Driving force and pathway in polyelectrolyte complex coacervation

RESEARCHARTICLE

CHEMISTRY

Driving force and pathway in polyelectrolyte complex

coacervation

ShenshengChen

andZhen-GangWang

a,1

EditedbyMatthewTirrell,TheUniversityofChicago,Chicago,IL;receivedJune10,2022;acceptedAugust1,2022

There is notable discrepancy between experiments and coarse-grained model studies

regarding the thermodynamic driving force in polyelectrolyte complex coacervation:

experiments find the free energy change to be dominated by entropy, while simulations

using coarse-grained models with implicit solvent usually report a large, even dominant

energetic contribution in systems with weak to intermediate electrostatic strength.

Here, using coarse-grained, implicit-solvent molecular dynamics simulation combined

with thermodynamic analysis, we study the potential of mean force (PMF) in the

two key stages on the coacervation pathway for symmetric polyelectrolyte mixtures:

polycation–polyanion complexation and polyion pair–pair condensation. We show that

the temperature dependence in the dielectric constant of water gives rise to a substantial

entropic contribution in the electrostatic interaction. By accounting for this electrostatic

entropy, which is due to solvent reorganization, we find that under common conditions

(monovalent ions, room temperature) for aqueous systems, both stages are strongly

entropy-driven with negligible or even unfavorable energetic contributions, consistent

with experimental results. Furthermore, for weak to intermediate electrostatic strengths,

this electrostatic entropy, rather than the counterion-release entropy, is the primary

entropy contribution. From the calculated PMF, we find that the supernatant phase

consists predominantly of polyion pairs with vanishingly small concentration of bare

polyelectrolytes, and we provide an estimate of the spinodal of the supernatant phase.

Finally, we show that prior to contact, two neutral polyion pairs weakly attract each other

by mutually induced polarization, providing the initial driving force for the fusion of the

pairs.

polyelectrolytecomplexcoacervation

entropy

thermodynamicdrivingforce

coarse-grainedsimulation

polarization

Polyelectrolyte (PE) complex coacervation refers to an associative liquid–liquid phase

separation (LLPS) upon mixing oppositely charged polyelectrolyte solutions, which results

in the formation of a coacervate phase containing most of the polyions and a dilute

supernatant phase (1, 2). This electrostatically induced LLPS underpins a number of

important biological phenomena such as membraneless organelles in cells (3–6) and ocean

life adhesion (7–11) and is also being exploited in novel biomedical and biomimetic

applications such as drug delivery (12–14) and underwater adhesion (10, 15–19). In

the 6 decades since the pioneering theoretical work by Overbeek and Voorn (20),

significant progress has been made on both the theory/simulation and experiment fronts

in understanding the many effects on this LLPS, such as chain connectivity (21–30),

excluded volume (22, 25, 31–35), charge sequence (36–40), ion pairing (22, 41, 42),

charge asymmetry (43–47), temperature (48–53), pH (54–57), and solvent quality

(58–60). We refer readers to several excellent recent reviews (61–65). Despite these

advances, however, some fundamental questions remain in regard to the thermodynamic

driving force and the mechanism for coacervation.

There has been a notable discrepancy between experimental results and simulation

studies concerning the thermodynamic driving force. Thermodynamically, polyelectrolyte

complex coacervation can be considered a two-stage process: first, the complexation of

polycations and polyanions into polyion pairs, and second, the condensation of the

polyion pairs into the coacervate phase (66–71), with the first stage dominating the free en-

ergy change for the overall process. Based on isothermal titration calorimetry (72), Tirrell

and coworkers determined the thermodynamic driving forces for both stages and found

that for each stage the free energy change is dominated by entropy with negligible enthalpic

contribution (71, 73). The entropy dominance was also demonstrated by Schlenoff and

coworkers in complex coacervation for some common polyelectrolytes (74, 75). On the

other hand, coarse-grained computer simulation studies of the complexation of two

oppositely charged polyelectrolyte generally found substantial energy contribution in

the free energy change (76–80). In a seminal work, Ou and Muthukumar (76), using

Significance

Coarse-grainedtheoryand

simulationusingcontinuum

electrostaticsmodelswithimplicit

solventarepowerfulmethods

forstudyingelectrolyte/

polyelectrolytesolutions.

Informationaboutthesolvent

degreesoffreedombecomes

hiddeninthecoarse-grained

representation,whichcanleadto

inconsistenciesininterpreting

experimentaldataifthis

informationisnotproperly

accountedfor.Hereweshowthat

thetemperaturedependencein

thedielectricconstantofwater

canbeexploitedtoextractthe

solvententropycontributionin

theelectrostaticinteractions,

therebyresolvingamajor

discrepancybetweenexperiments

andcoarse-grainedmodelstudies

concerningthethermodynamic

drivingforce(i.e.,entropyvs.

energy)forpolyelectrolyte

complexcoacervation.Forweakly

chargedsystems,thiselectrostatic

entropyconstitutesthemain

contributiontotheoverall

entropychange.

Author affiliations:

Division of Chemistry and Chemical

Engineering,CaliforniaInstituteofTechnology,Pasadena,

CA91125

Authorcontributions:S.C.andZ.-G.W.designedresearch,

performedresearch,analyzeddata,andwrotethepaper.

Theauthorsdeclarenocompetinginterest.

ThisarticleisaPNASDirectSubmission.

This article is distributed under

CreativeCommons

Attribution-NonCommercial-NoDerivatives License 4.0

(CCBY-NC-ND)

To whom correspondence may be addressed. Email:

zgw@caltech.edu.

This article contains supporting information online at

https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.

2209975119/-/DCSupplemental

PublishedAugust29,2022.

PNAS

2022 Vol.119 No.36 e2209975119

https://doi.org/10.1073/pnas.2209975119

1of9

Downloaded from https://www.pnas.org by 76.89.95.90 on August 12, 2023 from IP address 76.89.95.90.

Langevin dynamics simulation, found that the process is energy

driven for electrostatic strength

1.5

,where

is the

Bjerrum length and

is the spacing between charges along the

chain backbone. The electrostatic energy contribution remains

favorable up to

=2.5

. Similar results were reported in sim-

ulations from Rathee et al. (78) and from Singh and Yethiraj

(79). The discrepancy becomes especially obvious in the range

ξ<

for which simulations predicted significantly favorable

energetic contributions, while experiments found the energetic

contribution to be generally small and unfavorable (75, 81).

Furthermore, in the relevant range of

values (

<ξ<

) for

typical polyelectrolyte complex coacervation systems used in ex-

periments, the enthalpy change, while remaining relatively small,

changes sign from positive to negative with increasing

(75, 81);

this trend is opposite to that observed in the simulation of ref. 76.

Because of these differences, Schlenoff and coworkers (74, 75, 82)

raised serious concerns about the use of continuum electrostatics

to describe polyelectrolyte complex coacervation systems that

most simulations (and theories) rely on. This discrepancy between

experimental data and simulation results must be resolved in

order to continue using coarse-grained simulation/theory with

confidence to study the properties of polyelectrolyte complex

coacervation.

The large entropic driving force in polyelectrolyte complex

coacervation observed in experiments is commonly believed to

arise from the counterion release upon complexation between

oppositely charged polyelectrolyte chains (61–63, 76, 83). How-

ever, this counterion-release mechanism can only explain the en-

tropy dominance in the regime of strong electrostatic interactions

(

ξ>

2.5

), in which over

60%

of the counterions are bound

to the uncomplexed polyions (76). As the electrostatic strength

decreases, the counterion-release entropy becomes less important

and is only a minor contribution to the overall free energy change

when

ξ<

1.5

. Thus, there is a missing entropy contribution that is

not accounted for in the coarse-grained simulations. In this article,

we will show that this missing entropy is rooted in the electrostatic

interaction used in the coarse-grained models.

To appreciate the role of entropy in electrostatics, we note that

in implicit solvent models with dielectric continuum treatment of

electrostatics, the interaction energy between two charges

and

, separated by distance

in a solvent of electric permittivity

∑

, is in fact a potential of mean force (PMF)

with the solvent degrees of freedom averaged out. The dielec-

tric constant of water (the primary solvent used in coacervation

studies) is known to have significant temperature dependence

(84). The temperature-dependent dielectric constant has been

shown to be a major factor (50, 53) responsible for the lower

critical solution temperature behavior in some polyelectrolyte

complex coacervation systems (82, 85, 86). More importantly,

the temperature dependence of dielectric constant means the

electrostatic energy,

, has an entropic contribution (87, 88).

By interpreting

as a pure energy term in the thermodynamic

analysis of polyelectrolytes, previous works did not consider the

entropic contribution in this electrostatic energy. In this work,

we show that by properly accounting for the entropic contribu-

tion in

from the coarse-grained molecular dynamics simu-

lation, both stages of the polyelectrolyte complex coacervation

are found to be entropy driven with only minor contributions

from energy, consistent with known experimental results. Based

on the free energy calculation, we analyze the state of the dilute

phase and investigate the pathway of pair–pair condensation,

which is a key step in the pathway of the second stage in the

coacervation.

We start with the free energy

for the canonical system, which

is given in terms of the partition function

−

(

−

(

∑

−

[Γ]

)

[1]

where

,and

[Γ]

are the Boltzmann constant, temperature,

and energy for microstate

, respectively. In a fully microscopic

model, the microstate

includes all degrees of freedom, and the

energy

[Γ]

is the true energy, independent of temperature. How-

ever, in a coarse-grained model, such as implicit solvent models

commonly employed for studying electrolyte and polyelectrolyte

systems,

refers only to the reduced degrees of freedom, and

[Γ]

should be interpreted as a PMF, which is generally temperature

dependent. Regardless of the degree of coarse graining, the ther-

modynamic entropy

is given by

−

∂

−

(

)

−

∂

(

)

∂

−

∑

−

[Γ]

(

[Γ]

−

∂

[Γ]

∂

)

−

〈

∂

〉

[2]

i.e.,

−

〈

∂

〉

[3]

Thus, for a fully microscopic model, the last term is zero, and

we recover the well-known relation

−

. However,

for coarse-grained models, the last term is generally nonzero and

therefore must be included in the calculation of the true, physical

entropy. For implicit-solvent models of electrolyte and polyelec-

trolyte systems, the temperature dependence of the energy is in the

electrostatic interaction

through the dielectric constant. The

nonelectrostatic part of the interaction contributes little to the free

energy change in this study, as shown in

SI Appendix

,Fig.S1

.For

the electrostatic part,

〈

∂

〉

〈

∂

∑

∂

〉

〈

∑

〉

(

−

)

∂

−

∂

[4]

Substituting Eq.

into the last term of Eq.

yields

−

∂

[5]

Thus, when there is significant temperature dependence in the

dielectric constant, such as in the case of water, the last term in

Eq.

results in a substantial correction to the calculation of the en-

tropy and energy. We shall call this term the “electrostatic entropy”

(after dividing by the temperature), to indicate that this is the

entropy contained in the electrostatic interaction potential used

in coarse-grained model. The physical origin of this electrostatic

entropy is the reorganization of the solvent, particularly in the

orientation degrees of freedom of the dipolar solvent molecules.

In the Onsager theory for dielectric constant of liquids, the

temperature dependence is directly due to the orientation of the

2of9

https://doi.org/10.1073/pnas.2209975119

pnas.org

Downloaded from https://www.pnas.org by 76.89.95.90 on August 12, 2023 from IP address 76.89.95.90.

permanent dipoles (89). Although in real liquids such as water,

local structural correlations (such as due to hydrogen bonding)

result in modification of the dipolar contribution to the dielec-

tric constant (87, 90) and thus more complicated temperature

dependence than predicted by the Onsager theory, the dipolar

orientation remains the major root cause for the temperature

dependence of the dielectric constant. In

SI Appendix

, we provide

a simple explanation of the physical basis of the electrostatic

entropy based on the dipolar orientation of the solvent molecules.

Using the empirical temperature dependence of the dielectric

constant of water given by Malmberg and Maryott (84),

= 87.74

−

0.4008

+ 9.398

−

1.410

−

[6]

where

−

273.15

and

is the vacuum electric permittiv-

ity, we find, at room temperature 298.15 K,

(

)(

∂/∂

−

1.3576

,andso

−

1.3576

[7]

Since our interest in the thermodynamic analysis involves only

changes of thermodynamic functions between different states

under isothermal conditions, the more relevant equation is

−

+Δ

−

1.3576Δ

[8]

We note that this equation is unaffected whether or not the self-

energy of the ions is included in

since the self-energies of the

ions between the different thermodynamic states are unchanged

and will cancel out in taking the difference. From Eq.

,wealso

see that the true energy change

should be

=Δ

−

1.3576Δ

. Thus, by properly accounting for the entropic

contribution in the electrostatic interaction, both the entropic

and energetic contributions in driving force will be altered. For

comparison in our later discussion, we will use the notation

†

and

†

for the uncorrected energy and entropy change,

respectively.

We first examine the PMF

(

)

between two fully

charged polyions with opposite charges as a function of their

center-of-mass distance

in the absence of added salt (with

=60

=1.03

; see details in

Materials and Methods

Fig. 1 shows a large free energy decrease as the two chains

approach each other, reaching a magnitude of about 400

at close contact; this number is very close to that reported by

Ou and Muthukumar (76) under the same condition. Without

the entropic correction in the electrostatic interaction, Fig. 1

shows that complexation is energy driven as most of the free

energy decrease comes from the energy, i.e.,

−

†

The small entropy increase is likely due to counterion release upon

complexation (76, 78). The value of the energy change between

the complex and the two separate chains is also in agreement with

that reported in ref. 76, and the general conclusion of energy

dominance is consistent with previous simulation findings for

ξ<

1.5

(76, 78, 79). However, when the entropic contribution

to the electrostatic interaction is accounted for, the complexation

becomes entirely entropy driven, as shown in Fig. 1

.The

corrected energy contribution is small and unfavorable, consistent

with the endothermic behavior observed in experiments (71).

SI Appendix

,Fig.S2

shows that adding salt (

0.01

∼

0.1

)tothe

system weakens the overall driving force as measured by PMF,

and the energetic contribution becomes negligible at higher salt

concentrations. With the entropic correction, all our results are

consistent with experimental observations (71, 75).

Although we have only presented results for

≈

1.03

,Eq.

can be applied to previous simulation data based on continuum

Fig.1.

(

)PMFforthecomplexationofapolycation(orange)andapolyan-

ion (cyan) each with chain length

60. The small dots in

Inset

are the

corresponding counterions. (

) Components of the free energy without the

electrostaticentropycorrection.(

)Componentsofthefreeenergywiththe

electrostatic entropy correction according to Eq.

. The energy unit in all

curvesis

atroomtemperature(298.15K).

electrostatics to extract the electrostatic entropy in broader ranges

.In

SI Appendix

,Fig.S3

, we show that using the simulation

data in ref. 76, after making the entropic correction in the elec-

trostatic term, the polycation–polyanion complexation becomes

strongly entropy driven with only minor energetic contributions

in the most experimentally relevant ranges of

(

for aqueous

systems, in agreement with experiments. Furthermore, after the

correction, the trend in the enthalpy change (turning from posi-

tive to negative) with increasing

in the range of

<ξ<

is also

consistent with experiments (75, 81). Moreover, by comparing

the entropy change before and after the correction, we find that

the electrostatic entropy is the primary entropic driving force for

ξ<

1.7

The PMF of polycation–polyanion complexation can shed

useful light on the nature of the dilute phase in polyelectrolyte

complex coacervation, which has received considerably less

attention than the coacervate phase. In the calculation of phase

diagrams, most theories (20–22, 25, 26, 30–32, 91–98) assume

the polyions exist as dispersed chains in the dilute phase and

consequently predict unphysically low polymer concentration

in the supernatant phase. On the other hand, it has long been

suggested that the supernatant phase for symmetric polyelectrolyte

mixtures should consist primarily of polyion pairs under

physically relevant conditions (21, 66, 67, 99). This has been

explicitly demonstrated by field-theoretic simulation (100) and

theoretical calculation (43) for weakly charged polyelectrolytes

PNAS

2022 Vol.119 No.36 e2209975119

https://doi.org/10.1073/pnas.2209975119

3of9

Downloaded from https://www.pnas.org by 76.89.95.90 on August 12, 2023 from IP address 76.89.95.90.

in the absence of counterions and elaborated more recently for

the general polyelectrolyte complex coacervation in symmetric

mixtures (101). With the calculated PMF for polycation–

polyanion complexation, we can quantify the onset concentration

for polyion pairing from dispersed polycations and polyanions.

We consider the polycation–polyanion complexation as a

chemical reaction:

[9]

where

,and

denote polyanion, polycation, and

polyanion–polycation pair, respectively. The equilibrium constant

is given by

[10]

where

is the concentration of the

specie and

is the

corresponding internal partition function (102). The ratio of the

partition functions on the right-hand side of Eq.

is related to

the PMF by

∫

max

−

(

)

[11]

where

max

is the range of the attractive well of the PMF.

Noting that

(

)

increases steeply with

, the integral is

dominated by contributions near the minimum (

), and we

may expand

(

)

to linear order by writing

(

)

(0)

(

/σ

)

,with

being the local slope near

the minimum. For the PMF given in Fig. 1, Eq.

yields

an approximate value for the equilibrium constant

≈

π/

)

−

(

)(0)

≈

174

≈

150

.Ifwetake

the onset concentration for the complexation to be

∗

,thenweobtain

∗

≈

−

174

, an unfathomably

small number. The onset concentration increases by many orders

of magnitude with increasing salt concentration, but even for

the highest salt concentration we studied (

∼

0.1

∗

is still

unphysically low. Thus, except at very high salt concentrations,

a dilute mixture of polyanions and polycations will consist of

polyion pairs, practically devoid of uncomplexed polyelectrolyte

chains.

The value of the complexation free energy obtained from

our simulation is comparable to those measured in experiments.

On a per monomer basis, the

∼

400

free energy drop per

chain shown in Fig. 1 amounts to 17 kJ/mol per monomer.

This number falls in the range of the experimentally reported

values (10 to 30 kJ/mol) of the complexation free energy for

a series of polyanion and polycation pairs (71, 75). Given the

simplicity of the coarse-grained model employed in our work, this

level of agreement is quite heartening. In

SI Appendix

,webriefly

discuss the effects of chain length and length asymmetry in the

pairing free energy for polycation–polyanion complexation (see

SI Appendix

,Fig.S4

Since the dilute phase of a symmetric mixture of polyanions

and polycations is essentially devoid of uncomplexed chains, the

relevant species are the polycation–polyanion pairs, whose con-

densation eventually results in the coacervate. Here we consider

the elementary process of the fusion of such polyion pairs. In

Fig. 2

, we show the PMF between two polyion pairs, each

containing one polycation and one polyanion that are both fully

charged and have the same chain length

= 100

, at different

added salt concentrations

∼

0.15

Because the two polyion pairs have no net macromolecular

charges, the primary driving force for their fusion comes from the

tendency to decrease the interfacial free energy (43, 101). Con-

sequently, the free energy of fusion is considerably less than the

Fig.2.

Thermodynamics driving force for the condensation of two

polycation–polyanionpairsunderdifferentaddedsaltconditionsfor

100.

(

)PMF

)Correctedentropycontribution(Eq.

).(

)Correctedenergy

contribution. The energy unit in all curves is

at room temperature

(298.15K).

free energy of the polycation–polyanion complexation discussed

earlier; see Fig. 2

and compare with Fig. 1. A few representative

structures along the fusion pathway are shown in the insets of

Fig. 2

, where we see two isolated globules at long distances, going

through an elongated connected state, to the final fused larger

globule.

As in the case of polycation–polyanion complexation, we find

that the fusion between two polyion pairs is also driven by entropy

(when the corrected entropy expression is used), as shown in

Fig. 2

and

. The energy contribution is slightly unfavorable

for the salt-free system (still with counterions) and becomes less

with increasing salt concentration. Although the fusion of two

polyion pairs is only part of the overall coacervation process,

the trend observed in our simulation is consistent with the en-

thalpy change during the second stage of coacervation obtained

in experiments (71, 73), in that the measured enthalpy change is

small and slightly unfavorable, and its magnitude decreases with

added salt. Without the entropic correction in the electrostatic

interaction,

SI Appendix

,Fig.S5

shows the energetic contribution

is significant and comparable to the entropic contribution, which

is inconsistent with the slightly endothermic nature of the second

stage in coacervation reported in refs. 71, 73.

The PMF between two polyion pairs allows us to estimate

the spinodal of the supernatant phase. To this end, we write the

osmotic pressure of the supernatant phase due to the polymers as

a virial expansion,

Π=

(

...

[12]

4of9

https://doi.org/10.1073/pnas.2209975119

pnas.org

Downloaded from https://www.pnas.org by 76.89.95.90 on August 12, 2023 from IP address 76.89.95.90.

where

and

are the second and third osmotic virial coeffi-

cients, respectively.

is related to the PMF by (103)

−

∫

(

−

(

)

−

[13]

Fitting the PMF as a straight line with slope

in the range of

the interaction

≤

,where

≈

is the onset distance for

appreciable interaction (roughly the onset of contact between the

two polyion pair globules, which appears insensitive to the salt

concentrations we have examined),

is estimated to be

≈−

[

−

(0)

−

]

[14]

Invoking the spinodal condition

∂

/∂ρ

and ignoring the

contribution from the third virial coefficient [which is presumably

positive and not very large (66, 67)], we estimate the concentra-

tion of the polyion pairs at the spinodal to be

≈−

[15]

For the chain length and salt conditions studied here, the sec-

ond term on the right-hand side of Eq.

is only a small fraction

(

≤

) of the first term. Therefore, the spinodal concentration is

roughly

≈

πσ

(0)

[16]

In the supernatant phase at undersaturated concentrations, and

at the binodal, the solution consists primarily of isolated polyion

pairs, with much lower concentration of two-pair complexes and

higher-order clusters (101). From the PMF shown in Fig. 2

,we

can obtain the equilibrium constant for the dimerization from two

polyion pairs,

≈

π/

)

−

(0)

[17]

where

and

are the concentration of the polyion pairs

and that of the dimers of these pairs (hereafter termed two-

pair complex), respectively. As the concentration of polyion pairs

increases,

increases according to Eq.

. For sufficiently

high

, the concentration of the two-pair complexes becomes

comparable to that of the polyion pairs. This happens when

≈

πσ

(0)

. Up to a numerical factor of order 1,

this is just the spinodal condition given in Eq.

. This result

is not surprising: since the fusion of two polyion pairs does

not involve any kinetic barrier, the only free energy barrier to

coacervation is due to the translational entropy loss of the polyion

pairs in forming two-pair complexes and higher-order clusters, if

we consider the cluster size as the reaction coordinate to nucleation

(101, 104). When

≈

, the free energy barrier to cluster

growth vanishes—this is the condition of spinodal.

The results for the spinodal concentration are shown in Fig. 3

for the four salt concentrations studied. The spinodal concen-

tration increases with added salt, following the same trend as

the binodal shown in both theoretical and experimental phase

diagrams; the binodal for the present system, which is not available

from these two-pair calculations, is expected to lie below the

curve shown in Fig. 3. Between the binodal and the spinodal,

the system is in the metastable state, from which coacervation

proceeds by nucleation and growth, with vanishing barrier at the

spinodal. In practice, the metastable state can no longer exist when

Fig.3.

Spinodalconcentrationofthepolymersforthedilutebranchunder

differentaddedsaltconditions.Thepolymerconcentrationismeasuredusing

thetotalnumberofmonomers.

the nucleation barrier becomes a few

due to the enhanced

fluctuation near the spinodal (101, 105–107).

While the PMF becomes significant for

(the contact

distance), a close inspection of the curves in Fig. 2

reveals a

faint attraction at

. This attraction arises from the mutually

induced polarization of the two polyion-pair globules. To quantify

this induced polarization effect, we examine the fluctuation in the

net dipole of each globule, as well as the overall dipole of the two-

globule system.

For each pair, the electric dipole moment is given by

∑

,where

is the charge on monomer

and

is its

vector position, and the sum is over all monomers in pair

(

1, 2

). The total dipole of the system is then

.We

calculate the projection of these dipole moment onto the center-

of-mass vector between the pairs:

[18]

where

is the unit vector of the center of mass between the two

pairs defined as

−

)

−

,with

being the instantaneous center of mass position of pair

.Because

of symmetry,

. Thus, we quantify the fluctuation by the

second moment

and

. Furthermore, we calculate the

dipole correlation between the two pairs

[

−

[19]

In Fig. 4

and

, we show the fluctuation in the total

dipole moment and the dipole–dipole correlation between the two

polyion pairs. We see that the system dipole fluctuation becomes

appreciable at

/σ

, indicating mutually induced polariza-

tion well beyond the contact distance. This induced polarization

is more easily seen in the correlation

shown in Fig. 4

Thelargeincreaseinthecorrelationas

decreases below

indi-

cates that the dipoles from the two polyion pairs point in the same

direction. Furthermore, the polarization increases with increasing

salt concentration; this can be understood by the decrease in

the density of the polyion-pair globule (

SI Appendix

,Fig.S7

)and

the accompanying decrease in the interfacial tension with added

salt (93, 108), which makes the globule more deformable. The

globule shape anisotropy follows similar trend as the polarization

(

SI Appendix

,Fig.S6

Beyond the fusion between two polyion pairs, coacervation

involves the condensation into larger clusters which eventually

leads to macroscopic phase separation. It is therefore of interest

PNAS

2022 Vol.119 No.36 e2209975119

https://doi.org/10.1073/pnas.2209975119

5of9

Downloaded from https://www.pnas.org by 76.89.95.90 on August 12, 2023 from IP address 76.89.95.90.

Fig.4.

(

)Fluctuationofthelongitudinalcomponentofthepolarization

asafunctionofthecenter-of-massdistancebetweenthetwopairs,relative

tothatatinfinitedistance.Themarkersrepresentthesimulationdata,and

thecurvesarefittedtoguidetheeye.(

)Polarizationcorrelationbetweenthe

twopolyionglobules

asafunctionofthecenter-of-massdistance

betweenthetwopairsat

/σ

≥

14,wherethetwopairsweaklyattracteach

otherpriortocontact.Thecurvesarefittedtoguidetheeye.

to examine how the structural properties evolve as the cluster size

increases. To this end, we study and compare clusters with sizes

ranging from 1 to 10 polyion pairs.

Fig. 5

–

show the polymer radial density profile. For each

cluster, the polymer density within the cluster decreases with

increasing salt, consistent with the behavior of the density on the

coacervate branch of the phase diagram observed in experiments

and predicted by theories. Thus, the structure of the interior of

these clusters is similar to the bulk coacervate phase. Interestingly,

the polymer density in the interior of the clusters deceases

with increasing cluster size, as shown in the comparison in

Fig. 5

–

, where the four figures share the same ordinate coor-

dinate (

axis); this is seen more clearly in

SI Appendix

,Fig.S7

The higher density for smaller clusters is due to the effect

of the interfacial tension, which results in increased Laplace

pressure inside the cluster with decreasing cluster radius

according to

γ/

,where

is the surface tension. In

addition, visual inspection of the cluster morphology shows

that the smaller clusters tend to be more rounded, implying a

higher interfacial tension. Indeed, by examining the decay of

the density profile in Fig. 5

–

,wecanseethatthelarger

clusters have larger interfacial width. All these observations

are consistent with theoretical predictions from a recent

study (108).

We next examine the salt partition inside and outside the

clusters. Fig. 5

–

shows the density profile

of the small

ions (counterion + salt), normalized by the ion density outside the

cluster

. The salt is lower inside the cluster, suggesting a negative

tie line in the phase diagram, which is consistent with most

theories and experiments. Interestingly, the ion concentration

shows a peak just outside of the cluster at low salt concentrations

(salt-free and

=0.05

); this feature is in agreement with the

theoretical prediction in ref. 108.

In conclusion, by focusing on the elementary process of

the complexation between a polycation and a polyanion

and the subsequent fusion of two polyion pairs, we have

elucidated the driving force and pathway in polyelectrolyte

complex coacervation in symmetric polyelectrolyte mixtures.

By accounting for the electrostatic entropy, which is implicit

in the coarse-grained models, we resolve an apparent discrepancy

between experiments and previous coarse-grained simulations

regarding the thermodynamic driving force. Our analysis shows

that both the polycation–polyanion complexation and the pair–

pair condensation are driven almost entirely by entropy, in

agreement with experimental results. For fully charged PE with

moderately long chain lengths, we estimate that under most

conditions away from the critical point, the concentration of

uncomplexed polyions is vanishingly small; the supernatant

phase consists predominantly of polyion pairs. From the PMF

between two polyion pairs, we are able to estimate the spinodal

concentration of the supernatant phase. Finally, we show there

exists a weak attraction between two macromolecular neutral

polyion-pair globules prior to contact due to mutually induced

polarization; this action-at-a-distance provides the initial driving

force for the fusion of the polyion pairs, eventually leading to the

formation of macroscopic coacervate phase.

While the prevailing view in the polyelectrolyte community

holds that counterion release is the primary entropic contri-

bution to the thermodynamic driving force for polyelectrolyte

coacervation (61–63, 72, 76), our results show that a significant

entropic component in the free energy change comes from the

temperature dependence in the dielectric constant of water,

and this electrostatic entropy is the main contribution to the

overall entropy change in the regime of weak to intermediate

electrostatic strength. More generally, our results emphasize that

coarse-grained effective potentials inherently include entropic

contributions (109) due to solvent reorganization, and care must

be taken when interpreting experimental thermodynamic data

using coarse-grained simulations.

We end our discussion with a few remarks on the limitations

of the simulation model used here (and in many other coarse-

grained simulations of electrolyte and polyelectrolyte systems).

The treatment of ion–ion interactions using a uniform bulk

dielectric constant has two clear drawbacks. First, it does not

capture the discrete nature of the solvent molecules at short ion–

ion separations. Second, high polymer and salt concentration will

result in changes in the local dielectric permittivity. The first effect

would require a model with explicit solvent, while the second

would require a better description of the local polarization due

to polymer or ion enrichment. Although computational methods

for including both effects exist (110–112) (in addition to fully

atomistic models), accounting for these effects in the context of

polyelectrolyte complex coacervation is computationally demand-

ing and is beyond the scope of our work. A closely related effect

is the change in the local hydration structure of the ions due to

high temperature and strong binding by multivalent ions (75,

113–120), which can result in substantial entropy changes related

to water release (121). Such an effect cannot be described with

a simple dielectric continuum treatment. However, within the

framework of the primitive model for electrolytes/polyelectrolytes,

6of9

https://doi.org/10.1073/pnas.2209975119

pnas.org

Downloaded from https://www.pnas.org by 76.89.95.90 on August 12, 2023 from IP address 76.89.95.90.

ABCD

EFGH

Fig.5.

(

–

) Radialmonomerdensityprofile for neutralpolyion clusterswithvarying sizesatdifferentsalt concentrations.(

–

) Corresponding small-ion

densityprofileforeachcluster.

it is possible to capture many of the structural features of polyelec-

trolyte complex coacervation due to divalent ions (122–124) by

introducing a phenomenological binding energy (125). Despite

these considerations, we note that the implicit-solvent, dielectric

continuum description of charged systems has been very successful

in modeling a wide range of systems involving electrolyte and

polyelectrolyte solutions even at fairly high salt concentrations

(126). In view of the overall agreement between our simulation

results and experimental findings, we expect that for weak-to-

moderate electrostatic strengths with monovalent salt ions at

room temperature, these additional effects will modify but not

fundamentally alter the main results of this work.

Materials and Methods

The monomers and small ions in our simulation are represented as beads inter-

actingwiththeshifted-truncatedLennard-Jonespotentialof theform

WCA

(

[(

)

−

(

)

]

with a cutoff distance

cut

.Weset

and

. Neighboring beads along the polymer chains are subjected to

the FENE potential given by

FENE

(

bond

[

−

(

)

]

,with

bond

/σ

and

1.5

. All the beads have the same mass

and size

. This

setup reproduces a good solvent condition for neutral polymers. With

being

the energy scale in these potentials, these nonelectrostatic interactions should

be considered as entropic in nature. However, since they contribute minimally to

the PMF (

SI Appendix

, Fig. S1

), the precise designation of these nonelectrostatic

interaction energies is immaterial.

The polyelectrolytes are linear chains with every monomer carrying a unit

charge

−

. Each charged polymer bead has a corresponding counterion

bead carrying the opposite unit charge. Salt ions are modeled as monovalent

beads having the same identity as the counterions. The long-range electrostatic

interaction is given by

, where the Bjerrum length is given by

(

)

.Forwaterat

298.15K,

≈

0.7nm.Thisvalueisroughly

the diameter of common hydrated monovalent ions in water (127).Therefore,we

set

in our simulation.The average equilibrium bond length is

≈

0.97

;

thus, the electrostatic strength

≈

1.03. At this

, without considering

the electrostatic entropy, the complexation of two oppositely charged PEs was

reported to be energy driven in previous simulations (76, 78).

All our simulations are performed in the canonical ensemble with a Langevin

thermostat using the LAMMPS (Large-scale Atomic/Molecular Massively Parallel

Simulator) platform. The simulation time scale is given by

√

1. The positions and velocities of the beads are updated with an integration

time step

0.01

. For the PMF calculation of the two-chain systems, the

simulation box is set to 130

130

. This is the same setup as that

in refs.76 and 78.For the two-pair systems,the simulation box is 45

. Note that each polycation–polyanion pair exists as a neutral globule, which

allows us to use a smaller box size to study the pair–pair condensation PMF. Each

structural unit (a polyelectrolyte chain or polyion pair) is equilibrated for 10

(10

time steps) before performing the PMF calculations. The PMF calculations

are carried out by the adaptive bias force algorithm (128, 129) implemented

in LAMMPS (130). The coordinate of the PMF is taken to be the center-of-mass

distance between the two structural units (chains or pairs). The distance range

is 0

∼

for the two-chain case and 0

∼

for the two-pair case. To

improvetheefficiencyof thePMFcalculations(129),thedistancerangeisdivided

into 10 consecutive windows.Each window is further divided into bins with equal

width 0.5

for the two-chain case and 0.2

for the two-pair system. For each

window, we perform the simulation for 2

to get convergence. To avoid

nonequilibrium effects, we use the scheme documented by Miao et al. (131) to

set the sampling time prior to applying the adaptive bias force in each bin. The

PMF is calculated on the fly by integrating the biased force between neighboring

bins. The PMF in most windows reaches convergence before 2

,after

which the relative free energy change in each window is less than 1

for an

additional time duration of 10

Data,Materials,andSoftwareAvailability.

Allstudydataareincludedinthe

article and/or

SI Appendix

ACKNOWLEDGMENTS.

This research is supported by funding from Hong Kong

Quantum AI Lab Ltd. We thank the general computation time allocated by the

resources of the Center for Functional Nanomaterials, which is a US Department

of Energy Office of Science User Facility, at Brookhaven National Laboratory

under Contract DE-SC0012704. We thank Prof. M. Muthukumar for bringing our

attention to his forthcoming book.

PNAS

2022 Vol.119 No.36 e2209975119

https://doi.org/10.1073/pnas.2209975119

7of9

Downloaded from https://www.pnas.org by 76.89.95.90 on August 12, 2023 from IP address 76.89.95.90.

1. F. W. Tiebackx, Gleichzeitige Ausflockung zweier Kolloide.

Z. Chem. Ind. Koll.

8

, 198–201 (1911).

2. B. de Jong, H. R. Kruyt, Coacervation.

Proc.R.Acad.Amsterdam

32

, 849–856 (1929).

3. C. P. Brangwynne, P. Tompa, R. V. Pappu, Polymer physics of intracellular phase transitions.

Nat. Phys.

11

, 899–904 (2015).

4. T. J. Nott

et al.

, Phase transition of a disordered nuage prot

ein generates environmentally responsive

membraneless organelles.

Mol. Cell

57

, 936–947 (2015).

5. S. Alberti, A. Gladfelter, T. Mittag, Considerations and challenges in studying liquid-liquid phase

separation and biomolecular condensates.

Cell

176

, 419–434 (2019).

6. N. A. Yewdall, A. A. Andr

e, T. Lu, E. Spruijt, Coacervates as models of membraneless organelles.

Curr. Opin. Colloid Interface Sci.

52

, 101416 (2021).

7. B. P. Lee, P. B. Messersmith, J. N. Israelachvili, J. H. Waite, Mussel-inspired adhesives and coatings.

Annu.Rev.Mater.Res.

41

, 99–132 (2011).

8. W. Wei

et al.

, A mussel-derived one component adhesive coacervate.

Acta Biomater.

10

, 1663–1670

(2014).

9. J. H. Waite, Mussel adhesion—Essential footwork.

J. Exp. Biol.

220

, 517–530 (2017).

10. H. J. Kim, B. Yang, T. Y. Park, S. Lim, H. J. Cha, Complex coacervates based on recombinant mussel

adhesive proteins: Their characterization and applications.

Soft Matter

13

, 7704–7716 (2017).

11. R. J. Stewart, C. S. Wang, I. T. Song, J. P. Jones, The role of coacervation and phase transitions in the

sandcastle worm adhesive system.

Adv. Colloid Interface Sci.

239

, 88–96 (2017).

12. K. M. Jin, Y. H. Kim, Injectable, thermo-reversible and complex coacervate combination gels for

protein drug delivery.

J. Control. Release

127

, 249–256 (2008).

13. N. R. Johnson, Y. Wang, Controlled delivery of heparin-binding EGF-like growth factor yields fast and

comprehensive wound healing.

J. Control. Release

166

, 124–129 (2013).

14. W. C. Blocher, S. L. Perry, Complex coacervate-based materials for biomedicine.

Wiley Interdiscip.Rev.

Nanomed. Nanobiotechnol.

9

, e1442 (2017).

15. H. Shao, R. J. Stewart, Biomimetic underwater adhesives with environmentally triggered setting

mechanisms.

Adv. Mater.

22

, 729–733 (2010).

16. R. J. Stewart, C. S. Wang, H. Shao, Complex coacervates as a foundation for synthetic underwater

adhesives.

Adv. Colloid Interface Sci.

167

, 85–93 (2011).

17. M. Domp

et al.

, Thermoresponsive complex coacervate-based underwater adhesive.

Adv. Mater.

31

e1808179 (2019).

18. M. Domp

et al.

, Hybrid complex coacervate.

Polymers (Basel)

12

, 320 (2020).

19. M. Vahdati, F. J. Cedano-Serrano, C. Creton, D. Hourdet, Coacervate-based underwater adhesives in

physiological conditions.

ACS Appl. Polym. Mater.

2

, 3397–3410 (2020).

20. J. T. G. Overbeek, M. J. Voorn, Phase separation in polyelectrolyte solutions; theory of complex

coacervation.

J. Cell. Physiol. Suppl.

49

(suppl. 1), 7–22, (1957).

21. V. Yu Borue, I. Ya Erukhimovich, A statistical theory of globular polyelectrolyte complexes.

Macromolecules

23

, 3625–3632 (1990).

22. A. Kudlay, A. V. Ermoshkin, M. Olvera de la Cruz, Complexation of oppositely charged

polyelectrolytes: Effect of ion pair formation.

Macromolecules

37

, 9231–9241 (2004).

23. M. Castelnovo, J. F. Joanny, Complexation between oppositely charged polyelectrolytes: Beyond the

random phase approximation.

Eur.Phys.J.E

6

, 377–386 (2001).

24. J. Lee, Y. O. Popov, G. H. Fredrickson, Complex coacervation: A field theoretic simulation study of

polyelectrolyte complexation.

J. Chem. Phys.

128

, 224908 (2008).

25. S. L. Perry, C. E. Sing, PRISM-based theory of complex coacervation: Excluded volume versus chain

correlation.

Macromolecules

48

, 5040–5053 (2015).

26. J. Qin, J. J. de Pablo, Criticality and connectivity in macromolecular charge complexation.

Macromolecules

49

, 8789–8800 (2016).

27. M. Radhakrishna

et al.

, Molecular connectivity and correlation effects on polymer coacervation.

Macromolecules

50

, 3030–3037 (2017).

28. K. Shen, Z.-G. Wang, Polyelectrolyte chain structure and solution phase behavior.

Macromolecules

51

, 1706–1717 (2018).

29. A. M. Rumyantsev, E. B. Zhulina, O. V. Borisov, Complex coacervate of weakly charged

polyelectrolytes: Diagram of states.

Macromolecules

51

, 3788–3801 (2018).

30. P. Zhang, K. Shen, N. M. Alsaifi, Z.-G. Wang, Salt partitioning in complex coacervation of symmetric

polyelectrolytes.

Macromolecules

51

, 5586–5593 (2018).

31. A. Kudlay, M. Olvera de la Cruz, Precipitation of oppositely charged polyelectrolytes in salt solutions.

J. Chem. Phys.

120

, 404–412 (2004).

32. T. K. Lytle, C. E. Sing, Transfer matrix theory of polymer complex coacervation.

Soft Matter

13

7001–7012 (2017).

33. L. Li

et al.

, Phase behavior and salt partitioning in polyelectrolyte complex coacervates.

Macromolecules

51

, 2988–2995 (2018).

34. S. P. O. Danielsen, J. McCarty, J. E. Shea, K. T. Delaney, G. H. Fredrickson, Molecular design of

self-coacervation phenomena in block polyampholytes.

Proc.Natl.Acad.Sci.U.S.A.

116

, 8224–8232

(2019).

35. J. Huang, J. E. Laaser, Charge density and hydrophobicity-dominated regimes in the phase behavior

of complex coacervates.

ACS Macro Lett.

10

, 1029–1034 (2021).

36. C. W. Pak

et al.

, Sequence determinants of intracellular phase separation by complex coacervation of

a disordered protein.

Mol. Cell

63

, 72–85 (2016).

37. A. M. Rumyantsev

et al.

, Controlling complex coacervation via random polyelectrolyte sequences.

ACS Macro Lett.

8

, 1296–1302 (2019).

38. L. W. Chang

et al.

, Sequence and entropy-based control of complex coacervates.

Nat. Commun.

8

1273 (2017).

39. J. J. Madinya, L. W. Chang, S. L. Perry, C. E. Sing, Sequence-dependent self-coacervation in high

charge-density polyampholytes.

Mol. Syst. Des. Eng.

5

, 632–644 (2020).

40. N. A. Zervoudis, A. C. Obermeyer, The effects of protein charge patterning on complex coacervation.

Soft Matter

17

, 6637–6645 (2021).

41. J. B. Schlenoff, M. Yang, Z. A. Digby, Q. Wang, Ion content of polyelectrolyte complex coacervates and

the Donnan equilibrium.

Macromolecules

52

, 9149–9159 (2019).

42. S. P. O. Danielsen, S. Panyukov, M. Rubinstein, Ion pairing and the structure of gel coacervates.

Macromolecules

53

, 9420–9442 (2020).

43. A. M. Rumyantsev, I. I. Potemkin, Explicit description of complexation between oppositely charged

polyelectrolytes as an advantage of the random phase approximation over the scaling approach.

Phys.Chem.Chem.Phys.

19

, 27580–27592 (2017).

44. P. Zhang, N. M. Alsaifi, J. Wu, Z.-G. Wang, Polyelectrolyte complex coacervation: Effects of

concentration asymmetry.

J. Chem. Phys.

149

, 163303 (2018).

45. S. Friedowitz

et al.

, Looping-in complexation and ion partitioning in nonstoichiometric

polyelectrolyte mixtures.

Sci. Adv.

7

, eabg8654 (2021).

46. S. V. Bobbili, S. T. Milner, Closed-loop phase behavior of nonstoichiometric coacervates in

coarse-grained simulations.

Macromolecules

55

, 511–516 (2022).

47. S. Chen, P. Zhang, Z. G. Wang, Complexation between oppositely charged polyelectrolytes in dilute

solution: Effects of charge asymmetry.

Macromolecules

55

, 3898–3909 (2022).

48. S. Adhikari, M. A. Leaf, M. Muthukumar, Polyelectrolyte complex coacervation by electrostatic dipolar

interactions.

J. Chem. Phys.

149

, 163308 (2018).

49. J. D. Tang, S. R. Caliari, K. J. Lampe, Temperature-dependent complex coacervation of engineered

elastin-like polypeptide and hyaluronic acid polyelectrolytes.

Biomacromolecules

19

, 3925–3935

(2018).

50. S. Adhikari, V. M. Prabhu, M. Muthukumar, Lower critical solution temperature behavior in

polyelectrolyte complex coacervates.

Macromolecules

52

, 6998–7004 (2019).

51. Z. Ye, S. Sun, P. Wu, Distinct cation-anion interactions in the UCST and LCST behavior of

polyelectrolyte complex aqueous solutions.

ACS Macro Lett.

9

, 974–979 (2020).

52. A. Narayanan, J. R. Menefee, Q. Liu, A. Dhinojwala, A. Joy, Lower critical solution temperature-driven

self-coacervation of nonionic polyester underwater adhesives.

ACS Nano

14

, 8359–8367 (2020).

53. A. S. Ylitalo, C. Balzer, P. Zhang, Z.-G. Wang, Electrostatic correlations and temperature-dependent

dielectric constant can model LCST in polyelectrolyte complex coacervation.

Macromolecules

54

11326–11337 (2021).

54. F. Weinbreck, R. de Vries, P. Schrooyen, C. G. de Kruif, Complex coacervation of whey proteins and

gum arabic.

Biomacromolecules

4

, 293–303 (2003).

55. P. Jha, P. Desai, J. Li, R. Larson, pH and salt effects on the associative phase separation of oppositely

charged polyelectrolytes.

Polymers (Basel)

6

, 1414–1436 (2014).

56. M. Ghasemi, R. G. Larson, Role of electrostatic interactions in charge regulation of weakly

dissociating polyacids.

Prog. Polym. Sci.

112

, 101322 (2021).

57. A. R. Knoerdel, W. C. Blocher McTigue, C. E. Sing, Transfer matrix model of pH effects in polymeric

complex coacervation.

J. Phys. Chem. B

125

, 8965–8980 (2021).

58. S. Friedowitz, J. Qin, Reversible ion binding for polyelectrolytes with adaptive conformations.

AIChE J.

67

, e17426 (2021).

59. M. Lee, S. L. Perry, R. C. Hayward, Complex coacervation of polymerized ionic liquids in non-aqueous

solvents.

ACS Polym. Au

1

, 100–106 (2021).

60. L. Li

et al.

, Effect of solvent quality on the phase behavior of polyelectrolyte complexes.

Macromolecules

54

, 105–114 (2021).

61. C. E. Sing, Development of the modern theory of polymeric complex coacervation.

Adv. Colloid

Interface Sci.

239

, 2–16 (2017).

62. M. Muthukumar, 50th anniversary perspective: A perspective on polyelectrolyte solutions.

Macromolecules

50

, 9528–9560 (2017).

63. C. E. Sing, S. L. Perry, Recent progress in the science of complex coacervation.

Soft Matter

16

2885–2914 (2020).

64. A. M. Rumyantsev, N. E. Jackson, J. J. de Pablo, Polyelectrolyte complex coacervates: Recent

developments and new frontiers.

Annu. Rev. Condens. Matter Phys.

12

, 155–176 (2021).

65. V. M. Prabhu, Interfacial tension in polyelectrolyte systems exhibiting associative liquid–liquid phase

separation.

Curr. Opin. Colloid Interface Sci.

53

, 101422 (2021).

66. K. I. Tainaka, Study of complex coacervation in low concentration by virial expansion method.

I. Salt free systems.

J.Phys.Soc.Jpn.

46

, 1899–1906 (1979).

67. K. I. Tainaka, Effect of counterions on complex coacervation.

Biopolymers

19

, 1289–1298 (1980).

68. M. Girard, S. L. Turgeon, S. F. Gauthier, Thermodynamic parameters of beta-lactoglobulin-pectin

complexes assessed by isothermal titration calorimetry.

J.Agric.Food Chem.

51

, 4450–4455 (2003).

69. X. Feng, M. Leduc, R. Pelton, Polyelectrolyte complex characterization with isothermal titration

calorimetry and colloid titration.

Colloids Surf. A Physicochem. Eng. Asp.

317

, 535–542 (2008).

70. L. Aberkane, J. Jasniewski, C. Gaiani, J. Scher, C. Sanchez, Thermodynamic characterization of acacia

gum-beta-lactoglobulin complex coacervation.

Langmuir

26

, 12523–12533 (2010).

71. D. Priftis, N. Laugel, M. Tirrell, Thermodynamic characterization of polypeptide complex coacervation.

Langmuir

28

, 15947–15957 (2012).

72. A. B. Kayitmazer, Thermodynamics of complex coacervation.

Adv. Colloid Interface Sci.

239

, 169–177

(2017).

73. D. Priftis, K. Megley, N. Laugel, M. Tirrell, Complex coacervation of poly(ethylene-imine)/polypeptide

aqueous solutions: Thermodynamic and rheological characterization.

J. Colloid Interface Sci.

398

39–50 (2013).

74. C. B. Bucur, Z. Sui, J. B. Schlenoff, Ideal mixing in polyelectrolyte complexes and multilayers: Entropy

driven assembly.

J. Am. Chem. Soc.

128

, 13690–13691 (2006).

75. J. Fu, J. B. Schlenoff, Driving forces for oppositely charged polyion association in aqueous solutions:

Enthalpic, entropic, but not electrostatic.

J. Am. Chem. Soc.

138

, 980–990 (2016).

76. Z. Ou, M. Muthukumar, Entropy and enthalpy of polyelectrolyte complexation: Langevin dynamics

simulations.

J. Chem. Phys.

124

, 154902 (2006).

77. R. M. Elder, T. Emrick, A. Jayaraman, Understanding the effect of polylysine architecture on DNA

binding using molecular dynamics simulations.

Biomacromolecules

12

, 3870–3879 (2011).

78. V. S. Rathee, H. Sidky, B. J. Sikora, J. K. Whitmer, Role of associative charging in the entropy-energy

balance of polyelectrolyte complexes.

J. Am. Chem. Soc.

140

, 15319–15328 (2018).

79. A. N. Singh, A. Yethiraj, Driving force for the complexation of charged polypeptides.

J. Phys. Chem. B

124

, 1285–1292 (2020).

80. A. N. Singh, A. Yethiraj, Liquid-liquid phase separation as the second step of complex coacervation.

J. Phys. Chem. B

125

, 3023–3031 (2021).

81. N. Laugel

et al.

, Relationship between the growth regime of polyelectrolyte multilayers and the

polyanion/polycation complexation enthalpy.

J. Phys. Chem. B

110

, 19443–19449 (2006).

82. Q. Wang, J. B. Schlenoff, The polyelectrolyte complex/coacervate continuum.

Macromolecules

47

3108–3116 (2014).

83. A. S. Michaels, Polyelectrolyte complexes.

Ind. Eng. Chem.

57

, 32–40 (1965).

84. C.G.Malmberg,A.A.Maryott,Dielectricconstantofwaterfrom0

◦

to 100

◦

J. Res. Natl. Bur. Stand.

56

, 1–8 (1956).

85. S. Ali, M. Bleuel, V. M. Prabhu, Lower critical solution temperature in polyelectrolyte complex

coacervates.

ACS Macro Lett.

8

, 289–293 (2019).

86. Y. Ma, S. Ali, V. M. Prabhu, Enhanced concentration fluctuations in model polyelectrolyte coacervate

mixtures along a salt isopleth phase diagram.

Macromolecules

54

, 11338–11350 (2021).

87. H. Fr

ohlich,

Theory of Dielectrics: Dielectric Constant and Dielectric Loss

(Clarendon Press, Oxford,

1958).

8of9

https://doi.org/10.1073/pnas.2209975119

pnas.org

Downloaded from https://www.pnas.org by 76.89.95.90 on August 12, 2023 from IP address 76.89.95.90.

88. M. Murugappan,

Physics of Charged Macromolecules: Synthetic and Biological Systems

(Cambridge

University Press, Cambridge, United Kingdom, in press).

89. L. Onsager, Electric moments of molecules in liquids.

J. Am. Chem. Soc.

58

, 1486–1493 (1936).

90. J. G. Kirkwood, The dielectric polarization of polar liquids.

J. Chem. Phys.

7

, 911–919 (1939).

91. N. P. Shusharina, E. B. Zhulina, A. V. Dobrynin, M. Rubinstein, Scaling theory of diblock

polyampholyte solutions.

Macromolecules

38

, 8870–8881 (2005).

92. Z. Wang, M. Rubinstein, Regimes of conformational transitions of a diblock polyampholyte.

Macromolecules

39

, 5897–5912 (2006).

93. J. Qin

et al.

, Interfacial tension of polyelectrolyte complex coacervate phases.

ACS Macro Lett.

3

565–568 (2014).

94. P. Zhang, N. M. Alsaifi, J. Wu, Z.-G. Wang, Salting-out and salting-in of polyelectrolyte solutions:

A liquid-state theory study.

Macromolecules

49

, 9720–9730 (2016).

95. A. Salehi, R. G. Larson, A molecular thermodynamic model of complexation in mixtures of oppositely

charged polyelectrolytes with explicit account of charge association/dissociation.

Macromolecules

49

, 9706–9719 (2016).

96. M. Rubinstein, Q. Liao, S. Panyukov, Structure of liquid coacervates formed by oppositely charged

polyelectrolytes.

Macromolecules

51

, 9572–9588 (2018).

97. A. M. Rumyantsev, J. J. de Pablo, Liquid crystalline and isotropic coacervates of semiflexible

polyanions and flexible polycations.

Macromolecules

52

, 5140–5156 (2019).

98. G. M. C. Ong, C. E. Sing, Mapping the phase behavior of coacervate-driven self-assembly in diblock

copolyelectrolytes.

Soft Matter

15

, 5116–5127 (2019).

99. A. Veis, C. Aranyi, Phase separation in polyelectrolyte systems. I. Complex coacervation of gelatin.

J. Phys. Chem.

64

, 1203–1210 (1960).

100. K. T. Delaney, G. H. Fredrickson, Theory of polyelectrolyte complexation—Complex coacervates are

self-coacervates.

J. Chem. Phys.

146

, 224902 (2017).

101. P. Zhang, Z.-G. Wang, Supernatant phase in polyelectrolyte complex coacervation: Cluster formation,

binodal, and nucleation.

Macromolecules

55

, 3910–3923 (2022).

102. C. N. Likos, Effective interactions in soft condensed matter physics.

Phys. Rep.

348

, 267–439 (2001).

103. Z.-G. Wang, 50th anniversary perspective: Polymer conformation—A pedagogical review.

Macromolecules

50

, 9073–9114 (2017).

104. R. Wang, Z.-G. Wang, Theory of polymers in poor solvent: Phase equilibrium and nucleation

behavior.

Macromolecules

45

, 6266–6271 (2012).

105. K. Binder, Collective diffusion, nucleation, and spinodal decomposition in polymer mixtures.

J. Chem. Phys.

79

, 6387–6409 (1983).

106. K. Binder, Nucleation barriers, spinodals, and the Ginzburg criterion.

Phys.Rev.A

29

, 341–349

(1984).

107. Z.-G. Wang, Concentration fluctuation in binary polymer blends:

parameter, spinodal and

Ginzburg criterion.

J. Chem. Phys.

117

, 481–500 (2002).

108. P. Zhang, Z.-G. Wang, Interfacial structure and tension of polyelectrolyte complex coacervates.

Macromolecules

54

, 10994–11007 (2021).

109. A. Ben-Shaul, Entropy, energy, and bending of DNA in viral capsids.

Biophys.J.

104

, L15–L17 (2013).

110. D. J. Grzetic, K. T. Delaney, G. H. Fredrickson, Field-theoretic study of salt-induced order and disorder

in a polarizable diblock copolymer.

ACS Macro Lett.

8

, 962–967 (2019).

111. A. A. Gavrilov, Dissipative particle dynamics for systems with polar species: Interactions in dielectric

media.

J. Chem. Phys.

152

, 164101 (2020).

112. A. A. Gavrilov, E. Y. Kramarenko, Two contributions to the dielectric response of polar liquids.

J.Chem.

Phys.

154

, 116101 (2021).

113. J. B. Schlenoff, A. H. Rmaile, C. B. Bucur, Hydration contributions to association in polyelectrolyte

multilayers and complexes: Visualizing hydrophobicity.

J. Am. Chem. Soc.

130

, 13589–13597

(2008).

114. H. H. Hariri, A. M. Lehaf, J. B. Schlenoff, Mechanical properties of osmotically stressed polyelectrolyte

complexes and multilayers: Water as a plasticizer.

Macromolecules

45

, 9364–9372 (2012).

115. E. Yildirim, Y. Zhang, J. L. Lutkenhaus, M. Sammalkorpi, Thermal transitions in polyelectrolyte

assemblies occur via a dehydration mechanism.

ACS Macro Lett.

4

, 1017–1021 (2015).

116. Y. Zhang, F. Li, L. D. Valenzuela, M. Sammalkorpi, J. L. Lutkenhaus, Effect of water on the thermal

transition observed in poly(allylamine hydrochloride)–poly(acrylic acid) complexes.

Macromolecules

49

, 7563–7570 (2016).

117. R. Zhang, Y. Zhang, H. S. Antila, J. L. Lutkenhaus, M. Sammalkorpi, Role of salt and water in the

plasticization of PDAC/PSS polyelectrolyte assemblies.

J. Phys. Chem. B

121

, 322–333 (2017).

118. Y. Zhang

et al.

, Molecular origin of the glass transition in polyelectrolyte assemblies.

ACS Cent. Sci.

4

638–644 (2018).

119. P. Batys, Y. Zhang, J. L. Lutkenhaus, M. Sammalkorpi, Hydration and temperature response of water

mobility in poly(diallyldimethylammonium)-poly(sodium 4-styrenesulfonate) complexes.

Macromolecules

51

, 8268–8277 (2018).

120. P. Batys, S. Kivist

o, S. M. Lalwani, J. L. Lutkenhaus, M. Sammalkorpi, Comparing water-mediated

hydrogen-bonding in different polyelectrolyte complexes.

Soft Matter

15

, 7823–7831 (2019).

121. R. E. Bulo

et al.

, “Site binding” of Ca

ions to polyacrylates in water: A molecular dynamics study of

coiling and aggregation.

Macromolecules

40

, 3437–3442 (2007).

122. H. Dautzenberg, J. Kriz, Response of polyelectrolyte complexes to subsequent addition of salts with

different cations.

Langmuir

19

, 5204–5211 (2003).

123. S. Perry, Y. Li, D. Priftis, L. Leon, M. Tirrell, The effect of salt on the complex coacervation of vinyl

polyelectrolytes.

Polymers (Basel)

6

, 1756–1772 (2014).

124. D. Iyer, V. M. S. Syed, S. Srivastava, Influence of divalent ions on composition and viscoelasticity of

polyelectrolyte complexes.

J. Polym. Sci

59

, 2895–2904 (2021).

125. T. K. Lytle, C. E. Sing, Tuning chain interaction entropy in complex coacervation using polymer

stiffness, architecture, and salt valency.

Mol. Syst. Des. Eng.

3

, 183–196 (2018).

126. R. Triolo, J. R. Grigera, L. Blum, Simple electrolytes in the mean spherical approximation.

J. Phys.

Chem.

80

, 1858–1861 (1976).

127. J. Israelachvili,

Intermolecular and Surface Forces

(Academic Press, 2011).

128. E. Darve, D. Rodr

ıguez-G

omez, A. Pohorille, Adaptive biasing force method for scalar and vector free

energy calculations.

J. Chem. Phys.

128

, 144120 (2008).

129. J. Comer

et al.

, The adaptive biasing force method: Everything you always wanted to know but were

afraid to ask.

J. Phys. Chem. B

119

, 1129–1151 (2015).

130. G. Fiorin, M. L. Klein, J. H

enin, Using collective variables to drive molecular dynamics simulations.

Mol. Phys.

111

, 3345–3362 (2013).

131. M. Miao

et al.

, Avoiding non-equilibrium effects in adaptive biasing force calculations.

Mol. Simul.

47

, 390–394 (2021).

PNAS

2022 Vol.119 No.36 e2209975119

https://doi.org/10.1073/pnas.2209975119

9of9

Downloaded from https://www.pnas.org by 76.89.95.90 on August 12, 2023 from IP address 76.89.95.90.