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Artem M.
Rumyantsev
^{ab} and
Igor I.
Potemkin
*^{abc}
^{a}Physics Department, Lomonosov Moscow State University, 119991 Moscow, Russian Federation. E-mail: igor@polly.phys.msu.ru
^{b}DWI – Leibniz Institute for Interactive Materials, Aachen 52056, Germany
^{c}National Research South Ural State University, Chelyabinsk 454080, Russian Federation

Received
4th August 2017
, Accepted 15th September 2017

First published on 15th September 2017

A polyelectrolyte complex (PEC) of oppositely charged linear chains is considered within the Random Phase Approximation (RPA). We study the salt-free case and use the continuous model assuming a homogeneous distribution of the charges throughout the polyions. The RPA correction to the PEC free energy is renormalized via subtraction of polyion self-energy in order to find the correlation free energy of the complex. An analogous procedure is usually carried out in the case of the Debye–Hückel (DH) plasma (a gas of point-like ions), where the infinite self-energy of point-like charges is subtracted from the diverging RPA correction. The only distinction is that in the PEC both the RPA correction and chain self-energy of connected like charges are convergent. This renormalization allows us to demonstrate that the correlation free energy of the PEC is negative, as could be expected, while the scaling approach postulates rather than proving the negative sign of the energy of interactions between the blobs. We also demonstrate that the increasing concentration of oppositely charged polyions in the solution first results in the formation of neutral globules of the PEC consisting of two polyions as soon as the concentration reaches a certain threshold value, c_{gl}, whereas solution macroscopic phase separation (precipitation of globules) occurs at a much higher concentration, c_{coac}, c_{coac} ≫ c_{gl}. Partitioning of polyions between different states is calculated and analytical dependencies of c_{gl} and c_{coac} on the polyion length, degree of ionization and solvent polarity are found.

The spatial charge distribution in the “asymmetric” systems is essentially inhomogeneous. It allows developing mean-field theories (Poison–Boltzmann or multi-zone approximation) like for dilute solutions of PE micelles^{17–20} or microgels.^{21} On the other hand, the inhomogeneous charge distribution means strong spatial charge fluctuations which are equivalent to the strong charge fluctuations at a certain spatial coordinate evolving over time as long as the ergodic hypothesis is valid. Therefore, calculations of the fluctuation contribution to the free energy in such systems are sophisticated. In contrast, charge fluctuations are much smaller in the “symmetric” solutions and the mean-field treatment is not able to describe the Coulomb interactions properly (they are equal to zero at this level). The electrostatics due to the small charge fluctuations can be taken into account within the framework of the Random Phase Approximation (RPA), which is a linear response approximation analogous to the one-loop approximation used in quantum field theory.

Over 90 years ago, Debye and Hückel (DH) have considered the problem of electrolyte solutions (the simplest “symmetric” system) demonstrating deviations from the ideal gas behaviour.^{22} Using the Poison–Boltzmann equation linearization, they have calculated corrections to the osmotic pressure and to the free energy of the ideal gas, both are negative. However, independently of the valency of the ions, their condensation or ion pair formation cannot be described within such an approach. The condition for linearization is that the energy of the electrostatic interactions between the ions is much smaller than the energy of their thermal motion. Three decades later, the first theoretical attempt to describe saline solutions of oppositely charged PEs was undertaken by Overbeek and Voorn.^{23} They have treated non-electrostatic interactions within the Flory–Huggins lattice theory, while Coulomb interactions between all charges of the system (mobile low-molecular-weight ions and charged groups of the chains) have been described using the DH expression. Thus, connectivity of the charged groups into the chains was neglected.

In the late 1980s in pioneer works of Borue and Erukhimovich, the RPA was applied to semidilute PE solutions and complexes, where the charge connectivity into the chain was taken into consideration by the addition of the polyion conformational entropy term to the free energy functional.^{24,25} The beauty of the RPA for the oppositely charged PEs is that the condensation of the chains (complex formation) can properly be described at this level because polymer chains are known to be “poor in entropy”: even small attraction between charged groups is able to overcome the entropic penalty due to the complex formation. Possibility of microphase separation in semidilute PE solutions and mixtures of oppositely charged polyions was predicted within the framework of this approach.^{24,26–28} Later on, similar theory was applied for the description of PE multilayers,^{29} polyampholytes^{30} and overcharged interpolyelectrolyte complexes.^{31,32} Further development of the RPA included consideration of ion pairing,^{28,33,34} short wavelength integration cutoff,^{33,35} hard-core and finite-size ion effects.^{33,34,36,37} In our group, fluctuations of polymer and charge densities were examined independently (discrete charge distribution), which allowed getting a general expression for the free energy unifying different limiting cases.^{38} The RPA theory for rod-like polymer chains capable of liquid-crystalline ordering was developed in ref. 39–42. Treatment of single PE chain trapping own counterions as one-component plasma (i.e. point-like charges immersed into homogeneously and oppositely charged background) is also based on the RPA.^{35,43–46} Castelnovo and Joanny,^{47} and then Fredrickson and co-workers^{48} extended the fluctuation theory of complexation beyond the RPA.

Recently account for fluctuations by means of liquid state theory approaches allowed describing solutions^{49} and complexes^{50} of highly charged PEs. In the framework of this method it was theoretically shown that low-molecular-weight salt ions prefer a supernatant rather than a coacervate,^{50} in contrast to predictions of Vroon–Overbeek theory that neglects both connectivity of PE charges and excluded volume interactions.^{37,51,52} This finding on salt ion partitioning being in principal agreement with experiment was later confirmed using computer simulations.^{53,54}

In the present paper, we propose a procedure of physically motivated renormalization of the RPA results for the symmetric PEC allowing us to distinguish the correlation (free) energy of the PEC and self-energy of polyions in the salt-free case. Proper renormalization makes possible to find partitioning of polyions between different states (non-aggregated chains, neutral PEC globules and coacervates) in a stoichiometric mixture of polyanions and polycations. Threshold concentrations of globule formation and macroscopic phase separation are found analytically.

The model with a charge continuously smeared throughout the flexible chain is used for the theoretical treatment of the PEC.^{24,55} The RPA correction to the free energy is known to be convergent within the framework of this model.^{29–32} However, the knowledge of this positive expression, which allows us to find the electrostatic contribution to the osmotic pressure and the polymer volume fraction in the complex properly, is not enough for calculation of the correlation (free) energy. Nonzero self-energy of the chains caused by connectivity of the charged groups has to be calculated and subtracted from the RPA results. The self-energy takes into account repulsive interactions of the charges in each individual chain. Due to the subtraction procedure, the correlation energy, which corresponds to the attraction between oppositely charged chains, occurs to be negative as one would expect. In contrast to the scaling approach, where the interaction energy between the blobs was postulated to be negative,^{56} the RPA proposes a straightforward way for calculation of the correlation free energy defining the numerical factor as soon as one deals with the polymers in a Θ-solvent. In particular, the knowledge of the self-energy allows quantifying the association of single chains into a complex (calculation of the critical concentration of association).

Here it is necessary to mention that initially a renormalization procedure of the RPA results for PE solutions based on the subtraction of chains self-energy was proposed by Mahdi and Olvera de la Cruz who motivated it primarily by mathematical convenience.^{36} Lately Shen and Wang have explicitly distinguished self-energy of polyions in PE solution and found solution correlation energy via subtraction of the chain self-energy from the formal field-theoretic result.^{57} In the present paper, we apply the same renormalization procedure to the case of the PEC. It makes possible to argue for the first time the negative sign of the complex correlation energy within the RPA.

In order to make the calculation of the correlation free energy in the case of PECs more lucid and clear, we use analogy with well-known DH plasma wherein subtraction of infinite self-energy of point-like ions from the diverging RPA result is necessary to get a finite negative expression for the correlation free energy. An analogy between correlation functions in the case of the PE complex and the DH plasma also occurs to be very useful.

The paper is organized as follows. In the next section the DH plasma is discussed in detail, first without and then with volume interactions, and well-known results for ion–ion correlation functions and correlation free energy of plasma are reproduced. In Section 3, we move on to the case of the PEC and propose a renormalization procedure of the RPA free energy correction. It allows us to obtain the correlation free energy of the complex. Section 4 deals with the calculation of the critical concentrations of PEC formation and coacervation. Closed-form expressions for these concentrations are found.

F_{tot}{n_{+}(r), n_{−}(r)} = F_{tr} + F_{el-st} | (1) |

(2) |

(3) |

Correlated electrostatic attraction makes the osmotic pressure of the DH plasma lower than that of the ideal gas. The RPA can be used in order to find the free energy correction to the ideal gas term and to calculate the density–density correlation functions of the DH plasma. The concentration fluctuations are defined as δn_{k}(r) = n_{k}(r) − n_{0}, and Fourier transforms are given by

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

The correction to the electrostatic energy caused by ion–ion correlations (density fluctuations) can be calculated as follows:

(11) |

(12) |

(13) |

(14) |

(15) |

(16) |

(17) |

(18) |

Another way to get the correlation free energy correction is based on the calculation of Gaussian functional integrals

(19) |

In order to calculate the integrals, quadratic form

(20) |

(21) |

(22) |

(23) |

As we mentioned in Introduction, the above calculations of the correlation (free) energy are valid when this correction is small as compared to the energy of thermal motion, |F_{corr}| ≪ |F_{tr}|. This condition is fulfilled at a low concentration of the ions, n_{0} ≪ (a/l_{b})^{3}.

(24) |

(25) |

(26) |

(27) |

G_{++} = G_{self} + G_{vol} − G_{corr} | (28) |

G_{+−} = G_{vol} + G_{corr} | (29) |

Note that eqn (11)–(14) remain valid in the case of non-zero volume interactions as well, so that definitions of the self-energy and the correlation energy are the same. Indeed, relationship G_{++} − G_{+−} = G_{self} − 2G_{corr} is satisfied despite G_{vol} ≠ 0. In the general case, the term caused purely by the electrostatic interactions should be defined as

G_{corr} = (G_{+−})|_{B=C=0} | (30) |

In the next section we apply the proposed approach to describe complex formation between oppositely charged PE chains.

The total free energy in k_{B}T units takes the form:

F_{tot}{Φ_{+}(r), Φ_{−}(r)} = F_{tr} + F_{conf} + F_{el-st} + F_{vol} | (31) |

(32) |

(33) |

(34) |

(35) |

(36) |

Expansion of the free energy functional into the series in powers of the polymer density fluctuations allows calculating the inverse matrix of correlation functions

(37) |

(38) |

(39) |

G_{++} = G_{self} + G_{vol} − G_{corr} | (40) |

G_{+−} = G_{vol} + G_{corr} | (41) |

In order to find the correlation free energy and the self-energy, one should explicitly calculate these components of the correlation functions:

(42) |

(43) |

(44) |

(45) |

Using relationship similar to eqn (13) and introducing additional multiplier f^{2} for the transition from the polymer–polymer to charge–charge correlation function, one obtains

(46) |

Since u ∼ 1/T, the free and the internal electrostatic energies coincide, F_{self} = U_{self}, and the density of the self-energy can be represented as follows:

(47) |

It is necessary to emphasize that the self-energy term is not a mean-field energy of the charged chains with Gaussian statistics but the correction term caused by Gaussian correlations in the location of charges (see Appendix A for a detailed discussion). The value of the self-energy term is defined by the choice of the reference polymer system: it is the system without volume and electrostatic interactions, C = 0 and l_{b} = 0. Therefore, the chains in the reference system possess Gaussian statistics at any length scales owing to the conformational entropy term F_{conf} in the total free energy.

Thus, similar to the case of the DH plasma (eqn (23)), the correlation energy can be written as follows:

(48) |

Result similar to eqn (48) has been used earlier by Mahdi and Olvera de la Cruz who calculated the phase diagram of semidilute PE solution in the presence of salt.^{36} The authors justified subtraction of the second term in square brackets by mathematical convenience: this “… irrelevant subtracted term only facilitates the algebra and gives an electrostatic contribution that reduces to the simple Debye–Hückel electrostatic free energy in the correct limit.”^{36} In fact, subtraction of the self-energy does not influence results for PE solution since it is a constant term and can be omitted: the self free energy density is proportional to the polymer volume fraction Φ and does not contribute to the osmotic pressure.^{70} However, if we would like to calculate the correlation energy of the PEC or to compare two states of the PE chains – aggregated into PEC and free – it is necessary to properly calculate and comprehend calculation of the free energy of the system. Subtraction of the self-energy of the chains is very important for distinguishing the reference state of the system and, hence, proper calculation of energy of Coulomb interactions. Application of the renormalization procedure to the case of the PEC and the proof of the negative sign of the correlation free energy are the novelties of the above part of the present work.

The self-energy is proportional to the number of PE subchains in the system _{chains}, like the self-energy is proportional to the number of charges in the DH plasma. The difference between these two systems is the following: point-like charges are structureless and their self-energy is unchanged under any conditions, while the PE chain can adopt different conformations resulting in different mutual spatial locations of charges and different self-energy.

The same identification of the PE chain self-energy as the energy necessary to connect charges onto a single chain has been recently used by Shen and Wang, and our choice of the reference state coincides with theirs: “the zero energy of the electrostatics is taken to be the state where charges are dispersed into infinitesimal bits at infinity.”^{57} The dependence of the chain self-energy on its conformation was discussed in ref. 57 as well. The authors introduced chain self-energy in order to find correlation free energy of PE solution, and we use very similar renormalization with the same chain self-energy definition. The only difference is that we apply it to the case of the interpolyelectrolyte complexes rather than solution of similarly charged chains.

The positive value of the PRA correction (39) means that the number of like charges surrounding any selected charge on the chain in the PEC is more than the number of opposite charges, despite repulsion of like charges and attraction of opposite charges. This fact is caused by the binding of like charges into the chain. Since these bonds exist before complex formation, their impact on the charge–charge correlations was separated from the correlations induced by purely Coulomb attraction. The Coulomb energies caused by these correlations were treated as the self-energy and the correlation energy, respectively.

(49) |

In the case of infinitely long chains (N → ∞), we get , so that the polymer volume fraction within the complex reads

(50) |

(51) |

(52) |

A more rigorous method to derive the similar result for chain electrostatic energy is not to adopt a priori assumption on the blob size, ξ_{blob} = ξ_{el-st}. Indeed, let ξ_{blob} be the size of the blob, and correlations inside it are Gaussian. The number of monomer units in the blob g ∼ (ξ_{blob}/a)^{2} and the average distance between i-th and j-th units . The electrostatic energy of a single blob is given by

(53) |

(54) |

(55) |

(56) |

(57) |

(58) |

(59) |

(60) |

A more common procedure for the self-energy calculation of a swollen PE chain in both dilute and semidilute salt solutions can be found in the recent work of Shen and Wang, where the authors took into account coupling between the chain conformation and the screening of Coulomb interactions in the solution.^{57} Owing to the low polyion ionization degree, f ≪ 1, we have also neglected chain entanglements (knots), which are known to be localized, adopt a tight configuration and relax very slowly (or even remain frozen) in the case of highly charged polyelectrolytes.^{72}

Let us assume that the chains are pretty long, N(uf^{2}lnN/3)^{2/3} ≫ 1, i.e. the number of the electrostatic blobs within the chain exceeds unity by far. Free energy of the chains in each state can be divided into concentration-dependent entropic and concentration-independent energy contributions. The latter has the form:

_{sw} = 2.16(uf^{2})^{2/3}N(lnN)^{2/3} | (61) |

_{coac} = 2.17(uf^{2})^{2/3}NC^{1/9} | (62) |

The energy contribution for the case of the globule includes the energy per chain of the complex coacervate and the excess surface energy per chain, 2πR_{gl}^{2}γ. Here γa^{2}/(k_{B}T) = 0.11(uf^{2})^{2/3}C^{−7/18} is the surface tension coefficient of the globule.^{32,75} This result takes into consideration connectivity of PE charges^{32} and for this reason differs from that found in ref. 76, where combination of Vroon–Overbeek and Cahn–Hilliard theories neglecting polymer specificity of the system was used. Thus, free energy per chain in globule reads

_{gl} = 2.17(uf^{2})^{2/3}NC^{1/9} + 0.68(uf^{2})^{4/9}N^{2/3}C^{−5/54}. | (63) |

Denoted by M_{tot} the total number of polycations (polyanions) in the solution of volume V. Let the fraction (1 − s) of polyions forms coacervate, and the total number of single polycations and polycations in the globules (i.e. in supernatant) equals M = sM_{tot}. If all polyions in the supernatant are single chains, their free energy reads

(64) |

Z_{sn} = Z_{0}P_{comb}W_{vol}e^{2Mp(sw−gl)}. | (65) |

(66) |

(67) |

(68) |

Finally, free energy of chains in the coacervate can be written as follows:

(69) |

(70) |

The fractions of polyions in the coacervate, globules and non-aggregated state are shown in Fig. 2. In Fig. 3 we plot corresponding concentrations of polyions and globules in the supernatant. It is seen that increasing the polyion concentration first results in the formation of single globules. The single chains do not aggregate only at extremely small polymer concentrations. Their coexistence with the globules is observed also at extremely small polymer concentrations. The single globules dominate in the solution in a very wide concentration range, c = 10^{−20}–10^{−6}, above which the macroscopic coacervate is formed. These numerical results coincide well with analytical estimations of the critical concentrations of the globules c_{gl} and coacervate c_{coac} formation, which can be found from eqn (68)–(70):

(71) |

(72) |

Our principal result on the formation of single neutral interpolyelectrolyte globules prior to the macroscopic phase separation at increasing concentration of polyions in solution coincides with that recently reported by Delaney and Fredrickson.^{78} They proved this fact by means of both scaling analysis and theoretical-field methods. At that, analytical expressions for threshold concentrations, c_{gl} and c_{coac}, found in the present paper seem to be more precise as compared to scaling estimations^{78} where both the lnN correction as well as the surface effects were neglected, while strict theoretical-field analysis performed in their work did not allow finding analytical formulas.

The self-energy of polyions depends on the conformation of the chain, and different chain conformations can be used as a reference state. However, in the framework of the Borue–Erukhimovich model^{25} which was widely used in the past few decades, the reference state is the Gaussian coil. This choice of the reference state is caused not only by mathematical (the structure of the free energy functional containing the conformational entropy term) but also by physical reasons. Indeed, a PEC in a Θ-solvent is known to be a polymer globule, so that polyions within the globular complex/in coacervate adopt the conformation of a Gaussian coil on any length scales.

The self-energy of a PE chain within the complex equal to is the Coulomb energy due to the correlations of charges along the polyion in ideal coil conformation. It is important to stress that the mean-field free energy of charged polyions with Gaussian coil statistics should not be included into their self-energy because of the overall complex electric neutrality (see Appendix A). The physical meaning of the chain self-energy can be defined as follows. It is the work which is necessary to simultaneously assemble an equal number of polycations and polyanions in the same volume (volume of PEC) providing them conformations of Gaussian coil. This work is required not to get non-zero overall charge of this volume (as in the case of single polyions, eqn (60)), since the complex is electrically neutral, but to achieve connectivity of charges into Gaussian coils. In turn, energy of fluctuation-induced attraction between these already assembled polyions is the true correlation energy, F_{corr}.

For the first time within Borue–Erukhimovich polyelectrolyte complex model we split the RPA result for the energy of Coulomb interactions within the complex, F_{RPA}, into two parts: self-energy of the chains, F_{self}, and correlation energy of the complex, F_{corr}. We demonstrate that the correlation free energy of the complex within the framework of this model considering continuous charge distribution, F_{corr} = F_{RPA} − F_{self}, is negative, in accordance with general expectations. The proposed procedure for the RPA result renormalization (subtraction of the chain self-energy) is based on physical arguments and uniquely defined. This procedure and PE chain self-energy definition are akin to those used formerly by Mahdi and Olvera de la Cruz^{36} and later by Shen and Wang in the case of PE solutions.^{57}

Accounting for not only the mean-field term but also the correlation electrostatic free energy of the swollen single chain allows us to find the critical concentration c_{gl} of the single-globule PEC formation and concentration of precipitation of these globules, c_{coac}. Analytical expressions for these concentrations are found for the case of pretty long chains providing a large number of electrostatic blobs. It is shown that the single globules of the PEC exist in a wide range of polyion concentrations, c_{gl} ≪ c_{coac}, when complexation is already favourable owing to a high energy gain in the course of polyion neutralization, while precipitation is still unfavourable because of entropic reasons.

(73) |

^{G,ch}_{el-st} = ^{G,ch}_{MF} + ^{G,ch}_{corr} | (74) |

(75) |

(76) |

(77) |

(78) |

(79) |

(80) |

(81) |

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