Complex Coacervate of Weakly Charged Polyelectrolytes - American

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Complex Coacervate of Weakly Charged Polyelectrolytes: Diagram of States Artem M. Rumyantsev,† Ekaterina B. Zhulina,‡,§ and Oleg V. Borisov*,†,‡,§ †

Institut des Sciences Analytiques et de Physico-Chimie pour l’Environnement et les Matériaux, UMR 5254 CNRS UPPA, Pau, France ‡ Institute of Macromolecular Compounds, Russian Academy of Sciences, 199004, St. Petersburg, Russia § National Research University of Information Technologies, Mechanics and Optics, 197101 St. Petersburg, Russia ABSTRACT: The equilibrium structure of an interpolyelectrolyte complex (IPEC) coacervate consisting of weakly charged flexible polyanions and polycations with equal degree of ionization is considered. We study effects of salt concentration cs and solvent quality defined by second virial coefficient v on polymer density, surface tension, salt absorption, and correlations of charges and monomer units in equilibrium IPEC. Five different scaling regimes of IPEC are revealed, and a salt−solvent quality (cs−v) diagram of coacervate states is constructed. Boundaries between different regimes correspond to crossovers between low/high salt concentrations and poor/Θ-/good solvent conditions. Derived scaling laws for coacervate physical properties in each of regimes are in agreement with analysis based on the random phase approximation (RPA). This study summarizes and complements current theoretical comprehension of symmetrical IPEC of weakly charged polyions.

I. INTRODUCTION Phase separation in the solution of oppositely charged polyelectrolytes (PEs) into highly diluted and polymer enriched phases is usually referred to as coacervation. Coacervate is a liquid phase constituting interpolyelectrolyte complex (IPEC) between polyanions (PA) and polycations (PC). Though coacervate is overall electrostatically neutral, its stability, structure, and relatively high density are attributed to the Coulomb interactions between oppositely charged monomers, namely, the attraction induced by charge density fluctuations. The important role of coacervation in biological processes and numerous practical applications assures sustainable interest to this phenomenon among both theoreticians1−5 and experimentalists.4−10 In particular, understanding the underlying principles that govern coacervate structure and properties is important for design and applications of soft colloids with IPEC core surrounded by hydrophilic shell, such as complex coacervate core micelles,11−13 partially neutralized PE stars,14,15 and brushes.14,16 The first model aimed to explain the coacervation phenomenon was Voorn−Overbeek (VO) theory17 in which the account for short-ranged interactions within the coacervate by means of the Flory−Huggins (FH) lattice theory18 was combined with the Debye−Hückel (DH) treatment of correlation attraction between charges in polyions.19 This approach has been widely used for many years and allowed for at least qualitative agreement with the experimental data.1,6,20−22 However, recent more rigorous theories as well as simulation data show that success of the VO model is caused to a degree by fitting of experimental data with inappropriate © XXXX American Chemical Society

values of theoretical parameters and partial compensation of errors inherent to it.23−25 Indeed, the VO approach entirely neglects connectivity of charges in PE chains as well as finite sizes of both charges in polyions and salt ions. Electrostatic correlations between charges in PE chains are stronger than in DH plasma and enhance fluctuation-induced attraction, while excluded volume interactions are especially important at high polymer volume fractions and salt concentrations and hinder formation of dense coacervates. These inaccuracies may balance each other, resulting in a qualitatively reasonable prediction of coacervate density even when it is high enough. However, in this case VO theory fails to predict salt expulsion from the coacervate attributed to nonzero own volume of salt ions.25 Application of the random phase approximation (RPA) to coacervation was the next step in theoretical comprehension of this problem.26 RPA-based theories were the first that took into account the connectivity of charges in polymer chains. They allowed to properly consider the charge correlations within the coacervate and elucidated that their higher strength is a consequence of low entropy of PE chains. The initially proposed approach26−30 was later improved to account for ion pairing31 as well as hard-core repulsion between ions32 and generalized to the case of stiff (rodlike) molecules.33 The scaling consideration of IPECs was first introduced in the analysis of self-assembly in dilute solution of diblock copolymer with oppositely charged blocks.34 This study Received: February 14, 2018 Revised: April 16, 2018

A

DOI: 10.1021/acs.macromol.8b00342 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules focused on the case of Θ-solvent conditions and treated saltfree coacervate as an array of densely packed electrostatic blobs. The preferential neighborhood of oppositely charged electrostatic blobs provided fluctuation-induced attraction between PAs and PCs, while their repulsion was caused by the shortrange three-body interactions between monomers.34 Shortly afterward, this scaling theory was generalized to the case of saltfree good solvent where similar correlations were revealed between swollen electrostatic blobs,35 with their Coulomb attraction equilibrated by two-body repulsions between monomers. However, neither RPA nor scaling is able to provide proper treatment of highly charged PEs coacervate because of strong charge correlations in it, so that more sophisticated theoretical methods (as well as simulations23,36,37) are required to solve this problem. Recent theories of coacervate try to combine liquid state theory approaches with the account of connectivity of charges in polymer chains. 24 This is achieved via introduction of intramolecular correlation function into the Ornstein−Zernike equations. For simplicity, correlations between charges in linear polyions are assumed to be independent of the distance of the charge from the chain end; i.e., edge effects are neglected. Applicability to the case of strongly charged PEs and reasonable predictions on strong charge correlations with regard to excluded volume effects are the main advantages of this approach.38 Unfortunately, liquid state theory methods usually require numerical solution of integral equations and unable to produce analytical results. So-called renormalized Gaussian fluctuation field theory, a method which allows to find not only charge correlations but also conformations of PE chains in a self-consistent way (within the RPA chain structure is fixed by a priori assumptions), has been recently applied to coacervate.39 This work revealed that chains within the coacervate indeed adopt Gaussian conformations in a wide range of salt concentrations, in agreement with earlier findings of field-theoretic simulations undertaken in ref 3. A rigorous description of the supernatant where chains swell and lose ideal-coil statistics became the main advantage of these approaches over the conventional RPA, and convincing binodals of associative phase separation have been constructed.3,39 Another approach to treat dense IPECs of strongly charged polyelectrolytes was proposed in ref 40. Using transfer matrix formalism, the authors studied short-ranged correlations in the vicinity of polyion backbone accounting for possible adsorption of charged groups of e.g. PA or salt anions to charged units of PC. While modern theoretical approaches are aimed mostly on description of IPEC between strongly charged PEs, there is still no complete picture of coacervation between weakly charged polyions that incorporates wide ranges of solution salinity and solvent quality. In the present article, we try to fill this gap via constructing the scaling-type diagram of coacervate states in the salt concentration−solvent quality coordinates. In order to summarize the existing advances and to demonstrate the consistency of the results, we compare the scaling predictions with the results of the RPA in its simplest formulation. We demonstrate that for weakly charged polyelectrolytes there is no contradiction between the scaling and RPA approaches, so that the proposed diagram of states summarizes and complements a considerable number of theoretical works on polyelectrolyte coacervation.

The paper is organized as follows. In section II we discuss different regimes of coacervate depending on the solvent quality and salt concentration in the solution, with subsections corresponding to each of them. In each subsection, we first perform scaling analysis with derivation of power laws for coacervate density, correlation length, surface tension, and salt partitioning. Then RPA-based description of coacervate is discussed, which allows us to compare scaling and RPA predictions. In the end of section II we outline limitations of our consideration. In section III we summarize our findings in the diagram of coacervate states and discuss the boundaries between different regimes. Conclusions are formulated in section IV.

II. REGIMES OF COACERVATE GLOBULE AT DIFFERENT SALT CONCENTRATIONS AND SOLVENT QUALITIES We consider a symmetric coacervate of oppositely charged polyelectrolytes with equal degrees of ionization f of the polyions, f+ = f− = f, immersed in solution of 1:1 salt. Polyions are assumed to be flexible, with equal lengths of statistical segment, a. We introduce the dimensionless second and third virial coefficients, v = B/a3 and w = C/a6, for two-body and three-body interactions between monomers (of both PA and PC). In the following we implement w ≃ 1 in the scaling model while keeping w as an independent variable in the RPA approach. The strength of electrostatic interactions within and outside the coacervate depends on the medium polarity (dielectric permittivity ϵ) . We neglect dielectric mismatch between coacervate and solution due to low coacervate density, ϕ ≪ 1, and introduce the dimensionless Bjerrum length u = lB/a = e2/ϵakBT defining solvent polarity with ϵ being the dielectric constant of the solvent. Concentration of salt defined as the sum of anion and cation concentrations, cs = c+s + c−s , is expressed in dimensionless units, so that cs = 1 corresponds to solution volume per one salt ion equal to a3. All length scales below are dimensionless (expressed in a units). Coacervate is assumed to be homogeneous owing to the symmetry of polyions and has zero net charge. Polyelectrolytes are supposed to be rather long, with degree of polymerization N exceeding by far the number of monomers in the correlation blob in IPEC. We restrict our consideration to the case of weakly charged polyelectrolytes, f ≪ 1, so that effect of crosschain ion pairing41 can be entirely neglected. 1. Coacervate in Salt-Free Θ-Solvent Solution (Regime IΘ). 1.1. Scaling Analysis. According to the scaling model of symmetric coacervate in Θ-solvent,34 coacervate consists of oppositely charged electrostatic blobs with the size of each blob ξel‑st,Θ ≃ (uf 2)−1/3 (Figure 1). The sign “≃” denotes equality within the scaling level of accuracy when small corrections to the main term and all numerical coefficient are omitted. The energy of Coulomb interaction between two neighboring blobs is on the order of the thermal energy. The number of oppositely charged blobs surrounding any fixed blob of the coacervate is larger than the number of blobs with the same sign of charge due to charge correlations. At the meanfield level average charge density within the coacervate is zero. Because of local fluctuations in the net charge density, emerging correlation attraction is also termed fluctuation induced and can be considered as a Coulomb attraction between oppositely charged regions arising (and disappearing) because of fluctuations. Providing the coacervate stability, it is equilibrated by three-body repulsions between monomers, and the B


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Macromolecules

indicates an attractive nature of the Coulomb interactions within coacervate. The value of exponent 3/4 in Π corr dependence on ϕ shows that fluctuation-induced attraction within IPEC has long-range nature.42 The free energy density of three-body repulsion equilibrating the correlation attraction has conventional form - vol = wϕ3

if ϕ ≪ 1, and the corresponding osmotic pressure, Πvol = 2wϕ3, is positive. Equality of the attractive and repulsive contributions to the total osmotic pressure, Πvol + Πcorr = 0, allows one to find the equilibrium coacervate density:

Figure 1. Blob picture of coacervate in regime IΘ. Inside the electrostatic blob ξel‑st,Θ the chain segment exhibits Gaussian statistics, ν = 1/2. Chain of blobs ξel‑st,Θ is also Gaussian, so that polyions exhibit ideal coil statistics at all length scales.

ϕIΘ = (3π )−1/9 2−2/3(uf 2 )1/3 w−4/9

(1)

ξIΘ ≃ rp, Θ ≃ (uf 2 )−1/3 w1/9

The number of monomer units within the correlation volume is given by gIΘ ≃ ξ2IΘ ≃ (uf 2)−2/3. The average coacervate density and polymer volume fraction within the blob are the same: g ϕIΘ ≃ I3Θ ≃ ξI−Θ1 ≃ (uf 2 )1/3 ξIΘ (2)

γIΘ = 0.11(uf 2 )2/3 w−7/18

-corr

(3)

ξel‐st, Θ ≃ rD

cs* ≃ u−1/3f 4/3 w−2/9

d-corr 1 − -corr = − ∼ −ϕ3/4 dϕ 24 2 πrp,3 Θ

(12)

chain segments in neighboring blobs within the coacervate interact via screened rather than bare Coulomb forces. The higher the cs, the stronger the screening and the lower equilibrium density of the complex. Concentrations of salt inside the coacervate cins and outside it out cs are basically different because of the different correlations of charges in these regions. We restrict our consideration to the case of weak association of polyelectrolytes, (uf 2)−1/3 ≫ 1, and moderate salt concentrations, ucs1/3 ≪ 1, with still valid linear DH approximation (RPA). In this case the energy of salt ion interaction with the surrounding cloud of ions, both mobile and bound to polymer chains, is much lower than kBT, and the difference between salt concentrations cins and cout s in coacervate and the solution is low as well, δcs = cins − cout ≪ cins , cout s s . Thus, Debye radii within the IPEC and the solution are close to each other, and the value of rD in the solution was used for

(4)

(5)

is the creening radius of Coulomb interactions by polyelectrolyte chains (polymer screening radius) within coacervate and -corr is the correlation free energy per unit volume. Here and below all the free energies are expressed in kBT units. Equation 4 holds in the case of Gaussian statistics of polymer segments in IPEC at all length scales. The negative sign of the correlation osmotic pressure Πcorr = ϕ

(11)

At high salt concentration, cs > c*s , exceeding the threshold value

where rp, Θ = (48πuf 2 ϕ)−1/4

(10)

The results of the RPA (eqs 8−10) and scaling analysis (eqs 1−3) are in agreement with each other. 2. Coacervate in Salt-Added Θ-Solvent Solution (Regime IIΘ). In this section we study the effect of lowmolecular-weight 1:1 salt on the coacervate structure under Θsolvent conditions. At low salt concentrations cs salt ions hardly influence the IPEC properties. The threshold concentration c*s above which the coacervate characteristics considerably depend on cs can be found by equating the correlation length in saltfree coacervate, ξIΘ ≃ ξel‑st,Θ, to Debye radius in the surrounding solution, rD = (4πucs)−1/2:

The analysis of salt partitioning between coacervate and a saltadded solution in both low and high salt regimes, IΘ and IIΘ, is performed in the next section. 1.2. RPA Consideration. The contribution of the electrostatic interactions within coacervate cannot be calculated at the mean-field level because the average charge density of coacervate is zero. The second-order correction to the meanfield free energy caused by the fluctuations of PA and PC densities within the coacervate with given polymer volume fraction ϕ can be calculated by means of the RPA:26 F 1 = corr = V 6 2 πrp,3 Θ

(9)

and coincides with electrostatic blob size in salt-free solution. In the framework of the RPA consideration, IPEC surface tension coefficient was calculated in refs 30 and 44:

Each blob at the interface of the coacervate has a lower number of neighboring blobs, which results in extra ≃kBT energy per ξ2IΘ area of the interface. Therefore, dimensionless surface tension coefficient γ expressed in kBT/a2 units equals γIΘ ≃ ξI−Θ2 ≃ (uf 2 )2/3

(8)

This result derived earlier in refs 26, 29, 30, and 43 is consistent with the scaling estimation, eq 2. The advantage of the RPA over the scaling is the possibility to reveal the dependence of the polymer volume fraction on the third virial coefficient w and calculate exact numerical prefactor equal approximately to 0.49. The value of the correlation radius is on the order of the screening radius of Coulomb interactions within the complex:

coacervate structure is similar to that of the semidilute polymer solution under Θ-solvent conditions with the free energy of short-range ternary interactions kBT per blob. Therefore, the correlation length within the coacervate is equal to the electrostatic blob size in Θ-solvent: ξIΘ ≃ ξel‐st, Θ ≃ (uf 2 )−1/3

(7)

(6) C


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Macromolecules estimation of c*s . Below we use the salt concentration in the solution cout s ≡ cs which can be adjusted in experimental systems as an independent parameter. 2.1. Scaling Analysis. At high salt concentrations, cs ≫ cs*, the correlation radius in the complex is much larger than the Debye length, ξ ≫ rD, and the correlation blob contains a large number of Debye blobs (Figure 2). Here we use the term “Debye blob” for the segment of the chain with dimensions rD.

explained as follows: increasing strength of electrostatic interactions (growing u) is fully countervailed by the concomitant diminution of the Debye blob size and charge. Thus, one can find the correlation length within the coacervate at high salt concentrations ξIIΘ ≃ csf −2

(16)

which is also independent of u. The screened Coulomb attraction between oppositely charged neighboring correlation blobs is equilibrated by three-body repulsion of monomers. The structure of the coacervate at high salt concentrations resembles one in a salt-free solution, but the correlation length is given by eq 16 rather than equal to the size of the electrostatic blob (see Figure 2). One can check that condition ξIIΘ ≫ rD is fulfilled as soon as cs ≫ c*s . The coacervate density is equal to the polymer volume fraction within the correlation blob and diminishes under addition of salt: ϕIIΘ ≃ ξIIΘ−1 ≃ cs−1f 2

(17)

Finally, the coacervate surface tension is given by Figure 2. Blob picture of coacervate in regime IIΘ. Correlation blob ξ comprises of large number of Debye blobs rD. Polyions exhibit Gaussian statistics at all length scales, ν = 1/2.

γIIΘ ≃ ξIIΘ−2 ≃ cs−2f 4

Salt Partitioning. In order to find salt partitioning between coacervate and outer solution, it is necessary to study screening of the test point charge within the coacervate by writing down linearized Poisson−Boltzmann (PB) equation which takes into account screening by both polyions and ions of salt. Let R be the electrostatic screening radius within the coacervate. Fragments of PE chains (termed below as loops) with R end-to-end distance should be considered as Z-ions, and their charge in Θ-solvent equals Z = fgR ≃ f R2 owing to the Gaussian statistics of polyions at all length scales. The concentration of loops within the screening radius can be calculated as nZ ≃ fϕ/Z because fϕ is the total concentration of charges bound to polymer chains. Here ϕ and R are independent parameters. The linearized PB equation accounting for screening by both salt ions and PE loops treated as Zions reads

The chains exhibit the Gaussian statistics at all length scales, so that the numbers of monomer units in the correlation and Debye blobs are given by g ≃ ξ2 and gD ≃ r2D, respectively. Debye blobs interact with each other via screened Coulomb potential ∼exp(−r/rD)/r; that is, one can assume that they do not interact at all if the distance between them is large, r ≫ rD, and interact via nonscreened Coulomb potential ∼1/r if they are close to each other, r ≃ rD. For this reason, the electrostatic interactions in the framework of scaling can be considered as effective short-range two-body interactions between Debye blobs.45 We assume that similarly to the salt-free case, the correlation blobs in IPEC are surrounded predominantly by the oppositely charged counterparts, and thereby the fluctuation-induced electrostatic energy is dominated by the attraction between oppositely charged correlation blobs. We denote qD = efgD the total charge of the Debye blob. The energy of pairwise interaction between Debye blobs expressed in kBT units can be found as WD ≃

qD2 ϵkBTrD

≃ uf 2 rD3

Δψ − 4πu(cs in + nZ Z2)ψ = 0

p≃

ξ3

r ≃ D ξ

rD−2 + rp,−Θ4 R2 = R−2 (13)

(20)

At salt concentrations below the threshold, cs ≪ cs*, screening is primarily provided by polyions, and R ≃ rp,Θ since rD ≫ rp,Θ. The potential near an elementary test charge reads ψ (r ) =

(14)

e exp( −r /R ) e e ≈ − ϵ r ϵr ϵR

(21)

and infinitesimal at r → 0 terms are omitted. The first term in this expansion is the potential created by the test charge itself while the second term is the potential due to correlation cloud. Therefore, correlation correction to the energy of the test charge scales as −e2/ϵkBTR = −u/R. Equality of chemical potentials of salt ion (equivalent to the test charge) inside and outside the coacervate, ln(cs + δcs) − u/R = ln cs − u/rD, with the last term on the right-hand side being the correlation correction to the free energy of salt ion in the outer solution,

where Nb ≃ ξ2/r2D is the number of Debye blobs in one correlation blob. The total energy of screened electrostatic interactions between two neighboring correlation blobs Wc ≃ pNbWD ≃ uf 2 ξrD2

(19)

−1/4

Using rp,Θ ≃ (uf ϕ) (in accordance with eq 5) and after performing some transformations, one arrives at the equation defining the screening radius R: 2

The probability p for one such blob to interact with another one (in two-body collision) is equal to the volume fraction of Debye blobs within the correlation blob:

NbrD3

(18)

(15)

should be of the order of unity according to the correlation blob definition. Remarkably the derived result (15) for Wc is u-independent owing to rD ∼ u−1/2 dependence and can be D


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Macromolecules allows us to find the difference between salt concentrations inside coacervate and in the supernatant: δcsIΘ ≃ csu 4/3f 2/3

in Πcorr − Π out corr ≃ −

≃ −u1/2f 4 cs−3/2

(22)

Here we have neglected usual DH screening in the outer solution which results in extra energy −u/rD per salt ion since it is small as compared to −u/R at rD ≫ R. The difference between the salt concentrations inside and outside the coacervate being the linear function of cs shows that at cs ≪ cs* salt ions do not considerably influence coacervate structure. It is interesting to mention that salt ions of both positive and negative signs of charge are equally absorbed by the neutral coacervate, and the concept of the Donnan equilibrium is not applicable to this case. At high salt concentrations, cs ≫ c*s , screening caused by salt ions dominates over screening by polyions, rD ≪ rp,Θ, and the screening radius inside the coacervate is given by

R=

rD−4 + 4rp,−Θ4 − rD−2 2rp,−Θ4

⎛ r4 ⎞ ≈ rD⎜⎜1 − 4D ⎟⎟ rp, Θ ⎠ ⎝

(25)

is equilibrated by surplus osmotic pressure of salt ions trapped 1/2 4 −3/2 IIΘ by the complex, ΔΠid = Πinid − Πout , since id = δcs ≃ u f cs ideal gas contributions to the osmotic pressure due to translational motion of salt ions inside and outside the coacervate are equal to Πinid = cins and Πout id = cs, respectively. Contribution due to short-range repulsive monomer−monomer interactions within the coacervate is small in regime IIΘ: Πvol ≃ ξIIΘ−3 ≪ ΔΠid at cs ≫cs*. Similar arguments are valid for high salt regime of the coacervate in good solvent as well. 2.2. RPA Consideration. The correlation correction to the free energy of the coacervate in Θ-solvent in the presence of salt was calculated for the first time by Borue and Erukhimovich26 and later reproduced by Castelnovo and Joanny:27 -corr =

(23)

1 (1 − s) 2 + s 12πrp,3 Θ

(26)

Here s = r2p,Θ/(rinD)2 = cins πu/3f 2 ϕ is the dimensionless parameter which can serve as a measure of salt concentration in the system. Equation 26 is reduced to eq 4 in salt-free case, i.e., at s = 0. High salt concentrations correspond to the high values of dimensionless parameter s ≫ 1. For equilibrium coacervate in Θ-solvent the condition s ≃ 1 defines the threshold salt concentration within the RPA. It is equivalent to crossover cs ≃ c*s found via scaling considerations. Finally, for solution containing no polyelectrolyte, ϕ → 0, but only salt, the correlation correction given by eq 26 acquires well-known DH form:

The energy of interaction between a salt ion and the correlation cloud inside the complex equals −u/R, while corresponding energy in the outer solution is −u/rD. Combining eq 23 with the equality of chemical potentials, ln(1 + δcs/cs) = u/R − u/rD, one can find that this difference in correlation energies results in δcsIIΘ ≃ cs−3/2u1/2f 4

4 1 rD ≃ −u1/2f 2 cs−1/2ϕIIΘ 3 4 rD rp, Θ

(24)

The relative difference between the salt concentrations inside −5/2 , goes down sharply with and outside the IPEC, δcIIΘ s /cs ∼ cs increasing cs. The reason for it is the swelling of the coacervate upon addition of salt resulting in the decreasing screening of the salt ions by polyions and decreasing correlation energy gain. In turn, swelling of the coacervate is the consequence of (i) salt induced screening of interactions between PAs and PCs and (ii) exerting osmotic pressure of salt surplus trapped by the complex (δcs > 0). In the above consideration we have neglected the difference between rinD and rout D ≡ rD. One can check that this assumption is justified and account for rinD ≠ rout D would lead to only small corrections in the found δcs value. This fact can be explained as follows: It is nonzero polymer concentration within the coacervate providing extra correlation energy gain, which promotes absorption of salt ions: δcs = 0 at ϕ = 0 since solution of salt is stable with respect to the spinodal decomposition. Therefore, trapping of salt ions caused by PE screening is the main effect, while extra screening caused by additionally absorbed salt is a small second-order correction owing to δcs/cs ≪ 1. These arguments remain valid for each of regimes and will be used below for salt partitioning analysis. The balance of osmotic pressures inside and outside the coacervate can be verified as follows. Correlation induced (negative) osmotic pressure within the coacervate providing its stability scales as Πincorr ≃ −R−3, while the same in the outer −3 solution reads Πout corr ≃ −rD . Prefactors in these expressions are equal because of identical physical nature of these terms. The difference between them

-DH = −

1 12πrD3

(27)

In order to find the equilibrium IPEC density, one should write down free energy densities of the coacervate -in and outer solution -out , which take into account (i) translational entropy of salt ions, (ii) correlation correction to the free energy, and (iii) short-range ternary interactions of monomer units within the coacervate in Θ-solvent: ⎛ c in ⎞ -in = -tr + - vol + -corr = csin ln⎜ s ⎟ + wϕ3 ⎝ 2e ⎠ 1 (1 − s) 2 + s + 12πrp,3 Θ

(28)

⎛c ⎞ 1 -out = -tr + -DH = cs ln⎜ s ⎟ − ⎝ 2e ⎠ 12πrD3

(29)

Here s = r2p,Θ/(rinD)2 depends on the salt concentration cins within the coacervate, while Debye radius rD a function of the outer concentration of salt, r−2 D = 4πucs. According to the Lifshitz theory of globules,46 equilibrium between globular coacervate and the outer solution is defined by (i) equality of osmotic pressures inside and outside the globule and (ii) equality of salt ion chemical potentials: E


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Macromolecules ⎧ in 1 s2 + s + 1 π 3 3/2 ⎪ cs + 2wϕ − 24πrp,3Θ s + 2 = cs − 3 (ucs) ⎨ in ⎪ ln cs − 1 3 3in s(1 + s) = ln cs − π u3/2c1/2 s 2 s+2 2 12πrp, Θ 2cs ⎩

( )

Fa3 = S

⎜

⎛ − μs ⎜ ⎝

()

(30)

∞

⎞ ⎟

⎞

∞

∫−∞ cs(x) dx − Ns⎠

⎟

(37)

Here x-axis is normal to the flat interface of the globule with x → −∞ corresponding to globule interior and x → ∞ corresponding to the outer solution. The volume contribution written as

Substituting s = 0, one can reproduce eq 8 derived earlier for salt-free case, and this result remains asymptotically valid as long as s ≪ 1. Consideration of low salt limit at s ≪ 0 allows to find distribution of salt between the coacervate and the outer solution. Salt accumulation into the coacervate is favorable due to correlation effects: δcsIΘ = csin − cs ≃ csu 4/3f 2/3 w−1/9

⎛

∞

∫−∞ (- + -grad) dx − μϕ⎝∫−∞ ϕ(x) dx − Nϕ⎠

⎛ c (x) ⎞ 1 - = cs(x) ln⎜ s ⎟ + w(ϕ(x))3 + (1 − s(x)) 2 + s(x) ⎝ 2e ⎠ 12π(rp, Θ(x))3

(31)

(38)

At high salt concentrations, cs ≫ c*s corresponding to s ≫ 1, equilibrium coacervate density and salt partitioning in the system can be found via consequent linearization of equations with respect to the small parameter δcs/cins

reproduces eqs 28 and 29 inside and outside the complex, respectively. In fact, we suppose that the correlation contribution to - (the third term in eq 38) depends on the local values of the polymer and salt densities, ϕ(x) and cs(x). This assumption is justified by the thickness of the globule interfacial layer Δ being on the order of the correlation length and exceeding the screening length within the coacervate in regime IIΘ by far, ΔIIΘ ≃ ξIIΘ ≫ rD. The gradient contribution to the free energy functional

⎧ ⎪ δc (1 − β /2) + 2wϕ3 s ⎪ ⎤ βc in ⎡ 1 + s−1 + s−2 ⎪ = s ⎢ − 1⎥ ⎪ ⎥⎦ 3 ⎢⎣ 1 + 2s−1 ⎨ ⎪ ⎪ δc ⎡ 1 + s −1 ⎤ β ⎪ s = ⎢ − 1⎥ in ⎪ cs ⎥⎦ 1 − β /2 ⎢⎣ 1 + 2s−1 ⎩

-grad = (32)

3

(33)

One can find polymer volume fraction within the coacervate and salt redistribution: ⎛ 3 ⎞1/3 f 2 ⎜ ⎟ ⎝ 4π 2 ⎠ csw 2/3

ϕIIΘ = δcsIIΘ =

34/3 2 π

c 5/3 7/6 s

(34)

∂− μs = 0 ∂cs

(35)

−

−3/2 4 1/2 −2/3

f u

w

The correlation length can be obtained by equating of threebody interactions energy within the correlation volume, F ≃ wξ3ϕ3, to unity: ξIIΘ ≃

rp,2 Θ rD

≃ f −2 csw1/3

(39)

takes into account conformational entropy losses caused by inhomogeneous distribution of polymer (first term47,48) and extra free energy arising due to gradient in salt ions concentration (second term49), respectively. The dimensionless ratio k/a2 depends on the intermolecular interactions between salt ions and surrounding media.49 Since in our analysis salt ions are treated as point-like and their short-range (excluded volume) interactions are disregarded, it is natural to assume k = 0. Finally, two last integrals in eq 37 account for the normalization conditions for polymer and salt concentrations with μϕ and μs being chemical potentials of polymer units and salt ions, while Nϕ and Ns being the total numbers of monomers and ions of salt. Minimization of the free energy functional with respect to cs(x) and ϕ(x) = y2(x) yields

with β = πcsinu3 ≪ 1 and further series expansion with respect to s−1 ≪ 1. This allows to obtain an equation balancing contributions to the osmotic pressure caused by short-range monomer−monomer interactions (left-hand side) and by interplay between correlation effect and ideal gas pressure of salt ions (right-hand side): 6 rD3 1 1 1 ⎛⎜ rD ⎞⎟ 2wϕ ≈ ≈ = ⎜ ⎟ 24πrD3 s 3 24πrD3 ⎝ rp, Θ ⎠ 24πrp,6 Θ

a 2(∇ϕ(x))2 + k(∇cs(x))2 24ϕ(x)

(40)

2 ⎛ ∂⎞ a2 d y + 2y⎜ − μϕ ⎟ = 0 2 3 dx ⎝ ∂ϕ ⎠

(41)

The following boundary conditions have to be satisfied: (1) ϕ(x = −∞) = ϕ0 and cs(x = −∞) = cins = cs + δc0s apply at x → −∞, which corresponds to the globule interior0; (2) x → +∞ is the outer solution, so that ϕ (x = +∞) = 0 and cs(x = +∞) = cs. Since at x = ±∞ all derivatives of ϕ(x), y(x), and cs(x) should vanish, the condition (40) yields equality of salt ion chemical potential inside and outside the globule

(36)

These results obtained earlier by Castelnovo and Joanny27,28 are consistent with the scaling predictions. Surface Tension of Coacervate. In order to find the surface tension of coacervate in volume approximation, it is necessary to consider the case of infinitely large globule and write down and minimize the corresponding one-dimensional free energy functional, F(ϕ(x),cs(x)), which depends on both polymer volume fraction, ϕ(x), and salt concentration, cs(x):

μs =

∂∂cs

= in

∂∂cs

out

(42)

which can be written as the second equation of the system (30). One can multiply eq 40 by cs′(x), eq 41 by y′(x), sum them, and after integrating obtain F


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Macromolecules a2 (y′(x))2 = μϕ ϕ(x) + μs cs(x) − Π 0 6

-(ϕ(x), cs(x)) −

prefactor occurs to be very small, which is consistent with wellknown fact that coacervate interfacial tension is remarkably low.6,50 3. Coacervate in Poor Solvent (Subregimes I− and II−). 3.1. Scaling Analysis. Regimes IΘ and IIΘ corresponding to the Θ-solvent conditions are realized at relatively small |v| values, i.e., close to the Θ-point. It is natural to expect that binary interactions should affect the structure of the coacervate. Pairwise attraction between monomers starts to manifest itself when its energy per correlation volume is comparable to the thermal energy, i.e ≃ 1 in kBT units.

(43)

Substitution of boundary conditions leads us to the relationship defining equality of osmotic pressures inside and outside the coacervate globule (μϕ ϕ + μs cs − -)|in = (μϕ ϕ + μs cs − -)|out = Π 0

(44)

which coincides with the first equation of set (30). Solution of these equations defining chemical and mechanical equilibrium between globule and outer solution allows us to find ϕ0 = y02 and δc0s given by eqs 34 and 35, respectively. In order to find the surface tension, one should substitute -(ϕ(x), cs(x)) expressed from eq 43 into the initial free energy functional and distinguish excess interfacial energy γ=

a 3

∞

∫−∞ (y′(x))2 dx = a3 ∫y

vξ 3ϕ2 ≃ −1

Substituting ξ ≃ ξIΘ and ϕ ≃ ϕIΘ, one can find the boundary between regimes of salt-free coacervate in poor and Θ-solvent, I− and IΘ, respectively vIΘ /I − ≃ − v* ≃ −u1/3f 2/3

0

y′(y) dy (45)

0

vIIΘ /II − ≃ −f 2 cs−1

-(ϕ(x), cs(x)) − μs cs(x) − μϕ ϕ(x) + Π 0

Excess salt concentration δcs(x) should be found from eq 40 linearized with respect to δcs(x)/cs(x): 1 8πrp,3 Θ(1

⎡ s(1 + s) ⎤ − s 3/2 ⎥ ⎢ ⎦ − β /2) ⎣ s + 2

(47)

Here both rp,Θ and s depend on x. After expansion of Π0 term around cs(x) with respect to the small parameter cs − cs(x) = −δcs(x), we can get - − μs cs + Π 0 = wϕ3 + ≈ wϕ3 +

s 3/2 ⎡ 1 − s−1 − 2s−2 ⎤ ⎢ ⎥ − 1 12πrp,3 Θ ⎢⎣ 1 + 2s−1 ⎥⎦

⎞ s 3/2 ⎛⎜ 3 −2 s − s−3⎟ 3 ⎝ ⎠ 12πrp, Θ 2

(48)

- − μs cs − μϕ ϕ + Π 0 ≈ wϕ(ϕ2 − 3ϕ03) + 6wϕ0 2ϕ −

12πr p3, Θs 3/2

≈

wϕ03[t 6

3

2 1/2 2 w ϕ0 3

∫0

1

(54)

ξ− ≃ |v|−1

(55)

γ− ≃ ξ−−2 ≃ |v|2

(56)

rp, Θ ≃ (uf 2 ϕ−)−1/4 ≃ u−1/4f −1/2 |v|−1/4

2

− 4t + 3t ]

(57)

The boundary between I− and II− regimes is given by the equality between polymer screening radius rp,Θ and Debye radius rD and can be presented as follows:

(49)

Here we have neglected small with respect to s−1 and β terms and denoted t = y/y0. Finally, combining eqs 45, 46, and 49 allows us to find the result γIIΘ =

ϕ− ≃ |v|

Salt Partitioning. Distinction between coacervate subregimes I− and II− is the difference in screening of Coulomb interactions. At low salt concentrations, in subregime I−, it is primarily provided by polymer chains, while in subregime II− screening by salt ions dominates. This leads to different salt absorption by the complex. The salt distribution can be calculated in a way analogous to that performed for regime IIΘ (see eqs 19−21 and 23), since polymer loops have Gaussian statistics at all length scales. The only difference is the value of the polymer screening length (see eq 5) dependent on ϕ−:

Substituting μϕ found from boundary conditions for globule interior applied to eq 41, one can get the final result for radical expression in eq 46:

1

(53)

In poor solvent, both at low and high salt concentrations, the short-range two-body attractions are stronger than correlation attraction and governs the density of the coacervate being equilibrated by three-body repulsion. For this reason, I− and II− subregimes constitute the single regime of poor solvent, “−”. The coacervate density, surface tension, and correlation length in this regime coincide with those in a neutral polymer globule:

(46)

δcs(x) =

(52)

This boundary is valid at sufficiently low salt concentrations, cs < c*s . At high salt concentrations, cs > c*s , crossover IIΘ/II− between coacervate in Θ-solvent and in poor solvent is given by the same eq 51 but with ξ ≃ ξIIΘ and ϕ ≃ ϕIIΘ and reads

Here, we find surface tension without explicit calculation of polymer and salt density profiles passing from x- to y-integration. It becomes possible as soon as y′(x) is expressible as the functions dependent on y(x) = ϕ(x) but not explicitly on x, and from eq 43 we obtain 6 y′(x) = − a

(51)

vI − /II − ≃ −uf −2 cs 2

(58)

In subregime I−, at v < vI−/II−, the excluded volume attractive interactions within the globular coacervate are strong enough to provide high polymer volume fraction and polymer screening length is smaller than the Debye length, rp,Θ < rD. Therefore, the effective (apparent) screening length in this regime is R ≃ rp,Θ. The excess salt concentration within the coacervate is equal to

t 6 − 4t 3 + 3t 2 dt = 0.0267cs−2f 4 w−5/6 (50)

which is in agreement with the scaling estimations but additionally provides us with the exact value of numerical prefactor and also dependence of γ on w. The numerical G


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Macromolecules u cs ≃ u5/4f 1/2 cs|v|1/4 R

δcsI − ≃

vI − /II − ≃ −uf −2 cs 2w

(59)

At high salt concentrations, s ≫ 1, equality of osmotic pressures inside and outside the coacervate can be obtained similarly to the case of regime IIΘ:

On the contrary, situation v > vI−/II− corresponds to II− subregime where the salt concentration is high enough to provide predominant screening by salt ions. In turn, this results in screening radius given by eq 23 so that the mismatch between inner and outer salt concentrations equals

rD3 24πrp,6 Θ

−1 ⎡ ⎤ rD4 ⎞ u u u ⎢⎛⎜ II − δcs ≃ cs − cs ≃ cs ⎜1 − 4 ⎟⎟ − 1⎥ ⎥ R rD rD ⎢⎝ rp, Θ ⎠ ⎣ ⎦

≈ ucs

rD3 r p4, Θ

vIIΘ /II − ≃ −f 2 cs−1w1/3

(60)

δcsII − ≃ u1/2f 2 cs−1/2|v|w−1

(61)

(62)

The short-range binary monomer attraction dominates over correlation attraction as long as the second term on the lefthand side of eq 62 exceeds the first one, i.e., at v < vIΘ/I− with

1 |v| 2 w

(63)

(64)

This result for ϕ is valid in both I− and II− subregimes because in both of them the electrostatic correlation attraction is weak as compared to short-range binary attraction between monomers. Mean-field results for the correlation length ξ and the surface tension γ of a neutral globule are also valid for both for I− and II− subregimes of regime “−”. They are given by51 −1 2/3

ξ− ≃ |v| w γ− =

|v|2 8 6 w 3/2

vIΘ /I + ≃ −vIΘ /I − ≃ v* ≃ u1/3f 2/3

(72)

vIIΘ /II + ≃ −vIIΘ /II − ≃ f 2 cs−1

(73)

ξI + ≃ ξel‐st, + ≃ u−3/7f −6/7 v 2/7

(74)

ϕI + ≃ u 4/7f 8/7 v−5/7

(75)

γI + ≃ ξI +−2 ≃ u6/7f 12/7 v−4/7

(76)

Each swollen electrostatic blob comprises large number of thermal blobs, ξt < ξel‑st,+. Coulomb attraction between neighboring electrostatic blobs, Wel‑st ≃ (efgIV)2/ϵξI+ ≃ 1 is balanced by the short-range two-body repulsions with energy ≃1 per units. The number of monomers per blob is gI+ ≃ ϕI+ξ3I+. Equality of the Debye length rD to the size of the electrostatic blob in good solvent, ξI+ ≃ ξel‑st+, can be considered as a boundary

(65)

1

(66)

Equality of salt ion chemical potentials inside and outside the coacervate (given by the second equation of system 30) allows −1 one to get for I− regime δcs/cs ≃ sr−3 p,Θcs that after substitution of ϕ− into rp,Θ yields δcsI − ≃ u5/4f 1/2 cs|v|1/4 w−1/4

(71)

The same result could be obtained from the equality between correlation length in regimes IΘ and IIΘ and thermal correlation length ξt ≃ |v|−1, which is controlled by the strength of binary short-range repulsive interactions (solvent quality). Consider IPEC under good solvent conditions at low salt concentrations, regime I+. It can be envisioned as an array of densely packed oppositely charged swollen electrostatic blobs, ξel‑st,+ (instead of ξel‑st,Θ for Θ-solvent coacervate; see Figure 1), as it was for the first time described by Wang and Rubinstein:35

since coacervate density coincides with that of the neutral globule: ϕ− =

(70)

It is seen that scaling and RPA results for IPEC in poor solvent are in agreement, with RPA additionally predicting dependencies of the complex properties on w. 4. Coacervate in Salt-Free Good Solvent Solution (Regime I+). 4.1. Scaling Analysis. Excluded volume repulsive interactions within the coacervate start to play a considerable role at the same absolute values of second virial coefficient, |v|, in the regions of both poor and good solvent (see eq 51). Therefore, upper boundaries of regimes IΘ and IIΘ read

while other terms, -corr and -tr , remain unchanged. In the low salt limit, s ≪ 1, coacervate density is determined by the condition of vanishing osmotic pressure which leads to the equation

vIΘ /I − ≃ − v* ≃ −u1/3f 2/3 w 5/9

(69)

Salt partitioning in this regime should be found from the second equation of system 32 remaining valid under poor solvent conditions via its linearization with respect to s−1 ≪ 1:

Similarly to the case of Θ-solvent, the ratio δcs/cc under poor solvent conditions is independent of cs in low salt subregime I− and decreases under addition of salt in high salt subregime II−. 3.2. RPA Consideration. In order to generalize RPA-based analysis performed for Θ-solvent to the case of poor solvent, it is sufficient to modify the contribution to the free energy caused by excluded volume interactions:

1 − vϕ2 = 2wϕ3 24 2 πrp,3 Θ

− vϕ2 = 2wϕ3

Two-body interactions prevail over correlation attraction at 2 −1 one can r3Dr−6 p,Θ < |v|ϕ , and taking into account ϕ− ≃ |v|w deduce the boundary between regimes of poor and Θ-solvent for coacervate in salt-added solution:

≃ u1/2f 2 cs−1/2|v|

- vol ≃ vϕ2 + wϕ3

(68)

vI + /II + ≃ u−1/4f 3 cs−7/4

(67)

The boundary between high and low salt regimes, I− and II−, is given by s ≃ 1 which is equivalent to

(77)

between low and high salt regimes, I+ and II+, for coacervate in good solvent. H


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avoiding walk scaling exponent value ν = 3/5 within it (Figure 3). Therefore, the size of Debye blob and the number of

Salt Partitioning. Under good solvent conditions the relationship between the charge Z of polymer loop (chain segment with R dimensions) and the effective screening radius R is given by Z ≃ fgR ≃ f v−1/3R5/3. Bearing in mind that the average concentration of loops equals nZ ≃ fϕ/Z, it is easy to obtain an equation for the potential ψ of the test point charge Δψ − (rD−2 + rp,−+11/3R5/3)ψ = 0

(78)

which is analogous to eq 19 derived for Θ-solvent conditions. Here we have introduced the radius of screening by swollen polymer loops as rp, + ≃ (uf 2 ϕ)−3/11v1/11

(79)

According to the R definition R−2 = rD−2 + rp,−+11/3R5/3

Figure 3. Blob picture of coacervate in subregime II+a. Correlation blob ξ comprises of large number of Debye blobs with size rD, that in turn contain numerous thermal blobs with size ξt: ξt < rD < ξ. Chains exhibit swollen coil statistics with ν = 3/5 in the range ξt < r < ξ and ideal coil statistics with ν = 1/2 on other length scales. Thus, Debye blobs are swollen.

(80)

This equation remains valid until polymer loops within the screening radius R are swollen coils with ν = 3/5 and provides us with another representation of I+/II+ boundary: rD ≃ rp,+. In low salt regime I+, screening is caused by polymer loops and screening radius coincides with the blob size, R ≃ rp,+ ≃ ξI+. One can obtain expected linear dependence of δcs on cs: u δcsI + ≃ cs ≃ u10/7f 6/7 csv−2/7 (81) R

monomers within it are linked through rD ≃ v1/5g3/5 D . Since the charge of each Debye blob is equal to qD = efgD, the energy of Coulomb interaction between two Debye blobs reads WD ≃ −1/3 uf 2r7/3 . Owing to the estimation of the total number of D v Debye blobs within the correlation volume Nb ≃ g/gD ≃ ξ5/3/ r5/3 D , the probability of pairwise interaction between Debye blobs belonging to neighboring oppositely charged correlation blobs reads

Equation 80 allows us to study salt partitioning at high cs values as well, in one of subregimes of regime II+ when polyions demonstrate swollen coil statistics within the screening radius R. It will be done in the next section, after we propose blob picture for both subregimes of the regime II+. 4.2. RPA Consideration. Since in good solvent statistics of PAs and PCs cease to be Gaussian at small length scales, chain structure factor S(q) is not equal to Debye function at high q. Accounting for S(q) ∼ q−5/3 (characterizing swollen coil with ν = 3/5) instead of S(q) ∼ q−2 (corresponding to ideal coil with ν −1 = 1/2) within the correlation blob, i.e., at q > ξ−1 I+ ≃ rp,+, allows one to find correlation free energy density -corr ≃ rp,−+3 .26,52 Correlation attraction is equilibrated by two-body repulsion, which cannot be treated at mean-field level inside the IPEC being semidilute solution in good solvent.53 Scaling estimation yields - vol ≃ ϕ9/4v 3/4 . Countervailing of corresponding osmotic pressures, Πcorr + Πvol = 0, leads to the final result for coacervate density ϕI+ given by eq 75 which was found for the first time in the framework of the RPA by Borue and Erukhimovich.26 5. Coacervate in Salt-Added Good Solvent Solution (Regime II+). 5.1. Scaling Analysis. In order to find the scaling dependences for coacervate correlation length and density, we consider the blob structure of coacervate in regime II+. Correlation blob of ξ size is swollen and contains a large number of Debye blobs since rD < ξ. However, Debye blobs can be either swollen, at rD > ξt ≃ v−1, or Gaussian, at rD < ξt ≃ v−1. We denote the former case corresponding to higher v values as subregime II+a, while the latter case is denoted as subregime II+b. The boundary between these subregimes is given by

vII + a/II + b ≃ u1/2cs1/2

p≃

NbrD3 ξ3

⎛ rD ⎞4/3 ≃⎜ ⎟ ⎝ξ⎠

(83)

Therefore, the total Coulomb energy of attraction between neighboring oppositely charged correlation blobs is given by Wc ≃ pNbWD ≃ uf 2 rD2v−2/3ξ1/3

(84)

This screened electrostatic attraction is equilibrated by excluded volume repulsion of blobs with energy on the order of the thermal energy (i.e., ≃ 1 in kBT units) to give ξII + a ≃ f −6 cs 3v 2

(85)

Salt Partitioning. In subregime II+a total screening radius of electrostatic interactions R is close to rD, and polymer loops within it are swollen. Therefore, eq 80 should be applied to this case and one can get the following estimation for R: R≈

rD 1 + (rD/rp, +)11/3

⎡ ⎛ r ⎞11/3⎤ ⎢ ≈ rD⎢1 − ⎜⎜ D ⎟⎟ ⎥⎥ ⎝ rp, + ⎠ ⎦ ⎣

(86)

This result together with known ϕII+a value allows to find salt concentration difference: δcsII + a ≃

rD8/3 ⎛u u ⎞⎟ ⎜ − cs ≃ u 11/3 cs ≃ u 2/3f 10 cs−13/3v−10/3 ⎝R rD ⎠ rp, + (87)

(82)

B. Subregime II+b. In this subregime neighboring to saltadded Θ-solvent regime IIΘ, the excluded volume interactions are weaker than in regime II+a. It leads to Gaussian statistics of polyions within Debye blobs and gD ≃ r2D (Figure 4). Energy of interaction between Debye blobs equals WD ≃ uf 2r3D, while

which corresponds to the onset of swelling of the Debye blobs. A. Subregime II+a. This regime has the boundary with regime I+ (in fact vI+/II+ = vI+/II+a) and volume interactions within each Debye blob are strong enough to provide selfI


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5.2. RPA Consideration. At high salt concentrations effective RPA correction to the differential osmotic pressure within the coacervate taking into account both correlation attraction of polyions and osmotic pressure of salt surplus adsorbed by the complex reads Πcorr

33/5 rD18/5 1 ⎛⎜ rD ⎞⎟ ≃ − 3⎜ ≃ − ⎟ rD ⎝ rp, + ⎠ rp,33/5 +

(94)

r−3 p,+

and coincides with salt-free expression Πcorr ≃ at the I+/II+ boundary rD ≃ rp,+ (compare with eq 33 for the IIΘ regime).52 Analogous to regime I+, equality between correlation-induced attraction and short-range repulsion, Πvol ≃ ϕ9/4v3/4, allows to obtain IPEC density26 coinciding with ϕII+ found via scaling (eq 90). Thus, our scaling predictions for regimes I+ and II+ agree with RPA formulas derived by Borue and Erukhomovich26 for the corresponding low-salt and high-salt regimes in good solvent. However, the rest of our diagram and the scaling laws for other regimes crucially differ from their results because the effect of polymer three-body interactions within the coacervate was disregarded, and authors of ref 26 did not study coacervate under poor solvent conditions. 6. Range of the Theory Applicability. The range of validity of both scaling analysis and RPA consideration is specified by the same condition u ≪1 (95) R

Figure 4. Blob picture of coacervate in regime II+b. Correlation blob ξ comprises numerous thermal blobs with size ξt, which in turn contain a lot of Debye blobs with size rD: rD < ξt < ξ. Chains exhibit swollen coil statistics with ν = 3/5 in the range ξt < r < ξ, and ideal coil statistics with ν = 1/2 on other length scales. Thus, Debye blobs are Gaussian.

their number within the correlation volume is Nb ≃ g/gD ≃ ξ5/3v−1/3r−2 D . Since the probability of blob−blob interaction between Debye blobs scales as p ≃ Nbr3D/ξ3 ≃ rDv−1/3ξ−4/3, the total energy of Coulomb interaction between correlation blobs is given by Wc ≃ pNbWD ≃ v−2/3f 2 cs−1ξ1/3

(88)

The balance between electrostatic attraction and short-range repulsion of blobs, Wc ≃ 1, yields

ξII + b ≃ f −6 cs 3v 2

which implies small value of correlation energy per ion as compared to the thermal energy. This allows to apply linearized PB equation in the framework of scaling (in all regimes) and to use RPA (being linear response approximation) to describe weak perturbation of the correlation functions. In the regimes IIΘ, II−, and II+ the salt concentration is high so that R ≃ rD, and the applicability range coincides with that of the DH approximation, β ≪ 1, which can be written as

(89)

i.e., despite different hierarchy of blobs in subregimes II+a and II+b, the correlation length ξII+a ≃ ξII+b follows the same power law dependence. Correlation blob in both subregimes of regime II+ demonstrates the swollen coil statistics, so that ξII+ ≃ −1/3 5/3 v1/5g3/5 ξII+ , and the polymer volume II+ . Therefore, gII+ ≃ v fraction within the correlation blob ϕII+ ≃ g/ξ3II+ ≃ v−1/3ξII+−4/3 equal to the average coacervate density reads ϕII + ≃ f 8 cs−4v−3

cs < cs** ≃ u−3

(90)

For coacervate in water where u ≈ 1, the upper threshold salt concentration c*s * can be estimated as approximately 1 M. For the regimes with low salt concentration (IΘ, I+, and I−) one should substitute the found values of R into eq 95. The resulting limitation is equivalent to δcs/cs ≪ 1. The inequality

The surface tension in both subregimes reads γII + ≃ ξII +−2 ≃ f 12 cs−6v−4

(91)

However, distinction in statistics of the chain segment inside the Debye blob leads to difference in salt absorption. Salt Partitioning. In subregime II+b r ≃ rD as for II+a, but loops within the Debye blobs are Gaussian. Equation 23 can be used for the total screening radius R: ⎛ r4 ⎞ R ≈ rD⎜⎜1 − 4D ⎟⎟ r p,Θ ⎠ ⎝

uf 1/2 ≪ 1

(97)

makes it fulfilled in the regimes IΘ and I+, ensuring a large number of charged monomers within the correlation blob and weak association regime for IPEC.35 Thus, scaling results remain asymptotically valid at low f, but in aqueous media, u ≈ 1, strong deviations from the scaling laws are not expected even at f equal to several dozens of percent, that is, in the experimentally relevant range of parameters. For regime I− the proposed consideration is valid until

(92)

By substituting an appropriate ϕ = ϕII+ into polymer screening radius rp,Θ, one gets ⎛u r3 u⎞ δcsII + b ≃ ⎜ − ⎟cs ≃ u 4D cs ≃ u1/2f 10 cs−9/2v−3 rD ⎠ rp, Θ ⎝R

(96)

v > v** ≃ −u−5f −2 w (93)

(98)

−4 −2

Since u f ≫ 1, and physically reasonable u values cannot be much lower than unity, one can conclude that v** < −1. However, v > −1 is the natural restriction of our consideration exploiting virial expansion and entirely neglecting ionomer effect which plays a key role at ϕ close to unity.54,55 Therefore, condition v > v** is ensured.

Finally, it is instructive to mention that ξt < rD < ξ is subregime II+a, while rD < ξt < ξ in subregime II+b. The relationship rp,Θ ≃ rp,+ is fulfilled at the II+a/II+b boundary and indicates that screening by ideal polymer loops is changed by screening by swollen loops as v grows. J


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Figure 5. Diagram of states of coacervate in salt concentration−solvent quality coordinates, cs−v. Boundaries between regimes and subregimes are indicated by solid and dashed lines, respectively. Consideration applicability limits are cs < cs** ≃ u−3 and |v| < 1 (not shown in diagram).

Table 1. Scaling Laws for Coacervate Physical Properties (Density ϕ, Correlation Length ξ, Surface Tension γ, and Relative Excess Salt Concentration Ratio δcs/cs as a Measure of Salt absorption) in Different Regimes Shown in Figure 5. Right Column Displays Hierarchy of Lengths Inherent to Each of Regimes regime

ϕ

ξ

γ

δcs/cs

hierarchy of lengths

IΘ IIΘ I− II− I+ II+a II+b

u1/3f 2/3w−4/9 −2/3 f 2c−1 s w

u−1/3f−2/3w1/9 f−2csw1/3

u2/3f4/3w−7/18 −5/6 f4c−2 s w

|v|w−1

|v|−1w2/3

|v|2w−3/2

u4/7f 8/7v−5/7

u−3/7f−6/7v2/7

u6/7f12/7v−4/7

−3 f 8c−4 s v

f−6c3s v2

f12c−6v−4

u4/3f 2/3w−1/9 u1/2f4c−5/2 w−2/3 s u5/4f1/2|v|1/4w−1/4 u1/2f 2c−3/2 |v|w−1 s 10/7 6/7 −2/7 u f v u2/3f10c−16/3 v−10/3 s u1/2f10c−11/2 v−3 s

ξ ≃ rp,Θ < min(ξt,rD) rD < rp,Θ < ξ ≃ r2p,Θ/rD < ξt ξ ≃ ξt < rp,Θ < rD ξ ≃ ξt & rD < rp,Θ ξt < ξ ≃ rp,+ < rD ξt < rD < rp,+ < ξ rD < ξt, rp and ξt, rp < ξ

III. DIAGRAM OF COACERVATE STATES AND DISCUSSION The diagram of coacervate regimes in cs−v coordinates is shown in Figure 5. Power law dependences for coacervate properties and equations for boundaries between different regimes are summarized in Tables 1 and 2, respectively. The diagram is divided into two zones. Zone I of low salt concentrations cs comprises of subregimes IΘ, I−, and I+. Zone II of high salt concentrations cs comprises of subregimes IIΘ, II−, and II+. Though there is no universal equation v = v(cs) describing the whole boundary between these zones, all parts of the boundary, namely lines IΘ/IIΘ, I+/II+a, and I− /II−, are specified by the same condition, that is, equality between polymer-controlled and salt-controlled screening lengths, rp ≃ rD. The former is given by

Table 2. Boundaries between Different Regimes in the Diagram of States Shown in Figure 5a

rp ≃ (uf 2 ϕ)−ν /(2ν + 1)v(2ν − 1)/(2ν + 1) ⎧1/2 for ideal loops with ν = ⎨ ⎩ 3/5 for swollen loops

explicit equation

universal equation

IΘ/IIΘ IΘ/I− IΘ/I+

cs ≃ u−1/3f4/3w−2/9 v ≃ − u1/3f 2/3w5/9 v ≃ u1/3f 2/3w5/9

cs ≃ c*s v ≃ − v* v ≃ v*

IIΘ/II−

1/3 v ≃ − f 2c−1 s w

IIΘ/II+b

1/3 v ≃ f 2c−1 s w

I−/II−

v ≃ − uf−2c2s w

I+/II+a

v ≃ u−1/4f 3c−7/4 s

II+a/II+b

v ≃ u1/2c1/2 s

( ) v ≃ v*( ) v ≃ − v*( ) v ∼ v*( ) v ∼ v*( ) v ≃ − v*

cs cs*

rp,Θ ≃ rD rp,Θ ≃ ξt rp,Θ ≃ ξt cs cs*

cs cs*

−1

−1

cs cs*

cs cs*

equations for length scales

2

−7/4

1/2

r2p,Θ ≃ rDξt r2p,Θ ≃ rDξt rp,Θ ≃ rD rp,+ ≃ rD rD ≃ ξt and rp,Θ ≃ rp,+

a Threshold salt concentration cs* ≃ u−1/3f4/3w−2/9; threshold second virial coefficient v* ≃ u1/3f 2/3w5/9. The sign “∼” in the universal equation column means equality with accuracy to w in arbitrary power and appears for boundaries between good solvent regimes only where effect of three-body interactions is negligible.

⎪ ⎪

boundary

(99)

In the low salt zone I, screening of any charge is provided mainly by polyelectrolyte chains and leads to independence of ratio δcs/cs on the salt concentration cs in subregimes IΘ, I−, and I+ (see Table 1). Interaction of salt ion with the correlation shell in this zone does not considerably change the charge correlations within the coacervate and ensures the free energy gain arising upon salt ion absorption almost independent of cs. Other coacervate physical properties such as its density ϕ,

correlation length ξ, and surface tension γ are also independent of cs. On the contrary, in the zone II of high salt concentrations, screening is mostly due to salt ions that weaken correlation attraction between PAs and PCs. Together with excess osmotic pressure of trapped salt ions, this results in lower globule density and decreasing dependence of ratio δcs/cs on salt K


Article

Macromolecules concentration cs in (sub)regimes IIΘ, II−, and II+ (see Table 2). In these (sub)regimes (except for the II− regime where coacervate is stabilized predominantly by short-range binary attractive interactions) coacervate density ϕ is a decreasing function of cs. Zone I of low salt concentrations is wider at negative v values than at positive ones because two-body attraction between polymer units leads to higher ϕ and lower rp values. Depending on the strength of volume interactions and their effect on the coacervate properties, zones of Θ-solvent (regimes IΘ and IIΘ), poor solvent (subregimes I− and II−), and good solvent (regimes I+ and II+) are distinguished. In the former zone the coacervate physical properties are independent of the second virial coefficient v (see Table 1). The upper and lower boundaries of this zone (lines IΘ/I+, IIΘ/II+, and IΘ/I−, IIΘ/II−) are symmetric with respect to the cs-axis. In poor solvent zone, ϕ and δcs increase with growing |v| and depends on the value of w, while under good solvent conditions the coacervate density, surface tension, and excess salt concentration decrease upon an increase in the strength of binary repulsive interactions (i.e., increasing v values) and remain independent of w. The boundaries between low/high salt concentration zones (I/II) and poor/Θ-/good solvent (−/Θ/+), drawn together allow to distinguish all subregimes on the phase diagram except for II+a and II+b, where a delicate interplay between excluded volume interactions and salt screening leads to slightly different exponents in power law dependence of δcs on cs. An alternative way to define boundaries between different regimes and subregimes is to compare the characteristic length scales inherent to the system. The first one is the Debye length rD ≃ (ucs)−1/2 dependent only on salt concentration cs, and the second one is the size of the thermal blob ξt ≃ |v|−1w2/3, i.e., the length below which a single neutral polymer chain is not perturbed by two-body interactions. These characteristic lengths are independent of the coacervate structure and density ϕ. The third length is the polymer screening radius rp given by eq 99 and being the function of ϕ. Representation of the boundaries as the relationships involving these length scales is given in the right column in Table 2. The established hierarchy between these three lengths and correlation length ξ within the coacervate delineated in Table 1 corresponds to each subregime of the diagram. The dependence of the coacervate correlation length ξ on the salt concentration cs at different values of v is shown in Figure 6. Curve parts below and above black dashed line denoting the Debye radius value correspond to low and high salt zones, I and II, respectively. We note that δcs is positive in all the regimes due to stronger charge correlations within the coacervate compared to the outer salt solution. This result is valid for pointlike salt ions not interacting with each other and polymer via nonelectrostatic forces, the case which was considered herein. Recent theoretical studies24,39 and simulations2,10 of coacervates formed by strongly charged polyions demonstrated that the excluded volume of salt ions (especially in the case of bulky ions with dimensions on the order of the monomer size24) is crucial for explanation of experimentally detected salt expulsion from the dense coacervate at high salt concentrations.10 At the same time, at low cs salt ions are known to prefer coacervate phase rather than supernatant even in the case of strongly charged PEs.10,39,56

Figure 6. Correlation length ξ within the coacervate as a function of the salt concentration cs at different values of the second virial coefficient in the log−log plot: (i) v > v* (good solvent, red curve); (ii) v = 0 (Θ-solvent, green curve); (iii) v < −v* (poor solvent, blue curve). Dashed black line represents dependence of the Debye radius rD on cs. In Θ-solvent, ξ ≃ rD at cs ≃c*s where black dashed and solid green line intersect each other.

Our analysis is devoted to weakly charged chains forming coacervates of relatively low density. In this case, salt ions finite size effects are weak and can be neglected, at least in low salt regimes. For instance, in the regime IΘ we have found that correlation energy gain per salt ion due to absorption by the coacervate phase grows as ∼ϕ1/4 with increasing coacervate density ϕ (see eqs 21 and 5), while the penalty for excluded volume interactions with monomers grows steeper; it is linear in ϕ if two-body repulsive interactions are taken operative. Thus, correlation contribution to the salt ion chemical potential dominates over excluded volume term at low ϕ, and region of salt concentrations where δcs > 0 widens with decreasing ionization degree of polyions f providing lower density of the coacervate. This makes our predictions on salt partitioning at low f reliable. A particular case of high salt concentrations and bulky salt ions when their mutual short-range repulsion becomes significant requires separate consideration which is beyond the scope of performed analysis. It is also worth noting that we avoided the comparison of correlation free energies derived within RPA and the scaling approach because these methods adopt different reference states for the free energy. In the framework of scaling model (which ascribes ≃−kBT energy per correlation blob within the coacervate34) it is a system of blobs separated at infinite distance from each other, while within the RPA it is infinitely distant infinitesimal charges. Moreover, the RPA results require proper regularization (renormalization) via subtraction of polyions self-energy43,57 in order to get the negative correlation free energy of the complex.43 Finally, it was assumed that interactions of the PA uncharged monomers with each other and with uncharged monomers of the PC are identical. However, incompatibility (immiscibility) of polyions under appropriate conditions might lead to either macroscopic phase separation into polyanion/polycation-rich coacervate phases or intracoacervate microphase separation.26,58 This problem will be the subject of the forthcoming work.

IV. CONCLUSIONS In summary, the effects of salt concentration and solvent quality on the equilibrium structure of coacervate comprising of weakly charged polyelectrolytes with equal degrees of ionization have L


Article

Macromolecules

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been examined. Different scaling regimes were revealed, and the corresponding diagram of states was constructed. The power law dependences obtained with the scaling approach have been confirmed and supplemented by either existing or newly derived results based on the random phase approximation (RPA). The results of the scaling and RPA approaches are in agreement in the whole considered range of salt concentrations and solvent qualities. Three important length scales within the interpolyelectrolyte complex (IPEC) were distinguished: the Debye radius rD, the polymer screening radius rp, and the thermal blob size ξt. The established hierarchy of these three length scales and the correlation radius ξ within the coacervate signify each of the considered IPEC regimes.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (O.V.B.). ORCID

Artem M. Rumyantsev: 0000-0002-0339-2375 Oleg V. Borisov: 0000-0002-9281-9093 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank the ANR for funding of the MESOPIC Project ANR15-CE07-0005. This work was partially supported by Russian Foundation for Basic Research, Grant 17-03-01115a.

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REFERENCES

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