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Dilute polyelectrolyte solutions: recent progress and
open questions†
Carlos G. Lopeza, Atsushi Matsumotob, and Amy Q. Shenc
Polyelectrolytes are a type of polymers possessing ionic groups on their repeating units. Since coun-
terions can dissociate from the polymer backbone, polyelectrolyte chains are strongly influenced by
the electrostatic interactions. As a result, the physical properties of polyelectrolyte solutions are sig-
nificantly different from those of electrically neutral polymers. The aim of this article is to highlight
key results and some outstanding questions in the polyelectrolyte research from recent literature. We
focus on the influence of electrostatics on conformational and hydrodynamic properties of polyelec-
trolyte chains. A compilation of experimental results from the literature reveal significant disparities
with theoretical predictions. We also discuss a new class of polyelectrolytes called poly(ionic liquid)s
that exhibit unique physical properties in comparison to ordinary polyelectrolytes. We conclude this
review by listing some key research challenges to fully understand the conformation and dynamics of
polyelectrolytes in solutions.
1 Introduction
The field of the polymer science originated in 1920, with the
core concept based on the macromolecular hypothesis proposed
by Staudinger.1Since then, it has been well accepted that poly-
mers are made of molecules covalently bonded with each other.
The definition by IUPAC states that polymers are the substances
composed of macromolecules with their molecular weights larger
than a few thousand.2During the past century, polymer chemists
have made significant advances in creating new polymer species
and controlling the polymerisation process, while polymer physi-
cists have focused on understanding the properties of polymers by
using theoretical, computational, and experimental approaches.
As a result, polymer-based materials are prevalent and exten-
sively used in various industries and daily products.3
Despite the significant research progress in the physics of non-
ionic polymers, for which their properties (especially in dilute
solutions) are relatively well understood, less advancement has
been made for charged polymers. Polyelectrolytes, which re-
ceived their name from Raymond M. Fuoss in 1948,4are a
aInstitute of Physical Chemistry, RWTH Aachen University, Aachen, 52056, Germany,
European Union. Present Address: Department of Materials Science and Engineering,
Penn State University, University Park, Pennsylvania 16802
bDepartment of Applied Chemistry and Biotechnology, Graduate School of Engineering,
University of Fukui, 3-9-1 Bunkyo, Fukui City, Fukui 910-8507, Japan
cMicro/Bio/Nanofluidics Unit, Okinawa Institute of Science and Technology Graduate
University, 1919-1 Tancha, Onna-son, Okinawa 904-0495, Japan
E-mail:(CGL) lopez@pc.rwth-aachen.de; (AM) atsushi5@u-fukui.ac.jp; (AQS)
amy.shen@oist.jp
† Electronic Supplementary Information (ESI) available under DOI:
10.1039/cXsm00000x/
class of ion-containing polymers. Historically, polymers at rel-
atively low ion-contents (10–15%) were defined as ionomers,
while polymers containing very high ion-contents were defined as
polyelectrolytes.5A better definition was proposed based on the
physical properties of ion-containing polymers:6Polyelectrolytes
are polymers in which solution properties are governed by elec-
trostatic repulsion between dissociated groups along the chain,
while ionomers are polymers in which the bulk properties are gov-
erned by ionic interactions (i.e., dipole interactions between ion
pairs) in discrete regions of the polymer material where attrac-
tion dominates. In fact, some ion-containing polymers exhibited
a transition from ionomer-like to polyelectrolyte-like behaviours,
depending on, for example, the temperature. 7Polyelectrolytes
can be further divided into two groups depending on the nature of
ionic groups.8Weak polyelectrolytes are conventionally defined
as polymers with weak acidic or basic groups, in contrast to strong
polyelectrolytes, which are composed of polymers with definitive
strong acid or base groups9–12 Note that this differs from strongly
charged and weakly charged polyelectrolytes, which refer to sys-
tems with high and low density of ionic monomers along the back-
bone respectively. However, there is no universally agreed-upon
threshold that defines ’high’ versus ’low’ charge density. Vinylic
polyelectrolytes, which carry approximately one charge every 0.2-
0.3 nm (e.g., PSS), and polysaccharides that have one charge per
monomer unit, equating to roughly one charge every 0.5 nm, are
considered as strongly charged.
Regardless of the type of polyelectrolytes, their conformation
in solution is significantly influenced by the electrostatic interac-
tions. The modelling of charged polymers is rather complicated
Journal Name, [year], [vol.],
1–54 | 1
since ions on the backbone chain are covalently bonded. For ex-
ample, it is widely accepted that some counterions stay bound
(condensed) in the vicinity of the chain backbone due to strong
electrostatic attractions between polyions and counterions. This
phenomena is called the counterion condensation.13 As a result,
the effective charge fraction on a polyelectrolyte chain in solu-
tion is not always equal to the charge density (i.e., the number
of ionic monomers) of the chain. Many theoretical models with
different approaches have been reported in literature, and their
predictions could explain some properties of polyelectrolyte solu-
tions.9,10,14,15 However, many conflicting results between experi-
mental data and theoretical predictions have been reported, even
for linear polyelectrolyte chains in solutions.
The number of factors and lengthscales which control the so-
lution properties14,16,17 is larger for charged polymers than for
neutral polymers, making polyelectrolyte theory lag behind that
of neutral polymers. These highly complex systems are difficult
to study experimentally. For example, if we study the rheology of
a neutral polymer in solution, measurements varying the polymer
concentration, molar mass, solvent quality and temperature will
usually suffice to understand the given system. However, for a
polyelectrolyte, additional physical quantities, such as the charge
density, the counterion size and its valence, added salt concentra-
tion, and dielectric constant of the solvent, all become relevant
experimental variables. Moreover, it is not possible to define a
single solvent quality, and instead the chemical structure of the
polyelectrolyte backbone, side-chain ions and counterions have
to be considered separately. In concentrated solutions, the poly-
electrolyte systems exhibited fascinating dynamics18–29 which we
are only beginning to understand. Their ability to complex with
oppositely charged matter made them relevant to many indus-
trial formulations and biological systems, and is one of the main
aspects of polyelectrolyte research today. 30–34
It is also important to highlight an emerging category of poly-
electrolytes known as poly(ionic liquid)s or polymerized ionic liq-
uids. Within this context, poly(ionic liquid)s (PILs) denote poly-
mers in which ionic liquid structures are covalently integrated
into the repeating units.35,36 Here, ionic liquids (ILs) are molten
salts consisting of cations or anions which melt below 100◦C.37
Fig. 1 displays representative chemical structures of a PIL poly-
cation and counteranions: PIL ions are relatively large, asym-
metric, and charge delocalized, making the physical properties of
PILs different from those of ordinary ion-containing polymers.38
For example, PILs exhibited glass transition at relatively low tem-
peratures even if the charge density is high.39 This unique glass
transition behaviour of PILs has led to active research in manip-
ulating the bulk properties of PILs, e.g., their ionic conductivity
and viscoelasticity, which have been extensively investigated over
the past decade.40–53 The research community acquired a basic
understanding of the behaviour of PILs, which act as ionomers.
However, the solution properties of PILs, i.e., the behaviour of
PILs as polyelectrolytes, have been scarcely investigated.
In solution, PILs can release their counterions into the sol-
vent and possess charges on their chain backbone, showing sim-
ilar properties to those for ordinary polyelectrolytes, such as
poly(sodium styrenesulfonate).54 In contrast to ordinary poly-
Fig. 1 Representative chemical structures of a PIL poly-
caion and four different types of counteranions. Polycation:
PC+
4; poly(1-butyl-3-vinylimidazolium), Counteranions: TFSI−;
bis(trifluoromethanesulfonyl)imide, TfO−; trifluoromethanesulfonate,
PF−
6; hexafluorophosphate, BF−
4; tetrafluoroborate.
electrolytes, PILs can be dissolved in solvents with a wide range
of dielectric constants even if the charge density on PIL chains
is high.55 A good solubility of PILs can also be obtained in pure
ILs.55 The unique features of PILs in solutions have raised many
interesting questions in the polyelectrolyte research community.
For example, how do PILs behave in solvents with low dielec-
tric constants?56 According to the Manning model for the coun-
terion condenstation,13 the number of dissociated counterions is
predicted to decrease with decreasing solvent dielectric constant.
If so, PILs would behave as neutral polymers or as ionomers in
low dielectric solvents. Investigating the charge screening effects
exerted by ionic liquid ions on polymerized ionic liquid chains
within pure ionic liquids offers an interesting research avenue, es-
pecially considering the distinctive solvent characteristics of ionic
liquids.57
Our primary aims are 1) to establish scaling laws that eluci-
date the relationship between the properties of polyelectrolytes
and variables such as molar mass, salt concentration, and charge
fraction; 2) to test the theoretical models reviewed in Section 2;
and 3) to highlight experimental results which expand or chal-
lenge the current framework of polyelectrolyte physics. We pro-
vide a critical overview of several key questions in polyelec-
trolyte physics, with emphasis on the experimental literature
where new conclusions can be drawn by compiling and/or re-
analysing data. We hope this perspective will complement ear-
lier reviews in the field which have focused on developments
of theory, 9,15,58–62 simulations,9,63,64 and particular experimen-
tal methods, systems, or properties 65–81 This article narrows its
scope to exploring the intricacies of dilute polyelectrolyte solu-
tions.82 For experimental reviews dealing with non-dilute solu-
tions and gels, we direct readers to refs. [14,68,83–85].
This review article is structured as follows: Section 2 introduces
several theoretical models proposed in literature to represent the
conformation of dilute polyelectrolyte solutions; Section 3 dis-
cusses the properties of polyelectrolytes in the dilute regime, di-
viding our discussion into sixteen subsections along with research
2 | 1–54
Journal Name, [year], [vol.],
questions related to the conformation and dynamics of single
chains in solution; Section 4 concludes our review and highlights
open questions.
2 Theoretical Approaches to the Conformation and
Dynamics of Polyelectrolytes in Solution
Dilute polymer solutions are defined as those for which chains
do not overlap.14,86 Considering a chain made up of Nchemical
monomers with an end-to-end distance R, the overlap concentra-
tion (c∗) in units of number of repeating units per volume is:
c∗=N
R3.(1)
In this article, we will use the symbol cto refer to the concentra-
tion in number of monomers per unit volume. When plotting data
or quoting concentration values, it is more convenient to use con-
centrations in moles of monomers per volume. This is indicated
by the unit M, which denotes moles of monomers per dm3. Intrin-
sic viscosities are in units of M−1as opposed to the more common
units of dL/g. We also use Ninstead of molar masses. The fol-
lowing example illustrates the reason for these choices: suppose
we compare the properties of polystyrene sulfonate (PSS) with
Na+and Cs+counterions. Let us assume that the conformation
of the chains is not influenced by the choice of counterion. A plot
of c∗or [η] vs. Nwill overlap for both PSS salts when expressed
in units of M and M−1respectively, thus capturing the essential
physics that the chain conformation is unchanged. By contrast
a plot of c∗in g/L vs. Mwwill result in two separate curves for
NaPSS and CsPSS because the Cs+ion has a larger mass than the
Na+one. A few exceptions, such as Fig. 6, use the concentration
in mass per volume (cp) as part of a dimensionless product.
2.1 Conformation in dilute salt-free solutions
2.1.1 Scaling approach
The scaling approach to conformation and dynamics was primar-
ily developed by de Gennes et al. and Pfeuty and Dobrynin
et al.87–90 Consider a polyelectrolyte chain with a bare (non-
electrostatic) Kuhn length lK,0. Each Kuhn segment is made up of
gKchemical monomers with length b(lK,0=gKb). For chains with
characteristic lengthscales smaller than the Kuhn length, they are
rigid, and their end-to-end distance Rscales as
R≃bN for N≤gK.(2)
For chains with end-to-end distances larger than a Kuhn segment,
they adopt random walk statistics up to the thermal blob size ξT.
Each thermal blob contains gTmonomers, therefore, the end-to-
end distance of chains (between the lK,0and ξT) is given by:
R≃lK,0N
gK1/2
for gK≤N≤gT.(3)
If the end-to-end distance is larger than ξT, the polymer confor-
mation depends on the solvent quality exponent ν:
R≃ξTN
gTν
for gT≤N≤gel,(4)
where ν=1
3for poor solvents, ν=1
2for theta solvents, and ν=3
5
for good solvents. Here, ξTis given by Eq. 3 with N=gT.
The next relevant lengthscale is the size of the electrostatic
blob. This marks the distance at which the electrostatic energy
is of the order of the thermal energy kBT, where kBis the Boltz-
mann constant and Tthe absolute temperature. The Coulomb
energy of an electrostatic blob is
Uel ≃(gel f)2e2/(ε0εrξel),(5)
where fis fraction of monomers bearing a dissociated charge, i.e.,
the charge fraction, and gel is the number of chemical monomers
inside an electrostatic blob. The end-to-end distance of the elec-
trostatic blob is:
ξel =
l4/7
K,0b3/71
u f 21/3
for T≪θ, (6a)
l4/7
K,0b3/71
u f 21/3
for T=θ, (6b)
l6/7
K,0ξ−2/7
Tb3/71
u f 23/7
for T>θ, (6c)
where u=lB/b, with lBthe Bjerrum length. θis the theta temper-
ature. Eq. 6 was derived by using the equations of ξel =ξTgel
gTν
and ξT=plK,0bgT90 under the assumption of Uel =kBTfor T≥θ
and Uel =γξel for T≪θ, where γis the polymer/solvent interfa-
cial tension, given by γ≈τ2kBT
ξ2
T
with τ≡θ−T
θbeing the reduced
temperature.
On distances larger than the electrostatic blob size, the chain is
stretched, and its conformation is a pole of electrostatic blobs:
R≃ξel
N
gel
for N≥gel,(7)
where ξel is given by Eq. 6. The schematic illustration of a dilute
polyelectrolyte chain is shown in Fig. 2. In Dobyrnin et al.’s 1995
Fig. 2 Dilute polyelectrolyte chain adopts an extended configuration of
electrostatic blobs with length ξel inside which the conformation is a self-
avoiding walk of thermal blobs with length ξT. A random walk of Kuhn
segments with length lK,0forms the thermal blob.
model,89 the shortest lengthscale considered was the monomer
size, which was equated with the thermal blob size and the Kuhn
length, corresponding to ξT=lK,0=bin the above equations. The
updated scaling model of Dobrynin and Jacobs90,91 worked out
the solution properties of polyelectrolytes for arbitrary values of
ξTand ξel.
Journal Name, [year], [