Institute for Condensed Matter Physics of the National Academy of Sciences of
Ukraine,
79011 Lviv, Ukraine Leibniz-Institut für Polymerforschung Dresden e.V., 01069
Dresden, Germany UNAM - National Nanotechnology Research Center and Institute
of Materials Science & Nanotechnology, Bilkent University, 06800 Ankara,
Turkey Leibniz-Institut für Polymerforschung Dresden e.V., 01069 Dresden,
Germany

# Structural properties of
cyclic polyelectrolytes
in dilute good solvent

Andrzej Grzyb  Institute of Physics, University of Silesia in Katowice, 75
Pułku Piechoty 1, 41-500 Chorzów, Poland  Khristine Haydukivska  Institute of
Physics, University of Silesia in Katowice, 75 Pułku Piechoty 1, 41-500
Chorzów, Poland  Jarosław S. Kłos  Faculty of Physics, A. Mickiewicz
University, Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland  Aykut Erbaş
Institute of Physics, University of Silesia in Katowice, 75 Pułku Piechoty 1,
41-500 Chorzów, Poland  Jarosław Paturej  Institute of Physics, University of
Silesia in Katowice, 75 Pułku Piechoty 1, 41-500 Chorzów, Poland
<EMAIL_ADDRESS>
###### Abstract

Cyclic polymers display unique physical behaviors in comparison to their
linear counterparts. Theoretical, computational and experimental studies have
revealed that some of their distinctive properties are also observed in
charged variants of cyclic polymers, known as cyclic polyelectrolytes (PEs),
especially in terms of their structural responses to variations in the
strength of electrostatic interactions. In this study, we investigate the
impact of cyclic topology on the conformations of PE chains in dilute good
solvent using scaling analysis and coarse-grained bead-spring molecular
dynamics simulations. Our observations indicate that, in contrast to linear PE
chains, cyclic topology results in more compact conformations at low and
intermediate Bjerrum lengths. Moreover, two structural metrics, asphericity
and prolateness, which quantify deviations from spherical and flat molecular
shapes, exhibit non-monotonic behaviors for cyclic PEs. This stands in
contrast to linear PEs, where these shape characteristics exhibit a monotonic
trend with increasing Bjerrum length. A feasible analytical theory, developed
to account for ionic distributions around cyclic PE chains, suggests that the
fundamental difference between linear and cyclic chain conformations may be
attributed to topological effects influencing long-range electrostatic
interactions.

##  1 Introduction

Polyelectrolytes (PEs) are polymers that contain ionizable functional groups
along their backbones. When PE chains are dissolved in polar solvents,
ionizable groups can release counterions into solution 1, leaving a net charge
on the polymer backbone. Mobile counterions make PEs solutions highly
responsive to changes in pH, ionic strength, and electric fields 2. For the
same reason, while synthetic PEs are commonly found throughout various
applications including industrial water treatment 3, dispersant agents,
emulsifiers 4, superplasticizers 5, gene therapy and drug coating/delivery 6,
tissue engineering and cosmetics, biological PEs such DNA plasmids play a
dominant role in many biochemical processes 7, 8.

PE chains share properties with both electrolytes and polymers, and the
interplay of these properties determines their equilibrium configuration.
Electrostatic interactions, counterion entropy, and conformational free energy
of a polymer chain affect the degree of counterion delocalization from polymer
backbone 9. Ions localized near or on the chain, on the other hand, can
neutralize the backbone charges, and in turn, change the equilibrium structure
of the PE chain 10, 11. For instance, in linear PEs, the ionic strength of the
solution controls the polymer-counterion interactions and can determine
whether the chain adapts stretched, pearl-necklace or collapsed conformations
12, 13. The coupling between PE counterions and chain conformation can further
be affected by charge distribution along the polymer backbone, charge valence,
polymer chemistry, degree of polymerization, and macromolecular architecture.

Introducing structural complexity in a macromolecular architecture, for
instance, in the form of grafts, branching points, and closed loops, can
expand the possibilities for designing novel polymers with distinctive
physiochemical properties that their linear counterparts cannot achieve.
Cyclic polymers are an intriguing class of such polymers with loop topology.
In solutions of charge-neutral cyclic polymers, lack of ends enforces a
smaller radius of gyration, smaller hydrodynamic volume, lower melt viscosity,
higher thermostability, and a higher glass transition temperature as compared
to linear chains 14, 15, 16, 17, 18, 19, 20, 21, 22. The compactness of cyclic
polymers plays a crucial role in controlling the segregation properties of
colloidal particles in confinement 23. Additionally, cyclic topology strongly
influences specific viscoelastic properties of polymer melts, such as the
absence of plateau modulus that is common in melts of entangled linear chains
21. Cyclic polymers also exhibit significantly larger swelling ability and
maximum strain at break in their swollen networks 24 and enhanced lubrication
abilities25, 26.

Synthetic cyclic PEs, which are experimentally available through various
synthetic strategies 27, 28, 29, have also shown peculiar behaviour similar to
their neutral counterparts in experimental and theoretical studies.
Thermodynamics of ring PE solutions was investigated using analytical methods
30. The authors found that at semidilute concentrations, the electrostatic
contribution to osmotic pressure is larger for PE rings as compared to linear
chains. Molecular dynamics (MD) studies on counterion distribution around
highly-charged cyclic, linear, and branched PE architectures in polar solvent
demonstrated the effect of macromolecular complexity on the fraction of
counterions transiently localized to PE chains 31. Linear and cyclic PE chains
grafted on planar surfaces exhibit variation in surface morphologies in MD
simulations 32. While both linear and cyclic polymers exhibit a formation of
vertically phase-separated patched structures depending on the grafting
density, cyclic topology favours smaller patches. Monte-Carlo simulations 33
suggested that charged ring-shaped macromolecules experience a significantly
more pronounced segregation confined in nanochannels.

Liu et al. studied the motion of circular PEs in shear flow by means of
mesoscale hydrodynamic simulations and revealed coupling between shear rate,
electrostatic strength and polymer architecture and their impact on
conformation and dynamics of charged rings 34. The circular architecture plays
a pivotal role in defining deformation and orientation characteristics of
charged rings subjected to flow. If the electrostatic interaction strength is
weak (i.e. at high salt concentration), the PE chain shifts its shape from an
oblate ring at low shear rates to a prolate ring at high shear rates.
Conversely, when the electrostatic interaction strength is strong (i.e. at low
salt concentration), the transition occurs from a prolate coil to a prolate
ring. Last but not least, biological cyclic PEs such as bacterial genome and
plasmids somewhat prefer cyclic topology to optimize various vital biological
processes such as loop formation, replication, recombination, repair, gene
sequencing and coding 35, 36, 37, 38, 39, 40, 41.

In the present work, we study the impact of topology on conformations of PEs
in salt-free good solvent conditions using analytical approaches and bead-
spring coarse-grained MD simulations. We systematically investigate the role
of polymerization degree and the Bjerrum length (i.e. strength of
electrostatic interactions ) on cyclic PE chains size, shape and monomer
density and compared them to linear PEs of the same degree of polymerization.
We observe significant differences in the equilibrium conformation properties
of cyclic PEs, some of which can be described by our scaling arguments
suggested at the strongly-stretched chain limit.

The paper is organized as follows. In the next section, we outline the details
of the simulation model utilized in this study. We apply scaling analysis to
estimate conformations of cyclic PE chains. Then, we present numerical results
and discussed them in the context of our theoretical analysis. We draw
conclusions and remarks in the final section of the manuscript.

##  2 Simulation Model and Methodology

Linear PEs are long-chain polymers with repeating units that contain charged
functional groups. Ring polyelectrolytes are polymers with a closed-loop or
circular architecture. Like linear polyelectrolytes, they contain repeating
units with charged functional groups, which can be positively charged
(cationic) or negatively charged (anionic). In our study we utilize molecular
dynamics (MD) simulations, implementing the well-established Kremer-Grest
bead-spring model that was previously employed in investigations of linear
PEs. 42, 43, 44, 45. In the KG model, non-bonded interactions between all
beads are accounted by a shifted, truncated LJ potential 42:

$U^{\text{LJ}}\left(r_{ij}\right)=\begin{cases}4\epsilon\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}-\left(\frac{\sigma}{r_{c}}\right)^{12}+\left(\frac{\sigma}{r_{c}}\right)^{6}\right]&\text{for
}r\leq r_{c}\\\ 0&\text{for }r>r_{c}\\\ \end{cases}$ (1)

where $r_{ij}$ is the distance between two interacting beads $i$ and $j$,
$\sigma$ is the bead diameter and $r_{c}$ is the cutoff distance. The
parameter $\epsilon$ in Eq. (1) controls the strength of the LJ potential. The
connectivity between two adjacent chain monomers is maintained by the finite
extension nonlinear elastic (FENE) potential

$U^{\text{FENE}}\left(r\right)=\begin{cases}-0.5kr_{\text{max}}^{2}\ln{\left(1-\frac{r^{2}}{r_{\text{max}}^{2}}\right)},&\text{if
}r\leq r_{\text{max}}\\\ \infty,&\text{if }r>r_{\text{max}}\\\ \end{cases}$
(2)

where $k$ is the spring constant and $r_{\text{max}}$ is the maximum bond
length 42. To investigate linear and ring PEs monovalent positive charges are
assigned to each bead along the polymer backbone. Furthermore, an appropriate
number of monovalent, negative counterions are considered as well to maintain
system electroneutrality. The electrostatic interactions between charges were
introduced by Coulomb pairwise potential

$\frac{U^{\mbox{\tiny C}}_{ij}}{k_{B}T}=l_{B}\frac{z_{i}z_{j}}{r}$ (3)

where $k_{B}$ denotes the Boltzmann constant, $T$ the absolute temperature,
$l_{B}$ the Bjerrum length of the solvent, and $z_{i}$, $z_{j}=\pm 1$ is the
charge valence of $i$-th and $j$-th particles. The Bjerrum length is defined
as $l_{B}={e^{2}}/({4\pi\varepsilon_{0}\varepsilon_{r}k_{B}T})$, where
$\varepsilon_{0}$ stands for the electric permittivity of the vacuum,
$\varepsilon_{r}$ for the relative permittivity of the solvent and $e$ for
elementary charge. For instance, in water at room temperature, $T_{r}\approx
298$ K, the Bjerrum length is $l_{B}(\varepsilon_{r}=80,T=T_{r})\approx 7$Å.

The LAMMPS MD package 46 was used to carry out Langevin dynamics in NVT
ensemble in the reduced LJ units. Namely, $\epsilon=1k_{B}T_{r}$ and
$\sigma=l_{B}(\varepsilon_{r}=80,T=T_{r})$ were taken as the real units of
energy and length respectively. For the FENE potential the reduced parameters
were: $k^{*}=k\sigma^{2}/\epsilon=7$ and $R_{0}^{*}=2$.42 According to the
formula for reduced elementary charge,
$e^{*}=e/\left(4\pi\varepsilon_{0}\epsilon\sigma\right)^{1/2}$, and for the
reduced temperature, $T^{*}=k_{B}T/\epsilon$, we assigned them values
$e^{*}=9$ and $T^{*}=1$. The size $\sigma^{*}=1$ and monomeric mass $m^{*}=1$
were chosen to be the same for all monomers and counterions. The cutoff
distance $r_{c}^{*}$ in Eq. (1) for monomer-monomer interactions was equal to
$2.5$ and for monomer-counterion and counterion-counterion interactions
$2^{1/6}$. Furthermore, for monomer-counterion and counterion-counterion
interactions the LJ interaction strength parameter was set to
$\epsilon^{*}=1$. Our simulations mimic good solvent conditions corresponding
to the choice of $\epsilon^{*}=0.1$ for monomer-monomer interactions 45. Note
that the value of $\epsilon^{*}\approx 0.5$ between monomers yields nearly
$\theta$ conditions for charged polymers 45, 42. Coulomb interactions were
calculated via Particle-Particle-Particle Mesh (PPPM) Ewald method with the
error tolerance for force $10^{-5}$ and the real space cutoff distance
$r_{e}^{*}=30$. The dumping parameter in the Langevin equation of motion was
taken as $\gamma^{*}=\gamma/\tau=1$, where
$\tau=\sigma\left(m/\epsilon\right)^{1/2}$ is the LJ time unit and $m$ is the
assumed real mass unit. The reduced time step was $\delta t^{*}=\delta
t/\tau=0.005$.

Simulations were carried out in cubic boxes with periodic boundary conditions.
The simulation box was varied depending on the degree of polymerization $N$ of
PEs to obtain an equal concentration of cationic and anionic monomers for all
systems $\rho_{+}=\rho_{-}\approx 2.35\cdot 10^{-6}\sigma^{-3}$. Throughout
our study, we varied $l_{B}$ by adjusting the relative permittivity
$\varepsilon_{r}$ of the medium. Typically, about $10^{7}$ equilibration
integration steps were performed and followed by $10^{7}$ integration steps of
the production runs. The simulation snapshots were rendered using the Visual
Molecular Dynamics program 47.

##  3 Results and Discussion

###  3.1 Theoretical considerations

In the subsequent analysis, we thoroughly examine the conformations of linear
and cyclic PEs by employing an analytical approach. We start the discussion by
re-introducing scaling arguments that are able to characterize the
conformations of linear PEs 48, 49, 50, 51, 52, and we then extend these
analyses to cyclic polymers for the first time. Our analysis will be
restricted to a weak-to-intermediate scale of Coulomb couplings (i.e.,
$l_{B}<7$) for which chains are stretched. This is partially because at higher
Bjerrum lengths, ionic correlations become more dominant, preventing the use
of conventional scaling arguments. From a broad perspective, the scaling
analysis relies on the assumption of distinct length scales and incorporates
the notion of an electrostatic blob 48, 49. All the chemical details of chains
are described by Kuhn segment size, $b$, the number of Kuhn segments $N$, and
the fraction of charged monomers $f$.

####  3.1.1 Linear PE chains

In dilute solutions of neutral linear polymers, the average size of a chain
(i.e., end-to-end distance) is given by $R_{e}^{\rm n}\approx bN^{\nu}$, where
$\nu$ is the scaling exponent which takes values of $\nu\approx 3/5$ for good
solvent, $\nu=1/2$ for $\theta$ solvent, and $\nu=1/3$ for poor solvent. In
salt-free and dilute solutions of PEs at low-to-intermediate $l_{B}$ values,
the Debye screening length is much larger than the average distance between
individual molecules. In this case, intrachain electrostatic interactions
dominate, outweighing the interchain interactions. The backbone charges on the
chains interact through unscreened long-ranged electrostatic repulsive forces,
which results in a elongated chain size $R^{0}_{e}$, such that
$R^{0}_{e}\approx bN$.

In its simplest form, the conformation of a linear, elongated PE chain can be
conceptualized as a linear string of equally sized electrostatic blobs 48. The
average size of a blob $D_{e}^{0}$ represents the smallest correlation length
of electrostatic interactions and can be estimated using scaling arguments.
Blobs are subsections of chains consisting of $g_{e}^{0}$ monomers that remain
unperturbed by electrostatic interactions, and their statistical properties
resemble those of neutral chains in good solvents. Hence, the blob size
$D_{e}^{0}$ is related to the number $g_{e}^{0}$ of monomers within a blob
through the equation: $D^{0}_{e}=b(g_{e}^{0})^{\nu}$. The electrostatic
interactions of monomers inside each blob are approximately of the order of
thermal energy $k_{B}T$: $F_{e}=(qfg^{0}_{e})^{2}/(4\pi\epsilon
D^{0}_{e})\approx k_{B}T$48. By utilizing the definition of $l_{B}$, the
latter equation yields: $(fg^{0}_{e})^{2}l_{B}/D^{0}_{e}\approx 1$ which leads
to the following relationship $D^{0}_{e}=l_{B}(fg_{e}^{0})^{2}$. We can
rewrite equations for $g^{0}_{e}$ and $D^{0}_{e}$ as:

$g^{0}_{e}=B^{\frac{1}{\nu-1}},$ (4)

and

$D^{0}_{e}=bB^{\frac{\nu}{\nu-1}}.$ (5)

In Eqs. (4)-(5), we introduced a constant $B$ which depends on chain
statistics and is given by

$B=(uf^{2})^{\frac{\nu-1}{\nu-2}}=\begin{cases}(uf^{2})^{1/3}&\text{for
$\nu=1/2$}\\\ (uf^{2})^{2/7}&\text{for $\nu=3/5$},\end{cases}$ (6)

where $u\equiv l_{B}/b$.

At length scales larger than $D^{0}_{e}$, electrostatic interactions become
dominant. As a result, the electrostatic blobs repel each other, leading to
the formation of an extended chain consisting of electrostatic blobs of length
$R_{e}^{0}$. In this manner, the end-to-end distance $R^{0}_{e}$ of a PE chain
is imposed by the geometry of a linear array of blobs48, 50, 52:

$R^{0}_{e}=\frac{N}{g^{0}_{e}}D^{0}_{e}=bBN.$ (7)

The above equation reveals that while the PE chain exhibits an extended
overall configuration, within the electrostatic blobs, the chain is coiled.
The scaling model presented above is limited in that it assumes a constant
tension along the chain, which is inaccurate due to the variation in
electrostatic potential along the chain. Specifically, the electrostatic
potential is higher at the center of the chain compared to the chain ends. To
address this limitation, a more refined model53, 44 that incorporates the non-
uniform stretching of a PE chain is discussed below.

The charge distribution along a linear PE chain is an end effect, where the
charges are situated at the minima of the electrostatic potential. The reduced
electrostatic potential $\phi(z)$ at position $z$ along the deformation axis
of a PE chain is defined as a sum of all contributing charges along the chain,
$\phi(z)=l_{B}\sum_{i}q_{i}/|z\mathbf{\hat{e}}_{z}-\mathbf{r}_{i}|$, where
$\mathbf{\hat{e}}_{z}$ is a unit vector pointing along the deformation axis.
In the case of an infinitely long chain, the average potential remains
constant throughout the chain. However, for a finite chain, the charges at the
ends have fewer neighbours compared to the center, resulting in inhomogeneous
in the electrostatic potential 53. As a consequence, the nonuniform stretching
of the linear chain occurs, with the central segments of the chain
experiencing stronger electrostatic repulsion leading to their more
significant stretching compared to the sections near the chain ends 44, 53.
The conformation of a linear PE chain is characterized by a trumpet-like shape
and can be visualized as a linear array of electrostatic blobs with varying
sizes 53. The size $D_{e}(z)$ of blobs increases from the smallest blob in the
middle of the chain towards the chain ends, where the $D_{e}(z)$ is the
largest53, 44. The relation between the size $D_{e}(z)$ and the number
$g_{e}(z)$ of monomers per blob is:

$D_{e}(z)\approx bg^{\nu}_{e}(z).$ (8)

In Refs. 53, 44, the analysis focused on the conformations of linear PEs in a
$\theta$ solvent (i.e., $\nu=1/2$). In the following, we modify the
calculations of this analysis to implement them for good-solvent conditions
(i.e., $\nu=3/5$), for which excluded-volume interactions become more
pronounced. To do so, we effectively consider excluded-volume effects in our
calculations by utilizing the scaling relation given in Eq. (8). The reduced
potential $\phi(z)$ can be estimated from one-dimensional charge distribution
as follows 53, 44:

$\phi(z)\approx
fl_{B}\left(\int_{-R_{e}/2}^{z-D_{e}(z)/2}\frac{g_{e}(z^{\prime})}{D_{e}(z^{\prime})}\frac{dz^{\prime}}{z-z^{\prime}}+\int^{R_{e}/2}_{z+D_{e}(z)/2}\frac{g_{e}(z^{\prime})}{D_{e}(z^{\prime})}\frac{dz^{\prime}}{z^{\prime}-z}+\frac{g_{e}(z)}{D_{e}(z)}\right).$
(9)

By using Eq. (8), the ratio $g_{e}(z)/D_{e}(z)$ in Eq. (9) can be rewritten as

$\frac{g_{e}(z)}{D_{e}(z)}=\frac{1}{b}\left(\frac{D_{e}(z)}{b}\right)^{\frac{1-\nu}{\nu}}.$
(10)

By assuming a weak variation of the blob size $D_{e}(z)$ along the stretching
direction of the chain and utilizing the relationship given in Eq. (10), we
can simplify Eq. (9) as

$\phi(z)\approx
uf\left(\frac{D_{e}(z)}{b}\right)^{\frac{1-\nu}{\nu}}\left(\ln{\left[\frac{R_{e}^{2}-4z^{2}}{D_{e}^{2}(z)}\right]+1}\right).$
(11)

The terms inside the logarithm impose that the above equation is only valid in
the interval $-\frac{R_{e}-D^{0}_{e}}{2}<z<\frac{R_{e}-D^{0}_{e}}{2}$.

The derivative of Eq. (11) with respect to position provides an expression for
the electrostatic force acting on each point of the chain along its most
probable, strongly stretched conformation 53, 44. This force stretches the
segments inside the blob and is resisted by a blob tension
$t(z)=D_{e}(z)/g_{e}(z)$. The balance between these forces determines the
dependence of the electrostatic blob size on $z$ as

$\frac{3}{2}b^{\frac{2(1-\nu)}{\nu}}\frac{\rm{d}}{{\rm
d}z}D_{e}^{\frac{2(\nu-1)}{\nu}}(z)\approx
f\frac{{\rm{d}}\phi(z)}{{\rm{d}}z}.$ (12)

Solving Eq. (12) leads to an expression for $D_{e}(z)$ along the deformation
direction

$D^{\rm L}_{e}(z)\approx
D^{0}_{e}\left[\ln{\left(\frac{R^{2}_{e}-4z^{2}}{2R_{e}D^{0}_{e}}\right)-1}\right]^{\frac{\nu}{3(\nu-1)}},$
(13)

where $D^{0}_{e}=b(uf^{2})^{\nu/(\nu-2)}$ denotes the electrostatic blobs at
chain ends. Eq. (13) indicates that the smallest electrostatic blob is located
in the middle of the chain.

The mean chain size $R_{e}$ can be obtained from the conservation condition of
monomers 53, 44

$N=2\int_{0}^{\frac{R_{e}}{2}-\frac{D^{0}_{e}}{2}}\\!\\!\rho(z){\rm{d}}z\approx\frac{R_{e}(D^{0}_{e})^{\frac{1-\nu}{\nu}}}{b^{1/\nu}}\left(\ln{\frac{eR_{e}}{D^{0}_{e}}}\right)^{-1/3},$
(14)

where $\rho(z)=g_{e}(z)/D_{e}(z)$ is the projection of the monomer density
onto the direction of the end-to-end vector and can be written using Eqs. (10)
and (13) as

$\rho(z)\approx\rho_{0}[\ln{(R_{e}^{2}/4-z^{2})+C}]^{-1/3},$ (15)

In Eq. (15), parameters $\rho_{0}$ and $C$ are the monomer density away from
the ends and a constant, respectively. The expression provided in Eq. (15)
will be employed in our subsequent analysis to facilitate a comparison between
scaling predictions and the numerical results obtained through MD simulations
(see Fig. 7a).

By employing Eq. (15) and following an iterative approach to solving Eq. (14)
leads to an expression for the equilibrium size of a linear PE chains 44, 53:

$R_{e}\approx
b^{1/\nu}(D_{e}^{0})^{\frac{\nu-1}{\nu}}[\ln{eN/g_{e}^{0}}]^{1/3}=bBN[\ln{eN/g_{e}^{0}}]^{1/3}.$
(16)

The logarithmic term in Eq. (16) arises due to the non-uniform stretching of
linear PE chains. This logarithmic correction term is not observed in a simple
equi-sized blob model (in contrast to Eq. (7)). Note that, similarly to Eq.
(7), the solvent quality effects are controlled by the prefactor $B$ defined
in Eq. (6). For $\theta$-solvent condition with $\nu=1/2$ the scaling law of
Eq. (16) reduces to $R_{e}\approx bN(uf^{2})^{1/3}[\ln{eN/g_{e}^{0}}]^{1/3}$
which is the result known from previous studies on linear PEs and was derived
using force balance equation53, 44 and several other calculation techniques
48, 54, 55. For good solvent with $\nu=3/5$, Eq. (16) becomes $R_{e}\approx
bN(uf^{2})^{2/7}[\ln{eN/g_{e}^{0}}]^{1/3}$. Note that the logarithmic
correction is not affected by the solvent quality.

####  3.1.2 Cyclic PE chains

If the two ends of a linear chain are attached covalently, the resulting ring
polymer structure does not suffer from the non-uniform stretching discussed in
the previous section. In this section, the constitutive equations we employ
for linear chains will be modified to describe the equilibrium conformational
properties of cyclic PE chains. Fig. 1 illustrates the schematics of a cyclic
PE chain in its extended configuration, represented as a circular array of
electrostatic blobs. Due to the circular symmetry, the blob positions on the
cyclic chain are described in terms of a polar angle $\theta$.

Figure 1: Schematic representation of a cyclic PE chain in a stretched
conformational regime observed at low values of Bjerrum length
$l_{B}/\sigma<7$. $R_{s}$ is the spanning size of a ring whereas
$D_{e}(\theta)$ and $g_{e}(\theta)$ denote the size of electrostatic blob and
the number of monomers per blob, respectively.

Similar to Eq. (9), the electrostatic potential of circular charge
distribution can be estimated from

$\displaystyle\phi(\theta)\approx
fl_{B}\int_{0}^{\theta-\frac{D_{e}}{2}}d\theta^{\prime}\frac{g_{e}(\theta^{\prime})}{D_{e}(\theta^{\prime})}\frac{\sqrt{2}}{R_{s}\sqrt{(}1-\cos{[\theta-\theta^{\prime}]})}$
$\displaystyle+\int_{\theta+\frac{D_{e}}{2}}^{2\pi}d\theta^{\prime}\frac{g_{e}(\theta^{\prime})}{D_{e}(\theta^{\prime})}\frac{\sqrt{2}}{R_{s}\sqrt{(}1-\cos{[\theta^{\prime}-\theta]})}+\frac{g_{e}(\theta)}{D_{e}(\theta)},$
(17)

where $R_{s}$ is defined as the root mean-square distance between monomers $1$
and $N/2+1$. Upon assuming a weak variation of the blob size $D_{e}(\theta)$
with the angle $\theta$, and making use of Eq. (10) with angular dependence,
we simplify Eq. (17) to the following form

$\phi(\theta)\approx
uf^{2}\left(\frac{D_{e}(\theta)}{b}\right)^{\frac{1-\nu}{\nu}}\left(\frac{4}{R_{s}}\arctan{\cos{[D_{e}^{0}/4]}}+1\right)$
(18)

In order to obtain the electrostatic blob size $D_{e}(\theta)$, a force
balance similar to that resulting in Eq. (12) can be written as follows

$\frac{3}{2}b^{\frac{2(1-\nu)}{\nu}}\frac{d}{d\theta}D_{e}(\theta)^{\frac{2(\nu-1}{\nu})}\approx
f\frac{d\phi}{d\theta},$ (19)

which balances blob tension (i.e., $t(\theta)=D_{e}(\theta)/g_{e}(\theta)$)
with electrostatic force acting on each point of the chain along its most
probable strongly stretched circular contour.

The solution to Eq. (19) reads

$\frac{3}{2}b^{\frac{3(1-\nu)}{\nu}}D_{e}(\theta)^{\frac{3(\nu-1)}{\nu}}\approx\frac{3}{2}b^{\frac{3(1-\nu)}{\nu}}\frac{(D^{0}_{e})^{\frac{2(\nu-1)}{\nu}}}{(D_{e}(\theta))^{\frac{1-\nu}{\nu}}}+uf^{2}\left(1-\left(\frac{D^{0}_{e}}{D_{e}(\theta)}\right)^{\frac{1-\nu}{\nu}}\right)\left(\frac{4}{R_{e}}\arctan{\cos{[D_{e}^{0}/4]}}+1\right)$
(20)

where $D^{0}_{e}=b(uf^{2})^{\nu/(\nu-2)}$. The above equation can be further
simplified by employing the strong stretching approximation, where
$D^{0}_{e}/D_{e}(\theta)\approx 1$. In the following sections, MD data
simulations will verify this assumption a posteriori. This approximation
renders the contribution of the second term on the right-hand side of Eq. (20)
negligibly small and leads to

$D_{e}(\theta)\approx D^{0}_{e}.$ (21)

The above expression provides an approximately constant blob size along the
circular contour.

The total number of monomers in the ring is related to $R_{s}$ and $D^{0}_{e}$
as

$N\approx\frac{R_{s}}{2}\int^{2\pi}_{0}\rho(\theta)d\theta=\frac{\pi
R_{s}(D_{e}^{0})^{\frac{1-\nu}{\nu}}}{b^{1/\nu}}.$ (22)

In the above equation, the density $\rho(\theta)$ is given by

$\rho(\theta)=\frac{g_{e}(\theta)}{D_{e}(\theta)}\approx\frac{(D_{e}^{0})^{\frac{1-\nu}{\nu}}}{b^{1/\nu}}\approx\frac{N}{\pi
R_{s}}.$ (23)

From Eq. (22), the mean size of a cyclic PE chain is obtained as

$R_{s}\approx\pi b^{1/\nu}(D_{e}^{0})^{\frac{\nu-1}{\nu}}N\approx bBN.$ (24)

Notably, the above scaling relationship takes the form of Eq. (7), and the
influence of solvent quality is captured by the prefactor $B$ as in Eq. (6).
Importantly, due to the equal blob sizes of cyclic chains, Eq. (24) does not
incorporate any logarithmic corrections to the scaling as in the case of
linear PE chain (cf. Eq. (16)).

###  3.2 Comparison between theory and MD simulations

We performed MD simulations of PEs exploring two different molecular
architectures: linear chains and cyclic structures. In simulations, we
systematically change the number of monomers per PE chain (i.e.,
$N=124,244,304$, and $604$). Each system is studied under a range of Bjerrum
lengths ($0.125\leq l_{B}\leq 27$), encompassing both weak and strong
electrostatic coupling to examine the equilibrium conformational behaviour of
the PEs across these extremes.

####  3.2.1 Equilibrium size of linear and cyclic PEs

We begin our analyses by focusing on the radius of gyration $R_{g}$ of linear
and cyclic PE chains. In Fig. 2a, $R_{g}$ values calculated by averaging
simulation trajectories are plotted as a function of polymerization degree $N$
for various values of the Bjerrum length, $l_{B}$. Regardless of $l_{B}$,
increasing $N$ leads to larger molecular sizes, but linear chains are up to
three times larger than cyclic chains except for the completely collapsed
structures at $l_{B}/\sigma\gg 1$. For values of Bjerrum lengths of around
$l_{B}/\sigma\lesssim 7$, both linear and cyclic chains adopt stretched
conformations as can be seen also in Fig. 3, where we display the
representative MD snapshots of PEs for both types of topologies. In Fig. 2a,
we also plot the scaling law predictions for chain sizes given in Eqs. (16)
and (24) in the stretched regime. For a fair comparison, we first plot the
scaling prediction for cyclic chains (i.e., $R_{g}\propto N$) and then
multiply this prediction by the logarithmic correction term to describe the
size of linear chains (i.e., $R_{g}\propto N[\ln{N}]^{1/3}$). The logarithmic
correction terms, absent in the case of cyclic chains, serve to account for
the more elongated configurations that linear chains can adopt. At the strong
electrostatic coupling limit (i.e., $l_{B}/\sigma\gg 1$), both linear and
cyclic chains are collapsed (Fig. 3), and their sizes are statistically
indistinguishable. In this regime, the chains dimensions are characterized by
a scaling law of globular polymers $R_{g}\propto N^{1/3}$ 45 (see also Fig.
3). In the intermediate range of electrostatic couplings (i.e., for $7\lesssim
l_{B}/\sigma\lesssim 9$ ) the scaling exponent in the power law $R_{g}\propto
N^{\nu_{\rm eff}}$ smoothly crosses-over from value $\nu_{\rm eff}=1$ to
$1/3$. In this regime both topologies adapt pearl-necklace conformations (cf.
Fig. 3).

In Fig. 2b, the data given in Fig. 2a are plotted as a function of $l_{B}$ for
two values of $N$. These data are in agreement with the results obtained in
the previous numerical studies for linear PE chains11, 56. Overall, we observe
a non-monotonic behavior of the chain sizes, regardless of topology. As
$l_{B}\rightarrow 0$, the chains shrink in size due to weak electrostatic
coupling and resulting weaker repulsion between backbone charges. In the
stretched regime (i.e., for $l_{B}/\sigma\approx 1$) the majority of
counterions are distanced from the chain, resulting in a negligible
condensation effect and extended conformations for both types of
architectures. A notable distinction between linear and cyclic PEs becomes
evident in this regime. Linear PEs adopt one-dimensional rod-like
conformations, whereas cyclic PEs take on a two-dimensional, ring-like
structure that cover a broad surface area (see the snapshots in Fig. 3). For
$l_{B}\rightarrow\infty$, the chains are strongly collapsed and take a
globular conformation. In the globular regime, a great majority of the
counterions condense on the chain (cf. Fig. 3). For the intermediate values of
$l_{B}$ (i.e., for $7\lesssim l_{B}/\sigma\lesssim 9$), an overall
conformation is rather a combination of collapsed and stretched portions of
the linear chains. Such conformations are reminiscent of the transition
between the weak and strong coupling regimes and termed as pearl-necklace
conformation in the literature 45, 57, 43, 42, 58, (see also Fig. 3). This
transition takes place as a result of the partial condensation of counterions
onto the backbone, while the rest of the counterions become delocalized away
from the chain. The transformation is also evident in Fig. 2a and b, where the
fractal dimension characterizing the chain sizes undergoes changes in response
to variations in the Bjerrum length. Hence, our simulations validate that such
a transition takes place even when the chain backbone is in a good solvent
condition.

Figure 2:  a) The mean radius of gyration $R_{g}$ for linear (squares) and
cyclic PEs for stretched, necklace and globule-like conformational regimes
plotted as a function of degree of polymerization $N$ for different values of
Bjerrum length $l_{B}$ as indicated. Dashed lines represent theoretical
scaling laws in stretched ($R_{g}\propto N(\ln{N})^{1/3}$ for linear and
$R_{g}\propto N$ for cyclic architectures) and globule ($R_{g}\propto
N^{1/3}$) conformations. b) The variation of the rescaled mean-square radius
of gyration $R_{g}^{2}/N$ of linear (squares) and cyclic (circles) PEs with
$l_{B}$. The data are displayed for different values of $N$ as indicated.

While linear chains take a necklace conformation as predicted previously at
the intermediate values of Bjerrum length 45, (cf. Fig. 3), for cyclic chains,
we also observe a qualitatively similar behavior. However, conformational
fluctuations of cyclic chains are stronger than those observed for linear
chains. While we systematically and visually observe the necklace formation
for linear chains, cyclic chains exhibit a transition between necklace to
weakly collapsed conformations as can be seen in the snapshots in Fig. 3. The
collapsed states transiently emerge as multiple sub-globular sections
approaching each other and forming a more crumbled chain conformation as
compared to linear PEs. A visual inspection of simulation trajectories show
that this transition is observed for all of our chain sizes and simulation
replicas.

To quantitatively characterize the conformational fluctuations of chains in
different conformational regimes, we calculate probability distribution
functions of $R_{g}$ for linear and ring chains for three values of Bjerrum
length (Fig. 4). For both stretched and globule-like regimes, the distribution
functions are of a narrow width (Fig. 4). Nevertheless, cyclic chains exhibit
a slightly higher peak as compared to the linear chains. On the contrary, in
the intermediate coupling regime where necklace structures are observed, there
is an increase in the width of the distributions compared to the weak and
strong coupling regimes for both topologies. However, cyclic chains display a
slightly broader distribution, suggesting larger conformational fluctuations,
even though there is a stronger peak observed for linear chains. This analysis
is consistent with the visual observations made in the simulation snapshots
displayed in Fig. 3.

Figure 3:  Molecular dynamics snapshots of linear (top panel) and cyclic
(bottom panel) PEs displayed for different conformational regimes observed at
different intervals of Bjerrum lenght $l_{B}$. The degree of polymerization of
all PEs is $N=304$. PE monomers are depicted as green beads, whereas
counterions are presented in silver.

Figure 4:  Probability distribution functions $\rho_{s}$ of the radius of
gyration $\tilde{R}_{g}$ of linear (squares) and cyclic PEs normalized by its
root mean-square value $R_{g}$ and plotted for different conformational
regimes observed at different values of Bjerrum lenght $l_{B}$. In panel data
for a) stretched conformations ($l_{B}/\sigma=1$) are depicted. Panel b) shows
data for globular conformations ($l_{B}/\sigma=27$), while panel c) data for
necklace conformations ($l_{B}/\sigma=7.5$). In panel c) arrows indicate MD
snapshots of cyclic PEs. The middle snapshot shows conformation with size
comparable to the average $\approx R_{g}$. The left and right snapshots
correspond to the cases where molecular sizes are respectively smaller than
$R_{g}$ (crumpled conformation) and larger than $R_{g}$ (expanded
conformation). In all plots the degree of polymerization $N$ was fixed to
$N=304$.

In the previous sections, we described scaling model predictions for the mean
size of linear (Eq. (16)) and cyclic (Eq. (24)) PE chains at weak
electrostatic couplings. In Fig. 5, we compare the scaling predictions by
considering the Bjerrum length $l_{B}/\sigma\leq 2$, corresponding to a
regime, in which chains exist in stretched conformations. To construct the
scaling plot, we calculate the average electrostatic blob size $D_{e}$ using
the projected monomer density distributions along the linear direction,
$\rho(z)$, and the angular direction, $\rho(\theta)$. The detailed analysis of
monomer densities $\rho(z)$ and $\rho(\theta)$ are provided in the subsequent
subsection. The obtained mean blob sizes for both types of architectures are
listed in Table 1. We observe a strong concurrence between the theoretical
predictions and the simulation results. The data for linear and cyclic chains
collapse onto its own universal curve. Notably, the rescaled $y$-axes of the
data for linear chains (Fig. 5a) indirectly validates the predicted non-
uniform stretching along the chain, whereas the data for cyclic chains (Fig.
5b) indicates a uniform chain stretching. This observation highlights the
distinct behavior of the two architectures at the weak electrostatic coupling
regime. Overall, our simulations show that cyclic PE chains exhibit a distinct
conformational behavior, particularly from low to intermediate ionic strengths
corresponding to a Bjerrum length range of around $1<l_{B}/\sigma<7$. In the
next subsection, we characterize these differences by introducing further
geometric quantities characterizing the shape of PE in solution.

Figure 5:  The rescaled mean: a) end-to-end distance $R_{e}$ for linear PEs (squares) and b) spanning radius $R_{s}$ for cyclic PEs (circles) plotted as a function of degree of polymerization $N$ for different values of Bjerrum length in the stretched regime as indicated. The lines represent theoretical scaling laws predicted for linear (Eq. (16)) and for cyclic (Eq. (24)) architectures.  |  | $D_{e}/\sigma$
---|---|---
$N$ | $l_{B}/\sigma$ | linear | cyclic
124 | 0.5 | 1.84 | 1.58
244 | 0.5 | 1.73 | 1.52
304 | 0.5 | 1.60 | 1.50
604 | 0.5 | 1.63 | 1.49
124 | 1.0 | 1.73 | 1.49
244 | 1.0 | 1.66 | 1.45
304 | 1.0 | 1.67 | 1.44
604 | 1.0 | 1.58 | 1.40
124 | 2.0 | 1.54 | 1.59
244 | 2.0 | 1.50 | 1.57
304 | 2.0 | 1.51 | 1.52
604 | 2.0 | 1.49 | 1.52
|  |  |
Table 1: Mean values of electrostatic blob size $D_{e}$ for linear and cyclic
PEs obtained for different values of degree of polymerization $N$ and Bjerrum
length $l_{B}$.

####  3.2.2 Shape of linear and cyclic PEs

Figure 6:  a) Asphericity $A$ and b) prolatness $P$ shape factors for linear
(squares) and cyclic (circles) PEs plotted as a function of the Bjerrum lenght
$l_{B}$. The data are displayed for different degrees of polymerization $N$ as
indicated. In panel b), arrows are used to indicate molecular dynamics
snapshots of cyclic molecules, displaying their characteristic shapes in
different conformational regimes, namely the stretched, necklace, and globule-
like conformations.

Our simulations show that cyclic PE chains exhibit distinct conformational
fluctuations as the Bjerrum length is varied (Fig. 3). These conformational
fluctuations can affect the average shape of the molecules. Nevertheless,
characteristic chain sizes fall short in analyzing such geometrical shape
details. Alternatively, the average shape of a polymer chain can be described
by the average asphericity $A$ and the average prolateness $P$ 59, 60, 61. The
quantity $A$ measures a degree to which a polymer conformation deviates from a
perfect sphere. The parameter $P$ does provide information about the
flattening of a molecule 61, 62.

The average asphericity can be defined as 59, 60, 61

$A=\frac{1}{6}\left\langle\sum_{i=1}^{3}\frac{(\lambda_{i}-\bar{\lambda})^{2}}{\bar{\lambda}^{2}}\right\rangle,$
(25)

where $\langle\ldots\rangle$ denotes ensemble average, $\lambda_{i}$ (for
$i=1,..,3$) are the eigenvalues of the gyration tensor and
$\bar{\lambda}=(\lambda_{1}+\lambda_{2}+\lambda_{3})/3$ is the average
eigenvalue. The parameter $A$ is limited between the values of 0 and 1 and
takes the minimum value for spherically symmetric configurations
($\lambda_{1}=\lambda_{2}=\lambda_{3}$) and the maximum value for rod-like
configurations ($\lambda_{1}\neq 0$, $\lambda_{2}=\lambda_{3}=0$). For planar
symmetric objects, $A$ converges to the value of $1/4$ 60.

The parameter $P$ is defined as61, 62

$P=\left\langle\prod_{i=1}^{3}\frac{(\lambda_{i}-\bar{\lambda})}{\bar{\lambda}^{3}}\right\rangle.$
(26)

The values of $P$ are limited between the value of $-1/4$ for perfectly
oblate, disk-like shaped configurations ($\lambda_{1}=\lambda_{2}$,
$\lambda_{3}=0$) and the maximum value of $2$ for fully prolate molecular
configuration elongated along one axis ($\lambda_{1}\neq 0$,
$\lambda_{2}=\lambda_{3}=0$). In general, molecules with $P>0$ adapt prolate
conformations ($\lambda_{1}\gg\lambda_{2}\approx\lambda_{3}=0$), whereas
molecules with $P<0$ adapt oblate conformations
($\lambda_{1}\approx\lambda_{2}\gg\lambda_{3}=0$). The case of $P=0$ indicates
an ideally spherical shape. In other words, $P=0$ signals that the molecule is
perfectly symmetrical in all directions, with no elongation or flattening
along any axis.

In Fig. 6 we analyze the shape factors of linear and cyclic PEs by plotting
$A$ and $P$ as a function of $l_{B}$ for chains with different $N$. It is
found that an increase in electrostatic coupling induces a conformational
transition in linear PEs, shifting them from a rod-like shape (with
approximate values of $A\approx 1$ and $P\approx 2$) to a spherical shape
(with $A\approx P\approx 0$). Cyclic PEs exhibit distinct conformations in the
stretched and necklace regime observed up to $l_{B}/\sigma<9$. This is evident
from the asphericity parameter $A\approx 1/4$ which signals formation of
planar (two-dimensional) configurations. Furthermore, there is an observed
increase in the $P$ parameter with $l_{B}$ up to $l_{B}/\sigma<9$ suggesting a
conformational transition in the shape of cyclic PEs (see snapshots in Fig.
6b). In the stretched regime cyclic PEs adapt nearly perfectly oblate (e.g.,
$P\approx-1/4$ for $l_{B}/\sigma=1$) shapes. In the necklace regime cyclic PEs
become slightly prolate (e.g., $P\approx 1/4$ for $l_{B}/\sigma\approx 7)$ and
exhibit wavy shape due to formation of localized globular structures.
Similarly to linear PEs, cyclic molecules tend to take a spherical shape
characterized by $A\approx P\approx 0$ for larger values of $l_{B}$. In
summary, while shape transitions occur in both linear and cyclic chains as the
Bjerrum length increases, cyclic chains exhibit unique, non-monotonic shape
transitions, as clearly indicated by the asphericity and prolateness analyses
(cf. Fig. 6).

####  3.2.3 Monomeric distributions and pair correlation functions of linear
and cyclic PEs

Figure 7:  a) Projection of monomer density $\rho(z)$ of linear PEs on the
direction of the end-to-end vector $R_{e}$ plotted from the position of a
middle monomer. The solid lines are the best fit to Eq. (15). b) Angular
monomer density $\rho(\theta)$ of cyclic PEs. The dashed line represents the
density calculated from Eq. (23) for $N=604$. In both panels the symbols are
results of MD simulations for linear (squares) and cyclic (circles) PEs with
different degree of polymerization $N$ as indicated. All the data are obtained
for fixed Bjerrum lenght $l_{B}/\sigma=1$.

In the previous subsections, we present equations for the projected and
angular monomeric densities of linear and cyclic chains respectively (i.e.,
Eqs. (15) and 23). Accordingly, while linear chains exhibit a non-uniform
monomeric distribution due to non-uniform stretching, cyclic chains show a
rather uniform density profile. In Fig. 7a and b, we compare these predictions
with our MD results, namely the projection of monomer density, $\rho(z)$, and
the angular density $\rho(\theta)$, for linear (squares) and cyclic (circles)
PEs for various $N$. We restrict our analysis to the stretched regime (i.e.
for $l_{B}/\sigma=1$), for which we have analytical equations. The
corresponding theoretical predictions provided by Eqs. (15) (solid lines) and
(23) (dashed line) are also shown in Fig. 7. Overall, the fitting lines
exhibit a high level of agreement with the MD data. The $\rho(z)$ profiles are
in qualitative accordance with the findings of Ref.44, which studied linear
PEs in $\theta$ solutions. In general, Fig. 7a reveals that $\rho(z)$
increases with the distance $z$ from the chain center, indicating a nonuniform
stretching of linear PEs towards their ends. Considering that the monomer
density $\rho(z)$ is proportional to the electrostatic blob size $D_{e}(z)$
(cf. Eq. (15)), the data in Fig. 7a indicate the variation of $D_{e}(z)$ along
the direction of chain stretching. In the case of cyclic PEs, the variation of
blob size is absent. As displayed in Fig. 7 b) the monomer density
$\rho(\theta)$ remains approximately constant along the circular contour,
which suggests a uniform stretching of charged cyclic chains.

Next, we analyze the intrachain monomer-monomer correlation function
$g_{\rm{intra}}$(r), which is directly related to the probability of finding a
pair of monomers on the same chain separated by the distance $r$. This
quantity can further provide insights into internal polymer structure at
length scales smaller than the chain size. The $g_{\rm{intra}}$ function is
defined as follows:

$g_{\rm{intra}}(\vec{r})=\frac{1}{cN}\sum_{i\neq
j}\langle\delta(\vec{r}-\vec{r}_{ij})\rangle$ (27)

where $\langle\ldots\rangle$ denotes ensemble averaging. To calculate $g^{\rm
C}_{\rm{intra}}$ for cyclic PEs, we again consider their conformations under
the assumption of the strong stretching limit. Consequently, the fluctuations
of monomer density along the circular contour are negligible in comparison to
the average angular density $\rho(\theta)$. With this approximation, Eq. (27)
can be reformulated as:

$g^{\rm{C}}_{\rm{intra}}(\vec{r})=\frac{1}{cN}\frac{R_{s}}{2}\int_{0}^{2\pi}\,d\theta\int_{0}^{2\pi}\,d\theta^{\prime}\,\rho(\theta)\rho(\theta^{\prime})\left\langle\delta\left(\vec{r}-R_{s}\sin\left(\frac{\theta-\theta^{\prime}}{2}\right)\vec{e}\right)\right\rangle_{\rm
o},$ (28)

where $\vec{e}$ is a unit vector along a line that connects two points on the
rings and $\langle\ldots\rangle_{\rm o}$ represents averaging over all
possible orientations of the vector. As a result of this averaging procedure,
the following equation is obtained:

$g^{\rm{C}}_{\rm{intra}}(r)=\frac{R_{s}}{4\pi
r^{2}cN}\int_{0}^{2\pi}\,d\theta\,\rho(\theta)\rho(\theta+2\arcsin{[r/R_{s}]})$
(29)

The expression given by Eq. (29) can be further simplified using constant
monomer density $\rho(\theta)=N/(\pi R_{s})$ along the ring:

$g^{\rm
C}_{\rm{intra}}(r)=\frac{N(2\pi-2\arcsin{[r/R_{s}]})}{2\pi^{2}r^{2}cR_{s}}\theta(2\pi-2\arcsin{[r/R_{s}]}),$
(30)

where $\theta(x)$ is the Heaviside step function with the following
properties: $\theta(x)=1$ for $x\geq 0$ and $\theta(x)=0$ for $x<0$. We note
that the correlation function $g^{\rm L}_{\rm{intra}}$ for linear PEs in
$\theta$ solvent was calculated using the strong limit approximation in Ref.
44, and it takes the following analytical form:

$g^{\rm L}_{\rm{intra}}(r)=\frac{1}{2\pi
cNr^{2}}\int^{R_{e}/2}_{-R_{e}/2}dz\rho(z)\rho(z+r)=\frac{N}{2\pi
cr^{2}}\frac{R_{e}-r}{R^{2}_{e}}\theta(R_{e}-r).$ (31)

Figure 8:  The intra-chain monomer-monomer correlation functions
$g_{\text{intra}}(r)$ for: a) linear (squares) and b) cyclic (circles)
polyeletrolytes displayed for different degree of polymerization $N$ as
indicated. All pair correlations functions were rescaled by the overall
monomer concentration $c$. The lines represent theoretical scaling laws
predicted for linear (Eq. (31)) and for cyclic (Eq. (30)) architectures. All
the data are displayed for fixed Bjerrum lenght $l_{B}/\sigma=1$.

In Fig. 8, a comparison is presented between the intra-chain pair correlation
functions obtained through analytical calculations (lines) and MD data
(symbols) for linear and cyclic architectures with various values of $N$. In
both cases, the functions $g_{\rm intra}(r)$ exhibit a scaling law and decay
as $\propto r^{-2}$ for distances $r\ll R_{e}$ in the reference to linear PEs
(solid line), and for distances $r\ll R_{s}$ in the case of cyclic PEs (dashed
line). As noted above the average monomer density $\rho(z)$ for nonuniformly
stretched linear PEs varies logarithmically along the elongation axis (cf. Eq.
(15)). The summation in $g_{\rm intra}(r)$ (as seen in Eq. (27)) encompasses
all possible monomer pairs. Consequently, this leads to an additional
averaging of the monomer density $\rho(z)$ along the chain’s stretching
direction, where both ends and middle sections contribute to $g^{\rm L}_{\rm
intra}(r)$ (cf. Eq. (31)). As a result of this averaging of the logarithmic
function, $g^{\rm L}_{\rm intra}(r)$ closely resembles the correlation
function of a uniformly stretched chain, as previously observed for linear PEs
in $\theta$ solution. 44.

The primary distinction between the intra-chain functions for linear and
cyclic architectures is the non-monotonic behavior of $g_{\rm intra}^{C}(r)$,
which becomes evident at distances comparable to $R_{s}$. This distinct
property, a hump in $g_{\rm intra}^{C}(r)$, does not solely arise due to the
absence of chain ends, but also originates from the shape of cyclic PEs. These
molecules exhibit a slightly oval shape, as evidenced by the eigenvalue ratio
$\langle\lambda_{1}\rangle/\langle\lambda_{2}\rangle\approx 1.2$. This
deviation from a perfectly symmetric ring conformation, where
$\langle\lambda_{1}\rangle/\langle\lambda_{2}\rangle=1$, results in a non-
monotonic correlation function as displayed in Fig. 8b. It is worth noting
that when the architecture is arranged regularly on a planar circle, the
corresponding intra-chain correlation function lacks a pronounced hump and
resembles the linear correlation function $g^{\rm L}_{\rm intra}(r)$ shown in
Fig. 8a.

####  3.2.4 Structure factors of linear and cyclic PEs

To allow direct comparison between experiments and our results, we calculate
the structure factors for linear and cyclic topologies. The form factors,
$S(q)$, for all three electrostatic coupling regimes and various degrees of
polymerization are shown in Fig. 9. In particular, at low and intermediate
Bjerrum lengths, we observe a striking difference between linear and cyclic
form factors. For linear PEs (squares) in the stretched regime, as $q$
increases $S(q)$ exhibits a power-law decay with a slope of $-1$, which is
typical of rod-like molecules (Fig. 9a). This indicates that the chain is
stretched at all length scales up to a blob size $D_{e}$, which contains few
monomers. The simulation data for linear PE chains were also accurately
aligned with the theoretical prediction of the form factor $S_{\rm rod}(q)$
for rod-like molecules outlined in Eq. (A1) using a parameter-free fitting
approach.

In the stretched regime, cyclic PEs (circles) also exhibit a power-law decay
in $S(q)$ at high values of $q$. However, their behavior in the intermediate
range of $q$ is different, as shown in Fig. 9a. Specifically, cyclic chains
display sinusoidal undulations superimposed upon the underlying power-law
decay, characterized by a slope of $-1$. These undulations were fitted to the
theoretical prediction for the form factor, $S_{\rm rigid\,ring}(q)$, of a
rigid ring (see Eq. (A2)) without involving any adjustable parameters. It is
worth noting that similar undulation patterns were observed in $S(q)$ of
semiflexible and rigid neutral cyclic polymers in recent theoretical and
experimental studies 63, 64, 65.

The undulations observed at the weak coupling regimes become weaker as the PE
chains enter their necklace regimes at the intermediate values of the Bjerrum
length (i.e., $7\lesssim l_{B}/\sigma\lesssim 9$ (Fig. 9b). As the
electrostatic coupling increases, chains become more flexible due to
condensation of counterions on their backbones. At large $q$ values, the chain
remains stiff up to sections composed of several monomers, which results in a
slope of $-1$. At intermediate $q$ values, chains become coiled which is
indicated by a transition to a slope of 1/$\nu_{\rm eff}$, where $\nu_{\rm
eff}\approx 0.6$ at $l_{B}/\sigma=7.5$.

In the strong coupling regimes, for which $l_{B}/\sigma\gg 1$, the difference
between linear and cyclic chains vanishes since both chains exist in highly
collapsed conformation (Fig. 9c). At the large wave-vector limits, we observe
a strong decay with an exponent $-4$. This strong decay reflects the presence
of well-defined interfaces and boundaries within the sample 66. The form
factors for both topologies were accurately aligned through parameterless
fitting to the theoretical structure factor, $S_{\rm sphere}(q)$, of a
homogeneous sphere (cf. Eq. (A3)).

Figure 9:  Form factors $S(q)$ of linear PEs (squares) and cyclic PEs
(circles) for different conformational regimes observed at varying values of
the Bjerrum length $l_{B}$: a) stretched ($l_{B}/\sigma=1$), b) necklace
($l_{B}/\sigma=7.5$) and c) globule ($l_{B}/\sigma=27$). In panel a), the
abscissa is rescaled by the root-mean-square end-to-end distance $R_{e}$ for
linear chains and by the root means-square spanning distance $R_{s}$ for
cyclic chains. In panels b) and c), the abscissa is rescaled by the root mean-
square radius of gyration $R_{g}$ for both molecular architectures. The lines
represent theoretical scaling laws $S(q)\propto q^{-\alpha}$, where $\alpha$
is the scaling exponent (see text for details). The data are displayed for
various degrees of polymerization $N$ as indicated.

##  4 Conclusions

In this study, we apply scaling arguments that were originally formulated for
linear PE chains to construct a theoretical framework. This framework allows
us to estimate equilibrium chain sizes and monomer distributions for cyclic PE
chains, which we subsequently compare to the outcomes of our coarse-grained
molecular dynamics simulations. Our analysis demonstrates that, to a certain
extent, charged cyclic polymers exhibit conformational properties that are
similar to those of linear PE chains across a range of electrostatic coupling
regimes, encompassing weak, intermediate, and strong electrostatic
interactions (corresponding to small, intermediate, and large Bjerrum
lengths). As a result, in these three regimes, we observe stretched, pearl-
necklace, and globular-like conformations for both types of topologies.
Nevertheless, the resemblance between linear and cyclic architectures
diminishes when examining their shapes, the chain length dependence of their
characteristic sizes and structure factors, particularly in the weak and
intermediate coupling regimes. Notably, two metrics that describe the shape of
PE chains, namely asphericity and prolatness, exhibit substantial differences
between cyclic and linear chains. Specifically, they display a non-monotonic
dependence upon increasing Bjerrum length for cyclic chains, whereas for
linear chains, they exhibit a consistent, monotonous decrease as the
electrostatic strength is increased. In the stretched regime, the shape of
cyclic PEs resembles a flat, two-dimensional ring-like conformation with a
substantial surface area. In contrast, linear PEs adopt nearly rod-like, one-
dimensional conformations. Our analysis also reveals that cyclic PE chains are
much more compact in the stretched and necklace regimes as compared to linear
PEs. Furthermore, the logarithmic correction term present in the scaling law
describing the dependence of the average size of linear PEs on the degree of
polymerization is absent in the case of cyclic chains due to their circular
geometry. This distinction could potentially contribute to the unique
conformational behaviors observed in cyclic chains. Specifically, the cyclic
topology gives rise to undulations resulting from structures forming over
several monomers or more, as evidenced by the structure factor analysis. In
conclusion, our study demonstrates that long-range electrostatic interactions
can exert a significant influence on the conformational behavior of cyclic
chains, highlighting the need for further experimental investigations in the
future.

##  Appendix A Appendix

The form factor of a rigid rod of length $R_{e}$ is67

$S_{\rm
rod}(q)=\frac{2}{qR_{e}}\left(\int_{0}^{qR_{e}}\frac{\sin{x}}{x}dx-2\frac{\sin^{2}{[qR_{e}/2]}}{qR_{e}}\right).$
(A1)

The form factor of a rigid ring of diameter $R_{s}$ reads68, 69

$S_{\rm
rigid\,ring}(q)=\int_{0}^{\pi/2}d\varphi[J_{0}(R_{s}q\sin{\varphi})]^{2}\sin{\varphi},$
(A2)

where $J_{0}(x)$ is the Bessel function of the first kind with order 0.
The form factor of a sphere with radius of gyration $R_{g}$ is given by70

$S_{\rm
sphere}(q)=\left[\frac{3}{(\sqrt{5/3}qR_{g})^{3}}(\sin{[\sqrt{5/3}qR_{g}]}-\sqrt{5/3}qR_{g}\cos{[\sqrt{5/3}qR_{g}}])\right]^{2}.$
(A3)

This work has been supported by the National Science Center, Poland (Grant
Sonata Bis No. 2018/30/E/ST3/00428). We thank M. Lang for fruitful
discussions. We are also grateful to PL-Grid Infrastructure for a generous
grant of computing time.

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