Electrostatic Correlations and the Polyelectrolyte Self Energy
EElectrostatic Correlations and the Polyelectrolyte Self Energy
Kevin Shen and Zhen-Gang Wang a) Division of Chemistry and Chemical Engineering, California Institute of TechnologyPasadena, California, 91125, USA
We address the effects of chain connectivity on electrostatic fluctuations in polyelectrolyte solutions usinga field-theoretic, renormalized Gaussian fluctuation (RGF) theory. As in simple electrolyte solutions (Z.-G.Wang, Phys. Rev. E. , 021501 (2010)), the RGF provides a unified theory for electrostatic fluctuations,accounting for both dielectric and charge correlation effects in terms of the self-energy. Unlike simple ions, thepolyelectrolyte self energy depends intimately on the chain conformation, and our theory naturally providesa self-consistent determination of the response of intramolecular chain structure to polyelectrolyte and saltconcentrations. The theory captures the expected scaling behavior of chain size from the dilute to semi-dilute regimes; by properly accounting for chain structure the theory provides improved estimates of the selfenergy in dilute solution and correctly predicts the eventual N -independence of the critical temperature andconcentration of salt-free solutions of flexible polyelectrolytes. We show that the self energy can be interpretedin terms of an infinite-dilution energy µ el m, and a finite concentration correlation correction µ corr which tendsto cancel out the former with increasing concentration. I. INTRODUCTION
Polyelectrolytes are widely used for many applications,ranging from energy materials to solution additives (e.g.for food, cosmetics, and healthcare products). Polyelec-trolytes are also ubiquitous in biology, as many nat-urally occurring polymers – DNA/RNA, proteins, andsome polysaccharides – are charged. Consequently, un-derstanding the interplay of electrostatics with polyelec-trolyte functionality is central for understanding manybiological processes, and guiding the design of materialssuch as adhesives, drug-delivery microencapsulants, and micro/nanoactuators. The long-range nature of electrostatic interactionsgives rise to nontrivial correlation effects in polyelec-trolyte systems such as ion-condensation and complexcoacervation. A key challenge in the theoreticalstudy of polyelectrolytes is the proper description ofelectrostatic correlations and their consequences on thestructure and thermodynamics.The physical origin of electrostatic correlation is thepreferential interaction between opposite charges. Forsimple electrolyte solutions this correlation is manifestedin the “ionic atmosphere” (Fig. 1) first proposed byDebye and H¨uckel (DH). As the result of the favor-able interaction of an ion with its ionic atmosphere, thefree energy of the system is lowered. Theoretically, forpoint charges, the spatial extent of the ion atmosphere ischaracterized by the well-known inverse Debye screeninglength κ = λ − D = 4 πl b X i z i c i , (1.1)where the Bjerrum length l b = e / πεkT is the lengthscale at which two unit charges interact with energy a) Electronic mail: [email protected] kT and characterizes the strength of charge interactions.These charge correlations modify a host of propertiessuch as osmotic pressure, ionic activities, and mobilities.For dilute electrolyte solutions, the electrostatic free en-ergy density and the associated excess chemical potentialare given respectively by: βf el = − κ πβµ eli = − z i l b κ . (1.2)A salient feature of the DH theory is that electro-static correlations increase with increasing ion valency z i . Thus, polyelectrolytes, being inherently multivalent,have increased correlation effects compared to simpleelectrolyte systems. However, the magnitude of the mul-tivalency effect is unclear due to the spatial extent ofthe polyelectrolyte chains (which introduces new lengthscales) and the conformation degrees of freedom of thepolymers (Fig. 1). FIG. 1. (a) For simple ions, the self energy involves both asolvation contribution and, at finite concentrations, a correla-tion contribution due to the ionic atmosphere. (b) For poly-electrolytes, the ionic atmosphere is modified by neighboringmonomers along the same and other chains, and depends onthe chain structure.
The foundational (and still widely used) theory a r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n of polyelectrolyte coacervation put forth by Voorn andOverbeek completely ignores the contribution of chainconnectivity to electrostatic correlations, directly bor-rowing the DH expressions for charge correlations andhence treating the backbone charges as disconnected freeions. The idea of approximating the backbone chargeas disconnected ions has been adopted in recent theoriesof polyelectrolytes that replace the DH correlation usingmore advanced treatments of simple electrolytes. Another widely used theory is the thermodynamic per-turbation theory (TPT), which aims to capture the ef-fects due to chain connectivity through a leading or-der perturbation expansion around the simple electrolyteresults.
The first-order TPT (TPT-1) forms the ba-sis for a widely used density-functional framework forthe study of inhomogeneous polyelectrolyte systems.
While producing reasonable agreements for bulk proper-ties and inhomogeneous systems at higher densities, likethe VO theory and its variants, TPT-1 offers no insightinto how one can use some of the hallmark design fea-tures of polymers – backbone architecture and chargedistribution on the backbone – to control electrostaticinteractions in materials.An approach that tries to capture the effects of chainconnectivity from the outset is the one-loop expan-sion/Random Phase Approximation (RPA) based ona leading order treatment of fluctuations in the fieldtheoretic partition function.
RPA theories requirea chain structure as input, and for prescribed chainstructure provide explicit expressions describing howchain-connectivity generates extra charge correlations.However, for flexible chains, theories that use a fixed,Gaussian-chain structure for all concentrations (hereafterreferred to as f ixed g aussian- RPA ) overestimate the cor-relation effects, particularly at dilute concentrations. Asa result, for flexible chains fg-RPA predicts critical con-centrations that vanish with increasing chain length, incontradiction to simulation results. Previous work hastried to fix this deficiency by introducing an ad hoc meso-scopic wavevector cut-off while still keeping the Gaus-sian structure factor for the long-wavelength fluctuationcontributions. The RPA theories of polyelectrolytes typically onlyconsider fluctuations in the electrostatic interactions. Inorder to account for excluded volume fluctuations thatare important even for neutral polymer systems, avariational method was employed to treat the “doublescreening”, due to both excluded volume and electro-static interactions. The theory has been used to studyphase separation, adapted to account for counterioncondensation, and applied to study polyelectrolytecoil-globule transitions. Unlike the fg-RPA, this ap-proach allows the polyelectrolyte conformation to self-adapt. However, in the interest of obtaining analyticalexpressions, the double-screening theory pre-integratesthe salt degrees of freedom, resulting in the Debye-H¨uckeldescription of screened interactions between monomerunits, which does not feed back on the effective inter- action between the salt ions. We note that such an ap-proximation of using a screened Couloumb interaction formacroions is a common practice in many analytical andsimulation studies.
Another method that bypasses the fixed-structure as-sumption is the self-consistent
PRISM (sc-PRISM) in-tegral equation approach that employs a structure-dependent effective medium-induced potential; such anapproach has been applied to polyelectrolytes by Yethi-raj and coworkers.
The sc-PRISM has beenquite successful for studying polyelectrolyte solutionstructure, but there have been limited studies on thethermodynamics, perhaps due to the inconsistency be-tween the two different routes (i.e., virial vs. com-pressibility) to obtaining the equation of state from thestructure. Further, PRISM theories require the useof closures, the choice of which is guided by experienceand comparisons to experiment and theory. Finally, todate self-consistent PRISM studies have been limited toonly one polyelectrolyte species. To our knowledge theonly application of PRISM to complex coacervates ig-nored the self-consistent determination of chain struc-ture, and inconsistently borrowed thermodynamic ex-pressions from theories of monomeric solutions. A variational theory close in spirit to the sc-PRISMwas proposed by Donley, Rudnick and Liu to study theconcentration-dependence of polyelectrolyte chain struc-ture. In hindsight, the theory can be understood as asc-PRISM where the form of the effective intrachain in-teraction is motivated by RPA theory using heuristic ar-guments. Their theory yields good agreement with avail-able computer simulation data on the end-to-end distanceof the polyeletrolyte chain as a function of the polyelec-trolyte and salt concentrations.In this work, we study electrostatic fluctuations inpolyelectrolyte solutions using a field-theoretic renormal-ized Gaussian fluctuation theory (RGF). In this theory,the key thermodynamic quantity that captures the elec-trostatic fluctuations is the self-energy of an effective sin-gle chain. For simple electrolytes, the self-energy is theelectrostatic work required to assemble charge from aninfinitely dispersed state onto an ion and is given by βµ el chg = z Z d r d r h chg ( r − r ) G ( r , r ) h chg ( r − r )(1.3)where h chg is the charge distribution on an ion, and G ( r , r ) is a self-consistently determined Green’s func-tion characterizing electrostatic field fluctuations, whichcan be thought of as an effective interaction between twotest charges in an ionic environment. Eq. (1.3) is a uni-fied expression accounting for both the polarization ofthe dielectric medium (e.g. Born solvation and imagecharge interactions) and correlations due to the ionic at-mosphere.For polyelectrolytes, we will see that the self energy isanalogously the work required to assemble charge ontothe polyelectrolyte. However, because of the conforma-tional degrees of freedom, part of the work is due to theentropic change of deforming the chain. The internal en-ergy contribution to the self-energy resembles Eq. (1.3)and involves the single-chain structure factor, reflectingthe spatial extent of the polyelectrolyte chain. The RGFtheory prescribes a self-consistent determination of theeffective intrachain structure along with the effective in-teraction G ( r , r ) as an inherent part of the theory.The rest of this article is organized as follows. In Sec-tion II we present a full derivation of the RGF theory forpolyelectrolyte solutions. At this stage, our derivation isgeneral for arbitrary charged macromolecules. A key partof the variational calculation is the natural emergence ofa self-consistent calculation of single-chain averages andchain structure under an effective interaction G ( r , r ).We then specify to a bulk system and provide expres-sions for the osmotic pressure and electrostatic chemicalpotential of polyelectrolytes, the latter being identified asthe average self energy. In Section III we apply our theoryto study flexible, discretely charged chains, and demon-strate how the self-consistent procedure and single-chainaverages can be approximated with a variational proce-dure. In Section IV we present and discuss numericalresults for the intrachain structure, effective interaction,polyelectrolyte self-energy, osmotic coefficient, and criti-cal point. We compare our results with those from sev-eral existing theories, with particular attention paid tothe chain length dependence in the various properties.Finally, in Section V we conclude with a summary of thekey results and future outlook. II. GENERAL THEORYA. Field-Theoretic Formulation
We consider a general solution of polyelectrolytes(charged macromolecules with arbitrary internal connec-tivity and charge distribution) and salt ions in a sol-vent. We start with the microscopic density operatorfor species γ ˆ ρ γ ( r ) = n γ X A =1 N γ X j =1 δ ( r − r γAj ) , (2.1)where A refers to the A -th molecule of species γ , runningup to the total number n γ of molecules of species γ , and j refers to the j -th “monomer” out of a total number N γ ofmonomers in a molecule A of species γ . For monomericspecies, such as the small ions and solvent, N γ = 1 andthe index j only takes the value of 1.Following previous work on the self energy of simpleelectrolytes , individual charges (i.e. salt ions or chargedmonomers) are described by a short-ranged charge distri-bution ez h ( r − r ), for an ion located at r , where we havefactored out the elementary charge e and the (signed) va- lency z . A convenient choice for h is a Gaussian h = (cid:18) a (cid:19) / exp (cid:20) − π ( r − r ) a (cid:21) . (2.2)This distribution gives the ion a finite radius, not ofexcluded volume, but of charge distribution. Doing soavoids the diverging interactions in point-charge modelsand captures the Born solvation energy of individual ionsin a dielectric medium, as well as finite-size correctionsto the ion correlation energy. This feature allows ourcoarse-grained theory to capture the essential thermody-namic effects of finite-size ions at higher densities withouthaving to resolve the microscopic structure.For simple salt, the charge density of the A -th moleculeof species γ in unit of the elementary charge e is simplyˆ ρ chg γA = z γ h γ ( r − r A ). However, for polyelectrolytes weneed to sum over all monomers j of a particular macro-molecule A of species γ . In addition, to allow for arbi-trary charge distribution along the polymer backbone, weintroduce the signed valency z γ,j such that an unchargedmonomer in the polyelectrolyte chain has z γ,j = 0. Thecharge density due to the A -th molecule of the γ -thspecies is ˆ ρ chg γA ( r ) = X j z γj h γj ( r − r γAj ) , (2.3)where the sum runs over each monomer j of object A of species γ . With this definition, the charge density ofeach species γ is given byˆ ρ chg γ ( r ) = X A ˆ ρ chg γA . (2.4)Allowing for the presence of external (fixed) charge dis-tribution ρ ex , we then define a total charge densityˆ ρ chg = ρ ex + X γ ˆ ρ chg γ . (2.5)Treating the charged interactions as in a linear dielec-tric medium with electric permittivity ε (which can bespatially dependent), the Coulomb energy of the systemis written as H C = e Z d r d r ˆ ρ chg ( r ) C ( r , r )ˆ ρ chg ( r ) , (2.6)where C ( r , r ) is the Coulomb operator given by −∇ · [ ε ∇C ( r , r )] = δ ( r − r ) . (2.7)To complete the description of the system, we add theexcluded volume interactions and polyelectrolyte confor-mation degrees of freedom. For the excluded volume weuse the incompressibility constraint in its familiar expo-nential representation δ (1 − P γ ˆ φ γ ) = R D ηe iη (1 − P γ ˆ φ γ ) ,which introduces an incompressibility field η and the vol-ume fraction operator ˆ φ γ , which is related to the densityoperator via ˆ φ γ = v γ ˆ ρ γ , where v γ is the monomer vol-ume of species γ (for simplicity, we have implicitly as-sumed that all monomers on the polyelectrolyte chainhave the same volume). The conformation degrees offreedom of the polyelectrolyte is accounted for by an(arbitrary) chain connectivity bonded interaction hamil-tonian H B which depends on the relative positions ofthe monomers in a polyelectrolyte chain. To focus onthe electrostatic correlation, we ignore other interactions,such as the Flory-Huggins interaction.The canonical partition function is Q = Y γ n γ ! v n γ N γ γ Y A,j Z d r γAj Z D η exp( − βH ) , (2.8)where the effective “Hamiltonian” is βH = βH C − iη − X γ ˆ φ γ ! + βH B (2.9)and includes the incompressibility constraint. In Eq.(2.8) the species index γ runs over all species (sol-vent, simple salt ions, and polyelectrolyte). We use themonomer volume v γ instead of the usual cube of the ther-mal de Broglie wavelength to avoid introducing nonessen-tial notations; this merely produces an immaterial shiftin the reference chemical potential.We then introduce the scaled Coulomb operator C = βe C and the scaled permittivity (cid:15) = ε/ ( βe ) , which has units of inverse length and is related to theBjerrum length by l b = 14 π(cid:15) . For an inhomogeneous dielectric medium, (cid:15) will be moreconvenient to use than l b . The scaled permittivity alsoleads naturally to the scaled inverse Coulomb operator C − ( r , r ) = ∇ r · [ (cid:15) ( r ) ∇ r δ ( r − r )] , which is related to the Coulomb operator by Z d r C − ( r , r ) C ( r , r ) = δ ( r − r )To further simply notation, we henceforth use k B T asthe unit of energy and e as the unit of charge, so we set β = 1 and e = 1.Next, we use the Hubbard-Stratanovich (HS) trans-formation to decouple the Coulomb interaction, whichintroduces the electrostatic potential Ψ( r ) (non-dimensionalized by βe ) and renders the canonical par-tition function as Q = 1Ω C Y γ n γ ! v n γ N γ γ Z D Ψ D η exp( − Y ) , (2.10) where Ω C is a normalization factor from the HS trans-formation given byΩ C = Z D Ψ exp (cid:20) − Z d r d r Ψ( r ) C − ( r , r ) , Ψ( r ) (cid:21) = Z D Ψ exp (cid:20) − Z d r (cid:15) ( ∇ Ψ) (cid:21) = [det C ] / (2.11)and the canonical “action” is Y = 12 Z d r d r Ψ( r ) · C ( r , r ) − · Ψ( r ) (2.12) − Z d r ( iη − iρ ex Ψ) − X λ n γ ln Q γ [ η, Ψ] , where Q γ is the single-particle/polymer partition func-tion given shortly below.Transforming to the grand canonical ensemble by in-troducing the species fugacities λ γ , we obtain the grandcanonical partition functionΞ = 1Ω C Z D Ψ D ηe − L [Ψ ,η ] (2.13)with the grand canonical “action” LL [Ψ , η ] = 12 Z d r d r Ψ( r ) · C ( r , r ) − · Ψ( r ) (2.14) − Z d r ( iη − iρ ex Ψ) − X λ λ γ Q γ [ η, Ψ] . It is useful to notationally distinguish between thethree basic types of species: solvent ( s ), simplesalt ( ± ), polymer ( p ). Correspondingly, the single-particle/polymer partition functions of the three basictypes of species are: Q s = Z d r s e − iv s η (2.15) Q ± = Z d r ± e − iv ± η − iz ± h ± ∗ Ψ (2.16) Q p = Z DR e − H B − iv p R d r η ˆ ρ p − i R d r Ψˆ ρ chg p (2.17)where we have introduced R to denote collectively thepositions of all monomers in a single polyelectrolyte. Foreconomy of notation, it is to be understood that ˆ ρ chg p refers to the charge density of a single chain only, de-fined as in Eq. (2.3). We also write h ∗ Ψ to denote aconvolution, or spatial averaging by the distribution h : h ∗ Ψ = Z d r h ( r − r )Ψ( r ) . (2.18)Thus far Eq. (2.13) is the formally exact expressionfor the partition function of our system. It forms thestarting point for field-based numerical simulations suchas the Complex Langevin methods as well as approxi-mate analytical theories. B. Renormalized Gaussian Fluctuation Theory
For analytical insight, we seek to develop an approx-imate theory for evaluating Ξ, Eq. (2.13). The lowest-order saddle-point approximation would lead to a self-consistent mean-field theory, in which the saddle-pointcondition on Ψ results in a Poisson-Boltzmann (PB) leveldescription where correlations between fixed charges ρ ex and mobile charges ρ chg γ are included, but correlations be-tween mobile charges themselves are ignored. The stan-dard RPA theory accounts for the quadratic fluctuationsaround the saddle-point. In doing so, however, one is leftwith a fixed structure factor determined solely by thesaddle-point condition. For uniform systems of polyelec-trolytes, the chain structure factor that enters the RPAis independent of polyelectrolyte and salt concentrations.To circumvent this shortcoming, we approximate the par-tition function Eq. (2.13) as in our previous RGF theory,using a non-perturbative variational calculation.The RGF theory follows the Gibbs-Feynman-Bogoliubov (GFB) variational approach by introducinga general Gaussian reference action L ref . As written,Eq. (2.13) involves two fields η and Ψ to which we mayapply the variational method. Allowing fluctuations inboth fields will lead to the so-called double screening ofboth electrostatic and excluded volume. However, inthis work, we focus on the fluctuation effects due to elec-trostatics and thus perform the variational calculationonly for the Ψ field; the excluded volume interaction willbe treated at the mean-field level by the saddle-pointapproximation for the η field. For the Ψ field, we makethe following Gaussian reference action: L ref = 12 Z d r d r [Ψ( r ) + iψ ( r )] G − ( r , r )[Ψ( r ) + iψ ( r )](2.19) which is parametrized by a mean electrostatic potential − iψ ( r ) and a variance, or Green’s function G ( r , r ) whichwe will later show to correspond to an effective electro-static interaction that generalizes the familiar screened-Coulomb interaction. This reference action thus accountsfor the deviation χ = Ψ − ( − iψ ) = Ψ + iψ from the meanelectrostatic potential.Using L ref we rewrite the grand canonical partitionfunction Eq. (2.13) asΞ = 1Ω C Z D Ψ D ηe − L ref [Ψ] e − ( L [Ψ ,η ] − L ref [Ψ]) = Ω G Ω C Z D η D e − ( L [Ψ ,η ] − L ref [Ψ]) E ref (2.20)where h· · · i ref denotes an average over Ψ with respectto the reference action L ref , and Ω G the correspondingpartition function of L ref , defined analogously to Ω C inEq. (2.11) with G in place of C . For notational clarity,we will henceforth write h· · · i ref as h· · · i .To implement the GFB procedure, we begin withapproximating the field integral over Ψ with a leadingorder cumulant expansionΞ ≈ Z D η Ω G Ω C e −h L − L ref i ≡ Ξ GF B . (2.21)The first cumulant in the exponent can be readily eval-uated owing to the Gaussian nature of the fluctuatingfield, and is given by: h L − L ref i = 12 Z d r d r (cid:10) Ψ · C − · Ψ (cid:11) − Z d r ( iη − iρ ex h Ψ i ) − X j λ j h Q j i − Z d r d r G − ( r , r ) · h χ ( r ) χ ( r ) i = 12 Z d r d r [ C − − G − ] · G − Z d r d r ψ · C − · ψ − Z d r ( iη − ρ ex ψ ) − X γ λ γ h Q γ i (2.22)where we have used h Ψ i = − iψ and h χ ( r ) χ ( r ) i = G ( r , r ). The grand partition function Ξ GF B and variational grandfree energy W v are found to be:Ξ GF B = Z D ηe − W v [ G,ψ ; η ] (2.23) W v [ G, ψ ; η ] = −
12 ln (cid:18) det G det C (cid:19) + 12 Z d r d r [ C − − G − ] · G − Z d r d r ψ · C − · ψ − Z d r ( iη − ρ ex ψ ) − X γ λ γ h Q γ i . (2.24)In Eq. (2.24), the Ψ-field averaged single-particle/polymer partition functions are: h Q s i = Z d r s exp [ − iv s η ] (2.25) h Q ± i = Z d r ± exp [ − iv ± η − z ± h ± ∗ ψ ] · exp (cid:20) − z ± h ± ∗ G ∗ h ± (cid:21) (2.26) h Q p i = Z DR exp (cid:20) − H B − iv p Z d r η ˆ ρ p − Z d r ψ ˆ ρ chg p (cid:21) · exp (cid:20) − Z d r d r ˆ ρ chg p · G · ˆ ρ chg p (cid:21) . (2.27)For the small ions, the electrostatic fluctuations char-acterized by G ( r , r ) enter as an instantaneous self-interaction which defines the self-energy of the ion u ± ( r ) ≡ z ± h ± ∗ G ∗ h ± = 12 z ± Z d r d r h ± ( r − r ) G ( r , r ) h ± ( r − r ) . (2.28)Similarly, the calculation of the single-chain partitionfunction now features G ( r , r ) as an effective intra chaininteraction u inst p ( R ) ≡ Z d r d r ˆ ρ chg p ( r ) · G ( r , r ) · ˆ ρ chg p ( r ) . (2.29)In the limit where the polymer has only one monomer,ˆ ρ chg p ( r ) = z p h ( r − ˆ r ), and the polyelectrolyte expressionreduces to that of the simple electrolytes above. In thegeneral case, however, this instantaneous (hence the su-perscript ‘inst’) interaction is clearly conformation de-pendent and non-local ; Eq. (2.27) further suggests thatthere will also be chain conformation entropy contribu-tions.We emphasize that although our theory has thestructure of independent particles and chains, thesingle-particle/chain partition functions involve thefluctuation-mediated effective intra-particle/chain inter-action G ( r , r ) that is missing in self-consistent mean-field (SCMF) theories. For polymeric species, it is pre-cisely this intrachain interaction that is able to generatechain structures that adapt to the solution conditions.For transparency and notational simplicity, in the fol-lowing we specify to a system of solvent, salt, and onepolyelectrolyte species, but the expressions can be triv-ially extended to treat the general case with more saltand polyelectrolyte species.To proceed, we first make the saddle-point approxi-mation for the field η . Anticipating that the saddle-point value of η is purely imaginary, we define a realfield P = iη . The saddle-point condition is δW v δ P ( r ) = 0 (2.30)which yields1 − v s ρ s − v + ρ + − v − ρ − − v p ρ p = 0 . (2.31) The densities of the species are given by: ρ s ( r ) = − λ p v s δ h Q s i δ P ( r ) = λ s e − v s P ρ ± ( r ) = − λ ± v ± δ h Q ± i δ P ( r ) = λ ± e − v ± P− z ± h ± ∗ ψ − u ± ρ p ( r ) = − λ p v p δ h Q p i δ P ( r ) . (2.32)Eq. (2.31) is just the condition of incompressibility, and P can be solved to yield P = − v s log 1 − φλ s v s (2.33)where φ is the total volume fraction φ = P γ = s v γ ρ γ ofnon-solvent species. It is customary to set λ s = 1 /v s ,which gives P = − v s log (1 − φ ) (2.34)We note that, at the saddle-point level, our theory can beeasily adapted to accommodate other models of excludedvolume and hard sphere equations of state.With the excluded volume effects taken care of, wenow discuss the determination of the variational param-eters ( G, ψ ) describing the electrostatic fluctuations andinteractions. The self-consistency of the GFB procedurecomes from determining the values of (
G, ψ ) such that W v [ G, ψ ; P ] is stationary at fixed pressure field P , by apartial functional differentiation with respect to the vari-ational parameters G and ψ . , Performing the variation with respect to the mean elec-trostatic potential δW v δψ ( r ) = 0 (2.35)leads to a Poisson-Boltzmann type expression −∇ · (cid:15) ( r ) ∇ ψ = ρ ex + ρ chg+ + ρ chg − + ρ chg p , (2.36)where the species charge densities are given by: ρ chg ± ( r ) = − λ ± δ h Q ± i δψ ( r ) = λ ± z ± h ± ∗ e − v ± P− z ± h ± ∗ ψ − u ± ρ chg p ( r ) = − λ p δ h Q p i δψ ( r ) (2.37)Finally, the stationarity condition on GδW v δG ( r , r ) = 0 (2.38)leads to an integro-differential equation δ ( r − r ) = Z d r [ C − ( r , r ) + 2 I ( r , r )] · G ( r , r )(2.39)where the ionic strength term is given by2 I ( r , r ) = X γ λ γ δ h Q γ i δG ( r , r )= z Z d r h + ( r − r ) ∗ ρ + ( r ) ∗ h + ( r − r )+ z − Z d r h − ( r − r ) ∗ ρ − ( r ) ∗ h − ( r − r )+ λ p h Q p i D ˆ ρ chg p ˆ ρ chg p E . (2.40)In the last line above we have used the identity for the single-chain charge correlation D ˆ ρ chg p ˆ ρ chg p E = 1 h Q p i δ h Q p i δG ( r , r ) (2.41)of a single chain with partition function h Q p i to rewritethe differentiation with respect to Gλ γ δ h Q p i δG ( r , r ) = λ p h Q p ih Q p i δ h Q p i δG ( r , r )= λ p h Q p i D ˆ ρ chg p ˆ ρ chg p E . (2.42)In the case where the polymer only consists of onemonomer, the polymer contribution to the ionic strengthreduces to the simple electrolyte case.Equations (2.34), (2.36), (2.39), and (2.40) constitutethe central expressions of our self-consistent theory. Theself-consistent determination of polymer conformationoriginates from the fact that the Green’s function G ( r , r )Eq. (2.39) itself depends on the single-chain charge cor-relations D ˆ ρ chg p ( r )ˆ ρ chg p ( r ) E , which in turn comes from theaverage single-chain partition function h Q p i determinedby G ( r , r ), Eq. (2.27).Although the idea of a self-consistent determination ofchain structure is not new, it is gratifying that our deriva-tion of the RGF naturally prescribes how to perform theself-consistent calculation. C. Bulk Solution Thermodynamics: Self Energy andOsmotic Pressure
To demonstrate the nature of this self-consistent cal-culation, we now specify to a bulk solution with ρ ex = 0. For a bulk solution, the single-particle/polymer partitionfunctions simplify to: h Q ± i = V e − v ± P− z ± ψ q ± q ± = e − R drdr z ± h ± · G · z ± h ± = e − u ± h Q p i = V e − N p v p P− z totp ψ q p q p = 1 V Z DR e − H B − R drdr ˆ ρ chg p · G · ˆ ρ chg p (2.43)where z totp is the total charge carried by a chain. q γ = Q γ /V is the single-particle/chain partition function ex-cluding the translational degrees of freedom; for simpleions q ± is simply the Boltzmann weight given by the sim-ple ion self energy u ± in Eq. (2.28).Using Eq. (2.32), we evaluate the density and deter-mine the fugacities to be: λ ± = n ± h Q ± i = ρ ± exp[ − u ± ] exp[ − v ± P − z ± ψ ] λ p = n p h Q p i = ρ p /N p q p exp[ − v p N p P − z totp ψ ] (2.44)where ρ p = n p N p /V is the monomer density. Using thefugacity relation, the polymer contribution Eq. (2.42) tothe ionic strength is simply λ p h Q p i D ˆ ρ chg p ˆ ρ chg p E = n p D ˆ ρ chg p ˆ ρ chg p E = n p V · V D ˆ ρ chg p ˆ ρ chg p E = ρ p N p S chg p . (2.45)Recognizing that the single -chain structure D ˆ ρ chg p ˆ ρ chg p E scales as the density of a single chain N/V , we have pre-emptively regrouped a factor of
V /V in anticipation thatthe single-chain charge structure factor defined as S chg p ≡ V D ˆ ρ chg p ˆ ρ chg p E is independent of volume.The fugacity is related to the chemical potential by µ γ = ln( λ γ v γ ), whence we can identify the per-ion andper- chain chemical potentials as: µ ± = ln( ρ ± v ± ) + v ± P + z ± ψ + u ± µ p = ln (cid:18) ρ p v p N p (cid:19) + v p N p P + z totp ψ + u p (2.46)where u p = − ln q p . (2.47)The first three terms in both chemical potential expres-sions of Eq. (2.46) are the same as in a mean-field anal-ysis of a bulk solution. The physical content of the lastterm u p is the free energy of a chain interacting with itselfvia the effective potential G , and within our theory u p is easily identifiable as the per-chain , chemical potentialattributable to electrostatic fluctuations. We thus define µ elp ≡ u p and term it the (bulk) per-chain self-energy .It can be easily verified that all the polymer expressionsabove reduce to those of simple electrolytes in the single-monomer limit N p = 1, since then z totp = z p , H B canbe set to zero, ˆ ρ chg p ( r ) = z p h ( r − ˆ r ), and by translationinvariance R DR → R dr → V . Clearly, in this limit u p N p =1 −−−−→ u ± .In the bulk, it is also useful to define the self energyper monomer (note subscript ‘m’ for ‘monomer’) as µ elm ≡ u p N p . (2.48)Further, the self-consistent set of equations (2.34),(2.36), (2.39), and (2.40) are simple in the bulk case: theconstitutive equation (2.36) for ψ is just the global chargeneutrality constraint, while in Eq. (2.39) the structurefactors and G ( r , r ) become translation-invariant, allow-ing a simple Fourier representation. Further, becauseof the rotational symmetry, only the magnitude of thewavevector matters, and Eqs. (2.39) and (2.40) become:1 = (cid:15)k ˜ G ( k ) + 2 ˜ I ( k ) ˜ G ( k ) (2.49)˜2 I ( k ) = ρ + ˜ S chg+ ( k ) + ρ − ˜ S chg − ( k ) + ρ p N p ˜ S chg p ( k ) (2.50)which can be easily solved to obtain˜ G ( k ) = 1 (cid:15) [ k + ˜ κ ( k )] , (2.51)where we identify ˜ κ ( k ) = 2 ˜ I ( k ) /(cid:15) as the wave-vectordependent screening function, a generalization of the De-bye screening constant. In our spread-charge model, evensimple salt ions have some internal charge structure˜ S chg ± = z ± ˜ h ± ( k ) . (2.52)For point charges ˜ h ± = 1, recovering the same ionicstrength contribution as in DH theory. Therefore in theabsence of polymers, in the point charge limit for simpleelectrolyte, ˜ G ( k ) is precisely the DH screened Coulombinteraction. In bulk solution, a polyelectrolyte with discrete chargeshas a charge structure factor that can generally be di-vided into a self and non-self piece˜ S chg p ( k ) = X l z pl ˜ h pl ( k )+ X l X m = l z pl ˜ h pl ( k ) z pm ˜ h pm ( − k )˜ ω lm ( k ) . (2.53)The first sum is the l = m self piece, and the secondsum is over all other terms. The structure is character-ized by the intramolecular correlation ˜ ω lm between twomonomers l and m on the same chain. While ˜ ω lm ( k )has unknown analytical form, we know that ˜ ω lm ( k ) → k →
0, and ˜ ω lm ( k ) → k (cid:29) /a . We thus see thatin the large wavelength limit the polyelectrolyte chargescontribute collectively to the screening ˜ κ ( k ) as a high-valency object ∼ ( z tot ) where z tot = P l z pl is the totalvalency. In contrast, in the small wavelength limit thecharges screen as independent charges, which for histori-cal reasons we call the Voorn-Overbeek (VO) limit (onlyin the sense of treating the charges as disconnected fromeach other – the original VO theory used DH theory withpoint charges, while we leave open the possibility of giv-ing charges internal structure).It has been previously noted that the magnitude ofcollective screening by polyelectrolyte charges, shouldbe wave-vector dependent and described by the chargestructure (contained in ˜ I ): at different wavelengthsportions of chains screen as independent objects, and thesize of these screening portions is set by the structure.These discussions correctly identified that with increas-ing density, screening will be increasingly controlled byhigher-k structure. However, previous discussions oftensmear out the charges on a chain, thus treating simpleions and polymer charges on different footing and missingthe approach to the the VO-limit at high wavevectors.We now present the osmotic pressure Π. We can use λ γ h Q γ i /V = ρ γ /N γ to identify the ideal osmotic con-tribution. Then, using Fourier integrals to evaluate thedeterminants in W v Eq. (2.24), the osmotic pressure is:Π = − (cid:20) W v − W v V (cid:21) = − π Z ∞ k dk (cid:20) ln (cid:18) κ ( k ) k (cid:19) − ˜ κ ( k ) k + ˜ κ ( k ) (cid:21) − v s (cid:18) − φ ) (cid:19) + 1 − φv s + ρ ± + ρ p N p (2.54)where W v is the grand free energy of a pure solvent sys-tem.An important feature of the theory is the necessity ofself-consistently determining the chain charge structure˜ S chg p ( k ) Eq. (2.53) and ˜ G ( k ) Eq. (2.51). The Green’s function ˜ G ( k ) itself depends on the chain structure; thelatter is in turn determined by a chain interacting with it-self through G in the single-chain partition function h Q p i ,Eq. (2.27). The self-consistency is typically solved by it-eratively approximating G and the chain structure untilconvergence is achieved.The last piece required to implement our theory is anevaluation of the single-chain partition function and cor-responding intramolecular charge structure. The exactevaluation of single-chain partition functions is difficulteven for simpler pair interactions, and in general shouldbe done by numerical simulation. In the next section wedemonstrate how h Q p i can be approximately and simplyevaluated, but we point out that such an approximationis not itself inherent to the general theory. III. SELF-CONSISTENT CALCULATION OF FLEXIBLECHAIN STRUCTURE
Our discussion has heretofore been general for macro-molecules of arbitrary internal connectivity and chargedistribution. To illustrate one way of carrying out theself-consistent calculation and facilitate comparison toprevious theories, we specify to study flexible polyelec-trolyte chains with Kuhn length b , equally spaced (dis-crete) charges of the same valency z p , and overall chargedmonomer fraction f , such that the total polymer chargeis z totp = N f z p . Again, the discrete nature of the chargeswill be reflected in the charge structure factor and is im-portant at high wavevectors.Given this chain model, the expressions for the per-monomer chemical potential, density, and charge struc-ture factor ˜ S chg γ ( k ) are now: µ m = 1 N p ln ρ p v p N p − v p P + f z p ψ + u p N p (3.1) ρ p = λ p N p e − N p v p P− N p fz p ψ − u p (3.2)˜ S chg p ( k ) = z p N f [1 + (
N f − ω ( k )]˜ h p ( k ) (3.3)where we have re-expressed the sum over all monomer-monomer pair correlations ˜ ω lm with an average per-monomer structure ˜ ω . Following our previous discussion,one can check that when k →
0, ˜ S chg p ∼ ( N f z p ) as for a N f z p -valent object, and when ka (cid:29)
1, ˜ S chg p ∼ N f z p asfor N f independent charges of valency z p .Returning to the task of calculating the single-chainpartition function, we resort to a commonly used varia-tional technique. There are many variations reported inthe literature, but they all essentially re-duce to a Flory-type decomposition of the single-chainfree energy into entropic F ent and interaction F int contri-butions u p = − ln q p ≈ min ζ (cid:20) F ent ( ζ ) + F int ( ζ ) (cid:21) (3.4)where ζ indicates some conformational parameterizationof a reference chain. In Flory’s treatment of the excludedvolume of a single chain, for example, ζ would be theaverage end-to-end distance. For our case, we take ourreference chain to be a wormlike chain parameterized by an effective persistence length ζ ≡ l eff . Under this model, l eff controls the chain expansion α = (cid:10) R ee (cid:11) /R ee, α = 2 l eff b (cid:20) − l eff N b (cid:16) − e − Nb/l eff (cid:17)(cid:21) . (3.5)The variational parameter l eff also controls the per-monomer structure ˜ ω ( k ; l eff ) in the charge structure fac-tor ˜ S chg p . While exact expressions of the worm-like-chain(WLC) structure factor exist in literature, to facilitatecalculations we use a simple analytical form:˜ ω ( k ) = exp[ − kl eff / k N bl eff / − exp[ − kl eff / kN b/π (3.6)which interpolates between the appropriate asymptoticlimits of ˜ ω ( k ; l eff ): ˜ ω ( k ) ∼ , k < p /N bl eff /k N bl eff , p /N bl eff < k < /l eff π/kN b, k > /l eff (3.7)Like a previously proposed expression, our expres-sion interpolates between Gaussian-chain behavior ˜ ω ∼ /N bl eff k at low wavevector and rodlike behavior ˜ ω ∼ π/kN b at high wavevector, with a crossover set by l eff .An important feature of the WLC chain structure cap-tured by our expression is that the magnitude of ˜ ω athigh wavevector is negligibly affected by l eff , reflectingthe intuition that while electrostatics can greatly de-form overall chain structure, smaller scale structure isless affected, consistent with blob-theory arguments and simulation observations. As long as we work inthe regime where electrostatic blobs have only g ∼ ( b/f l b ) / ∼ O (1) monomer each, the WLC structurepersists down to the monomer length scale so that smallerlength-scale structures do not need to be resolved.We are thus able to write a one-parameter ( l eff ) modelfor the single-polyelectrolyte free-energy Eq. (3.4) withentropic and interaction terms given by: F ent = − N ln (cid:18) − α N (cid:19) − α ) (3.8) F int = 14 π Z k dk ˜ G ( k ) ˜ S chg p ( k ; l eff ) (3.9)The first term of the entropic free energy is a finite exten-sibility approximation that lies between the elastic freeenergies obtained from integrating the worm-like-chain(WLC) and freely-jointed-chain (FJC) force-extensionrelationships. The second term − α ) is a termthat resists chain compression, first deduced by Floryand used by several subsequent authors. We also note that the expression for the interactionenergy Eq. (3.9) is an improvement upon typical scal-ing estimates of the Coulomb energy. When the effec-tive interaction G is a bare Coulomb interaction, for anextended structure ( R ∼ N ) the usual scaling estimate0gives an energy of N /R ∼ N . The structure factor ofan extended chain is roughly ˜ S chg ( k ) ∼ N / (1 + kN/π ),and Eq. (3.9) gives an interaction energy of ∼ N ln N with the correct logarithmic correction. An interesting consequence of this decomposition ofthe single-chain partition function is that the electro-static fluctuation contribution to the self energy is de-composed into two contributions: 1) entropic work ofdeforming the chains and 2) average interaction en-ergy. The presence of an entropic contribution to thefluctuation-induced excess chemical potential is a specialfeature of flexible chains.With the structure factor specified, we solve for self-consistency iteratively: for given Green’s function ˜ G weminimize the single-chain free energy Eq. (3.4) to ap-proximate l eff and estimate the charge structure factor˜ S chg p ( k ) via Eq. (3.3) and (3.6), which we then use asinput to update ˜ G using Eq. (2.51). We stop when therms relative error of l eff between iterations is below 10 − .Results of our calculations are presented in the followingsection. IV. NUMERICAL RESULTS AND DISCUSSION
For numerical calculations, we consider fully-chargedchains with f = 1 , z p = 1, monovalent salt, set all ionsizes to be the same diameter σ = 2 a = 1, and set theKuhn length b = σ . We also study systems with Bjerrumlength l b ∼
1, ensuring that electrostatic blobs only have g ∼ ( b/f l b ) / ∼ v γ are chosen to reproduce the divergence in theexcluded volume free energy at the closest packing num-ber density of hard spheres with diameter σ . We start byexamining salt-effects on the structure of isolated chains,then finite-concentration effects on chain size in salt-freepolyelectrolyte solutions. Subsequently, we present theeffective self-interaction G , demonstrate the presence ofcharge oscillations, and compare to screening predictionsunder the DH and fixed-Gaussian structure approxima-tions. Next, we examine the polymer self-energy of salt-free polyelectrolyte solutions and compare our predic-tions to alternative theories for electrostatic correlations.Finally, we study the consequences of our theory on theosmotic coefficient and phase separation behavior. A. Chain Structure
The scaling behavior of linear homopolyelectrolytes iswell-known for both the single-chain and semidiluteregimes. In the single-chain, salt-free limit, scaling theory pre-dicts that the long range electrostatic forces elongate flex-ible chains into a “cigar” of electrostatic blobs with chainsize scaling linearly with chain length as R ∼ N . At fi-nite concentrations of salt, the screened Coulomb interac- N R −10 −5 N ρ ∗ ± ( M ) Increasing Salt ∼ N / ∼ N ∼ N − ρ ± = 10 − ∼ − M FIG. 2. End-to-end distance in the single-chain limit, withparameters l b = 0 . , f = 1 , z p = 1, at different salt concentra-tions ρ ± . The blue and green dashed lines represent, respec-tively, the R ∼ N and R ∼ N / scalings in the zero and highsalt limits. The inset shows the crossover salt concentration ρ ∗± ∼ N − . tion effectively acts as an excluded volume interaction forchain segments separated by distances greater than κ − ,where κ is the inverse Debye length of the added salt.Consequently, while short chains still exhibit the “cigar”scaling, sufficiently long chains behave as self-avoidingwalks, with the crossover determined by the salt con-centration. These expectations are borne out in Fig. 2,where we plot results for the chain size as function ofchain length for several different salt concentrations ρ ± .For our theory, in the single-chain limit the polyelec-trolyte does not contribute to the screening ˜ κ ( k ), and theGreen’s function reduces to a modified screened Coulombinteraction ˜ G = 4 πl b / ( k + ˜ κ ( k ) ). Since in the single-chain limit ˜ G is independent of polymer conformation,the self-consistent calculation only requires us to mini-mize the free energy of a single effective chain Eq. (3.4).Because the salt concentrations considered are still di-lute enough for finite ion-size effects to be negligible, theDH expression κ = 4 πl b z (2 ρ ± ) = 8 πl b z ρ ± is a goodestimate of the screening length, and the crossover con-dition κR > ρ ∗± ∼ N − , where we have used that in the dilute saltlimit R ∼ N . In the inset of Fig. 2 we locate the crossoverby the intersection of fits to the asymptotic scaling lim-its, and verify this scaling expectation of the crossoverconcentration.At finite concentrations of polyelectrolyte, there isa new scaling regime for the polymer size when themonomer concentration ρ p becomes sufficiently high.This concentration is usually taken to be at the physicaloverlap, ρ ∗ p ∼ /N , with new scaling behavior given bythe semidilute prediction of ideal random walk statistics R ∼ N / . We plot our results for salt-free polyelectrolyte solu-tions in Fig. 3 at several polymer concentrations. For1 N R −10 −5 N ρ ∗ p ( M ) ρ p = 10 − ∼ − M IncreasingConcentration ∼ N − . ∼ N / ∼ N FIG. 3. End-to-end distance of a polyelectrolyte in salt-freesolutions at finite monomer concentration ρ p , with parameters l b = 0 . , f = 1 , z p = 1. The blue and green dashed lines rep-resent, respectively, the R ∼ N scaling in the dilute regime,and the R ∼ N / scaling in the semidilute regime. The insetshows the crossover monomer concentration ρ ∗ p ∼ N − . sufficiently high concentrations, we recover the ideal ran-dom walk scaling. Further, the crossovers happen be-low physical overlap, in accord with limited simulationdata, and is attributed to the fact that the Coulombinteractions are long-ranged and that chains repel eachother even below physical overlap.The crossover appears to be extremely gradual (morethan two decades). Nevertheless, for given concentrationwe can approximately locate the crossover chain lengthby again finding the intersection of the asymptotic limits.We plot these results in the inset of Fig. 3 and find thatthe crossover concentration goes as ρ ∗ p ∼ N − .To understand this apparently strong N -dependency,we will have to first understand the nature of screeningin solutions with finite concentrations of polyelectrolyte,and we give a more detailed discussion in Section IV C.We do mention, however, that if one uses the most con-servative estimate of screening where only counterionscontribute to the screening length, chains are expectedto interact at concentrations a factor of 1 / (4 πl b f ) belowphysical overlap, which is several orders of magnitudefor parameters studied in this paper (4 πl b f ≈ B. Effective Interaction G ( k ) For simple electrolytes, the field fluctuations and ef-fective screened interaction are well-described by theDebye-H¨uckel screened Coulomb function. The Voorn-Overbeek approximation of neglecting chain connectiv-ity takes Debye-H¨uckel as its starting point, and de-scribes electrostatic fluctuations in polyelectrolyte so-lution by a screened Coulomb with screening constant κ = 4 πl b ( ρ + + c p N ) = 8 πl b ρ p , where the monomernumber density ρ p is related to the chain number den-sity c p by ρ p = c p N , and ρ + = ρ p in salt-free solutionof polyelectrolyte (recall f = 1 , z p = 1). The oppositelimit is to treat polyelectrolytes as point charges of va-lency z tot = N . In this case, the electrostatic fluctua-tions are characterized by the screened Coulomb with arenormalized screening constant κ = 4 πl b ( ρ + + c p N ) =4 πl b ρ p (1+ N ). Clearly, the effect of chain connectivity onscreening should lie somewhere between these two limits.In Fig. 4, we plot the Fourier-transformed Green’s func-tion G for salt-free solutions of 1) chains with adaptablestructure (our theory, RGF) and for 2) chains with fixedGaussian-chain structure (fg-RPA), and compare to thetwo aforementioned limits.For sufficiently dilute systems, the Green’s function G for both our theory and RPA fall on the screenedCoulomb line with screening constant κ – informationabout the chain connectivity is reflected only through thetotal charge z tot . We argue that this is the correct limit-ing law – for sufficiently dilute systems translational en-tropy opposes any ion condensation and the counterionscan be considered a constant background charge. As longas the polymers are sufficiently far apart, they appear toeach other essentially as point charges with valency z tot ,and can be treated using results from the one-componentplasma (OCP) theory once one scales the charges by z tot .In the dilute limit, the OCP is known to be governed bythe DH expressions – when treating polyelectrolytes asa single z tot -valent object the OCP theory gives a screen-ing length λ that scales as ∼ ( l b ρN ) − / , which is con-sistent with our screening constant κ = 4 πl b ρ p (1 + N )for N (cid:29) G ( k ) at finite wavenumbers that de-pends on the concentration. This peak can be shownto lead to attractive wells in G ( r − r ), which allow forpositively charged chains to assume random walk statis-tics; the random-walk conformation would not be pos-sible with a purely repulsive screened Coulomb interac-tion. This is the reason our RGF is able to reproducethe Gaussian-chain scaling in semidilute solution. Onthe other hand, the fg-RPA assumes a Gaussian-chainstructure for all concentrations , and there is no feedbackof G onto the chain structure.The peak in G ( k ) is also associated with decreasedscreening compared to DH expectations using the dilutelimit κ N as the screening strength. The onset of a peak is2 −3 −2 −1 −2 ka G ( k ) / π l b DH V O our theoryfg-RPA ρ = − M ρ = − M ρ = − M ρ = DH N FIG. 4. Green’s function G/ πl b characterizing electrostaticfield fluctuations, at different polyelectrolyte concentrations,from our RGF theory (blue solid), fg-RPA (green dashed), DHprediction with VO screening strength κ = 8 πl b ρ p ( DH VO ,red dotted), and DH prediction using the N -valent screeningstrength κ N = 4 πl b ρ p (1 + N ) ( DH N , brown dot-dashed). Re-sults are for salt-free solutions ( l b = 1 , f = 1 , z p = 1 , N = 100)at different monomer densities ρ p . Relative to the RGF, thefg-RPA over-predicts screening, has a delayed crossover, andpredicts a peak at higher wavenumber (smaller wavelength). actually also present for simple electrolyte solutions andcorrects for the over-prediction of correlations within theDH approximation at higher concentrations. It can begenerally shown that the peak sets in at lower concentra-tions for larger ion sizes: at higher concentrations, onlysub-portions of spatially extended charged objects screenindependently, hence decreasing the effective valency andscreening strength.Polyelectrolytes have their charge greatly extendedacross space, and correspondingly their peak sets in at amuch lower concentration than simple electrolytes. TheRGF, which predicts an adaptable chain size that be-comes expanded relative to the ideal Gaussian chain, pre-dicts an earlier onset of a peak in G ( k ) and less screen-ing (larger G ( k ) values) than the fg-RPA theory at allconcentrations. However, the peaks of the two theoriesdo approach each other with increasing concentration, asexpected of the semidilute regime. C. Electrostatic Self Energy and Correlations
We now examine the self energy per polyelectrolytemonomer Eq. (2.48), which depends on both the chainstructure and Green’s function. We first give the total self energy, where the zero energy of the electrostatics istaken to be the state where charges are dispersed intoinfinitesimal bits at infinity in vacuum. This perspec-tive highlights the energetic consequences of connectingcharges onto a chain, which is especially important in di-lute solution. Further, this reference energy includes the energy of assembling charge onto each charged monomer,thus ensuring we account for both dielectric effects of sol-vation and interactions between charges.To study correlation effects due to finite polymer con-centration, we argue that the most natural definition in-volves subtracting out the infinite-dilution energy. Weare then able to distinguish a dilute limit following DH-like scaling, and a crossover to less effective screening dueto the overlap of polyelectrolyte chains in space.
1. Total Self Energy
One key feature of semidilute solutions is that the selfenergy should become independent of chain length forsufficiently long chain lengths. This is confirmed in Fig.5a, where we plot the total self energy of salt-free solu-tions of polyelectrolytes. With increasing chain length,the self energies begin overlapping over greater concen-tration ranges, in agreement with our expectations forsemidilute solutions.Figure 5a also shows that the fg-RPA theory greatlyover-estimates the self-energy in dilute solution, andrapidly grows with chain length N . Having shown in Sec-tion IV B that in dilute solution the Green’s function G becomes insensitive to chain structure, we conclude thatthe origin of this huge over-estimate of the self energy isthe fg-RPA chain structure.To understand the magnitude of the fg-RPA’s over-estimate of the self energy, we consider the infinite-dilution limit. The self energy of a simple ion is the theBorn solvation energy given by l b / a , representing thework done against the dielectric background to assem-ble a charge into a region of size a . For polyelectrolytes,we expect the infinite-dilution per-monomer energy to be higher than the Born solvation energy of an isolated ion,due to the additional work (including chain elasticity) re-quired to assemble multiple charges at finite separationfrom each other.As confirmed in Section IV A, the chain size of an iso-lated flexible polyelectrolyte scales linearly with chainlength R ∼ N . Elementary calculation of the energy ofa line of charges gives an energy that scales as ∼ N ln N ;as mentioned earlier in the context of our expression forthe interaction energy Eq. (3.9), this is in general true ofcharges arranged in a structure that scales as R ∼ N forlarge N . Thus for flexible polyelectrolytes we expect thatat infinite dilution, the per-monomer energy associatedwith connectivity grows logarithmically ∼ ln N .In contrast, for a fixed-Gaussian structure R ∼ N / and the infinite-dilution (chain) self energy scales as ∼ N /R ∼ N /N / ∼ N / , leading to a per-monomer selfenergy that grows as ∼ N / . This is the origin of therapidly diverging self-energy in fg-RPA, and is attributedto the artificially compact conformation imposed by afixed-Gaussian-chain structure. The dilute solution selfenergy predicted by the fg-RPA leads to an artificiallyhigh driving force for phase separation into denser states.3In contrast, our theory allows the chain conformation torelax, significantly reducing the self energy and increasingthe stability of the single-phase region of a polyelectrolytesolution relative to fg-RPA theory.For a constant dielectric background, the screening dueto correlations (as a result of finite polymer concentra-tion) reduces the amount of work required to assemblecharge onto a chain, which we have defined as the selfenergy. The infinite-dilution self energy, then, containsinformation about the amount of correlation energy at-tributable to chain connectivity and provides an upperbound for its magnitude. ρ p ( M ) µ e l m fg−RPAour theory TPT−1 (N →∞ )MSA − − − − N = 2 N = 16 (a) Total Self Energy −10 −10 −10 −1 −10 −2 ρ p ( M ) µ c o rr m − − − − N =2 IncreasingChainlength
TPT-1N =16 , MSAOurtheory,1:1 salt ∝ − ρ / p N=16
VO/DH = − (2 π l b ) / ρ / p (b) Correlation Self Energy FIG. 5. (a) Per-monomer total self energy µ elm of salt-freepolyelectrolyte solutions, with parameters l b = 1 , f = 1 , z p =1, comparing fg-RPA (dashed, green), RGF (solid, blue),MSA (dotted, black) and TPT-1 (dot-dashed, red) results.(b) Per-monomer correlation energy µ corr m = µ elm − µ elm, . Inaddition to MSA and TPT-1, we also plot the VO predictionusing point-charge DH expression (dashed, brown), and ourRGF theory for 1:1 salt (solid, purple). The fg-RPA correla-tion energy scales as N / , and is omitted because it is notwell-represented by the scales of the figure.
2. Electrostatic Correlation Energy
To isolate the correlation self energy µ corr m associatedwith finite concentrations of polyelectrolyte, for constant-dielectric backgrounds, it is most natural to subtract outthe infinite-dilution self energy µ elm, µ corr m = µ elm − µ elm, (4.1)In the simple-electrolyte case, this is simply taking thesingle-ion Born solvation energy to be the reference en-ergy, and is the usual reference used for studying simpleelectrolytes. For polyelectrolytes, however, one must becareful to subtract the infinite-dilution energy of an en-tire chain, not just the sum of the Born solvation energyof each of the charged monomers.In Fig. 5b we plot the correlation energy µ corr m . Forcomparison we also plot the correlation energy resultsfrom the liquid-state integral equation Mean SphericalApproximation (MSA) theory of simple electrolytes, TPT-1 chain perturbation theory, the VO approxima-tion, and our theory applied to 1:1 electrolytes. Herewe focus on the behavior for our theory, and postponecomparison until Section IV C 3.At sufficiently low concentrations, our theory predictsa per-monomer correlation energy µ corr m that scales as ∼ ( N ρ m l b ) / = ( N c p l b ) / , where we remind the read-ers that ρ m and c p are the monomer and chain numberdensities, respectively. Comparison with the DH pointcharge result ∼ ( z c l b ) / indicates that in sufficientlydilute solution, the correlations follow DH-scaling, withchains screening as N -valent ions – the entire chain be-haves as a fundamental, N -valent screening unit. This isin accord with the dilute limit, renormalized-DH behav-ior of electrostatic fluctuations described by G ( k ).At first sight this N -dependence may seem unusuallystrong. Examination of Fig. 5b shows that the dilutescaling quickly crosses over to a weaker concentrationdependence at higher concentrations. The presence of acrossover is a generic feature of finite-sized charges, andis also present in the MSA theory for simple electrolytesand the RGF theory applied to simple electrolytes. How-ever, the location of the crossover in simple electrolytesdepends on the ion size a , which is much smaller than thesize of a polyelectrolyte. As hinted by our examinationof the chain structure and Green’s function, the dilutesolution DH behavior only persists while the chain size R < ξ , where ξ is some length scale that we attempt toidentify below.In general the screening function ˆ κ in our RGF theoryis wavelength-dependent but, as demonstrated above inthe dilute solution limit, DH behavior describes the ther-modynamics, with an N -dependent screening constant κ = 4 πl b ρ p (1 + N ), suggesting that the relevant lengthscale may be given by ξ − = κ . Combined with thedilute solution scaling R ∼ N , the condition R/ξ DH > ρ p < N − ≡ ρ ∗ DHp , which is the crossoverscaling observed in Section IV A for the chain size. We4note that this crossover scaling is in contrast to the phys-ical overlap condition or “minimal” screening argumentsthat predict a crossover that scales as ρ ∗ p ∼ N − . Weleave the resolution of this discrepancy to future research.Nevertheless, even if our RGF estimate of the locationof crossovers is not accurate, the theory still reproducesthe asymptotic limits in both dilute and semi-dilute so-lutions for the electrostatic correlations, both set by thescale of the infinite-dilution self energy. Thus the range ofuncertainty in the intermediate concentrations is limitedand therefore we expect the theory to be able to rea-sonably describe the thermodynamic properties in thisconcentration range. The same cannot be said for thefg-RPA theory, which severely overpredicts correlationsin the dilute limit.
3. Comparison to Other Theories
We now compare our predictions for the correlationenergy to other theories. Note that in Fig. 5b the fg-RPA results are not discussed because they are not well-represented on the axes used: the over-estimation of thecorrelation energy is too great.As can be seen in Fig. 5b, classic VO theory approxi-mates correlations with a solution of disconnected pointcharges using Debye-H¨uckel theory. DH theory predictsa self energy that scales linearly with the Debye screen-ing constant κ ∼ ρ / for all concentrations, withoutcrossovers. Compared to our theory for polyelectrolytes,the VO theory underestimates correlations for most con-centrations.We also plot the MSA theory as an example of a liquid-state integral equation theory for the restricted primitivemodel of simple electrolytes, which accounts for ion sizethrough a hard-core model. At low concentrations theintegral equation theory matches the DH point chargetheory; it is only at higher concentrations ( κa > make the same VO ap-proximation of treating the correlations with a solutionof disconnected charges, and are expected to more or lesscoincide with the MSA results presented here. While forhigh concentrations the MSA correctly reduces correla-tions relative to point-charge VO, the lack of any chain-length information means that the chain-length depen-dent crossover is completely neglected.Thermodynamic perturbation theory (TPT-1) is an at-tempt at correcting for chain correlations by perturb-ing about liquid-state results for simple electrolytes. TPT-1 predicts a perturbation that grows with chainlength as ∼ ( N − /N , yielding a modest multiva-lency effect of chain-connectivity. However, the pertur-bation rapidly becomes insensitive to chain length. Asshown in Fig. 5b, the TPT-1 results for chain lengths N = 16 and N = 2 are indistinguishable on the scale of the plot. Further, because the TPT-1 theory uses cor-relations of a simple electrolyte system, it is unable tocapture the crossover behavior in the electrostatic cor-relations at low concentrations, an essential consequenceof polymer chain connectivity. Instead, the crossover ob-served in TPT-1 theory is tied to the monomeric length-scale a . D. Thermodynamics and Critical Point Behavior
The theory presented in this work is applicable to thestudy of the thermodynamics of general polyelectrolytesolutions, which will be the subject of future work. Belowwe illustrate its application to the osmotic coefficientsand the critical properties for a fully charged ( f = 1)salt-free polyelectrolyte solution with monovalent coun-terions.With increasing chain length N , we expect the osmoticcoefficient to become independent of N in semidilute so-lutions. The osmotic coefficient is defined as the ratio ofthe actual osmotic pressure of a solution to its ideal value(given by van’t Hoff’s law). For a salt-free polyelectrolytesolution with monovalent counterions, the osmotic coef-ficient is Φ = Π ρ p + c p = Π ρ p (1 + 1 /N ) (4.2)In Fig. 6, we plot the RGF theory’s predictions of Φfor salt-free polyelectrolyte solutions at l b = 1, whichcan be seen to reproduce the expected convergence inthe large- N limit. Our result for N = 16 is in goodquantitative agreement with reported simulation data ofsalt-free polyelectrolyte solutions for that chain length. Importantly, though not obvious from the figure, itcan be shown that at the presented l b = 1, the solutionremains stable against increased chain length. Our re-sults are in contrast to those from the fg-RPA, where atany l b , increasing the chain length will eventually turnthe osmotic coefficient negative and drive the system tophase separation. Lastly, although the TPT-1 involvesa modest chain-length correction, it far underpredicts thedependence of correlations on chain length, reflecting itsbehavior for the self energy.In Fig. 7, we plot the chain-length dependence ofthe critical Bjerrum length ( ∼ inverse temperature) l cb and critical monomer density ρ cp , for chain lengths upto N = 10 . The predicted critical point appears in-sensitive to chain length for chain lengths N ≈
30; thisinsensitivity to chain length is in agreement with previoussimulations and theories.
Previous literature suggested the origin of this criti-cal behavior as either due to counterion condensation orother strong correlations on small length scales that can-not be accounted for by weak fluctuation theories.
Incontrast, our theory suggests that, in fact, accounting forchain conformational change is sufficient to explain the5 ρ p (M) Φ − − − − N=16, TPT−1N=128, TPT−1N=16, Chang & Yethiraj 2005N=16 (our theory)N=16000 (our theory)N=2 (our theory) FIG. 6. Osmotic coefficient Φ = Π /ρ p (1 + 1 /N ) of salt-freepolyelectrolyte solutions as a function of the monomer concen-tration ρ p , with parameters l b = 1 . , f = 1 , z p = 1. Our RGFresults for N = 16 (solid, green) are shown in comparison withexisting simulation data (circle, black). In RGF theory, asthe chain length increases the osmotic coefficients approacheach other at high concentration ( N = 16000 and N = 2 are nearly indistinguishable). TPT-1 underpredicts the chain-length dependence, with N = 16 (solid, black) nearly indis-tinguishable from N = 128 (squares, black). chain-length independence of the critical point of salt-free solutions of flexible polyelectrolytes. The conforma-tional change is particularly important in dilute solution,where the unscreened Coulomb interactions can signifi-cantly distort the chain structure, and in doing so changethe thermodynamics of phase separation.One might notice that the predicted critical Bjerrumlength for long chains l cb ≈ f = 1) and hence expectcounterion condensation to play a role in further stabi-lizing the dilute phase. Although counterion condensa-tion will undoubtedly further reduce the self energy, themagnitude of such reduction is still bounded by the sameinfinite-dilution self energy. Because the infinite-dilutionself energy per monomer has only a weak dependence onthe chain length (logarithmic vs. N / from fg-RPA), theenergy range in the relevant range of the concentrationwhere counterion condensation can play a role for large N in our theory is rather limited; we do not expect counte-rion condensation to substantially affect our conclusions.(In contrast, the magnitude of the counterion conden-sation contribution to the correlation energy would bemuch greater if the correlation energy followed the fg-RPA behavior. But even with the inclusion of counte-rion condensation, the fg-RPA theory does not predictthe correct behavior of the critical properties without in-troducing additional modifications.) Nevertheless, forquantitative prediction it would be important to accountfor counterion condensation and this is planned for futurework. N l c b / σ critical Bjerrum length l bc / σ critical density ρ pc (M) ρ c p ( M ) FIG. 7. Chain-length dependence of the critical Bjerrumlength ( ∼ /T c ) and critical concentration of salt-free poly-electrolyte solutions, with parameters f = 1 , z p = 1. V. CONCLUSIONS
In this work, we have extended the field-theoreticrenormalized Gaussian fluctuation (RGF) variationaltheory of simple electrolyte systems to systematicallyaccount for electrostatic fluctuations in polyelectrolytes.The key results of our theory can be summarized as fol-lows:1. Our theory derives a self-consistent procedurewhereby electrostatic fluctuations characterized by G Eq. (2.39) are coupled to the intrachain struc-ture and vice versa. The theory provides a uni-fied framework for simultaneously describing thechain structure and thermodynamics, in dilute and semidilute solutions.2. Our theory correctly predicts the crossover from the R ∼ N scaling in the chain size to the R ∼ N / scaling as a function of increasing salt in the single-chain limit. For finite concentrations of polyelec-trolyte, the theory also predicts the dilute-limitscaling R ∼ N and the semidilute scaling R ∼ N / in salt-free solutions.3. The self-consistent procedure allows the determina-tion of the full concentration and wavenumber de-pendence of the effective interaction G , and henceclarification of the nature of screening. We con-firm the screening behavior both at long lengthscales, where the polyelectrolytes screen as polyva-lent point charges, and smaller length scales, wherethe charges on the polyelectrolyte chains behaveas disconnected units (the VO limit). The onsetand location of a peak in G is determined by thechain size, which is more accurately described byan adaptive chain structure.64. Our theory features prominently the role of thepolyelectrolyte self-energy u p , Eq. (2.47), which isthe free energy of an independent chain interactingwith itself through G , and is the work required toassemble charge onto a chain. The infinite-dilutionself-energy bounds the magnitude of connectivitycontributions to the correlation energy, the lattertending to cancel out the former with increasingconcentration.5. We clarify that the correlation energy µ corr m is thedifference of the self energy from its infinite-dilutionvalue; µ corr m characterizes finite-concentration ef-fects and reduces the self energy. In sufficientlydilute solutions, µ corr m follows a universal renormal-ized DH scaling µ corr m ∼ − ( ρ p N ) / , independent ofchain structure. Above some chain-size dependentcrossover, chain connectivity results in a weakerconcentration dependence.6. By predicting the correct infinite-dilution chainstructure and self energy, for salt-free solutions ourtheory captures the N -insensitivity both of the os-motic coefficient in semidilute solution and the crit-ical properties in the large- N limit.We note that our physical picture of the self-energy corroborates the self-energy explanation used bysome authors for “strong correlation” complexation. These works treated polyelectrolyte complexation in thezero temperature limit where Coulomb interactions dom-inate. They identified the driving force for polyelec-trolytes to aggregate into denser states as driven by a lossof an infinite-dilution self energy (which was estimatedusing the Coulomb energy of line charges ∼ N log N perchain) upon entering a dense, neutralized state. Our the-ory works at finite temperature, and for a given con-centration is able to quantify how much of the infinite-dilution self energy is lost. Being a weak coupling the-ory, our theory will require further modification to in-clude structures due to strong correlation effects, such ascounterion condensation and ion-pairing.The response of chain conformation to changing den-sity is a key feature in our theory that is not presentin theories of polyelectrolytes assuming fixed chainstructures, such as fg-RPA or Ultra-Soft-Restricted-Primitive Model. Such theories predict, for Gaussianchains, spuriously strong N -dependencies of correlationenergies, and this behavior is due the failure of the as-sumed Gaussian structure in dilute concentrations. Thefixed-Gaussian structure assumption artificially confinesflexible chains to a radius that is too small by a factorof √ N , thus raising the infinite-dilution self energy bythe same factor. Thus, the fg-RPA theory predicts ahigher infinite-dilution energy (which is a positive quan-tity), and there is correspondingly more electrostatic en-ergy for the correlations (which contribute a negative en-ergy) to reduce. By a self-consistent determination of thesingle-chain structure, our theory avoids the artificiallyhigh energies appearing in the fg-RPA theory. For context, our theory’s requirement of self-consistency between an effective interaction G and asingle-chain structure is similar in spirit to sc-PRISMproposals and the procedure proposed by Donley etal. Indeed, if one uses the PRISM equations with theso-called “RPA” closure, the smeared charge distri-butions h to regularize the electrostatic interactions, anda commonly employed estimate of the medium-inducedpotential, one will recover the same total effectiveinteraction as our Green’s function G .However, in contrast to sc-PRISM our theory comesfurnished with expressions for thermodynamics, Eq.(2.24), (2.46). We also emphasize that, in addition tothe intra chain structure presented in this work, our the-ory should also be able to predict an inter chain struc-ture that goes beyond that predicted by the aforemen-tioned procedure of using sc-PRISM with the “RPA”closure. This is a well-known feature, where a Gaussian-fluctuation free energy predicts structure factors that gobeyond the so-called “RPA” expressions. The cor-rect procedure involves calculating the solution responseto an external perturbation, and our preliminary deriva-tions find that corrections to the “RPA” structure factorsarise from the perturbation response of G and η . Theseresults will be reported in future work.While in this paper we have focused on linear ho-mopolyelectrolyte solutions, our theory is applicable togeneral polyelectrolyte systems such as polyelectrolytecoacervates, dendrimers, and gels. Promisingly, our the-ory gives a systematic framework for studying the impactof arbitrary polyelectrolyte architectures on electrostaticcorrelations, not achievable by many commonly-used the-ories of polyelectrolyte thermodynamics (i.e. TPT-1 andother theories that use the VO disconnected charges ap-proximation). This feature of our theory is critical foradvancing the theoretical design of novel polyelectrolytematerials, for which polymer architecture is a particu-larly important design parameter.Finally, our theory retains many of the advantages ofthe original Gaussian variational theory applied to sim-ple electrolytes, providing a systematic framework forstudying inhomogeneities in the dielectric medium andconcentration profiles. Study of inhomogeneous systemswill be the subject of future work. ACKNOWLEDGMENTS
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