Liquid Crystalline and Isotropic Coacervates of Semiflexible

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Cite This: Macromolecules XXXX, XXX, XXX−XXX

Liquid Crystalline and Isotropic Coacervates of Semiflexible Polyanions and Flexible Polycations Artem M. Rumyantsev† and Juan J. de Pablo*,†,‡ †

Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, United States Center for Molecular Engineering, Argonne National Laboratory, Lemont, Illinois 60439, United States

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‡

ABSTRACT: Polyelectrolyte complex coacervation plays an essential role in compartmentalization in living cells and is important in multiple scenarios of prebiotic evolution. This has fueled considerable interest in associative phase separation, which occurs in solutions of oppositely charged biological polyelectrolytes, including those containing double-stranded DNA. The high bending rigidity of DNA duplexes has an effect on the properties of the polymer-rich phase. In this paper, we combine scaling approaches and the random phase approximation (RPA) to develop a theory of coacervates formed from semiflexible (rigid) polyanions and flexible polycations. At low stiffness of the polyanion, coacervates are isotropic liquids with two different correlation lengths, equal to the mesh sizes of the polyanion and polycation interpenetrating semidilute solutions. When the polyanion stiffness exceeds a threshold, the coacervate undergoes liquid crystalline ordering (LCO). The formation of a nematic phase is induced by anisotropic excluded volume and anisotropic Coulomb interactions between semiflexible polyanions. Our theoretical predictions of LCO within the coacervate formed from flexible and semiflexible polyelectrolytes are consistent with experimental studies.

1. INTRODUCTION Over the past decade, considerable progress has been made in understanding of phase separation phenomena in solutions of oppositely charged polyelectrolytes (PEs). The attraction between polyanions and polycations results in the formation of highly diluted supernatant and condensed polymer-rich macroscopic phases. For many pairs of synthetic and biological PEs, this latter phase can be either liquid (called a PE coacervate) or solid (referred to as a PE complex), depending on the solvent type and salt concentration.1−3 Addition of salt is believed to diminish the number of cross-chain (intrinsic) ion pairs,4 thereby weakening the attraction between PEs and favoring coacervate formation. Coacervates are thermodynamic equilibrium fluid phases, highly homogeneous, and optically transparent.5−8 In contrast, at low salt concentrations, formation of glassy and heterogeneous (e.g., porous) PE complexes occurs.2,8,9 Inhomogeneity manifests itself in the opaqueness of the material and is attributed to the fact that relaxation processes within the glassy complex become kinetically arrested. It was recently reported by the Schlenoff group that the dried PDADMA/PSS complex immersed in pure water exhibits gradual swelling over a period of approximately 1 year.8 Molecular dynamics simulations of this system also support the concept of a glassy state for the salt-free complex.10 The transition between a liquid coacervate and an amorphous complex is currently thought to be a continuous, glass transition-type process.4,8,10 It is worth noting that both coacervates and complexes are isotropic (disordered). © XXXX American Chemical Society

Interest in complex coacervation is partly fueled by its relation to compartmentalization processes within living cells, which are a result of liquid−liquid phase separation11−13 and are related to mechanisms of prebiotic evolution.14−18 A number of recent works have been devoted to coacervation between oppositely charged polypeptides19−23 as well as between positively charged polypeptides and oligo/polynucleotides.3,12,24,25 In the latter case, hybridization of the DNA chains was found to greatly affect the coacervate state and structure: when mixed with polylysine at low salt concentrations, single-stranded (ss) DNA forms a liquid coacervate phase, while double-stranded (ds) DNA yields solid complexes that precipitate.3 This fact was attributed to the high linear charge density and high stiffness of the DNA duplex.3 At higher salt concentrations, however, dsDNA is also capable of forming liquid coacervates.3 The limited flexibility of dsDNA may have interesting consequences. Liquid coacervates are similar to a semidilute polymer solution,23,26,27 and solutions of neutral semiflexible polymers are prone to exhibit liquid crystalline order (LCO).28,29 One can therefore expect formation of a liquid crystalline (LC) coacervate if at least one of the polyions is sufficiently stiff. These simple arguments have been experimentally confirmed.25 At relatively high salt concentrations, around 0.8 M, a Received: April 17, 2019 Revised: June 5, 2019

A

DOI: 10.1021/acs.macromol.9b00797 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

The paper is organized as follows. In section 2, the structure of the isotropic coacervate is considered. The instability of the isotropic state with respect to LC nematic ordering is studied in section 3. Section 4 provides our concluding remarks.

secondary phase transition was observed into isotropic and anisotropic (cholesteric) regions within coacervate droplets formed from short stiff 22-bp dsDNA oligonucleotides and poly(L)lysine. Interestingly, formation of solid complexes was reported at lower salt concentrations, rather than liquid coacervates.25 However, substitution of over 50% of dsDNA with the nucleotide triphosphate promoted formation of a liquid coacervate and LC subcompartments (different from the cholesteric ones) in it, even at low solution salinity. Parallel packing of 22-bp dsDNA oligomers within the cores of complex coacervate core micelles occurs via their complexation with poly(L)lysine-b-poly(ethylene glycol) and provides yet another observation of an equilibrium LC (columnar) coacervate phase.24 In this state, the ends of the short dsDNA rods are attracted to each other by stacking interactions.30 Replacement of dsDNA by ssDNA leads to the disappearance of the LCO24,25flexible chains do not offer shape anisotropy (i.e., rigidity) and do not undergo orientational ordering.28,29 Since coacervate droplets serve as a model for membranefree protocellular systems, it is of interest to consider how LCO might influence processes that are important for prebiotic evolution.12,17,31 First, the isotropic−anisotropic phase separation within coacervates containing chains of limited flexibility provides extra compartmentalization. Second, different structures and densities of ordered and disordered coacervate phases promote selectivity of small molecule uptake, leading to molecular crowding. Third, for solutions of dsDNA oligonucleotides, it has been shown that orientational ordering enhances their ligation into long dsDNA chains.31−34 This effect, caused by the spatial proximity of stacking ends of oligomers in nematic or columnar mesophases,32,34 could also take place in LC coacervates. More importantly, the growth of the dsDNA length enhances selectivity for ssDNA complementary oligomers and improves the stability of the LC phase, which in turn broadens the range of external conditions over which ligation reactions are accelerated by ordering.31−33 This loop of positive feedback allows one to view LCO-assisted ligation as an autocatalytic reaction, providing steady elongation of the DNA duplexes until they acquire their own catalytic activity, thereby representing an important early stage of prebiotic evolution.31,32 Theoretical treatments of the structure of coacervates formed from flexible and semiflexible chains are not available. Despite significant achievements in the area of complex coacervation over recent years,35−48 past studies have been devoted to isotropic (and homogeneous but not microstructured49) coacervate states, and the problem of orientational ordering remains unexplored. Here we mention work by Fredrickson and co-workers,50 where the stability regions of isotropic and nematic states of coacervates formed from oppositely charged rodlike PEs were delineated. However, the case of coacervates formed from flexible polycations and semiflexible/rodlike polyanions considered here is also of particular interest because it corresponds to the majority of the experimentally explored biopolymeric coacervates consisting of rigid dsDNA and flexible polypeptides. In this work, our theoretical considerations are restricted to isotropic and nematic (the simplest LC) phases of liquid coacervates. Despite other mesophases not being studied here, our analysis identifies the conditions needed for the LCO in complex coacervates containing flexible and semiflexible PEs. It also provides basic insight into underlying physics of orientational ordering in these systems.

2. ISOTROPIC COACERVATE We consider an isotropic complex coacervate of flexible polycations and semiflexible polyanions and denote by f+ and f− the fractions of ionic monomers. The size of the monomer unit along the backbone for both chains is given by d. Their diameter also equals d, and the monomer volume in this section is assumed to be d3. For the flexible polycation, the statistical segment length is also on the order of d, while that for the rigid polyanion equals l and considerably exceeds d, l ≫ d. We consider Θ solvent conditions for both polyions, so that their correlation attraction, which provides the driving force for formation of the coacervate, is equilibrated by three-body repulsions, with w being the dimensionless (expressed in d6 units) third virial coefficient for the three-body interactions between any monomers. The correlation attraction between polyanions and polycations within the coacervate can be calculated in the framework of the random phase approximation (RPA). To do this, the statistics of the polyions at all length scales must first be elucidated. While semiflexible chains exhibit rodlike statistics at r < l and ideal coil at r ≫ l, irrespective of the coacervate density and f+/f− ratio, the conformations of the flexible chains change depending on the f+ and f− values. At relatively low polycation charge density, f+ ≤ f−, their charge is compensated at the length scales above the electrostatic blob size De,+ ≃ (uf2+)−1/3, leading to Gaussian statistics for flexible polycations at all length scales (all lengths are given in d units). Here u = e2/ϵdkBT is the Bjerrum length in a solvent with dielectric constant ϵ. Once f+ ≫ f−, at intermediate lengths De,+ < r < ξ+, flexible polycations can be viewed as rods of the electrostatic blobs,51 with ξ+ being the polycation correlation length. One can realize this by considering the limit where f− → 0, which corresponds to a (semi)dilute solution of flexible polycations.52 Similar conformations of flexible polycations within a coacervate with flexible polyanions have been recently predicted for f+ ≫ f−.53 For this reason, the cases f+ ≤ f− and f+ ≥ f− are examined separately. Special attention is paid to the interrelation between the polycation and the polyanion correlation lengths, which are in general different, ξ+ ≠ ξ−. The coacervate structure corresponding to different scaling regimes can be seen in Figure 1. At low f+ values, f+ ≤ f−, in Regimes I and II, flexible polycations have ideal-coil conformations. The difference between these regimes is that

Figure 1. Schematic representation of the coacervate structure in different scaling regimes. Increasing the fraction of ionic groups in the flexible polycations, f+, leads to successive transitions between Regimes I, II, and III. B

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Macromolecules ξ+ < ξ− in Regime I and ξ+ > ξ− in Regime II. At high f+ values, f+ ≥ f−, in Regime III, polycations are locally stretched because of the Coulomb repulsion of charges in them and ξ+ is much larger than ξ−. 2.1. Coacervate with Ideal-Coil Statistics for Flexible Polycations, f+ ≤ f− (Regimes I and II). 2.1.1. Correlation Attraction between Polyions. Scaling Arguments. To understand the correlation attraction between polyions within the coacervate, we consider a Debye−Hückel (DH)-type screening of a test point-like charge within the coacervate. Screening by nonpoint-like objects depends not only on their concentration but also on their shape:54 for linear chains in fractal conformations,55 the screening radius R is a function of the Flory exponent ν (i.e., the fractal dimension df of the conformations, df = 1/ν). Following ref 27, one can estimate the charges of the fragments of the coil-like polycation and the rodlike polyanion within the screening radius R: Z+ ≃ f+R2 and Z− ≃ f−R. The latter result is valid for R < l, providing ν = 1 for the polyanion within the screening radius. These parts of the polyions can be considered as Z+ and Z− multivalent ions, and their concentrations are nZ+ ≃ f+ϕ+/Z+ and nZ− ≃ f−ϕ−/Z−. Thus, a linearized Poisson−Boltzmann (PB) equation defining the electrostatic potential ψ(r) near the test point-like charge can be written as Δψ − 4πu(n Z+Z+2 + n Z−Z −2)ψ = 0

oppositely charge polyions within the coacervate, the relevant osmotic pressure can be approximated by27 −3 l o o− rcoil , ϕ ≪ ϕ* pcorr ≃ −R−3 ≃ o m o −3 o o− rrod , ϕ ≫ ϕ* n

rν ≃ (uf 2 ϕ)−ν /(2ν + 1) ≃ (uf 2 ϕ)−1/(2 + df )

(6) −1/2

This result reproduces the Debye length rD ≃ (uf ϕ) for point-like screening objects (ν → ∞ equivalent to df = 0), rcoil for ideal coils (ν = 1/2, see eq 3), rrod for rods (ν = 1, see eq 3), and the screening radius by coils swollen in an athermal solvent (ν = 3/5, see refs 56 and 27). For the symmetric coacervate formed by chains of the same fractal dimension and equal charge densities, R ≃ rν and the correlation osmotic pressure predicted by eq 5 2

pcorr ≃ −(uf 2 ϕ)3ν /(2ν+ 1) ≃ −(uf 2 ϕ)3/(2 + df )

(7)

is in agreement with a result recently derived in the framework of the RPA.55 RPA Considerations. Rigorous calculation of the correlation osmotic pressure can be performed by means of the RPA (see Appendix A). With the assumptions that (i) both polycation and polyanion are infinitely long and (ii) exhibit ideal-coil and rodlike statistics at all length scales, respectively, one can obtain

(1)

(2)

pcorr = −

(3)

1 1 − 3 3 24 2 πrcoil 12π 2rrod

(8)

with

Here the numerical coefficients are not known but will be determined later in the framework of the RPA. Different power-law dependencies of the screening radii R on the volume fraction of coils and rods, ϕ+ and ϕ− emphasize the importance of the screening object’s shape.54 With due regard to the coacervate zero net charge, ϕ+ f+ = ϕ− f−, eq 2 can be 2 −3 written as R−2 ≃ r−4 coil R + rrod R, which has the following asymptotic solutions l r , ϕ ≪ ϕ* o f2 o coil *≃ + ϕ R≃o m o o uf −4 o rrod , ϕ ≫ ϕ* n

(5)

Here the pressure is expressed in kBT/d units and the free energy in kBT units. Our scaling analysis can be verified as follows. A similar reasoning applied to the screening of objects with an arbitrary Flory exponent ν leads to Z ≃ f R1/ν and the screening radius

It is useful to introduce a screening radius by the polycation coils rcoil and by the polycation rods rrod rcoil ≃ (uf+2 ϕ+)−1/4 rrod ≃ (uf−2 ϕ−)−1/3

(Regime I) 3

The expression in parentheses defines the value of the total screening radius R−2 ≃ 4πu(f+2 ϕ+R2 + f−2 ϕ−R )

(Regime II)

rcoil = (48πuf+2 ϕ+)−1/4 rrod = (4π 2uf−2 ϕ−)−1/3

(9)

being the screening radii by coils and by rods found within the RPA. Equation 9 is consistent with the scaling findings and serves to specify the numerical coefficients in eq 3. Equation 8 for the correlation osmotic pressure not only reproduces the asymptotic scaling laws, eq 5, valid in the limiting cases of rodand coil-dominated screening, rcoil ≫ rrod and rcoil ≪ rrod amounting to ϕ ≫ ϕ* and ϕ ≪ ϕ* but also remains precise in the entire intermediate range between them. 2.1.2. Equilibrium Coacervate Density. The Coulomb attraction between PEs is balanced by their short-range repulsion, which for weakly charged chains f± ≪ 1, forming a low-density coacervate, ϕ ≪ 1, can be considered at the level of the virial expansion. The second and third virial coefficients of interacting rods or the statistical segments of semiflexible chains with Kuhn segment l in isotropic solution scale as Brod ≃ dl2 and Crod ≃ d3l3 to within ln(l/d) terms.28,29,58 Two-body and three-body interactions in the isotropic state can therefore be treated as those for a system of disconnected monomers.28,29,58 This is also applicable to flexible chains with l ≃ d. In a Θ solvent, only three-body interactions should be taken into account. For an isotropic coacervate consisting of an interpenetrating semidilute solutions of flexible and semiflexible chains, the free energy density of all three-body interactions can be written as - vol ≃ wϕ3. Here w ≃ 1 is the

(4)

In the former case, coacervate density is very low and screening by coils dominates the behavior of the system. The regime of rod-dominated screening is entered when coacervate density surpasses a threshold value ϕ*. It should be noted that the scaling analysis above provides us only with the characteristic length scale for screening, R, but it is unable to predict the exact law for the potential ψ(r) near the test charge, which is not simply of the DH type, ψ(r) ≈ exp(−r/R)/r. Specifically, for a coacervate formed from flexible polyions with an equal content of ionic groups, f+ = f−, the DHtype decay of ψ(r) is modulated by the oscillations with the period equal to R and R ≃ rcoil.56,57 Since R is the only length associated with the screening of Coulomb interactions and the correlation attraction between C

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Regime I. In this regime, the polyanion network mesh size ξ− is much larger than ξ+, as shown in Figure 2. Their ratio

dimensionless third virial coefficient for monomers, and ϕ = ϕ+ + ϕ− is the total polymer volume fraction within the coacervate. The equality pcorr + pvol = 0

ij f yz ξ− ≃ u1/4f +−1/4 f−3/4 ≃ jjjj − zzzz j f̃ z ξ+ k +{

(10)

between the repulsive part of the osmotic pressure pvol = ϕ

∂- vol − - vol = 2wϕ3 ∂ϕ

u1/2f+1/2 f− [6w(f+ + f− )]1/2

≃ (uf+ f− )1/2

Figure 2. Scaling depiction of coacervate in Regime I: ξ− > ξ+. Polycations have Gaussian conformations at all length scales because the polycation blob is much smaller than the electrostatic blob, ξ+ < De,+.

The blob picture of the coacervate in Regime I shows that the Coulomb attraction between polyions can be considered as an adsorption of polycations on the polyanion. To verify our findings one can envision a coacervate elementary cell as a cylindrical capacitor with the outer radius (thickness of the polycation shell surrounding polyanion) equal to ξ− and apply arguments recently used by Rubinstein et al.53 for treatment of the asymmetric coacervates (f+ ≠ f−) of flexible PEs. The electric field within the capacitor created by the polyanion follows E(r) ≃ − uρ−/r, with r being the distance from the capacitor axis and ρ− being the linear charge density of the plates. The energy of the electric field per elementary cell, which coincides with the mesh of the polyanion semidilute solution, equals

(12)

1/3

(13)

A crossover between Regimes I and II given by f− ≃ f+̃ takes place when the linear charge density of the polyanion ρ− ≃ f− is equal to that of the string of polycation electrostatic blobs, ρe,+ ≃ f+ge,+/De,+ ≃ f+̃ . The latter equation clarifies the physical meaning of f+̃ and can be viewed as its definition; ge,+ ≃ D2e,+ is the number of monomers per electrostatic blob. The Gaussian statistics of the polycations at all length scales −1 and the fact that ϕ+ ≃ ϕ at f+ ≤ f− leads to ξ+ ≃ ϕ−1 + ≃ ϕ . The polycation correlation length can be eventually written as ξ+ ≃ (uf+ f− )−1/2

FCoul ≃ ξ−

∫d

ξ−

E 2 (r ) iξ y r dr ≃ uρ−2 ξ− lnjjj − zzz u kd{

(17)

and becomes smaller as ξ− decreases. However, as ξ− decreases, the number of short-range repulsive (three-body) interactions between polycation monomers grows as well as the free energy cost for the shell compression. The latter is equal to the number of the polycation blobs within the coacervate elementary cell

(14)

Fvol ≃

Similarly, for polyanions, equation ϕ− ≃ f+ϕ/f− and the relationship ξ− ≃ ϕ−1/2 that follows from their rodlike conformations can be used to find ξ− ≃ u−1/4f +−3/4 f−1/4

(16)

(11)

with the last equality owing to f+ ≤ f−. Regime I is implemented at ϕ ≫ ϕ* or equivalently at f− ≫ f+̃ with ij f yz f+̃ ≃ jjj + zzz juz k {

≫1

increases with increasing polyanion charge density f−.

and the correlation attractive contribution defines the equilibrium ϕ value in the isotropic coacervate at any f+ ≤ f−. Note that the supernatant is assumed to be highly dilute, owing to the long length of the PEs and salt-free conditions;57,59 its osmotic pressure is therefore asymptotically zero and there is coexistence between a supernatant and a coacervate phase. We now consider two limiting cases, ϕ ≫ ϕ* and ϕ ≪ ϕ* (Regimes I and II), corresponding to the correlation attraction due to density fluctuations of semiflexible polyanions and flexible polycations, eq 5. In these I and II regimes, Coulomb interactions are screened primarily by rods and coils, respectively (cf. screening of the test charge). Further analysis is primarily aimed at revealing the microscopic structure of the coacervate being a double-semidilute solution with ξ+ ≠ ξ− in both regimes.53 Note that here we use the nomenclature of Rubinstein et al.,53 who used this terminology to describe two interpenetrating semidilute solutions with different correlation lengths. 2.1.3. Coacervate with Larger Correlation Length of Semiflexible Polyanions, ξ− > ξ+ (Regime I). At ϕ ≫ ϕ*, when screening by rods prevails and pcorr = −1/12π2r3rod = −uf2−ϕ−/3, the coacervate density reads ϕ=

3/4

ξ−3 ≃ ξ−3ϕ+3 ξ+3

(18)

The value of ξ− is defined by the balance of the free energy contributions, FCoul ≃ Fvol. In view of cell electroneutrality, ξ3−ϕ+ f+ ≃ ξ−ρ−, and neglecting logarithmic factors, one can find a shell thickness: ξ− ≃ u−1/4f−3/4 ρ1/4 + − . Finally, using ρ− ≃ f−, we get eq 15. In our analysis, for simplicity, the density of the polycation shell was assumed to be uniform. One can expect a higher local

(15)

One can also show that the screening radius within the coacervate equals R ≃ rrod ≃ ξ+, confirming the leading role of semiflexible polyanions on the electrostatic correlations in D

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Macromolecules shell density, ϕ+(r), in the vicinity of the semiflexible polyanions, and lower ϕ+(r) values at the periphery of the elementary cell, where the electric field created by the polyanion is partly screened by the shell. Accounting for the shell inhomogeneity, however, does not change the scaling laws for its average density and thickness, eqs 15 and 12.53,60 2.1.4. Coacervate with Larger Correlation Length of Flexible Polycations, ξ+ > ξ− (Regime II). When f+ ≤ f− ≤ f+̃ and when the coacervate density is much lower than ϕ*, the correlation osmotic pressure can be written as 3 pcorr = −1/24 2 πrcoil = −(uf+2 ϕ+)3/4 /(12π )1/4 , indicating that in Regime II the attraction between chains originates mainly from fluctuations of the flexible polycation density. The polymer volume fraction in the coacervate is given by ij yz uf+2 f− 1 jj zz ϕ = 2/3 j zz j 2 (3π )1/9 jj w 4/3(f+ + f− ) zz k {

2.2. Coacervate with Flexible Polycations Stretched at Intermediate Length Scales, f+ ≥ f− (Regime III). Once f+ exceeds f−, ϕ+ < ϕ−, and the short-range repulsions between stiff chains overcome those between flexible coils, leading to coacervate swelling. The coacervate average density becomes lower than the polymer volume fraction within the electrostatic blob, eq 19, and in Regime III, the system cannot be treated as an array of densely packed blobs. In this case, polycations exhibit rodlike statistics at intermediate length scales, De,+ ≤ r ≤ ξ+, and their conformations are akin to those in a salt-free semidilute solution of flexible PEs,52 as shown in Figure 4.

1/3

≃ (uf+2 )1/3 (19)

and is equal to that within the electrostatic blob, ϕ ≃ ϕe,+ ≃ D−1 e,+ . Therefore, the polycation correlation length is equal to the electrostatic blob size ξ+ ≃ De, + ≃ (uf+2 )−1/3

(20)

Figure 4. Scaling representation of the coacervate in Regime III: ξ+ > De,+ > ξ−. Flexible polycations exhibit Gaussian statistics within electrostatic blobs, at r < De,+, and outside coacervate elementary cell, at r > ξ+. At intermediate lengths, De,+ < r < ξ+, they form rods of electrostatic blobs.51

while the mesh size of the polyanion semidilute solution is given by ξ− ≃ u−1/6f +−5/6 f−1/2

(21)

The structure of the solution is schematically depicted in Figure 3, and the ratio between the correlation lengths ij f yz ξ− ≃ u1/6f +−1/6 f−1/2 ≃ jjjj − zzzz j f̃ z ξ+ k +{

Scaling Analysis. In Regime III, polycations can be viewed as charged rods with an effective linear charge density f+̃ given by eq 13. The test charge is screened by the rods of polyanions and effective rods of polycations (the latter are the strings of polycation electrostatic blobs). The charges of these rod fragments are Z− ≃ f−R and Z+ ≃ f+̃ R, and their concentrations are nZ− ≃ f−ϕ−/Z− and nZ+ ≃ f+ϕ+/Z+. From the PB eq 1 for the test charge and identifying the screening radius R one arrives at

1/2

≪1 (22)

is consistent with this blob picture. The screening length within the coacervate is on the order of the electrostatic blob size, R ≃ rcoil ≃ ξ+ ≃ De,+. The number of rodlike polyanion fragments within the polycation correlation volume ξ3+ can be estimated as ξ2+/ξ2− ≃ −1 u−1/3f1/3 + f− and is well above unity, as illustrated in Figure 3. However, the free energy of their three-body repulsions at f+ < f− is still lower than that for polycation monomers. The latter is on the order of kBT per one blob of ξ+ dimension. The polycation electrostatic blob is the coacervate elementary electroneutral cell in Regime II.

ij uf f (f ̃ + f )ϕ yz − zz ≃ (uf+̃ f− ϕ)−1/3 R ≃ jjjj + − + zz j z f+ + f− (23) k { ̃ Here the inequality f− < f+ < f+ has been invoked. Estimating the correlation osmotic pressure −1/3

pcorr ≃ −R−3 ≃ −uf− f+̃ ϕ

(24)

and equating it to the repulsive pressure pvol ≃ wϕ (caused mainly by short-range three-body repulsions between rods), one can derive the following scaling law for the coacervate density 3

ϕ ≃ (uf+̃ f− )1/2 ≃ u1/3f+1/6 f−1/2

(25)

The polyanion correlation length, given by ξ− ≃ becomes ξ− ≃ u−1/6f +−1/12 f −−1/4

ϕ−1/2 − , (26)

The electroneutrality constraint of the coacervate elementary cell, written as ξ3+ϕ− f− = ξ+ f+̃ , defines the mesh size of the polycation network

Figure 3. Scaling representation of the coacervate in Regime II: ξ+ ≃ De,+ > ξ−. Coacervate consists of densely packed polycation electrostatic blobs. Polycations are ideal coils at all length scales.

ξ+ ≃ u−1/3f+1/12 f −−3/4 E

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Table 1. Coacervate Physical Properties (density ϕ, polycation and polyanion correlation lengths, ξ+ and ξ−) and Hierarchy of Length Scales in Different Scaling Regimesa regime

range

ϕ

ξ+

hierarchy of lengths

f+̃ < f− f+ < f− < f+̃ f− < f+

ξ−

I II III

1/2 u1/2f1/2 + f− 1/3 2/3 u f+ 1/2 u1/3f1/6 + f−

u−1/2f−1/2 f−1/2 + − −1/3 −2/3 u f+ −3/4 u−1/3f1/12 + f−

u−1/4f−3/4 f1/4 + − −1/6 −5/6 1/2 u f+ f− u−1/6f−1/12 f−1/4 + −

ξ+ ≃ R < ξ− ξ− < ξ+ ≃ R ≃ De,+ ξ− < De,+ < R < ξ+

R is the screening radius of Coulomb interactions in coacervate, and D+ ≃ u−1/3f−2/3 is the size of the polycation electrostatic blob; f+̃ ≃ u−1/3f1/3 + + .

a

The low value of the ratio between the correlation lengths ij f yz ξ− ≃ u1/6f +−1/6 f−1/2 ≃ jjjj − zzzz j f̃ z ξ+ k +{

1/2

≪1 (28)

seen in Figure 4 coincides with the one in Regime II, eq 22. The screening radius given by eq 23 equals R ≃ u−1/3f−1/6 f−1/2 + − and greatly exceeds the size of the electrostatic blob, De,+ < R < ξ+. This result justifies the assumption that in Regime III polyanions screen the test charge in a manner analogous to that of the rods with f+̃ linear charge density. Interestingly, the correlation attraction between polyions at f− < f+ can be effectively viewed as an adsorption of semiflexible polyanions onto a highly charged string of polycation electrostatic blobs.53 Straightforward calculations of Coulomb and short-range (three-body) repulsion free energies, FCoul ≃ uf2+̃ ξ+ln(ξ+/De,+) and Fvol ≃ ξ3+ϕ3− (cf. eqs 17 and 18), allow one to reproduce the scaling laws for the correlation lengths and the coacervate density, eqs 25−27. RPA Considerations. To calculate the correlation osmotic pressure pcorr of oppositely charged PEs in the framework of the RPA, the conformational statistics within the coacervate must be known. In Regime III, a scaling analysis indicates that polycations can be considered as rodlike chains with f+̃ linear charge density, i.e., f+̃ fraction of ionic monomers. The pcorr in the semidilute solution of oppositely charge rodlike PEs with unequal charge densities, f− ≠ f+̃ , derived in Appendix B pcorr = −

Figure 5. Diagram of coacervate states in f+ vs f− coordinates; log−log plot. Boundaries f+ ≃ u−2 and f− ≃ u−1 confine the range of the theory’s applicability. For aqueous systems, u ≃ 1, the vertical solid and dashed lines coincide, u−2 ≃ u−1/2 ≃ 1; so do the horizontal solid and dashed lines, u−1 ≃ u−1/2 ≃ 1.

The changes of the internal coacervate structure are depicted in Figure 6a. They are accompanied by simultaneous changes of the correlation lengths (mesh sizes) of the interpenetrating semidilute solutions of flexible and semiflexible PEs, ξ+ and ξ−, illustrated in Figure 6b. The nonmonotonic change of ξ− in Regimes II−III is remarkable. Increasing f− results in a lower number of polyanions required for coacervate electroneutrality. In Regime II, when ϕ is independent of f−, this fact explains the increasing behavior of ξ− ≈ f1/2 − . However, in Regime III, the higher the f− the denser the coacervate (see Table 1); this causes a polyanion correlation length decrease, ξ− ≈ f−1/4 − . The polyanion and polycation correlation lengths of the coacervate are equal to each other at the I/II crossover: ξ+ ≃ ξ− at f− ≃ f+̃ , i.e., at f+ ≃ uf3− (see Figures 6b and 7b). Increasing the ionization of flexible polycation causes consecutive coacervate transitions between scaling Regimes I, II, and III if f− < u−2. Coacervate density increases monotonically: ϕ ∼ f1/2 at f+ ≤ uf3− (Regime I), then ϕ ∼ + f2/3 until f < f (Regime II), and finally ϕ ∼ f1/6 + + − + at higher fractions of positively charged groups (Regime III). The dependence of the coacervate density and the correlation lengths on f+ in this case are shown in Figure 7. At u−2 < f− < u−1, increasing f+ triggers transition between Regimes I and II (see Figure 5). 2.3.1. Limits of the Theory’s Applicability. Both scaling analysis and RPA considerations remain valid until correlations between charged monomers are weak,27,37,45,56 that is, the energy of interaction of the test charge with the surrounding screening cloud of the PEs should not exceed the thermal energy, u/R ≪ 1.27 Substitution of the screening radius R for

uf+̃ f− ϕ (29)

3

is in agreement with the scaling prediction of eq 24. A pressure balance yields the coacervate density ϕ=

(uf− f+̃ )1/2 6 w1/2

≃ u1/3f−1/2 f+1/6

(30)

which agrees with the scaling estimates. However, in Regime III, the RPA cannot provide the exact numerical coefficients because the f+̃ definition, eq 13, is solely based on scaling arguments. 2.3. Scaling Regimes of Isotropic Coacervates. The results of our considerations are summarized in Table 1 and in the diagram of coacervate states shown in Figure 5. The crossover between Regimes III and II reads f− ≃ f+, and that between the II and I Regimes reads f− ≃ f+̃ ≃ u−1/3f1/3 + . Increasing the ionization of the semiflexible macromolecules results in the growth of the polymer volume fraction ϕ within the coacervate in Regime III until f− is lower than f+ (see Figure 6a). The coacervate density saturates with further growth of f− and remains almost unchanged (at the scaling level of accuracy) in a wide region f+ < f− < f+̃ , where Regime II operates. Once f− exceeds the second crossover value f+̃ , a tightening of the coacervate proceeds (Regime I). F

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Figure 7. Dependence of coacervate properties on the fraction f+ of ionic monomers in the flexible polycations at f− < u−2; log−log plot: (a) average coacervate density ϕ; (b) correlation lengths of polyanions ξ− and polycations ξ+.

Figure 6. Dependence of coacervate properties on the fraction f− of ionic monomers in semiflexible polyanions; log−log plot: (a) average coacervate density ϕ; (b) correlation lengths of polyanions ξ− and polycations ξ+. Electrostatic blob size and the polymer volume −1/3 1/3 ̃ and ϕe,+ ≃ u1/3f2/3 f+ . fraction in it are De,+ ≃ u−1/3f−2/3 + + ; f+ ≃ u

The restriction l > R is needed to ensure rodlike statistics for semiflexible polyanions within the screening radius, which is adopted in both scaling and RPA calculations (see also Appendix A). Fulfillment of these constraints limits the applicability of the results of section 2 to the isotropic coacervate as well as the analysis of its instability with respect to LCO performed in section 3.

each regime results in the following dimensionless energy values

l u3/2f 1/2 f 1/2 , Regime I o o o + − o o o o u o 4/3 2/3 ≃m u f+ , Regime II o o R o o o o u 4/3f 1/6 f 1/2 , Regime III o o + − (31) n which should be lower than unity. The electric field near the polyanions in Regime I and the polycations in Regime III has local cylindrical symmetry, and this can trigger Manning-type condensation of the counterions at high linear charge densities. This effect can be neglected at uρ− ≃ uf− ≪ 1 in Regime I and at uf+̃ ≃ u2/3f1/3 + ≪ 1 in Regime III. In Regime I, counterion condensation sets in when the polycation blob contains only one charged monomer, as in the case of PE adsorption on a planar charged surface.60 Combining these restrictions with those given by eq 31 one can finally find the requirements f+ ≪ u−2 and f− ≪ u−1 that define the range of the accuracy of our analysis. These limitations are shown in Figure 5 and imply that the linear charge density of PEs should be low.

3. INSTABILITY OF ISOTROPIC COACERVATE WITH RESPECT TO NEMATIC ORDERING The analysis above has shown that the coacervate internal structure is analogous to a double-semidilute solution of flexible and semiflexible polymers. One can expect that at high enough coacervate density ϕ and stiffness of the rigid polyanions l/d the coacervate should undergo an LCO, as observed in semidilute solutions of semiflexible neutral polymers.61−66 The formation of a nematic coacervate phase is not merely due to the anisotropic excluded volume repulsions between the locally anisometric semiflexible polyanions responsible for the orientational ordering in solutions of neutral stiff polymers61 but also their anisotropic Coulomb interactions.50,67,68 Below, we show that electrostatic forces favor the ordered, parallel orientation of polyanions. Our analysis is restricted to the case of f+ ≤ f− (Regimes I and II of G

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Macromolecules the isotropic coacervate) when the flexible polycations do not exhibit local anisotropy of shape and hence interactions. In what follows we focus on the calculation of spinodals rather than binodals because (i) they can be found in analytical form, (ii) they require no additional assumptions and simplifications, and (iii) they provide us with a reasonable estimate of the boundary between the isotropic and the nematic phases. Moreover, the detailed spinodal analysis below allows us to predict how ordered the nematic phase is. 3.1. Spinodal Equation. To describe the LCO of the polyanions, a normalized orientational distribution function f(n) = f(θ) (with ∫ f(θ)dΩ = 1) is introduced. Here θ = ∠(u;n) is the angle between the director u and the vector of local orientation of the semiflexible polyanion n directed along the chain backbone. The order parameter S=

1 (3 cos2 θ − 1) 2

=

chains.28,61,71−74 A polyanion of contour length L can be considered as L/l rods, each of length l and diameter d, interacting via pairwise excluded volume interactions (first term) and pairwise attractive interactions (second term) rep attr + Fvol Fvol = Fvol 4lϕ = 5 − f (n1)f (n1)B(γ ) dΩ n1 dΩ n2 πd lϕ − 5 − u0(1 + ϰS2) 2d

∫

Here 5 is the total number of polyanion chains in the coacervate, and following classic works,61,69 the monomer volume is assumed to be πd3/4.75 In the repulsive term, B(γ) = 2l2d|sin γ| is the second virial coefficient of interacting rods, with their mutual orientation defined by the angle between their axes, γ = ∠(n1;n2).69 Substitution of the Onsager trial function allows for analytical calculation of the Frep vol term and a series expansion about α = 0: Ä É lϕ ÅÅ lϕ 2I (2α) 1 2 ÑÑÑÑ rep = 5 − 2 2 |α → 0 ≈ 5 − ÅÅÅÅ1 − α ÑÑ Fvol d ÇÅÅ 360 ÖÑÑ d sinh α Ä ÉÑ Ñ lϕ− ÅÅÅ 5 = 5 ÅÅÅ1 − S2 ÑÑÑÑ (38) d ÅÇ 8 ÑÖ

∫ 12 (3 cos2 θ − 1)f (θ) dΩ (32)

provides a measure of the orientational order: S = 0 in the isotropic state, with f i = 1/4π = const(θ), and S = 1 for a fully ordered state, where fo(θ) = (δ(θ) + δ(π − θ))/2π sin θ. To perform a spinodal analysis of the isotropic phase instability it is convenient to use the Onsager trial function for the orientational distribution of polyions69 f (θ ) =

α cosh(α cos θ) 4π sinh α

2 The second term Fattr vol contains isotropic and anisotropic (∼S ) attractive contributions, with the latter being reminiscent of the Mayer−Saupe theory of nematic ordering in low molecular weight LCs.65 A phenomenological parameter ϰ is used to ensure that the anisotropic contribution to the attractive forces is small when compared to the isotropic: the exact numerical value of ϰ ≪ 1 only weakly affects the system’s behavior.28,74 Since we consider a Θ solvent for the polyanion, in the isotropic state, excluded volume effects and short-range attractions should compensate for each other, yielding Fvol = 5lϕ−(2 − u0)/2d = 0 at S = 0. The resulting value of u0 = 2 is in fact the definition of the Θ temperature.28 In the weakly disordered state, S ≪ 1, the change in the free energy density for short-range interactions due to ordering equals

(33)

where parameter α describes the anisotropy. At α = 0, this distribution corresponds to the isotropic state, and at α → ∞ it corresponds to a completely ordered state (see ref 70 for the explicit S(α) function). The weakly anisotropic state is described by α ≪ 1, where the distribution function ÄÅ ÉÑ ÑÑ 1 ÅÅÅ α2 2 Å f (θ ) ≈ (3 cos θ − 1)ÑÑÑÑ = fi + δf (θ ) ÅÅ1 + ÑÑÖ 4π ÅÅÇ 6 (34) is the sum of the isotropic one, f i, and the weak orientational perturbation δf(θ). Substitution of this f(θ) into eq 32 shows that the orientational perturbation around the disordered state causes the change of the order parameter δS = S ≈

α2 15

δ - vol = (35)

Fvold3 4i5 y ≈ − jjj + ϰzzzϕ−2S2 πk8 V {

(39)

It can be seen that accounting for orientationally dependent attractions between rods produces a small correction to the main contribution coming from their excluded volume anisotropic repulsions, since ϰ ≪ 5/8. To calculate the second, Coulomb correlation term in eq 36, we use the method proposed for solutions of rodlike PEs by Potemkin et al.,67,76−78 who took into consideration the effect of rod orientation on their correlation attraction at the level of the RPA. In ref 50, Fredrickson and co-workers generalized this approach to the coacervate formed from oppositely charged rodlike polyions, where the PEs of both charges undergo orientational ordering. This method is applicable not only at high salt concentrations, when Coulomb interactions become effectively short-range and can be treated as pairwise (see earlier works79−82 where effective electrostatic second virial coefficients were derived), but also in the absence of salt. In the latter case, electrostatic forces are essentially long range and many body, and it is important to note that the RPA is capable of properly describing collective electrostatic interactions.50,67 By means of the RPA calculations performed in

and the dependence of S on α is quadratic. Result 35 is consistent with the series expansion of the explicit S(α) function.70 A stability analysis of the isotropic coacervate with respect to nematic ordering can be carried out through the orientationaldependent part of the coacervate free energy, Ftot, and permits calculation of the free energy associated with the δf(θ) perturbation. The free energy can be written as Ftot = Fvol + Fcorr + Forient

(37)

(36)

with contributions from anisotropic volume interactions, anisotropic Coulomb correlation interactions, and orientational entropy. The first two interaction terms favor orientational ordering, while the third accounts for the reduction of the orientational freedom of polyanions, which hinders alignment. The first term can be written in the framework of the Onsager approach to a solution of rods69 generalized by Khokhlov and Semenov to the case of semiflexible polymer H

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Macromolecules Appendix C, it is found that the orientational perturbation of the disordered state diminishes the Coulomb correlation energy and the corresponding term at S ≪ 1 reads δ -corr

i 5 8 2 = − uf−2 ϕ−S2jjjj1 + 6 3πρ k

yz zz z {

−1

(40)

3

(41)

indicates83 whether the screening of the test charge within the coacervate and the correlation attraction between PEs in it are mainly provided by density fluctuations of semiflexible polyanions (ρ ≫ 1 and rrod ≪ rcoil) or flexible polycations (ρ ≪ 1 and rrod ≫ rcoil), see also section 2.1.1. In the former case, corresponding to Regime I of the isotropic coacervate, Coulomb interactions between highly charged rigid polyanions are almost bare, with little screening by polycations. For this reason, the orientation-induced change in the correlation free energy density 5 1 δ -corr = − uf−2 ϕ−S2 ≃ − 3 S2 6 rrod

(44)

pers Forient =5

L l

∫

(∇f (θ ))2 dΩ 4f (θ )

(45)

fj ≈ δ - orient

4d α 4 10d ϕ− = ϕ−S2 πl 90 πl

(46)

pers ≈ δ - orient

4d α 4 30d = ϕ S2 ϕ πl − 30 πl −

(47)

δ -tot = δ -vol + δ -corr + δ -orient ≤ 0 (42)

(48)

The spinodals of the coacervate, which represent the main result of our analysis, are given by É−1 ÄÅ −1Ñ Ñ ÅÅ πuf−2 ijj ÅÅ ij 8 2 yzz ÑÑÑ 8ϰ yz l zz + freely jointed: = 4ÅÅÅϕ−jj1 + zz ÑÑÑ jj1 + ÅÅ k 3πρ { ÑÑÑ 5 { 3 k d ÅÇ Ö É ÄÅ 1 − Ñ −1Ñ ÅÅ πuf−2 ijj Å i 8 2 yzz ÑÑÑ 8ϰ zy l zz + persistent: = 12ÅÅÅÅϕ−jjj1 + zz ÑÑÑ jj1 + ÅÅ k 3πρ { ÑÑÑ 5 { 3 k d ÅÇ Ö

coincides with that in the salt-free solution of rodlike PEs, where screening by small counterions is also negligible. In the opposite limit, ρ ≪ 1 (Regime II), orientationally dependent interactions between rods are much weaker, because they are strongly screened by polycations r3 2 ρ 5π uf−2 ϕ−ρS2 ≃ − 3 S2 ≃ − coil S 6 16 2 rrod rrod

∫ f (θ)ln(4πf (θ))dΩ

The 3-fold difference between these results is remarkable, pers fj δ - orient = 3δ - orient . The coacervate is unstable with respect to nematic ordering when

67

δ -corr = −

L l

Equation 44 is consistent with the orientational entropy of a solution of separate rods, while the Fpers orient expression is the Lifshitz orientational entropy.29 The orientation of the persistent chain costs more entropy as compared to the freely jointed chain.71 This fact is supported by the series expansion of the corresponding free energies in the limit of weak orientations, S ≪ 1

Similar to the δFvol contribution, this term is proportional to the squared order parameter and favors the formation of the nematic phase. The dimensionless parameter ρ equal to ij r yz ρ = jjj coil zzz j rrod z k {

fj Forient =5

(43)

(49)

Note that eq 40 was derived under the assumption that the polycation density within the coacervate is homogeneous, which is necessary for the RPA calculations performed in Appendix C. In Regime I, nonuniform density of the polycation shell (see section 2.1.3) may lead to a stronger screening of Coulomb interactions between semiflexible polyanions. One can therefore expect to have slightly lower numerical coefficients in eqs 40 and 42, i.e., a slightly weaker tendency for orientational ordering at ρ ≫ 1. Finally, the last term in the total free energy, eq 36, is responsible for the orientational entropy of the polyanions. It depends on the flexibility mechanism inherent to these chains. Polyanions can be either freely jointed or persistent (wormlike); they are schematically shown in Figure 8a and 8b, respectively. The appropriate forms of the free energy read28,74

In these equations, the equilibrium polyanion density within the isotropic coacervate is a function of the total coacervate density, ϕ− = f+ϕ/( f+ + f−), defined from zero osmotic pressure within the coacervate, eq 10 1 1 + = 2wϕ3 3 2 3 24 2 πrcoil 12π rrod

(50)

For simplicity, the final spinodals given by eq 49 were derived using the Onsager trial function which approximates the distribution function f(θ), eq 33. The more rigorous approaches, such as (i) an expansion of the f(θ) function into the series of the Legendre polynomials at S ≪ 150,67 and (ii) minimization of the free energy functional Ftot{f(θ)} with respect to f(θ) followed by bifurcation analysis of the resulting integral equation,84 could show that these spinodals are general and independent of the form of the trial function. 3.2. Stability Diagrams of Isotropic Coacervate. Numerical solution of eqs 49 and 50 leads to the spinodals for the disordered coacervate with respect to the nematic order for polyanions with different types of flexibility. They are shown in Figure 9b in the form of polyanion stiffness vs polycation ionization coordinates, l/d vs f+. The polymer volume fraction ϕ within the isotropic coacervate at the spinodal is shown in Figure 9a. It does not depend on the flexibility mechanism. Each spinodal separates the region of the isotropic coacervate (meta)stability implemented at low chain

Figure 8. Schematic representation of (a) freely jointed and (b) persistent (worm-like) polyanions. Both types of chains are prone to LCO because of the local shape anisotropy, l/d ≫ 1. Chains should be considered as semiflexible when their contour lengths considerably exceed the Kuhn segment length, L ≫ l. I

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jointed chains and invoke a trivial generalization to the case of persistent polyanions. The higher propensity of freely jointed chains (and separated rods with the same orientational entropy, eq 44) to undergo orientational ordering probably accounts for the LC phases that have been detected in coacervates formed from short double-stranded oligonucleotides24,25 but, to our knowledge, not seen yet for long dsDNA. Short rods capable of end-to-end aggregation (stacking) are very similar to freely jointed chains, with the flexibility localized at the rod junction points,30,33 while dsDNA is a highly persistent macromolecule. 3.2.1. Driving Force for Nematic Ordering. Both shortrange (excluded volume) and long-range Coulomb interactions promote mesophase formation, and it is interesting to evaluate their relative contributions. For this purpose, in Figure 9c we plot spinodals that would be valid if the anisotropic part of only the volume interactions (blue dashed curve) or the electrostatic interactions (magenta dashed curve) was taken into account. The former curve can be calculated assuming δ -corr = 0, i.e., setting the dimensionless Bjerrum length to zero in eq 49, u = 0. The latter can be found from the same equation at ϕ− = 0, which would correspond to omitting the short-range contributions δ - vol to eq 48. Figure 9c shows that Coulomb interactions are responsible for LCO in the low-density coacervate (at low f+ values) when interactions between semiflexible polyanions are weakly screened by coiled polycations. In the limiting case f+ → 0, when ρ → ∞, one can find the spinodals freely jointed: persistent:

12 l = d πuf−2

l 36 = d πuf−2

(51)

which are similar to those derived for rodlike PEs in (semi)dilute salt-free solution where their electrostatic interactions are also almost bare.67 Substituting f− = 0.3 and u = 1, we get l/d = 42.4 for freely jointed chains and a much (3-fold) higher value l/d = 127 for persistent chains, in agreement with the numerical data shown in Figure 9b and 9c. From the analogy between low-density coacervates and (semi)dilute solutions of charged rods, one can expect the nematic coacervate formed primarily due to Coulomb interactions to be akin to a nematic phase in a solution and hence to be relatively weakly ordered, S ≪ 1.67,76 As polycation ionization and coacervate density grow, as seen in Figure 9a, anisotropic short-range interactions become dominant. The corresponding contributions to the spinodals are given by

Figure 9. (a) Coacervate density and (b) spinodals with respect to the nematic ordering: comparison of chains with freely jointed and persistent flexibility mechanisms. (c) Contributions to the spinodal coming from short-range and Coulomb interactions; empty circle at the spinodal is the crossover between electrostatically and sterically induced instability. Curves are plotted in the range f+ ≤ f− for f− = 0.3, u = 1, ϰ = 0.1, and w = 1.

l 4 ij 8ϰ yz jj1 + zz = d ϕ− k 5 {

−1

freely jointed:

stiffness l/d from the region of the coacervate instability at high l/d ratio, where the formation of the nematic phase, either being the only phase or coexisting with the isotropic phase, is ensured. This theoretical result is in line with the experimentally observed impact of DNA hybridization on the appearance of the mesophases: with all else being equal, coacervates formed from stiff DNA duplexes are liquid crystalline, while those containing single-stranded flexible oligonucleotides are isotropic.24,25 Equations 49 predict that the spinodal for the persistent polyanion is three times higher than that for the freely jointed. For this reason, we now plot spinodals only for the freely

l 12 ij 8ϰ yz jj1 + zz = d ϕ− k 5 {

−1

persistent:

(52)

For freely jointed chains, this contribution is shown by a blue dashed line in Figure 9c. Equations 52 are particular results of our spinodal analysis that are valid not only for coacervates but also for semidilute solutions. At ϰ = 0, when short-range attractions are absent but short-range-excluded repulsion is operative, they can be compared with the rigorous binodal positions of the isotropic− J

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0.3 into eq 53 results in f*+ ≃ 0.09, which is a reasonable estimate for the crossover position. It agrees with numerical data shown in Figure 9c: the intersection point of the dashed spinodals indicates the parity of electrostatic and excluded volume contributions to coacervate instability. The corresponding f+ value is the crossover coordinate, and the crossover point at the total (solid) spinodal is denoted by an empty circle. 3.2.2. Effect of Polyanion Charge Density and Solvent Polarity. The influence of the linear charge density f− of the semiflexible polyaion on the coacervate density ϕ and the spinodal position l/d is presented in Figure 10. To compare

nematic phase transition in a solution of semiflexible neutral chains in an athermal solvent. In neutral solutions, the transition between disordered and nematic phases is known to be first order:28 the solution is isotropic at ϕ ≤ ϕi, anisotropic at ϕ ≥ ϕa, and two phases form in the intermediate range of polymer volume fractions, ϕi < ϕ < ϕa. The values are ϕi = 3.25 d/l and ϕa = 4.86 d/l for freely jointed chains,28,61 and ϕi = 10.48 d/l and ϕa = 11.39 d/l for persistent chains, as shown by Khokhlov and Semenov.28,71 The predictions of eq 52 for the spinodal points at ϰ = 0, ϕfs j = 4 d/l and ϕpers = 12 s d/l, are in good agreement with the binodal position. In other words, the binodals and spinodals are quite close to each other, at least in the case of ordering induced by steric effects, which is realized at intermediate f+ values. Therefore, the calculated spinodals provide useful analytical guidelines available in analytical form that can reliably predict the boundaries between the disordered and the ordered coacervate states, though the region of phase coexistence at the Θ temperature is wider.28,29 In the phase coexistence region, the anisotropic (nematic) phase of the neutral athermal solution is highly ordered, with Sa = 0.88 and Sa = 0.49 for freely jointed and persistent chains, respectively.28 One can anticipate comparable values of the order parameter S in the nematic coacervate phase in the vicinity of the spinodal at considerably high f+ when shortrange anisotropic interactions play a major role (see Figure 9c). The weaker ordering of persistent chains is a consequence of their higher orientational entropy.28 Finally, the nematic coacervate should be denser than the isotropic, and the difference in ϕ is expected to be considerably higher for freely jointed chains.28 This fact is supported by the correlation attraction between the chains, which increases as a function of the order parameter (see eq 40 and Appendix C) as well as the lower steric repulsion between semiflexible chains at high S. Crossover Position for LCO Trigger. Simple scaling arguments can be used to find the boundary between LCO induced by Coulomb and excluded volume interactions. The crossover position is given by the ratio between electrostatic and volume orientationally dependent free energy terms -corr uf 2 i 8 2 yzz ≃ − jjjj1 + z 3πρ z{ ϕ− k - vol l uf 2 ϕ−1 , ρ ≫ 1 (Regime I) o o − − ≃o m o o o uf 2 ϕ−−1ρ , ρ ≪ 1 (Regime II) n − −1

Figure 10. (a) Isotropic coacervate density ϕ and (b) spinodals with respect to nematic ordering at various linear charge densities f− of the semiflexible freely jointed polyanion; empty circles at the spinodals show the crossover for electrostatically/sterically induced instability given by f+ = f*+ . Curves are plotted as a function of f+/f− for f− = 0.25, 0.3, and 0.35; other parameters are u = 1, w = 1, and ϰ = 0.1.

(53)

equal to unity. Using the results from Table 1 for isotropic coacervate properties, one can find -corr/- vol ≃ u1/2f−5/2 f +−3/2 ≫ 1 for Regime I, where Coulomb interactions dominate. They also continue to dominate in the part of Regime II (neighboring Regime I). Indeed, in Regime II, the ratio -corr/- vol ≃ uf −4 f +−2 yields the crossover position f +* ≃ u1/2f−2

curves with different f− values, an f+/f− coordinate is chosen for the abscissa. The higher the f−, the stronger the Coulomb correlation attraction between PEs and hence the higher the polymer volume fraction ϕ within the coacervate (see Figure 10a). Figure 10b shows that at higher fraction f− of charged units in the polyanions a lower stiffness l/d is required to induce an orientational instability of the isotropic coacervate; l/d values are lower both in the region of electrostatically and sterically induced LCO. In the region f+ ≤ f*+ of relatively weakly charged polycations this is a direct consequence of the stronger anisotropic Coulomb interaction between rods, see eq

(54)

3 which is above the crossover fI/II + ≃ uf− between Regimes I and II/III II, but below the crossover f+ ≃ f− between Regimes II and III, fI/II ≪ f+* ≪ fII/III . Note that the crossover given by f+* + + passes through the point (u −1/2 ;u −1/2 ), which is the II/III intersection point of fI/II shown in Figure 5. Thus, + and f+ short-range repulsion between rods overcomes electrostatics at f+ ≥ f+*, i.e., in part of Regime II. Substitution of u = 1 and f− =

K

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Macromolecules 51. At f+ > f*+ , these interactions are strongly screened by coils, owing to the fact that ρ ≪ 1 in Regime II. However, l/d still decreases with increasing f−, because the latter makes the coacervate denser and amplifies the effect of anisotropic excluded volume interactions in it. Since f*+ ∼ f2−, the region of electrostatically triggered orientational ordering in strongly charged systems is wider: a shift of the crossover point f+*/f− ≃ u1/2f− at the spinodals from ∼0.25 to ∼0.35 can be seen in Figure 10b. Decreasing the dielectric constant of the solvent ϵ (i.e., increasing the Bjerrum length u ∼ ϵ−1) affects coacervate density and spinodal position in a way similar to growing f− (see Figure 11). This is because Coulomb interactions in

dsDNA, equal to 5.88 e/nm, and (ii) the high polymer volume fraction within the coacervates (up to 50%), resulting in a low polarity as compared to that of the supernatant.15,17,42,85 The persistent length of dsDNA in solution, l ̃ ≈ 50 nm, is due to both its duplex nature and its electrostatic stiffening.3,29 Just as the strength of the stiffening depends on the salt concentration in the solution,29 so it depends on the polymer volume fraction within the coacervate. Note that in the framework of our theory this point was neglected and l was assumed to be independent of ϕ. Since Coulomb interactions within the coacervate are considerably screened, the persistent length value for the neutralized (via methylphosphonate substitution) double helix, l ̃ ≈ 30 nm, provides a reasonable lower estimate for dsDNA rigidity within the coacervate.3 ̃ ≈ 30: such a high Using d ≈ 2 nm, one can find l/d ≈ 2l/d stiffness of dsDNA ensures coacervate liquid crystallinity. In coacervates of short 22-bp dsDNA oligomers,24,25 the length to the diameter ratio of each rod equals 4. However, their end-toend stacking attraction may lead to considerably higher l/d values, sufficient to trigger formation of a LC phase. In the framework of our model, f− ≈ 12 (dozen of charges per effective monomer of 2 nm size) corresponds to the linear charge density of dsDNA. The presented here theory is consistent for coacervates of weakly charged PEs, f± ≪ 1, and cannot therefore be directly (quantitatively) applied to dsDNA-based coacervates. It is nevertheless capable of providing a qualitative physical description of these systems and serves to clarify the crucial role of Coulomb interactions in both coacervation and LCO, especially at high f− (see Figure 10b). Moreover, a recent comparison between the RPA and the field-theoretic simulations has shown that the former provides reasonable predictions of the coacervate density even at high f±, i.e., beyond the region of rigorous accuracy.59,86 Experimental observation of LC compartments in coacervates containing dsDNA (l/d ≫ 1), but not single-stranded counterparts (l/d ≃ 1),24,25 supports our main theoretical findings. It is challenging to develop an accurate quantitative theory of LCO in complex coacervates formed from highly charged flexible and semiflexible PEs. One can try to elaborate approaches that are successful for the isotropic state of coacervates formed from flexible highly charged chains. Among these are field-theoretic simulations,59,86 renormalized Gaussian fluctuation theory,47 and methods based on liquid state theory.41,87 Breaking spherical symmetry of the correlation functions (structure factors), however, substantially complicates the task. For dsDNA-containing systems, accounting for duplex chirality, which may lead to the formation of cholesteric rather than nematic mesophases,25,88 is also important. 3.3.2. Nematic Ordering in Regime III. The nematic ordering considerations above were restricted to Regimes I and II of the isotropic coacervate, where flexible polycations are ideal coils at all length scales. In Regime III, the rodlike statistics of these chains at intermediate length scales, De,+ ≤ r ≤ ξ+, lead to their local anisotropy. Since their parallel orientation may diminish the Coulomb correlation energy of the coacervate, the orientational ordering of flexible polycations should also be taken into account. Therefore, to develop a consistent theory of nematic ordering at f+ ≥ f− one should introduce two independent order parameters, S− and S+, defining the local orientation of both PEs.50 One can expect that in Regime III LCO is quite favorable owing to the combined effect of (i) weakly screened anisotropic Coulomb

Figure 11. (a) Isotropic coacervate density ϕ and (b) spinodals with respect to the liquid crystalline ordering, l/d vs f+/f−, for freely jointed chains in solvents of different polarity, i.e., for u = e2/ϵdkBT equal to 1, 1.5, 2, and 3; other parameters are f− = 0.3, w = 1, and ϰ = 0.1. Empty circles at the spinodals represent the crossovers between electrostatically and sterically induced instability given by f+ = f*+ .

media of low polarity are stronger. The considerable shift toward higher f+/f− values of the crossover position, f*+ ∼ u1/2, when solvent polarity decreases is remarkable: f+*/f− ≈ 0.35 at u = 1 and f+*/f− ≈ 0.55 at u = 3, as indicated by the empty circle position at the spinodals in Figure 11b. 3.3. Discussion. 3.3.1. Theory and Experiment. The results shown in Figures 10 and 11 indicate that in salt-free dsDNA containing coacervates formation of LC phases can only approximately be treated in the framework of the Onsager approach. In these systems, anisotropic Coulomb interactions are important and should also be taken into consideration. This is a consequence of (i) the high linear charge density of L

DOI: 10.1021/acs.macromol.9b00797 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules interactions between locally stretched flexible polycations and (ii) anisotropic excluded volume interactions between semiflexible polyanions, which are the major polymer component of the stoichiometric coacervates at low f− when ϕ− ≃ ϕ ≫ ϕ+. The Regimes I and II considered in section 3 are particularly interesting, because they correspond to the coacervates formed from dsDNA and positively charged polypeptides/synthetic PEs, where the linear charge density of the stiff polyions is well above that of the flexible chains, f− ≥ f+. 3.3.3. Effect of Salt on the Liquid Crystalline Order. Our analysis was focused on salt-free solutions of flexible polycations and semiflexible polyanions. Increasing the salt concentration can hinder or promote LCO, depending on the state of the polymer-rich phase in the salt-free solution. If this phase is a liquid coacervate,89 addition of salt (i) diminishes the density of the coacervate and (ii) screens Coulomb anisotropic interactions between semiflexible polyanions, thereby suppressing the orientational order and favoring formation of an isotropic phase. When the polymer-rich phase in the salt-free solution is a solid precipitant (which is likely kinetically frozen),3,25,89 increasing salt concentration can trigger a continuous precipitant-to-coacervate transition. In the vicinity of the transition region, the resulting coacervate phase can be sufficiently dense to exhibit LCO.25 In this case, further addition of salt should lead to coacervate swelling accompanied by a transition from LC to isotropic phase of the coacervate. At very high salt concentrations, a solution of oppositely charged PEs is usually uniphase, irrespective of the state of the polymer-rich phase in the absence of salt.3,25,89

by steric and by Coulomb interactions. Increasing the content of ionic groups in the polyanion, f−, and decreasing the solvent polarity both favor formation of a nematic coacervate, reducing the l/d threshold and expanding the region of electrostatically triggered LCO. The mechanism of polyanion flexibly also affects mesophase formation: coacervates formed by freely jointed semiflexible polyanions (or the corresponding disjointed rods) become unstable with respect to nematic ordering at 3-fold lower polyanion stiffness l/d than those formed from persistent (worm-like) molecules. In contrast to the continuous transition between the liquid coacervate and the solid complex, the one between the isotropic and nematic coacervates is first order. It is accompanied by the coexistence of these two phases and by a change of the system’s symmetry: a more ordered liquid crystalline phase is less symmetric than the isotropic phase. The theoretical predictions pertaining to LCO within the coacervate are corroborated by recent experimental studies. We hope these findings will provide insight into the structuring and compartmentalization (via isotropic−anisotropic phase separation) within oligo- and polynucleotide coacervates systems, whose study is important to elucidate the principles behind prebiotic evolution.

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APPENDIX A

Correlation Osmotic Pressure in Coacervate of Flexible and Rodlike Polyelectrolytes

4. CONCLUSION In this paper, we studied the structure of the coacervates that form from flexible polycations and semiflexible polyanions in a Θ solvent for both chains. A scaling approach, the random phase approximation (RPA), and a theory of nematic ordering of partially flexible polymers have been combined to provide a description of the structure of the isotropic liquid coacervate and to find the conditions leading to intracoacervate liquid crystalline ordering (LCO). In the isotropic (disordered) state, the coacervates can be considered as interpenetrating semidilute solutions of polycaions and polyanions, each with an appropriate correlation length, ξ+ and ξ−. If the fraction of ionic groups in polycations, f+, is low, their conformations are Gaussian. At higher f+ values, the charge of the polycation cannot be neutralized by polyanions at small length scales. As a consequence, polycations adopt extended conformations at intermediate lengths, De,+ < r < ξ+, higher than the electrostatic blob size, De,+, but lower than the mesh size of their solution, ξ+. The scaling laws derived here for the coacervate density ϕ, as well as both correlation lengths ξ+ and ξ−, are consistent with more rigorous RPA considerations in the whole range of f+ and f− values, i.e., in all scaling regimes of the isotropic coacervate. Formation of nematic coacervates occurs when the polyanions are sufficiently stiff, i.e., when the ratio l/d between their Kuhn segment length l and the diameter d exceeds a threshold value. LCO within the coacervate is induced by the anisotropic short-range excluded volume interactions between semiflexible polyanions and the anisotropic long-range Coulomb forces acting between them. The latter are the main driving force for the mesophase formation in low density coacervates, while the former plays a key role at higher ϕ. A crossover was found between the orientational ordering caused

The correlation osmotic pressure in a solution of oppositely charged flexible and semiflexible PEs can be calculated by means of the RPA, considering independent fluctuations of polycation and polyanion densities. We denote by N+ and L− the contour lengths of the polyions. In a Θ solvent, their structure factors can be written as approximations of the Debye function29,49 and the Neugebauer result,55,90 respectively −1 f coil (q) ≈ 1 +

q2N+ 12

(A1)

−1 f rod (q) ≈ 1 +

qL− π

(A2)

Semiflexible polyanions are considered as rodlike molecules at all length scales. This is reasonable until the screening radius within the coacervate, R ≃ min{rrod;rcoil}, is lower than the Kuhn segment length l, because the main contribution to the RPA integral comes from high wave vectors, qR ≥ 1. The RPA correction to the free energy density can be calculated in a standard way55,91 -RPA = FRPA 1 2

ij

∫ lnjjjjj1 + k

d3 1 = V 2

ij det[G −1(q)] yz d3q zz = zz 3 z ij u = 0 { (2π ) k

∫ lnjjjjj det[G−1ij(q)]|

yz d3q 48πuf 2 ϕ+ 4π 2uf−2 ϕ− + 2 2 + −1 zzzz −1 q (q + πL− ) q (q + 12N+ ) z (2π )3 { 2

(A3)

Here the G−1 ij (q) matrix is given by M

DOI: 10.1021/acs.macromol.9b00797 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

Equation A9 provides a reliable estimate of the asymptotic behavior for pcorr in the limits of high and low ρ. The deviation from the exact integral value is less than 10% in the entire range of ρ. The correlation osmotic pressure is approximately equal to the sum of the individual contributions from coils and rods.

Gij−1(q)

4πuf+ f− jij 4πuf+2 zyz q2N+ yzz 1 ijj jj zz jj1 + zz − jj 2 + zz 2 j z jj q zz 12 { N+ϕ+ k q j zz = jjj zz jj 2 4 uf f π qL− yzzzzz 4πuf− jj 1 ij + − jj jj1 + zzzz − + jj L−ϕ− k π {zz{ q2 q2 k (A4)

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Correlation Osmotic Pressure in Coacervate of Rodlike Polyelectrolytes with Unequal Linear Charge Densities

Equation A4 takes into account the structure of the PEs and the Coulomb interactions between them. The integral A3 is divergent at q → ∞. The correlation energy correction, -corr , is the difference between the calculated RPA result and the selfenergy of coiled and rodlike polyions,57,92 -corr = -RPA − -self . The latter can be written as57,67 -self

1 = 2

ij 4π 2uf 2 ϕ yz d3q 48πuf 2 ϕ+ jj − − + 2 2 + −1 zzzz jj 2 − 1 j q (q + πL− ) q (q + 12N+ ) z (2π )3 k {

∫

Consider a solution of oppositely charged rodlike polyions of L± length, which contain f± fractions of ionic monomers. Their structure factors are given by −1 f rod, (q) ≈ 1 + ±

at

∂-corr ∂− -corr = ϕ RPA − -RPA ∂ϕ ∂ϕ

1 2 d3q

∫

=−

ÄÅ 1 ÅÅÅÅ ÅÅ 3 Å 12π 2rcoil ÅÇ

∫0

∞

-RPA =

(A7)

(48πuf2+ϕ+)−1/4

The values of the screening radii, rcoil = and rrod = (4π2uf2−ϕ−)−1/3, are only due to coiled polycations and rodlike polyanions, respectively. The dimensionless parameter ρ defines whether the screening is primarily due to coils or rods83 3

1/4 1/4 2 u f− ϕ− f+3/2 ϕ+3/4

pcorr

(B2)

1 2

ij

∫ lnjjjjj1 + k

4π 2uf+2 ϕ+ 2

q (q +

πL+−1)

+

4π 2uf−2 ϕ− yzzz d3q z q (q + πL−−1) zz (2π )3 { (B3) 2

ÉÑ 3 ÄÅ ÅÅ ij 1 1 yzzÑÑÑÑ d q 1 ÅÅ j ÅÅ zÑ j ÅÅ 1 + (qr )3 − lnjj1 + (qr )3 zzÑÑÑ (2π )3 = − 12π 2r 3 rods rods {Ñ rods k ÑÖ ÅÇÅ uf+ f− ϕ

1 = 2

=−

∫

(B4)

3 −1/3

Here rrods = (4π uf+ f−ϕ) is the radius of joint screening by rodlike polyanions and rodlike polycations. 2

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(A8)

APPENDIX C

Correlation Free Energy Induced by an Orientational Perturbation of Rods in Coacervate of Flexible and Rodlike Polyelectrolytes

At ρ ≫ 1 (i.e., ϕ ≫ ϕ*, see eq 3), screening by rods dominates. In this case, I(ρ) ≈ ρ, and the correlation osmotic pressure, pcorr ≈ − 1/12π2r3rod, is caused by density fluctuations of rodlike polyanions.50,55,67 When ρ ≪ 1 and I(ρ) ≈ π /2 2 , the coil screening provides the main contribution to the 3 fluctuation-induced attraction, pcorr ≈ −1/24 2 πrcoil . This result is consistent with that corresponding to coacervates formed from flexible PEs.91,93,94 Because the I(ρ) integral cannot be calculated analytically, we approximate it with an interpolating function, I(ρ) ≈ ρ + π /2 2 , and find pcorr

can be written as

The integral in eq B3 is divergent. It can be regularized via subtraction of the polyion self-energy57,92 in the same manner as in eq A5. However, according to eq A6, the converging correlation osmotic pressure can be calculated directly from the diverging -RPA . For infinitely long rods, L± → ∞, we find

ÉÑ ÑÑ z 2dz ÑÑ = − 1 I(ρ) + ρ ÑÑ 4 3 ÑÑÖ 1 + ρz + z 12π 2rcoil

ij r yz 1 ij π 5 yz ρ = jjj coil zzz = jjj zzz j rrod z 2 k 81 { k {

G−1 ij (q)

Following the method used in Appendix A, one can find the RPA free energy correction

(A6)

ÉÑ ÄÅ ÅÅ (r q)−3 + (r q)−4 1 1 zyzÑÑÑÑ ÅÅ jij rod coil Ñ ÅÅ j z ÅÅ 1 + (r q)−3 + (r q)−4 − lnjj1 + (r q)3 + (r q)4 zzÑÑÑ ÅÅÇ coil rod coil rod k {ÑÑÖ

(2π )3

(B1)

π

4πuf+ f− zyz jij 4πuf+2 qL y 1 ji jj zz jj1 + + zzz − jj 2 + zz L+ϕ+ k π { zz q2 jjj q zz jj zz j j z 2 −1 j 4 uf f π uf 4 π 1 zzz Gij (q) = jj + − − jj zz − + jj L−ϕ− zzzz q2 q2 jj jj qL y zz ij jj jj1 + − zzz zzzz jj j π { z{ k k

For infinitely long chains, N+ → ∞ and L− → ∞, one arrives

pcorr =

qL±

The inverse structure factor matrix

(A5)

The correlation term -corr is convergent due to the regularization of the -RPA via a self-energy subtraction.57,92 Owing to coacervate electroneutrality, ϕ± = ϕf∓/( f+ + f−), the self-energy density is linear in the polymer volume fraction ϕ, and does not contribute to the osmotic pressure: pcorr = ϕ

APPENDIX B

Consider a coacervate of flexible polycations and rodlike polyions. For simplicity, we assume both PEs to be long, N+ → ∞ and L− → ∞. The Coulomb correlation energy in the arbitrary (in general non-isotropic) state can be written as50,67 -RPA =

y 1 ij π 1 1 jj ≈− + ρzzzz ≈ − − 2 3 j 3 3 12π rcoil k 2 2 24 2 πrcoil 12π 2rrod {

1 2

ji

∫ lnjjjjj1 + k

48πuf+2 ϕ+ zyz d3q 4πuf−2 ϕ−L− zz + t q zz (2π )3 q4 q2 {

Here the structure function tq =

(A9) N

∫

(C1)

50,67

sin 2(qnL−/2) f (θ)dn (qnL−/2)2

(C2) DOI: 10.1021/acs.macromol.9b00797 Macromolecules XXXX, XXX, XXX−XXX

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takes into account not only the rodlike shape of the polyanions but also their orientational distribution function, f(θ); θ = ∠(u;n) is the angle between the director u and the local chain orientation n. For the isotropic state, as L− → ∞, one can find tiq = π/qL− and arrive at eq A3 written for the case of infinitely long polyions. In the weakly perturbed anisotropic state, when f(θ) = f i + δf(θ), the structure function can be written as the sum of the isotropic value, tiq, and orientationally induced perturbation, which is linear in the order parameter S tq = tqi [1 + 5SP2(cos θ )]

(C3)

Here P2(cos θ) = (3 cos θ − 1)/2 is the Legendre polynomial of degree 2. The correlation energy associated with the orientation of rods reads (C4)

Note that it is convergent even for L− → ∞. This is because the diverging self-energy of infinitely long rods (see eq A5, first summand), which contributes to -RPA , is independent of their orientations and vanishes at subtraction, eq C4. After a series expansion of the integrand in eq C1 about the isotropic state, tq = tiq, and some algebra, one finds δ -corr = −

25S2 32π 3

∫0

π

∫0

(rrodq)−6 q2 dq

∞

[1 + (rrodq)−3 + (rcoilq)−4 ]2

2π[P2(cos θ)]2 sin θ dθ

(C5)

Here δ -corr ∼ S2 because a linear S term in the series expansion about the disordered state vanishes, ⟨P2(cos θ)⟩|i = Si = 0. We finally find that, as a consequence of the orientational perturbation, the free energy decreases δ -corr = −

5S2 3 8π 2rrod

∫0

∞

z 2 dz

(1 + z

i 5 8 2 ≈ − uf−2 ϕ−S2jjjj1 + 6 3πρ k

3

yz zz z {

+

1 zρ

4/3

2

)

−1

(C6)

Here the integral is approximated by the formula reproducing the asymptotic behavior in the limits of ρ → 0 and ρ → ∞ (i.e., regimes of coil- and rod-dominated screening). The maximal error in eq C6 is below 7%. According to eq C6, orientational ordering of the semiflexible polyions strengthens their correlation attraction.67

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REFERENCES

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2

δ -corr = δ -RPA = -RPA(tq) − -RPA(tqi )

Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Artem M. Rumyantsev: 0000-0002-0339-2375 Juan J. de Pablo: 0000-0002-3526-516X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful to Prof. Kenneth Schweizer, Dr. Nicholas Jackson, Boyuan Yu, and Phillip M. Rauscher for helpful discussions. This work is supported by the Department of Energy, Office of Science, Basic Energy Sciences, Materials Science and Engineering Division. O

DOI: 10.1021/acs.macromol.9b00797 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

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DOI: 10.1021/acs.macromol.9b00797 Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.macromol.9b00797 Macromolecules XXXX, XXX, XXX−XXX