lattice composed of two merged ionic layers, represented as clear and shaded circles: (a) small separation, h < 0.43az; (b) large separation, h > 0.43az. Lattice parameters of each configuration are given by eqs A1.3 and A1.4.
kinetic repulsion, which results in overall disruption of attraction. This is one more illustration of the fact that the real source of the attraction is the 2D rather than the 3D ion correlation. We see that simple knowledge of the 2D nature of correlation is enough to predict the scaling behavior of the attraction, which holds surprisingly well for a broad range of system parameters. In the next section we calculate the maximum electrostatic pressure curve analytically by considering more detailed models of the surface charge distribution. 3. Electrostatic Correlation Attraction in the High Correlation Limit The stronger the counterion condensation at the surface, the stronger the electrostatic correlations. This is illustrated most vividly by the complete absorption of counterions at the surface in the hypothetical case Γ2 f ∞ (or T f 0), corresponding to the strongest possible correlation attraction. In this limit the electrostatic correlation pressure can be calculated analytically. The diffuse double layer becomes vanishingly thin, eq 2.9 and indistinguishable from the complete adsorbed, strictly 2D distribution. Mutual repulsion of mobile ions within this layer leads to their periodic arrangement on the surface, while electrostatic interaction between two such layers results in their complementary adjustment, plus against minus. We treat this situation with two different models: lattice and disk. 3.1. Zero Temperature Correlation Attraction in the Lattice Model. The true ground state of this system corresponds to two perfect planar lattices complementarily shifted with respect to each other. The electrostatic energy and force per unit cell or per ion (readily converted to the force per unit area or pressure Pel) can be calculated by summing up the Coulombic interactions of one ion with all ions on the opposing surface. This calculation, presented in Appendix A, predicts that the ground state structure undergoes a first-order phase transition at the surface separation h ≈ 0.43az. At h near zero the combined lattice is triangular, so that all the ions are as far from each other as possible at the given surface charge density, while each individual surface lattice is orthogonal (Figure 4a). At larger separations each lattice tends to be triangular, while the combination of the two becomes hexagonal (Figure 4b). The resulting maximum electrostatic correlation attraction energy and pressure, in reduced units, are shown in Figure 5. The maximum attractive pressure is indeed kTns at h ) 0, as predicted by eq 2.15, and its decay scales with az, while the maximum value of the attractive energy per ion at h ) 0 is about 1.63Wc. 3.2. Zero Temperature Correlation Attraction in the Disk Model. Two complementary perfect ion lattices form the true ground state at the zero temperature. But is perfect lattice order essential for interaction? To answer this question we consider the simplest possible charge configuration which we expected
Figure 5. Reduced electrostatic energy (a) and pressure (b) in the limit of T ) 0 or maximum correlation (Γ2 f ∞) calculated according to the lattice (dashed lines) and disk (solid line) models. The lowest energy of interaction in the lattice model is from the orthogonal lattices in Figure 4a at short separation (dotted line) and the triangular lattices in Figure 4b at longer separation (short dash line).
Figure 6. Charge distribution in the disk model.
to reproduce the main features of the effect: we calculated the interaction between a point charge zqe and a disk. The disk is electroneutral overall, with uniform negative charge zqe smeared over it and a ring of equal positive charge along its edge (Figure 6). To mimic maximum correlation, the ring of charge at the edge is infinitely thin. The disk radius A is found by equating areas of lattice unit cell and disk: πA2 ) zqe/σ ) az2, so that A ) az/xπ. A positive point charge is located on the disk axis at distance h from the plane. In this simple geometry, the force and potential energy of electrostatic interaction at Γ2 f ∞ can be found analytically as a function of h (Appendix B).
9982 J. Phys. Chem., Vol. 100, No. 23, 1996 The results are very similar to those of the lattice model (Figure 5). This is reasonable, since the leading order of planar charge momentum in both cases is the same. The higher momenta are much better compensated in the case of the lattice, which results in the faster exponential decay (≈exp[-2πh/az]) of the potential for h g az/2π in that case. This result justifies the simplifications made in the disk model, such as representation of nearest neighbors by a smeared ring of charge, neglect of second nearest-neighbor coordination shells, and replacement of the complementary charge distribution on the second surface with a point charge at the center. The disk model, crude but adequate, allows formulation of an approximate but explicit result for the attractive energy and pressure at the maximum, and also at lower 2D coupling strengths, as we shall see in the next section. 4. Disruptive Effect of Planar Ion Thermal Motion on Attraction 4.1. Two Types of Ion Thermal Motion in the Diffuse Layer. In the limit of highest correlation, Γ2 f ∞, all neutralizing ions are in a perfect surface lattice. This configuration corresponds to maximum attraction. Increasing thermal motion of ions (reducing Γ2) will gradually destroy this attraction. The theoretical description of this effect is a very complex problem of statistical mechanics which to date has been approached only numerically. Understanding the predominantly 2D character of the diffuse counterion atmosphere allows us to greatly simplify the problem. We consider separately two types of ion motion: one perpendicular to the surface, the other parallel to it. The first results in the repulsive kinetic pressure Pkin described in the next paper. The second also leads to an effective repulsion relative to the “zero temperature” case by disrupting the planar order. Such separation implies that the two motions are independent and their influence on the correlation attraction is additive. This is essentially a perturbation theory expansion relative to the “zero temperature” ground state with respect to the small parameter 1/4πΓ2. The parallel and perpendicular motions lead to different terms in this expansion, since their average amplitudes behave differently with Γ2. The planar motion within the adsorbed mobile ion layer then makes a good approximation for that within the thin diffuse layer, which we consider in section 5. 4.2. Correlation Attraction Is Only Slightly Reduced by Planar Ion Thermal Motion. An important insight from the numerical studies on surfaces with adsorbed mobile ions, illustrated by Figure 3, is that the reduced attractive pressure Pel/(nskT) for moderate Γ2 is not much weaker than the limiting value at Γ2 f ∞. Even at Γ2 ) 0.5, the reduced pressure is only about 2-fold less than the maximum. This striking result indicates the very high stability of the correlation attraction, and therefore its likely importance for many experimental systems with adsorbed mobile ion layers and low coupling strength, including membranes, colloids, and mica surfaces. But our current study is aimed at interpreting DNA interaction, which involves diffuse double layers. The main force balance in this case occurs between correlation attraction Pel and kinetic repulsion Pkin, and results in disruption of the equilibrium bound state at much higher coupling strength Γ2,cr ≈ 2, as will be shown in the next paper. Thus we are mostly concerned with systems of higher Γ2. The strong correlation allows us to consider planar thermal motion of the ions in the harmonic approximation. 4.3. Harmonic Oscillator Analysis of Planar Thermal Motion. Motion in a highly correlated system has the character of oscillation around equilibrium. This means that the average
Rouzina and Bloomfield thermal displacement 〈δa〉 of an ion from its equilibrium position at the center of the correlation hole of size az is small: 〈u〉 ) 〈δa/az〉 < 1. Then we can use the harmonic approximation, leaving out all terms in the expansion of the potential energy of planar motion ∆W(u) with respect to u except for the first:
∆W(u,h) )
WcK(h) 2 u 2
(4.1)
where
K(h) )
[
]
2 1 d ∆W(h) Wc du2
(4.2)
u)0
is the rigidity constant in units of Wc/az2. The additional free energy of the system associated with this motion then can be calculated as
∆Fel(h) ) -kTln[Z(h)/Z(∞)]
(4.3)
where Z(h) is the partition function
Z(h) ) ∑e-∆W(u,h)/kT ) ∑e-β(h)u
2
(4.4)
In this case Z(h) is easily calculable:
Z(h) )
1 2 -βu2 ) (1 - e-β)(1/β) ∫ u
