- Email: [email protected]
Interfaces in solutions of randomly charged rods
Physica A 259 (1998) 235–244
Interfaces in solutions of randomly charged rods B.-Y. Ha, Andrea J. Liu ∗ Department of Chemistry and Biochemistry, University of California, Los Angeles, CA 90095, USA Received 24 April 1998
Abstract We study the interfacial properties of systems containing like-charged rods with charge uctuations along their length. We show that it is possible to describe clusters of rods in terms of the sum of a bulk free energy and a surface free energy. It is not obvious that this is valid for electrostatic systems, since the interactions are long-ranged, and the e ective interactions c 1998 Elsevier Science B.V. All rights reserved. between rods are not pairwise-additive. PACS: 61.20.Qg; 61.25.Hq; 87.15.Da Keywords: Interfaces; Charge uctuations; Polyelectrolytes; Polyampholytes
Solutions of charged particles with associated counterions often tend to contain wellde ned clusters of the particles. In the case of sti charged polyelectrolyte chains, for example, multivalent counterions mediate attractive interactions between the likecharged chains, and lead to clustering of the chains [1,2] into bundles. The existence of structure at some mesoscopic length scale is characteristic of complex uids [3], and is often treated at a phenomenological level by considering the interplay of bulk and interfacial free energies. Although this approach has proved extremely useful for most complex uids, it is not clear that it is applicable to systems with clusters of charged chains, for several reasons. First, interfacial free energies are often calculated using the Landau square-gradient approach, with the interface assumed to be wide compared to the interaction range. In electrostatic systems, however, the interaction range can be quite long. Second, the e ective interactions between particles are usually assumed to be pairwise-additive. Unfortunately, however, we and others [4,5] have found that the e ective interactions between charged rods (mediated by counterions) are not pairwiseadditive at all. Thus, it is worthwhile to determine whether a bundle of rods can be described in terms of the sum of a bulk free energy and an interfacial tension. Similar concerns have been addressed in the literature on small-molecule ionic uids, using ∗
Corresponding author. Tel.: (310) 825-8266; fax: (310) 206-4038; e-mail: [email protected].
c 1998 Elsevier Science B.V. All rights reserved. 0378-4371/98/$19.00 Copyright PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 2 6 1 - 1
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Fig. 1. A bundle of 16 randomly charged rods, with a dimensionless charge qj (s) on each monomer s of each rod j. Each monomer carries a net charge q with a variance .
either the linearized [6–8] or the nonlinear [9,10] Poisson–Boltzmann equation. Most recently, the problem of treating nonuniform densities in electrolytes has been studied by Lee and Fisher [11] within Debye–Huckel theory augmented with ion pairing. In this paper, we show that the decomposition of the free energy into a bulk term and an interfacial term is indeed valid for large bundles of charged rods. We do this for a speci c geometry, where N rods are placed on a square lattice to form a square bundle (see Fig. 1a). The charge distribution along the length of each rod is assumed to be nonuniform. This nonuniformity could arise from dissociation of some of the charges due to pH, in the case of a polyacid or a polybase. Alternatively, the nonuniformity could arise from uctuations in the density of condensed counterions along the rod [12,13], or from charge uctuations in a polyampholyte (i.e. a chain that contains charged groups of both signs, like most proteins). We characterize the charge distribution by two quantities: the net charge on the rod, and the variance in the e ective charge of a monomer on the rod. We consider two di erent interfacial free energies. First, we show that the free energy of a bundle can be written in the large N
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237
limit as the sum of a bulk free energy and a surface free energy that is proportional to the area of the outer surface of the bundle. Second, we bring two square bundles together and consider the free energy of the interface that divides them. The two halves of the resulting rectangular bundle are allowed to have di erent properties; for example, they may have di erent net charges, or di erent variances in the local charge. We nd that the two halves attract each other in the large N limit unless the uctuations are vanishingly small, and that there is a well-de ned free energy for the interface between the two halves. Although our calculations apply to a highly arti cial geometry, they do serve to prove the basic point that interfacial tensions in our system are nonzero and nite. At rst glance, our results are quite surprising. Consider the case where one-half of the bundle has N rods that each carry a net charge qI ¡0, while the other half has N rods with a net charge of qII ¡0. Suppose that there is no added salt in the system. The total charges of the two halves are NqI and NqII , respectively. If we calculated the free energy by summing over all pairs of rods using the form of the two-rod interaction calculated for rods with uctuating charge distributions [14], we would obtain (1) a strong repulsion between the two halves of the bundle that increases as√ N 2 with increasing N ; and (2) an interfacial free energy that scales as N instead of N in the limit N → ∞. In √ other words, the interfacial free energy would not scale with the area of the interface, N . However, when we solve the N -rod problem explicitly, without assuming pairwise-additivity, we nd (1) that the two halves attract each other more strongly as N increases, unless the charge uctuations I and II are extremely small. We also nd (2) that there is a well-de ned interfacial free energy that scales with the area of the interface as N → ∞. The origin of our surprising results lies in the breakdown of pairwise additivity of the e ective interactions between rods. Basically, the long-ranged repulsion between rods is highly screened in the explicit calculation because charge uctuations can be correlated over all rods. If we assume pairwise additivity, the uctuations can only be correlated pair by pair, leading to much less screening. The model that we study is shown in Fig. 1, and consists of N rods parallel to the z-direction arranged on a square lattice with lattice constant a. The total length of each rod is L, which is assumed to be much larger than any other length scale in the problem. Each rod consists of M cylindrical monomers of length b, as shown in Fig. 1b. Each monomer s on rod j carries a random charge qj (s), in units of the elementary charge e. The system can be described by the following Hamiltonian: N
M
1 X X qi (s)qj (s0 ) ; H = ‘B 2 |ri (s) − rj (s0 )| 0 ij
(1)
ss
where = 1=kB T and ‘B = e2 =kB T is the Bjerrum length, namely the length scale at which the electrostatic energy is comparable to the thermal energy kB T . This Hamiltonian includes interactions between charges on the same rod, i = j, and between charges on di erent rods, i 6= j. To describe the charge uctuations along the
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rods, we assume that the uctuations obey a Gaussian distribution in the absence of electrostatic interactions: (qj (s) − qj )2 1 exp − : (2) W[qj (s)] = p 2j 2j = qj , and charges at di erent sites are iniThus, the average charge on rod j is hqj (s)iD W E tially assumed to be uncorrelated such that (qj (s) − qj )(qj0 (s0 ) − qj0 0 )
W
= j jj0 (s−
s0 ). Here, j is the variance in the charge per monomer. Note that we have made a continuum approximation in that qj (s) can assume continuous instead of discrete values. Each rod j is now characterized by two quantities: the mean charge qj and the variance j . In the numerical calculations that follow, we will restrict application of our results to the case of negatively charged rods with positive condensed counterions of valency Z. In that case, all the charge uctuations arise from the condensed counterions and we have qj = −fj0 +Zfjc , where −fj0 is the chemical charge per monomer on rod j and fjc is the number of condensed counterions per monomer on rod j. Note that qj ¡0 since not all counterions are condensed. The variance in the monomeric charge is j = Z 2 fjc . The partition function is then the sum over all realizations of the charge variables, weighted by the distribution of the variables and the Boltzmann factor: Z
(3) Z = e− H q = Dqj (s)W[qj (s)]e− H : It is convenient to introduce matrices 0Q and Q(k), de ned by the matrix elements Qij = j−1 ij + 2 ln(L=Rij ) ;
0
Qij (k z ) = j−1 ij + 2K0 (|k z |Rij ) ;
(4)
where ≡ ‘B =b is the Manning parameter, k z is the one-dimensional wavevector conjugate to the coordinate z along the length of the rods, and K0 (x) is the zeroth-order modi ed Bessel function of the second kind. To avoid singularities, we cut o the electrostatic interaction at length scales smaller than the monomer size b. Once we assume that the charge uctuations are Gaussian, we can solve for the free energy of the interacting system exactly. It is given by N
b 1X −1 0 −1 qi qj [j−1 ij − −1 Qij ] + F = i j 2 ij 2 N
−
1X ln j − 2 j
N X j
! j
Z∞ ‘B −∞
Z∞ −∞
dk z ln[det Q(k z )] 2
dk z K0 (|k z |d) : 2
(5)
This is a generalization of our earlier results for the case of qi = q and = [4]. The rst term arises from the net charge repulsion, which is screened due to the charge
uctuations whereas the second term describes charge- uctuation attractions. The last term is the self-energy which should be subtracted from the total free energy.
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The free energy in Eq. (5) holds for any system of parallel rods characterized by {qj } and {j } and separated by {Rij }. We will now consider the speci c case of a square bundle of rods on a square lattice with lattice constant a, where all rods have the same net charge q and the same variance . The free energy of this bundle √ can be written as the sum of a bulk free energy NFbulk , and a surface free energy 4 N Fsurface , where Fbulk is the free energy per rod of an in nite bundle (N → ∞). To calculate Fbulk , we consider an in nite lattice and take advantage of its periodicity to recast the problem in discrete Fourier space: " Fbulk (q; ) = lim
N →∞
# Z b X dk z q2 + ln[1 + 2K0 (|k z |; k⊥ )] ; 0K −1 + 2 2N 2 0 k (6)
where k⊥ is the wave vector conjugate to the vector r⊥ = (x; y) in the xy-plane. Since √ discrete values: k = (2= N) we have a periodic lattice in the xy-plane, k⊥ assumes ⊥ √ (nx ; ny ) where nx ; ny can assume the values 0; 1; : : : ; N − 1. We denote the discrete Fourier transform of K0 (|k z |Rij ) as √
K0 (|k z |; k⊥ ) ≡
N −1 X
K0 (|k z |a|j⊥ |) cos(k⊥ · j⊥ ) ;
(7)
jx jy =0
√ where j⊥ = ( jx ; jy ) and jx ; jy = 0; 1; : : : ; N − 1. Finally, the function 0K0 is simply 0 K0 ≡ K0 (|k z | = b=L; k⊥ = 0). In order for the bulk free energy to be well-de ned, the free energy per rod Fbulk must approach a constant as N → ∞. To establish this, we examine the behavior of K0 (|k z |a|j⊥ |) for large N . Recall that N sets the maximum value of j⊥ . But for large √ x; K0 (x) ∼ (1= x)e−x . It can therefore be shown that the sum K0 (|k z |; k⊥ ), de ned in Eq. (7), approaches a nite value as NR→ ∞. Furthermore, as N → ∞, we can replace P 2=a the sum 1=N k [: : :] with an integral, 0 (dk=(2)2 )[: : :], up to a correction of order 1=N . From these arguments, we conclude that the second term in Eq. (6) approaches a nonzero constant as N → ∞. It is clear from inspection that the rst term in Eq. (6) also approaches a constant for large N , with corrections of order 1=N . Thus, the bulk free energy per rod, Fbulk , approaches a constant in the limit N → ∞, as it should. In other words, the free energy of a lattice of rods is extensive. In order to determine whether there is a well-de ned surface free energy, we now calculate the free energy of the N -rod bundle explicitly, and subtract the contribution from the bulk free energy: √ 4 N Fsurface = Fbundle − NFbulk :
(8)
In order for the surface free energy to be well-de ned, we must show that Fsurface approaches a constant as N → ∞. Instead of directly subtracting Fbulk from Fbundle ,
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however, we de ne another free energy: √ 4 N Fdi = Fbundle − Fper ;
(9)
where Fper is the free energy of an in nite lattice of rods constructed by replicating the N -rod bundle periodically (i.e., the (N +1)th rod has the same charge con guration as the rst rod). For large N , we recover the bulk free energy: Fper ≈ NFbulk (1 + O(1=N )) :
(10)
In that limit, we therefore also nd Fdi → Fsurface . Thus, we can use Eq. (9) to calculate the surface free energy. To calculate Fbundle − Fper , we de ne a new matrix Qij (k z ) for the periodic system, using the same form as in Eq. (4). In other words, Qij (k z ) applies to the in nite, periodic system made by replicating the N -rod bundle, while Qij (k z ) applies to the N -rod bundle. The surface free energy Fdi can then be written in the succinct form √
4 N Fdi
b = 2
Z∞ −∞
dk z ln det[1 + Q−1 (k z ) · (Q(k z ) − Q(k z ))] : 2
(11)
Now note that Q → Q as N → ∞. For large N , we can therefore expand Fdi in powers of Q−1 · (Q − Q) and use the fact that Fdi = Fsurface in that limit. After subtracting the self-energy, we obtain X Z∞ 2 2 1 b dk z K0 (|k z |; k⊥ ) K0 (|k z |; k⊥ )kˆ ⊥ · ∇k ; Fsurface = N 2 2 1 + 2K0 (|k z |; k⊥ ) k −∞
(12) where kˆ ⊥ = k⊥ =|k⊥ |. Following arguments similar to those below Eq. (7), we use the fact that K0 (|k z |; k⊥ ) approaches a nite value as N → ∞ to show that Fsurface also approaches an N -independent value in that limit. This shows that the surface energy of the bundle is well de ned and that the bundle free energy can indeed be written as the sum of a bulk free energy and a surface free energy for a suciently large bundle. Note that Fsurface in Eq. (12) depends on the charge uctuations, , but not on the net charge q. The charge uctuations control the attractive interactions, while the net charge controls the repulsive interactions. Thus, Fsurface originates from the attractive interactions, which are highly screened by the condensed counterions and are very short-ranged (of order Z 2 ‘B ). It is instructive to study Fsurface in the continuum limit where the lattice spacing a and the monomer size b both vanish. In that limit, we nd Fsurface ∼ 4e2 =ba2 . If we rewrite Fsurface in the form kB T=D2 , where D is then an e ective interaction range, we nd D−2 = 4‘B Z 2 n, where n = fc =ba2 is the number density of condensed counterions. In other words, D is the Debye length corresponding to an ion density given by the number of condensed counterions per unit volume [15]. This is an intuitively reasonable result.
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Fig. 2. Top view of a rectangular bundle consisting of two square bundles characterized by di erent values of net charge (qI and qII ) and variance (I and II ). White circles correspond to rods in bundle I while grey circles correspond to rods in bundle II .
The second interfacial free energy that we will consider corresponds to an interface that separates two bundles that have di erent properties. We will assume that rods in the left-half of the combined bundle, corresponding to 16j6N , have net charge qj = qI and variance j = I . Rods in the right half of the combined bundle, corresponding to N + 16j62N , have net charge qj = qII and variance j = II , as shown in Fig. 2. The total free energy can be written as the sum √ (13) F = NFI + NFII + N FI; II ; where FI is the free energy of an isolated half-bundle with net charge qj = qI and variance j = I . The cross-term FI; II describes the coupling between the two halves of the bundle. In the large N limit, the free energies FI and FII are given by Fbulk (qI ; I ) and Fbulk (qII ; II ), respectively, where the function Fbulk (q; ) is given in Eq. (6). We turn now to the coupling term FI; II . This can be calculated by computing explicitly the entire free energy of the bundle, F, and then subtracting NFI and NFII , as in Eq. (13). To calculate F, it is useful to partition the matrix Q(k) in Eq. (4) into 4 sub-matrices as follows: ! QI; II (k) QI; I (k) ; (14) Q(k) = QI;T II (k) QII; II (k) where Q; (k) refers to the isolated th half-bundle which does not interact with the other half-bundle. The interaction between the two halves is taken into account by QI; II and its transpose, QI;T II . att Like the bulk free energy, FI; II consists of two pieces, FI;rep II and FI; II . The rst term, rep FI; II , is positive and arises from the screened repulsions between the negatively charged rods. The second term, FI;attII , is negative and arises from attractions due to correlated
uctuations on the rods. We rst consider the repulsive contribution. Note that the repulsive interaction between two halves with di erent values of q and is always
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B.-Y. Ha, A.J. Liu / Physica A 259 (1998) 235–244
smaller than the repulsion between identical halves that have q and given by their arithmetical means: q = 12 (qI + qII ) and = 12 (I + II ). Thus we can show that √ 2 N q2 (0K00 − 0K0 ) rep (15) FI; II ¡ 0 (1 + 2 0K00 )(1 + 2 K0 ) √ where 0K00 is the same as 0K0 except that jx now runs from 0 to 2 N√− 1. We now observe that both 0K0 and 0K00 scale as O(N ), so that FI;rep II scales as 1= N as N → ∞. Thus, this contribution to the interfacial free energy per rod vanishes in the large N limit and can be dropped. The attractive contribution, FI;attII , can be conveniently written in terms of the submatrices in Eq. (14) as FI;attII
b = √ 2 N
Z∞ −∞
dk z −1 −1 T ln det[1 − QII; II (k z )QI; II (k z )QI; I (k z )QI; II (k z )] : 2
(16)
This contribution to the interfacial free energy can be neglected if the largest eigenvalues of QI; I and QII; II diverge as N → ∞. On the other hand, if all the eigenvalues of these √ matrices are nite and nonzero as N → ∞, then the free energy FI;attII should scale as N in the large N limit. To gain some insight into the scaling of FI;att II with N , we have calculated the determinant in Eq. (16) numerically for the special case of qI = qII and I = II . To carry out the numerical calculations, we have chosen the parameters T = 300 K, = 80; a = 20 A In addition, each rod is assumed to consist of M = 105 monomers. The and b = 1:7 A. mean charge and variance taken to be q = 0:120 and = 1:761. Our numerical results att for FI;att II are given in Fig. 3, and demonstrate that FI; II approaches a constant at large −1 N . This suggests that the largest eigenvalues of QII; II (k z )QII; I (k z )QI;−1 I (k z )QI; II (k z ) must √ −1 −1 scale as 1= N . Since QII; II (k z )QII; I (k z )QI; I (k z )QI; II (k z ) is small, we can therefore expand the free energy in powers of this product of matrices. As N becomes very large, we nd Z∞ 2 I II (K0I; II (|k z |; k⊥ ))2 b X dk z ; (17) FI; II ∼ − N 2 (1 + 2I K0 (|k z |; k⊥ ))(1 + 2II K0 (|k z |; k⊥ )) k
K0I; II
0
arises due to the coupling between the of the bundle, and is the where √ two halves P same as K0 except that jx now runs from 1 to N . Since k [: : :] ∼ N , the coupling term approaches a negative constant, in √ accord with our numerical results. This proves that the cost of the interface scales as N . In addition, the total free energy cost is negative, implying that two large systems always prefer to aggregate to form a larger bundle. In the limit where the rods are very close together (a → 0) and the monomer size vanishes (b → 0), the charge uctuations can explore the entire volume of the bundle. If we set the uctuations to have the same size in the two halves √ of the bundle (I = II ) then we nd that the free energy per unit volume, fI; II = N FI; II =V , reduces
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243
att (de ned in Eq. (16) to the interfacial free energy per monomer, as a function Fig. 3. The contribution FI;II of the number of rods (2N ) in the bundle. We nd that this contribution approaches a constant value as N diverges. This motivates the expansion used in Eq. (17). The parameters used for the numerical calculations and the lattice spacing of rods in the bundle is are T = 300 K and = 80. The monomer size is b = 1:7 A a = 20. The number of monomers per rod is M = 105 . The average charge per monomer is q = 0:120 and the variance in the monomer charge is = 1:761.
to fI;II ∼ − 3 . Here, 2 ≡ 4Z 2 fc = 4‘B Z 2 n where n is the concentration of charges. This is the usual form of the Debye screening parameter, and our result is equivalent to the Debye–Huckel limiting law fDH ∼ − 3 [15]. This is not a coincidence, since fI; II arises from charge uctuations that extend across the interface. These are precisely the uctuations that give rise to screening in the Debye–Huckel case. In summary, we have shown that it is valid to describe the free energy of a large cluster of charged rods as the sum of a bulk and surface term. We obtained these results by explicit calculation of the N -rod system. If we had assumed that the e ective interactions between rods are pairwise additive, then we would have found that it is not possible to separate the free energy into the sum of bulk and surface terms. Thus, our result underlines the importance of non-pairwise-additive, many-body interactions in charged systems. We gratefully acknowledge the support of the National Science Foundation through grant DMR-9619277. References [1] V.A. Bloom eld, Biopolymers 31 (1991) 1471; V.A. Bloom eld, Curr. Opinion in Struc. Biol. 6 (1996) 334 and references therein. [2] J.X. Tang, S. Wong, P. Tran, P. Janmey, Ber. Buns. Phys. Chem. 100 (1996) 1; J.X. Tang, T. Ito, T. Tao, P. Traub, P.A. Janmey, Biochemistry 36 (1997) 12 600 and references therein.
244 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
B.-Y. Ha, A.J. Liu / Physica A 259 (1998) 235–244 W.M. Gelbart, A. Ben-Shaul, J. Phys. Chem. 100 (1996) 13 169. B.-Y. Ha, A.J. Liu, Phys. Rev. Lett. (in press). R. Podgornik, V.A. Parsegian, Phys. Rev. Lett. 80 (1998) 1560. L. Onsager, N.N. Nicholas, J. Chem. Phys. 2 (1934) 258. J. Groenwold, J. Chem. Phys. 107 (1997) 9668. B. Duplantier, R.E. Goldstein, V. Romero-Rochin, A.I. Pesci, Phys. Rev. Lett. 65 (1990) 508. H.W.N. Lekkerkerker, Physica A 167 (1990) 184. D.J. Mitchel, B.W. Ninham, D.F. Evans, J. Phys. Chem. 88 (1985) 6344. B.P. Lee, M.E. Fisher, Phys. Rev. Lett. 76 (1996) 2906. G.S. Manning, J. Chem. Phys. 51 (1969) 954. F. Oosawa, Biopolymers 6 (1968) 134; Polyelectrolytes, Marcel Dekker, New York, 1971. B.-Y. Ha, A.J. Liu, Phys. Rev. Lett. 79 (1997) 1289. See, for example, L.D. Landau, E.M. Lifshitz, Statistical Physics, Part 1, Pergamon Press, Oxford, 1980, pp. 239–242.
Interfaces in solutions of randomly charged rods B.-Y. Ha, Andrea J. Liu ∗ Department of Chemistry and Biochemistry, University of California, Los Angeles, CA 90095, USA Received 24 April 1998
Abstract We study the interfacial properties of systems containing like-charged rods with charge uctuations along their length. We show that it is possible to describe clusters of rods in terms of the sum of a bulk free energy and a surface free energy. It is not obvious that this is valid for electrostatic systems, since the interactions are long-ranged, and the e ective interactions c 1998 Elsevier Science B.V. All rights reserved. between rods are not pairwise-additive. PACS: 61.20.Qg; 61.25.Hq; 87.15.Da Keywords: Interfaces; Charge uctuations; Polyelectrolytes; Polyampholytes
Solutions of charged particles with associated counterions often tend to contain wellde ned clusters of the particles. In the case of sti charged polyelectrolyte chains, for example, multivalent counterions mediate attractive interactions between the likecharged chains, and lead to clustering of the chains [1,2] into bundles. The existence of structure at some mesoscopic length scale is characteristic of complex uids [3], and is often treated at a phenomenological level by considering the interplay of bulk and interfacial free energies. Although this approach has proved extremely useful for most complex uids, it is not clear that it is applicable to systems with clusters of charged chains, for several reasons. First, interfacial free energies are often calculated using the Landau square-gradient approach, with the interface assumed to be wide compared to the interaction range. In electrostatic systems, however, the interaction range can be quite long. Second, the e ective interactions between particles are usually assumed to be pairwise-additive. Unfortunately, however, we and others [4,5] have found that the e ective interactions between charged rods (mediated by counterions) are not pairwiseadditive at all. Thus, it is worthwhile to determine whether a bundle of rods can be described in terms of the sum of a bulk free energy and an interfacial tension. Similar concerns have been addressed in the literature on small-molecule ionic uids, using ∗
Corresponding author. Tel.: (310) 825-8266; fax: (310) 206-4038; e-mail: [email protected].
c 1998 Elsevier Science B.V. All rights reserved. 0378-4371/98/$19.00 Copyright PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 2 6 1 - 1
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B.-Y. Ha, A.J. Liu / Physica A 259 (1998) 235–244
Fig. 1. A bundle of 16 randomly charged rods, with a dimensionless charge qj (s) on each monomer s of each rod j. Each monomer carries a net charge q with a variance .
either the linearized [6–8] or the nonlinear [9,10] Poisson–Boltzmann equation. Most recently, the problem of treating nonuniform densities in electrolytes has been studied by Lee and Fisher [11] within Debye–Huckel theory augmented with ion pairing. In this paper, we show that the decomposition of the free energy into a bulk term and an interfacial term is indeed valid for large bundles of charged rods. We do this for a speci c geometry, where N rods are placed on a square lattice to form a square bundle (see Fig. 1a). The charge distribution along the length of each rod is assumed to be nonuniform. This nonuniformity could arise from dissociation of some of the charges due to pH, in the case of a polyacid or a polybase. Alternatively, the nonuniformity could arise from uctuations in the density of condensed counterions along the rod [12,13], or from charge uctuations in a polyampholyte (i.e. a chain that contains charged groups of both signs, like most proteins). We characterize the charge distribution by two quantities: the net charge on the rod, and the variance in the e ective charge of a monomer on the rod. We consider two di erent interfacial free energies. First, we show that the free energy of a bundle can be written in the large N
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237
limit as the sum of a bulk free energy and a surface free energy that is proportional to the area of the outer surface of the bundle. Second, we bring two square bundles together and consider the free energy of the interface that divides them. The two halves of the resulting rectangular bundle are allowed to have di erent properties; for example, they may have di erent net charges, or di erent variances in the local charge. We nd that the two halves attract each other in the large N limit unless the uctuations are vanishingly small, and that there is a well-de ned free energy for the interface between the two halves. Although our calculations apply to a highly arti cial geometry, they do serve to prove the basic point that interfacial tensions in our system are nonzero and nite. At rst glance, our results are quite surprising. Consider the case where one-half of the bundle has N rods that each carry a net charge qI ¡0, while the other half has N rods with a net charge of qII ¡0. Suppose that there is no added salt in the system. The total charges of the two halves are NqI and NqII , respectively. If we calculated the free energy by summing over all pairs of rods using the form of the two-rod interaction calculated for rods with uctuating charge distributions [14], we would obtain (1) a strong repulsion between the two halves of the bundle that increases as√ N 2 with increasing N ; and (2) an interfacial free energy that scales as N instead of N in the limit N → ∞. In √ other words, the interfacial free energy would not scale with the area of the interface, N . However, when we solve the N -rod problem explicitly, without assuming pairwise-additivity, we nd (1) that the two halves attract each other more strongly as N increases, unless the charge uctuations I and II are extremely small. We also nd (2) that there is a well-de ned interfacial free energy that scales with the area of the interface as N → ∞. The origin of our surprising results lies in the breakdown of pairwise additivity of the e ective interactions between rods. Basically, the long-ranged repulsion between rods is highly screened in the explicit calculation because charge uctuations can be correlated over all rods. If we assume pairwise additivity, the uctuations can only be correlated pair by pair, leading to much less screening. The model that we study is shown in Fig. 1, and consists of N rods parallel to the z-direction arranged on a square lattice with lattice constant a. The total length of each rod is L, which is assumed to be much larger than any other length scale in the problem. Each rod consists of M cylindrical monomers of length b, as shown in Fig. 1b. Each monomer s on rod j carries a random charge qj (s), in units of the elementary charge e. The system can be described by the following Hamiltonian: N
M
1 X X qi (s)qj (s0 ) ; H = ‘B 2 |ri (s) − rj (s0 )| 0 ij
(1)
ss
where = 1=kB T and ‘B = e2 =kB T is the Bjerrum length, namely the length scale at which the electrostatic energy is comparable to the thermal energy kB T . This Hamiltonian includes interactions between charges on the same rod, i = j, and between charges on di erent rods, i 6= j. To describe the charge uctuations along the
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rods, we assume that the uctuations obey a Gaussian distribution in the absence of electrostatic interactions: (qj (s) − qj )2 1 exp − : (2) W[qj (s)] = p 2j 2j = qj , and charges at di erent sites are iniThus, the average charge on rod j is hqj (s)iD W E tially assumed to be uncorrelated such that (qj (s) − qj )(qj0 (s0 ) − qj0 0 )
W
= j jj0 (s−
s0 ). Here, j is the variance in the charge per monomer. Note that we have made a continuum approximation in that qj (s) can assume continuous instead of discrete values. Each rod j is now characterized by two quantities: the mean charge qj and the variance j . In the numerical calculations that follow, we will restrict application of our results to the case of negatively charged rods with positive condensed counterions of valency Z. In that case, all the charge uctuations arise from the condensed counterions and we have qj = −fj0 +Zfjc , where −fj0 is the chemical charge per monomer on rod j and fjc is the number of condensed counterions per monomer on rod j. Note that qj ¡0 since not all counterions are condensed. The variance in the monomeric charge is j = Z 2 fjc . The partition function is then the sum over all realizations of the charge variables, weighted by the distribution of the variables and the Boltzmann factor: Z
(3) Z = e− H q = Dqj (s)W[qj (s)]e− H : It is convenient to introduce matrices 0Q and Q(k), de ned by the matrix elements Qij = j−1 ij + 2 ln(L=Rij ) ;
0
Qij (k z ) = j−1 ij + 2K0 (|k z |Rij ) ;
(4)
where ≡ ‘B =b is the Manning parameter, k z is the one-dimensional wavevector conjugate to the coordinate z along the length of the rods, and K0 (x) is the zeroth-order modi ed Bessel function of the second kind. To avoid singularities, we cut o the electrostatic interaction at length scales smaller than the monomer size b. Once we assume that the charge uctuations are Gaussian, we can solve for the free energy of the interacting system exactly. It is given by N
b 1X −1 0 −1 qi qj [j−1 ij − −1 Qij ] + F = i j 2 ij 2 N
−
1X ln j − 2 j
N X j
! j
Z∞ ‘B −∞
Z∞ −∞
dk z ln[det Q(k z )] 2
dk z K0 (|k z |d) : 2
(5)
This is a generalization of our earlier results for the case of qi = q and = [4]. The rst term arises from the net charge repulsion, which is screened due to the charge
uctuations whereas the second term describes charge- uctuation attractions. The last term is the self-energy which should be subtracted from the total free energy.
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The free energy in Eq. (5) holds for any system of parallel rods characterized by {qj } and {j } and separated by {Rij }. We will now consider the speci c case of a square bundle of rods on a square lattice with lattice constant a, where all rods have the same net charge q and the same variance . The free energy of this bundle √ can be written as the sum of a bulk free energy NFbulk , and a surface free energy 4 N Fsurface , where Fbulk is the free energy per rod of an in nite bundle (N → ∞). To calculate Fbulk , we consider an in nite lattice and take advantage of its periodicity to recast the problem in discrete Fourier space: " Fbulk (q; ) = lim
N →∞
# Z b X dk z q2 + ln[1 + 2K0 (|k z |; k⊥ )] ; 0K −1 + 2 2N 2 0 k (6)
where k⊥ is the wave vector conjugate to the vector r⊥ = (x; y) in the xy-plane. Since √ discrete values: k = (2= N) we have a periodic lattice in the xy-plane, k⊥ assumes ⊥ √ (nx ; ny ) where nx ; ny can assume the values 0; 1; : : : ; N − 1. We denote the discrete Fourier transform of K0 (|k z |Rij ) as √
K0 (|k z |; k⊥ ) ≡
N −1 X
K0 (|k z |a|j⊥ |) cos(k⊥ · j⊥ ) ;
(7)
jx jy =0
√ where j⊥ = ( jx ; jy ) and jx ; jy = 0; 1; : : : ; N − 1. Finally, the function 0K0 is simply 0 K0 ≡ K0 (|k z | = b=L; k⊥ = 0). In order for the bulk free energy to be well-de ned, the free energy per rod Fbulk must approach a constant as N → ∞. To establish this, we examine the behavior of K0 (|k z |a|j⊥ |) for large N . Recall that N sets the maximum value of j⊥ . But for large √ x; K0 (x) ∼ (1= x)e−x . It can therefore be shown that the sum K0 (|k z |; k⊥ ), de ned in Eq. (7), approaches a nite value as NR→ ∞. Furthermore, as N → ∞, we can replace P 2=a the sum 1=N k [: : :] with an integral, 0 (dk=(2)2 )[: : :], up to a correction of order 1=N . From these arguments, we conclude that the second term in Eq. (6) approaches a nonzero constant as N → ∞. It is clear from inspection that the rst term in Eq. (6) also approaches a constant for large N , with corrections of order 1=N . Thus, the bulk free energy per rod, Fbulk , approaches a constant in the limit N → ∞, as it should. In other words, the free energy of a lattice of rods is extensive. In order to determine whether there is a well-de ned surface free energy, we now calculate the free energy of the N -rod bundle explicitly, and subtract the contribution from the bulk free energy: √ 4 N Fsurface = Fbundle − NFbulk :
(8)
In order for the surface free energy to be well-de ned, we must show that Fsurface approaches a constant as N → ∞. Instead of directly subtracting Fbulk from Fbundle ,
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however, we de ne another free energy: √ 4 N Fdi = Fbundle − Fper ;
(9)
where Fper is the free energy of an in nite lattice of rods constructed by replicating the N -rod bundle periodically (i.e., the (N +1)th rod has the same charge con guration as the rst rod). For large N , we recover the bulk free energy: Fper ≈ NFbulk (1 + O(1=N )) :
(10)
In that limit, we therefore also nd Fdi → Fsurface . Thus, we can use Eq. (9) to calculate the surface free energy. To calculate Fbundle − Fper , we de ne a new matrix Qij (k z ) for the periodic system, using the same form as in Eq. (4). In other words, Qij (k z ) applies to the in nite, periodic system made by replicating the N -rod bundle, while Qij (k z ) applies to the N -rod bundle. The surface free energy Fdi can then be written in the succinct form √
4 N Fdi
b = 2
Z∞ −∞
dk z ln det[1 + Q−1 (k z ) · (Q(k z ) − Q(k z ))] : 2
(11)
Now note that Q → Q as N → ∞. For large N , we can therefore expand Fdi in powers of Q−1 · (Q − Q) and use the fact that Fdi = Fsurface in that limit. After subtracting the self-energy, we obtain X Z∞ 2 2 1 b dk z K0 (|k z |; k⊥ ) K0 (|k z |; k⊥ )kˆ ⊥ · ∇k ; Fsurface = N 2 2 1 + 2K0 (|k z |; k⊥ ) k −∞
(12) where kˆ ⊥ = k⊥ =|k⊥ |. Following arguments similar to those below Eq. (7), we use the fact that K0 (|k z |; k⊥ ) approaches a nite value as N → ∞ to show that Fsurface also approaches an N -independent value in that limit. This shows that the surface energy of the bundle is well de ned and that the bundle free energy can indeed be written as the sum of a bulk free energy and a surface free energy for a suciently large bundle. Note that Fsurface in Eq. (12) depends on the charge uctuations, , but not on the net charge q. The charge uctuations control the attractive interactions, while the net charge controls the repulsive interactions. Thus, Fsurface originates from the attractive interactions, which are highly screened by the condensed counterions and are very short-ranged (of order Z 2 ‘B ). It is instructive to study Fsurface in the continuum limit where the lattice spacing a and the monomer size b both vanish. In that limit, we nd Fsurface ∼ 4e2 =ba2 . If we rewrite Fsurface in the form kB T=D2 , where D is then an e ective interaction range, we nd D−2 = 4‘B Z 2 n, where n = fc =ba2 is the number density of condensed counterions. In other words, D is the Debye length corresponding to an ion density given by the number of condensed counterions per unit volume [15]. This is an intuitively reasonable result.
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Fig. 2. Top view of a rectangular bundle consisting of two square bundles characterized by di erent values of net charge (qI and qII ) and variance (I and II ). White circles correspond to rods in bundle I while grey circles correspond to rods in bundle II .
The second interfacial free energy that we will consider corresponds to an interface that separates two bundles that have di erent properties. We will assume that rods in the left-half of the combined bundle, corresponding to 16j6N , have net charge qj = qI and variance j = I . Rods in the right half of the combined bundle, corresponding to N + 16j62N , have net charge qj = qII and variance j = II , as shown in Fig. 2. The total free energy can be written as the sum √ (13) F = NFI + NFII + N FI; II ; where FI is the free energy of an isolated half-bundle with net charge qj = qI and variance j = I . The cross-term FI; II describes the coupling between the two halves of the bundle. In the large N limit, the free energies FI and FII are given by Fbulk (qI ; I ) and Fbulk (qII ; II ), respectively, where the function Fbulk (q; ) is given in Eq. (6). We turn now to the coupling term FI; II . This can be calculated by computing explicitly the entire free energy of the bundle, F, and then subtracting NFI and NFII , as in Eq. (13). To calculate F, it is useful to partition the matrix Q(k) in Eq. (4) into 4 sub-matrices as follows: ! QI; II (k) QI; I (k) ; (14) Q(k) = QI;T II (k) QII; II (k) where Q; (k) refers to the isolated th half-bundle which does not interact with the other half-bundle. The interaction between the two halves is taken into account by QI; II and its transpose, QI;T II . att Like the bulk free energy, FI; II consists of two pieces, FI;rep II and FI; II . The rst term, rep FI; II , is positive and arises from the screened repulsions between the negatively charged rods. The second term, FI;attII , is negative and arises from attractions due to correlated
uctuations on the rods. We rst consider the repulsive contribution. Note that the repulsive interaction between two halves with di erent values of q and is always
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smaller than the repulsion between identical halves that have q and given by their arithmetical means: q = 12 (qI + qII ) and = 12 (I + II ). Thus we can show that √ 2 N q2 (0K00 − 0K0 ) rep (15) FI; II ¡ 0 (1 + 2 0K00 )(1 + 2 K0 ) √ where 0K00 is the same as 0K0 except that jx now runs from 0 to 2 N√− 1. We now observe that both 0K0 and 0K00 scale as O(N ), so that FI;rep II scales as 1= N as N → ∞. Thus, this contribution to the interfacial free energy per rod vanishes in the large N limit and can be dropped. The attractive contribution, FI;attII , can be conveniently written in terms of the submatrices in Eq. (14) as FI;attII
b = √ 2 N
Z∞ −∞
dk z −1 −1 T ln det[1 − QII; II (k z )QI; II (k z )QI; I (k z )QI; II (k z )] : 2
(16)
This contribution to the interfacial free energy can be neglected if the largest eigenvalues of QI; I and QII; II diverge as N → ∞. On the other hand, if all the eigenvalues of these √ matrices are nite and nonzero as N → ∞, then the free energy FI;attII should scale as N in the large N limit. To gain some insight into the scaling of FI;att II with N , we have calculated the determinant in Eq. (16) numerically for the special case of qI = qII and I = II . To carry out the numerical calculations, we have chosen the parameters T = 300 K, = 80; a = 20 A In addition, each rod is assumed to consist of M = 105 monomers. The and b = 1:7 A. mean charge and variance taken to be q = 0:120 and = 1:761. Our numerical results att for FI;att II are given in Fig. 3, and demonstrate that FI; II approaches a constant at large −1 N . This suggests that the largest eigenvalues of QII; II (k z )QII; I (k z )QI;−1 I (k z )QI; II (k z ) must √ −1 −1 scale as 1= N . Since QII; II (k z )QII; I (k z )QI; I (k z )QI; II (k z ) is small, we can therefore expand the free energy in powers of this product of matrices. As N becomes very large, we nd Z∞ 2 I II (K0I; II (|k z |; k⊥ ))2 b X dk z ; (17) FI; II ∼ − N 2 (1 + 2I K0 (|k z |; k⊥ ))(1 + 2II K0 (|k z |; k⊥ )) k
K0I; II
0
arises due to the coupling between the of the bundle, and is the where √ two halves P same as K0 except that jx now runs from 1 to N . Since k [: : :] ∼ N , the coupling term approaches a negative constant, in √ accord with our numerical results. This proves that the cost of the interface scales as N . In addition, the total free energy cost is negative, implying that two large systems always prefer to aggregate to form a larger bundle. In the limit where the rods are very close together (a → 0) and the monomer size vanishes (b → 0), the charge uctuations can explore the entire volume of the bundle. If we set the uctuations to have the same size in the two halves √ of the bundle (I = II ) then we nd that the free energy per unit volume, fI; II = N FI; II =V , reduces
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att (de ned in Eq. (16) to the interfacial free energy per monomer, as a function Fig. 3. The contribution FI;II of the number of rods (2N ) in the bundle. We nd that this contribution approaches a constant value as N diverges. This motivates the expansion used in Eq. (17). The parameters used for the numerical calculations and the lattice spacing of rods in the bundle is are T = 300 K and = 80. The monomer size is b = 1:7 A a = 20. The number of monomers per rod is M = 105 . The average charge per monomer is q = 0:120 and the variance in the monomer charge is = 1:761.
to fI;II ∼ − 3 . Here, 2 ≡ 4Z 2 fc = 4‘B Z 2 n where n is the concentration of charges. This is the usual form of the Debye screening parameter, and our result is equivalent to the Debye–Huckel limiting law fDH ∼ − 3 [15]. This is not a coincidence, since fI; II arises from charge uctuations that extend across the interface. These are precisely the uctuations that give rise to screening in the Debye–Huckel case. In summary, we have shown that it is valid to describe the free energy of a large cluster of charged rods as the sum of a bulk and surface term. We obtained these results by explicit calculation of the N -rod system. If we had assumed that the e ective interactions between rods are pairwise additive, then we would have found that it is not possible to separate the free energy into the sum of bulk and surface terms. Thus, our result underlines the importance of non-pairwise-additive, many-body interactions in charged systems. We gratefully acknowledge the support of the National Science Foundation through grant DMR-9619277. References [1] V.A. Bloom eld, Biopolymers 31 (1991) 1471; V.A. Bloom eld, Curr. Opinion in Struc. Biol. 6 (1996) 334 and references therein. [2] J.X. Tang, S. Wong, P. Tran, P. Janmey, Ber. Buns. Phys. Chem. 100 (1996) 1; J.X. Tang, T. Ito, T. Tao, P. Traub, P.A. Janmey, Biochemistry 36 (1997) 12 600 and references therein.
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