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Simple Model for Attraction between Like-Charged Polyions Jeferson J. Arenzon Instituto de F´ısica, Universidade Federal do Rio Grande do Sul Caixa Postal 15051, 91501-970, Porto Alegre, RS, Brazil E-mail: [email protected] Ju˝rgen F. Stilck 9 9 Instituto de F´ısica, Universidade Federal Fluminense 9 Av. Litorânea, s/no, 24210-340, Niterói, RJ, Brazil 1 E-mail: [email protected]ff.br n Yan Levin∗ a J Instituto de F´ısica, Universidade Federal do Rio Grande do Sul 9 Caixa Postal 15051, 91501-970, Porto Alegre, RS, Brazil 1 E-mail: [email protected] (January 8, 1999) ] t Wepresent a simple model for the possible mechanism of appearance of attraction between like f o charged polyions inside a polyelectrolyte solution. The attraction is found to be short ranged, and s exists only in presence of multivalent counterions. The attraction is produced by the correlations t. in thecondensedlayersofcounterionssurroundingeachpolyion, andappearsonlyif thenumberof a condensed counterions exceeds the threshold, n>Z/2α, where α is the valence of counterions and m Z is thepolyion charge. - d Polyelectrolytesolutionsandchargedcolloidalsuspen- various simple models which might help shed additional n o sions present an outstanding challenge to modern statis- light on the origin of this attraction. c tical mechanics. One of the reasons for the great dif- [ ficulty in achieving an understanding of these complex Our discussion will be restricted to rodlike polyelec- systemsis due tothe intricateroleplayedby the counte- trolytes, a good example of which is an aqueous solu- 2 tion of DNA segments. Let us consider the simplest v rions (microions) which counterbalance the much bigger model of such a polyelectrolyte solution. The polyions 8 charge of the polyions (macroions). Since the number shallberepresentedbyrigidcylindersofnetcharge Zq 5 density of counterions is so much greater than that of − 3 polyions it is the counterions that dominate the ther- distributed uniformly, with separation b, along the ma- 6 jor axis of a macromolecule. The counterions shall be modynamicpropertiesofpolyelectrolytesolutionsatlow 0 treated as small rigid spheres of charge αq located at densities. The heterogeneity combined with the long- 8 the center. The counterions can be monovalent, diva- range Coulomb force makes polyelectrolytes almost im- 9 lent,ortrivalent,forsimplicityhowever,weshallrestrict / possible to study by the traditional methods of liquid t our solution to contain only one of the above types of a state theory. It is, however,exactly this complexity that counterions. An appropriate number of coions is also m is responsible for the richness of the behaviors encoun- presentinthe solutionto keepanoverallchargeneutral- tered in polyelectrolyte solutions and charged suspen- - d sions. ity. The solvent shall be modeled by a uniform medium n of a dielectric constant D. It has been argued by Man- o One of the mostfascinating results of the subtle inter- ning that in the limit of very large Z and infinite dilu- c play of various interactions is the appearance of attraction, a certain number of counterions will condense onto : v tion between two like-charged macromolecules inside a the polyions, thus renormalizing their effective charge. i solution or a suspension. This attraction is purely elec- From a simple phenomenological argument Manning de- X trostatic and is not a result of some additional short- termined the number of condensed counterions to be r range van der Waals force. The attraction has been ob- n =(1 1/αξ)Z/αfor ξ >1/αandn =0 for ξ <1/α, a c c servedinsimulationsofstronglyasymmetricelectrolytes whereξ−=q2/Dk Tb[6]. Notethatthisresultsubtlyde- B [1], as well as a number of experiments [2]. The ex- pendsontheorderofthelimitstobetaken,firstZ →∞ actmechanismresponsibleforthisunusualphenomenais and then the infinite dilution. If the limits are inter- still not understood, although some theories attempting changed no condensation will appear. Recently we have toexplainitsbasis,havebeenrecentlyadvanced[3,4]. At extendedtheManningtheorytofiniteconcentrationsand this time,webelieve,thereexists agreatneedto explore finitepolyionsizes. Inthiscaseitispossibletoshowthat ∗ Corresponding author 1 the counterionassociationstillpersists,however,instead sates for its abstract nature. Evidently, if we can un- of a fixed number of condensed counterions associated derstand under what conditions the attraction can arise witheachpolyion,wenowfinda distributionofclusters, between two macromolecules in this idealization, in the each composed of one polyion and 1 m Z/α associ- future it might pave the way to a more complete, physi- ≤ ≤ atedcounterions. The distributionofcluster sizes is well cally realistic model. localized, and in the limit of large polyion charge and With this disclaimer in mind, we proceed to rewrite infinite dilution approaches a delta function centered on the Hamiltonian as β = ξH, where β = 1/k T and ξ B the value proposed by Manning, nc [7]. is the Manning paramHeter defined earlier. The adimen- Can the condensed layer of counterions be responsi- sional reduced Hamiltonian is ble for the observed attraction between two like charged polyions? Toanswerthisquestionweproposethefollow- 1 (1 ασij)(1 ασi′j′) ingsimplemodel. Considertwoparallel,rodlikepolyions, H = − − . (2) with Z monomers, inside the polyelectrolyte solution. 2(i,j)X6=(i′,j′) |i−i′|2+(1−δjj′)x2 p The separation between two macromolecules is d. If the attractionisproducedbysomesortofcharge-correlation Usingatransformationofoccupationvariablesdefined mechanism, we expect that it should be short-ranged. by σ′ = 1 σ it is easy to see that the Hamiltonian We shall, therefore, restrict our attention to distances ij − ij exhibits the following symmetry such that d < ξ , where ξ is the Debye screening D D length. As was mentioned earlier, the strong electro- H(Z,α, σ )=(α 1)2H(Z,α′, σ′ ), (3) static attraction between the polyions and the counte- { } − { } rions favors the formation of clusters composed of one polyion and some number of associatedcounterions. For where n′ =Z n and α′ =1+1/(α 1). − − the purpose of this exposition, we shall neglect the poly- The partition function is dispersity of cluster sizes and assume that both polyions have n<Z/α condensed counterions. It is important to ′ ′ rememberthatweareconcernedwiththeinteractionbe- Q= exp( β )= exp( ξH), (4) − H − tween the two polyions inside a polyelectrolyte solution. {Xσij} {Xσij} As was stressed before, an isolated polyion can confine counterions only if it is extremely long, while inside a where the prime indicates that the occupation numbers solution the cluster formation can take place with the are subjected to the constraints of the number of con- polyions of any size [7]. densed counterion conservation. The symmetry relation The associated counterions are free to move along the (3) leads to invarianceof the partition function, and any length of the polyions. We define the occupation vari- thermodynamic quantity derivedfromit, Q(Z,n,ξ,α)= ables σij, with i =1,2,...,Z and j =1,2, in such a way Q(Z,Z n,[α 1]2ξ,1+1/[α 1]). Thispropertyisquite that σij =1 if a counterion is attached at i’th monomer useful w−hen pe−rforming the c−alculations, since it reduces ofthej’thpolyionandσij =0otherwise. Sincethenum- the ranges of parameters needed to study. berofcondensedcounterionsisfixedbythermodynamics It is convenient to rewrite the partition function in [7],thevaluesofoccupationvariablesobeytheconstraint terms of the variables y = exp( ξ/i),i= 1,2,...,Z 1, Zi=1σi1 = Zi=1σi2 =n. Weshallassumethattheonly associated with the intirapolyion−interactions and z−i = Peffect of thePcounterion association is a local renormal- exp( ξ/√x2+i2),i = 0,1,...,Z 1, related to the in- izationofthemonomercharge. TheHamiltonianforthis terpo−lyion interactions. The parti−tion function, for given model takes a particularly simple form, values of Z and n, may be expressed as H= 21D Z 2 q2(1−rα(iσ,ijj;)i(′1,j−′)ασi′j′), (1) Q= Nc Z−1yuij Z−1zvik, (5) i,Xi′=1j,Xj′=1 j k Xi jY=1 kY=0 where the sum is restricted to (i,j) = (i′,j′), 6 r(i,j;i′,j′) = b i i′ 2+(1 δjj′)x2 is the distance where the sum runs over all the allowed configurations between the monpo|m−ers |located−at (i,j) and (i′,j′), δjj′ of counterions. The exponents uij and vik are quadratic is the Kronecker delta, and x = d/b. Clearly, the above polynomials in α with integer coefficients. For not too model is a great over-simplification of physical reality. bigvaluesofZ,itispossibletogeneratethe wholesetof The molecularnature ofthe solventis not takeninto ac- integers in the expressionabove, thus obtaining the par- count. Thecounterionsareassumedtobeconfinedtothe tition function exactly. The force between the polyions surfaceofthepolyions,whilethepolyionsthemselvesare may now be calculated through treatedascompletelyrigid. Nevertheless,webelievethat the simplicity of this phenomenologicalmodel, which al- 1 ∂lnQ F = . (6) lows us to perform exact analytic calculations, compen- bβ ∂x 2 −0.6 6 −0.8 4 ) 0 Db 2 x−1.0 n / 2 l q ( 0 F −1.2 −2 −1.4 −6.3 −5.8 −5.3 −4.8 −4.3 −3.8 −3.3 α −4 ln ( n/Z − 1/2) 0 1 2 3 x FIG.3. Log-log plot near n=Z/2α for the α=2 data in FIG.1. ForceversusdistancebetweenpolyionsforZ =20, fig. 2. The fit exponent is 0.32. α = 2, ξ = 2.283 (corresponding to polymethacrylate) and n=5,...,10(from toptobottom)intheMonteCarlo simu- To simulate the model weuse astandardMonte Carlo lation. The lines are only guides tothe eye. with particle-hole exchange, not restricted to nearest- neighbor pairs. This non local diffusion comes from the factthatthecounterionsmayassociate/dissociateatany pointalongthepolyion. Afterthermalizing,boththeen- ergy and the force are measured, results being time and sample averaged. In Fig.2 it is clear that the relevant parameter is the fraction of neutralized charge, αn/Z, Wefindthatatshortdistancesandforα>1theforce the curves for several values of Z (here from 20 to 200) between the polyions may become attractive (negative). collapsing on an universalfunction, deviations occurring The exact results are in full agreement with the Monte only for very small Z. Carlo simulations which can also be extended to much As expected, the attractive interactionis short-ranged larger values of Z, Fig. 1. For α = 1 the force is always and at larger distances the force becomes repulsive. To repulsive,whichisinfull agreementwiththe experimen- further explore this point we consider the limit of small tal evidence on absence of attraction if only monovalent x. In this case the variable z vanishes, while for finite 0 counterions are present [2]. When the polyion charge ξ and Z all the other variables y and z remain posi- j j is completely neutralized, n = Z/α, the force becomes tive definite. If we define v =min (v ), it is possible to i i,0 purely attractive, as can be seen in Figs. 1 and 2. Fur- rewrite the partition function as Q=zvW(1+P), with thermore,theexactsolutionandthesimulationsindicate 0 thata criticalnumber ofcondensedcounterionsis neces- l Z−1 Z−1 sary for attraction to appear, see Fig. 2. W = yuij zvik, (7) j k Xi=1 jY=1 kY=1 3 and ξ = 2.283 α = 2 1 Nc Z−1 Z−1 2 α = 3 P = W  z0vi0−v yjuij zkvik, (8) i=Xl+1 jY=1 kY=1   0 x wherewesupposev =vforthefirstloftheN configu- i0 c rations. As x 0, the function P vanishes, and we may 1 use the appro→ximation Q Wzv. We then notice that ≈ 0 v corresponds to the configurations which maximize the number of favorablehorizontalinterpolyioninteractions, 0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 v =Z 2nα (9) α n/Z − for α 2. Using the above approximation for the parti- FIG. 2. Distance below which the polyion interaction be- tion fu≥nction we obtain comes attractive versus the number of condensed counterion in the Monte Carlo simulation. Averages are over 100 sets 1 1 ∂W ξv F + . (10) and Z ranges from 20 to 200. ≈ bβ (cid:18)W ∂x x2(cid:19) 3 Itisnotdifficultto seethatthe derivativeofW vanishes condensed counterions is above the threshold n = Z/2α as x 0, so that for small x we can Taylor expand the the favorable counterion-monomer interactions will out- → first term of Eq. 10. To leading order we find numbertheunfavorablemonomer-monomerinteractions, thus producing a net attraction at short distances. Us- ξv ingthethresholdvalueofnwefindthattheattractionis f bβF hx, (11) ≡ ≈ x2 − dominated by the zero temperature correlations as long as ξ > 2/α. On the other hand applying the Manning whereh<0(asvvanishes)isafunctionoftheremaining condensation criterion to the threshold number we see parameters of the model. We notice that as x 0 the that the attraction is possible only if ξ > 2/α. We thus → forcevanishes ifv =0,is largeattractiveifv isnegative, cometoanimportantconclusion: Ifthe attractionexists andislargerepulsiveifv ispositive. Inparticular,inthe it is produced by the zero temperature correlations. limit v 0− the equilibrium distance is In view of the current interest in the possible mecha- → nismsresponsiblefortheattractionbetweenlikecharged 1/3 1/3 ξv 2nα objectsitisworthwhiletomakesomefurther comments. x 1 . (12) 0 ≈(cid:18) h (cid:19) ∼(cid:18) Z − (cid:19) A most common approach used to study this difficult problemreliesonthe fieldtheoretic methodologysimilar This scaling behavior is also observed in Monte Carlo to the one developed in Quantum Field Theory to study simulations, Fig. 2 and 3. Casimir forces. The partition function is mapped onto an effective field theory which is then studied using a 5 loop expansion. Due to the underlying difficulty of this process the expansion is usually terminated at the first loop level which is equivalent to the so called Gaussian approximation. The attraction, in this approach, arises 1 0 as a result of the correlations in the Gaussian fluctua- 0.8 tions [3]. Clearly from the above discussion this is not F 0.6 themechanismresponsiblefortheattractionbetweenthe 0 x two charged rods which was found by us. The problem 0.4 −5 withthefieldtheoreticapproachesuseduptonowisthat 0.2 the ground state energy was not properly treated. 00 1 2 ξ 3 4 5 Insteadofallowingforastaggeredconfiguration,which we found to be responsible for the attraction observed, −10 the fieldtheoretic approachesneglectthe discrete nature 0.5 1.5 2.5 x of charges and uniformly smear the condensed counterions over the polyion. Clearly at zero temperature this can only result in a repulsion! The attraction, then, ap- FIG. 4. Temperature dependence of the force between pears only as a finite temperature correction, produced polyionsforZ =20,n=6,α=2andseveralvaluesofξ: 2.28 (circles), corresponding to polymethacrylate, 4.17 (squares), by the correlations in the Gaussian fluctuations of the correspondingtoDNAand20(triangles)forverylowtemper- counterion charge densities on the two polyions, the ef- atures. The inset shows the distance x0 at which the attrac- fect which is much weaker than the one found by us to tion first appears as a function of ξ. Wesee that for divalent be responsible for the attraction. In view of our results counterions, ξ = 1 serves as a division point for the low and we must conclude that the currently used field theoretic high temperatures regimes. Clearly for ξ > 1 the attraction approaches [3] must be reexamined. is driven by thezero temperature mechanism! We have presented a simple model for the possible mechanism of appearance of attraction between the like We wouldliketo stressthatfora fixednumber ofcon- charged polyions inside a polyelectrolyte solution. The densed counterions the attraction is insensitive to the attraction is found to be short ranged, and is possible temperature (see figure 4). In fact the force between only in a presence of multivalent counterions. The at- two lines at zero temperature (ξ = ) is almost ex- tractionisproducedbythecorrelationsinthecondensed ∞ actly the same as at finite temperature, as long as the layers of counterions surrounding each polyion and ex- average electrostatic energy of interaction between two tends uniformly from zero temperature. The attraction condensed counterions is greater than the thermal en- appears only if the number of condensed counterions ex- ergy,α2q2/Dd>k T,where d is the averageseparation ceeds the threshold, n > Z/2α. Using the counterion B between the condensed counterions. The mechanism for condensation theory to estimate the number of associ- attractionbetweentwolike-chargedrodsisnowclear[5]. ated counterions, we see that the attraction is possible Atzerotemperaturethecounterionsonthetworodswill only for polyelectrolytes with ξ > 2/α. This is the fun- take on staggered configuration, i.e. if the site of the damental result which has, evidently, gone unnoticed in first rod is occupied by a counterion the parallel site the previous studies of this interesting phenomena. of the second rod will stay vacant. If the number of We are grateful to Prof. J.A.C. Gallas for kindly pro- 4 vidingtimeonhisAlphastation. Thisworkwaspartially Bunsen-Ges. Phys. Chem. 100, 1 (1996); supported by the Brazilian agencies CNPq and CAPES. [3] B.-Y. Haand A.J. Liu, Phys. Rev. Lett. 79, 1289 (1997); [4] F. Oosawa, Biopolymers 6, 134 (1968); J. Ray and G.S. Manning, Langmuir 10, 2450 (1994); J. Barrat and J. Joanny Adv. 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