Electrostatic Correlations and the Polyelectrolyte Self Energy

0.91 MB·

by  Kevin Shen

Electrostatic Correlations and the Polyelectrolyte Self Energy
Kevin Shen and Zhen-Gang Wanga)
Division of Chemistry and Chemical Engineering, California Institute of Technology
Pasadena, California, 91125, USA
We address the effects of chain connectivity on electrostatic fluctuations in polyelectrolyte solutions using
a field-theoretic, renormalized Gaussian fluctuation (RGF) theory. As in simple electrolyte solutions (Z.-G.
Wang, Phys. Rev. E. 81, 021501 (2010)), the RGF provides a unified theory for electrostatic fluctuations,
7 accountingforbothdielectricandchargecorrelationeffectsintermsoftheself-energy. Unlikesimpleions,the
1 polyelectrolyte self energy depends intimately on the chain conformation, and our theory naturally provides
0 a self-consistent determination of the response of intramolecular chain structure to polyelectrolyte and salt
2 concentrations. 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 self
n
a energy in dilute solution and correctly predicts the eventual N-independence of the critical temperature and
J concentrationofsalt-freesolutionsofflexiblepolyelectrolytes. Weshowthattheselfenergycanbeinterpreted
1 intermsofaninfinite-dilutionenergyµel andafiniteconcentrationcorrelationcorrectionµcorr whichtends
m,0
to cancel out the former with increasing concentration.

]
h
c
I. INTRODUCTION kT andcharacterizesthestrengthofchargeinteractions.
e
m These charge correlations modify a host of properties
Polyelectrolytesarewidelyusedformanyapplications, such as osmotic pressure, ionic activities, and mobilities.
-
t rangingfromenergymaterials1 tosolutionadditives(e.g. For dilute electrolyte solutions, the electrostatic free en-
a for food, cosmetics, and healthcare products).2 Polyelec- ergydensityandtheassociatedexcesschemicalpotential
t
s trolytes are also ubiquitous in biology, as many nat- are given respectively by:
.
t urally occurring polymers – DNA/RNA, proteins, and
a κ3
m some polysaccharides – are charged. Consequently, un- βfel =
−12π
derstanding the interplay of electrostatics with polyelec-
- l κ
d trolyte functionality is central for understanding many βµel = z2 b . (1.2)
n biologicalprocesses,3 andguidingthedesignofmaterials i − i 2
o such as adhesives,4 drug-delivery microencapsulants,5–7
A salient feature of the DH theory is that electro-
c and micro/nanoactuators.8
static correlations increase with increasing ion valency
[ The long-range nature of electrostatic interactions
z . Thus, polyelectrolytes, being inherently multivalent,
1 gives rise to nontrivial correlation effects in polyelec- hiave increased correlation effects compared to simple
v trolyte systems such as ion-condensation9 and complex electrolytesystems. However, themagnitudeofthemul-
4 coacervation.10–12 A key challenge in the theoretical
tivalency effect is unclear due to the spatial extent of
1 study of polyelectrolytes is the proper description of
the polyelectrolyte chains (which introduces new length
2 electrostatic correlations and their consequences on the
scales) and the conformation degrees of freedom of the
0
structure and thermodynamics.
polymers (Fig. 1).
0
The physical origin of electrostatic correlation is the
.
1 preferential interaction between opposite charges. For
0 simpleelectrolytesolutionsthiscorrelationismanifested
7 in the “ionic atmosphere” (Fig. 1) first proposed by
1 Debye and Hu¨ckel (DH).13 As the result of the favor-
:
v able interaction of an ion with its ionic atmosphere, the
i free energy of the system is lowered. Theoretically, for
X
pointcharges,thespatialextentoftheionatmosphereis
r characterized by the well-known inverse Debye screening
a
length
κ2 =λ−D2 =4πlb zi2ci, (1.1) FIG. 1. (a) For simple ions, the self energy involves both a
i
X solvationcontributionand,atfiniteconcentrations,acorrela-
where the Bjerrum length l = e2/4πεkT is the length tion contribution due to the ionic atmosphere. (b) For poly-
b
scale at which two unit charges interact with energy electrolytes, the ionic atmosphere is modified by neighboring
monomers along the same and other chains, and depends on
thechainstructure.
a)Electronicmail: [email protected] The foundational (and still widely used)14–17 theory
2
of polyelectrolyte coacervation put forth by Voorn and action between the salt ions. We note that such an ap-
Overbeek12 completely ignores the contribution of chain proximationofusingascreenedCouloumbinteractionfor
connectivity to electrostatic correlations, directly bor- macroions is a common practice in many analytical and
rowing the DH expressions for charge correlations and simulation studies.42–44
hencetreatingthebackbonechargesasdisconnectedfree Another method that bypasses the fixed-structure as-
ions. The idea of approximating the backbone charge sumption is the self-consistent PRISM (sc-PRISM) in-
as disconnected ions has been adopted in recent theories tegral equation approach45,46 that employs a structure-
of polyelectrolytes that replace the DH correlation using dependent effective medium-induced potential; such an
more advanced treatments of simple electrolytes.18 approach has been applied to polyelectrolytes by Yethi-
Anotherwidelyusedtheoryisthethermodynamicper- raj and coworkers.43,47–50 The sc-PRISM has been
turbation theory (TPT), which aims to capture the ef- quite successful for studying polyelectrolyte solution
fects due to chain connectivity through a leading or- structure, but there have been limited studies on the
derperturbationexpansionaroundthesimpleelectrolyte thermodynamics,51 perhaps due to the inconsistency be-
results.19–24 The first-order TPT (TPT-1) forms the ba- tween the two different routes (i.e., virial vs. com-
sis for a widely used density-functional framework for pressibility) to obtaining the equation of state from the
thestudyofinhomogeneouspolyelectrolytesystems.25–28 structure.52–54 Further, PRISM theories require the use
While producing reasonable agreements for bulk proper- of closures, the choice of which is guided by experience
ties and inhomogeneous systems at higher densities, like and comparisons to experiment and theory.43 Finally, to
the VO theory and its variants, TPT-1 offers no insight date self-consistent PRISM studies have been limited to
into how one can use some of the hallmark design fea- only one polyelectrolyte species. To our knowledge the
tures of polymers – backbone architecture and charge only application of PRISM to complex coacervates ig-
distribution on the backbone – to control electrostatic nored the self-consistent determination of chain struc-
interactions in materials. ture, and inconsistently borrowed thermodynamic ex-
pressions from theories of monomeric solutions.55
An approach that tries to capture the effects of chain
connectivity from the outset is the one-loop expan- A variational theory close in spirit to the sc-PRISM
sion/Random Phase Approximation (RPA) based on wasproposedbyDonley,RudnickandLiu56 tostudythe
a leading order treatment of fluctuations in the field concentration-dependence of polyelectrolyte chain struc-
theoretic partition function.29–31 RPA theories require ture. In hindsight, the theory can be understood as a
a chain structure as input, and for prescribed chain sc-PRISM where the form of the effective intrachain in-
structure provide explicit expressions describing how teraction is motivated by RPA theory using heuristic ar-
chain-connectivity generates extra charge correlations. guments. Theirtheoryyieldsgoodagreementwithavail-
However, for flexible chains, theories that use a fixed, ablecomputersimulationdataontheend-to-enddistance
Gaussian-chainstructureforallconcentrations(hereafter of the polyeletrolyte chain as a function of the polyelec-
referredtoasfixedgaussian-RPA)overestimatethecor- trolyte and salt concentrations.
relation effects, particularly at dilute concentrations. As In this work, we study electrostatic fluctuations in
a result, for flexible chains fg-RPA predicts critical con- polyelectrolytesolutionsusingafield-theoreticrenormal-
centrationsthatvanishwithincreasingchainlength,32 in ized Gaussian fluctuation theory (RGF). In this theory,
contradiction to simulation results.33 Previous work has the key thermodynamic quantity that captures the elec-
triedtofixthisdeficiencybyintroducinganadhocmeso- trostaticfluctuationsistheself-energyofaneffectivesin-
scopic wavevector cut-off while still keeping the Gaus- gle chain. For simple electrolytes, the self-energy is the
sian structure factor for the long-wavelength fluctuation electrostatic work required to assemble charge from an
contributions.34 infinitely dispersed state onto an ion and is given by57
The RPA theories of polyelectrolytes typically only
z2
coordnesridteor flaucccotuuanttiofnosr ienxctlhuedeeldecvtorolusmtaeticfluinctteuraatcitoinosnst.hIant βµechlg = c2hg dr0dr00hchg(r−r0)G(r0,r00)hchg(r00−r)
Z
are important even for neutral polymer systems,35 a (1.3)
variational method36 was employed to treat the “double where hchg is the charge distribution on an ion, and
screening”,37 due to both excluded volume and electro- G(r,r0) is a self-consistently determined Green’s func-
static interactions. The theory has been used to study tion characterizing electrostatic field fluctuations, which
phase separation,38 adapted to account for counterion canbethoughtofasaneffectiveinteractionbetweentwo
condensation,39,40 and applied to study polyelectrolyte test charges in an ionic environment. Eq. (1.3) is a uni-
coil-globule transitions.41 Unlike the fg-RPA, this ap- fied expression accounting for both the polarization of
proach allows the polyelectrolyte conformation to self- the dielectric medium (e.g. Born solvation and image
adapt. However, in the interest of obtaining analytical charge interactions) and correlations due to the ionic at-
expressions, the double-screening theory pre-integrates mosphere.
thesaltdegreesoffreedom,resultingintheDebye-Hu¨ckel For polyelectrolytes, we will see that the self energy is
description of screened interactions between monomer analogously the work required to assemble charge onto
units, which does not feed back on the effective inter- the polyelectrolyte. However, because of the conforma-
3
tional degrees of freedom, part of the work is due to the lency z. A convenient choice for h is a Gaussian
entropicchangeofdeformingthechain. Theinternalen-
ergy contribution to the self-energy resembles Eq. (1.3) 1 3/2 π(r r)2
0
h= exp − . (2.2)
and involves the single-chain structure factor, reflecting 2a2 − 2a2
(cid:18) (cid:19) (cid:20) (cid:21)
thespatialextentofthepolyelectrolytechain. TheRGF
theory prescribes a self-consistent determination of the This distribution gives the ion a finite radius, not of
effective intrachain structure along with the effective in- excluded volume, but of charge distribution. Doing so
teraction G(r,r) as an inherent part of the theory. avoids the diverging interactions in point-charge models
0
The rest of this article is organized as follows. In Sec- andcapturestheBornsolvationenergyofindividualions
tionIIwepresentafullderivationoftheRGFtheoryfor in a dielectric medium, as well as finite-size corrections
polyelectrolyte solutions. At this stage, our derivation is to the ion correlation energy.57 This feature allows our
generalforarbitrarychargedmacromolecules. Akeypart coarse-grainedtheorytocapturetheessentialthermody-
of the variational calculation is the natural emergence of namiceffectsoffinite-sizeionsathigherdensitieswithout
a self-consistent calculation of single-chain averages and having to resolve the microscopic structure.
chain structure under an effective interaction G(r,r). Forsimplesalt,thechargedensityoftheA-thmolecule
0
We then specify to a bulk system and provide expres- of species γ in unit of the elementary charge e is simply
sions for the osmotic pressure and electrostatic chemical ρˆcγhAg = zγhγ(r−rA). However, for polyelectrolytes we
potentialofpolyelectrolytes,thelatterbeingidentifiedas need to sum over all monomers j of a particular macro-
theaverageselfenergy. InSectionIIIweapplyourtheory molecule A of species γ. In addition, to allow for arbi-
to study flexible, discretely charged chains, and demon- trarychargedistributionalongthepolymerbackbone,we
strate how the self-consistent procedure and single-chain introducethesignedvalencyzγ,j suchthatanuncharged
averages can be approximated with a variational proce- monomer in the polyelectrolyte chain has zγ,j = 0. The
dure. In Section IV we present and discuss numerical charge density due to the A-th molecule of the γ-th
results for the intrachain structure, effective interaction, species is
polyelectrolyte self-energy, osmotic coefficient, and criti-
cal point. We compare our results with those from sev- ρˆcγhAg(r)= zγjhγj(r−rγAj), (2.3)
eral existing theories, with particular attention paid to j
X
the chain length dependence in the various properties.
where the sum runs over each monomer j of object A
Finally,inSectionVweconcludewithasummaryofthe
of species γ. With this definition, the charge density of
key results and future outlook.
each species γ is given by
ρˆchg(r)= ρˆchg. (2.4)
γ γA
II. GENERAL THEORY A
X
Allowing for the presence of external (fixed) charge dis-
A. Field-Theoretic Formulation
tribution ρ , we then define a total charge density
ex
We consider a general solution of polyelectrolytes ρˆchg =ρ + ρˆchg. (2.5)
ex γ
(chargedmacromoleculeswitharbitraryinternalconnec- γ
X
tivity and charge distribution) and salt ions in a sol-
vent. We start with the microscopic density operator Treating the charged interactions as in a linear dielec-
for species γ tric medium with electric permittivity ε (which can be
spatially dependent), the Coulomb energy of the system
is written as
nγ Nγ
ρˆγ(r)= δ(r rγAj), (2.1) e2
A=1j=1 − HC = drdr0ρˆchg(r) (r,r0)ρˆchg(r0), (2.6)
XX 2 C
Z
whereAreferstotheA-thmoleculeofspeciesγ,running where (r,r) is the Coulomb operator given by
0
uptothetotalnumbern ofmoleculesofspeciesγ,andj C
γ
referstothej-th“monomer”outofatotalnumberNγ of [ε (r,r0)]=δ(r r0). (2.7)
−∇· ∇C −
monomers in a molecule A of species γ. For monomeric
species, such as the small ions and solvent, N = 1 and To complete the description of the system, we add the
γ
the index j only takes the value of 1. excluded volume interactions and polyelectrolyte confor-
mation degrees of freedom. For the excluded volume we
Following previous work on the self energy of simple
electrolytes57,individualcharges(i.e. saltionsorcharged use the incompressibility constraint in its familiar expo-
monomers)aredescribedbyashort-rangedchargedistri- nential representation δ(1− γφˆγ)= Dηeiη(1− γφˆγ),
butionezh(r r),foranionlocatedatr,wherewehave whichintroducesanincompressibilityfieldηandtPhevol-
factoredoutt−he0elementarychargeeand0the(signed)va- umefractionoperatorφˆ ,whPichisrelaRtedtothedensity
γ
4
operator via φˆ = v ρˆ , where v is the monomer vol- where Ω is a normalization factor from the HS trans-
γ γ γ γ C
ume of species γ (for simplicity, we have implicitly as- formation given by
sumed that all monomers on the polyelectrolyte chain
have the same volume). The conformation degrees of ΩC = DΨexp 1 drdr0 Ψ(r)C−1(r,r0),Ψ(r0)
freedom of the polyelectrolyte is accounted for by an −2
Z (cid:20) Z (cid:21)
(arbitrary) chain connectivity bonded interaction hamil- 1
tonian H which depends on the relative positions of = DΨexp dr(cid:15)( Ψ)2 =[detC]1/2 (2.11)
B −2 ∇
the monomers in a polyelectrolyte chain. To focus on Z (cid:20) Z (cid:21)
theelectrostaticcorrelation,weignoreotherinteractions, and the canonical “action” is
such as the Flory-Huggins interaction.
1
The canonical partition function is Y = drdr0Ψ(r) C(r,r0)−1 Ψ(r0) (2.12)
2 · ·
1 Z
= dr ηexp( βH), (2.8)
Q γ nγ!vγnγNγ A,jZ γAjZ D − − dr(iη−iρexΨ)− nγlnQγ[η,Ψ],
Y Y Z λ
X
where the effective “Hamiltonian” is
where Q is the single-particle/polymer partition func-
γ
tion given shortly below.
βH =βHC iη 1 φˆγ +βHB (2.9) Transforming to the grand canonical ensemble by in-
− − !
γ troducing the species fugacities λ , we obtain the grand
X γ
canonical partition function
and includes the incompressibility constraint. In Eq.
(2.8) the species index γ runs over all species (sol-
1
vent, simple salt ions, and polyelectrolyte). We use the Ξ= Ψ ηe−L[Ψ,η] (2.13)
Ω D D
monomervolumev insteadoftheusualcubeofthether- C Z
γ
maldeBrogliewavelengthtoavoidintroducingnonessen- with the grand canonical “action” L
tial notations; this merely produces an immaterial shift
in the reference chemical potential. L[Ψ,η]= 1 drdr0Ψ(r) C(r,r0)−1 Ψ(r0) (2.14)
We then introduce the scaled Coulomb operator 2 · ·
Z
C =βe2 dr(iη iρ Ψ) λ Q [η,Ψ].
ex γ γ
C − − −
Z λ
and the scaled permittivity X
It is useful to notationally distinguish between the
(cid:15)=ε/(βe2),
three basic types of species: solvent (s), simple
salt ( ), polymer (p). Correspondingly, the single-
which has units of inverse length and is related to the ±
particle/polymer partition functions of the three basic
Bjerrum length by
types of species are:
1
l = .
b
4π(cid:15) Qs = drse−ivsη (2.15)
For an inhomogeneous dielectric medium, (cid:15) will be more Z
convenient to use than lb. The scaled permittivity also Q = dr e−iv±η−iz±h±∗Ψ (2.16)
leads naturally to the scaled inverse Coulomb operator ± ±
Z
C−1(r,r0)=∇r·[(cid:15)(r)∇r0δ(r−r0)], Qp =Z DRe−HB−ivpRdrηρˆp−iRdrΨρˆcph1g (2.17)
which is related to the Coulomb operator by
where we have introduced to denote collectively the
R
positionsofallmonomersinasinglepolyelectrolyte. For
dr1C−1(r,r1)C(r1,r0)=δ(r r0) economy of notation, it is to be understood that ρˆchg
− p1
Z refers to the charge density of a single chain only, de-
To further simply notation, we henceforth use kBT as fined as in Eq. (2.3). We also write h Ψ to denote a
the unit of energy and e as the unit of charge, so we set convolution, or spatial averaging by the∗distribution h:
β =1 and e=1.
Next, we use the Hubbard-Stratanovich (HS) trans-
formation to decouple the Coulomb interaction, which h∗Ψ= dr0h(r−r0)Ψ(r0). (2.18)
Z
introduces the electrostatic potential Ψ(r) (non-
dimensionalized by βe) and renders the canonical par- Thus far Eq. (2.13) is the formally exact expression
tition function as for the partition function of our system. It forms the
starting point for field-based numerical simulations such
1 1
= Ψ ηexp( Y), (2.10) as the Complex Langevin methods58 as well as approxi-
Q ΩC γ nγ!vγnγNγ Z D D − mate analytical theories.
Y
5
B. Renormalized Gaussian Fluctuation Theory which is parametrized by a mean electrostatic potential
iψ(r)andavariance,orGreen’sfunctionG(r,r)which
0
−
For analytical insight, we seek to develop an approx- we will later show to correspond to an effective electro-
imate theory for evaluating Ξ, Eq. (2.13). The lowest- static interaction that generalizes the familiar screened-
order saddle-point approximation would lead to a self- Coulombinteraction. Thisreferenceactionthusaccounts
consistentmean-fieldtheory,59 inwhichthesaddle-point forthedeviationχ=Ψ ( iψ)=Ψ+iψ fromthemean
− −
conditiononΨresultsinaPoisson-Boltzmann(PB)level electrostatic potential.
description where correlations between fixed charges ρ
ex Using L we rewrite the grand canonical partition
andmobilechargesρchgareincluded,butcorrelationsbe- ref
γ function Eq. (2.13) as
tween mobile charges themselves are ignored. The stan-
dardRPAtheoryaccountsforthequadraticfluctuations
aroundthesaddle-point. Indoingso,however,oneisleft
1
with a fixed structure factor determined solely by the Ξ= Ψ ηe−Lref[Ψ]e−(L[Ψ,η]−Lref[Ψ])
Ω D D
saddle-point condition. For uniform systems of polyelec- C Z
trolytes, the chain structure factor that enters the RPA =ΩG η e−(L[Ψ,η]−Lref[Ψ]) (2.20)
isindependentofpolyelectrolyteandsaltconcentrations. ΩC Z D D Eref
Tocircumventthisshortcoming,weapproximatethepar-
titionfunctionEq. (2.13)asinourpreviousRGFtheory,
using a non-perturbative variational calculation. where denotes an average over Ψ with respect
h···iref
The RGF theory follows the Gibbs-Feynman- to the reference action L , and Ω the corresponding
ref G
Bogoliubov (GFB) variational approach by introducing partition function of L , defined analogously to Ω in
ref C
a general Gaussian reference action L . As written, Eq. (2.11) with G in place of C. For notational clarity,
ref
Eq. (2.13) involves two fields η and Ψ to which we may we will henceforth write as .
h···iref h···i
apply the variational method. Allowing fluctuations in
To implement the GFB procedure,60 we begin with
both fields will lead to the so-called double screening of
approximating the field integral over Ψ with a leading
bothelectrostaticandexcludedvolume.36,37 However,in
order cumulant expansion
thiswork,wefocusonthefluctuationeffectsduetoelec-
trostatics and thus perform the variational calculation
onlyfortheΨfield; theexcludedvolumeinteractionwill
Ω
be treated at the mean-field level by the saddle-point Ξ η Ge−hL−Lrefi ΞGFB. (2.21)
approximation for the η field. For the Ψ field, we make ≈Z D ΩC ≡
the following Gaussian reference action:
1 The first cumulant in the exponent can be readily eval-
Lref = 2 drdr0[Ψ(r)+iψ(r)]G−1(r,r0)[Ψ(r0)+iψ(r0)] uated owing to the Gaussian nature of the fluctuating
Z
(2.19) field, and is given by:
1 1
L Lref = drdr0 Ψ C−1 Ψ dr(iη iρex Ψ ) λj Qj drdr0G−1(r,r0) χ(r)χ(r0)
h − i 2 · · − − h i − h i− 2 ·h i
Z Z j Z
(cid:10) (cid:11) X
1 1
= drdr0[C−1 G−1] G drdr0ψ C−1 ψ dr(iη ρexψ) λγ Qγ (2.22)
2 − · − 2 · · − − − h i
Z Z Z γ
X
wherewehaveused Ψ = iψ and χ(r)χ(r) =G(r,r). ThegrandpartitionfunctionΞ andvariational grand
0 0 GFB
h i − h i
free energy W are found to be:
v
ΞGFB = ηe−Wv[G,ψ;η] (2.23)
D
Z
1 detG 1
Wv[G,ψ;η]= ln + drdr0[C−1 G−1] G
− 2 detC 2 − ·
(cid:18) (cid:19) Z
1
drdr0ψ C−1 ψ dr(iη ρexψ) λγ Qγ . (2.24)
− 2 · · − − − h i
Z Z γ
X
6
In Eq. (2.24), the Ψ-field averaged single-particle/polymer partition functions are:
Q = dr exp[ iv η] (2.25)
s s s
h i −
Z
1
Q = dr exp[ iv η z h ψ] exp z2h G h (2.26)
h ±i ± − ± − ± ±∗ · −2 ± ±∗ ∗ ±
Z (cid:20) (cid:21)
1
hQpi= DR exp −HB−ivp drηρˆp1− drψρˆcph1g ·exp −2 drdr0 ρˆcph1g·G·ρˆcph1g . (2.27)
Z (cid:20) Z Z (cid:21) (cid:20) Z (cid:21)
For the small ions, the electrostatic fluctuations char- The densities of the species are given by:
acterized by G(r,r) enter as an instantaneous self-
0
λ δ Q
interaction which defines the self-energy of the ion57 ρs(r)= p h si =λse−vsP
− v δ (r)
s
1 P
u (r) z2h G h λ δ Q
± ≡2 ± ±∗ ∗ ± ρ±(r)=− v± δh (±r)i =λ±e−v±P−z±h±∗ψ−u±
=12z±2 Z dr1dr2h±(r−r1)G(r1,r2)h±(r2−(2r.)2.8) ρp(r)=−±λvppPδδhQ(rp)i. (2.32)
P
Eq. (2.31) is just the condition of incompressibility, and
Similarly, the calculation of the single-chain partition
can be solved to yield
function now features G(r,r) as an effective intrachain
0 P
interaction
1 1 φ
= log − (2.33)
uipnst(R)≡ 12 drdr0ρˆcph1g(r)·G(r,r0)·ρˆcph1g(r0). (2.29) P −vs λsvs
Z where φ is the total volume fraction φ = γ=svγργ of
In the limit where the polymer has only one monomer, non-solvent species. It is customary to sePt λ6s = 1/vs,
ρˆchg(r) = z h(r ˆr), and the polyelectrolyte expression which gives
p1 p −
reduces to that of the simple electrolytes above. In the
1
general case, however, this instantaneous (hence the su- = log(1 φ) (2.34)
perscript ‘inst’) interaction is clearly conformation de- P −vs −
pendent and non-local; Eq. (2.27) further suggests that Wenotethat,atthesaddle-pointlevel,ourtheorycanbe
there will also be chain conformation entropy contribu- easilyadaptedtoaccommodateothermodelsofexcluded
tions. volume and hard sphere equations of state.
We emphasize that although our theory has the With the excluded volume effects taken care of, we
structure of independent particles and chains, the now discuss the determination of the variational param-
single-particle/chain partition functions involve the eters (G,ψ) describing the electrostatic fluctuations and
fluctuation-mediated effective intra-particle/chain inter- interactions. The self-consistency of the GFB procedure
action G(r,r) that is missing in self-consistent mean- comes from determining the values of (G,ψ) such that
0
field (SCMF) theories. For polymeric species, it is pre- W [G,ψ; ] is stationary at fixed pressure field , by a
v
P P
cisely this intrachain interaction that is able to generate partialfunctionaldifferentiationwithrespecttothevari-
chain structures that adapt to the solution conditions. ational parameters G and ψ.60,61
For transparency and notational simplicity, in the fol- Performingthevariationwithrespecttothemeanelec-
lowing we specify to a system of solvent, salt, and one trostatic potential
polyelectrolyte species, but the expressions can be triv-
δW
ially extended to treat the general case with more salt v =0 (2.35)
δψ(r)
and polyelectrolyte species.
To proceed, we first make the saddle-point approxi-
leads to a Poisson-Boltzmann type expression
mation for the field η. Anticipating that the saddle-
point value of η is purely imaginary, we define a real (cid:15)(r) ψ =ρ +ρchg+ρchg+ρchg, (2.36)
field =iη. The saddle-point condition is −∇· ∇ ex + − p
P where the species charge densities are given by:
δW
v =0 (2.30) δ Q
δP(r) ρc±hg(r)=−λ± δhψ(r±)i =λ±z±h±∗e−v±P−z±h±∗ψ−u±
which yields
δ Q
ρchg(r)= λ h pi (2.37)
1 v ρ v ρ v ρ v ρ =0. (2.31) p − pδψ(r)
s s + + p p
− − − − −−
7
Finally, the stationarity condition on G Forabulksolution,thesingle-particle/polymerpartition
functions simplify to:
δW
δG(r,vr0) =0 (2.38) hQ±i=Ve−v±P−z±ψq±
leads to an integro-differential equation Qqp± ==eV−e12−RNdprvdprP0−z±zptho±tψ·Gqp·z±h± =e−u±
h i
δ(r−r0)= dr1 [C−1(r,r1)+2I(r,r1)]·G(r1,r0) qp =1 e−HB−12 drdr0ρˆcph1g·G·ρˆcph1g (2.43)
Z V DR
(2.39) Z R
where ztot is the total charge carried by a chain. q =
where the ionic strength term is given by p γ
Q /V is the single-particle/chain partition function ex-
γ
cluding the translational degrees of freedom; for simple
δ Q
2I(r,r0)= λγ h γi ionsq issimplytheBoltzmannweightgivenbythesim-
γ δG(r,r0) ple ion±self energy u in Eq. (2.28).
X Using Eq. (2.32),±we evaluate the density and deter-
=z+2 dr1 h+(r−r1)∗ρ+(r1)∗h+(r1−r0) mine the fugacities to be:
Z
n ρ
+z−2 dr1 h−(r−r1)∗ρ−(r1)∗h−(r1−r0) λ± = Q± = exp[ u ]exp[±v z ψ]
Z h ±i − ± − ±P− ±
+λphQpi ρˆcph1gρˆcph1g . (2.40) λp = Qnp = q exp[ vρpN/Np ztotψ] (2.44)
D E h pi p − p pP− p
In the last line above we have used the identity for the
where ρ = n N /V is the monomer density. Using the
single-chain charge correlation p p p
fugacityrelation, thepolymercontributionEq. (2.42)to
the ionic strength is simply
1 δ Q
ρˆchgρˆchg = h pi (2.41)
p1 p1 Q δG(r,r)
D E h pi 0 λphQpi ρˆcph1gρˆcph1g =np ρˆcph1gρˆcph1g
of a single chain with partition function hQpi to rewrite D E= npDV ρˆchgEρˆchg
the differentiation with respect to G V · p1 p1
= ρp SchgD. E (2.45)
λ δhQpi =λ hQpi δhQpi Np p
γ p
δG(r,r) Q δG(r,r)
0 h pi 0
=λ Q ρˆchgρˆchg . (2.42) Recognizing that the single-chain structure ρˆcph1gρˆcph1g
ph pi p1 p1 scales as the density of a single chain N/V, weDhave preE-
D E
emptivelyregroupedafactorofV/V inanticipationthat
In the case where the polymer only consists of one
thesingle-chainchargestructurefactordefinedasSchg
monomer,thepolymercontributiontotheionicstrength p ≡
reduces to the simple electrolyte case. V ρˆchgρˆchg is independent of volume.
p1 p1
Equations (2.34), (2.36), (2.39), and (2.40) constitute
DThe fugacEity is related to the chemical potential by
the central expressions of our self-consistent theory. The
µ = ln(λ v ), whence we can identify the per-ion and
γ γ γ
self-consistent determination of polymer conformation
per-chain chemical potentials as:
originatesfromthefactthattheGreen’sfunctionG(r,r)
0
Eq. (2.39) itself depends on the single-chain charge cor- µ =ln(ρ v )+v +z ψ+u
relations ρˆcph1g(r)ρˆcph1g(r0) ,whichinturncomesfromthe µ± =ln ±ρpv±p +±vPN ±+ztotψ±+u (2.46)
average sDingle-chain partEition function hQpi determined p (cid:18) Np (cid:19) p pP p p
by G(r,r), Eq. (2.27).
0
where
Althoughtheideaofaself-consistentdeterminationof
chainstructureisnotnew,itisgratifyingthatourderiva-
u = lnq . (2.47)
p p
tionoftheRGFnaturallyprescribeshowtoperformthe −
self-consistent calculation. 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 last
C. Bulk Solution Thermodynamics: Self Energy and termu isthefreeenergyofachaininteractingwithitself
p
Osmotic Pressure via the effective potential G, and within our theory u
p
is easily identifiable as the per-chain, chemical potential
To demonstrate the nature of this self-consistent cal- attributable to electrostatic fluctuations. We thus define
culation, we now specify to a bulk solution with ρ =0. µel u and term it the (bulk) per-chain self-energy.
ex p ≡ p
8
Itcanbeeasilyverifiedthatallthepolymerexpressions Inbulksolution,apolyelectrolytewithdiscretecharges
abovereducetothoseofsimpleelectrolytesinthesingle- has a charge structure factor that can generally be di-
monomer limit N = 1, since then ztot = z , H can vided into a self and non-self piece
p p p B
be set to zero, ρˆchg(r) = z h(r ˆr), and by translation
p1 p −
invariance dr V. Clearly, in this limit S˜chg(k)= z2h˜2 (k)
DR → → p pl pl
up Np=1 uR. R Xl
I−n−−th−→e bu±lk, it is also useful to define the self energy + zplh˜pl(k)zpmh˜pm( k)ω˜lm(k). (2.53)
−
per monomer (note subscript ‘m’ for ‘monomer’) as l m=l
XX6
u
µel p. (2.48) The first sum is the l = m self piece, and the second
m ≡ N
p sum is over all other terms. The structure is character-
ized by the intramolecular correlation ω˜ between two
Further, the self-consistent set of equations (2.34), lm
monomers l and m on the same chain.49 While ω˜ (k)
(2.36),(2.39),and(2.40)aresimpleinthebulkcase: the lm
has unknown analytical form, we know that ω˜ (k) 1
constitutiveequation(2.36)forψisjusttheglobalcharge lm →
as k 0, and ω˜ (k) 0 as k 1/a. We thus see that
neutrality constraint, while in Eq. (2.39) the structure → lm → (cid:29)
in the large wavelength limit the polyelectrolyte charges
factors and G(r,r) become translation-invariant, allow-
0 contribute collectively to the screening κ˜2(k) as a high-
ing a simple Fourier representation. Further, because
valency object (ztot)2 where ztot = z is the total
of the rotational symmetry, only the magnitude of the ∼ l pl
valency. In contrast, in the small wavelength limit the
wavevector matters, and Eqs. (2.39) and (2.40) become: P
chargesscreenasindependentcharges, whichforhistori-
1=(cid:15)k2G˜(k)+2I˜(k)G˜(k) (2.49) cal reasons we call the Voorn-Overbeek (VO) limit (only
in the sense of treating the charges as disconnected from
ρ
˜2I(k)=ρ S˜chg(k)+ρ S˜chg(k)+ p S˜chg(k) (2.50) eachother–theoriginalVOtheoryusedDHtheorywith
+ + − − Np p point charges, while we leave open the possibility of giv-
ing charges internal structure).
which can be easily solved to obtain
It has been previously noted that the magnitude of
1 collective screening by polyelectrolyte charges, should
G˜(k)= , (2.51)
(cid:15)[k2+κ˜2(k)] be wave-vector dependent and described by the charge
structure (contained in I˜):30,56 at different wavelengths
where we identify κ˜2(k) = 2I˜(k)/(cid:15) as the wave-vector portionsofchainsscreenasindependentobjects,andthe
dependent screeningfunction,ageneralizationoftheDe- size of these screening portions is set by the structure.
byescreeningconstant. Inourspread-chargemodel,even These discussions correctly identified that with increas-
simple salt ions have some internal charge structure ing density, screening will be increasingly controlled by
higher-k structure. However, previous discussions often
S˜chg =z2h˜2(k). (2.52) smear out the charges on a chain, thus treating simple
± ± ± ionsandpolymerchargesondifferentfootingandmissing
For point charges h˜ = 1, recovering the same ionic the approach to the the VO-limit at high wavevectors.
strength contribution±as in DH theory. Therefore in the We now present the osmotic pressure Π. We can use
absence of polymers, in the point charge limit for simple λ Q /V = ρ /N to identify the ideal osmotic con-
γ γ γ γ
electrolyte, G˜(k) is precisely the DH screened Coulomb tribhutioin. Then, using Fourier integrals to evaluate the
interaction. determinants in W Eq. (2.24), the osmotic pressure is:
v
Π= Wv−Wv0 = 1 ∞k2 dk ln 1+ κ˜2(k) κ˜2(k)
− V − 4π2 k2 − k2+κ˜2(k)
(cid:20) (cid:21) Z0 (cid:20) (cid:18) (cid:19) (cid:21)
1 1 φ ρ
1+log(1 φ) + − +ρ + p (2.54)
− vs(cid:18) − (cid:19) vs ± Np
where W0 is the grand free energy of a pure solvent sys- function G˜(k) itself depends on the chain structure; the
v
tem. latterisinturndeterminedbyachaininteractingwithit-
selfthroughGinthesingle-chainpartitionfunction Q ,
An important feature of the theory is the necessity of h pi
Eq. (2.27). Theself-consistencyistypicallysolvedbyit-
self-consistently determining the chain charge structure
eratively approximating G and the chain structure until
S˜chg(k) Eq. (2.53) and G˜(k) Eq. (2.51). The Green’s
p
9
convergence is achieved. aneffectivepersistencelengthζ l . Underthismodel,
eff
The last piece required to implement our theory is an l controls the chain expansion≡α2 = R2 /R2
eff ee ee,0
evaluationofthesingle-chainpartitionfunctionandcor-
(cid:10) (cid:11)
reevsapluoantdioinngoifntsrinamgleo-lcehcualianrpcahratritgieonstfruunccttuiroen.s Tishdeiffiexcaucltt α2 =2lebff 1− Nleffb 1−e−Nb/leff (3.5)
even for simpler pair interactions, and in general should (cid:20) (cid:16) (cid:17)(cid:21).
bedonebynumericalsimulation.48Inthenextsectionwe The variational parameter leff also controls the per-
demonstratehow Qp canbeapproximatelyandsimply monomer structure ω˜(k;leff) in the charge structure fac-
evaluated, but wehpoiint out that such an approximation torS˜chg. Whileexactexpressionsoftheworm-like-chain
p
is not itself inherent to the general theory. (WLC) structure factor exist in literature,65 to facilitate
calculations we use a simple analytical form:
exp[ kl /2] 1 exp[ kl /2]
III. SELF-CONSISTENT CALCULATION OF FLEXIBLE ω˜(k)= − eff + − − eff (3.6)
CHAIN STRUCTURE 1+k2Nbl /6 1+kNb/π
eff
which interpolates between the appropriate asymptotic
Our discussion has heretofore been general for macro-
limits of ω˜(k;l ):66,67
molecules of arbitrary internal connectivity and charge eff
distribution. To illustrate one way of carrying out the
1, k< 6/Nbl
self-consistent calculation and facilitate comparison to eff
previous theories, we specify to study flexible polyelec- ω˜(k)∼6/k2Nbleff, 6/pNbleff <k<1/leff (3.7)
trolyte chains with Kuhn length b, equally spaced (dis- π/kNb, kp>1/leff
crete)chargesofthesamevalencyz ,andoverallcharged
p
monomer fraction f, such that the total polymer charge Like a previously proposed expression,66 our expres-
isztot =Nfz . Again,thediscretenatureofthecharges sion interpolates between Gaussian-chain behavior ω˜
willpbe reflectped in the charge structure factor and is im- 6/Nbl k2 at low wavevector and rodlike behavior ω˜ ∼
eff
∼
portant at high wavevectors. π/kNb at high wavevector, with a crossover set by l .
eff
Given this chain model, the expressions for the per- An important feature of the WLC chain structure cap-
monomer chemical potential, density, and charge struc- tured by our expression is that the magnitude of ω˜ at
ture factor S˜chg(k) are now: high wavevector is negligibly affected by l , reflecting
γ eff
the intuition that while electrostatics can greatly de-
1 ρ v u form overall chain structure, smaller scale structure is
µ = ln p p v +fz ψ+ p (3.1)
m N N − pP p N less affected,62 consistent with blob-theory arguments68
p p p
and simulation observations.69 As long as we work in
ρp =λpNpe−NpvpP−Npfzpψ−up (3.2) the regime where electrostatic blobs have only g
S˜chg(k)=z2Nf[1+(Nf 1)ω˜(k)]h˜2(k) (3.3) (b/f2l )2/3 (1) monomer each, the WLC structur∼e
p p − p b ∼ O
persistsdowntothemonomerlengthscalesothatsmaller
where we have re-expressed the sum over all monomer- length-scale structures do not need to be resolved.
monomer pair correlations ω˜ with an average per- Wearethusabletowriteaone-parameter(l )model
lm eff
monomerstructureω˜. Followingourpreviousdiscussion, for the single-polyelectrolyte free-energy Eq. (3.4) with
onecancheckthatwhenk 0, S˜chg (Nfz )2 asfora entropic and interaction terms given by:
→ p ∼ p
Nfz -valent object, and when ka 1, S˜chg Nfz2 as
p (cid:29) p ∼ p 3 α2
for Nf independent charges of valency zp. Fent = Nln 1 3ln(α) (3.8)
Returning to the task of calculating the single-chain − 2 − N −
(cid:18) (cid:19)
partition function, we resort to a commonly used varia- 1
F = k2dkG˜(k)S˜chg(k;l ) (3.9)
tional technique. There are many variations reported in int 4π2 p eff
the literature,47,49,50,56,62–64 but they all essentially re- Z
duce to a Flory-type decomposition of the single-chain Thefirsttermoftheentropicfreeenergyisafiniteexten-
freeenergyintoentropicF andinteractionF contri- sibility approximation that lies between the elastic free
ent int
butions energies obtained from integrating the worm-like-chain
(WLC) and freely-jointed-chain (FJC) force-extension
relationships.70,71 The second term 3ln(α) is a term
up =−lnqp ≈mζin Fent(ζ)+Fint(ζ) (3.4) that resists chain compression, first−deduced by Flory
(cid:20) (cid:21) and used by several subsequent authors.72–75
where ζ indicates some conformational parameterization We also note that the expression for the interaction
ofareferencechain. InFlory’streatmentoftheexcluded energy Eq. (3.9) is an improvement upon typical scal-
volume of a single chain, for example, ζ would be the ing estimates of the Coulomb energy. When the effec-
average end-to-end distance. For our case, we take our tive interaction G is a bare Coulomb interaction, for an
reference chain to be a wormlike chain parameterized by extended structure (R N) the usual scaling estimate
∼
10
gives an energy of N2/R N.68 The structure factor of
an extended chain is roug∼hly S˜chg(k) N2/(1+kN/π), 106 100 ∼ N3/5
)
wanitdhEthqe. c(o3r.r9e)ctgilvoegsarainthminticercaocrtrieocntieonn∼e.7r6gy of ∼ NlnN 105∗ρ(M±10−5 ∼N−2
An interesting consequence of this decomposition of 104 10−11001 103 105
the single-chain partition function is that the electro- N
static fluctuation contribution to the self energy is de- R103
composed into two contributions: 1) entropic work of ∼ N1
deregfoy.rmTinhge tphreesecnhcaeinosfaannden2t)roapviecracgoentrinibtuertaiocntiotno tehne- 102 ρ± = 10−8 ∼ 10−2M
fluctuation-inducedexcesschemicalpotentialisaspecial 101 Increasing Salt
feature of flexible chains.
With the structure factor specified, we solve for self- 101 102 103 104 105 106 107
consistency iteratively: for given Green’s function G˜ we N
minimize the single-chain free energy Eq. (3.4) to ap-
proximate l and estimate the charge structure factor
eff
S˜chg(k) via Eq. (3.3) and (3.6), which we then use as FIG. 2. End-to-end distance in the single-chain limit, with
p
input to update G˜ using Eq. (2.51). We stop when the parameterslb=0.7,f =1,zp=1,atdifferentsaltconcentra-
tions ρ . The blue and green dashed lines represent, respec-
rmsrelativeerrorofleffbetweeniterationsisbelow10−10. tively,t±heR N andR N3/5 scalingsinthezeroandhigh
Resultsofourcalculationsarepresentedinthefollowing ∼ ∼
salt limits. The inset shows the crossover salt concentration
section. ρ∗ N−2.
±∼
IV. NUMERICAL RESULTS AND DISCUSSION tioneffectivelyactsasanexcludedvolumeinteractionfor
chain segments separated by distances greater than κ 1,
−
For numerical calculations, we consider fully-charged where κ is the inverse Debye length of the added salt.
chains with f = 1,z = 1, monovalent salt, set all ion Consequently, while short chains still exhibit the “cigar”
p
sizes to be the same diameter σ = 2a = 1, and set the scaling, sufficiently long chains behave as self-avoiding
Kuhnlengthb=σ. WealsostudysystemswithBjerrum walks,77 with the crossover determined by the salt con-
lengthl 1, ensuringthatelectrostaticblobsonlyhave centration. These expectations are borne out in Fig. 2,
b
g (b/f2∼l )2/3 1 Kuhn monomers each. To facilitate where we plot results for the chain size as function of
b
∼ ∼
comparison with the restricted primitive model, the pa- chain length for several different salt concentrations ρ .
rametersv arechosentoreproducethedivergenceinthe For our theory, in the single-chain limit the polyele±c-
γ
excluded volume free energy at the closest packing num- trolytedoesnotcontributetothescreeningκ˜(k),andthe
berdensityofhardsphereswithdiameterσ. Westartby Green’sfunctionreducestoamodifiedscreenedCoulomb
examiningsalt-effectsonthestructureofisolatedchains, interaction G˜ = 4πl /(k2 +κ˜(k)2). Since in the single-
b
then finite-concentration effects on chain size in salt-free chain limit G˜ is independent of polymer conformation,
polyelectrolyte solutions. Subsequently, we present the the self-consistent calculation only requires us to mini-
effective self-interaction G, demonstrate the presence of mize the free energy of a single effective chain Eq. (3.4).
chargeoscillations,andcomparetoscreeningpredictions Becausethesaltconcentrationsconsideredarestilldi-
under the DH and fixed-Gaussian structure approxima- luteenoughforfiniteion-sizeeffectstobenegligible, the
tions. Next, we examine the polymer self-energy of salt- DH expression κ2 = 4πl z2(2ρ ) = 8πl z2ρ is a good
b b
free polyelectrolyte solutions and compare our predic- estimate of the screening length±, and the cro±ssover con-
tionstoalternativetheoriesforelectrostaticcorrelations. dition κR > 1 predicts a crossover salt concentration
Finally, we study the consequences of our theory on the ρ N 2, where we have used that in the dilute salt
∗ −
osmotic coefficient and phase separation behavior. li±mi∼tR N. IntheinsetofFig. 2welocatethecrossover
∼
by the intersection of fits to the asymptotic scaling lim-
its, and verify this scaling expectation of the crossover
A. Chain Structure concentration.
At finite concentrations of polyelectrolyte, there is
The scaling behavior of linear homopolyelectrolytes is a new scaling regime for the polymer size when the
well-known for both the single-chain68 and semidilute monomer concentration ρ becomes sufficiently high.
p
regimes.77 This concentration is usually taken to be at the physical
In the single-chain, salt-free limit, scaling theory pre- overlap, ρ 1/N2, with new scaling behavior given by
∗p ∼
dictsthatthelongrangeelectrostaticforceselongateflex- the semidilute prediction of ideal random walk statistics
iblechainsintoa“cigar”ofelectrostaticblobswithchain R N1/2.77
size scaling linearly with chain length as R N.68 At fi- W∼e plot our results for salt-free polyelectrolyte solu-
∼
niteconcentrationsofsalt,thescreenedCoulombinterac- tions in Fig. 3 at several polymer concentrations. For