attractive forces between circular polyions of the same charge

Physica A 311 (2002) 23–34www.elsevier.com/locate/physa

Attractive forces between circular polyions of thesame charge

Silvia Martins, J%urgen F. Stilck∗

Instituto de F��sica, Universidade Federal Fluminense, Av. Litoranea s=n, 24210-340 Niter�oi, RI, Brazil

Received 14 December 2001

Abstract

We study two models of ring-like polyions which are two-dimensional versions of simplemodels for colloidal particles (model A) and for rod-like segments of DNA (model B), bothin solution with counterions. The counterions may condense on Z sites of the polyions, and wesuppose the number of condensed counterions on each polyion n to be 0xed. The exact freeenergy of a pair of polyions is calculated for not too large values of Z , and for both modelswe 0nd that attractive forces appear between the rings even when the condensed counterionsdo not neutralize the total charge of the polyions. This force is due to correlations betweenthe condensed counterions and, in general, becomes smaller as the temperature is increased. Formodel A divergent force may appear as the separation between the rings vanishes, and this makesan analytical study possible for this model and for vanishing separation, showing a universalbehavior in this limit. Attractive forces are found for model A if the valence of the counterionsis larger than one. For model B, no such divergences are present, and attractive forces are foundfor a 0nite range of values of the counterion valence, which depends of Z; n, and the temperature.c© 2002 Published by Elsevier Science B.V.

PACS: 05.70.Ce; 61.20.Qg; 61.25.Hq

Keywords: Polyions; Colloidal particles; Attractive forces

1. Introduction

The interaction between colloidal particles in suspensions are a statistical mechanicalproblem which has attracted much attention from both theoretical and experimentalpoints of view for more than one century now, but there are still issues which arenot totally understood [1]. Besides the di?culty of properly taking into account the

∗ Corresponding author. Tel.: +55-21-2620-6735; fax: +55-21-2620-3881.E-mail address: [email protected] (J.F. Stilck).

0378-4371/02/$ - see front matter c© 2002 Published by Elsevier Science B.V.PII: S 0378 -4371(02)00837 -3

24 S. Martins, J.F. Stilck / Physica A 311 (2002) 23–34

long-range Coulomb interactions, it is necessary to consider the fact that each colloidalparticle in solution may contribute with quite a large number of counterions, and theircontribution to the thermodynamic properties of the suspension is quite important [2].One particularly important issue related to these systems is the possibility of phaseseparation, and although it has been argued that phase separation may occur even insituations where repulsive interactions between pairs of colloidal particles are present[3], recently it was found that for aqueous solutions with monovalent counterions nophase separation is expected [4] and this seems to be consistent with conclusionsdrawn from simulations [5]. For multivalent counterions, however, correlations betweencounterions which are condensed on the colloidal particles may produce short-rangeattractive eEective interactions between the particles, even if their bare charge is nottotally neutralized [6]. This interesting phenomenon of eEective attractive interactionsbetween objects with charges of the same sign has attracted much attention recently,particularly for systems of rod-like polyions, such as segments of DNA [7].The theoretical approaches used so far to address the problem of attraction between

rod-like polyions of the same charge may be classi0ed in two groups. The 0rst for-mulates the system as a 0eld-theoretic problem and then studies its properties in ahigh-temperature approximation [8]. The second approach, which is also approximate,supposes the system of polyions and condensed counterions to form a Wigner crystalat zero temperature, and then proceeds considering the eEect of temperature on thesystem [9–13]. Some time ago, a very simple model has been proposed for an isolatedpair of rod-like polyions in a solution with counterions which may be exactly solved,with the aid of a computer, for not too large rods [10]. In this model, the only eEect ofthe condensation of a counterion on the site where an ion is located is to renormalizethe local charge. The valence of each counterion is �, so that each counterion has acharge �q, and the polyion consists of a linear array of Z charges −q with a distance bbetween 0rst neighbors. The solution has an eEective dielectric constant D. The meanvalue of the number n of condensed counterions may be determined by the Manningcriterion [14] as

n=(1− 1

�

)Z�; (1)

where

=q2

DbkBT(2)

is the Manning parameter. For su?ciently low temperature (¿ 1=�), counterions will,therefore, condense on the polyions. It should be remarked that the expression abovefor the number of condensed counterions is an approximation for an isolated polyionand in the limit Z → ∞. One may wonder if it is still valid for 0nite values of Z and,supposing that the Manning value for n is the mean value of a distribution of numbersof condensed counterions it is relevant to ask if this distribution is broad or narrow.It turns out that expression (1) is actually a good approximation even for rather smallvalues of Z and that in usual physical conditions the distribution of values of n is quitenarrow [15]. Now if we consider two polyions separated by a distance d, we may use

S. Martins, J.F. Stilck / Physica A 311 (2002) 23–34 25

expression (1) in the limits d → ∞ and d → 0. The total number of counterionscondensed on both rods in the limit d→ ∞ will be

n∞ = 2n (3)

and if we now consider the other limit d→ 0 (where both chains merge into one withZ charges −2q) the number of condensed charges will now be given by

n0 =(1− 1

2�

)2Z�

(4)

and, therefore, the change in the number of condensed counterions is given by

Kn= n0 − n∞ = n∞1

�− 1(5)

and for not too high temperatures it may be a reasonable approximation to ignore thischange, as is done in the simple model we described.In this paper, we study two models for the interaction between two ring-like charge

structures which are inspired by studies of colloidal particles and of rod-like polyions,respectively. The models are similar to the one for rod-like polyions described above,and we study the thermodynamic properties of these models by solving them exactlyfor not too large particles. In the 0rst model (model A), the bare colloidal particlesare supposed to be formed by Z point charges −q placed regularly on circles, andwe consider two of these ring-like particles placed in a solution with counterions.The number of counterions condensed on each particle is equal to n and we proceedcalculating the force between them as a function of the separation between the rings,for 0xed values of the valence � of the counterions and the Manning parameter . Thesecond model is a two-dimensional version for a model which was recently proposedfor rod-like DNA segments in which, the negative charge of the polyion is supposedto be concentrated at the axis of a cylinder, while the counterions are condensed at thesurface of this cylinder [16]. Thus, in model B each ring-like polyion is formed by apoint charge −Zq in the center of the circle, while n counterions may condense on Zequally spaced sites on the circle. The basic point we address in our calculation is thepossibility of attractive forces between polyions of the same total charge, and for bothmodels we 0nd that such forces are indeed possible.In the next section we present both models in detail and the results we obtained for

them, which are either analytical or numerically exact. Final comments and discussionsmay be found in Section 3.

2. De�nition of the models and their solution

We consider two circular polyions of radius r with an even number Z of pointcharges −q which are placed as follows:

• Model A uniformly distributed on the circle.• Model B in the center of the circle.


j=1 j=2

i=0

i=1

i=0

i=1

i=7 i=7

r r

θ1 db

Fig. 1. A possible con0guration of two ring polyions with Z = 8 and n = 3 condensed counterions. Fullsmall circles indicate adsorption sites with condensed counterions, whereas empty small circles are emptycondensation sites.

These polyions are in a ionic solution of �-valent counterions, also supposed to bepoint charges (charge �q). We will admit that n of the counterions are condensed onZ sites on the circle. In model A, since the counterions condense on top of the 0xednegative charges, the only eEect of a condensed counterion is to renormalize the chargeon the condensation site, which will be equal to q(� − 1). In model B, the charge ata condensation site on the circle is equal to zero if no counterion is condensed onit and �q otherwise. The distance between the centers of the polyions is 2r + d. Thede0nitions above are illustrated in Fig. 1.The positions of the condensation sites on the rings will be referred by two indices

j = 1; 2, indicating the ring, and i= 0; 1; : : : ; Z − 1, indicating the position on the ring.For model A, the electrostatic interactions between the charges may then be describedby the Hamiltonian

HA =q2

2Dr

Z−1∑i; i′=0

2∑j; j′=1

(1− ��i; j)(1− ��i′ ; j′)�(i; j; i′; j′)

; (6)

where the sum is restricted to (i; j) �=(i′; j′) and the occupation variables �i; j as-sume the value 0 if no counterion is condensed on site (i; j) and 1 otherwise. Dis the dielectric constant and the ratio of the distance between two sites and r isgiven by

�(i; j; i′; j′) =

2|sin �i; i′2 | if j = j′ ;

√(x + 2− cos �i − cos �i′)2 + (sin �i − sin �i′)2 otherwise ;

(7)

where x = d=r and

�i; i′ =2�(i − i′)

Z; (8)

�i = �i;0 : (9)


Model B may be described by the Hamiltonian

HB =q2�2

2Dr

Z−1∑i; i′=0

2∑j; j′=1

�i; j�i′ ; j′�(i; j; i′; j′)

− Zq2�Dr

Z−1∑i=0

�i;1 + �i;2�(i)

+Z2q2

r1

2 + x; (10)

where the interaction between the central charge and the condensed counterions on thesame ring was not included, since it is constant, and

�(i) =√1 + (2 + x)2 − 2(2 + x) cos �i : (11)

The partition function Q is calculated summing over the possible positions of thecondensed counterions

QA;B =∑{�i; j}

′exp (−�HA;B) ; (12)

where the prime denotes the constraint of having n counterions condensed on eachring.It is convenient to de0ne adimensional Hamiltonians H through

�HA;B = ′HA;B ; (13)

where ′ = �q2=Dr is related to the Manning parameter de0ned in expression (1) forrod-like polyions. Then

HA =12

∑(i; j)�=(i′ ; j′)

(1− ��ij)(1− ��i′j′)�(i; j; i′; j′)

(14)

and

HB =�2

2

∑(i; j)�=(i′ ; j′)

�i; j�i′ ; j′�(i; j; i′; j′)

− Z�∑i

�i;1 + �i;2�(i)

+Z2

2 + x: (15)

We may de0ne the statistical weights

yi = exp(−′ai

); i = 1; 2; : : : ; Z=2 (16)

and

zi = exp(−′bi

); i = 1; 2; : : : ; m (17)

related to the intra- and interpolyion interactions between charges at the condensationsites, respectively, and

ti = exp(−′ci

); i = 0; 1; : : : ; Z=2 ; (18)

which correspond to the interaction between a central charge of a ring and the con-densed counterions on the other ring. The denominators in the arguments of the expo-nentials are the adimensional distances between the interacting charges. For a pair of


charges in the same polyion

ai = 2∣∣∣∣sin �i2

∣∣∣∣ : (19)

The expressions for the denominators bi are straightforward to obtain from (7) butwill be omitted here for brevity. m + 1 = Z2=4 + Z=2 + 1 is the number of diEerentdistances between condensation sites located on distinct rings. Finally, the adimensionaldistances between condensation sites and the central charge of the other ring are ci=�i,according to Eq. (11).Using these statistical weights, we may rewrite the partition function as

QA;B =Nc∑i=1

!A;B(i) ; (20)

where the sum runs over the

Nc =[

Z!(n!(Z − n)!)

]2(21)

allowed con0gurations of condensed counterions and

!A(i) =Z=2∏j=1

yuA(i; j)j

m∏k=0

zvA(i; k)k : (22)

The exponents uA(ij) and vA(ik) are quadratic polynomials in � with integer coe?-cients. For model B we obtain

!B(i) =Z=2∏j=1

yuB(i; j)j

m∏k=0

zvB(i; k)k

Z=2∏l=0

twB(i; l)l ; (23)

where uB(ij) and vB(ik) are integer coe?cients multiplied by �2 and wB(i; l) are neg-ative integer coe?cients multiplied by Z�. The whole set of integer coe?cients can becalculated with the help of a computer program, and then we can obtain the partitionfunction exactly for both models and not too large values of Z .The force between the rings is given by

FA;B =−9'A;B

9d ; (24)

where 'A;B =−kBT lnQA;B is the Helmholtz free energy. It is convenient to de0ne anadimensional force fA;B = (Dr2=q2)FA;B, and thus

fA;B =1

′QA;B

9QA;B

9x : (25)

For model A, we 0nd that the force between the rings becomes attractive for shortdistances and �¿ 1. For the other cases the force is always repulsive. As x → 0,


0 2x

10

0

10

20

30

f

α = 0.9α = 1α = 1.1

1 3 4

Fig. 2. The force between the rings as a function of the distance x for model A. The results presented arefor Z = 10; n = 2 and = 2.

the force is 0nite only for �=1. The 0nite repulsive force observed in the limit x → 0when �=1 may be easily understood. In this limit, the probability of having no chargescondensed on the pair of sites (0; 1) and (0; 2) vanishes, since this would lead to adivergence in the energy. Thus, this pair of sites does not contribute at all to the force,and the contributions of the other pairs is either positive or zero. This argument alsoexplains the decreasing value of the force at vanishing distance as n is increased, asmay be seen in Fig. 2.To study the behavior of the attractive force for model A at short distances (�¿ 1),

we consider the limit of small x. In this case, we have

limx→0

z0 = 0 ; (26)

where z0 = exp − ′=x, is related to the interaction between the charges in positions(0; 1) and (0; 2). All the other variables yi and zi are 0nite. We can, therefore, de0nethe variable

v=mini(vA(i; 0)) ; (27)

where the index 0 refers to the pair of sites (0; 1) and (0; 2). We then rewrite thepartition function as

QA = zv0W (1− P) ; (28)


where

W =l∑i=1

Z=2∏j=1

yuA(i; j)j

m∏k=1

zvA(i; k)k (29)

and

P =1W

Nc∑i=l+1

zvA(i;0)−v0

Z=2∏j=1

yuA(i; j)j

m∏k=1

zvA(i; k)k : (30)

In these equations, we considered the 0rst l con0gurations to have an exponentvA(i; 0) = v. P vanishes when x → 0 since vA(i; 0)¿v for i6 l + 1, so we mayapproximate

QA ≈ zv0W : (31)

The possible values of vA(i; 0) are, 1 if there are no condensed counterions in bothsites, 1− � when there is one condensed counterion on one of the sites, and (1− �)2

if there is a condensed counterion on each site. As we discussed before, the forcebecomes attractive only for �¿ 1, so we can conclude that the minimum value isv= 1− �. Using the above approximation for the partition function we obtain

f =vx2

+1′W

9W9x : (32)

Taylor expanding the function g(x)=(1=′W )9W=9x; g(x) ∼ g0+g1x+ · · · and keepingonly the 0rst term, we get

f =vx2

+ g0 : (33)

For the distance x0 of vanishing force, we may then 0nd the following scaling relation:

x0 ≈(�− 1g0

)1=2

: (34)

In Fig. 3 curves of x0 as a function of �− 1 are shown for some values of Z; n; and, and the collapse of these curves at small values of x0 may be seen. Also, in theinset the limiting behavior found in Eq. (34) is evident.We proceed showing results for model B. In this case, it should be stressed that

only in the rather unphysical limit of in0nite temperature we have divergence in theforce at vanishing values of x. These divergences are present for 0nite temperaturesboth in model A and in the similar model for rod-like polyions [10,17] and are due topairs of sites with nonzero charge with vanishing distance between them. The absenceof these divergences in model B prevents us from performing an asymptotic analysissimilar to the one done for model A, which would also prove the existence (or not) ofattractive forces in model B. Simulations performed in a cylindrical version of model Bfound attractive forces at su?ciently low separation and enough condensed counterioncharges [16].In Fig. 4, the force between the rings for model B is shown as a function of the

separation x. For su?ciently low temperatures and not too low or too high values


Fig. 3. The equilibrium distance x0 as a function of � − 1 for ′ = 2 (model A). The inset shows data forZ = 8; n = 1 in the region of � close to 1. The scaling behavior is apparent and the numerical 0t resultedin a value 0:51± 0:01 for the slope, in accordance with expression (34).

Fig. 4. The force as a function of the distance for model B. The results shown are for Z = 10; n = 2, and�=1:6. Curves for three diEerent values of the inverse temperature ′ are shown: Dashed line—′ =2; fullline—′ = 1:23; dot–dashed line—′ = 0. In the inset, values of the force at vanishing x (f0 = limx→0 f)are shown as a function of the reduced temperature , = 1=′.

of �, attractive forces are found for low separation. As mentioned above, for 0nitetemperatures the forces are always 0nite, since in this model the energy of the con0g-urations where the pair of sites (0; 1) and (0; 2) is occupied diverges as x → 0. Theforce at vanishing separation f0 = limx→0 f(x) is shown as a function of the reducedtemperature , = 1=′ in the inset of the 0gure, and increases monotonically, so that,as expected, it diverges to +∞ as the temperature becomes in0nite (1=′ → ∞). We


Fig. 5. The equilibrium distance x0 as a function of �−1 for diEerent values of Z; n, and ′ (model B). Fullthin line—Z=8; n=2, and ′=1; full thick line—Z=8; n=2, and ′=0:05; long dashed line—Z=10; n=2,and ′ = 1; short dashed line—Z = 8; n = 3, and ′ = 1; dot–dashed line—Z = 10; n = 3, and ′ = 1.

obtained similar results for other values of Z and n, so that the data in Fig. 4 are anexample of a general trend.Fig. 5 shows results for the equilibrium distance x0 as a function of the valence �

of the counterions for some choices of Z; n, and ′. As mentioned above, we werenot able to perform a simple analytical study of model B in the limit of vanishing x,and as expected no simple scaling behavior is found for model B in this limit, similarto the ones observed for model A and the other model proposed for rod-like polyions[10,17]. Thus, the minimum value of � for the existence of an attractive interactionvaries with Z; n, and ′, and so does the initial slope of the curve x0 × �. Anotherfeature of these curves is the existence of a maximum value of the valence of thecounterions, above which no attractive force is found. Thus, as is apparent in Fig. 5,for each curve there are two values of � for which x0 =0. Results for f0 as a functionof � which will not be presented here for brevity also show this feature: The forceis attractive in the interval between the limiting values of �, and, therefore, passesthrough a minimum in some point inside the interval. It should be stressed, however,that the larger limiting value of � we found for model B assumes values which aretoo large to be physically realizable, but since the model itself is rather arti0cial, onemay wonder of this eEect may not be seen in more realistic models such as the onediscussed in Ref. [16] and in experiments. For model A and the model which wasproposed for rod-like polymers, no such maximum value of � is found, and the curvesof x0 × � pass through a maximum but approach zero asymptotically as �→ ∞.

3. Final comments and discussions

In an appropriate range of the parameters, both models studied here display attrac-tive interaction forces, which may be explained by correlations between the counterions


condensed on the ring-like polyions. As the temperature is increased, in general, thecorrelations become weaker and, therefore, the attractive force decreases. In simpli-0ed models which were exactly solved before [10,17], as well as in model A in thiswork, the attractive forces increase without bond as the distance between the polyionsis lowered, since one or more pairs of sites with opposite charges are separated by adistance which vanishes. This eEect allowed an exact analysis of the models at van-ishing separation, but also has the drawback of being not very reasonable physically.In particular, in the case of model A, it leads to the conclusion that it is su?cient tohave only one condensed counterion on each polyion, with a valence slightly largerthan one, to ensure the appearance of attractive forces. As was already found in simu-lations of cylindrical polyions [16] and other more realistic models for DNA segments[12], attraction between like-charged polyions is not restricted to this rather arti0cialsituation where sites with opposite charges become very close. In model B we 0ndanother example of this kind: The forces are 0nite for 0nite temperature, and a valencewhich is larger than one by a 0nite amount is needed to ensure attractive forces.In both models studied here, when the parameters are such that attractive forces

are found, the equilibrium distance x0 as a function of � has a maximum. At lowtemperature, the maximum is sharp and located close to � = Z=n, which correspondsto neutral polyions. As the temperature is increased, the maximum becomes broaderand moves towards larger values of �. With increasing temperature, the region ofattractive force in the � × x plane shrinks and 0nally disappears as the temperaturebecomes in0nite. An example of this behavior is seen in the two curves for diEerenttemperatures for model B with Z = 8 and n= 2 in Fig. 5. Also, in Fig. 4 one noticesthat for in0nite temperature (′ = 0), the force is always repulsive.

Acknowledgements

It is a pleasure to acknowledge Yan Levin for many suggestions in this work and fora critical reading of the manuscript. Partial 0nancial support by the brazilian agenciesCNPq and FAPERJ is gratefully acknowledged.

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attractive forces between circular polyions of the same charge

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