conformational characteristics of single flexible polyelectrolyte chain

DOI 10.1140/epje/i2009-10532-5

Regular Article

Eur. Phys. J. E 30, 341–350 (2009) THE EUROPEANPHYSICAL JOURNAL E

Conformational characteristics of single flexible polyelectrolytechain

C.G. Jesudason1,a, A.P. Lyubartsev2,b, and A. Laaksonen2,c

1 Department of Chemistry, Science Faculty, University of Malaya, 50603 Kuala Lumpur, West Malaysia2 Department of Physical, Inorganic and Structural Chemistry, Arrhenius Laboratory, Stockholm University, S-10691 Stock-

holm, Sweden

Received 16 January 2009 and Received in final form 13 July 2009Published online: 29 November 2009 – c© EDP Sciences / Societa Italiana di Fisica / Springer-Verlag 2009

Abstract. The behaviour of a flexible anionic chain of 150 univalent and negatively charged beads connectedby a harmonic-like potential with each other in the presence of an equal number of positive and freecounterions, is studied in molecular dynamics simulations with Langevin thermostat in a wide range oftemperatures. Simulations were carried out for several values of the bending parameter, corresponding tofully flexible polyion, moderately and strongly stiff polyion as well as for the case when bend conformation ispreferable to the straight one. We have found that in all cases three regimes can be distinguished, which canbe characterized as “random coil”, observed at high temperatures; “extended conformation” observed atmoderate temperatures (of the order of 1 in reduced units), and compact “globular conformation” attainedat low temperatures. While the transition between high-temperature random and extended conformationsis gradual, the transition from the extended coil to the globular state, taking place at a temperature ofabout 0.2 in reduced units, is of abrupt character resembling a phase transition.

PACS. 82.35.Rs Polyelectrolytes – 82.20.Wt Computational modeling; simulation

1 Introduction

Despite the great interest shown in studies of polyelec-trolytes during the last few decades [1–7], see also “Hand-book of Polyelectrolytes” [8], backed by many experimen-tal investigations [8], the properties of polyelectrolytes arestill poorly understood from the theoretical point of view.One of the problems encountered is that ideas developedfor uncharged polymers are generally not applicable forstrongly charged polyelectrolytes due to the long-rangecharacter of electrostatic interactions and the presence ofmobile ions as an additional component, and so a directtranslation of such concepts from neutral polymers hasnot as yet provided effective descriptions [3,4]. At best,interactions between monomers of polyions are theoreti-cally described within the framework of the Debye-Huckelapproximation, without explicit description of counter-ions [7]. This approximation, except for the case of weakpolyelectrolytes, is too crude. It is not capable of describ-ing, for example, the experimentally observed attractionbetween equally charged polyions in the presence of multi-valent counterions [9], as well as the related phenomenon

a e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]

of contraction of a single polyelectrolyte chain, also ob-served in the presence of multivalent counterions [10].As such, there is not enough understanding of even themost fundamental properties of strongly charged polyelec-trolytes, such as the general dependence of the gyrationradius (or end-to-end distance) on the length of the chainat different conditions.

On the other hand, in recent times, numerical databased on Monte Carlo or molecular dynamics simula-tions concerning polyelectrolyte structure and topologyat various thermodynamically defined regimes are begin-ning to appear, which would provide in time a sound ba-sis for theoretical development. From the physical pointof view, it should be useful to present data and inter-pretations of some of the most basic properties of poly-electrolyte systems hoping that it would further encour-age theoretical formulations aimed at forming a com-prehensive scheme for the various possible structuresand phases. Many previous theoretical studies dealt withmono- or multicharged (i.e. containing both positive andnegative charges) polymer strands interacting by thescreened Coulomb (Debye-Huckel) potential of the formUDH = q1q2 exp(−κr)

4πε0εr[11–14]. In other studies, conforma-

tional properties of a single polyectrolyte chain have beenstudied with explicit counterions, and the interactions be-tween polyion monomers and ions were described by the

342 The European Physical Journal E

true Coulombic potential [5,6,15–24]. The properties as-sessed were radius of gyration [5], ion-polyion distributionfunctions [5,16], shape of the polyelectrolyte coil [20] andthe effect of electric field on it [17], as well some otherproperties.

In several recent investigations, properties of flexiblepolyelectrolyte chains were studied within the frameworkof lattice models [25–27]. In these works, an interestinggeneral observation was made (which in fact was also no-ticed in some earlier simulations [28,29]) that, upon thechange from high to low reduced temperature, the poly-mer undergoes conformational changes, from neutral-likepolymer to an extended conformation and then to a com-pact globular one, with the maximum coil size observed atthe values of the reduced temperature of about 1, whichholds even with addition of salt ions. In the work [30],it was also demonstrated, with the help of entropy sam-pling and Wang-Landau algorithm, that the transition toa compact globular conformation has an abrupt characterwith typical features of a phase transition.

While the above-cited works cover well the temper-ature dependence of some basic conformational proper-ties of a single polyelectrolyte chain, a possible limitationmay be that they were done within lattice models. It maytherefore be interesting to find out how the propertiesobserved in lattice models hold for continuous polymermodels. For this purpose, we have carried out a series ofsimulations of a single flexible polyelectrolyte presentedas a chain of beads connected by harmonic bonds, withoptional bending potential between the bonds. Since theMetropolis Monte Carlo approach faces difficulties in con-formational sampling of a polyion chain surrounded by alarge amount of counterions with a close contact with thepolyion, we employed in this work a molecular dynamicsalgorithm with Langevin thermostat, where all the parti-cles are moving simultaneously according to forces actingon them, which allows us to sample the conformationalspace faster. The main point of interest is the tempera-ture behaviour of the size of the polyelectrolyte coil. Wealso describe the energetic behaviour and follow the inter-nal coil structure.

2 Model, method and parameters

The simulated system consisted of one polyionic chaincomposed of 150 negatively charged beads linked by har-monic bond potentials, and an equal number of free, posi-tively charged counterions. A cubic cell of the size L = 200(dimensionless units) with periodic boundary conditionsis used. The potentials for the interactions are defined asfollows:

1) Non-bonded short-range repulsive Lennard-Jones po-tential ΦLJ:

ΦLJ = 4ε

((σ

r

)12

−(σ

r

)6

+ s

), (1)

for r < rcut = 21/6σ and Φ = 0 otherwise. Here ε =σ = 1 and s = 1/4. This potential goes smoothly to

zero exactly at rcut distance. All pairs of beads notconnected by a direct bond interact with this potential.

2) Bonded harmonic potential for neighboring beads ofthe polyion:

Uh =kh

2(r − Rh)2, (2)

where kh is the force constant and Rh is the “equilib-rium” length between two beads Pi and Pi+1.

3) Bending interaction potential Ub

Ub(φ) =kb

2(φ − φ0)2, (3)

where φ is the angle between vectors connecting con-secutive bonded centers Pi with Pi+1 and Pi with Pi+2.

4) Non-bonded Coulombic interaction which acts be-tween all the charged particles in the system (beadsof the polyion and mobile ions):

UCoul =q1q2

r. (4)

The energy and forces originating from (4) are com-puted by the Ewald summation method according tothe P3M technique [31] with an error tolerance in thetotal electrostatic energy less than 10−6 for a series ofrandom configurations, typically about 16 or more byadjusting or tuning the parameters connected to themethod.

Reduced units were used, with the unit length equal tothe equilibrium length distance Rh, unit mass equal to themonomer mass (which was also equal to the mass of ions),unit charge equal to the absolute value of the charge of amonomer or counterion. The LJ interaction parametersbetween any pair of units (monomers or ions) (1) wereset to ε = σ = 1; and the LJ potential was cut off at adistance r = 1.12246 for each of these interactions. Thusthe effective size of each monomeric unit and each ion wasequal to 1 in reduced units.

The reduced temperature T was defined as the ratio ofthermal energy kBTK to the electrostatic energy of unitcharges at unit distance, and is related to the real (Kelvin)temperature TK by T = 4πRhε0εr

e2 kBTK , with the standardassignment of variables (e, kB , ε0, εr) being the electroncharge, Boltzmann constant, permittivity in vacuum andrelative dielectric permittivity, respectively. Since in aque-ous solutions the product εrTK changes very little withthe temperature, the reduced temperature reflects essen-tially not the real temperature, but the effective strengthof electrostatic interactions. The strength of a polyioncharge is often defined in terms of “reduced charge den-sity” ξ = lB/Rh with Bjerrum length lB = e2

4πε0εrkBTK

and distance between the neighbouring charges Rh. It iseasy to ensure that the reduced temperature is just aninverse of the reduced charge density: T = 1/ξ. This rela-tionship sets up correspondence between the parametersof our model system and real polyelectrolytes.

In order to maintain a relatively fixed chain length incase of simulation at high temperature, the harmonic bond

C.G. Jesudason et al.: Conformational characteristics of single flexible polyelectrolyte chain 343

constant was scaled with the temperature as kh = 100T .In this case, the equipartition theorem gives average vari-ation of the bond length as

√T/kh ≈ 0.1, being indepen-

dent of temperature. This allows us to eliminate a pos-sible effect of bond extension with temperature on theconformational properties of the polyelectrolyte coil, andconcentrate on purely electrostatic effects.

A series of simulation for 4 systems with differentbending potential was carried out:

Case 1) No bending potential.Case 2) Weak bending potential with kb = 10 and φ0 = 0.Case 3) Stiff bending potential with kb = 120 and φ0 = 0.Case 4) Bending potential with 180◦ equilibrium angle:

kb = 5 and φ0 = π.

While Cases 2) and especially 3) favor rod-like struc-tures, Case 4) favors structures with 1-3 neighbours alongthe chain in contact.

In each series, the temperature was changed between0.1 and 10; moreover a simulation with switched offCoulombic potential was carried out in order to elucidatethe role of electrostatic interactions.

The different Cases explore, in the absence of electro-static interactions, the behavior patterns over a wide con-formational range: Case 1) is a fully flexible chain, Case 4)corresponds to the situation of a directed bond with in-termolecular forces keeping the polyanion collapsed in aconfined space whereas Case 2) and 3) refer to a rod-likestructure at zero temperature.

All simulations were carried out by molecular dynam-ics with Langevin thermostat [32]; this method impliesa synthetic algorithm where the Newtonian equations ofmotion of the system are modified to a form

mivi = Fi − γvi + Fr, (5)

where Fi is the conservative force acting on particle iwhich is defined by the gradient of the potential energy,γ is a thermostat friction parameter, and Fr is the ran-dom noise term, the intensity of which is determined bythe fluctuation-dissipation theorem. The (dimensionless)time step was equal to 0.01, and the value of the langevinthermostat coupling parameter γ was set to 0.1. The sim-ulations were carried out using ESPResSo package [33].

In the beginning of each simulation, the coordinatesare chosen randomly and a special algorithm that capsthe potential is applied during the warm-up phase. There-after each run consists of about 1 million initial time stepswhere no average collection is done followed by a pro-duction stage. The lengths of simulations were between5 and 10 millions of MD steps. In order to control con-vergence and determine statistical error, each simulationrun consisted of “windows” of 40000 MD steps each witha computation of different averages within each window,and their subsequent averaging. The start of final aver-aging was determined from visual inspection of averagesobtained within each window, in particular noting the val-ues of the highly fluctuating and sensitive quantity Rg, theradius of gyration. These values are observed for each run

0

20

40

60

80

100

120

140

0 2 4 6 8 10

Re

Temperature

Case 1Case 2Case 3Case 4

Fig. 1. End-to-end distances for polyelectrolyte.

5

10

15

20

25

30

35

40

45

0 2 4 6 8 10

Rg

Temperature


Fig. 2. Radius of gyration for polyelectrolyte.

until no systematic drop or rise is observed and that thevalues keep fluctuating about some value. It was foundthat whenever the Rg variable reached the equilibrium, sodid the other thermodynamical variables.

3 Results and discussion

3.1 Radii of gyration and end-to-end distance

The average (more exactly, root-mean-square) end-to-end distance and the gyration radius (defined as R2

g =(∑Nc

i=1(ri−rCM )2)/Nc) of the polyelectrolyte coil as func-tions of temperature are displayed in figs. 1 and 2, respec-tively. One can clearly see the maximum of both curves ata temperature T ∗ ≈ 1, observed also in a number of previ-ous works [28,29]. These curves appear typical if comparedto those given in fig. 9 of [30] for a chain on a cubic lattice.The reason of the non-monotonous behavior of the size ofa polyelectrolyte coil with the change of temperature hasbeen discussed in earlier works. At high temperatures, thecoil behaves essentially as a non-charged polymer. Whenthe temperature is becoming lower, the size of the coil isgrowing due to repulsion between its monomers. When thetemperature falls even lower, the counterions condense on


the polyion and screen the electrostatic repulsion. More-over, the counterions cause effective attraction betweenpolyion’s monomers which leads to condensation of thecoil into a compact globule.

The fact that the polyion size maximum is attained atT ∗ ≈ 1 (which corresponds to the reduced charge densityξ ≈ 1) seems to be related to the Manning condensationtheory [34,35]. According to the Manning theory, at ξ > 1the counterions condense on the polyion maintaining thusthe total charge density at the level ξ = 1, and effectiveinteraction between the polyions can be then describedby the Debye-Huckel theory. Still, the temperature be-haviour of a polyelectrolyte chain cannot be understoodwithin the Manning theory alone (putting aside the factthat the Manning theory is strictly speaking applicableonly to a rod-like polyelectrolyte). Since no changes aresupposed to happen in the counterion cloud around thepolyion at ξ > 1 (since all additional counterions are justcondensing on the polyion), the polyion should just stopexpansion after T decreases below 1. The observed tran-sition to a globular state can be understood only if someattraction mechanism exists between different fragmentsof the polyelectrolyte chain. The appearance of an effec-tive attraction between likewise charged polyions due tocounterions correlation has been described both in simu-lations [36] and theoretically [37], and the conditions forappearance of the net attraction correlate with conditionsof contraction of the polyion coil observed at low temper-atures [30].

In a recent paper Winkler et al. [5] have discussedthe behaviour of the polyelectrolyte coil size as a func-tion of the interaction strength, which also showed non-monotonous behavior when the interaction strength pa-rameter was about 1. Although the current work is basedon results for different temperatures for fixed Bjerrumlength, one can also interpret the results as for a fixed tem-perature with varying charge interaction strengths, which,given the fact that both the effective size of the monomerand the monomer-monomer distance is equal to 1 in ourmodel, can be expressed as 1/T . In work [5], the collapseof the polyelectrolyte coil at large values of the interactionstrength parameter was discussed in terms of the ion pairformation and effective attraction of the ion pair dipoles.On the other hand, it was noted in [5] (which is also thecase of the current work), that the collapse of a freelyjoint polyelectrolyte chain occurs at a higher temperaturethan the critical temperature of the primitive electrolytemodel (T ∗ = 0.057). According to our data, the collapse ofthe polyelectrolyte coil occurs at a temperature of about0.15–0.2 (the same value was observed in previous simu-lations with a lattice model [30]). Clearly, the fact thatpolyelectrolyte monomers are bound by bonds favors thetransition to the compact state at higher temperature incomparison with a simple (primitive) electrolyte model.

Another interesting observation from figs. 1 and 2 isthat the maximum value of both Rg and Re with respectto temperature variation follows a trend: with increasingstiffness as measured by the decrease of φ0 and kb, i.e.for the sequence Case 4) → 1) → 2) → 3), the maximumshifts from right to left in temperature (i.e. from higher

to lower temperature), and upwards to higher Rg and Re

values. Such behaviour can be well understood from thefact that the increase of the stiffness of the chain favorsmore rod-like, elongated conformations with larger gyra-tion radius and end-to-end distance.

Further, it is also noticed that Rg and Re graphsare similar in shape. According to standard theories andassumptions concerning a long-enough chain length, forpolymers that are free to arrange themselves according toa random walk, there exists the relationship R2

e = 6R2g giv-

ing the factor√

6 ≈ 2.45 between Rg and Re. Comparisonof figs. 1 and 2 shows the R2

e/R2g factor about 10 around

the maximum point for completely flexible chain (case 1))and even 11 for the stiff chain (case 3)). For higher tem-peratures (T ∗ = 10) this factor becomes 7–8, that is closerto the theoretical value for a random walk. This observa-tion is well in line with the fact that the factor relating R2

e

and R2g is increasing for stretched conformations reaching

the theoretical value R2e/R2

g = 12 for rod-like polymers.The relationship between Rg and Re with the variation ofthe interaction strength was also discussed in paper [5], interms of critical exponents. Finally, discussing the scalingrelationships, we note that polyelectrolytes can be charac-terized by several length scales with scaling laws changingfrom one typical length to another; and even our lengthN = 150 may be too short to reproduce the limiting value,especially for extended conformations. To date, data forlonger chains N � 150 are hardly available to make de-cisions concerning the applicability of random walk mod-els in determining the relationship between Re and Rg.One area therefore for theoretical investigation and sim-ulational verification is the creation of scaling theories ofintermediate lengths which can relate Re and Rg moreaccurately and which can be investigated in simulations.

At low temperatures, the most abrupt change of thecoil size occurs when the temperature falls below 0.2 in allcases. For flexible chains (Cases 1) and 4)) at T ∗ = 0.1,the coil reaches the state of a dense globule when nofurther contraction is possible because of the steric re-pulsion of monomers and ions. Even stiffer chains showcontraction thought not to the completely compact state.Also, the onset of contraction for stiffer chains is ob-served at lower temperature. It was also found that, es-pecially for Case 3) (stiff rod-like system), the maximumwas rather extended and flat, covering the range 0.3–0.8of reduced temperatures, with relatively high fluctuationsabout these points during the simulation (a typical statis-tical error is depicted as a vertical bar). These fluctuationsreflect the competition, becoming stronger at lower tem-peratures, between the bending forces favoring a stretchedconformation, and attractive, counterion-mediated elec-trostatic forces. Correspondingly, the chain fluctuates be-tween stretched, almost rod-like conformation and morecompact ones when different parts of the chain becomeclose to each other, attracted by the interaction with thecounterions.

At high temperatures, one can see that the curves (ex-cept perhaps Case 3)) are tending to converge, reflectingthe fact that both bending and electrostatic interactions


1

10

100

1 10 100

log

Dc(

i)

log Particle interval |i-j|

T=0.10T=1.0

T=10.0

Fig. 3. Distance correlation functions Dc(i) for polyelectrolytefor Case 1), with f(x) = x reference line.

become less important. It is expected that in the limit ofvery high temperatures all the curves converge to the val-ues obtained by simulations of fully flexible chain (Case 1))with electrostatic interactions switched off: Re(0) = 24.1and Rg(0) = 9.9. It is seen from the figures that even atT = 10, the curves are more than twice above these limit-ing values, reflecting the fact that even rather weak elec-trostatic interactions can cause a rather strong expansionof a polyelectrolyte coil.

3.2 Distance correlation function

Let us define the (Euclidean) distance between atoms iand j of the polyanion chain as d(i, j). Then the distancecorrelation function Dc(i) for the interparticle labeled dif-ference i for the chain length of N is defined as

D(i) =

⟨N−i∑j=1

d(j, j + i)N − i

⟩, (6)

where 0 < i < N − 1.The above-defined distance correlation function for

Case 1) for high, medium and low temperatures is givenin fig. 3 in terms of a logarithmic plot. The plot seemsto be consistent with the graph for Re reflecting the factthat D(N − 1) ≈ Re (a small deviation may occur due tothe difference between arithmetic and root-mean-squareaverages). The largest slope of the curve is for the mostextended regime corresponding to T = 1.0. Here the chainis typically nearly straight, with a high Re value consti-tuting about 2/3 of the contour length of the polyion, andthis explains the high linearity of the correlation function,especially at lower i indexes. The slight curvature at veryhigh i is due to smooth bending of the entire structurewhich cannot be “observed” locally at low separation dis-tances, but only at “global” distances.

At high T = 10, the structure of the coil becomes morereduced. Most noticeably it is seen at low values of the in-dex i which reflects the local flexibility of the chain. It is

remarkable that the slope of the D(i) curve is increasingat larger (above 6–8) values of index i. Such behaviour isknown as “blob structure” when a locally flexible chainforms blobs which at a large scale form an elongated con-formation. Such blob structure of weakly charged polyelec-trolytes was discussed previously within analytical theo-ries (see, e.g., [38]). At very large values of the index i theslope of the T = 10 curve is again decreasing reflectingweaker electrostatic repulsion at such distances.

At low temperature T = 0.1 the graph is also inter-esting. At very low i, the dependence is linear and coin-cides with the case T = 1 implying the same, close torod-like, short-range structure. However at higher i, theslope rapidly falls to nearly zero. In the snapshots onecan see how relatively straight pieces of the polyion foldinto a ball-shaped structure. Recently, possibilities of for-mation of compact coil- or ring-like structures of stronglycharged polyelectrolytes were discussed [23]. If a kind ofcoil structure with a repeat in the same direction is statis-tically significantly present in the system, then one wouldexpect a periodic variation in D(i) with i, which is notobserved and so it is inferred that the folding is randomand interpenetrating. The broad inference here is that thefolding patterns can be elucidated from such correlationfunctions.

In previous decades, there were many discussionsabout the contribution of the electrostatic interaction intothe persistence length of a polyelectrolyte chain [1,2,39].This concept usually implies the so called “worm-like”chain, which behaves rod-like at distances shorter thanthe persistence length and as a random flexible chain onlarger distances. For a bead-spring chain (as in the presentmodel), the persistence length can be determined via bondvector correlation function [28] (which can be evaluateddirectly from the distance correlation function D(i) ati = 2). Some other ways of definition of the persistencelength are considered in a recent paper [40]. Our data onthe distance correlation function show that the confor-mational properties of a polyelectrolyte cannot always bedescribed in terms of the persistence length, if one under-stands the persistence length as a parameter characteriz-ing the local flexibility of the chains. For example, dataat T = 10 in fig. 3 show a rather flexible local structureimplying a short (in fact, unchanged in comparison withuncharged polymer) persistence length; but on larger dis-tances the slope of the D(i) curve is more typical for alonger persistence length. Previous simulation of flexiblepolyelectrolyte chains have also shown that, in general,a polyion cannot be characterized as a worm-like chain,and that different definitions of the persistence length canlead to strongly different values [3,20]. The existence ofseveral characteristic distances in polyelectrolyte solutionmakes it difficult to describe the structure using a singleparameter or within a simple scaling law.

3.3 Energetics

The electrostatic energy for all 4 series of simulations isdisplayed in fig. 4. An interesting observation is that at low


-200

0

200

400

600

800

1000

0 2 4 6 8 10

Eel

ectr

osta

tic

Temperature


Fig. 4. Electrostatic energy for polyelectrolyte.

temperatures, one can see a convergence in electrostaticenergy for all the Cases 1)-4). That is, in this limit theelectrostatic energy is not affected by the rigidity of thepolyion. This observation may give some ground for usingresults of studies of charged polyelectrolyte rods to under-stand the behavior of flexible unordered polyelectrolytes.

Note that the transition to a globular state, which inall studied cases is observed at T ≈ 0.2, is very close tothe point when the electrostatic energy becomes negative(T = 0.25). One can speculate that at lower temperaturesthe counterions not only screen the electrostatic repulsionbetween the polyion monomers, but induce the effectiveattraction between them. The negative electrostatic en-ergy brings negative virial contribution to the pressure;thus it favors the overall contraction of the system size(which is of course opposed by the steric repulsion whenthe system reaches the state of a compact globule).

For higher temperatures, one can see a clear tendencyof decreasing the electrostatic energy with increase of thechain rigidity. The explanation is that the bending termleads to an increase of the rigidity which decreases the pos-itive contribution of the electrostatic monomer-monomerinteractions.

The general form of the curves, including actual valuesof the specific energy, are similar to those observed in pre-vious simulation within the lattice model [30], includingthe point when the electrostatic energy is becoming neg-ative. Note that in paper [30] it was demonstrated thatthe specific electrostatic energy (per monomer) is not de-pendent on the length of the polyion at low temperatures,while in the current paper it appears independent of thechain rigidity for a continuous model in the same temper-ature range.

3.4 Radial distribution functions

The monomer-monomer and monomer-ion radial distri-bution functions are displayed in figs. 5, 6, 7 and 8. Theapparently high values of the RDFs is due in part to thebox volume of 8 × 106 units and the fact that a singlechain with bounded beads was featured in the system. Wetherefore use log10 scales in the plots.

0.1

1

10

100

1000

10000

100000

0 0.5 1 1.5 2 2.5 3 3.5 4

RD

F P

-P

Distance

T=0.1T=0.8

T=10.0

a)

0.1

1

10

100

1000

10000

0 0.5 1 1.5 2 2.5 3 3.5 4

RD

F P

-I

Distance

T=0.1T=0.8

T=10.0

b)

Fig. 5. RDF for monomer-monomer pairs of the polyion (a),and for monomers of polyanion-counterions (b), for Case 1).

Figure 5(a,b) depicts Case 1) (chain without rigidity)for the monomer-monomer (a) and monomer-counterion(b) interactions. The low-temperature globule-like struc-ture with random winding (as discussed above in sect. 3.2)implies at low distances a more uniform distribution ofmonomers with the maximum shifted to the right becausethe low kinetic energies do not lead to closer proximity tothe negatively charged atoms on the polyanion (which re-quires overcoming the strongly repulsive Lennard-Jonesforce). At a moderate temperature (T = 0.8), the ex-tended structure is very labile but nevertheless relatively“straight-chained” and therefore has a regularity that isdisplayed in the zig-zag periodicity with maxima (minima)separated by the monomer-monomer distance Rh = 1.The high temperature T = 10 structure (which is close toa random coil) is midway between these other two, andalso, because of the higher kinetic energy, leads to themaxima located to the left of the others.

The ion-monomer RDF at high temperature (fig. 5(b))is rather flat and close to unity except at the distances ofoverlap. Free ions do not feel the electrostatic attractionof the polyion due to high temperature. When the temper-ature becomes lower, the maximum of the ion-monomerRDF increases sharply. Ions condense around the polyion,and, at low temperature (T = 0.1), their concentrationaround the polyion reaches levels which are able to induce


0.001

0.01

0.1

1

10

100

1000

10000

100000

0 0.5 1 1.5 2 2.5 3 3.5 4

RD

F P

-P

Distance

T=0.1T=0.5

T=10.0

a)

0.1

1

10

100

1000

10000

0 0.5 1 1.5 2 2.5 3 3.5 4

RD

F P

-I

Distance

T=0.1T=0.5

T=10.0

b)


counterion-mediated attraction between various volumeelements of the polyion, which then causes the collapseof the polyion chain.

RDFs for Case 2) (weak bending interaction along thechain) is shown in fig. 6(a,b). At low temperatures, onecan see a sequence of maxima corresponding to second,third, and so on neighbours along the chain. This structurebecomes more diffuse at larger distances. The quasiperi-odic structure becomes much more pronounced for Case 3)of strong bending, see fig. 7(a,b). In the latter Case, thepolyion has a rod-like shape and the monomer-monomerRDF has correspondingly a set of maxima separated byranges of zero intensity. The ion-monomer RDF shows lesssensitivity to the rigidity of the polyion, showing similarbehaviour in all Cases 1)-3).

Case 2) (fig. 6) is the same as Case 3) except for asmaller bending constant kb = 10. Here, there appears atlow temperature the same tendency toward contraction bythe external ionic atmosphere, whereas this is relieved atintermediate temperatures only to be subjected less dras-tically to the same external forces that favor contractionat T = 10.0.

Additional insight into the structure of the polyelec-trolyte solution can be acquired from the running coor-dination numbers, which are obtained by integration of

0.1

1

10

100

1000

10000

100000

0 0.5 1 1.5 2 2.5 3 3.5 4

RD

F P

-P

Distance

T=0.1T=0.2T=0.5

T=10.0

a)

0.1

1

10

100

1000

10000

0 0.5 1 1.5 2 2.5 3 3.5 4

RD

F P

-I

Distance

T=0.1T=0.2T=0.5

T=10.0

b)


the RDF with factor 4πr2 up to a certain distance, andwhich show the average number of certain particles aroundthe given particle. The running coordination numbers forCases 1) and 3) are displayed in fig. 9 and fig. 10.

The polyion-polyion running coordination number fora flexible chain at T = 0.1 increases first to value 2 (ac-counting for the two neighbouring monomers), and thenbegin a fast growth at distances more than 2. This cor-responds to a globular state of the polyelectrolyte chainwhen polyion monomers of different fragments of the chainare separated by counterions. The polyion-counterion co-ordination number shows a similar behavior, demonstrat-ing that there is about an equal number of monomers andcounterions in the globule (that is, almost all counterionsare condensed to the globule).

For the medium temperature T = 0.8, correspondingto the maximum coil size in Case 1), the growth of thecoordination number is slowest, showing increase by 2 af-ter each unit of distance. This implies essentially strechedconformation of the chain with only occasional contribu-tion from other, non-neighbouring parts of the polyion.For high temperature (T = 10), the running coordinationnumber is becoming again higher, which reflects a smallersize of the polyelectrolyte coil and more monomer pairswithin the specified distance.


0.1

1

10

100

1000

10000

100000

0 0.5 1 1.5 2 2.5 3 3.5 4

RD

F P

-P

Distance

T=0.1T=2.0

T=10.0

a)

0.1

1

10

100

1000

10000

0 0.5 1 1.5 2 2.5 3 3.5 4

RD

F P

-I

Distance

T=0.1T=2.0

T=10.0

b)


For a stiff polyion (Case 3), fig. 10) and T = 0.5, therunning coordination number has a form of an (almost)stepwise function, which reflects essentially the rod-likecharacter of the polyion, where neigbouring monomerscontribute with “2” to the coordination number aftereach unit length. The average end-to-end distance for thiscase is 140, which, for the chain length 150, means alsostretched conformation with a small overall bending. Evenfor lower temperature, T = 0.2, the running coordinationnumber is nearly stepwise, though one can notice a con-tribution from non-neighbouring monomers. Such contri-bution becomes noticeable for T = 0.1, when monomers,long separated along the chain, occur frequently on a shortdistance from each other, due to the effective attractionmechanism caused by the counterions. This behaviour iswell in line with the results seen in the behaviour of theend-to-end distance, gyration radius and RDFs.

The monomer-counterion running coordination num-bers (figs. 9b, 10b) fall very quickly with the increase ofthe temperature. Already at medium temperatures 0.5–0.8 they are several times lower than the values observedat lowest temperature, showing that only a smaller part ofcounterions are present in close vicinity to the polyion. AtT = 10 the curve for the monomer-counterion coordina-tion number goes very close to zero (reaching the value of

0 2 4 6 8

10 12 14 16 18 20

0 0.5 1 1.5 2 2.5 3 3.5 4

Coo

rd.N

um.

P-P

Distance

T=0.1T=0.8

T=10.0

a)

0

5

10

15

20

0 0.5 1 1.5 2 2.5 3 3.5 4

Coo

rd.N

um.

P-I

Distance

T=0.1T=0.8

T=10.0

b)

Fig. 9. Running coordination number for polyion monomer-monomer pairs (a), and for monomer of polyanion-counterions(b), for Case 1).

about 0.1 at distance 4), showing that there are no coun-terions which are bound to the polyion in this regime. Theobserved results are well in line with previous estimationof the fraction of condense ions for shorter (N > 24) poly-electrolyte chains [41].

4 Conclusion

We have presented some basic results on the behaviour ofa single polyionic chain neutralized by the proper numberof counterions in a wide range of reduced temperatureswhich relates to the reduced polyion charge density ξ asξ = 1/T . Simulations were carried out for several valuesof the bending parameter, corresponding to fully flexiblepolyion, moderately and strongly stiff polyion as well asfor the case when bend conformation is preferable to thestraight one. In all cases we clearly observed three regimeswhich can be classified as “random coil”, observed at hightemperatures; “extended conformation” observed at mod-erate temperatures (of the order of 1 in reduced units), and“globular conformation” observed at low temperatures.While the transition between random and extended coilis gradual, the transition from extended coil to globular


0

2

4

6

8

10

12

0 0.5 1 1.5 2 2.5 3 3.5 4

Coo

rd.N

um.

P-P

Distance

T=0.1T=0.2T=0.5

T=10.0

a)

0

1

2

3

4

5

6

0 0.5 1 1.5 2 2.5 3 3.5 4

Coo

rd.N

um.

P-I

Distance

T=0.1T=0.2T=0.5

T=10.0

b)

Fig. 10. Running coordination number for polyion monomer-monomer pairs (a), and for monomer of polyanion-counterions(b), for Case 3).

behavior has a more abrupt character, exhibiting morethan doubled contraction of the gyration radius and theend-to-end distance when the reduced temperature dropsfrom 0.2 to 0.1. The question whether the transition to theglobular state for a strongly charged polyelectrolyte chainis a first-order phase transition is debated in the litera-ture [41–44]. The simulation data gathered for a restrictedset of temperature points and for finite chain lengths donot allow to make a definite conclusion on whether theobserved transition is a phase transition, that is whetherthe coil size changes continuously or discontinuously inthe thermodynamic limit. Previous simulations within alattice model of the work [30] showed a stronger fall ofthe polyion size when simulations were done within en-tropic sampling —the Wang-Landau algorithm. It mightbe possible that ergodicity and sampling problems makeobtaining the true canonical averages difficult, which wasobserved in ref. [30] in comparison with the Metropolisand entropic sampling simulations. But it might be a realdifference between the lattice and continuous models. Theuse of entropic sampling —Wang-Landau techniques— forcontinuous polyelectrolyte models may shed light on thisquestion.

C.G.J. would like to thank A. Laaksonen and A. Lyubartsevfor a conducive research environment and their kind hospital-ity over the 6 month period (2007) where this research wasformulated, executed and completed. The local computing fa-cilities here and especially at University of Malaya financedunder IRPA grant (2003-2006) 09-02-03-1031, and a sabbaticalallowance from UM are also acknowledged. The work has beensupported by the Swedish Research Council (Vetenskapsradet).

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conformational characteristics of single flexible polyelectrolyte chain

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