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Chemical Physics Letters 420 (2006) 29–34

Ribbonlike nanostructures from stiff polyanionsand short cationic chains

Pavel V. Komarov a,*, Lubov V. Zherenkova b, Pavel G. Khalatur b,c

a Department of Theoretical Physics, Tver State University, Sadoviy per. 35, Tver 170002, Russiab Department of Physical Chemistry, Tver State University, Tver 170002, Russia

c Institute of Organoelement Compounds of the Russian Academy of Science, Moscow 117823, Russia

Received 27 September 2005; in final form 3 December 2005Available online 4 January 2006

Abstract

AMonte Carlo simulation is used to study the formation of finite-size nanostructures in a dilute salt-free solution of stiff polyanions inthe presence of short cationic diblock chains consisting of neutral tails and charged heads. At strong electrostatic interaction, the systemis found to undergo the self-organization resulting in the formation of planar aggregates that look like a ‘ladder’ of polyanions sand-wiched between cationic chains. We investigate the stability of different morphologies and find that these aggregates are thermodynam-ically stable.� 2005 Elsevier B.V. All rights reserved.

1. Introduction

The interaction of polyelectrolytes with oppositelycharged multivalent counterions and various macroionscan have a dramatic effect on the aggregation processes.In particular, strongly charged polyelectrolyte chains cancondense at a critical concentration of multivalent ions.Usually, this process is followed by macroscopic phase sep-aration. In recent years, it has been shown that some bio-polymers can condense in the presence of multivalentions into soluble aggregates – bundles – of a well-definedsize. Such a phenomenon was observed for double-stranded long DNA dilute solutions and short DNA frag-ments, for F-actin, fd-viruses, etc. [1–5].

More complex aggregates can be formed in a mixture ofanionic polyelectrolytes with flexible-chain cationic copoly-mers (e.g., diblock polyelectrolytes, cationic surfactants,etc.). Depending on the concentration of oppositelycharged polymers and chain flexibility, polyanions were

0009-2614/$ - see front matter � 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2005.12.024

* Corresponding author. Fax: +7 0822331274.E-mail addresses: pv_komarov@mail.ru, kpv@germany.ru (P.V. Ko-

marov).

found to form various three-dimensional structures, whichinclude spherical and cylindrical micelles, bilayers, andinverse cylindrical micelles in cubic, hexagonal, lamellar,and inverted hexagonal phases [6–11].

However, many challenges still need to be overcomebefore exhaustive theoretical understanding the aggrega-tion mechanism can be achieved. Of our special interesthere is how to control the spatial organization of the aggre-gates formed in an aqueous solution, in particular, how toget a planar configuration of the aggregates consisting ofstiff polyelectrolyte molecules. One anticipates that thechemical structure of the diblock cationic molecules shouldaffect the final structure of the aggregates. At present time,self-assembled planar configurations of stiff molecules areobtained on flat surfaces.

With this purpose, we study the structural properties ofstrongly charged polyelectrolytes in a dilute solution con-taining short cationic chains using a Monte Carlo simula-tion. Since the formation of ordered structures occursunder complex conditions, computer simulations of self-organization in a relatively simple model system arebelieved to be useful for understanding the key featuresof such processes. Also, it is of principle interest to examine

30 P.V. Komarov et al. / Chemical Physics Letters 420 (2006) 29–34

the influence of the length and charge of cationic chains onstructural and thermodynamic properties of polyelectrolytesystems.

2. Model and method

We study a dilute salt-free solution containing stronglycharged rodlike polyanions (subsystem A), which modelstiff polymer fragments such as short DNA chains, andoppositely charged diblock molecules. The latter are repre-sented by short rodlike chains consisting of interaction sitesof two types, namely, one charged head unit (subsystem B)and neutral tail units (subsystem C). Both molecules con-sist of interaction sites of diameter r connected by bondsof fixed length r. The parameter r is used to set the lengthscale. The solvent is a continuous medium characterized byits dielectric constant e.

We assume that each interaction site corresponds to asegment of real polymer chain or to a fragment of somebiological macromolecule. Therefore, it contains severalchemical groups and can carry more than one elementarycharge. For instance, one can mention three stronglycharged biological macromolecules: DNA, F-actin, and fil-amentous bacteriophage M13 virus [11]. The diameter ofdouble helix DNA is about 20 A. The chain fragment ofsuch length, which can be considered as an ‘effective mono-mer’, includes �6 base pairs and the surface charge densityof DNA is �1/106 e A�2. Therefore, if we consider DNAas an object of simulation, one interaction site in our modelshould have the size of 20 A and its total charge is Q = 12e.The diameter of F-actin is �75 A and the surface chargedensity is about 1/625 e A�2. In this case, the model param-eters should be estimated as r = 75 A and Q = 28e. For thefilamentous bacteriophage M13 virus with the charge den-sity of �1/256 e A�2, one has: r = 65 A and Q = 52e [11].

In the model, the polyanion chain consists of NA inter-action sites (‘effective monomers’). For computational effi-ciency, chains with relatively short length, NA = 10, areused for most of the calculations. The length of the cationicneutral tail, NC, is varied between 1 and 4. Every effectivemonomer of the polyanion carries the negative charge Q.The head monomer of the cationic chain carries the samepositive charge Q. The number density of interaction sitesbelonging to polyanions is qA = 0.001 r�3. The value ofqA is sufficiently low to correspond to a diluted polyelectro-lyte solution. According to the charge electro-neutralitycondition, the following equalities should be met: qB = qAand qC = NCqA, where qB and qC are the number densitiesof interaction sites in the B and C subsystems, respectively.

The total potential energy of the system is given by thesum of short-range repulsive interactions (‘soft-spherepotential’) and electrostatic interactions

U ¼X

i; j; i 6¼ j

a; b

kBT 0

r

rabij

!12

þX

i; j; i 6¼ j

a; b

kBT 0

C

rabij;

ð1Þ

where rabij ¼j rai � rbj j is the distance between non-bondedsites ith and jth (a,b = A,B,C), the coupling parameter isdefined as C = Q2/4pe0ekBT0, e0 denotes the permittivityof free space, and e is the relative permittivity of the med-ium. The charges on polyions and counterions are ex-pressed in units of the elementary charge e. In this study,we set T0 = 298 K. Note that the coupling parameter C ismeasured in units of r. In reality, C can be tuned by chang-ing the solvent permittivity e or temperature.

For the results reported below, the value of C is variedbetween 1r and 100r. We take quite wide range of C inorder to investigate the effect of strong electrostatic interac-tions. It should be noted that for the biopolymers men-tioned above, the coupling parameters of thecorresponding chains should be equal to 51.1r (forDNA), 74.2r (for F-actin), and 295.4r (for M13 virus) inaqueous solutions under normal conditions.

The Monte Carlo calculations were performed in thecanonical ensemble for the model system of nA polyanionsand nANA cationic chains. The molecules were enclosed ina cubic box with periodic boundary conditions. As before[12], we use the simulation technique employing the stan-dard Metropolis algorithm [13]. Two different equiprobabletypes of displacements for molecules were used, viz., (i) bothends of a randomly chosen molecule were moved in randomdirections and then the chain length was adjusted to the pre-scribed value, r(NA � 1) or rNC, using the SHAKE algo-rithm [14,15], (ii) a randomly chosen molecule wasdisplaced along its long axis over a random distance withinthe range from 0 to r. The first mechanism provided thereorientation of chains. The maximum displacements wereadjusted so that �50% of the trial moves were accepted.The transition probability from an old configuration O toa new one N is PðO ! NÞ ¼ minf1; expðDU=kBT Þg, whereDU is the difference between the old and new total systemenergies (1). The long-range Coulomb interactions weretaken into account using Ewald’s summation technique[16,17]. At least 200,000 MC steps (MCS) per a monomerwere performed for each set of the parameters after the sys-tem came to an equilibrium state where no systematicenergy drift was observed.

3. Results

To study the spatial organization of positively and neg-atively charged chains, we calculated the partial staticstructure factors, S(q), which characterize the structure ofthe system as a whole. The structure factor is given by

SaðqÞ ¼1

naN a

Xij

expðiq½rai � raj �ÞD E

; ð2Þ

where q is the wave vector. The normalization for Sa(q) isSa(0) = kBTqavT, where qa is the partial density, vT denotesthe compressibility, and the index a corresponds to poly-ions (a = A) or charged (B) sites on the cations.

Fig. 1a presents the structure factor SA(q) at different C,for NC = 2. At small C, the magnitude of polyanion–poly-

P.V. Komarov et al. / Chemical Physics Letters 420 (2006) 29–34 31

anion correlations appears to be relatively insensitive to C.In the q ! 0 limit, the value of SA(q) increases as C isincreased, which indicates the growth of the compressibilityand means that polyanions in the simulation box show thetendency towards aggregation. At CJ 30r, there is a broadpeak in SA(q) which narrows and moves slowly to lowerwave vectors as the electrostatic interaction is increased.The characteristic scale of this structure is determined bythe Bragg condition r* = 2p/q* and is estimated as 2.5r,which corresponds to the average distance between neigh-boring polyanions in the aggregates. Under these condi-tions, the cation–cation structure factor SB(q) (not shown)also demonstrates a broad peak. For sufficiently high C,we find a prominent peak in the cation–cation partial struc-ture factor at roughly the same wave number as the peak inSA(q). It is easy to understand that this is a manifestation ofthe fact that the cation–cation correlations near the chargedchain sites is very high, i.e., the cation distribution isstrongly correlated with that of the polyions.

The local structure of the system was examined via thepair-correlation functions, g(r), that provide an informa-tion on charge distribution and order in the system.Fig. 1b shows the pair correlation functions gA(r) calcu-lated for polyanions. It is seen that at C J 25r, the sharpfirst peak begins to develop. For C = 36r, we observe sev-eral well-pronounced peaks separated by approximately

Fig. 1. (a) Partial static structure factors, SA(q), and (b) the partialcorrelation functions, gA(r), for polyanions, at different C(NC = 2 andqA = 0.001).

equal distance. This distance is almost the same as thatfound from SA(q). Thus, under these conditions, a wellordered crystal-like structure appears in the subsystem A.

Obviously, the structural reorganizations describedabove occur due to the effective attraction between polya-nions mediated by condensed cationic chains. At relativelysmall C, the repulsion between the like charged rods tendsto keep them apart. However, the repulsive force decreaseswith increasing C due to the enhanced condensation of thecationic chains. Interchain attraction can also arise fromthe charge fluctuations along the backbone of the chainsas condensed counterions come on and off the chains. So,sharing cationic clouds causes the attractive interactionbetween the like charged stiff chains.

Earlier, Ray and Manning [18] suggested that two like-charged objects (e.g., rigid rods) can share counterions insuch a way to form what is analogous to a covalent bond.A similar picture is observed in our simulations for thehigh-C regime when two or more polyanions stick togetherdue to the cation condensation on the oppositely chargedchains. One may say that the counterions bound on thesurface of stiff-chain polyanions play the role of the specific‘glue’. We should like to stress once more that the effectiveelectrostatic attraction is evident only for systems havingvery large Cs, when one can overcome the Coulombicrepulsion. It is now well established that the like-chargeaggregation is due to the strong counterion mediated corre-lations. This phenomenon has been widely studied forspherical monovalent and multivalent counterions [19–23]. Moreover, Limbach et al. [24] have shown thatstrongly charged stiff chains can form polyelectrolyte bun-dles due to the counterion condensation. New effects thatcan be expected for the system of rodlike polyions shouldbe related to the condensation of anisotropic counterions.

Fig. 2 shows orientational order parameter of the sys-tem, K, and the average number of polyanions in the aggre-gates, L, as a function of C. The order parameter isdetermined from the largest eigenvalue of the symmetricmatrix with elements

Fig. 2. Orientational order parameter, K (solid squares, dashed line), andthe average number of polyanion chains in the aggregates, L (open circles,solid line), as functions of C (NC = 2 and qA = 0.001).

32 P.V. Komarov et al. / Chemical Physics Letters 420 (2006) 29–34

kst ¼1

2nA

XnAi¼1

3bi;sbi;tb2i

� dst

" #; ð3Þ

where s, t = x,y,z; bi,s and bi,t are the projections of thevector of the ith polyanion on the axes s and t, respectively;and dst is the Kronecker delta. The order parameter K indi-cates a very sharp transition from a disordered state to anorientationally ordered state, which arises by spontaneoussymmetry breaking. As seen, there is a narrow range of Cwhere K sharply increases. The order parameter jumps atCT � 30r, which evidently is the transition threshold. It isseen that at C < CT, we deal with a system of single-chainnon-aggregating polyanions, which repel each other be-cause they still have a sufficiently high net charge. Slightlyabove CT, the aggregates are formed and they include prac-tically all the polyanions that are present in the simulationbox. Such a behavior is evident from the analysis of theaverage aggregation numbers, L. Naturally, the structurefactor of the system also reflects these features.

It is interesting to examine the internal structure of theaggregates and their stability. Fig. 3 presents a typicalsnapshot of the aggregate formed above the thresholdvalue CT. The aggregate shown in Fig. 3 contains 9 poly-ions. At C > CT, the polyanions and the cationic chainsself-assemble into ribbonlike aggregates. One can see thatthe polyanions are closely packed in a ladder-like manner(Fig. 3a) forming a quasi-two-dimensional structure. Asmentioned above, the polyanions form the aggregate bysharing the cloud of the opposite charged heads of the cat-

Fig. 3. A typical snapshot of two different projections of the ribbonlikeaggregate formed at C > 30r (nA = 9 and NC = 2). Head monomers of thecations are in gray, the neutral tails are in white, and the polyanionmonomers are in black.

ionic chains (Fig. 3b), but repulsive interaction between theneutral cationic tails does not allow the polyanions to foldinto a bundle. As a result, the aggregates can grow onlyalong the ribbon plane. Also, the ribbonlike aggregatesdemonstrate another remarkable feature; they are twisted(Fig. 3b). This behavior has a simple explanation. It isdue to entropy effects related to the thermal motion ofchain polyanios.

A deeper insight into possible structures formed bycharged rods in the presence of anisotropic counterionsmay help to understand general processes of the interac-tions between stiff-chain polyelectrolytes in a dilute solu-tion. To this end, the following three initial configurationsof the polyanions were considered: (i) planar ribbonlikeconfigurations; (ii) multilayer configurations (bunches);and (iii) cylindrical configurations. They are shown inFig. 4a. The number of neutral sites in each cationic chainwas fixed at NC = 2. In order to investigate the stability ofthese model structures, we performed a comparison of theiraverage reduced potential energies, U/NkBT (N � the totalnumber of interaction sites). A series of calculations wascarried out at the largest coupling parameter (C = 100r)during 200,000 MCS. It should be emphasized that underthese conditions, the strong electrostatic attraction domi-nates and the main contribution to the free energy comesfrom the potential energy. Indeed, the number of moleculesentering the initial aggregates was unchanged during thesimulation. Furthermore, the potential energy decreases asthe system evolves. Obviously, the most stable systemshould have the lowest potential energy at the end of thesimulation. As seen from Fig. 4a, the ribbonlike aggregatesare the most stable as compared to other configurationsstudied here.

The system with spherical counterions (NC = 0) was alsoinvestigated under the same conditions. The results areshown in Fig. 4b. It is seen that in this case, a bunch con-figuration is energetically preferable. We have mentionedthat Limbach et al. observed the formation of similar struc-tures [24] (see also Ref. [12]). They concluded that thesestructures can spontaneously disintegrate since the con-comitant entropy increase overcompensates the energeticeffects [24]. In this connection, it should be kept in mindthat the presence of chain counterions, which have lowertranslational entropy as compared to their single-site coun-terparts, provides higher structural stability of the resultingaggregates. The mechanisms responsible for the stabiliza-tion of the charged finite-size aggregates have also dis-cussed in Ref. [25].

Dashed arrows in Fig. 4 demonstrate the evolution ofthe system towards the equilibrium state. In the case ofchain counterions (Fig. 4a), the final morphology corre-sponds to ribbonlike structures, while in the case of sin-gle-site counterions (Fig. 4b) we observe bunches. Thestabilization of the aggregates is due to the interplaybetween the cation-mediated attractive forces and thelong-range Coulomb interactions together with a (small)contribution from the translational entropy of mobile cat-

Fig. 4. Comparison of the average reduced potential energies for the three systems with different initial configurations of the polyanion chains [(i) theribbon, (ii) the bunch, (iii) the cylinder] in the presence of (a) cationic copolymer chains (NC = 2) and (b) spherical single-site counterions (NA = 9 andC = 100r).

P.V. Komarov et al. / Chemical Physics Letters 420 (2006) 29–34 33

ionic chains. Short-range repulsive interaction between theneutral tails of the cationic chains assists the stabilizationof the ribbonlike morphologies. Indeed, we did not observesuch structural features in the presence of spherical single-site cations. In this case, each of the initial configurationscomes to the bunch morphology. Thus the quasi-planarribbon aggregates can be stable due to the repulsionbetween neutral tails of cationic chains.

Fig. 5 demonstrates the effect of the total number of pol-yanion chains in ribbonlike aggregates, L, on the averagereduced potential energy at a few values of NC. As seen,at small L the excess internal energy a little bit decreaseswith increasing L and then does not change. It means thatthe stability of ribbonlike structures hardly depends on thenumber of the polyanion chains in the aggregate over theentire range of the parameters studied. Also, Fig. 5 showsthat the stability of the aggregates of any size decreaseswith increasing NC. However, for all the values of NC con-

Fig. 5. Average reduced potential energy as a function of the total numberof polyanion chains, L, in the ribbonlike aggregate, at different NC

(qA = 0.001 and C = 100r).

sidered in this calculation (NC > 0), the aggregate energyremains negative.

4. Conclusion remarks

The salt-free polyelectrolyte solution containing stiff-chain polyanions and short cationic chains has been stud-ied. The formation of ribbonlike structures from oppositelycharged chains has been observed in the dilute regime. Theaggregates are spontaneously formed with increasing theelectrostatic coupling parameter C and remain thermody-namically stable. The formation of the aggregates isinduced by the condensed cationic chains that can bridgepolyanions. Ribbonlike morphology of the aggregates hasbeen found to be the most preferable energetically, whilein the presence of spherical single-site counterions the moststable configuration is a bunch. Hence, in order to ‘unroll’the bunch, the counterions with neutral tails should beincorporated into the polyelectrolyte solution instead ofthe spherical counterions. The stability of the ribbonlikeaggregates has been predicted to become less with increas-ing the length of the neutral cationic tail.

Aggregation of rodlike polyanions mediated by cationicchains is expected to serve as a good model for the theoret-ical study of self-assembling processes in various polyelec-trolyte systems, in particular, the self-organization of theDNA fragments. Also, the planar structures observed in thisstudy could be useful for technical applications, e.g., formaking nanodevices. For instance, one might obtain nano-sized metallic petals by metallization of such aggregates.

Acknowledgment

The financial support from the Russian Foundation forBasic Research (Project No. 05-03-32952) is highlyappreciated.

34 P.V. Komarov et al. / Chemical Physics Letters 420 (2006) 29–34

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