rod-like polyelectrolyte adsorption onto charged surfaces in monovalent and divalent salt solutions

Rod-Like Polyelectrolyte Adsorption onto Charged Surfacesin Monovalent and Divalent Salt Solutions

HAO CHENG, MONICA OLVERA DE LA CRUZ

Department of Materials Science and Engineering, Northwestern University, 2225 North Campus Drive,Evanston, Illinois 60208

Received 10 November 2003; revised 17 February 2004; accepted 1 March 2004DOI: 10.1002/polb.20206Published online in Wiley InterScience (www.interscience.wiley.com).

ABSTRACT: We analyze the adsorption of strongly charged polyelectrolytes onto weaklycharged surfaces in divalent salt solutions. We include short-range attractions betweenthe monomers and the surface and between condensed ions and monomers, as wellcorrelations among the condensed ions. Our results are compared with the adsorptionin monovalent salt solutions. Different surface charge densities (�), and divalent (m)and monovalent (s) salt concentrations are considered. When the Wigner-Seitz cellsdiameter (2R) is larger than the length of the rod, the maximum amount of adsorptionscales like nmax � �4/3 in both monovalent and divalent solutions. For homogeneouslycharged surfaces, the maximum adsorption occurs at s* � �2 when s* � �, where � isthe monomer concentration, the counterpart for divalent salt solution, m* roughlyscales as �2.2 when m* � �. The effective surface charge density has a maximumabsolute value at m� � m*. A discrete surface charge distribution and short-rangeattractions between monomers and surface charge groups can greatly enhance surfacecharge inversion especially for high salt concentration. The critical salt concentrationfor adsorption in divalent salt solution roughly scales as mc � b�1.9, where b is thedistance between two neighboring charged monomers. © 2004 Wiley Periodicals, Inc. JPolym Sci Part B: Polym Phys 42: 3642–3653, 2004Key words: polyelectrolyte adsorption; charge inversion; ion condensation; correla-tion; attraction

INTRODUCTION

The modification of surfaces through polyelectro-lyte adsorption in solution has great potentialapplications in many fields.1,2 In particular, ad-sorption of strongly charge biomolecules, such asDNA and RNA, is important in biotechnology.3,4

Often the adsorption involves short-range attrac-tions between the surface and polyelectrolytes.5,6

These short-range correlations need to be addedto classical models that include long-range corre-

lations only such as the Poisson-Boltzmann ap-proach.7–9 Moreover, corrections to the Poisson-Boltzmann equation need to be included to ac-count for the strong lateral correlations amongthe adsorbed chains. Lateral correlations werefirst included analytically by Rouzina and Bloom-field10 in multivalent ions adsorption. Their effectin weakly charged flexible chain adsorption hasbeen recently analyzed by using the scaling the-ory.11,12 The adsorption can be described usingWigner-Seitzs cells on the weakly charged sur-faces with the adsorbed chains located at theircenters. The size of the Wigner-Seitz cells is de-termined by the average separation between theadsorbed chains. Recently, we extended this ap-

Correspondence to: M. Olvera de la Cruz (E-mail:[email protected])Journal of Polymer Science: Part B: Polymer Physics, Vol. 42, 3642–3653 (2004)© 2004 Wiley Periodicals, Inc.

3642

proach to analyze strongly charged rod-like poly-electrolyte adsorption in monovalent salt solu-tions by equating the chemical potentials of thefree and the adsorbed rods and the free and thecondensed counterions.13

Important aspects of surface polyelectrolyteadsorption include the effect of a discrete distri-bution of charges on the surface14 and specificinteractions in the adsorption such as the pres-ence of divalent salts in the medium.6 In mono-valent salt solutions the condensed counterionsare hydrated, and have a large fluctuation alongthe chain.15,16 Therefore, one can assume an av-erage effective monomer charge in calculations.In divalent counterion solutions, however, it hasbeen demonstrated that part of the condenseddivalent counterions along DNA form “inner-sphere” complexes in which the metal ions canbind to phosphate groups directly, without watermolecules between them.17 Moreover, althoughdivalent counterions do not precipitate DNA(Mn2� can precipitate DNA at high temperaturepossibly due to structural changes18,19), mostmonomer charges are compensated by divalentions,20 resulting in correlations among the con-densed ions along the rods. In this article, westudy the adsorption of strongly charged rodsonto oppositely weakly charged surfaces with adiscrete charge distribution in divalent salt solu-tions. We include short-range correlations to cal-culate the fraction of monomers with condenseddivalent counterions, and the interactions be-tween the adsorbed chains and the surface. Thecations can be free in the bulk solution or con-densed along the bulk free chains and along theadsorbed chains. We equate the chemical poten-tial of all the species to determine the equilibriumnumber of ions in the different states and numberof the adsorbed chains self-consistently.

Most adsorption studies treat the surfacecharges as continuum. In this article we considerthe discrete nature of the surface charge distribu-tion by including the resulting short-range attrac-tion between the adsorbed rods and the surfacecharge groups. This short-range attraction mightbe useful to determine the possibility of polyelec-trolyte adsorption onto like-charged surfaces viashort-range attractions at specific sites. For ex-ample, it has been demonstrated that double-stranded DNA can adsorb onto like-charged sur-faces, such as mica, in solutions containing cer-tain small divalent cations such as Ni(II), Co(II),and Zn(II).6 Because the divalent cations thatlead to DNA adsorption do not precipitate DNA in

the solution,5 it is possible that the binding ad-sorption is a two-step process. The divalent ionsfirst bind strongly to mica at specific binding siteson the surface, decreasing or even inverting thenegative surface charge, and then DNA adsorbs.We study the effect of short-range attractions be-tween surface charge groups and rod-like poly-electrolytes only in the case where the surface isweakly positively charged in divalent counterionsolutions. We do not analyze here rod adsorptiononto strongly oppositely charged surfaces in thepresence of divalent ions, which may lead to rodprecipitation on the surface.5

In this article we determine the maximumpolyelectrolyte adsorption, the adsorption–de-sorption transition or critical salt concentrationfor polyelectrolyte adsorption, and how to controlpolyelectrolyte adsorption, which are the mostimportant concerns in experiment. We also deter-mine the degree of charge inversion. Indeed, poly-electrolyte adsorption often leads to charge inver-sion of the surface–polyelectrolyte com-plex.12,21–23 Recently, some studies have analyzedthe effect of discrete surface charge distributionson the counterion adsorption from the bulk.24,25

Messina et al.14 determined the effect of a discretecharge distribution on charge inversion by coun-terion adsorption in salt-free solution. We findhere that for weakly charged surface, the discretedistribution of surface charges and short-rangeattraction can greatly enhance surface charge in-version by polyelectrolyte adsorption, and the en-hancement is more remarkable with the increasesof salt concentration.

MODEL

We extend the scaling theory of weakly chargedpolyelectrolyte adsorption by Dobrynin et al.12 tostrongly charged polyelectrolyte adsorption in-cluding the effect of ion condensation and consid-ering the entropic energies of free counterionsand free chains in the bulk solution. In this articlewe only consider the adsorption in dilute saltsolutions. The association of the divalent saltcoins to the divalent ions is neglected.26 More-over, we investigate dilute polyelectrolyte solu-tions and adsorption onto weakly charged sur-faces.

Strongly Charged Rod-Like Polyelectrolytes inSolution

We outline here the model for strongly chargedrod-like poleylectrolytes in the divalent salt solu-

POLYELECTROLYTE ADSORPTION ONTO CHARGED SURFACES 3643

tion. The model for solutions of strongly chargedrods in monovalent salt is described in our previ-ous article.13

We consider monodispersed rod-like polyelec-trolyte solutions. The degree of polymerization ofthe rods is N. Each monomer carries a negativeelementary charge. The number concentration ofmonomers in the solution is �, and b is the dis-tance between two neighboring charge groupsalong the chain. The valence of the chain counte-rions is 2, and their effective size is a. The addeddivalent salt has the same cations as the originalcounterions of the polyelectrolytes and its numberconcentration is denoted by m. Therefore, the to-tal number concentration of cations is m � �/2.The concentration of small monovalent anionicions in the system is 2m.

The structure of condensed multivalent ions ofvalence z depends on both z and the Bjerrumlength lB � e2/�kBT, where e is the elementarycharge, � is the solvent permittivity and kBT isthe thermal energy. Divalent ions can form nearlyperiodic structures along strongly charged rods atlow effective temperatures (ł/z2lB ��1), when therod charges are nearly neutralized.10,26 For dis-crete rod charge distributions, the periodic struc-ture is correlated with the position of the chargesalong the rods. At room temperature, condenseddivalent ions do fluctuate and are exchanged withthe free counterions in the bulk solution. How-ever, because we are considering here the effect ofshort-range attractions among the condensed ionsand the charges along the rods, which inhibitsfluctuations and enhance the fraction of con-densed divalent counterions, we assume for sim-plicity that the condensed divalent counterionsform a periodic structure along the chain at roomtemperature (Fig. 1). That is, in our model we donot differentiate between condensed ions forminginner and outer complexes. Instead, all the con-densed ions can be exchanged with the free ionsin the bulk solution, although the fluctuations ofthe ions along the rods are neglected. With thissimplification, as explained below (see eq 2), asingle parameter denoted by d (which is an effec-tive distance between the centers of the con-densed metal ion and the contacted negativecharge group along the rod), can be used to com-pute the effect of short range correlations in theanalysis. That is, in the limit of fc � 1, one recov-ers with this approach the cohesive energy ex-pected for strongly correlated condensed ions.10,26

Moreover, our approach allows us to include thelong-range electrostatic self-energy (eqs 3 and 4)

of the rod with condensed ions self-consistently. Aneutral cluster is created by one condensed coun-terion and two neighboring monomers. For sim-plicity, we neglect the electrostatic interaction be-tween this neutral cluster and other charges inthe system; the addition of interactions betweenneutral clusters in the model may lead to poly-electrolyte precipitation in the bulk, which is notobserved experimentally for strongly chargedpolyelectrolytes in divalent counterion solutions.5

The fraction of monomers with condensed coun-terions for rods in solution denoted by fc isdetermined by minimizing an appropriate freeenergy. In this study, all the free energies are inunits of kBT.

The total free energy of the polyelectrolyte so-lution is given by

Fslt � Fsl

sn � Fslsr � Fsl

fi � Fslfc. (1)

Fslsn, the short-range electrostatic energy of neu-

tral clusters, is given by

Fslsn

V �fc

2��, (2)

where V is the volume of the bulk solution, fc�/2 isthe number of neutral clusters in unit volume,and � is the electrostatic energy of one neutralcluster. The electrostatic attraction energy be-tween the metal ion and two monomers separatedby a distance d is �4lB/d, and the repulsion en-ergy between two monomers in the neutral clus-ters is lB/b giving � equal to lB/b�4lB/d. Notice

Figure 1. Strongly charged rods in solution. One con-densed divalent counterion can form a neutral clusterwith two neighboring monomers.

3644 CHENG AND OLVERA DE LA CRUZ

that if fc tends to be one and d � 2b, the cohesiveenergy of the resulting correlated distribution ofcondensed ions is the same as the one derived inref. 26, where we extended the strongly correlatedapproach of reference 10 for continuously chargedsurfaces to charged rods with a discrete chargedistribution, where � � �z(z�1)lB/d, up to a pro-portionality constant of the order of one.26 There-fore, this short-range energy can also account forthe electrostatic correlations expected when thefraction of condensed ions increases.

The second term in eq 1 is the intrachain self-repulsion energy, which is nonzero only if fc 1,and is given by

Fslsr

V ��

NlB

b/1 � fc��

1�i�j�N1�fc�

exp(�j � i�b/1 � fc�)

j � i . (3)

Here, b/(1 � fc) is the average distance betweenmonomers, which do not form neutral clusterswith counterions. The inverse Debye screeninglength is . For short rods and dilute salt solution,Nb � 1, eq 3 is approximately equal to

Fslsr

V � �lB�

b 1 � fc�2 ln N1 � fc�/e. (4)

Notice that in eq 4 we are neglecting the screen-ing effect. Therefore, we are only analyzing smallsalt concentrations.

In eq 3 and 4 we assume an effective averagedistance, b/(1 � fc), between uncompensatedmonomers. Clearly, this approximation breaksdown for small values of fc, because in this limitthe condensed ions have large concentration fluc-tuations along the rods. Smaller values fc areobtained when w � 0 in eq 2. In this case, and inthe limit of low ionic strength solutions, Nb � 1,eq 3 is approximately equal to the self-energy ofrods in monovalent salts,10 given by �(lB/b)(1�fc)

2

ln N/e, which gives similar results to eq 4. Themajor change between monovalent and divalentions is the strength of the short range correlationin eq 2, which is nonzero for divalent ions.

The translational entropic energy of the freeions gives the contribution

Fslfi

V � �m ��

2 �fc�

2 � ln ��m ��

2 �fc�

2 �a3/e�, (5)

The contribution of the free anions is not includedsince they do not condense along the chains andgive a constant contribution to the total free en-ergy. Fsl

fc is the translational entropic energy ofthe chains,

Fslfc

V ��

N ln �v/e, (6)

where v is the monomer volume.For salt-free or dilute salt solution, the linear-

ized electrostatic contribution to the Helmholtzenergy from the free ions is very small13 (thelinearized contribution from the chains shouldnot be included to avoid spurious results27). Forsemidilute salt solution, however, this term canbe even larger than other terms. However, in bothcases if we neglect this energy term, our finalresults are not affected as we shown in ref. 13.

Rod-Like Polyelectrolyte Adsorption onto WeaklyOppositely Charged Surfaces

Strongly charged rods can adsorb onto a weaklyoppositely charged surface and form a periodicstructure due to strong correlations betweenthem as shown in Figure 2. In Figure 2, thecharges in the substrate are the charge groups ofthe surface. We denote the surface charge groupsthat are in direct contact with monomers as con-tact points and assume that the distance fromtheir centers to the charge of contact monomer isalso d. The total energy on the surface is given by

Fsfl � Fsf

sn � Fsfsr � Fsf

bc � Fsfla � Fsf

gr. (7)

The first term is the short-range attraction en-ergy of contact points with monomers and theelectrostatic energy of neutral clusters,

Fsfsn � �

A�R2

lB

d � �A

�R2

Nfs

2 �. (8)

Here, A is the area of the charged surface, R is thecell radius, A/�R2 is the number of the rods on thesurface, and � is the number of contact points perchain equal to Nb�1/2 � 1. In our study, somecations are still condensed along the negativelycharged rods after the rods adsorb onto weaklyoppositely charged surfaces. We denote by fs thefraction of monomers with condensed counterionsfor rods on the surface.


Fsfsr is the intrachain self-repulsion energy for

adsorbed chains,

Fsfsr �

A�R2

lB

b N1 � fs�2 ln N1 � fs�/e. (9)

The repulsion energy between different rods onthe surface is12

Fsfbc �

12

A�R2 �

R

1 � fs�N

1 � fs�N

�R2 2�rlB

r exp (�r)dr

�A

�R4 1 � fs�2N2

lB

exp(�R). (10)

Here, the inverse Debye screening length, isgiven by

2 � 4�lB�2m � 4�m ��

2 �Nfs

2A

�R2V �fc

2 ��,

�� A

�R2

NV , (11)

where �� is the monomer concentration in solu-tion after some rods absorb onto the charged sur-face. The contribution of monomer charges toscreening is not included because for stronglycharged polyelectrolytes most of monomercharges are compensated by divalent counterions,and this contribution is only effective for longinteraction distances. However, the small freeions contribute to screening both at small andlarge distances.28 Therefore, the contributionfrom monomer charges does not affect greatly ourcalculation for salt-free solution and is negligiblefor salt solutions. The long-range attraction en-ergy between rods and surface charge groups is

Fsfla � �

A�R2��

0

N � Nfs � ��2�rlB

r � exp (�r)dr

� ��1/2/2

�2�rlB

r �exp(�r)dr� � �A

�R2 2��lB

�N � Nfs � �� exp ��1/2

2 �� (12)

where � is the surface charge density, and ��1/2/2is half the distance between two neighboringcharge groups. We assume the monomers that donot touch contact points are in a mean electro-static field created by the charge groups of thesurface. However, the monomers that touch con-tact points are excluded from this field within thedistance of ��1/2/2 because we have already con-sidered this short-range attraction in eq 8. Therepulsion energy among the charge groups of thesurface is Fsf

gr � A��2lB/. This term can be ne-glected if the volume of the bulk solution islarge.13

Note that the charge distribution on the sur-face is discrete, and polyelectrolytes may havesignificant orientational order on the surface.Further studies including this effect are needed toobtain more accurate results when the surfacecharge density increases or in salt-free conditions.In this article the long-range electrostatic ener-gies in eq 10 and 12 are treated as a continuum.

Figure 2. Adsorbed polyelectrolytes form nanopat-terns on weakly charged surfaces. The charges in thesubstrate are the surface charge groups and ��1/2/2 ishalf of the distance between two neighboring chargegroups.


The total energy in solution after adsorption isalso given by eq 1–6 with some minor changes:monomer concentration reduce to �� from �; thecation concentration becomes

m ��

2 �Nfs

2A

�R2V �fc��

2 .

The summation of the energies on the surface andin solution is the system total energy. To get thethermodynamic equilibrium state, we minimizethe total energy with respect to fs, fc and thenumber of adsorbed chains on the surface (nc� A/�R2). This is equivalent to equilibrate thechemical potentials of the free counterions in thebulk solution with the condensed counterionsalong chains in the bulk solution and on the sur-face, and equilibrate the chemical potentials offree and adsorbed chains. We get the followingequations

�1

V��

�Ft

�fc�

12 ln��m �

�

2 �Nfs

2A

�R2V �fc��

2 �a3��

12 � � 2

lB

b 1 � fc�ln N1 � fc�/�e � 0, (13)

�1

AN�R2

�Ft

�fs�

12 ln��m �

�

2 �Nfs

2A

�R2V �fc��

2 �a3�

�12 � � 2

lB

b 1 � fs� ln N1 � fs�/�e

� 2NR2

lB

1 � fs�exp � R� � 2��

lB

� 0 (14)

�Ft

�A

�R2

�12 fc � fs�N ln��m �

�

2 �Nfs

2A

�R2V

�fc��

2 �a3� � ln��v �12fc � fs�N� �

lB

d � �lB

bN

�1 � fs�2 ln N1 � fs�/e � 1 � fc�

2 ln N1 � fc�/e�

� 2N2

R2

lB

1 � fs�

2 exp(�R)�1 �R

4 �� 2��

lB

�N � Nfs � �� exp��1/2

2 �� 0

(15)

In the limit of large or infinite bulk solutions,

m ��

2 �Nfs

2A

�R2V �fc��

2 � m ��

2 �fc�

2

and �� . Then we can obtain the followingequations,

ln��m ��

2 �fc�

2 �a3� � �

� 4lB

b 1 � fc�ln N1 � fc�/�e � 0, (16)

ln��m ��

2 �fc�

2 �a3� � � � 4lB

b 1 � fs�ln

� N�1 � fs)/�e � 4NR2

lB

1 � fs� exp(�R�

� 4��lB

� 0, (17)

fc � fs�N ln��m ��

2 �fc�

2 �a3� � 2 ln �v

� 4N2

R2

lB

1 � fs�

2 exp(�R)�1 �R

4 � � 2lB

bN

�1 � fs�2 ln N1 � fs�/e � 1 � fc�

2 ln N1 � fc�/e�

� fc � fs�N� � 2lB

d � � 4��lB

�N � Nfs � ��

� �exp� ��1/2

2 �� 0 (18)

Although is also a function of fs, fc and nc, inref. 13 we showed that the derivatives of withrespect to fs, fc and nc are zero for infinite largebulk solutions. Therefore, we do not include thederivatives of in eqs. 13–18.

RESULTS AND DISCUSSION

We analyze our results for parameter values use-ful to describe adsorption of short (rod-like) DNAin aqueous solutions where the distance betweencharges b � 0.17 nm, and the diameter of themonomers D � 2 nm. For the distance betweencondensed counterions and the charges groupsalong DNA (d) we use experimental result,17 d� 0.5 nm. We use a typical size of hydrated diva-lent metal ions a � 0.45 nm. The Bjerrum lengthin water at room temperature is lB � 0.714 nm, v� (D/2)2b � 0.17 nm3, � � 10�5 M and N � 100.


Polyelectrolyte adsorption can be controlled ex-perimentally by adjusting the solution pH value,ionic strength, and the surface charge.1,2 In thisarticle the effect of different salt concentrations(m) and surface charge densities (�) are consid-ered. The analysis of changes in the solution pH,which may involve chemical reactions, requiresadditional experimental data. Therefore, we as-sume all the charge groups are ionized and do notinclude pH effect. In some figures we also analyzethe effect of different rod charge densities of thechain (lB/b). However, without specification, thevalue of b is 0.17 nm.

The fraction of monomers with condensed ionsfc increases as salt concentration m increases(Fig. 3). The smaller a and b and the larger N and�, the more counterions condense along polyelec-trolytes in solution. Setting N � 300 (150 basepairs), the calculated value is in excellent agree-ment with experimental data, 0.86 � 0.08.20 Thestrongly charged polyelectrolytes retain most oftheir condensed divalent counterions after poly-electrolyte adsorption onto the weakly chargedsurfaces. Strongly charged polyelectrolyte adsorp-tion in divalent salt solutions is similar to adsorp-tion in monovalent salt solutions, with highervalues of fc and fs. When the surface charge den-sity is increased (from 0.008e/nm2 to 0.08e/nm2),

more polyelectrolyte counterions are releasedfrom the adsorbed chains (Fig. 3) and gain en-tropic energy in the bulk solution. In Figure 3 thecurves are ended at some points: for � � 0.008e/nm2 the plot ends when all the adsorbed chainsdesorb from the surface, and for � � 0.08e/nm2

the curve is ended when the diameters of the cellsbecomes smaller than the length of the rod be-cause our model is only valid for the diameter ofthe cells larger than the rod length. As the saltconcentration increases, the value of fc � fs be-comes smaller because the repulsion of the sur-face to the condensed counterions decreases whenthe salt concentration increases.

The amount of adsorbed polyelectrolytes on thesurface n � A/�R2 initially increases (i.e., thepattern cell radius R decreases) with the additionof salt. The reduced Debye screening length de-creases the repulsion from the rods in other cellsand does not greatly affect the electrostatic at-traction between charged monomer and surfacecharge groups in the Wigner cell when �1 islarger than R. This accounts for the increase ofpolyelectrolyte adsorption with increasing salt.With further addition of salt, however, �1 keepsdecreasing and the adsorbed polyelectrolytes be-gin to desorb from the surface to gain entropicenergy in the bulk solution when the surfacecharge groups cannot provide enough attractionenergy due to the ion screening. Therefore, thereis a maximum adsorption as a function of saltconcentration. The maximum adsorption nmax isassociated with a minimum cell radius Rmin givenby nmax � A/�Rmin

2 . In Figure 4(a) we show theminimum cell radii Rmin for surfaces as a functionof surface charge density, and in Figure 4(b) (thesolid line) we show the salt concentration m* cor-responding to the maximum adsorptions as afunction of surface charge density. We find thatRmin approximately scales like Rmin � ��2/3 in therange of surface charge density considered here;that is, the maximum adsorption scales as nmax

� �4/3. However, m* does not obey a power law inthis range of surface charge density. If we con-sider uniformed charged surfaces instead, m*roughly scales like �2.2 when m* � �, as discussedbellow. We find here that in monovalent salt so-lutions (following the model of ref. 13, which ne-glects shortrange attractions and correlationsamong condensed ions) the maximum polyelectro-lyte adsorption onto homogeneously charged sur-faces also increases with surface charge density,and it scales with a power law slightly larger than

Figure 3. Fractions of monomers with condensedions in bulk rods, fc (solid curve), and in the adsorbedrods, fs (the first to lowest dashed curves are for �� 0.008e/nm2, 0.01e/nm2, 0.02e/nm2, 0.04e/nm2, 0.06e/nm2, and 0.08e/nm2, respectively), as a function of saltconcentration m. Both fc and fs increase as m increasesbut the difference becomes smaller. Also, fs becomesmaller when the surface charge density increases. Thevalues of the parameters are b � 0.17 nm, v � 0.17nm3, d � 0.5 nm, N � 100, � � 10�5 M and a � 0.45nm.


4/3. When s* � �, s* roughly scales as �2, like inweakly charged polyelectrolyte adsorption.12

Charge inversion induced by polyelectrolyteadsorption has attracted much attention in recentyears. Two extensively discussed reasons forcharge inversion are the lateral correlations be-tween adsorbed polyelectrolytes29 and the releaseof condensed polyelectrolyte counterions.30 In thisarticle both effects, lateral correlations and par-tial release of condensed counterions after chainadsorption, contribute to charge inversion. More-over, the effects of the discrete surface chargegroup distribution and the short range attractionalso contribute to charge inversion in our work13

as found in other models.14 The effective charge ofsurface–polyelectrolyte–ion complex is defined as

�* � � �N � fsN

�R2 (19)

The sign of the effective surface charge is oppo-site to that of the bare surface, that is, the surface–polyelectrolyte–ion complex shows charge inver-sion. The absolute value of the surface charge den-sity initially increases as the salt concentrationincreases, and has a maximum absolute value forsurfaces of � � 0.008e/nm2 and � � 0.01e/nm2. Thecurve for � � 0.02e/nm2 is monotonous in the con-sidered range of salt concentration. The maximumabsolute values of the effective surface densities forthis surface or other surfaces with higher surfacecharge densities are expected at higher salt concen-tration (Fig. 5). It is natural to believe that themaximum polyelectrolyte adsorption corresponds tothe maximum charge inversion. However, Figure4(b) shows that for fixed surface charge density themaximum charge inversion appears at m�, which islower than the salt concentration of maximum ad-sorption m*. The reason is that the fraction ofmonomers with condensed ions in the surface fsincreases, and it approaches fc as the salt concen-tration increases due to screening of the surface andcondensed counterion repulsions.

To show the effect of short-range attractionsbetween monomers and discrete surface chargegroups, we consider polyelectrolyte adsorptiononto surfaces with uniform charge distributions

Figure 5. Effective surface charge density (�*) as afunction of salt concentration (m) at different surfacecharge density (�) conditions for b � 0.17 nm, v � 0.17nm3, d � 0.5 nm, N � 100, � � 10�5 M and a � 0.45nm.

Figure 4. (a) The minimum cell pattern radius Rmin

for different surface charge density conditions scaleslike Rmin � ��2/3 in the considered range of surfacecharge density. (b) For fixed surface charge density themaximum charge inversion appears in a lower saltconcentration, m� compared with the salt concentra-tion, m* corresponding for maximum adsorption; theparameters’ values are b � 0.17 nm, v � 0.17 nm3, d� 0.5 nm, N � 100, � � 10�5 M and a � 0.45 nm.


and compare it with the adsorption onto surfaceswith discrete surface charge distributions. For asurface with a homogeneous charge density, thelong-range attraction energy between rods andsurface charge simply is

Fsfla � �

A�R2 2��

lB

N1 � fs�. (20)

The short-range attraction of contact points is notincluded in the calculations for adsorption ontouniformly charged surfaces. Figure 6 shows thatthe absolute value of the effective surface chargedensity is smaller if we do not consider the con-tact points and assume the bare charge of thesurface is uniformly distributed. The charge den-sity difference between these two states is smallfor salt-free conditions in the range of parametersstudied here, and is intensified by the increase ofsalt concentration. However, even in salt free so-lutions, the charge density difference can be verylarge if the short-range attraction increases suchas for a smaller distance between surface chargegroups and monomer charges. Greberg et al.showed that for charged surfaces in salt solution(no polyelectrolyte) the size of the surface counte-rions could be an important factor for charge in-version.29 Some other kinds of strong short-rangeattractions are also possible to increase the sur-face charge inversion.31 Local electrostatic attrac-tions, which are enhanced in surfaces with a dis-crete surface charge distribution, can easily dom-

inate the long-range electrostatic repulsionsbetween the adsorbed rods, especially for smallDeybe screening lengths. Messina et al. demon-strated the enhancement of charge inversion dueto a discrete charge distribution by analyzing amacroion using Molecular dynamics simula-tions.14

To compare strongly charged polyelectrolyteadsorption in divalent salt solution with the ad-sorption in monovalent salt solution, which wepresented in previous article,13 we assume thesurface charge distribution is homogeneous. InFigure 7, we compare the adsorption in monova-lent and divalent salt solutions at the same Debyescreening length (42.1 nm, s � 0.05 mM, m� 0.0165 mM). The pattern radius R scales like R� ��0.44 for monovalent salt solution and ��0.46

for divalent salt solution. [Note: Fig. 7(a) is dif-ferent with Fig. 4(a), which is obtained at differ-ent salt concentration; Fig. 7(a) is obtained at onesalt concentration, and the conditions are farfrom maximum adsorption.]32 The absolute valueof the effective surface charge density P�*P in-creases with the bare surface charge density at apower of 0.60 for monovalent salt solution, but theincrease does not obey a power law for divalentsalt solution in the considered range of surfacecharge density. Figure 7(a) shows that for solu-tions with same ionic strength (low salt concen-tration level), more polyelectrolytes can be ad-sorbed onto the charged surface in divalent saltsolution than in monovalent salt solution. This isdue to a smaller repulsion between adsorbedchains (due to higher counterion condensation) indivalent salt solution than in monovalent saltsolution. For the larger correlations between ad-sorbed chains, the absolute value of effective sur-face charge density in monovalent salt solution islarger than that in divalent salt solutions. Similarresults can be obtained for different values of �1

if the salt solution is dilute.We investigate polyelectrolytes with differ-

ent rod charge densities and same degree ofpolymerization. The results for adsorption indivalent salt solution do not show importantdifferences with the ones for adsorption inmonovalent salt solution. Therefore, we onlydiscuss the critical adsorption (adsorption–desorption transition) here. The adsorption–desorption transition line for polyelectrolytes isobtained by setting the radii of the cell infinite,R � , in eq 17–18

Figure 6. Effective surface charge density (�*) as afunction of salt concentration (m) for uniform and dis-crete distributions of surface charges for b � 0.17 nm,v � 0.17 nm3, d � 0.5 nm, N � 100, � � 10�5 M, a� 0.45 nm, and � � 0.02e/nm2. The discrete distribu-tion can greatly enhance charge inversion.


2NR2

lB

1 � fs�exp(�R) � 0 and

2NR2

lB

1 � fs�

2exp(�R)�1 �R

4 � � 0

In Figure 8 we plot the critical salt concentra-tion for adsorption (mc) as a function of the sur-face charge density. There is no adsorption if the

salt concentration is higher than this criticalvalue. With the increase of rod charge densitymore counterions condense along the rod gener-ating lower rod effective charges. For weaklycharged surfaces, polyelectrolytes with a highercharge density have a larger amount of chainadsorption at low salt concentration due to asmaller repulsion between adsorbed chains. How-ever, the result reverses at the critical adsorptionmc. For uniformly weakly charged surfaces, thecritical salt concentration scales like mc � b (asshown in Fig. 9; for different surface charge den-

Figure 9. The critical salt concentration for adsorp-tion mc as a function of the inverse line charge densityb for � � 0.005e/nm2 (uniform), v � 0.17 nm3, N � 100,� � 10�5 M, and a � 0.45 nm.

Figure 7. Comparison of polyelectrolyte adsorption indivalent (solid line) and monovalent salt (dashed line)solutions at the same Debye screening length (42.1 nm, s� 0.05 mM, m � 0.0165 mM). In both figures of (a) and (b)logarithmic coordinates are used, the surfaces are uni-formly charged and b � 0.17 nm, v � 0.17 nm3, N � 100,and � � 10�5 M; the values for the ion sizes a are 0.45 nmand 0.4 nm for divalent and monovalent counterions,respectively. (a) In the considered surface charge densityrange, R obey the power law R��0.46 for divalent saltsolutions and R��0.44 for monovalent salt solutions. (b)The absolute value of effective surface charge densityincreases with the bare surface charge density. The slopefor the dashed line is 0.6.

Figure 8. The critical salt concentration for adsorp-tion mc as a function of surface charge density � forpolyelectrolytes with different rod charge densities(lB/b � 4.2, 3, and 2) and v � 0.17 nm3, d � 0.5 nm, N� 100, � � 10�5 M and a � 0.45 nm.


sities mc also increase with b linearly). Highersalt concentrations are needed to deplete all theadsorbed chains in the presence of short rangeattractions, as the solid, dashed and dotted linesshow in Figure 8. When mc � �, mc scales as �

where is approximately 1.9. Netz et al. consid-ered the adsorption–depletion transition forweakly charged polyelectrolyte adsorption in saltsolution.33 Their transition point definition isslightly different. They neglected the monomer–monomer electrostatic repulsion and got sc � �2/3

b�2/3 for fixed surface charge density. In theirmodel, the negative power of b results from thecondition that there is no ion condensation inweakly charged polyelectrolytes. The smallerpower law dependence of � results from neglect-ing the electrostatic repulsion (lateral correla-tions) among adsorbed polyelectrolytes, which isimportant in strongly charged polyelectrolytesadsorbed onto weakly charged surfaces.

CONCLUSIONS

Short-range attractions between adsorbed poly-electrolytes and charge groups on the surfacesincrease the degree of adsorption of stronglycharged rods onto weakly charged surfaces. Inthis article we consider adsorption in divalent saltsolutions and also include the short-range attrac-tions between the condensed divalent ions andthe monomers, as well as the correlations be-tween condensed divalent counterions and thecorrelations between adsorbed chains.

Most of the charges of strongly charged poly-electrolytes are compensated by divalent counte-rions due to the correlations between the con-densed counterions and due to the short-rangeattractions between the divalent counterions andthe monomers. The adsorbed chains release partof their condensed counterions. The unreleaseddivalent counterions are indispensable to de-crease the electrostatic repulsion among the ad-sorbed chains on the surface. Polyelectrolyte ad-sorption is a competition among the attractionsbetween the surface charge groups and the poly-electrolytes, the repulsions between the adsorbedchains and the entropy of the chains. In stronglycharged polyelectrolyte, it is also important toinclude the release of the condensed counterionsupon adsorption. There is a maximum in the de-gree of polyelectrolyte adsorption (nmax � A/�Rmin

2 ) as a function of salt concentration. Whenthe Wigner-Seitz cells diameter is larger than the

length of the rod, the maximum amount of ad-sorption scales as nmax � �4/3 for both monovalentand divalent solutions. For homogeneouslycharged surfaces, the divalent salt concentrationcorresponding to the maximum adsorptions, m*,roughly scales like �2.2 when m* � �. In monova-lent salt solutions, however, the salt concentra-tion for maximum adsorption, s*, roughly scaleslike �2 when s* � �.

As the salt concentration increases the fractionof monomers with condensed counterions for rodsin bulk solution, fc increases. Furthermore, thedifference between fc and the fraction of mono-mers with condensed counterions in adsorbedrods, fs, becomes smaller due to the screening ofthe repulsions between the surface and the rodcondensed counterions. These two effects give amaximum absolute value of the effective surfacecharge density at a salt concentration m�, whichis lower than m*. Besides the correlations be-tween the adsorbed chains and the release of thecondensed counterions, we find that the short-range attractions between the polyelectrolytesand the surface charge groups also contribute toincrease the degree of surface charge inversion.This effect and the discrete distribution of surfacecharge groups both intensify the degree of chargeinversion. Any of the above four effects to chargeinversion can dominate in experiments for differ-ent adsorption conditions.

We compare polyelectrolyte adsorption inmonovalent salt solution to adsorption in divalentsalt solution at the same Debye screening length(low salt concentrations). More strongly chargedpolyelectrolytes adsorb onto charged surfaces indivalent salt solutions than in monovalent solu-tions. This is due to a reduction of the repulsionbetween the adsorbed chains (due to higher coun-terion condensation) in divalent salt solutions.Instead, large lateral correlations between ad-sorbed chains in monovalent salt solutions resultin large absolute values of effective surfacecharge densities.

The adsorption–desorption transition lines forpolyelectrolytes with different line charge densi-ties show that the critical salt concentration foradsorption in divalent salt solutions roughlyscales like mc � b�1.9 in the valid range of themodel. On the other hand, the dependence on linecharge density found in weakly charged polyelec-trolytes in monovalent salt solution scales as b

with � 0 because there is no ion condensation inthe weakly charged polyelectrolyte adsorptionmodel.33 The adsorption–desorption transition


lines might be useful for selective polyelectrolyteadsorption from solutions that contain polyelec-trolytes with different charge densities.

Our work only considers weakly charged sur-faces and strongly charged rods. Larger degrees ofadsorption are expected in highly charged sur-faces and/or in the case of weakly charged rods. Insuch cases, the Wigner cell sizes decrease andcorrelations among the chains may induce orien-tational order among the chains. For a low degreeof overlapping electrostatics may lead to a her-ringbone structure described in ref. 12.

This work was supported primarily by the NanoscaleScience and Engineering Initiative of the NationalScience Foundation under NSF Award Number EEC-0118025.

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rod-like polyelectrolyte adsorption onto charged surfaces in monovalent and divalent salt solutions

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