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\DOI{10.5802/crchim.326}
\datereceived{2024-01-31}
\daterevised{2024-06-07}
\dateaccepted{2024-07-05}
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\dateposted{2025-01-23}
\begin{document}

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%\makeatletter
%\def\TITREspecial{\relax}
%\def\cdr@specialtitle@english{French Network on Solvation (GDR 2035 SolvATE)}
%\def\cdr@specialtitle@french{R\'eseau th\'ematique sur la solvatation (GDR 2035 SolvATE)}
%\makeatother

\title{Ion-specific effects in polyelectrolyte solutions: chain--chain
interactions, chain rigidity and dynamics}

\alttitle{Effets specifiques des ions et solutions de polyelectrolytes : 
structure et dynamique}

\author{\firstname{Claire} \lastname{Hotton}\CDRorcid{0000-0002-8088-4170}}
\address{Laboratory of Physical Chemistry of Electrolytes  and
Interfacial Nanosystems (PHENIX), Sorbonne Universit\'e, CNRS, 75005
Paris, France}
\email[C. Hotton]{claire.hotton@universite-paris-saclay.fr}

\author{\firstname{Yasine} \lastname{Sakhawoth}}
\addressSameAs{1}{Laboratory of Physical Chemistry of Electrolytes  and
Interfacial Nanosystems (PHENIX), Sorbonne Universit\'e, CNRS, 75005
Paris, France}
\email[Y. Sakhawoth]{yasine.sakhawoth@gmail.com}

\author{\firstname{Anne-Laure} \lastname{Rollet}\CDRorcid{0000-0001-6150-768X}}
\addressSameAs{1}{Laboratory of Physical Chemistry of Electrolytes  and
Interfacial Nanosystems (PHENIX), Sorbonne Universit\'e, CNRS, 75005
Paris, France}
\email[A.-L. Rollet]{anne-laure.rollet@sorbonne-universite.fr}

\author{\firstname{Juliette} \lastname{Sirieix-Pl\'enet}\CDRorcid{0000-0002-9873-8534}}
\addressSameAs{1}{Laboratory of Physical Chemistry of Electrolytes  and
Interfacial Nanosystems (PHENIX), Sorbonne Universit\'e, CNRS, 75005
Paris, France}
\email[J. Sirieix-Pl\'enet]{juliette.sirieix-plenet@sorbonne-universite.fr}

\author{\firstname{Lingsam} \lastname{Tea}\CDRorcid{0000-0003-2732-9028}}
\addressSameAs{1}{Laboratory of Physical Chemistry of Electrolytes  and
Interfacial Nanosystems (PHENIX), Sorbonne Universit\'e, CNRS, 75005
Paris, France}
\email[L. Tea]{lingsam.tea@sorbonne-universite.fr}

\author{\firstname{Sophie} \lastname{Combet}\CDRorcid{0000-0002-8672-4514}}
\address{Laboratoire L\'eon-Brillouin (LLB), UMR12 CEA-CNRS,
Universit\'e Paris-Saclay, F-91191 Gif-sur-Yvette CEDEX, France}
\email[S. Combet]{sophie.combet@cea.fr}

\author{\firstname{Melissa} \lastname{Sharp}\CDRorcid{0000-0002-4629-2868}}
\address{European  Spallation  Source  (ESS)  AB,  Box  176, S-22100
Lund, Sweden}
\email[M. Sharp]{melissa.sharp@esss.se}

\author{\firstname{Ingo} \lastname{Hoffmann}\CDRorcid{0000-0001-7178-6467}}
\address{Institut Laue Langevin (ILL), Grenoble F-38042, France}
\email[I. Hoffmann]{hoffmann@ill.fr}

\author{\firstname{Fr\'ed\'eric} \lastname{~Nallet}}
\address{Centre de Recherche Paul-Pascal, UMR Universit\'e de
Bordeaux--CNRS 5031, 115 Avenue du Dr Albert Schweitzer, 33600 Pessac,
France}
\email[F. Nallet]{frederic.nallet@u-bordeaux.fr}

\author{\firstname{Natalie} \lastname{Malikova}\CDRorcid{0000-0002-3692-1958}\IsCorresp}
\addressSameAs{1}{Laboratory of Physical Chemistry of Electrolytes  and
Interfacial Nanosystems (PHENIX), Sorbonne Universit\'e, CNRS, 75005
Paris, France}
\email[N. Malikova]{natalie.malikova@sorbonne-universite.fr}

\shortrunauthors

\keywords{\kwd{Polyelectrolytes}
\kwd{Ion-specific effects}
\kwd{Inter-chain interactions}
\kwd{Self-diffusion}
\kwd{Collective diffusion}
\kwd{PFG-NMR}
\kwd{Neutron spin echo}}

\altkeywords{\kwd{Poly\'electrolytes}
\kwd{Effets ioniques sp\'ecifiques}
\kwd{Interactions entre cha\^{\i}nes}
\kwd{Diffusion individuelle}
\kwd{Diffusion collective}
\kwd{R\'esonance magn\'etique nucl\'eaire \`a \`a gradient de champ modul\'e}
\kwd{Diffusion des neutrons \`a \'echo de spin}}

\begin{abstract}
In this article, ion-specific effects in aqueous solutions of polyelectrolytes (PEs) are
addressed. We focus on ionene cationic chains, featuring simple
structure, absence of side groups, and very regular chain charge
density. Ion-specific effects in ionene solutions are demonstrated
using a series of monovalent (halide) counterions. The study combines
both static and dynamic measurements by small angle neutron scattering,
neutron spin echo (NSE), and pulsed field gradient NMR (PFG-NMR). 
Ion-specific effects are a phenomenon at
high PE concentration, and the nature of the counterion is
seen to influence drastically  ionene chain--chain interactions 
{via what we refer to as ``ion-specific screening''.} The origin lies in
the closer approach of large, highly polarisable counterions to the
chain backbone, leading to more constricted counterion clouds. Equally
affected is the local chain rigidity as well as collective and
self-diffusion coefficients at larger scales.  Small, nonpolarisable,
strongly hydrating counterions, here F\tralicstex{\textsuperscript{-}}{$^-$} ions, lead to locally rigid
chains. For such chains, the nm-scale collective dynamics as seen by
NSE is the fastest while self-diffusion seen at the
\tralicstex{\textmu}{$\upmu$}m scale by PFG-NMR is the slowest. In other words, the loss of
charge on the chain due to ion-specific screening has the opposite
effect on collective diffusion and self-diffusion of the chains.
\end{abstract}

\begin{altabstract}
Les effets sp\'ecifiques des ions dans les solutions aqueuses de
poly\'electrolytes sont abord\'es ici.  Nous nous concentrons sur les
cha\^{\i}nes cationiques de type ion\`ene, qui se caract\'erisent par
une structure simple, l'absence de groupes lat\'eraux et une densit\'e
de charge de cha\^{\i}ne tr\`es r\'eguli\`ere. Les effets sp\'ecifiques
aux ions dans les solutions ioniques sont mis en \'evidence via une
s\'erie de contre-ions monovalents (des halog\'enures). L'\'etude
combine des mesures statiques et dynamiques par diffusion de neutrons
aux petits angle, par \'echo de spin, ainsi que par r\'esonance
magn\'etique nucl\'eaire \`a gradient de champ modul\'e (PFG-NMR). Les
effets sp\'ecifiques aux ions se manifestent \`a haute concentration de
poly\'electrolyte et la nature du contre-ion influence sensiblement les
interactions cha\^{\i}ne--cha\^{\i}ne, par le biais de ce que nous
appelons un ``effet d'\'ecran sp\'ecifique aux ions". L'origine de cet
effet d'\'ecran r\'eside dans le rapprochement des contre-ions
volumineux, hautement polarisables, du squelette de la cha\^{\i}ne, ce
qui conduit \`a des nuages de contre-ions plus compacts. La rigidit\'e
locale de la cha\^{\i}ne, ainsi que,  \`a plus grande \'echelle, les
coefficients de diffusion collective ou individuelle sont \'egalement
affect\'es. Les petits contre-ions non polarisables et fortement
hydratants, ici les ions F\tralicstex{\textsuperscript{-}}{$^-$}, conduisent \`a des cha\^{\i}nes
localement rigides. Pour de telles cha\^{\i}nes, la dynamique
collective \`a l'\'echelle du nanom\`etre, telle qu'observ\'ee par
\'echo de spin en diffusion dynamique des neutrons, est la plus rapide,
tandis que la diffusion individuelle observ\'ee \`a l'\'echelle du
microm\`etre par PFG-RMN est plus lente. En d'autres termes, la
neutralisation partielle des charges port\'ee par la cha\^{\i}ne due
\`a l'effet d'\'ecran sp\'ecifique \`a l'ion a un effet oppos\'e sur la
diffusion collective et la diffusion des cha\^{\i}nes individuelles.
\end{altabstract}

\maketitle

\vspace*{.5pc}

\twocolumngrid

\end{noXML}

%\let\rmmu\upmu

\section{Introduction}\label{sec1}

Charged polymer chains, also referred to as ``polyelectrolytes'' (PEs), are
omnipresent among natural compounds (e.g., polysaccharides, 
nucleic acids) or indeed among the many synthetic substances used in food,
cosmetic, and packaging industries as well as in water treatment and
several other fields~\cite{DautzBOOK}. The behaviour of charged polymer
chains in solution, and we shall restrict ourselves here to aqueous
solutions, is a world of its own. Two possible reference situations can
be thought of as systems of departure: (a) electrolyte solutions and
(b) uncharged polymer chains in solution. In the first case, we depart
from a solution of atomic or molecular ions, both positive and
negative, and we add connectivity between one type of these ions, be it
positive or negative. This creates significant charge density
inhomogeneities in the solution. This is by now not the common way of
thinking about PE solutions, but it was indeed the point
of departure for Fuoss et~al.~back in the 1950s~\cite{Edelson_Fuoss50}.
In the second case, we start with the already quite complex case of a
macromolecular chain in solution, where the quality of the solvent
(good/theta/bad) decides the chain conformation. 
Next we add charge to
a fraction of the monomers on the chain, but importantly we also need
to introduce a population of counterions into the surrounding solution
(and partially condensed onto the chain if the necessary conditions
are met). Either way, the transition (adding connectivity or adding
charge) is far from trivial and the consequences in terms of the
structure, interactions, and dynamics of the (macro)molecular species
present in such a solution are considerable. This is naturally
reflected in the very different properties of the solution at the
macroscopic scale, its phase diagram, rheology, osmotic pressure, heat
of dilution, and other thermodynamic properties~\cite{DautzBOOK}.

Historically, it is indeed through the macroscopic properties that the
peculiar behaviour of PE solutions was uncovered, and the rheological
studies of Fuoss et~al.~(the Fuoss law) remain a reference in this
respect (e.g., \cite{Fuoss51}). Tremendous progress has been made since
the Fuoss studies and we recommend two reviews, dating from very
different times, to trace this 
progress~\cite{Eisenberg77,Muthukumar17}.  A breakthrough came in the
1970s with the advent of scattering techniques, in particular neutron
scattering, the seminal experimental work of Cotton, Jannink 
et~al.~\cite{Cotton72,Daoud75}, and the accompanying theoretical
developments by de Gennes, Pfeuty et~al.~(the scaling
approach)~\cite{deGennes76,Pfeuty77,Pfeuty78}. From this point onwards,
the molecular level description of solutions of neutral polymers and
later on PE solutions indeed began to emerge. 

{\advance\baselineskip by .08pt

Overall, the scaling-based theory of PE solutions is highly successful,
especially when dealing with the interpretation of neutron and X-ray
scattering experiments. There, it predicts correctly the scaling of the
PE chain correlations probed via the universally observed maximum in
the scattering data, the so-called \emph{polyelectrolyte peak}. This
feature of the scattering curves is completely absent for solutions of
neutral polymers and reflects the electrostatic repulsion present in
solutions of \emph{charged} chains. De Gennes' theory of
PE solutions was later broadened by Dobrynin and
Rubinstein to include all cases of solvent quality and concentration
regimes~\cite{Dobrynin95}. The initial picture of dilute and
semidilute concentration regimes was enriched (entangled and
nonentangled subregimes) and extended, especially in the high
concentration range~\cite{Muthukumar96,Nishida01,Lorchat14}. The
competition between electrostatic repulsion along a PE chain and
hydrophobic (solvophobic) collapse has led to the prediction of the
so-called \emph{pearl-necklace} conformation of PE chains for the case
of bad solvent~\cite{Dobrynin96}, and these have indeed been observed
experimentally~\cite{Spiteri07,Essafi09} and by
simulations~\cite{Holm03}.

Both the conformation of individual chains (the chain form factor) and
the chain--chain correlations (the structure factor) modify the
scattering curves. These individual contributions often cannot be
distinguished easily, contrary to solutions of inorganic colloids for
example. Polymer/PE solutions pose two problems in this respect: (1) the
conformation of polymer/PE chains changes as a function of
concentration and (2) polymer chains are penetrable objects, that is,
individual chains can get entangled. A very elegant method, the
zero average contrast (ZAC) method, is accessible in neutron scattering
to highlight each of the contributions for cases where deuteration of
the polymer/PE chains is possible~\cite{Boue94}. In the absence of such
measurements, the main feature to account for in the scattering
spectra remains the position of the scattering peak.  As follows from
the above paragraphs, the position of the PE peak depends not only on the PE
concentration but also on the chain conformation (consequence of the
effective chain charge and the solvent quality). 

}

In our past publications, we have shown that for identical PE
concentration, solvent quality, and valence of the counterions, the
\emph{chemical nature} of the counterion by itself can also influence
the shape of the scattering curve, the position, and shape of PE peak
and the extent to which this peak is indeed visible or
not~\cite{Malikova12,Malikova15}.  {The origin lies in the hydration
of the given ion, which leads to a different degree of screening of the
chain charge.} When the chemical nature of an ion is to ``blame'' for a
given observation, biochemists and physical chemists refer to such
instances as \emph{ion-specific
effects}~\cite{Salis14,kunzBOOK09,Marcus09,Jungwirth06,Collins97,Xie13}. 
It has been observed that ion-specific effects manifest themselves more
strongly for anions than cations~\cite{Morita14,Schwierz13}. Solutions
and gels based on cationic chains with compensating \emph{anions}, such
as ionenes~\cite{Malikova12,Malikova15,Hotton23}, show indeed stronger
ion-specific effects than anionic PEs, such as the widely studied
polystyrene
sulfonate~\cite{Waigh01,Combet05,Spiteri07,Essafi11,Combet12}. 
Needless to say, any purely electrostatic theory, such as the
scaling approach of de Gennes, the Manning theory of counterion
condensation~\cite{Manning69a,Manning69b}, or indeed the
Poisson--Boltzmann approach~\cite{Combet05} cannot account for these
effects as hydration properties of solvated ions do not come into
consideration. Attempts in what seems the correct direction are 
theories accounting for local dielectric heterogeneities around the
ions and the PE chains~\cite{Muthukumar04}. The orientation of the
dipole moment of water molecules is indeed closely linked to the
hydration and the polarisability of the hydrated species. 

Ionenes, the focus of this study, are a group of water soluble cationic
PEs with pH independent charge, based on quaternary
ammonium charged centers linked by simple hydrocarbon chains. Ionenes
have already several applications including ion exchange
resins~\cite{Raskop07}, water treatment in the oil industry~\cite{Lucas09},
humidity sensors~\cite{Erdmenger10}, organic templates in the synthesis
of mesoporous silica~\cite{Berezovska06}, and anti-microbial
agents~\cite{Kiss12}.  Within the realm of PEs, ionenes
present the advantage of a regular and tunable separation of charges
on the backbone, as opposed to \mbox{statistically} distributed charges for
other PEs. In our initial scattering studies on ionene aqueous
solutions, we explored the transition from hydrophilic to hydrophobic
polyelectolyte behaviour as the ionene charge density decreases. This
transition was indeed found, however later than expected: the
hydrophobicity of the hydrocarbon backbone of ionenes becomes
``visible'' when only 15\%  of the monomers are charged, not
before~\cite{Malikova15}. Dramatic \emph{ion-specific effects} in
ionene aqueous solutions have been initially observed in thermodynamic
properties~\cite{Luksic12,Serucnik12,Cebasek13} and later on the
microscale by scattering for the particular case of two halide ions,
F$^{-}$ and Br$^{-}$~\cite{Malikova12,Malikova15}. In this
contribution, we show how this generalises for an entire series of
halide counterions and what consequences it has for the dynamics of the
PE chains. We bring information on the chain dynamics at the
microscopic (nm) scale, by the neutron spin echo (NSE) technique, and
also on the mesoscopic ($\upmu$m) scale, by pulsed field gradient NMR
(PFG-NMR). 

\section{Experimental methods}

\subsection{Ionenes: synthesis and structural overview}

Ionenes and their precursors were synthesised using a procedure adapted
from those described previously~\cite{Malikova12,Cebasek13,Malikova15}.
The details of the synthesis are provided in the SI file (part 1). The
synthetic route leads invariably to ionenes with bromide counterions.
The molecular weights of ionenes were determined by size exclusion
chromatography (SEC) as described in~\cite{Sakhawoth17}. The range
of molecular weights is 20,000--60,000~g/mol, which corresponds to
\mbox{100--300}~nm in terms of chain length. SEC measurements on cationic
PEs are very difficult~\cite{Laymann08} and, for us, were
successful only for 6,9-ionenes. In the following, we consider that the
above range of molecular weights applies also to ionenes of other
charge densities, which were synthesised under identical conditions. We
have indeed confirmation that the molecular weights of ionenes with
different charge densities are of the same order of magnitude from the
NMR signal of amine end groups, which allows estimation of the degree of
ionene polymerisation~\cite{Williams08}.

Counterion exchange was performed by dialysis starting from Br-ionenes.
Dialysis tubes (Sigma-Aldrich, MWCO\ ${=}$\ 12,000~g${\cdot}$mol$^{-1}$)
were filled with 0.02~M solutions of Br-ionenes and first dialysed
against 0.05~M solution of the desired NaX (3 weeks) to exchange anions
and then dialysed against water (2 weeks) to remove sodium ions. All
ionene solutions for neutron scattering and NMR measurements were
prepared gravimetrically.  Deuterated water (Eurisotop, 99.9\%D) was
used for neutron scattering samples as well as NMR samples. The pH of
the solutions was close to neutral, and thus we estimate the effects of
any dissolved carbonic acid as very small.

The general chemical formula of ionenes is
\mbox{[--(CH$_3$)$_2$N$^+$--(CH$_2$)$_x$--(CH$_3$)$_2$N$^+$--(CH$_2$)$_y$--]$_n$}
for an $x,y$-ionene chain with Br$^-$ or other counterions 
(Figure~\ref{ionene-param}). Ionenes with X$^-$ counterions are referred to as
X-ionenes in the rest of the manuscript. Values $x$ and $y$ represent
the number of --CH$_2$-- (methylene) units between adjacent charged
centers (quaternary ammonium centers) and can be varied accurately by
synthesis~\cite{Rembaum72,Noguchi72,Williams09}. By increasing $x$ and
$y$, the ionene chain is less charged. In this manuscript, we 
discuss ionene chains for which $x$, $y=\text{3,3}$ and 6,9 (referred to as
3,3-ionenes and 6,9-ionenes). The simple \mbox{structure} (absence of bulky
side groups) and finely tunable and \emph{regular} charge density are
very interesting structural features of ionenes. Other PEs, including
styrene, often present charge on (bulky) side groups and charged
monomers are distributed statistically along the chains. In order to
draw a parallel between ionenes on one side and polystyrene-based and
other PEs on the other side, we may consider the structure of ionenes as a
sequence of charged and uncharged ``monomers'' as depicted in 
Figure~\ref{ionene-param}. 

\begin{figure}
\includegraphics{fig01}
\caption{\label{ionene-param}\emph{Left}: (a) Schematic view of an
$x,y$-ionene chain. (b) Schematic view of a 3,3-ionene ($x=3$, $y=3$)
chain, showing the definition of a charged and an uncharged monomer.
\emph{Right}: Ionene structural parameters: $a$ is the charge
separation on the chain, $f_{\mathrm{chem}}$ the fraction of charged
monomers, and $\xi$ the Manning charge density parameter, defined as
$\xi = L_{\mathrm{B}}/a$, where $L_{\mathrm{B}}$ is the Bjerrum length
(7.14~\AA~in water at room temperature). While 4,5-ionenes are at the
$\xi=1$ limit (onset of Manning-type condensation), only 3,3-ionenes
have sufficient charge density to induce significant condensation
\unskip\break($\xi>1$) and decrease the chemical charge
($f_{\mathrm{chem}}$) to an effective charge ($f_{\mathrm{eff}}$).}
\end{figure}

\subsection{Neutron scattering}

Small angle neutron scattering (SANS) measurements were carried out on
the PACE spectrometer at LLB-Orph\'ee, Saclay, France. Using up to
three different combinations of incident neutron wavelength ($\lambda$)
and  sample to detector distance, a wavevector ($q$) range of 
0.01--0.45~\AA$^{-1}$ was covered ($q=4\uppi\sin(\theta/2)/\lambda$).  The
detector efficiency was taken into account by normalisation of data
with a flat (incoherent) signal from bulk light water. Ionene solutions
 {(hydrogenated chains in D$_2$O solvent)} were loaded into  quartz
cells with a path length of 1 or 2~mm. Due to the isotropic nature of our
samples, data were grouped in concentric rings, each corresponding to a
given $q$ value. The measured scattered intensities  were corrected for
transmission, sample thickness, and incoherent and solvent background to 
yield the coherent scattered intensity $I_{\mathrm{coh}}$. We checked
the reproducibility of neutron scattering spectra by measuring samples
from different synthesis batches. 

\begin{figure*}
\includegraphics{fig02}
\vspace*{-5pt}
\caption{\label{SANS}Coherent neutron scattering intensity normalised by
ionene monomer concentration ($I_{\mathrm{coh}}$/$c_{p}$) in arbitrary
units versus scattering wavevector ($q$) for room temperature aqueous
solutions (in D$_2$O) of 3,3 X-ionenes at two monomer concentrations,
0.4~M (left) and 2~M (right). These monomer concentrations correspond to
volume fractions around 2\% and 10\%, respectively.}
\vspace*{-5pt}
\end{figure*}

Neutron Spin Echo (NSE)~\cite{Mezei_notes, ActuChim_MM22} experiments
were carried out on the IN15 spectrometer in ILL, Grenoble, France.
Samples  {(hydrogenated chains in D$_2$O solvent)} were measured in
1 mm or 2 mm flat quartz cells. Using a combination of two neutron
wavelengths, 6~\AA~and 9~\AA, and detector angles of 
3\textdegree, 5\textdegree, 7\textdegree, 10\textdegree, 13\textdegree,
19\textdegree~at 6~\AA~and 3\textdegree, 6\textdegree\ at 9~\AA, we
achieved an accessible $q$ range of 0.04~\AA$^{-1}$ to 0.5~\AA$^{-1}$
and time range of 0.07--11~ns at 6~\AA~and 0.25--36~ns at 
9~\AA. Both static  ($I_{\mathrm{coh}}$) and dynamic data ($I(q,t)$) in
NSE were corrected for contributions from the quartz cell and the
solvent (D$_2$O) background.  

\subsection{Pulsed field gradient NMR}
The single pulse $^{1}$H NMR spectra were recorded using a Bruker
Avance III 300~MHz NB spectrometer operating at 7.05~T. 
The lock was
obtained with a sealed 2~mm capillary filled with {D$_2$O} inserted
inside the NMR tube. The chemical shift was referenced to
CHCl$_3$.  The PFG\_NMR experiments were
performed using a BBFO probe equipped with a 55~G${\cdot}$cm$^{-1}$ gradient
coil. We used an NMR pulse sequence combining bipolar gradient pulses
and stimulated echo. This sequence was repeated with 16 gradients of
increasing strength from 2 to 50~G${\cdot}$cm$^{-1}$ for a duration of 1.5~ms. The
diffusion time was approximately 200~ms, which corresponds to a diffusion
over a length scale of 1.5~$\upmu$m.  The self-diffusion coefficients are
obtained by nonlinear least-square fitting of the echo attenuation,
using the Bruker TopSpin software. All PFG-NMR data were measured at
room temperature on ionene solutions in {D$_2$O}.  


\section{Results and discussion}

SANS data of 3,3 X-ionenes with four
different halide counterions 
($\mathrm{X}^{-} = \mathrm{F}^{-}$, Cl$^{-}$, Br$^{-}$, I$^{-}$), at a
moderate and a high monomer concentration, are shown in Figure
\ref{SANS}. From our previous scattering studies on ionenes, we know
that both of these concentrations are in the semidilute regime. The
overlap concentration for ionenes synthesised using our protocols was
estimated to be below $c_p=0.07$~M~\cite{Malikova12}.  {In addition,
viscosity data (see SI part~2) confirms that both F-ionenes and
Br-ionenes are in the same concentration regime, that is, semidiluted.} In
the semidiluted regime, the position of the peak reflects the
mesh size formed by the interpenetrating chains. At 0.4~M monomer
concentration, the spectra of all systems show a well-defined
PE peak (a clear maximum) at an almost identical position
in the wavevector $q$ (a slight shift towards higher $q$ values is
noticed for the F-ionene; see later). 
At high monomer concentration
(2~M), the four systems feature very different scattering curves. Note
that the increase in intensity in the small $q$ region ($q <
0.03$~\AA$^{-1}$) is due to large-scale heterogeneities in the 
system---a repeatedly observed feature for PE solutions/gels, which remains
poorly understood. The changes that interest us most are thus confined
to the $q$ region roughly between 0.03~\AA$^{-1}$ and 0.4~\AA$^{-1}$.
In this central region, a clear PE peak remains visible
only for the F-ionene. Its position is shifted to higher $q$ values in
comparison to data at 0.4~M, as expected, due to a denser mesh size at
this higher concentration. In the sequence F$^-$\ $\rightarrow $ Cl$^-$\ 
$\rightarrow $ Br$^-$\ $\rightarrow $ I$^-$, the central part of the
spectrum gains in intensity and the intensity dip to the left of the PE
peak seen for F-ionene {gradually}\unskip\break disappears.
As a consequence, the peak
becomes highly asymmetric (in the case of Cl-ionene, still a slight
maximum is observed). The peak disappears completely for the Br-ionenes
and I-ionenes and instead a plateau is seen in the spectra, resembling
a signal we would expect from a neutral polymer.  

Figure~\ref{PE_peak} summarises the position of the PE peak for the
four 3,3 X-ionenes seen in their SANS spectra. Up to 0.1~M
concentration, the position of the peak is very close for all
systems; for $c_p > 0.1$~M, the situation changes. Only the F-ionene
follows the predicted $c_p^{1/2}$ law and the three other systems
depart significantly from this description. Combining information from
Figures~\ref{SANS} and~\ref{PE_peak}, we observe that at 0.4~M, the PE peak
position for 3,3 F-ionene is already somewhat higher than for all
other systems; all curves present a well-defined PE peak. For higher
concentrations, the \emph{strong} departure from the $c_p^{1/2}$ law in
Figure~\ref{PE_peak} for Cl-, Br-, and I-ionenes is mainly a
consequence of the PE peak \emph{disappearing} from the scattering
signal. For a poorly defined asymmetric peak, the determination of its
position is increasingly difficult (see the right side of Figure~\ref{SANS}). 
Overall, clearly the ion-specific effect is a high
concentration phenomenon and we can place the critical concentration at
around 0.1~M. This corresponds to a charge concentration in the system
of 0.05~M according to $c(X^-)=c(N^+)=c_p f_{\mathrm{chem}}$. 

\begin{figure}
\includegraphics{fig03}
\vspace*{-5pt}
\caption{\label{PE_peak}Position of the polyelectrolyte peak in SANS spectra ($q^*$)
versus ionene  monomer concentration ($c_p$) for 3,3-ionenes with
different counterions. Typical error bars are represented  on the 3,3-F
data set. Dashed line is a guide to the eye representing a $c_p^{1/2}$
scaling law, expected in the semidilute concentration regime.}
\vspace*{-5pt}
\end{figure}

In order to further investigate the ion-specific effect and its
consequence on the scattering spectra, we carried out measurements for
a series of ionenes with mixed counterion clouds at a constant monomer
concentration. This was done for ionenes of different charge densities
(3,3, 6,9, and 12,12).  {(Note that based on our previous
scattering results, we know that only the 12,12-ionenes begin to show
the signature of backbone hydrophobicity}~\cite{Malikova15}.  {All
ionene chains with higher charge densities behave as hydrophilic.)} The
most striking changes in the spectra are observed for the most highly
charged chains (3,3-ionenes), and this system is shown in
Figure~\ref{SANS_2M_33BrF}. Additional data for 6,9-Br/F and
12,12-Br/F systems are included in the SI file (part 3). The two
extreme systems ($\mathit{xF}=1$ and $\mathit{xF}=0$) are naturally identical to the 
F- and Br-ionene data in Figure~\ref{SANS}. All intermediate systems
place themselves logically between the two extremes, with gradual
changes with increasing/decreasing $\mathit{xF}$. The complete disappearance of
the PE peak (absence of a maximum) is only present for the pure Br system.
As expected, the intermediate curves \emph{cannot} be obtained by a
linear combination of the curves corresponding to the $\mathit{xF}=1$ and $\mathit{xF}=0$
extremes~\cite{Yasine-PhD}. For a given $\mathit{xF}$, each chain is surrounded
by a mixed counterion cloud at the given ratio; there are no chains in
a ``pure F'' or ``pure Br'' environment.  Interestingly, as we decrease
the charge density of the ionene chains (6,9- and 12,12-ionenes), the
difference in scattered intensity between the pure F and pure Br
extremes is diminished (see SI). This is probably due to a decreasing
overall counterion concentration in the system as we move from 3,3- to
12,12-ionenes.  

\begin{figure}
\includegraphics{fig04}
\caption{\label{SANS_2M_33BrF}Coherent neutron scattering intensity normalised by ionene
monomer concentration ($I_{\mathrm{coh}}/c_{p}$) in arbitrary units
versus scattering wavevector ($q$) for room temperature aqueous
solutions (in D$_2$O) of 3,3-ionenes with mixed Br--F counterion clouds.
The fraction of F$^-$ counterions ($\mathit{xF}$) is shown in the legend. All
systems are at 2~M monomer concentration.}
{\vspace*{-2pt}}
\end{figure}

Having scattering data for ionene solutions across a whole series of
halide counterions gives a very strong argument for the previously
suggested origin in terms of a decreased effective charge of the ionene
chains as we move towards larger, more polarisable ions with a lower
hydration energy, that is, as we descend the halogen series in the periodic
table. For completeness, radii of halide ions in solution,
polarisability, and hydration energies are summarised for the four
halide ions used in Table~\ref{halide-prop}. For the larger ions, the
counterion atmosphere around the ionene backbone is more constricted
as has also been clearly shown by previous molecular dynamics
simulations on ionene solutions~\cite{Druchok16}. As soon as counterion
clouds of adjacent chains do not overlap, the repulsion between the
chains is no longer present and the PE peak in the scattering spectra
disappears.  {This is a concentration-dependent phenomenon,
accentuated at high PE (and thus counterion) concentration, which we
can refer to as ``ion-specific screening''. It seems important to
distinguish this from counterion condensation in the Manning sense of
the word.} As we have seen by osmotic pressure measurements in ionene
solutions in the past~\cite{Malikova12} and as was equally observed for
other systems~\cite{Qu06}, the counterions indeed still contribute to
the osmotic pressure. This is contrary to what has been seen for
\mbox{counterions} in \unskip\break\mbox{solutions}
of hydrophobic polystyrene-based
PEs, where counterions are condensed as part of the
``pearls'' in the pearl-necklace conformation~\cite{Essafi05}. 

\begin{table}
\caption{\label{halide-prop}Ionic radii in solution ($R_s$),
polarisabilities ($\alpha$), and hydration free energies ($\Delta
G_{\mathrm{hyd}}$) for halide ions\vspace*{-2pt}}
\begin{tabular}{ccccc}
\thead
Ion & \multicolumn{1}{c}{\parbox[t]{1cm}{\centering$R_s$~\cite{Marcus88} (\AA)}} & 
\multicolumn{1}{c}{\parbox[t]{1cm}{\centering$\alpha$~\cite{Lamoureux06} (\AA$^3$)}} & 
\multicolumn{1}{c}{\parbox[t]{1.4cm}{\centering$\Delta G_{\mathrm{hyd}}$~\cite{Lamoureux06} (kcal/mol)}} &
\multicolumn{1}{c}{\parbox[t]{1.4cm}{\centering$\Delta G_{\mathrm{hyd}}$ ($k_{\mathrm{B}}T$/ion)}}\vspace*{2pt}\\
\endthead
F$^-$ & 1.24 & 1.20 & ${-}$112.1\0 & ${-}$189.3 \\
Cl$^-$ & 1.80 & 3.65 & ${-}$82.4 &  ${-}$139.1 \\
Br$^-$ & 1.98 & 4.96 & ${-}$76.1 & ${-}$128.5 \\
I$^-$ & 2.25 & 7.30 & ${-}$67.0 & ${-}$113.1
\botline
\end{tabular}
\vspace*{-3pt}
\end{table}


 {For completeness, we note that the interpretation of the 
 SANS on ionene solutions, here and in our previous
publications}~\cite{Malikova12,Malikova15},  {is based on the
assumption that the scattered signal is dominated by the ionene
monomer--monomer correlations and that contributions of the counterions
can be neglected. The underlying estimation of the relative intensities
is provided in the SI (part 4). It shows that contributions of halide
ions to the scattered intensity increase as we move from F$^-$ to
I$^-$. Importantly, for the purposes of the \emph{qualitative} trends
that we discuss here, and which are common to ionenes of all charge
densities, the above assumption is indeed\break reasonable.}



In the following, we are interested in exploring the rigidity of the
ionene chains as a function of the above  {ion-specific screening}.
We have explored this for the case of F$^-$ and Br$^-$ counterions
using the NSE technique~\cite{Mezei_notes,ActuChim_MM22}.  {Neutron spin echo gives
access to the microscopic dynamics of the chain on the length scale of
nm and a timescale of ps-ns. These are sufficiently short length scales
to avoid the influence of large heterogeneities, which lead to the
observation of a slow mode, a very common feature in dynamic light
scattering studies on PE
solutions}~\cite{Sedlak_chapter_in_book,Ngai96}. The measured data in
NSE is the intermediate scattering function $I(q,t)$,
which is formally the spatial Fourier transform of the van Hove
correlation function $g(r,t)$. We measure the dynamic data on the
coherently scattered signal arising from the contrast between
hydrogenated ionene chains in a deuterated solvent (D$_2$O), the same
systems as those used previously for the small angle scattering
experiments. The NSE data were collected for ionene chains with
intermediate chain charge densities, 6,9-ionenes, at 0.4~M and 2~M
monomer\break concentrations. 

An example of NSE data is shown in Figure~\ref{NSE_Iqt}, where a series
of $I(q,t)$ curves for 6,9-Br ionene at 0.4~M is presented, each curve
corresponding to a given $q$ value. Following a standard analysis, the
data was modeled using  mono-exponential decay to obtain a value of a
characteristic relaxation time $\tau$ at a given $q$ value. Under a
simple diffusion model, this characteristic time is converted into an
effective diffusion coefficient $D_{\mathrm{eff}}$ following the
relation $1/\tau=D_{\mathrm{eff}}q^2$. Note that for large $q$ values,
the background corrections (quartz cell and solvent) lead to an
unphysical long-time asymptote of slightly less than~0. For these cases,
the fitting parameters were relaxed to allow for a nonzero (slightly
negative) constant background. 

\begin{figure}
\includegraphics{fig05}
\vspace*{-5pt}
\caption{\label{NSE_Iqt}Intermediate scattering function $I(q,t)$  for
6,9-Br ionene at $c_p=0.4$~M as measured by NSE. Each curve corresponds
to a given $q$ value in the range from 0.04 to 0.36~\AA$^{-1}$. Lines
are mono-exponential fits for a selection of the curves.}
\vspace*{-5pt}
\end{figure}

Figure~\ref{NSE_Iq_Deff} summarises the coherently scattered intensity
(as measured by polarisation analysis on NSE) and the effective
diffusion coefficients $D_{\mathrm{eff}}$ resulting from the
mono-exponential fitting of the $I(q,t)$ curves. Data for all four
systems studied by NSE are shown: 6,9-Br and 6,9-F ionenes, each at 0.4~M
and 2~M monomer concentration. The position of the PE peak in the
coherently scattered intensity observed by NSE
(Figure~\ref{NSE_Iq_Deff}, left) reflects what has been \mbox{observed}
already by SANS (Figures~\ref{SANS} and~\ref{SANS_2M_33BrF}). The data
for Br- and F-ionenes at low monomer concentration show a peak at the
same $q$ position while at high concentration, the scattered intensity
is radically different.  {For completeness, the relative
contributions of coherent and incoherent scattering, as determined by
polarisation analysis in NSE, are presented in SI (part 5).}

\begin{figure*}
\vspace*{-1pt}
\includegraphics{fig06}
\vspace*{-1pt}
\caption{\label{NSE_Iq_Deff}Coherently scattered intensity (polarisation analysis in NSE,
left) and effective diffusion coefficient $D_{\mathrm{eff}}$ (right)
as a function of the wavevector $q$ for all systems studied by NSE.
Note that 10~\AA$^2$/ns ${=}$\ $10^{-10}$~m$^2$/s.}
\vspace*{-2pt}
\end{figure*}

In Figure~\ref{NSE_Iq_Deff} (left), we also indicate the high $q$
intensity dependence. Assuming that the chain--chain correlations do not
contribute in a significant way to the scattered intensity in the high
$q$ region (this is an approximation, as we are not working under the
``zero average contrast'' conditions here), the power law reflects the
conformation of the individual chains. While a $q^{-1}$ behaviour is
characteristic of a rod-like conformation (and this is the case of the
low concentration data for both Br- and F-ionenes), a $q^{-2}$
behaviour corresponds to the signal of a Gaussian chain, in other words a
neutral polymer in a $\rmTheta$ solvent ($q^{-1.7}$ indicates Gaussian
chains with an excluded volume contribution). It is clear that compared
to 0.4~M data, both 2~M data sets have a higher decay exponent
($q^{-1.42}$) at high $q$ values, which indicates a less rod-like
conformation. At the same time, we need to note that the high $q$
dependence in the SANS spectra (Figures~\ref{SANS} 
and~\ref{SANS_2M_33BrF}) does not agree with the NSE-determined decay
exponents. The SANS data features much higher exponents (close to
$q^{-2}$ already for the 0.4~M data sets and even higher for the 2~M data
sets). It is important to realise that the decay exponent at high $q$
is very sensitive to the incoherent background subtraction. This is
done in very different ways in SANS and in NSE. In SANS data reduction,
the density of H atoms is first estimated from the concentrations of
the hydrogenated chains in the deuterated solvent and compared to H
atom density of pure water. The background constant to subtract is
determined from the ratio of these two H atom densities. In NSE, the
decomposition of the total scattered signal into coherent and
incoherent contributions is measured \emph{directly} using polarisation
analysis, which relies on the spin flip of incoherently scattered
intensity from H nuclei~\cite{Squires_book}. As a result, we consider
the high $q$ exponents determined from NSE as more reliable and we do
not conclude on the chain conformation from the high $q$ SANS signal. 

There are several points to note regarding the effective diffusion
coefficients in Figure~\ref{NSE_Iq_Deff} (right), all of the order of
10$^{-10}$~m$^2$/s. We may consider the figure in two parts, below and
above the $q^*$ position or the high $q$ end of the observed plateau,
depending on the given system. Both theory~\cite{Hayter80} and previous
experiments~\cite{Nallet83,Kanaya89,Lee09} on PE solutions
show a rapid decrease in $D_{\mathrm{eff}}$ for $q<q^*$ and then a
constant value for $q>q^*$. The constant value at high $q$ reflects the
rod-like conformation of a charged chain. On the contrary, the signal
in a semidilute solution of a neutral polymer chain should show
$D_{\mathrm{eff}}$ increasing linearly with $q$ in the high $q$
region~\cite{Adam77}.   {The behaviour of $D_{\mathrm{eff}}$ in the
case of charged rigid chains stems from the theoretical considerations of
de Gennes and collaborators}~\cite{Hayter80}.  {First,
$D_{\mathrm{eff}}$ is expressed as  $D_{\mathrm{eff}}=k_{\mathrm{B}}T\mu(q)/S(q)$,
where $\mu(q)$ is the mobility and $S(q)$ is the static scattering
signal}~\cite{HansenMcDonald_book}.  This is indeed a mathematical
expression of the so-called ``de Gennes narrowing'', which in a
simplified way states that a slowing down of dynamics takes place in
$q$ regions where peaks in structural correlations are present. Clever
modelling by de Gennes indicates that for locally rigid chains,
$\mu=\mu(0)/(ql_{\mathrm{p}})$, where $l_{\mathrm{p}}$ is the persistence length, that is, $\mu
\propto q^{-1}$ for $ql_{\mathrm{p}}\geq1$. Taking into account that for locally
rigid chains $S(q) \propto q^{-1}$ in that same high $q$ range, the $q$
dependence of $D_{\mathrm{eff}}$ cancels out and a constant
$D_{\mathrm{eff}}$ in the high $q$ region is recovered.\looseness=-1

From the high $q$ behaviour, the most rigid chain is indeed that of
F-ionenes at 2~M concentration, as this data set features the most
constant values beyond $q^*$. The remaining three systems (Br 2~M, Br
0.4~M, and F 0.4~M) all feature a more significant increase in
$D_{\mathrm{eff}}$ beyond $q^*$. Although both 0.4~M data sets exhibited clear
rod-like behaviour in the coherently scattered intensity
(Figure~\ref{NSE_Iq_Deff}, left), these chains do not show the highest
rigidity as seen from  $D_{\mathrm{eff}}$. One difficulty in the
interpretation is probably the intra- and inter-chain correlations
mixing in both the static and  dynamic signals. Measurements under
ZAC would probably help disentangle the inter- and intra-chain
correlations contributing to the scattered signal, at least for the
static signal. NSE measurements under ZAC seem challenging. 

 {Overall, we view the trends in} Figure~\ref{NSE_Iq_Deff} (right)
 as consistent with the theoretical predictions outlined previously,
which indicate above all a very different rigidity for the ionene
chains with Br$^-$ and F$^-$ ions at 2~M concentration. The striking
difference between Br and F is related to the very different scattered
intensities: as $q$ decreases below 0.2~\AA$^{-1}$ (a) the 2~M F system
passes through a structural maximum and its dynamics increases very
fast for all smaller $q$ values; (b) the scattered intensity for the 2~M Br system
continues to grow below 0.2~\AA$^{-1}$ to reach a plateau, and the dynamics
in 2~M Br is significantly suppressed in comparison to 2~M F in the
entire region below 0.2~\AA$^{-1}$, but begins to rise slowly once the
plateau is reached.

The mesoscopic dynamics of ionene chains in {D$_2$O} solutions was measured
by PFG-NMR. Figure~\ref{PFG-NMR-cp} summarises the $D_{\mathrm{NMR}}$
data for 3,3-ionenes with the four different halide counterions as a
function of monomer concentration $c_p$. For comparison, this figure
also features data from sodium polystyrene sulfonate (PSS), measured by
PFG-NMR in H$_2$O, from reference~\cite{Oostwal93}. The ionene and PSS
data fall into the same range of $D_{\mathrm{NMR}}$ of the order of
10$^{-11}$~m$^2$/s. At room temperature, the ratio of D$_2$O and H$_2$O
viscosities is 1.25, and this conversion factor would have to be used
for a detailed quantitative comparison of the two data sets. The ionene
chains correspond to molecular weights of 20--60~kDa, which fall well
within the range of molecular weights for the PSS
data~\cite{Oostwal93}. However, the size polydispersity (PID) of ionene
chains (due to the poly-addition reaction as opposed to radical
polymerisation for PSS) is significantly higher: PID(ionene)\ ${=}$\ 1.8--2.0
and PID(PSS)\ ${=}$\ 1.25--1.5. Given these differences, we do not dwell on a
detailed quantitative PSS--ionene comparison. The general trend as a
function of $c_p$ is similar for the two types of PE chains, with a
decrease in $D_{\mathrm{NMR}}$ for $c_p$ above approximately 0.1--0.2~M. 

\begin{figure}
\includegraphics{fig07}
{\vspace*{-3pt}}
\caption{\label{PFG-NMR-cp}Self-diffusion coefficients of ionene chains for 3,3-ionenes
with four different halide counterions as indicated versus monomer
concentration $c_p$ (solvent ${=}$ D$_2$O) as measured by PFG-NMR. Crosses
correspond to measurements for polystyrene sulfonate with Na$^+$
counterions (NaPSS) at three different molecular masses as indicated.
NaPSS data from reference~\cite{Oostwal93} (solvent ${=}$ H$_2$O).}
{\vspace*{-8pt}}
\end{figure}

Let us concentrate on the $D_{\mathrm{NMR}}$ data in
Figure~\ref{PFG-NMR-cp} for different counterions, especially in the
higher $c_p$ region (above 0.1~M). The ordering of
$D_{\mathrm{NMR}}$ data for 3,3-ionenes does not follow in a simple way
the halide anion series, contrary to what was seen in the SANS data.
The order here is $D(\mathrm{F})<D(\mathrm{Cl})>D(\mathrm{Br})
\gg D(\mathrm{I})$. We see two phenomena behind
this non-monotonous behaviour: (a) change in chain conformation (less
rod-like as we move down the halide series towards larger anions) and
(b)~inter-chain aggregation (the system becomes less soluble as we move
down the halide series). A change from rod-like to globular chain
conformation leads to an \emph{increase} in {self}-diffusion
coefficient of the chain~\cite{Koene83B}; the inter-chain aggregation
leads to its \emph{decrease} (diffusion of larger objects). For the 
3,3 I-ionenes, the inter-chain aggregation is indeed pronounced; 3,3 I-ionene
shows a significantly lower $D_{\mathrm{NMR}}$ than all the other
systems (roughly lower by a factor of 2) throughout the entire
concentration range. This scenario is consistent with the evolution of
viscosity ($\eta$) of aqueous solutions of a neighbouring cationic PE,
poly(diallyldimethylammonium), along the halide series~\cite{Lezov22}.
Indeed, $\eta$ decreases between {Cl$^-$} and {Br$^-$} solutions of
this PE (chains change from a rod-like to a more globular
conformation), and the system becomes insoluble with {I$^-$} (no
{F$^-$} data are available in ref~\cite{Lezov22}). We attach importance
to the fact that 3,3 F-ionenes show somewhat lower $D_{\mathrm{NMR}}$
than Br- and Cl-ionenes in the moderate to high $c_p$ range. To ensure
that this difference is real, we have explored changes in
$D_{\mathrm{NMR}}$ for ionenes with mixed Br/F counterion clouds. 
The data is summarised in Figure~\ref{PFG-NMR-xF} for two different
ionene charge densities. Indeed, as we move from pure Br to pure F
ionene chains, the chain dynamics clearly \emph{decreases}. 


\begin{figure}
\includegraphics{fig08}
\caption{\label{PFG-NMR-xF}Self-diffusion coefficients of ionene chains for systems with
mixed Br/F counterion atmospheres at a monomer concentration of 2~M as
a function of  F$^-$ counterion fraction $\mathit{xF}$ as measured by
PFG-NMR. Two different ionene chain charge densities are presented.}
\end{figure}

\section{Conclusion}

The combination of SANS, NSE,
and PFG-NMR provides us with several pieces of information on the state of
the counterion atmosphere around positively charged PE chains in
aqueous solution as a function of the counterion chemical nature, that is,
\emph{ion-specific effects}. A series of monovalent halide counterions
is explored. {A strong ion-specific effect is observed clearly at
high PE monomer concentrations and is consistent with the picture of an
increasingly more compact counterion atmosphere around the PE chain, a
phenomenon we refer to as ``ion-specific screening''. A stronger
counterion screening of the charge on the PE chains takes place as we
move along the halide series towards larger, more polarisable and more
weakly hydrated counterions. This can indeed be seen as another example
of the Collins' concept of ``matching water affinities'' in which ions are
characterised by their ``softness'' with consequences for favourable
ion pairing between ``soft'' (weakly hydrated) anions and cations on
one hand and ``hard'' (strongly hydrated) anions and cations on the
other}~\cite{Collins97}. {Within the Collins' classification, the
quaternary ammonium groups on ionene PE chains are ``soft'' cations, and
thus more favourable ion-paring is in place with ``soft'' halide ions,
that is, larger and more polarisable anions. More effective screening of
the PE chain charge is demonstrated by a reduced chain--chain repulsion
in the system as shown by the disappearance of the PE structural peak
in the scattering data.} Importantly, the ion-specific effect has
consequences for the chain rigidity and local and mesoscopic chain
dynamics as shown further by NSE and PFG-NMR. 

{In principle, it is important to distinguish between the
{self}-diffusion coefficient measured by PFG-NMR (due to the
position encoding method using magnetic field gradients, PFG-NMR indeed
measures {self}-diffusion) at the $\upmu$m scale and the effective
diffusion coefficient $D_{\mathrm{eff}}$ of the PE chains as measured
by NSE (arising from the coherently scattered signal) at the nm scale.
Depending on the length scale ($q$ value), $D_{\mathrm{eff}}$ obtained by
NSE indeed represents different quantities: (a) for length scales below
the mesh size ($q$ values above $q*$), it represents the local individual
chain dynamics}~\cite{Hayter80};  {(b)~for length scales above the
mesh size ($q$ values below $q*$), $D_{\mathrm{eff}}$ merges with
$D_{\mathrm{collective}}$ in the hydrodynamic limit $q\rightarrow0$,
(e.g., \cite{Martin72}).}  {Let us now look at the different scales in
turn, starting from the most local scale}:

\begin{itemize}

\item Local scale below PE mesh size, probed by NSE: Here, the
notion of \emph{collective} motion seen via the coherent signal in NSE
no longer applies; the scale probed is too small. NSE probes the
dynamics of individual chains and informs us on their rigidity. We
observe that PE chains retaining a strong chain--chain repulsion (i.e.,
with F$^-$ counterions) show increased  rigidity of the PE chains
at high monomer concentration.

\item nm scale larger than the PE mesh size, probed by NSE: Here,
NSE dynamic data probe \emph{collective} dynamics in the PE
network, as the coherent signal is dominated by the \emph{inter-chain}
correlations at these length scales. For locally rigid chains (i.e.,
with F$^-$ counterions), the collective dynamics at this scale is very
fast. For more flexible chains  (i.e., with Br$^-$ or Cl$^-$
counterions), the dynamics here is suppressed. We see this as a case of
``de Gennes narrowing'', as the Br$^-$ and Cl$^-$ systems show an
intense scattering signal at this spatial scale. The effective
\emph{collective} diffusion coefficients at the nm scale are all of the
order of 10$^{-10}$ m$^2$/s. 

\item $\upmu$m scale, probed by PFG-NMR: On this largest scale,
PFG-NMR probes the {self}-diffusion of the individual PE chains.
Locally rigid charged chains, with fast collective dynamics seen at
the nm scale (i.e., with F$^-$ counterions), diffuse consistently slightly
\emph{slower} than locally more flexible chains with slow nm scale
dynamics. The {self}-diffusion coefficients at the $\upmu$m scale
are all of the order of 10$^{-11}$~m$^2$/s.  
\end{itemize}

Dynamics in PE solutions has been probed extensively by
DLS~\cite{Sedlak_chapter_in_book}. In the semidilute PE concentration
regime, this techniques gives access to the \emph{collective} dynamics
($D_{\mathrm{coll}}$) at the mesoscopic scale; the closest length scale
would be the PFG-NMR scale. Concentrating on the fast mode in the
DLS signal, charged rigid PE chains show consistently \emph{faster}
collective dynamics compared to flexible neutral chains. This is again
a demonstration of de Gennes narrowing, which leads to large values of
$D_{\mathrm{coll}}$ for repulsive systems at length scales much larger
than the PE mesh size (expressed in the scattering language, as $q<q*$, which is
where DLS operates). 
For charged chains at high monomer concentration
in the absence of salt, the order of magnitude for DLS-determined
\emph{collective} diffusion coefficients is 
10$^{-10}$~m$^2$/s~\cite{Sedlak_chapter_in_book}. This is the same
order of magnitude as the \emph{collective} diffusion coefficients
measured here by NSE. Furthermore, the order of magnitude difference between
{self}-diffusion and collective diffusion coefficients in PE
solutions has been noted before ($D_{\mathrm{self}} \ll
D_{\mathrm{coll}}$)~\cite{Sedlak_chapter_in_book,Oostwal93}. Our new
data sets (NSE, PFG-NMR) on ionene PE solutions are consistent with
these observations. However, the effects of the counterion specificity
on $D_{\mathrm{self}}$ measured here by PFG-NMR suggest that
$D_{\mathrm{self}}$ of charged chains is \emph{lower} than that of
neutral chains. In other words, the loss of charge on the chain, due to
counterion-specific screening of the chain charge, has the opposite
effect on $D_{\mathrm{self}}$ and $D_{\mathrm{coll}}$. This seems to be
indeed in line with very recent DLS and PFG-NMR data on PSS~\cite{Buvalaia23}. 
Overall, the more strongly charged chains
adopt a more extended conformation, resulting in a lower
{self}-diffusion coefficient while the \emph{collective}
diffusion in these more repulsive systems is enhanced (de Gennes
narrowing).

\section*{Declaration of interests}
{\vspace*{-3pt}}

The authors do not work for, advise, own shares in, or receive funds
from any organisation that could benefit from this article, and have
declared no affiliations other than their research organisations.

\section*{Acknowledgements}

The authors thank Sa\v{s}o \v{C}eba\v{s}ek at the University of
Ljubljana for help with sample preparation, Matija Tom\v{s}i\v{c} at
the University of Ljubljana for help throughout NSE measurements at the
ILL (Grenoble), and Fran\c{c}ois Ribot at the College de France (Paris) for
discussions and access to NMR spectrometers. The NSE data from ILL is
available under
doi:\href{http://doi.org/10.5291/ILL-DATA.9-11-1624}{10.5291/ILL-DATA.9-11-1624}.

\back{}

\section*{Supplementary data}
Supporting information for this article is available on the journal's
website under \printDOI\ or from the author.  

\CDRsupplementaryTwotypes{supplementary-material}{\cdrattach{crchim-326-suppl.pdf}}

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