IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 22 (2010) 414102 (9pp) doi:10.1088/0953-8984/22/41/414102

Polyelectrolyte-mediated bridginginteractions: columnar macromolecularphasesMatjaz Licer1 and Rudolf Podgornik2,3

1 Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana,SI-1000 Ljubljana, Slovenia2 Department of Physics, Faculty of Mathematics and Physics, and Institute of Biophysics,School of Medicine, University of Ljubljana, SI-1000 Ljubljana, Slovenia3 Department of Theoretical Physics, J Stefan Institute, SI-1000 Ljubljana, Slovenia

Received 19 October 2009, in final form 8 December 2009Published 30 September 2010Online at stacks.iop.org/JPhysCM/22/414102

AbstractWe present a mean-field theory for charged polymer chains in an external electrostatic field inthe weak and strong coupling limits. We apply the theory to describe the statistical mechanicsof flexible polyelectrolyte chains in a hexagonal columnar lattice of stiff cylindrical macroions,such as DNA, in a bathing solution of a uni-univalent salt (e.g. NaCl). The salt effects are firstdescribed in the Debye–Huckel framework. This yields the macroion electrostatic field in thescreened Coulomb form, which we take to represent the mean field into which the chains areimmersed. We introduce the Green’s function for the polyelectrolyte chains and derive thecorresponding Edwards equation which we solve numerically in the Wigner–Seitz cylindricalcell using the ground state dominance ansatz. The solutions indicate the presence ofpolyelectrolyte bridging, which results in a like-charge attraction between stiff macroions. Thenwe reformulate the Edwards theory for the strong coupling case and use the standardPoisson–Boltzmann picture to describe the salt solution. We begin with the free energy whichwe minimize to obtain the Euler–Lagrange equations. The solutions yield self-consistentlydetermined monomer density and electrostatic fields. We furthermore calculate the free energydensity as well as the total osmotic pressure in the system. We again show that bridgingimplicates like-charge attractions of entropic origin between stiff cylindrical macroions. Byanalyzing the osmotic pressure we demonstrate that, in certain parts of the parameter space, aphase transition occurs between two phases of the same hexagonal symmetry.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Systems composed of flexible polyelectrolytes in thebackground of an ordered macromolecular phase, as, forexample, seen in the case of polyplexes, have been attractinga lot of theoretical and experimental attention in recentdecades due to their numerous technological and medicalapplications [1]. Polyion–macroion interactions have beenshown to play an important role in DNA self-assemblybehavior [2–5], DNA packing in chromatin [6] and within viralcapsids [7]. In all these cases the poly-counterions conferattractive interactions between stiffer DNA molecules that

usually condense into an ordered phase with locally hexagonalsymmetry.

The DNA condensation mechanisms in such systemsare determined at least to some extent [3] by electrostaticand entropic free energy terms that we can successfullystudy using mean-field theories. Experimentally thesesystems have been studied intensely using a wide range ofmethods ranging from osmotic stress coupled to x-ray densitydetermination [8] and light scattering [3, 4], to AFM single-molecule studies [9]. Several experiments have confirmed theexistence of high density columnar DNA mesophases withhexagonal symmetry [8, 10–14].

0953-8984/10/414102+09$30.00 © 2010 IOP Publishing Ltd Printed in the UK & the USA1

J. Phys.: Condens. Matter 22 (2010) 414102 M Licer and R Podgornik

In a recent paper DeRouchey and co-workers investigatedin detail the phase behavior of a columnar DNA phase usingsynchrotron small-angle x-ray scattering (SAXS) [8]. Theyreport on a discontinuous phase transition from a compacthexagonal crystalline phase to loose bundles and finally to anisotropic phase with increasing salt concentration. Their freeenergy model describes polyelectrolytes as rigid rods which donot, as they themselves already point out, suit the experimentalsituations in all respects. In the present paper we thusconcentrate on the opposite limit of flexible polyelectrolytesin order to supplement the calculation of polyelectrolyte-mediated interactions presented in [8]. Our approach is basedon the mean-field formulation of the statistical mechanics offlexible polyelectrolyte chains in external electrostatic fieldsintroduced before [15]. The results of this theory applied toan array of cylindrical macroions with hexagonal symmetryshow that competition between electrostatic and entropiccontributions to free energy can lead to a discontinuous phasetransition in systems such as the one observed in [8] drivenprimarily by polyelectrolyte bridging between stiff (DNA)macromolecules.

Polyelectrolyte bridging refers to a situation wherepart of the same polyelectrolyte chain is adsorbed to onecylindrical macroion and the rest to a neighboring cylindricalmacroion, pulling them together via the entropic effects ofthe connectivity of the chain. (For a detailed account anda complete list of references on the polyelectrolyte bridginginteraction, see [16, 17].) The bridging interaction is thusstrictly attractive.

In what follows we formulate a mean-field theory offlexible polyelectrolytes in the electrostatic field of oppositelycharged orientationally and positionally ordered rod-likemacroions in two distinct limits: first of all we treatthe electrostatic field of rod-like macroions as completelydecoupled from the local polyelectrolyte, which leads us to aformulation of the weak coupling model. Later on, by self-consistently including the effect of the local polyelectrolytedensity on the mean electrostatic field in the system, weformulate the strong coupling model of interactions in thissystem. We analyze the corresponding osmotic pressure andshow that it can exhibit a non-monotonic dependence on themacroion density, possibly giving rise to a first-order transitionbetween two phases of equal symmetry.

2. Model description: the weak coupling case

Our system consists of a hexagonal lattice of uniformlycharged, infinitely long and stiff rod-like macroions, flexiblepolyelectrolyte (PE) chains and a uni-univalent salt solutioninto which PE chains and stiff macroions are immersed.The stiff rod macroion approximation is legitimate forall macroion–polyelectrolyte systems where the macroionpersistence length is much larger than its radius and wheremacroion persistence length exceeds that of the polyelectrolyteby orders of magnitude. This holds, for example, in the caseof a system with DNA macroions (with a radius of 1 nmand persistence length of 50 nm) and poly-lysine and poly-arginine polyelectrolyte chains with much smaller persistence

length [8]. We hold the coupling between the polyelectrolytedensity and the external electrostatic field to be weak in thosecases where the external field dictates the shape of the PEdensity profiles, but not vice versa. In other words, in the weakcoupling limit PE monomers have no impact on the externalelectrostatic field even though they are themselves charged.It is set only by the fixed charges on the macroions. Thisapproximation clearly works only in the case where the PEchains are extremely weakly charged and should be viewed likea first-order perturbation theory on the complete electrostaticcoupling [15] which is then treated appropriately on the strongcoupling limit (see below).

We describe the macroion electrostatic field screenedby salt ions on the Debye–Huckel level, corresponding toa linearized Poisson–Boltzmann theory, which yields thefamiliar form for the screened electrostatic potential of acylindrical macroion !(r) = A0 K0("r), where r is the off-axis separation from the cylindrical macroion and K0("r) isthe modified Bessel function of the second kind. A0 =

µe04#$$0kBT = µ

e0

ee0

lB is the dimensionless interaction parameter,µ is the linear charge density of the cylindrical macroiondefined as charge e per Kuhn’s length b, lB = e2/4#$$0kBT isthe standard Bjerrum length and " stands for the inverse Debyescreening length " = %!1

D . e0 is the unit charge and $ is thestatic permittivity of water.

The statistics of polyelectrolyte chain conformations inan external field is described by the standard polyelectrolyteGreen function [18]:

G N (r", r; !) =! r

r"Dr(n)e! 3

2b2

" N0 dn( &r(n)

&n )2!'e" N

0 dn !(r(n)) (1)

where the integral over n in the exponent runs along the contourof the PE chain of N monomers, b is the chain’s Kuhn lengthwhile ' = 1/kBT stands for the inverse thermal energy. Thefirst term in the exponent describes the chain connectivityand entropy while the second one describes the Boltzmannweights of the chain in an external electrostatic potential !.The notation

"Dr(n) stands for the functional integral over

all possible polyelectrolyte chain conformations. The Greenfunction of the PE chain of N monomers can be interpretedas the probability density that the chain’s end-to-end vectorree = r(N) ! r(0) will span the difference r ! r" or, whichamounts to the same thing, as the probability density that achain of N monomers has its last monomer at r provided itsfirst monomer is at r". The polymer Green’s function has tosatisfy the Edwards equation [18, 19]:

! b2

6#2G N + 'e!G N = !&G N

& N. (2)

Since the Edwards equation is formally analogous to thequantum Schrodinger equation, being Wick-rotated in the timedomain, we can proceed by similar means to solve it. We defineeigenfunctions (k of the Edwards operator through

#!b2

6#2 + 'e!

$(k = Ek(k .

After performing an eigenfunction expansion of the Green’sfunction G N (r", r) = b3 %

k ($k (r")(k(r) e!NEk and truncating

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J. Phys.: Condens. Matter 22 (2010) 414102 M Licer and R Podgornik

Figure 1. Schematic depiction of our model with its Wigner–Seitzcell. Cylindrical macroions in a columnar phase are shown as bluecylinders of radius a, electrolyte ions are depicted as yellow spheres,while the polyelectrolyte chain is green. The transparent cylinderrepresents the Wigner–Seitz cell with surface normal n and radiusrWS. The radius of the cylindrical macroions, a, is always taken as1 nm.

the series after its first term—the ground state dominanceapproximation (GSDA) [18]—we obtain the followingapproximate relations for the Green’s function and the freeenergy:

G N (r, r") % ($0 (r)(0(r")e!E0 N (3)

F % kBT NE0. (4)

It is then straightforward to show that the polyelectrolytemonomer density may be expressed as [19]

)(r) = N(20 (r). (5)

Note that the ground state dominance approximationonly makes sense for chains that are much longer than theirpersistence lengths [20]. We feel no need to go deeper intothe mathematical specifics of the GSDA since this has alreadybeen adequately done elsewhere [15].

We solve the Edwards equation (2) describing our systemin a polyelectrolyte and salt-filled cylindrical Wigner–Seitz(WS) unit cell surrounding a lattice macroion placed in thecell’s axis of symmetry as shown in figure 1.

Each cell is, by assumption, electroneutral and thecoupling between them is provided solely by the appropriateboundary conditions for the mean electrostatic field. The WScell approach thus effectively decouples the partition functionof the entire system into a product of disjoint single macroionfactors, thus reformulating a many-body problem in the mean-field framework. This approach should work fine if thepolyelectrolyte chain spans only a few WS cells, which is whatwe assume here. If a single polyelectrolyte chain connectsmany different macroions a collective approach is in order, asformulated in [21].

In cylindrical geometry we may only keep the radial partof the Laplace operator in the Edwards equation since oursystem is translationally invariant along the principal axis ofthe Wigner–Seitz cell. The Edwards equation thus assumes the

form (see [21])

( ""0 (r) + 1

r( "

0(r) + 6b2

(A0K0("r) + E0)(0(r) = 0. (6)

E0 is the ground state energy eigenvalue of the Edwardsoperator ! b2

6 #2 + 'e!.The boundary conditions pertaining to equation (6) in the

Wigner–Seitz cell are as follows. We simulate the hexagonalpacking symmetry of the macroions by demanding that thenormal derivative of the density field (0 should vanish on thesurface of the Wigner–Seitz cell:

&(0

&n

&&&&r=rWS

= 0.

To fix the number of PE chains in the system the groundstate eigenfunction must be normalized:

"dr (2

0 (r) = 1.Furthermore we require that the PE chain is not allowed topenetrate into the macroion, thus setting the PE chain densityon the macroion surface to zero:

(0|r=a = 0.

The surface charge density of the macroion is assumed to be* = *DNA & 1e0 nm!2, corresponding approximately to thesurface charge density of DNA with a radius of a = 1 nm,carrying a linear charge density µ = 1e0/1.7 A.

2.1. Solutions of the Edwards equation

We solved equation (6) numerically in a cylindrical Wigner–Seitz cell using a combination of standard shooting andminimization algorithms and we now proceed to the results.Figure 2 shows the polyelectrolyte density profile )(w) '(2(w) as a function of reduced radial coordinate w = "rfor different values of " . The value of the dimensionlessinteraction parameter is taken as A0 = 10. The monomerdensity profile clearly exhibits a transition from depleted(monotonic profiles) to surface bound states (non-monotonicprofiles) as we reduce the inverse screening length " .

Surface bound states indicate the presence of polyelec-trolyte bridging [16] which grows stronger as we reduce theconcentration of the uni-univalent electrolyte in the cell. Thisis due to the fact that, at short screening lengths, the effectof electrostatics is diminished, the system is effectively dis-charged and polyelectrolyte adsorption to charged macroionscannot be the dominant mechanism of the free energy mini-mization. In this case free energy is minimized through en-hanced entropic steric effects which prefer the chain to be asfar from the macroion surface as possible.

A similar effect occurs when we fix the values ofparameters A0 and " and observe the PE monomer densityprofiles while increasing the cell’s radius, which correspondsto the changes in the concentration of the macroions. Athigh macroion concentrations the dominant mechanism offree energy minimization is entropy maximization whichleads to a surface-depleted regime where the chain is mostlyconcentrated at the periphery of the WS cell, i.e. in the spacebetween the macroions. At lower macroion concentrations the

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J. Phys.: Condens. Matter 22 (2010) 414102 M Licer and R Podgornik

Figure 2. Polyelectrolyte density profile )(w) ' ( 2(w) as afunction of reduced radial coordinate w = "r for different values of" . The value of the dimensionless interaction parameter is A0 = 10,corresponding to DNA. The profile clearly exhibits a transition fromdepleted to surface bound states as we reduce the inverse screeninglength " . Surface bound states indicate the presence of bridginginteractions which grow stronger as we reduce the molarity of salt.

electrostatic interactions become strong enough to significantlyincrease the monomer concentration close to the macroionsurface which leads to attractive bridging interactions betweenneighboring macroions. This is shown in figure 3 whichdepicts monomer density profiles and ground state energyeigenvalues at different cell diameters for fixed parametervalues A0 = 3 and " = 0.1 nm!1. The cell diameter variationshave a pronounced effect on the density profiles which aremonomodal (implying that no bridging is present) up until weincrease the dimensionless diameter over the ‘critical’ value ofwWS & 0.8. At that point the monomer density distributionsbecome bimodal and the free energy curve goes through aminimum. Since the forces between macroions are related topolyelectrolyte osmotic pressure

p$ = !#

&F(w)

&V

$

w=wWS

= !kBT N& E0(wWS)

&V(7)

non-monotonic free energy behavior, such as the one shownin figure 3, indicates the presence of attractive bridginginteractions due to the interplay between the chain entropy andelectrostatic interactions between the chain and the chargedmacroion.

3. Strong coupling case: self-consistent field theoryfor chains in cylindrical cells

The theory presented above does not take into account the factthat polyelectrolyte monomers are themselves charged and assuch modify the electrostatic field in the WS cell as well. Inwhat follows we reformulate the theory to describe cases wherethis coupling is too large to be ignored: in our descriptionwe remain close to the Poisson–Boltzmann theory which wegeneralize to include polyelectrolyte chain contributions tothe electrostatic field in a self-consistent manner. We arriveat a new set of self-consistent field equations which are

Figure 3. Polyelectrolyte monomer density profiles and ground stateenergy eigenvalues at different cell diameters for fixed parametervalues A0 = 3 and " = 0.1 nm!1. There is no bridging present athigh macroion concentrations (corresponding to small WS celldiameter) but if the macroion density is decreased so that thecorresponding WS cell diameter is wWS > 0.8 (dotted line) themonomer density distribution becomes bimodal and the eigenenergycurve goes through a minimum (vertical dotted line), indicating thepresence of attractive bridging interactions.

solved numerically in different regions of the parameter space.The solutions indicate the presence of a discontinuous phasetransition which was unobtainable within the framework of theweak coupling theory. In our brief outline of the derivationof the self-consistent field equations we follow closely thederivation for planar geometries thoroughly presented in [15].

The polyelectrolyte chain is again described using theEdwards model in the ground state dominance approximation.The contour length of the chain is Nb, where N is the numberof monomers of Kuhn length b. In addition the total number ofchains in the system is assumed to be N . They are all identical.All electrostatic interactions in the system are mediated via aCoulomb kernel u(r, r") = (1/4#$$0)|r ! r"|!1. The systemis described as monodisperse and we assume a homogeneouslinear charge distribution along each polyelectrolyte chain anda homogeneous surface charge density * on all columnarmacroions. The Hamiltonian of the system then is

'H = 32b2

N'

+=1

! N

0dn

#&r+(n)

&n

$2

+ '

2

! !dr dr" )(r))(r")u(r, r")

+ '

!dr )(r)!(r), (8)

where + runs over all the chains in the system, r+(n) isthe +th chain contour parameterized with the parameter n,)(r) is the density of all volume charges (electrolytes andpolyelectrolytes) in the system and ! is the electrostaticpotential due to macroion surface charges. The first termin Hamiltonian equation (8) describes the chain entropiccontribution due to its connectivity, the second term representsall the electrostatic interactions between volume charges whilethe third term stands for electrostatic interactions of volumecharges with the macroion electrostatic field. To get rid of

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J. Phys.: Condens. Matter 22 (2010) 414102 M Licer and R Podgornik

the nonlocal character of the above Hamiltonian we perform aHubbard–Stratonovich transformation. We may then formallyproceed to write the system partition function in the followinguseful form (for a detailed derivation see [15]):

, = -(')

!D!e!'S! , (9)

where"D! is a functional integral measure over fluctuating

electrostatic fields !, -(') = (2#)N/2(

det['u!1(r, r)] andthe action functional S! is

S! = HPB[i!] ! kBTN ln#! !

G N (r, r"; !)d3r d3r"$

.

(10)The first term in equation (10) is a Wick-rotated version of thePoisson–Boltzmann free energy [15] and

HPB[!] = 2kBT n0

!dr cosh('e0!(r)) +

)ds *!(s)

! $$0

2

!dr(#!(r))2 (11)

is the usual Poisson–Boltzmann Hamiltonian with n0 asthe electrolyte number density. G N (r, r"; !) is again thepolyelectrolyte Green function defined in equation (1). Thereare several approximate approaches to calculating the freeenergy from the partition function (9) and we use one of them,the so-called saddle point approximation. We approximatethe functional integral in (9) by its largest contribution whichcorresponds to the exponent of the action functional calculatedat its saddle point SSCF

. , namely

!D! e!'S! % exp[!'SSCF

. ].

This leads to the consideration of the saddle pointconfigurations of the fluctuating electrostatic field, ., definedas solutions of

/S!//.(r)|!=. = 0. (12)

As is well known from the polymer theory the self-consistentfield calculation coincides with the saddle point approximation.In fact, they represent the same approximation and itsramifications have been studied in detail [22].

Along with the Edwards equation (2) the saddle pointequation yields the following system of self-consistent field(SCF) equations for the mean electrostatic field . and thepolymer density field ( in the ground state dominanceapproximation equation (5)

$$0#2(.(r)!2kBT n0 sinh['e0.(r)]!eN N |((r)|2 = 0 (13)

! l2

6#2

(((r) + 'e.(r)((r) ! E0((r) = 0. (14)

Here e again stands for the net charge on each monomer and#2

( = r!1(&/&r)(r&/&r) for the radial part of the Laplaceoperator. It is straightforward to see that, in the absenceof polyelectrolyte, equation (13) simplifies to the standardPoisson–Boltzmann equation for the uni-univalent salt, whileequation (14) has the form of the Edwards equation where

the electrostatic potential .(r) has yet to be determined self-consistently with the polyelectrolyte density field ((r).

The boundary conditions to equations (13) and (14) areas follows. To simulate the hexagonally packed macroionswe stipulate that radial derivatives of the electrostatic and PEdensity fields vanish at the Wigner–Seitz cell surface, so that

&.

&n(r = rWS) = 0 and

&(

&n(r = rWS) = 0.

Since the chain cannot penetrate into the macroion, PE chaindensity on the macroion surface must also vanish, leading to

((r = a) = 0.

In addition, in order to ensure the electroneutrality of each cellthe following standard boundary condition must hold on themacroion surface of surface charge density * :

! $$0&.

&n(r = a) = *.

To fix the number of chains in each cell, we must make surethe PE density field solutions to equations (13) and (14) areproperly normalized, namely that

"dr|((r)|2 = 1.

3.1. Solutions of the SCF equations

Before we set out to solve the SCF equations numerically itis useful to rewrite them in dimensionless form. Without anyloss of generality we may assume in what follows that e = e0.We now introduce dimensionless distance w = %!1

D r = "r ,dimensionless macroion surface charge density 'e0%D*/$$0

and dimensionless parameter %$2 = ('e20/$$0)N (N/S)%D,

which measures the ratio between the number of PE chainsand the amount of salt in the cell. At room temperature we cancalculate the dimensionless macroion surface charge densityto be 'e0%D*/$$0 & 9 nm · %D(*/e0), assuming the staticpermittivity of water to be $ = 80. Following the experimentaldata [23] we have set Kuhn’s length of the polyelectrolyte to1 nm. Introducing furthermore the dimensionless electrostaticfield . ) 'e0., we can rewrite equations (13) and (14) in thedimensionless form:

d2(

dw2+ 1

w

d(

dw+ 6(%D/b)2(E0 ! .)( = 0 (15)

d2.

dw2+ 1

w

d.

dw! sinh . + %$2(2 = 0. (16)

The transformations of the appropriate boundary conditionsare straightforward and we do not state them explicitly at thispoint.

In what follows we obtain the solutions of the SCFequations (15) and (16) numerically using a combination ofiterative shooting and minimization methods. Figure 4 showspolyelectrolyte density profiles in cylindric Wigner–Seitz cellsfor different screening lengths %D at parameter values * =6*DNA, %D = 0.7:0.2:1.3 nm and %$2 = 1. The macroionrod is not depicted in the images but is located in the axisof symmetry of each density profile. Since this is the case ofextremely high macroion surface charge density the transition

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J. Phys.: Condens. Matter 22 (2010) 414102 M Licer and R Podgornik

Figure 4. Polyelectrolyte density profiles )(w) ' ( 2(w) incylindrical Wigner–Seitz cells for different screening lengths %D atparameter values * = 6*DNA, %D = 0.7:0.2:1.3 nm and %$2 = 1. Themacroion rod is not depicted in the images but is located in the axisof symmetry of each density profile. As we lower the salt content ineach cell the surface bound states become more explicit and themaximum of the monomer density distribution shifts closer to themacroion surface.

from depleted (profile at %D = 0.7 nm) to surface bound states(profiles at %D = 0.9, 1.1 and 1.3 nm) is very clear.

Self-consistent electrostatic field solutions in the systemare shown in figure 5 along with a numerical solutionof the cylindrical Poisson–Boltzmann equation without thepolyelectrolyte at parameter values * = 0.1*DNA, %D =0.26 nm and %$2 = 0.01:1:5.01. We have varied thepolyelectrolyte content %$2 in the cell from 0.01 to 5.01but held the macroion charge density and salt concentrationfixed. Low values of %$2 mean low polyelectrolyte densitieswhile high values of %$2 mean high densities. The bluesquares in figure 5 show the Poisson–Boltzmann electrostaticfield solutions while black lines represent .-field solutionsto the full SCF system. Since Poisson–Boltzmann theoryis the limiting case of the self-consistent field theory forlow polyelectrolyte densities, we expect the two solutions tocoincide in the low polyelectrolyte density regime (%$2 ' 0),as they in fact do. Figure 5 shows that Poisson–Boltzmanntheory gives quantitatively correct electrostatic field profilesonly as long as the concentration of polyelectrolytes is lowenough (%$2 ' 0.01) while at higher densities the differencesbetween the two theories grow in favor of the self-consistentfield approach.

4. Osmotic pressure

The interaction between like-charged macroions on differentlattice sites has two components [16]: (i) counterion

Figure 5. Electrostatic field solutions of the complete SCF system(lines) alongside with numerical solution to the cylindricalPoisson–Boltzmann equation (squares) at parameters * = 0.1*DNA,%D = 0.26 nm and %$2 = 0.01:1:5.01. As expected, both theoriesyield the same results in the absence of polyelectrolytes (where%$2 ' 0).

and salt-screened electrostatic repulsion and (ii) attractivepolyelectrolyte-mediated bridging interactions of entropicorigin. We show below that the competition between thesemechanisms leads to non-monotonic behavior of free energyand osmotic pressure.

Using the saddle point approximation we can calculate theground state eigenenergy E0 and the fields . and ( and thenobtain the system’s free energy as the logarithm of the partitionfunction in the form

F(w) = !kBT ln , = kBTN (N/S)E0(w)+HPB[.(r)] (17)

where HPB[.(r)] is the standard Poisson–Boltzmann Hamil-tonian defined in (11). The dimensionless osmotic pressure isthen given by p$ = (e0%D)2/($$0(kBT )2) · p, where p is thestandard osmotic pressure given by the negative derivative ofthe free energy with respect to the volume of the Wigner–Seitzcell.

Figure 6 shows osmotic pressure profiles for threedifferent macroion surface charges, two different Debyescreening lengths and fixed polyelectrolyte density. Saltconcentration obviously has a profound impact on the sign ofthe forces between the macroions. At high salt concentrationsthe forces between the macroions stay repulsive for alldistances (see the uppermost curve on figure 6). At lowconcentrations the pressure profiles for %D = 1.21 nm areall non-monotonic with long range negative tails implyingattractive bridging forces of order 10!2 pN nm!2 = 10 atmbetween like-charged macroions.

In certain regions of the parameter space we encounterosmotic pressure profiles of van der Waals type, indicatinga discontinuous phase transition between phases of equalsymmetry. We have analyzed these pressure profiles usingthe Maxwell construction and determined the spinodal and thecoexistence regions in the phase diagrams by calculating thedependence of the critical pressure on the macroion surfacecharge density. The critical pressure is defined via the criticaliso-ionic strength line that separates the regime of monotonic

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J. Phys.: Condens. Matter 22 (2010) 414102 M Licer and R Podgornik

Figure 6. Osmotic pressures at three different macroion surfacecharges (* = 0.9, 1, 1.3 *DNA, see legend), two different Debyescreening lengths (%D = 0.91 nm (top curve at each * ) and%D = 1.21 nm (bottom curve at each * )) and fixed polyelectrolytedensity (%$2 = 0.35). Pressure profiles for %D = 1.21 nm are allnon-monotonic with long range negative tails implying attractivebridging forces of order 10!2 pN nm!2 = 10 atm betweenlike-charged macroions. The magnitude of the attractive bridgingforces is stated for each macroion charge density at %D = 1.21 nm.

osmotic pressures from the one with a van der Waals type loopbehavior in complete analogy with the van der Waals theoryof first-order transitions. An example of a van der Walls-likeiso-ionic strength curve (a curve along constant values of %D)is seen in figure 7 showing osmotic pressure in the vicinity of aphase transition at fixed parameters * = *DNA, %D = 1.21 andat various values of %$2. From top to bottom the values of %$2

are 0.65, 0.63, 0.61 and 0.55. Note that at fixed Debye lengthhigher values of %$2 signify higher densities of polyelectrolyte.The two phases in this transition, corresponding to the samepacking symmetry of the macromolecules, could correspond(possibly) to the hexagonal and line hexatic phases as observedin [24]. This identification is, of course, very tentative.

Figure 7 indicates that at small macroion interaxialspacings or at high polyelectrolyte densities (the uppermostcurve with %$2 = 0.65) the largest contribution to the osmoticpressure stems from the steric confinement of the PE chainsbetween the macroions. As we lower the polyelectrolytedensities (as %$2 varies from 0.65 to 0.55) we observe acoexistence region of tightly packed and loosely packed phasesdetermined by the Maxwell construction for %$2 ! 0.59.A tightly packed structure could possibly correspond to ahexagonal crystalline phase and a loosely packed structure to aline hexatic phase seen in DNA in a different context [24]. Bothphases are schematically presented as an inset to figure 8 whichshows outlines of coexistence curves obtained from Maxwellconstructions on a set of van der Waals-like iso-ionic pressure–volume curves for different values of macroion surface chargedensity. Illustrations of the two phases of equal symmetry areadded to their respective regions of phase space.

It is clear from figure 8 that macroion surface chargedensity has a profound impact on the phase diagram. Thecritical point of the coexistence curve at * = 0.9*DNA

(lowermost curve square symbol) is located at a much lower

Figure 7. Osmotic pressure in the vicinity of a discontinuous phasetransition at fixed parameters * = *DNA, %D = 1.21 and at variousvalues of %$2. From top to bottom the values of %$2 are 0.65, 0.63,0.61 and 0.55. At high macroion and polyelectrolyte densities(uppermost curve at %$2 = 0.65) the system is dominated by stericpolyelectrolyte repulsion while at lower densities a hexagonalcrystalline–line hexatic phase transition occurs which allows forenhanced bridging interactions, yielding like-charge macroionattractions.

Figure 8. Outlines of calculated coexistence curves obtained fromMaxwell constructions on a set of van der Waals-like iso-ionicstrength pressure–volume curves for different values of macroionsurface charge density. We have varied surface charge density from* = 0.9*DNA to * = 1.3*DNA with step 0.1 *DNA at fixed Debyelength %D = 1.21 nm. Dotted lines serve strictly as guides to theeyes.

osmotic pressure than the one at * = 1.3*DNA (uppermostcurve with diamond symbol). This tells us that at lowermacroion charge densities entropic contributions can beneglected in comparison to steric interactions at much lowerpressures. At higher macroion charge densities bridging,which arises from electrostatic adsorption of the chain tothe macroion surface, remains present also for pressureswhich would, at lower macroion charge densities, alreadyenforce steric domination—yielding monotonic repulsiveforces. We can extract a similar conclusion from dimensionless

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J. Phys.: Condens. Matter 22 (2010) 414102 M Licer and R Podgornik

Figure 9. Dimensionless critical pressure dependence on themacroion charge density p$

c (*/*DNA). We show calculated criticalpressures for surface charge densities 0.9, 1.0, 1.1, 1.15, 1.20 and1.30 *DNA. We show also values for pressuresp = $$0(kBT )2 p$/(e0%D)2. Error bars are due to the fact that duringcalculations it was impossible to ‘hit’ exactly the critical parametersduring the shooting algorithm so we were forced to estimate criticalvalues from numerically neighboring solutions.

critical pressure dependence on the macroion charge densityp$

c (*/*DNA) shown in figure 9. The critical iso-ionic strengthline separates the monotonic pressure–volume curves fromthose showing a van der Waals loop. In figure 9 we showcalculated critical pressures for surface charge densities 0.9,1.0, 1.1, 1.15, 1.20 and 1.30 *DNA. We see that the dimensionalcritical pressure p = $$0(kBT )2 p$/(e0%D)2 monotonicallyrises for more than 0.085 pN nm!2 as we increase the macroioncharge density from 0.9*DNA to 1.3*DNA.

Note that, due to a relatively minor change in surfacecharge density, the critical pressure in the system increasesby almost two orders of magnitude. This implies yet againthat the DNA molecule—highly charged as it is—may beespecially suitable for self-assembly mechanisms that are dueto polyelectrolyte bridging interactions.

5. Discussion

Measurements and theoretical descriptions of the bridginginteractions in macroion–polyelectrolyte complexes all in-dicate that electrostatic approaches that disregard polyelec-trolyte configurational entropic effects are probably insuffi-cient. It is these entropic contributions to free energy that giverise to attractive bridging interactions between like-chargedmacroions while electrostatic contributions—if treated in theself-consistent field framework—invariably give rise to longerrange Poisson–Boltzmann repulsions. The combination ofthese short range bridging attractions and longer ranged elec-trostatic repulsions can give rise to a first-order transition be-tween two macroion phases of equal symmetry but differentdensities.

Though our calculation was not designed specifically forany particular system composed of stiff macroions and flexiblePE chains the behavior that we observe can be in many aspectsconnected with the observed salient features of the DNA–PE

mixtures that have been recently extensively studied [3, 4, 8].We could thus argue that at least some part of the attractiveinteractions responsible for self-assembly in these systems isdue exactly to polyelectrolyte bridging mechanisms.

The theoretical approaches described in this paper haveseveral well-known drawbacks. First of all, any mean-field or self-consistent field approaches completely disregardelectrostatic fluctuation and correlation effects [22] which canlead to sign reversal and oscillatory force behavior for highervalence counterions. Our analysis also does not includepolydispersity or intra-chain charge–charge correlations whichresult in electrostatic stiffening of the chain. The groundstate dominance approach has limited our investigation tochains with contour lengths much longer than their persistencelength, thereby neglecting all free tail effects that havebeen theoretically investigated elsewhere. We furthermoreassumed soft electrostatic adsorption of chains and ignoredthe possibility of polyelectrolyte grafting or specific adsorptionto the surface of macroions. This assumption invariablyyields a polyelectrolyte depleted zone in the vicinity of themacroion which obviously is not always the case, e.g. in thepolyelectrolyte brushes.

In the weak coupling limit we employed a linearizedDebye–Huckel version of the Poisson–Boltzmann theorywhich is by definition unable to describe any nonlinear effects.Our work does, however, consist of the first full numericalsolution of the Edwards equation in a cylindrical Wigner–Seitzcell in the presence of salt and macroions whereas analyticalapproximations and variational approaches in similar systemshave already been employed by one of the authors.

In the strong coupling limit we have self-consistentlycoupled the nonlinearized Poisson–Boltzmann theory for bulkelectrolytes with the Edwards functional integral descriptionof polyelectrolyte density on a hexagonal DNA lattice. Wehave solved the self-consistent field equations numericallyand demonstrated a salt-induced phase transition betweenhexagonal and line hexatic DNA phases.

Further work should be done by including Flory-typesteric (4 terms to free energy to describe intra-and inter-chainexcluded-volume interactions.

Acknowledgments

RP would like to acknowledge the financial support bythe Slovenian Research Agency under contract no. P1-0055 (Biophysics of Polymers, Membranes, Gels, Colloidsand Cells) and J1-0908 (Active media nanoactuators withdispersion forces).

References

[1] Kabanov A V, Felgner P and Seymour L W (ed) 1998Self-Assembling Complexes for Gene Delivery (New Jersey:Wiley)

[2] De Rouchey J and Rau D C 2009 in preparation[3] Toma A C, de Frutos M, Livolant F and Raspaud E 2009

Biomacromolecules at press[4] Korolev N, Berezhnoy N V, Eom K D, Tam J P and

Nordenskiold L 2009 Nucl. Acids Res. at press[5] Koltover I, Salditt T and Safinya C R 1999 Biophys. J. 77 915[6] Schiessel H 2003 J. Phys.: Condens. Matter 15 699

8

J. Phys.: Condens. Matter 22 (2010) 414102 M Licer and R Podgornik

[7] Leforestier A and Livolant F 2009 Proc. Natl Acad. Sci.106 9157

[8] DeRouchey J, Netz R R and Raedler J O 2005 Eur. Phys. J. ESoft Matter 16 17

[9] Minko S and Roiter Y 2005 Curr. Opin. Colloid Interface Sci.10 9

[10] Koltover I, Salditt T, Radler J and Safinya C R 1998 Science281 78

[11] Podgornik R, Strey H H, Gawrisch K, Rau D C,Rupprecht A and Parsegian V A 1996 Proc. Natl Acad. Sci.93 4261

[12] Strey H H, Parsegian V A and Podgornik R 1997 Phys. Rev.Lett. 78 895

[13] Podgornik R, Strey H H and Parsegian V A 1998 Curr. Opin.Colloid Interface Sci. 3 534

[14] Ewert K K, Evans H M, Zidovska A, Bouxsein N F,Ahmad A and Safinya C R 2006 J. Am. Chem. Soc.128 3998

[15] Podgornik R 1992 J. Phys. Chem. 96 884[16] Podgornik R 2004 J. Polym. Sci. B 42 3539[17] Podgornik R and Licer M 2006 Curr. Opin. Colloid Interface

Sci. 11 273Claesson P M, Poptoshev E, Blomberg E and Dedinaite A 2005

Adv. Colloid Interface Sci. 114 173[18] Doi M and Edwards S F 1986 The Theory of Polymer Dynamics

(Oxford: Oxford University Press)[19] de Gennes P G 1988 Scaling Concepts in Polymer Physics

(Ithaca, NY: Cornell University Press)[20] Wang Q 2005 Macromolecules 38 8911[21] Podgornik R and Saslow W M 2005 J. Chem. Phys.

122 204902[22] Fredrickson G 2006 The Equilibrium Theory of Inhomogeneous

Polymers (Oxford: Oxford University Press)[23] Manning G S 2006 Biophys. J. 91 3607–16[24] Strey H H, Wang J, Podgornik R, Rupprecht A, Yu L,

Parsegian V A and Sirota E 2000 Phys. Rev. Lett. 84 3105

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