Selfassembly of spherical interpolyelectrolyte complexes from oppositely charged polymers
Vladimir A. Baulin, and Emmanuel Trizac
Received Xth XXXXXXXXXX 20XX, Accepted
Xth XXXXXXXXX 20XX
First published on the web Xth XXXXXXXXXX 200X
DOI: 10.1039/b000000x
The formation of interpolyelectrolyte complexes from the association of oppositely charged polymers in an electrolyte is studied. The charged polymers are linear oppositely charged polyelectrolytes, with possibly a neutral block. This leads to complexes with a charged core, and a more dilute corona of dangling chains, or of loops (flowerlike structure). The equilibrium aggregation number of the complexes (number of polycations and polyanions ) is determined by minimizing the relevant free energy functional, the Coulombic contribution of which is worked out within PoissonBoltzmann theory. The complexes can be viewed as colloids that are permeable to microionic species, including salt. We find that the complexation process can be highly specific, giving rise to very localized size distribution in composition space ().
1 Introduction
Electrostatic interactions are instrumental in determining the structure and function of living organisms, biopolymers and drug delivery systems. Charged macromolecules can selfassemble and aggregate into compact intermolecular complexes. This ability of oppositely charged polymers to form finite size complexes determines their biological function, which for example is important in gene transfection and compactization of DNA ^{1, 2, 3}, that provide promising alternatives to viral vectors ^{4}. Such macromolecular systems, where electrostatic forces are usually stronger than van der Waals or hydrogen bonds, exhibit rich behavior and structural variability. The structures formed by opposite charges are usually more stable than neutral block copolymers micelles dissociating upon dilution or slight change in the external conditions. The concept of stabilization of intermolecular complexes by interaction of oppositely charged polymers is realized in interpolyelectrolyte or polyion complexes (PIC) and polyion complex micelles (PIC micelles) that can be used for drug delivery ^{5, 6, 7}. High stability of PICs opens the possibility to use them as functional devices where the responsiveness to external stimuli can be connected with a function, e.g. recognition at the molecular level ^{8}, pHsensitive switching devices ^{9} or drug delivery carriers transporting charged objects through the cell membrane ^{10, 7}.
Polyion complexes have enhanced ability to undergo structural changes subject to external conditions, compared to neutral block copolymer assemblies. In addition to the response in change of temperature and solvent quality ^{11}, the structure of the charged complexes can be very sensitive to changes in salt concentration ^{12, 13, 14, 15}, pH ^{15, 9, 16, 3}, charge ratio ^{17, 18}, addition of ions ^{19}, or mixing ratio ^{2, 20}. Tuning the molecular architecture and global properties of PICs would allow for precise control of their functional properties. Understanding the physics and fundamental features of the selfassembly of PICs is thus a challenging task. Oppositely charged polymers in symmetric solutions can precipitate into uniform macroscopic phase of polymers and ions ^{21, 22}. The physics of aggregated chains of opposite charge in precipitate is somehow similar to polyampholites, polymers containing both positive and negative charges dispersed along the chain ^{23}. However, if the distribution of charges along the chain is not random ^{24} or one of the charged polymers is a diblock copolymer with a neutral block ^{8, 3}, oppositely charged polymers can form finite size complexes composed of a dense polyelectrolyte core and a swollen corona which protects the cores from aggregation by steric repulsion.
In this paper, we explore electrostatic properties and equilibrium structures of spherical complexes formed by oppositely charged polymers. The stability of finite size aggregates results from the balance of the electrostatic attraction between opposite charges in the core of the complexes and the steric repulsion of backbone segments forming a corona around the core. The description of such complexes is similar to polyelectrolyte micellization ^{25, 26, 27, 28}, combined with thermodynamics of aggregation in bidisperse solution ^{29, 30, 31, 32}. The steric repulsion in the corona should be strong enough to stabilize the complexes of finite size. This is possible when the hydrophilic blocks forming the corona of the complexes are long enough to oppose the electrostatic attraction in the core. In addition, the bare charge of the complexes can be screened by the ions of salt and counterions, thus changing the electrostatic forces and affecting the equilibrium properties of the complexes. The interplay of those effects will be studied considering two geometries, as sketched in Fig. 1:

(case A) the building blocks are a linear uniformly charged polyelectrolyte and a diblock copolymer composed of a neutral block and a charged block of opposite charge. The charged blocks aggregate with the linear polyelectrolytes to form a complex with a core surrounded by a corona of neutral segments. This case is that of a “hairy” and neutral surrounding outside the charged core.

(case B) the building blocks are two linear polyelectrolytes of opposite charge with a large asymmetry of the distances between the charges. The core of such complexes is composed of charged blocks of both signs, and is surrounded by the corona of loops of the segments between the charges. The corona can be neutral (segments between neighboring charges along the chain) or slightly charged (tails or longer segments between distant charges). Compared to case A, the dangling “hair” are replaced by loops, that may bear an electric charge.
The paper is organized as follows. In section 2, we first consider case A, with a charged core decorated by neutral dangling hair. The electrostatics of the complexes is taken into account through the full PoissonBoltzmann (PB) equation which is solved numerically and compared with the analytical expression of the linearized DebyeHückel (DH) equation. Particular attention will be paid to the counterion uptake, where a significant quantity of charge can be “trapped” inside the core, thereby reducing the electric field created outside the complex. A second mechanism for charge reduction is ascribable to the nonlinearity of PoissonBoltzmann framework: nonlinear screening effectively modifies the total core charge, leading in general to a reduced effective (or renormalized) quantity seen from a large distance^{33}. At this level of description, the dangling hair are not taken into account. The more complex situation where the charged core is surrounded by charged loops (case B) will be addressed in section 3. In turn, these results will be used in section 4 to discuss the complexation behaviour of oppositely charged polymers. Conclusions will finally be drawn in section 5. An appendix summarizes the main notations employed.
2 Charged core surrounded by neutral corona (case A)
2.1 The model and its three relevant charges
The simplest structure of a thermodynamically stable polyion complexes of finite size is a spherical core, containing all bare charges, surrounded by a neutral corona (Figure 1A). Such a complex can be formed, for example, by diblock copolymers containing neutral blocks ^{7, 3, 8, 6, 34, 24, 17, 35}. The electrostatic interactions between oppositely charged polyelectrolytes drive the formation of a dense core which is stabilized by the steric repulsion of neutral blocks forming swollen corona around the core. Even in such simple geometry, it is possible to tune the structure of the complex. Its size can be controlled by the lengths of the blocks, the density of charges, pH, the charge asymmetry, the salt concentration and solution properties.
Assuming that the linear polyelectrolyte is positively charged while the blocks of the diblock copolymer bear a negative charge, a linear polyelectrolyte is described by the number of charges on the chain , and the distance between the charges while the block copolymer is described by the number of charges , the distance between the charges , and the length of a neutral block, . Hence, the length of the polyelectrolyte chain is and the total length of a block copolymer is . Here, the lengths are expressed in units of a Kuhn length (assumed common to both cationic and anionic chains).
If all polyelectrolyte charges of both signs are buried in the core and the neutral blocks form the corona, the total ”bare” charge of the core formed by linear polyelectrolytes and block copolymer chains is
(1) 
We will assume in the subsequent analysis that this charge is uniformly spread over the globule of radius , and therefore occupies a volume . The counterions and the salt ions can penetrate into the core of radius , and thus, screen the bare charge of the polymers. The resulting charge of the ”dressed” core, , is the charge of the core screened by small ions; assuming spherical symmetry, we have
(2) 
where the total charge density reads (in units of the elementary charge )
(3) 
In the above relation, is the inverse temperature, is the electrostatic potential and denotes the Heaviside step function, equal to inside and outside the core. The first term on the right hand side of Eq. (3) stems from the polymeric matrix, that contributes to the charge density as a spherical uniform background. We assume here that the system is in osmotic equilibrium with a salt reservoir with equal densities of labile cations and anions; the canonical situation, where the salt density in the system would be a priori prescribed, is amenable to a very similar treatment as the one presented here. From the reservoir ionic concentration, we define the Debye length through , where is the Bjerrum length and is the solvent dielectric permittivity. In Eq. (3), the last two terms are for the labile microions concentration. The corresponding exponential relation between the density profiles and the local electrostatic potential is typical of the PB (PoissonBoltzmann) approximation ^{36} that will be adopted in the remainder. Within such a meanfield simplification, the electrostatic problem at hand is the following:
(4) 
The charge of the ”dressed” core should be smaller than the ”bare” charge ^{37}, Eq. (1), because of counterion penetration inside the globule. However, the latter charge may be large enough to trigger significant nonlinear screening effects, that translate into an effective (or renormalized ^{33, 38}) charge that can be much smaller than . To be more precise, in the weakly coupled limit where (where one also has ), it is possible to solve analytically Eq. (4), since it reduces to for . The resulting DH (DebyeHückel) potential reads, for
(5) 
where is a saltdependent geometric prefactor; the complete solution will be provided below in section 2.2. To define the renormalized charge , it is sufficient to note that beyond the linear DebyeHückel regime, for an arbitrary charge , Eq. (4) again takes the form , but at large distances where becomes small. We consequently have, within the non linear PB framework:
(6) 
By construction, in the DebyeHückel regime, while upon increasing .
Introducing the dimensionless electrostatic potential and dimensionless distance , Eq. (4) in spherical polar coordinates is written in a dimensionless form
(7) 
There are then two dimensionless governing parameters, and . The solution of this nonlinear equation gives the charge density and the distribution of small ions around the complex.
2.2 Two limiting cases: weak charges and saltfree situation
Equation (7) can be solved analytically in the DH approximation when the electrostatic potential is small, . In this case, the charge density can be linearized, , and the solution can be written in the form
(8) 
where . As a consequence, the geometrical constant that enters into the electrostatic potential at large distances (5) is
(9) 
Our results for a charged polyelectrolyte complex in the presence of salt can be compared with the salt free regime ^{39}. In this case, it is essential to enclose the complete system in a confining boundary, otherwise the counterions “evaporate” –their energy loss upon leaving the globule vicinity is outbeaten by the entropy gain of exploring a large volume– and the problem becomes trivial. We therefore define , the WignerSeitz radius^{38} of a large sphere containing the system. We note that the ratio defines the volume fraction of globules in our system. In the present case where counterions only are present, the density of charges is
(10) 
where is a normalization parameter to ensure total electroneutrality (it does not have any physical significance as such, unless a particular “gauge” or reference has been chosen for the potential). A possible choice among others is . The corresponding dimensionless PB equation reads
(11) 
with rescaled distance . The dimensionless charge and are here independent control parameters.
2.3 Results
The solutions of the PB equation (7) for different dimensionless bare charge are presented in Figure 2a). The comparison with DH approximation (8) shows as expected that this approximation is valid for weakly charged objects. Such a comparison is a test for the numerical procedure used to solve Eqs. (4), and of course, strong deviations are observed between DH and PB solutions for larger than a few units. It is noteworthy in Fig. 2 that the electric potential inside the core tends to a plateau when is high enough. The corresponding labile ion local charge indeed tends to compensate for the background charge, resulting in a vanishing total local charge density. This requirement implies that one has
(12) 
which gives in Fig. 2a) for , and likewise in Fig. 2b) for , as can be seen.
The solution of the saltfree equation (11) is shown by a dotted line in Figure 2a). The absolute value of the corresponding electric potential cannot be compared to its counterpart found with salt, but the variations of can be. It can be seen on the figure, panel a), that the amplitude of is as expected larger without salt (for the same value of ). This illustrates the weaker screening without salt. In addition, panel b) shows that the small limit coincides with the saltfree limit, as it should (here, the saltfree solution has been shifted by the constant required to have the same potential at as in the case). The saltfree results reported here depend very weakly only on packing fraction.
The solution of the PB equation provides also the distribution of small ions around the core of the complex . It is shown for different charges of the core in Figure 3a) and fixed salt concentration in Figure 3b). As the charge is increased the concentration of small ions of opposite charge inside the core increases inducing stronger screening effect. The concentration of ions of both signs in the core increases with increase of the bulk concentration of salt. This is shown in Figure 3b), where the bare charge of the core is fixed, while the concentration of salt in the solution is changed. If the salt concentration is low, the redistribution of small ions around the core is almost due to counterions and the concentration profile of small ions obtained from Eq. (7) approaches the corresponding salt free solution given by Eq. (11). In addition, the increase of salt concentration results in higher concentration and induces stronger redistribution of the small ions around the core.
The presence of small ions inside the complex impinges on the radial distribution of charges . The background “core” contribution to this quantity is a step function, while due to the penetration of labile ions, is small in the globule center, where charge neutralization is most efficient, and increases upon increasing (see Fig. 4). In addition, the charge density in the center of the core vanishes at high concentrations but also for large enough .
The screening of the bare charge by small ions penetrating into the core can be measured by the charge of the ”dressed” core, introduced above in Eq. (2). The analytical expression for can be obtained within the DH approximation: , where is given by Eq. (8). Thus,
(13) 
from which it follows that
(14) 
It can be checked that this relation is consistent with the more familiar DH result for the potential of a spherical colloid having bare charge and radius ^{33}:
(15) 
which imposes that
(16) 
From Eq. (14), we have that within DH approximation, up to a saltdependent prefactor. However, upon increasing , nonlinear effects become prevalent and invalidate the DH approach, see Fig. 5, obtained by solving the nonlinear PB theory. The corresponding slower than linear increase of with is illustrated in Fig. 5a) and b), see also panel c) for the saltfree case. Increasing salt concentration screens out the charge, i.e. it leads to a decrease of , and flattens the curves in panel a), where the DH prediction (14) holds for low . In essence, increasing salt concentration ultimately leads to the DH limit where is independent, as can be inferred from Eq. (8), where is seen to decay with the increase of . A similar conclusion is drawn from expression (12): the DH limit is reached in the high salt limit. This trend is clearly seen in Fig. 5b), where all curves tend to collapse onto the DH behaviour for . On the other hand, is a strongly nonlinear function for large in the no salt case. More precisely, it has been shown in Ref 39 that in the strongly nonlinear saltfree regime, one has (or equivalently ). This prediction is successfully put to the test in Fig. 5c), which also shows that a change in the packing fraction does not affect the features discussed.
Finally, the behavior of the charged globule at large distances is encoded in the effective charge , defined in Eq. (6), and therefore extracted from the farfield of the numerical solution to the nonlinear equation (7). Such a quantity would rule the interactions between two distant globules. The corresponding plots of are shown in Figure 6a) as a function of the globule charge for fixed salt concentration, and as a function of salt density for fixed background charge in Fig. 6b) and c). As is invariably the case in such meanfield approaches, the effective, or renormalized, charge increases upon increasing the bare charge ^{33, 38}. It also increases with salt concentration ^{33} and as imposed by the very definition of , we find that in the DH limit (enforced either from considering low or large ). In addition, Fig. 6b) shows that the ratio is independent of bare charge , except at very small salt concentrations. This reflects the fact that even for large , counterion uptake is such that is significantly reduced, and such that the colloid included internal salt ions can be treated by linearized meanfield theory. Indeed, it can be seen in Fig. 6b) that is close to its DH counterpart, given by , see Eq. (14).
3 Charged core surrounded by charged corona (case B)
Thermodynamically stable polyelectrolyte complexes can also be formed by the complexation of two linear polyelectrolytes of opposite charge with a large asymmetry of the distances between the charges , which can form flowerlike structures ^{24}. The core of such complexes with a partially compensated charges is surrounded by a corona of long loops of a polymer with a longer distance between the charges (Fig. 1B). The loops of size , are neutral, but some larger loops and the tails can be charged.
Thus, we can generalize the discussion of the previous chapter to the case where the charged core is surrounded by a charged corona. We assume spherical symmetry in the distribution of the charges around the core, i.e. the charge in the corona depends only on the distance from the center of the core . Since the charges in the loops and tails are attached to the core by polymer chains, the electrostatic interaction of those charges with the core is balanced by a weak entropic force due to polymer chain extension. If the electrostatic force is not very strong, a polymer chain carrying the charge can be envisioned as a Gaussian coil and the probability of radial distribution of charges is , where is the length of the polymer chain in the corona and is the Kuhn segment length. This approximation is valid for small charges that do not perturb significantly the statistics of the chains. The resulting density of charges is the sum of three terms: the bare charge of the core, the charge of the counterions plus salt molecules, and the charge of the corona:
(17)  
where , positive or negative, is a parameter controlling the total charge of the corona and is the sign of this charge ( when and when ). It was assumed here that all loops have the same length and that ions in the corona are monovalent. This expression for leads to a Poisson equation similar to Eq. (7).
Solution of Poisson’s equation gives the radial distribution of the potential which is shown in Figure 7a) and 7b) for different charges of the corona. The charged corona influences the distribution of small ions around the core, a quantity that is displayed in Fig. 7c) and 7d). Weakly charged coronas clearly do not modify the monotonous decrease of with distance that was observed in Fig. 2, but this is no longer the case when is increased. Indeed, a point where reaches an extremum [maximum in panel a) and minimum in panel b)] can be observed. From Gauss theorem, this coincides with the point where the total integrated charge over a sphere having the corresponding radius vanishes. The physical phenomenon occurring in panel a) where the charges in the corona are of the same sign as the bare core (assumed positive), is that the positive corona induces a migration of negative microions inside the core and its vicinity, that change the sign of the uptake charge , which is now negative. Adding the corona charge to , though, leads to a positive charge. Hence the charge inversion evidenced by the potential extremum. The density peak of negative microions is clearly visible in panel c). On the other hand, when the bare core and the corona bear charges of opposite signs [panels b) and d)], a conjugate mechanism takes place: small labile cations are “sucked” inside by the core, which leads to an integrated charge in a running sphere that is positive for small spheres, and becomes negative once it includes the corona. In all cases, the mechanism can be viewed as a corona induced local charge inversion.
We do not repeat the full analysis of the difference between bare, uptake, and effective charge in the present case, but we show how the total charge of the corona, , depends on the parameter in Fig. 8.
4 Complexation of oppositely charged polymers
The previous sections, devoted to the electrostatics of polyelectrolyte complexes, have left aside the energetical aspects, to which we turn our attention hereafter. Once the total free energy of a given complex is known, it becomes possible to study the equilibrium behaviour, in particular the size distribution, of an initial “soup” of individual polycations and polyanions.
4.1 The total free energy and equilibrium complex size distribution
The size of the thermodynamically stable complexes of oppositely charged polymers is determined by the interplay of the steric repulsion of the chains in the corona and electrostatic attraction in the core. Thus, the formation of stable aggregates requires either long neutral blocks at least in one of the polyelectrolyte or a large asymmetry in the distances between the charges along the chain, e.g. . If the corona is composed of neutral blocks, the blocks should be long enough to stabilize the attraction in the core, if the corona is composed of loops between the charges, the segment should be long and flexible enough to form a loop in the corona in the micelle.
Consider then a spherical polyion complex made up of two polyelectrolytes of opposite charge. Each complex is defined by the number of polycations, , and the number of polyanions, . If we assume dense packing of the monomers in the core, the radius of the core can be expressed in terms of the number of chains and as , where is the Kuhn segment, are the lengths of the charged blocks (in units of ). The bare charge of the core is then also expressed in terms of and , see Eq. (1).
The distribution function of the polyion complexes is the number concentration of the aggregates with given aggregation numbers and . The total free energy of the solution of polyelectrolytes of opposite charge, their counterions and salt molecules is
(18) 
where is the volume of the system, is the free energy of the complex expressed in units of , is a molecular volume associated with the de Broglie length. Minimization of this free energy^{29, 30} with respect to along with two conservation of mass conditions, fixing the total concentrations of polyanions, , and polycations, ,
(19) 
gives the equilibrium distribution of the aggregates by their size^{32}
(20) 
The free energy of the complex can be written as the sum of an electrostatic contribution, and a term accounting for the steric repulsion of tails/loops in the corona
(21) 
These two contributions are detailed below.
4.2 The electrostatic contribution
The electrostatic contribution is related to the semigrand potential , relevant to discuss the present situation which is canonical for the colloids (polymers), and grandcanonical for the salt entities (in osmotic equilibrium with a salt reservoir of density ). The semigrand potential accounts for electrostatic attraction between polyelectrolytes and small ions in the system as well as the entropic contribution of small ions around the complexes. We have ^{40}
(22) 
where the first term is the electrostatic energy of the ionic distribution, and the second term is the entropy associated with the translational movements of small ions. We note that the integral in Eq. (22) diverges for large systems (as it would also for neutral systems), so that we consider in the following the excess semigrand potential with respect to reservoir
(23) 
Thus, the excess potential finally takes the form
(24)  
For a spherical globule, this equation can be written in the dimensionless form as
(25)  
where is the rescaled distance and we used the equality . The analysis of the free energy of the charged complexes suggests that the complexes with charged corona of the same sign as the bare core have larger free energy than their neutral counterparts, and thus, are less favorable (Figure 9a)). However, if the charges of the core and the corona are opposite, the electrostatic energy can be lower (Figure 9b)).
The above applies for spherical globules, but leaves aside the particular cases (, ) and conversely (, ), where the object to be considered is no longer a complex, but a polyanion or polycation respectively. We then need to adapt the previous arguments to these cases of an isolated charged chain in a salt solution. The polyelectrolyte chain is approximated as a cylinder of radius with uniform linear charge density , again treated within Poisson Boltzmann theory. Introducing dimensionless distance , where , the corresponding PB equation in cylindrical coordinates yields, for an infinite cylinder
(26) 
Here is the socalled Manning parameter (dimensionless line charge^{41}). Once this equation has been solved, the electrostatic contribution to and for isolated chains of both signs follows from a similar calculation as that of Eq. (4.2):
(27)  
which was calculated per chain length expressed in units of . In the following we assume that the Kuhn segment length of the polymer is of the order of . Upon using the free energy of the infinite cylindrical macroion configuration, we neglect end effects, the consideration of which would be technically more involved.
Isolated chains, corresponding to , and , configurations are penalized by a large electrostatic energy (see Figure 10). Indeed, these quantities bear a large selfterm, notwithstanding the solvation phenomenon, that manifests itself in the fact that decrease, for fixed charge , upon addition of salt (i.e. increase of ).
4.3 Steric repulsion of loops and tails
In the following, we consider a neutral and spherical polyelectrolyte complex made up of the charged core formed by oppositely charged polyelectrolytes and surrounded by the corona. We consider long tails and large loops in the corona, thus, the corona of the complex is then approximated by a star polymer (Fig. 1A) or a flower structure (Fig. 1B).
The electrostatic contribution (4.2) is balanced by the steric repulsion between the tails or loops in the corona. If the corona consists of long neutral blocks (star polymer, case A), this contribution is approximated by the free energy of a star polymer containing arms of length , which is the length of a neutral block. This approximation is valid when the core is much smaller than the corona and the arms are long enough to use the scaling expression^{42}
(28) 
In this expression, are the universal exponents of the star polymers and their numerical values are calculated in Ref. 43.
If the corona consist of long neutral loops and tails (flower structure, case B), the free energy contribution is similar to (28), but the exponent is different,
(29) 
This exponent is calculated as follows. If the loops are formed by a single chain with stickers joined together, , where the first term is the contribution of the center with vertices, the second term is the contribution of the two tails and the last term is the contribution of loops. Each loop contributes with the Flory exponent in the dimension of the space and is known numerically ^{44}. If the loops are formed by chains with stickers and chains with stickers and all stickers are condensed on the core, the exponent is given by
Isolated nonaggregated polycations and polyanions are linear polymers, thus their entropy contribution is . Here we neglect the surface tension and hydrophobic interactions between polycations and polyanions in the core of the complexes, thereby assuming that the electrostatic attraction of opposite charges is the leading contribution; hydrophobic interactions may however be dominant for neutral complexes.
4.4 Results
Eq. (20) defines the equilibrium distribution function of the complexes as a function of the geometry of the chains, asymmetry of the charges along the chain and salt concentration. We assume that the conformation of the polyelectrolyte complex is a spherical aggregate with a core formed by charged blocks surrounded by neutral corona (Figure 1A)). Since the electrostatic contribution of the core and isolated chains in the solution is the main contribution to the free energy, one might expect that the thermodynamically stable complexes would be narrowly distributed in size and have the minimal possible charge. Thus, the equilibrium of the free energy would require the compensation of the charges inside the core, such that the formed PIC micelles are almost neutral. However, in our description we allow for deviations from zero charge, because other contributions to the total free energy, the entropy of mixing, the salt concentration and the steric repulsion in the corona, may shift the equilibrium.
The distribution function of the complexes, Eq. (20) is calculated for each combination of (, ) and the results are shown in Fig. 11a). As an example, we plot the normalized for a mixture of a linear polymer with the charge and oppositely charged diblock copolymer with a charged block, , and neutral block of length (case A). The two polymers share the same distance between the charges along the chain: . It can be noted that all three distributions reported lie around the “electroneutrality line” . In the vicinity of that line, the precise location of the support of the distribution function stems from a subtle balance of effects, as embodied in the free energy (18). We observe in Fig. 11b) that upon increasing the salt concentration, the equilibrium size distribution is shifted towards smaller complex sizes and becomes more peaked. For instance, for , the peak corresponds to and (for which the complex has charge ). A similar trend is observed while changing the length of the neutral block, which controls the repulsion in the corona. Long tails in the corona favor smaller complexes, and shift the equilibrium accordingly. Since the complexes are close to neutrality, the salt concentration mostly affects the electrostatic energy of free chains [Eq. (4.2)], and the shift of the aggregation numbers along the electroneutrality line is mainly due to the chains in the solution. In addition, increasing the bulk concentration of polymers, and , increases the aggregation numbers, see Fig. 12.
Fig. 13 shows the size distribution function of the complexes formed by equally charged (”matched” in terms of Ref. 8) polymers (18,18), (44,44) and (78,78) and ”unmatched” polymers, (18,78) and (78,18). The polymer concentrations are chosen in such a way that the complexes are formed close to the origin, which may indicate the onset of aggregation. Aggregation of long polymers, (78,78), occurs at smaller concentrations than aggregation of short polymers, (18,18), due to the entropy of mixing, which strongly depends on the total length of polymers. We find that unmatched complexes [see the cases (18,78) and (78,18)], can also be formed if the aggregation numbers are close to the electroneutrality line (the opposite charges are compensated). On the other hand, Ref ^{8} has put forward a chain recognition mechanism where matched cases are more prone to form large complexes, but the system considered there is somewhat different, involving the equilibrium between three types of individual chains together with two and three component complexes.
5 Discussion and conclusions
We have developed a framework to study the formation of polyelectrolyte complexes from an initial arbitrary mixture of charged polymers, where both polycations and polyanions are present in an electrolyte solution. Two situations were addressed, as sketched in Fig. 1: for a given polycation type, the polyanion is either a diblock copolymer with a long neutral tail (case A), or a polyanion having a different intercharge spacing along the backbone (case B). Coulombic attraction between oppositely charged polymers leads to the formation of complexes, with an a priori unknown composition. The numbers of chains of both types in a given complex were denoted and . These complexes were envisioned as forming hairy structures, where the hair/corona is either made up of dangling neutral chains (case A) or of loops (case B), while the core of much smaller spatial extension contains most of the charges of the polymeric backbones. We started by focusing on the electrostatic aspects, treated at the level of PoissonBoltzmann theory. This part of the work thereby extends a previous study performed for saltfree systems ^{39}. In a second step, the resulting electrostatic free energy of the complexes was used, together with the entropic repulsion between tails/loops in the corona, to provide us with a free energy functional for an arbitrary mixture of complexes having a given size distribution . Upon minimizing this functional under the appropriate constraints of mass conservation for both polycationic and polyanionic species, we obtained the equilibrium composition of our mixture. Whereas this optimal distribution turns out to give a negligible weight to configurations that depart from complex global charge neutrality –a property that may have been anticipated–, it exhibits the non trivial feature of a high selectivity: out of an initial random soup of polycations and polyanions, well defined complexes with precise composition (, ) may emerge, particularly when the salt density is increased.
The problem under study here is characterized by a large number of dimensionless parameters, and we furthermore made simplifying assumptions in the description, such as equating the Kuhn lengths for both positively and negatively charged polymers. We chiefly focused on the effect of changing the salt concentration, which is an experimentally simple control parameter. The pH dependence of the core charge of the complexes has not been addressed, but it can be incorporated for instance via the Henderson–Hasselbalch equation ^{45}. In addition, the Coulombic aspects were treated at meanfield level, which is adequate provided the bare charge of the complex core, , is smaller than a bound of order ^{46}, which decreases when increasing the valence of the mobile microions (assumed here monovalent, i.e. ). Finally, we have neglected the structure of the core, by homogeneously smearing out its charge. This certainly leads to overestimate their free energy, due to the neglect of the corresponding negative correlation energy ^{36}.
Summary of main notations used
elementary charge  
Kuhn length, assumed equal for both polycationic and polyanionic chains  
total charge of a chain, in units of  
distance between charges along a linear polymer, in units of Kuhn length  
length of a neutral polymer block, in units of the Kuhn length  
number of positive/negative chains in an aggregate (core)  
Bjerrum length defined from temperature and solvent dielectric  
permittivity  
radius of a spherical aggregate/core  
bare charge of a spherical core (due to polymers)  
“uptake” charge of a core (due to polymers and salt ions inside the core)  
effective (or renormalized) charge of a spherical core, relevant at large  
distances from the core center  
reduced charge, defined as  
salt density in the reservoir  
Debye length, defined through  
total density of charge at a distance from core center  
parameter controlling the charge of the corona (case B)  
Manning parameter defined as , where is the linear charge  
of a linear polymer 
Acknowledgments
We would like to thank A. Chepelianskii, F. Closa and E. Raphael for useful discussions.
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