Current address: ]Department of Physics, College of Science and Humanities-
Shaqra, Shaqra University, Riyadh, Saudi Arabia

# Osmotic Swelling Behavior of Surface-Charged Ionic Microgels

Mohammed O. Alziyadi  [  Department of Physics, North Dakota State University,
Fargo, ND 58108-6050, USA  Alan R. Denton<EMAIL_ADDRESS>Department of
Physics, North Dakota State University, Fargo, ND 58108-6050, USA

###### Abstract

In recent years, ionic microgels have garnered much attention due to their
unique properties, especially their stimulus-sensitive swelling behavior. The
tunable response of these soft, permeable, compressible, charged colloidal
particles is increasingly attractive for applications in medicine and
biotechnologies, such as controlled drug delivery, tissue engineering, and
biosensing. The ability to model and predict variation of the osmotic pressure
of a single microgel with respect to changes in particle properties and
environmental conditions proves vital to such applications. In this work, we
apply both nonlinear Poisson-Boltzmann theory and molecular dynamics
simulation to ionic microgels (macroions) in the cell model to compute density
profiles of microions (counterions, coions), single-microgel osmotic pressure,
and equilibrium swelling ratios of spherical microgels whose fixed charge is
confined to the macroion surface. The basis of our approach is an exact
theorem that relates the electrostatic component of the osmotic pressure to
the microion density profiles. Close agreement between theory and simulation
serves as a consistency check to validate our approach. We predict that
surface-charged microgels progressively deswell with increasing microgel
concentration, starting well below close packing, and with increasing salt
concentration, in qualitative agreement with experiments. Comparison with
previous results for microgels with fixed charge uniformly distributed over
their volume demonstrates that surface-charged microgels deswell more rapidly
than volume-charged microgels. We conclude that swelling behavior of ionic
microgels in solution is sensitive to the distribution of fixed charge within
the polymer-network gel and strongly depends on bulk concentrations of both
microgels and salt ions.

##  I Introduction

Microgels are soft colloidal particles, made of a cross-linked polymer network
gel, that become swollen in a good solvent Fernández-Nieves  _et al._ (2011);
Lyon and Serpe (2012); Karg  _et al._ (2019); Scotti  _et al._ (2022).
Depending on the procedure for their chemical synthesis Pelton and Hoare
(2011), microgels can be designed to swell/deswell in response to external
parameters, such as temperature, $p$H, and ionic strength Sierra-Martin  _et
al._ (2011). Swelling behavior depends on mechanical (elastic) properties of
the gel and polymer-solvent interactions Flory (1953); Wyss  _et al._ (2011).
When microgels are dispersed in a polar solvent (e.g., water), dissociation of
counterions from the polymer chains can turn the particles into colloidal
macroions with a fixed (immobile) charge Fernández-Nieves  _et al._ (2003);
Likos (2011). Swelling behavior of ionic microgels is influenced by
electrostatic interactions between fixed charge, counterions, and coions (from
added salt) Romeo  _et al._ (2012). Microgels and, in particular, ionic
microgels have been extensively investigated due to their applications in
biomedical, food, pharmaceutical, and petroleum industries Pashkovski (2011);
Malmsten (2011); Ben  _et al._ (2011); Tavakoli and Tang (2017); Hamidi  _et
al._ (2008).

The swelling behavior of microgels, in contrast to that of bulk gels Ohmine
and Tanaka (1982); Ricka and Tanaka (1984); Siegel and Firestone (1988);
Barrat  _et al._ (1992), is fundamentally determined by the single-particle
osmotic pressure, defined as the difference in pressure between the interior
and exterior regions Denton and Tang (2016); Denton and Alziyadi (2019). In
thermodynamic equilibrium, this single-particle osmotic pressure must vanish,
achieved only through a delicate balance between electrostatic, elastic, and
mixing entropic contributions. In concentrated suspensions of microgels,
equilibrium particle sizes respond also to self-crowding, as each particle
adjusts its internal degrees of freedom to the local environment, with
relevance for macroscopic phase behavior.

Swelling properties of microgels (ionic and nonionic) have been investigated
by several complementary experimental methods, including light scattering
Fernández-Nieves  _et al._ (2000); Mohanty and Richtering (2008); Holmqvist
_et al._ (2012); Braibanti  _et al._ (2016), neutron scattering Stieger  _et
al._ (2004); Liétor-Santos  _et al._ (2010); Nöjd  _et al._ (2018); Zhou  _et
al._ (2023), osmometry Pelaez-Fernandez  _et al._ (2015); Scotti  _et al._
(2021), and dielectric spectroscopy Mohanty  _et al._ (2016). Compressible
microgels and suspensions thereof have been modeled by a variety of
theoretical methods, including mean-field theories Borrega  _et al._ (1999);
Cloitre  _et al._ (2003); Levin (2002); Riest  _et al._ (2012); Colla  _et
al._ (2014), integral-equation theories Moncho-Jordá (2013), effective
interaction theories Denton (2003); Gottwald  _et al._ (2005); Weyer and
Denton (2018); Aguirre-Manzo  _et al._ (2019), and cell model-based theories
Denton and Tang (2016); Denton and Alziyadi (2019); Brito  _et al._ (2019).
Simulation studies have applied various molecular dynamics Hedrick  _et al._
(2015); Kobayashi and Winkler (2014); Ghavami and Winkler (2017); Mohanty  _et
al._ (2014); Gnan  _et al._ (2017) and Monte Carlo Schneider and Linse (2002);
Quesada-Pérez and Martín-Molina (2013); Hofzumahaus  _et al._ (2018); Urich
and Denton (2016); Weyer and Denton (2018) methods. From past work, swelling
behavior is known to depend sensitively on single-particle properties,
including the distribution of cross-links in the polymer network comprising
the gel Mohanty  _et al._ (2012); Boon and Schurtenberger (2017). In general,
the lower the cross-link concentration, the softer and more compressible the
particles, although the spatial distribution of cross-links is also relevant.

Less well understood is the connection between swelling and the distribution
of fixed charge on the polymer network. In chemical synthesis protocols
developed to produce ionic microgels, the fixed charge distribution can be
controlled, to some extent, by adjusting the concentration and chemical
species of the initiator for the polymerization reaction in the synthesis. In
some microgel systems, initiator molecules themselves are charged and tend to
deposit near the particle’s periphery, yielding a surface-localized fixed
charge Scotti  _et al._ (2016, 2017); Gasser  _et al._ (2019); Scotti  _et
al._ (2021); Zhou  _et al._ (2012). How in detail the fixed charge
distribution affects osmotic pressure and equilibrium swelling of ionic
microgels is a largely unresolved question.

To address the question of how fixed charge distribution influences swelling
of microgels, we have developed theoretical and computational modeling methods
for ionic microgels and applied them to macroions with fixed charge confined
to their surface. Comparison of predictions for this idealized model with
corresponding results from our earlier studies of microgels with fixed charged
uniformly distributed throughout their volume Denton and Tang (2016); Denton
and Alziyadi (2019) contrasts the sensitivity of swelling to fixed charge
distribution between two extreme distributions. Our approach is based on
fundamental statistical mechanical relations for the electrostatic component
of the single-microgel osmotic pressure that accurately account for counterion
and coion correlations. Building on our previous work, this paper describes
the swelling of ionic microgels as a competition between electrostatic and gel
contributions to the osmotic pressure and demonstrates the sensitivity of
swelling to the fixed charge distribution.

In Sec. II, we define the primitive and cell models of ionic microgels, which
are the foundation for our study. In Sec. III, we review our theory of
microgel swelling, deriving the electrostatic component of the single-microgel
osmotic pressure from a partition function and approximating the gel
contribution using the Flory-Rehner theory of polymer networks. Section IV
outlines our computational implementation of the swelling theory via Poisson-
Boltzmann theory and molecular dynamics simulation. Section V presents
numerical results for counterion density profiles, single-microgel osmotic
pressure, and equilibrium swelling ratios of surface-charged microgels. We
compare and contrast the swelling behavior with that of volume-charged
microgels and discuss the influence of added salt. Finally, Sec. VI summarizes
our results and concludes with suggestions for future work.

##  II Models

###  II.1 Primitive Model

Theoretical descriptions of polyelectrolyte solutions and charge-stabilized
colloidal suspensions are often based on the primitive model, which
approximates the solvent as an implicit medium – a dielectric continuum
characterized entirely by a dielectric constant $\epsilon$ – and considers
explicitly only the charged species Deserno and Holm (2001); Denton (2014).
Within this coarse-grained model, we consider an aqueous suspension of $N_{m}$
ionic microgels, each composed of a microscopic cross-linked network of
polymer chains, dispersed in a solvent (water) of volume $V$ at temperature
$T$, as depicted in Fig. 1.

We model ionic microgels as spherical macroions of dry radius $a_{0}$, swollen
radius $a$, and fixed charge $-Ze$, resulting from dissociation of $Z$
monovalent counterions (of charge $e$) from the polymer chains. The swollen
microgels are assumed to be permeable to solvent, counterions, and coions (see
Fig. 1). Assuming that a microgel in its dry (unswollen) state comprises
randomly close-packed spherical monomers, the number of monomers per microgel
is related to the monomer radius $r_{\rm mon}$ by $N_{\rm
mon}=0.63(a_{0}/r_{\rm mon})^{3}$.

In experiments, the distribution of fixed charge can be controlled, to some
extent, via the chemical synthesis protocol. In some systems, e.g., poly-N-
isopropilacrylamide (pNIPAM) microgels, the fixed charges originate from the
initiator species for the polymerization reaction in the synthesis, which are
believed to end up localized near the particle surface Scotti  _et al._ (2016,
2017); Gasser  _et al._ (2019); Zhou  _et al._ (2012). We consider here the
ideal case that the fixed charge is strictly confined to and uniformly
distributed over the particle surface, with number density described by a
Dirac delta-function,

$n_{f}(r)=\frac{Z}{4\pi a^{2}}\,\delta(r-a),$ (1)

where $Z$ is the microgel valence and $r$ is the radial distance from the
microgel center. Elsewhere Wypysek  _et al._ (2019); Alziyadi and Denton
(2021), we investigated swelling of ionic microcapsules, whose fixed charge is
spread throughout a spherical shell of a hollow microgel. In contrast, the
microgels studied here are not hollow, but are completely filled with
hydrogel. The microions are modeled as monovalent point charges of valence
$z_{\pm}=\pm 1$. In Donnan equilibrium with a 1:1 electrolyte (salt)
reservoir, the suspension contains $N_{s}$ dissociated salt ion pairs, which
contribute equal numbers of coions and additional counterions to the solution.
Electroneutrality of the suspension dictates the total numbers of counterions
and coions as $N_{+}=Z+N_{s}$ and $N_{-}=N_{s}$, respectively, for a total of
$N_{\mu}=Z+2N_{s}$ microions.

Figure 1: Schematic drawings of (top) primitive model of a suspension of
spherical ionic microgels (large cyan spheres) and microions (small red and
blue spheres) dispersed in a dielectric continuum solvent and (bottom) cell
model of a surface-charged microgel of swollen radius $a$ confined to a
spherical cell of radius $R$ with counterions (red) and coions (blue).

###  II.2 Cell Model

The cell model, originally proposed by Wigner and Seitz to calculate
electronic properties of solids Ashcroft and Mermin (1976); Kittel (2004), can
also be applied to polyelectrolyte solutions, as first recognized by Marcus
Marcus (1955), charge-stabilized colloidal suspensions Deserno and Holm
(2001), and hydrogels Claudio  _et al._ (2009). In this context, the cell
model reduces a system of many macroions dispersed in an electrolyte solution
to a single macroion confined to a cell of like shape, along with
stoichiometric numbers of counterions and coions (Fig. 1). This relatively
simple model, by explicitly including microions, retains microion-microion and
microion-macroion interactions, but sacrifices macroion-macroion interactions
and correlations, aside from a mean-field contribution to the total
electrostatic energy.

In the cell model of an ionic microgel suspension, a swollen, spherical
microgel (macroion) of radius $a$, with radially symmetric fixed charge
density $n_{f}(r)$, is centered in a spherical cell of radius $R$, which
determines the suspension volume fraction $\phi=(a/R)^{3}$. From spherical
symmetry, the electrostatic potential and microion densities $n_{\pm}(r)$
depend on only the radial distance $r$ from the center of the cell. Further,
electroneutrality requires that the electric field vanishes at the cell
boundary ($r=R$).

The model system is governed by a Hamiltonian $H$, which we assume can be
separated Weyer and Denton (2018), according to

$H=H_{g}+H_{e},$ (2)

into a “gel” Hamiltonian $H_{g}$, associated with mutual interactions among
monomers of the cross-linked polymer chains and the solvent molecules, and an
“electrostatic” Hamiltonian $H_{e}$, which includes the Coulomb energies of
interaction between ions (fixed and mobile). The gel Hamiltonian incorporates
elastic and mixing degrees of freedom of a polymer network in solution. The
electrostatic Hamiltonian further decomposes, according to

$H_{e}=U_{m}(a)+U_{m\mu}(\\{{\bf r}\\};a)+U_{\mu\mu}(\\{{\bf r}\\}),$ (3)

whose three terms account, respectively, for the electrostatic self-energy of
a microgel and macroion-microion and microion-microion pair interaction
energies, which depend on the microion coordinates, $\\{{{\bf r}}\\}=\\{{\bf
r}_{1},...,{\bf r}_{N_{\mu}}\\}$. Note that the first two terms on the right
side of Eq. (3) are functions of the microgel radius, and thus vary with
swelling of the polymer network.

A surface-charged spherical macroion has electrostatic self-energy, in thermal
($k_{B}T$) units,

$\beta U_{m}(a)=Z^{2}\frac{\lambda_{B}}{2a},$ (4)

where the Bjerrum length, $\lambda_{B}\equiv\beta e^{2}/\epsilon$, is the
distance at which two elementary charges $e$ have a Coulomb energy equal to
the thermal energy $k_{B}T\equiv 1/\beta$. The microgel-microion interaction
energy can be expressed as

$U_{m\mu}(\\{{\bf r}\\};a)=\sum_{i=1}^{N_{+}}v_{m+}({\bf
r}_{i};a)+\sum_{i=1}^{N_{-}}v_{m-}({\bf r}_{i};a),$ (5)

where $v_{m\pm}$ are the macroion-counterion and macroion-coion electrostatic
pair potentials:

$\beta v_{m\pm}(r;a)=\left\\{\begin{array}[]{cc}\vspace{2mm}\mp
Zz\lambda_{B}/r,&\hskip 5.69054ptr>a\\\ \vspace{2mm}\mp
Zz\lambda_{B}/a,&\hskip 5.69054ptr\leq a.\\\ \end{array}\right.$ (6)

The fact that the macroion-microion pair potentials depend on the microgel
radius $a$ has direct implications for swelling of ionic microgels, as
discussed below in Sec. III. In passing, we note that, in the full primitive
model of a suspension of many microgels, the electrostatic Hamiltonian would
include, in addition to the three terms in Eq. (3), a fourth term accounting
for microgel-microgel electrostatic interactions,

$U_{mm}(a)=\sum_{i<j=1}^{N_{m}}v_{mm}(r_{ij};a).$ (7)

For spherical microgels, however, the dependence of the bare Coulomb microgel-
microgel pair potential $v_{mm}(r;a)$ on microgel radius arises only when the
microgels overlap. For other shapes of microgels, pair interactions do not
have this symmetry property, and would depend on the orientations of the
particles at all separations.

In the cell model, microgel-microgel electrostatic interactions are implicitly
accounted for through the boundary condition that the electric field vanishes
at the cell edge. The focus on a single microgel of course necessitates
neglect of microgel-microgel correlations. In an alternative approach,
microion degrees of freedom are averaged over (traced out of the partition
function), reducing the multi-component mixture to an effective one-component
model of pseudo-macroions that interact via an effective screened-Coulomb
(Yukawa) pair potential Denton (2003); Gottwald  _et al._ (2005); Weyer and
Denton (2018). In this multi-centered model, effective pair interactions
influence swelling at densities well below close packing, corresponding to
nearest-neighbor separations comparable to the Debye screening length.
Remarkably, as shown in ref. Weyer and Denton (2018) for volume-charged
microgels, the cell and one-component models predict very similar swelling
behavior up to close-packing densities.

##  III Theory of Microgel Swelling

Swelling of microgels is governed by the single-microgel osmotic pressure
$\pi_{m}$, defined as the difference in pressure between the interior and
exterior of the microgel. This single-microgel property is to be distinguished
from the osmotic pressure of a microgel suspension $\pi_{s}$, which is a bulk
(macroscopic) property. In a canonical ensemble description, in which the
suspension is assumed to be closed (i.e., has a fixed number of particles),
the single-microgel osmotic pressure is thermodynamically defined as a
derivative of the Helmholtz free energy per microgel $F$ with respect to the
microgel volume $v_{m}=4\pi a^{3}/3$. Within the spherical cell model,

$\pi_{m}=-\left(\frac{\partial F}{\partial
v_{m}}\right)_{N_{\mu},R,T}=-\frac{1}{4\pi a^{2}}\left(\frac{\partial
F}{\partial a}\right)_{N_{\mu},R,T},$ (8)

revealing the connection between $\pi_{m}$ and the variation of the free
energy with swollen radius. In contrast, the suspension osmotic pressure
depends on the variation of the free energy with cell volume or radius,

$\pi_{s}=-\left(\frac{\partial F}{\partial
V}\right)_{N_{\mu},T}=-\frac{1}{4\pi R^{2}}\left(\frac{\partial F}{\partial
R}\right)_{N_{\mu},T}.$ (9)

The cell theorem Marcus (1955); Wennerström  _et al._ (1982) further relates
$\pi_{s}$ to the total microion density at the cell boundary:

$\beta\pi_{s}=n_{+}(R)+n_{-}(R).$ (10)

In passing, we note that the results derived below can be equally well
obtained within the semi-grand canonical ensemble, in which the suspension is
free to exchange microions with a salt reservoir, which describes the
condition of Donnan equilibrium Denton and Tang (2016); Denton and Alziyadi
(2019).

Separation of the Hamiltonian into electrostatic and gel terms [Eq. (2)]
implies that the canonical partition function ${\cal Z}$ factorizes, according
to ${\cal Z}={\cal Z}_{e}{\cal Z}_{g}$, into electrostatic and gel factors.
Within the spherical cell model, the electrostatic partition function takes
the form

${\cal
Z}_{e}(N_{\mu},a,R,T)\propto\prod_{i=1}^{N_{\mu}}\int_{0}^{R}dr_{i}\,r_{i}^{2}e^{-\beta
H_{e}},$ (11)

where the position integrals cover possible configurations of all microions
within the cell. Factorization of the partition function implies decomposition
of the free energy,

$F=-k_{B}T\ln{\cal Z}=F_{g}+F_{e},$ (12)

into (1) a gel contribution, $F_{g}=-k_{B}T\ln{\cal Z}_{g}$, associated with
short-range monomer-monomer interactions, mixing entropy of the polymer-
solvent mixture, and conformational entropy of the polymer chains, and (2) an
electrostatic contribution, $F_{e}=-k_{B}T\ln{\cal Z}_{e}$, due to long-range
(Coulomb) ion-ion interactions. Correspondingly, the single-microgel osmotic
pressure separates, via

$\pi_{m}(\alpha)=\pi_{g}(\alpha)+\pi_{e}(\alpha),$ (13)

into gel and electrostatic contributions, $\pi_{g}(\alpha)$ and
$\pi_{e}(\alpha)$, both being functions of the microgel linear swelling ratio
$\alpha\equiv a/a_{0}$.

In thermodynamic equilibrium, a microgel swells until its osmotic pressure
vanishes, i.e., $\pi_{m}(\alpha)=0$. As we showed in previous work Denton and
Alziyadi (2019), this condition ensures continuity of the radial component of
the total pressure tensor across the microgel surface in equilibrium. Note
that the individual contributions to $\pi_{m}$ may be nonzero, as long as
their sum vanishes in equilibrium.

As an approximation for the gel contribution to the free energy, we adopt the
mean-field Flory-Rehner theory of polymer networks Flory (1953); Flory and
Rehner (1943a, b), which combines polymer-solvent interactions and mixing
entropy with elastic free energy of the network. For a microgel of swollen
radius $a$ and dry radius $a_{0}$, composed of $N_{\rm mon}$ monomers and
$N_{\rm ch}$ distinct chains (cross-linked into a network) in a solvent, the
gel free energy takes the form

$\displaystyle\beta F_{g}(\alpha)$ $\displaystyle=$ $\displaystyle N_{\rm
mon}[(\alpha^{3}-1)\ln(1-\alpha^{-3})+\chi(1-\alpha^{-3})]$ (14)
$\displaystyle+$ $\displaystyle\frac{3}{2}N_{\rm ch}(\alpha^{2}-\ln\alpha-1),$

where $\chi$ is the Flory solvency parameter, associated with polymer-polymer,
polymer-solvent, and solvent-solvent interactions. The corresponding gel
contribution to the single-microgel osmotic pressure can be written as

$\displaystyle\beta\pi_{g}(\alpha)v_{m}$ $\displaystyle=$ $\displaystyle-
N_{\rm mon}[\alpha^{3}\ln(1-\alpha^{-3})+\chi\alpha^{-3}+1]$ (15)
$\displaystyle-$ $\displaystyle N_{\rm ch}(\alpha^{2}-1/2).$

The electrostatic contribution to the single-microgel osmotic pressure can be
derived in principle from the electrostatic contribution to the free energy
via Colla  _et al._ (2014)

$\pi_{e}=-\frac{1}{4\pi a^{2}}\left(\frac{\partial F_{e}}{\partial
a}\right)_{N_{\mu},R,T}.$ (16)

However, this expression is only as accurate as the approximation for the free
energy. Alternatively, from Eqs. (4) and (5), $\pi_{e}$ can be expressed as

$\pi_{e}=-\frac{1}{4\pi a^{2}}\left(\frac{\partial U_{m}(a)}{\partial
a}+\left\langle\frac{\partial U_{m\mu}(a)}{\partial
a}\right\rangle\right)_{N_{\mu},R,T},$ (17)

where angular brackets denote an ensemble average over configurations of the
microions. Applying the latter approach to ionic microgels requires specifying
a model for the fixed charge distribution. Assuming a surface-charged microgel
and using the appropriate macroion-microion interaction potentials [Eq. (7)],
the second term on the right side of Eq. (17) takes the explicit form

$\beta\left\langle\frac{\partial U_{m\mu}(a)}{\partial
a}\right\rangle=\frac{Z\lambda_{B}}{a^{2}}\left\langle\sum_{i(r_{i}\leq
a)}z_{i}\right\rangle,$ (18)

where the sum over $i$ ($i=\pm$ for counterions or coions) includes only
microions inside the microgel. Finally, substituting the self energy [Eq. (4)]
and the expression from Eq. (18) into Eq. (17), we obtain the electrostatic
component of the single-microgel osmotic pressure:

$\beta\pi_{e}v_{m}=\frac{Z\lambda_{B}}{3a}\left(\frac{Z}{2}-\left\langle{N_{+}}\right\rangle+\left\langle{N_{-}}\right\rangle\right),$
(19)

where

$\left\langle N_{\pm}\right\rangle=4\pi\int_{0}^{a}dr\,r^{2}n_{\pm}(r)$ (20)

represent mean numbers of counterions and coions inside the microgel, ensemble
averaged over microion configurations Denton and Alziyadi (2019). The first
term on the right side of Eq. (19) derives from the microgel self-energy,
while the remaining terms are associated with the microion distribution.
Equations (13), (15), and (19), combined with the condition
$\pi_{m}(\alpha)=0$, determine a microgel’s equilibrium swollen size. In
previous work Denton and Alziyadi (2019), we showed that our expression for
the electrostatic component of the single-microgel osmotic pressure in the
cell model is equivalent to the jump at the microgel surface in the radial
component of the electrostatic pressure tensor Trizac and Hansen (1997); Widom
_et al._ (2009).

Note that Eq. (19), akin to the cell theorem for the suspension osmotic
pressure [Eq. (10)], is an exact theorem within the cell model, fully
accounting for microion correlations. In multi-center models of microgel
suspensions, these theorems cease to be exact, as they neglect osmotic
pressure contributions from microgel-microgel pair interactions and
correlations. Nevertheless, since these pair interactions depend on microgel
size only for overlapping microgels, Eq. (19) gives a reasonable approximation
for dilute suspensions, and should be accurate even for concentrations
approaching close-packing Weyer and Denton (2018).

##  IV Computational Methods

###  IV.1 Poisson-Boltzmann Theory in the Cell Model

Practical implementation of the theory described in Sec. III to model
equilibrium swelling of ionic microgels requires estimates for the mean
numbers of counterions and coions inside a microgel. Within the cell model, a
convenient method for computing $\left\langle N_{\pm}\right\rangle$ is
Poisson-Boltzmann (PB) theory. Based on a mean-field approximation that
neglects microion-microion correlations, this theory can be rigorously
derived, e.g., via calculus of variations Denton and Alziyadi (2019) and
classical density-functional theory Löwen  _et al._ (1992).

Defining $\psi(r)\equiv\beta e\phi(r)$ as the reduced (dimensionless) form of
the electrostatic potential $\phi(r)$, where $r$ is the radial distance from
the center of the cell, the Poisson equation can be expressed as

$\nabla^{2}\psi(r)=-4\pi\lambda_{B}[n_{+}(r)-n_{-}(r)-n_{f}(r)],$ (21)

where the right-hand side includes the total charge density, including the
mobile microions and the fixed charge on the microgel. For a suspension in
Donnan equilibrium with a salt reservoir of average salt density $n_{0}$, the
mean-field Boltzmann approximation for the microion equilibrium densities
takes the form

$n_{\pm}(r)=n_{0}\,\text{exp}[\mp\psi(r)].$ (22)

Combining Eqs. (21) and (22) yields the (nonlinear) Poisson-Boltzmann
equation:

$\psi^{\prime\prime}(r)+\frac{2}{r}\psi^{\prime}(r)=\kappa^{2}\sinh\psi(r)+4\pi\lambda_{B}n_{f}(r),$
(23)

where $\kappa\equiv\sqrt{8\pi\lambda_{B}n_{0}}$ is the Debye screening
constant. For a closed, salt-free suspension, with fixed counterion density,
the PB equation takes the alternate form

$\psi^{\prime\prime}(r)+\frac{2}{r}\psi^{\prime}(r)=-\tilde{\kappa}^{2}\exp[-\psi(r)]+4\pi\lambda_{B}n_{f}(r),$
(24)

where $\tilde{\kappa}\equiv\sqrt{4\pi\lambda_{B}\tilde{n}_{+}}$ is the
effective screening constant within the suspension and $\tilde{n}_{+}$ is the
counterion number density at $\psi=0$.

In the case of a microgel with fixed charge confined to its surface, we
substitute Eq. (1) into Eq. (23) to obtain

$\psi^{\prime\prime}(r)+\frac{2}{r}\psi^{\prime}(r)=\kappa^{2}\sinh\psi(r)+\frac{Z\lambda_{B}}{a^{2}}\,\delta(r-a),$
(25)

which is the PB equation in the spherical cell model for a suspension of
surface-charged microgels in Donnan equilibrium. This equation is subject to
the boundary conditions that the electric field must vanish at the center of
the cell, $\psi^{\prime}(0)=0$ (by symmetry), and on the cell boundary,
$\psi^{\prime}(R)=0$ (by electroneutrality). Applying these boundary
conditions, we numerically solved Eq. (25) for $\psi(r)$ inside and outside
the microgel and matched the solutions in the two regions at the microgel
surface to ensure continuity of the potential and the correct discontinuity of
the electric field at $r=a$:

$\lim_{\delta\to
0}\left[\psi^{\prime}(a+\delta)-\psi^{\prime}(a-\delta)\right]=\frac{Z\lambda_{B}}{a^{2}}.$
(26)

From our solution for $\psi(r)$, we computed the microion density profiles
from Eq. (22) and the electrostatic component of the single-microgel osmotic
pressure from Eqs. (19) and (20).

Previously, we showed Denton and Alziyadi (2019) that a careful analysis of
the pressure tensor within the cell model implementation of PB theory relates
the electrostatic component of the single-microgel osmotic pressure to the
discontinuity in the radial component of the electrostatic pressure tensor at
the microgel surface:

$\beta\pi_{e}=\frac{Z}{8\pi a^{2}}\lim_{\delta\to
0}\left|\psi^{\prime}(a+\delta)+\psi^{\prime}(a-\delta)\right|.$ (27)

We further showed that this expression is equivalent to our exact statistical
mechanical relation [Eq. (19)].

###  IV.2 Molecular Dynamics Simulations

An independent means of computing the microion density distributions is
molecular simulation. To validate our PB theory predictions, we implemented
and performed canonical ensemble molecular dynamics (MD) simulations to
compute $\left\langle N_{\pm}\right\rangle$ within the cell model. Using the
LAMMPS package lam ; Plimpton (1995), we simulated a collection of counterions
(and coions), interacting via a hybrid pair potential, combining a long-range
Coulomb pair potential with a short-range, repulsive, cut-off Lennard-Jones
pair potential. To mimic the influence of a central, spherical ionic microgel,
we applied a radial “external” force Denton and Tang (2016),

$F_{\rm
ext}(r)=\mp\frac{Ze^{2}}{4\pi\epsilon}\left\\{\begin{array}[]{l@{\quad}l}r/a^{3},&r\leq
a,\\\\[8.61108pt] 1/r^{2},&r>a,\end{array}\right.$ (28)

which is attractive/repulsive ($\mp$) for counterions/coions.

The microions were initialized on the sites of an FCC lattice with random
velocities consistent with a constant temperature, and confined to a spherical
region by a repulsive Lennard-Jones wall force. During the simulations,
trajectories of the mobile microions were computed by numerically integrating
Newton’s equations of motion using the velocity-Verlet algorithm. The average
temperature was held constant via a Nosé-Hoover thermostat. After
equilibrating the system for $10^{6}$ time steps (fs), we continued each run
for another $10^{8}$ steps, during which time we collected statistics.

In all MD simulations and PB calculations, the following system parameters
were kept fixed: Bjerrum length $\lambda_{B}$, cell radius $R$, microgel dry
radius $a_{0}$, cross-link fraction, dry volume fraction $\phi_{0}$, valence
$Z$, and Flory solvency parameter $\chi$. For a given value of $Z$, the total
number of dissociated monovalent counterions was $N_{+}=Z$ to maintain
electroneutrality. Figure 2 shows a typical snapshot from one of our
simulations of an ionic microgel and counterions.

Figure 2: Snapshot from MD simulation of an ionic microgel (blue sphere) and
mobile counterions (red spheres).

##  V Results and Discussion

To explore the dependence of equilibrium swelling behavior of ionic microgels
on fixed charge distribution, we implemented the models, theory, and
computational methods described in Secs. II-IV. We consider here aqueous
suspensions at temperature $T=293$ K, with a uniform Bjerrum length,
$\lambda_{B}=0.714$ nm, assuming equal dielectric constants inside and outside
of the microgels. As representative system parameters, we chose a dry microgel
radius of $a_{0}=10$ nm, corresponding to $N_{\rm mon}=2\times 10^{5}$
monomers of radius $r_{\rm mon}\simeq 0.15$ nm; number of chains $N_{\rm
ch}=100$, corresponding to average cross-link fraction $x=0.5N_{\rm ch}/N_{\rm
mon}=2.5\times 10^{-4}$; valence $Z=1000$; and Flory solvency parameter
$\chi=0.5$.

In previous work Denton and Tang (2016), we developed a similar modeling
approach and applied it to spherical ionic microgels carrying fixed charge
uniformly distributed over their volume. We numerically solved the PB equation
and performed MD simulations to compute average microion number density
profiles, and thus average numbers of interior microions $\left\langle
N_{\pm}\right\rangle$. From the resulting electrostatic component of the
single-microgel osmotic pressure $\pi_{e}$, we determined equilibrium swelling
ratios. Here, we apply a comparable approach to microgels with fixed charge
uniformly distributed over their surfaces.

Figure 3 shows our results for average counterion number density profiles
inside and outside of a surface-charged microgel of swollen radius $a=25$ nm
(swelling ratio $\alpha=2.5$) in the spherical cell model with no salt.
Excellent agreement between predictions of PB theory and data from our MD
simulations over a range of dry volume fractions serves to validate our
implementations of both methods. Consistent with the boundary conditions on
the PB equation for the electrostatic potential, the counterion density
profiles are relatively flat near the microgel center ($r=0$) and the cell
edge ($r=R$), where the electric field vanishes. Furthermore, the counterion
density peaks at the surface of the microgel, where the electric field is
highest. The cusp in the $n_{+}(r)$ curve at $r=a$ is a consequence of the
strict confinement of the fixed charge to the particle surface.

With increasing dry volume fraction, the maxima of the $n_{+}(r)$ curves (at
$r=a$) increase. This trend is to be expected, since the counterion density
closely tracks the electrostatic potential [Eq. (22)], whose magnitude
$|\psi(r)|$ attains a maximum at the surface and whose surface value
$|\psi(a)|$ increases with increasing $\phi_{0}$. This increase in peak height
reflects a stronger attraction of counterions to the surface at higher
microgel volume fractions. Furthermore, the average number of interior
counterions $\left\langle N_{+}\right\rangle$ [Eq. (20)] also increases with
increasing $\phi_{0}$, a trend that results largely from the corresponding
decrease in exterior volume. Interestingly, inside and away from the microgel
surface, the counterion density profile changes relatively little. This
behavior is expected, as the environment of the interior counterions is weakly
affected by a change in volume outside of the microgel as $\phi_{0}$ changes.
Note that at low volume fractions (dilute conditions) the counterion density
near the microgel center is higher than at the cell edge, while at higher
volume fractions this relation is reversed. In previous work Wypysek  _et al._
(2019); Alziyadi and Denton (2021), we found that distributing the fixed
charge over the shell thickness of a spherical microcapsule smooths and
broadens the counterion density profile.

Figure 3: Counterion number density profiles around a surface-charged ionic
microgel of dry radius $a_{0}=10$ nm, swollen radius $a=25$ nm, and valence
$Z=1000$ in a salt-free aqueous solvent at room temperature
($\lambda_{B}=0.714$) in the spherical cell model. For dry volume fractions
$\phi_{0}=0.01$, 0.02, and 0.03, predictions of PB theory (solid curves) agree
closely with MD simulation data (symbols). The corresponding swollen volume
fractions are $\phi=0.15625$, 0.3125, and 0.46875.

As discussed in Sec. IV, the electrostatic component of the single-microgel
osmotic pressure $\pi_{e}$ involves contributions both from the self-energy of
the microgel fixed charge and from the density distributions of the mobile
microions. Numerically integrating the microion density profiles over the
volume of the swollen microgel yields the average numbers of interior
microions $\left\langle N_{\pm}\right\rangle$ [Eq. (20)]. In the salt-free
case, substituting the average number of interior counterions $\left\langle
N_{+}\right\rangle$ into Eq. (19) gives $\pi_{e}$ for a microgel of a given
swollen radius. Repeating these calculations for different swollen radii, we
computed $\pi_{e}$ over a range of microgel swelling ratios and dry volume
fractions. Figure 4 shows our results for $\pi_{e}$ vs. $\alpha$ for several
values of $\phi_{0}$, together with the gel component of the single-microgel
osmotic pressure [Eq. (15], which is independent of $\phi_{0}$. Again, our
predictions from PB theory are in very close agreement with our simulation
data, as is to be expected, since $\pi_{e}$ is directly determined by the
value of $\left\langle N_{+}\right\rangle$.

For a given microgel swollen radius, as the dry volume fraction increases
(from $\phi_{0}=0.01$ to 0.05), the electrostatic component of the single-
microgel osmotic pressure steadily decreases (red curves and symbols in Fig.
4). This trend can be attributed to the progressive neutralization of ionic
microgels with increasing concentration of the suspension, as counterions
become increasingly confined within the microgel, i.e., as $\left\langle
N_{+}\right\rangle$ increases. (Note that in the cell model, increasing
$\phi_{0}$ implies decreasing cell radius $R$.) Furthermore, for these system
parameters, at a given concentration, $\pi_{e}$ also monotonically decreases
with increasing swelling ratio (i.e., with increasing swollen radius for a
fixed cell radius). At sufficiently large $\alpha$, the self-energy and
counterion contributions to the electrostatic osmotic pressure balance and
$\pi_{e}=0$. Thus, concentrating the suspension (increasing $\phi_{0}$) or
enlarging the microgels (increasing $\alpha$) weakens electrostatic effects
(lowering $\pi_{e}$), which tends to promote deswelling and expulsion of
solvent. Conversely, decreasing $\phi_{0}$ or $\alpha$ enhances electrostatic
effects, which favors swelling and absorption of solvent.

The gel component of the single-microgel osmotic pressure $\pi_{g}$ [Eq.
(15)], associated with mixing entropy and elasticity, counters the
electrostatic component. Recall that the total single-microgel osmotic
pressure $\pi_{m}$ – the sum of the electrostatic and gel components [Eq.
(13)] – vanishes when a microgel attains its equilibrium swollen size. An
uncharged (nonionic) microgel is in equilibrium when $\pi_{g}$ (dashed black
curve in Fig. 4) itself vanishes (at $\alpha\simeq 2.3$ in this case). With
increasing swelling ratio, $\pi_{g}$ monotonically decreases, becoming
negative when $\alpha>2.3$, but with a qualitatively different $\alpha$
dependence. While a nonionic microgel would be out of equilibrium beyond this
swelling ratio, an ionic microgel can be stabilized by the electrostatic
component of the osmotic pressure, here arising from the fixed surface charge.
The red dots in Fig. 4 mark the equilibrium size of the microgel, at which
$\pi_{g}(\alpha)+\pi_{e}(\alpha)=0$, for different dry radii.

Figure 4: Electrostatic and gel components of single-microgel osmotic pressure
vs. swelling ratio $\alpha$ for surface-charged microgels in salt-free aqueous
suspensions of dry volume fraction $\phi_{0}=0.01-0.05$ in the spherical cell
model for same system parameters as in Fig. 3. The electrostatic component
$\pi_{e}$ from PB theory (solid red curves) agrees closely with MD simulation
data (open red symbols). The gel component $\pi_{g}$ (dashed black curve) is
computed from Flory-Rehner theory [Eq. (15)]. At equilibrium swelling, the
total microgel osmotic pressure vanishes: $\pi_{m}=\pi_{e}+\pi_{g}=0$ (filled
symbols).  Figure 5: Equilibrium swelling ratio $\alpha$ vs. dry volume
fraction $\phi_{0}$ for salt-free aqueous suspensions of surface-charged (red
circles) and volume-charged (blue squares) microgels in the spherical cell
model for same system parameters as in Fig. 3, illustrating stronger
deswelling of surface-charged microgels. Over this range of $\phi_{0}$, the
swollen volume fraction, $\phi=\alpha^{3}\phi_{0}$, varies (nonlinearly) from
about 0.28 to 0.67 for surface-charged microgels.

The resulting equilibrium swelling behavior is summarized in Fig. 5, which
shows progressive deswelling with increasing $\phi_{0}$. Over this range of
dry volume fraction, the swollen volume fraction, $\phi=\alpha^{3}\phi_{0}$,
varies (nonlinearly) from about 0.28 to 0.67 for surface-charged microgels,
remaining below close-packing of spheres. Thus, the deswelling of these ionic
microgels is driven mainly by electrostatic effects, rather than by steric
interparticle interactions. With increasing $\phi_{0}$, increasingly more
counterions reside, on average, inside the microgel, lowering the contribution
of microgel-counterion electrostatic interactions to the osmotic pressure [Eq.
(18)], and thus inducing deswelling.

Our predictions of progressive deswelling of ionic microgels with increasing
concentration are qualitatively consistent with experimental observations of
deswelling, e.g., of pNIPAM and poly(vinylpyridine) microgels Pelaez-Fernandez
_et al._ (2015); Scotti  _et al._ (2016, 2017); Gasser  _et al._ (2019); Zhou
_et al._ (2012); Scotti  _et al._ (2021). In particular, pNIPAM microgels,
whose fixed charge is localized near the particle periphery, similar to our
surface-charged model, exhibit such a deswelling response Scotti  _et al._
(2016, 2017); Gasser  _et al._ (2019); Zhou  _et al._ (2012). A direct
quantitative comparison between theory and experiment may be possible, but is
complicated by the challenge of precisely measuring the fixed charge and by
the neglect in our model of any variation of fixed charge with microgel
concentration.

Fernández-Nieves et al. Scotti  _et al._ (2016, 2017); Gasser  _et al._ (2019)
have suggested a model of ionic microgels in which counterions that are
closely associated with fixed charge, i.e., quasi-bound to the surface with an
electrostatic potential energy $\gtrsim k_{B}T$, may, at sufficiently high
microgel concentration, be released (i.e., become free) and thereby contribute
to the osmotic pressure of the suspension. Such a release of counterions would
alter the effective fixed charge of the microgels. Since our coarse-grained,
cell-model-based approach focuses, for simplicity, on a single microgel
carrying a fixed charge that is independent of system properties, it does not
directly describe release of quasi-bound counterions in response to changes of
concentration. A multi-center model of microgel suspensions that includes
microgel-microgel interactions may be a basis for incorporating a
concentration-dependent fixed charge, using concepts of charge renormalization
with a thermal definition of effective fixed charge Denton (2010). While such
an approach is beyond the scope of the present study, it may be worth
exploring in future. Nevertheless, our theoretical predictions and simulations
do indicate stronger aggregation of counterions near the charged microgel
surface with increasing concentration (Fig. 3), which might be related to
quasi-binding of counterions in a more molecular-scale model. Figure 3 further
shows that the counterion number density at the cell edge increases with
concentration, suggesting increasing overlap of the counterion clouds
surrounding neighboring microgels, consistent with the conceptual picture
presented in refs. Scotti  _et al._ (2016, 2017); Gasser  _et al._ (2019).

We note that our results for counterion number density profiles are
qualitatively similar to those reported in a related study of surface-charged
ionic microgels Gasser  _et al._ (2019), in which the PB equation was
numerically solved in the cell model. However, our conclusions regarding
equilibrium swelling differ significantly from those in ref. Gasser  _et al._
(2019), where the electrostatic component of the single-microgel osmotic
pressure was determined from the microion number density at the center of the
cell, implying [from Eq. (22)]

$\beta\pi_{e}=n_{+}(0)+n_{-}(0)=2n_{0}\cosh[\psi(0)].$ (29)

The latter approximation, although valid in planar geometry (e.g., for a flat
film), neglects nonuniformity of the elements of the pressure tensor in
curvilinear geometry. In spherical geometry, for example, the normal (radial)
element of the pressure tensor, $P_{m}(r)$, varies with radial distance. As
described in detail elsewhere Denton and Alziyadi (2019); Trizac and Hansen
(1997); Widom  _et al._ (2009), this property follows from the requirement
that the divergence of the pressure tensor vanish in mechanical equilibrium.
In particular, for a surface-charged microgel in the spherical cell model, the
normal element of the electrostatic pressure tensor, $P_{e}(r)$, is higher at
the microgel surface than at the center (see Fig. 14 of ref. Denton and
Alziyadi (2019)). As a consequence, Eq. (29) actually represents the normal
element of the electrostatic pressure at the microgel center, $P_{e}(0)$,
rather than the electrostatic component of the single-microgel osmotic
pressure $\pi_{e}$. Furthermore, $P_{e}(0)$ is generally significantly lower
than $\pi_{e}$.

In equilibrium, the normal element of the gel contribution to the pressure
tensor, $P_{g}(r)$, must counteract the electrostatic contribution,
$P_{e}(r)$, to mechanically stabilize the fixed charge against the electric
field ($-\psi^{\prime}$). For a fixed charge density $n_{f}(r)$, a simple
mechanical argument yields the gel contribution as Denton and Alziyadi (2019)

$P_{g}(r)=-\int_{r}^{R}du\,n_{f}(u)\psi^{\prime}(u).$ (30)

In general, $P_{g}(r)=0$ outside the microgel, where $n_{f}=0$, and is
spatially varying inside. For our particular model of a surface-charged
microgel, with $n_{f}(r)$ given explicitly by a Dirac delta-function [Eq.
(1)], Eq. (30) yields Denton and Alziyadi (2019)

$P_{g}(r)=-\frac{Z}{8\pi a^{2}}\lim_{\delta\to
0}[\psi^{\prime}(a+\delta)+\psi^{\prime}(a-\delta)]=-\pi_{e}.$ (31)

Thus, for a surface-charged microgel, $P_{g}(r)$ jumps discontinuously at the
surface from zero to a value equal to the negative of the electrostatic
component of the single-microgel osmotic pressure, and remains constant at
that value inside the microgel. In contrast, $P_{e}(r)$ and $P_{m}(r)$ vary
with radial distance due to the spherical geometry.

To illustrate the difference between our approach and that of ref. Gasser  _et
al._ (2019), Fig. 6 plots the normal element of the pressure tensor in the
spherical cell model for the case $\phi_{0}=0.01$, $\alpha=2.5$, separately
showing the electrostatic and gel contributions, computed using the methods
from ref. Denton and Alziyadi (2019). Panel (a) is for the case of microgels
with dry and swollen radii $a_{0}=10$ nm and $a=25$ nm, respectively; panel
(b) is for $a_{0}=40$ nm and $a=100$ nm, which is closer to the experimental
system studied in ref. Gasser  _et al._ (2019). The thick blue vertical line
has a length equaling our result for $\pi_{e}$, while the thick red vertical
line length represents the value predicted by Eq. (29). Evidently, the two
approaches make substantially different predictions. In fact, when we use Eq.
(29), instead of our Eq. (19) [equivalently, Eq. (27)] to compute equilibrium
swelling ratios for the case of $a_{0}=10$ nm, we find that $\alpha$ varies
only slightly with $\phi_{0}$, actually increasing from 2.355 at
$\phi_{0}=0.01$ to 2.358 at $\phi_{0}=0.05$. In contrast, our approach, which
is consistent with nonuniformity of the pressure tensor Denton and Alziyadi
(2019), predicts $\alpha$ decreases from 3.03 at $\phi=0.01$ to 2.37 at
$\phi_{0}=0.05$ for this system (Fig. 5).

Figure 6: Normal (radial) element of pressure tensor, $P_{m}(r)$, and
electrostatic and gel contributions, $P_{e}(r)$ and $P_{g}(r)$, respectively,
where $P_{m}(r)=P_{e}(r)+P_{g}(r)$, for a surface-charged microgel in the
spherical cell model with dry volume fraction $\phi_{0}=0.01$, swelling ratio
$\alpha=2.5$, and dry and swollen radii (a) $a_{0}=10$ nm, $a=25$ nm and (b)
$a_{0}=40$ nm, $a=100$ nm. Thick red and blue vertical lines have lengths
representing the value of $\pi_{e}$ predicted, respectively, by Eq. (29)
Gasser  _et al._ (2019) and our Eq. (19) or, equivalently, (27).

Of particular interest here is a direct comparison between equilibrium
swelling of surface- and volume-charged microgels. As illustrated in Fig. 5,
with increasing concentration, surface-charged microgels deswell significantly
more rapidly than volume-charged microgels that have otherwise the same
particle properties. This trend can be attributed in part to the lower self-
energy of surface-charged microgels, $\beta U_{m}=Z^{2}\lambda_{B}/(2a)$,
compared with $\beta U_{m}=3Z^{2}\lambda_{B}/(5a)$ for volume-charged
microgels Denton and Tang (2016). However, the counterion density distribution
also clearly plays a role in lowering the electrostatic osmotic pressure,
which suppresses swelling.

Intermediate between the extremes of surface- and volume-charged microgels are
microgels with the same fixed charge uniformly distributed over the volume of
a peripheral spherical shell. In previous work on ionic microcapsules (hollow
microgels) Wypysek  _et al._ (2019); Alziyadi and Denton (2021), we derived an
expression for the self-energy of a charged spherical shell with inner and
outer radii $R_{i}$ and $R_{o}$, respectively:

$\beta
U_{m}(R_{o},\gamma)=\frac{3Z^{2}\lambda_{B}}{10R_{o}}~{}\frac{2-5\gamma^{3}+3\gamma^{5}}{(1-\gamma^{3})^{2}},$
(32)

where $\gamma\equiv R_{i}/R_{o}\leq 1$. With increasing shell thickness (i.e.,
decreasing $\gamma$), $U_{m}$ from Eq. (32) increases, exceeding $U_{m}$ for a
surface-charged microgel [Eq. (4)], and approaching the result for a volume-
charged microgel as $\gamma\to 0$. Thus, spreading the fixed charged over the
volume of a spherical shell would increase electrostatic self-energy and
thereby tend to promote swelling.

Figure 7: Equilibrium swelling ratio $\alpha$ vs. dry volume fraction
$\phi_{0}$ for surface-charged microgels of valence $Z=1000$ and dry radius
$a_{0}=10$ nm in aqueous suspensions with system salt concentrations $c_{s}=0$
(red circles), 10 mM (green squares), and 20 mM (black triangles). Over this
range of $\phi_{0}$, the swollen volume fraction, $\phi=\alpha^{3}\phi_{0}$,
varies (nonlinearly) from about 0.28 to 0.67 for surface-charged microgels and
the fixed charge concentration $c_{f}$ varies from about 4 to 20 mM.

Finally, we explore the response of ionic microgels to changes in salt
concentration. For two different system salt concentrations, $c_{s}=10$ and 20
mM, we numerically solved the PB equation [Eq. (23)] for both counterion and
coion density profiles $n_{\pm}(r)$, from which we computed the electrostatic
component of the single-microgel osmotic pressure from Eq. (19) and
equilibrium swelling ratios from Eq. (13). Figure 7 shows the resulting
variation of equilibrium swelling ratio with microgel concentration for the
same system parameters as in Fig. 3. Over this range of dry volume fraction,
$0.01<\phi_{0}<0.05$, the average concentration of fixed charge,

$c_{f}=\frac{3Z}{4\pi}~{}\frac{\phi_{0}}{(a_{0}[{\rm
nm}])^{3}}~{}\frac{1~{}{\rm mM}}{6.022\times 10^{-4}~{}{\rm nm}^{-3}}~{},$
(33)

varies from about 4 to 20 mM. Since $c_{f}$, and thus also the counterion
concentration, is comparable in magnitude to the salt concentration, some
significant response of swelling to variation of $c_{s}$ may be expected. At
higher salt concentrations, the nonlinear PB equation becomes so numerically
stiff that our differential equation solver unfortunately fails. However, even
for $c_{s}\leq 20$ mM, we already observe a significant influence of added
salt on microgel swelling.

In relatively dilute suspensions ($\phi_{0}<0.03$), microgels typically
respond to increasing salt concentration by deswelling. This response is
conventionally attributed to enhanced screening of the microgel fixed charge
upon increasing the overall concentration of microions and the attendant
suppression of electrostatic-driven swelling. Another interpretation is
suggested, however, by close inspection of the expression for the
electrostatic component of the single-microgel osmotic pressure [Eq. (19)].
Adding salt ion pairs to the suspension leads on average to microgels
preferentially absorbing counterions over coions (i.e., $\left\langle
N_{+}\right\rangle$ rises more than $\left\langle N_{-}\right\rangle$). Dilute
suspensions provide ample free volume outside of the microgels in which coions
can minimize their energetically costly interactions with microgel surfaces.
With the electrostatic component of the osmotic pressure being thus weakened,
the gel component favors a lower swelling ratio.

In more concentrated suspensions ($\phi_{0}>0.03$), we observe an interesting
reversal in the swelling response, with crowded microgels swelling slightly
upon adding salt. This unusual response implies that highly crowded microgels,
which already harbor a significant concentration of counterions, tend to
engulf more of any added coions than added counterions. This trend is perhaps
not surprising, considering that in crowded suspensions, where free volume
outside of the microgels is limited, exterior coions are constrained to be
relatively close to the like-charged microgel surfaces, where they must pay a
higher energetic cost of interaction. In contrast, by moving inside a
microgel, a coion can avoid the charged surface, while also reducing energy by
gaining proximity to oppositely-charged counterions. Swelling of microgels
further favors coions remaining inside the microgels. This result illustrates
the complex nature of the electrostatic component of the single-microgel
osmotic pressure, which involves a delicate balance between the mean numbers
of counterions and coions inside the microgels.

Whether the swelling trend observed with increasing salt concentration would
continue to higher microgel concentrations is an interesting question.
Unfortunately, our cell-model approach is unable to provide a clear answer,
since $\phi_{0}=0.05$ already approaches a swollen volume fraction at which
elastic interactions between neighboring microgels would start to become
important. This question could be addressed, however, within a multi-center
model that includes microgel-microgel interactions Weyer and Denton (2018).
Finally, we note that, despite the qualitatively different swelling behavior
of ionic microgels as compared with that of bulk ionic gels, our approach has
in common with theories of bulk gels Ohmine and Tanaka (1982); Ricka and
Tanaka (1984); Siegel and Firestone (1988); Barrat  _et al._ (1992) that it
associates swelling with osmotic pressure – albeit that of a single microgel –
which is intimately related to microion distributions inside and outside of
the gel.

##  VI Conclusions

Within the primitive and cell models of permeable, compressible, ionic
microgels, we investigated the dependence of single-particle osmotic pressure
and equilibrium swelling on the distribution of fixed charge over the cross-
linked polymer network making up the particles. Applying an exact theorem for
the electrostatic component of the single-particle osmotic pressure –
implemented within Poisson-Boltzmann theory and validated via molecular
dynamics simulations – and modeling the gel component using Flory-Rehner
theory of polymer networks, we computed equilibrium swelling ratios of
microgels (with monovalent counterions) whose fixed charge is confined to and
uniformly distributed over their surfaces.

Our study demonstrates that the osmotic pressure and swelling properties of
ionic microgels are strongly dependent on the fixed charge distribution
through its influence on the microgel electrostatic self-energy and the
microion density profiles. Our results further indicate that surface-charged
microgels deswell significantly more strongly with increasing concentration
(i.e., crowding) than microgels that carry a fixed charge uniformly
distributed over their volume. Results for these two idealized models may
serve to bracket the swelling behavior to be expected of ionic microgels with
intermediate fixed charge distributions. Finally, we found that, with
increasing salt concentration, swelling of surface-charged microgels in dilute
suspensions is suppressed, due to enhanced screening of bare Coulomb
electrostatic interactions, but that the effect is weaker, or can even invert,
in more concentrated suspensions.

Although our study is restricted to suspensions of relatively idealized model
microgels with uniform cross-link densities and fixed charges evenly
distributed over their surfaces or volumes, our results vividly reveal the
importance of fixed charge distribution for single-particle osmotic pressure
and equilibrium swelling. Further increasing the salt concentration or varying
counterion and microgel valences may significantly modify electrostatically-
driven swelling. Steric interactions associated with nonzero microion size
also may be important. While such effects are beyond the scope of the present
mean-field modeling approach, they would be well worth exploring in future.

Our conclusions regarding the influence of fixed charge distribution on
swelling are qualitatively consistent with experimental measurements of ionic
microgels and can be further tested in experiments (e.g., light or neutron
scattering) on microgels produced via synthesis protocols that yield varying
distributions of fixed charge. Thus, our results may provide insights into
swelling behavior and guide practical approaches to controlling swelling of
microgels for practical applications. In future work, we will explore
structural properties of bulk suspensions of surface-charged ionic microgels
and of mixtures thereof with size and charge asymmetry.

###### Acknowledgements.

We thank Mariano Brito, Gerhard Nägele, Jan Dhont, and Andrea Scotti for
helpful discussions. MA acknowledges Shaqra University for financial support.
ARD acknowledges support of the National Science Foundation (Grant No.
DMR-1928073).  DATA AVAILABILITY The data that support the findings of this
study are available from the corresponding author upon reasonable request.

## References

  * Fernández-Nieves  _et al._ (2011) A. Fernández-Nieves, H. Wyss, J. Mattsson, and D. A. Weitz, eds.,  _Microgel Suspensions: Fundamentals and Applications_ (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2011).
  * Lyon and Serpe (2012) L. A. Lyon and M. J. Serpe, eds.,  _Hydrogel Micro and Nanoparticles_ (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2012).
  * Karg  _et al._ (2019) M. Karg, A. Pich, T. Hellweg, T. Hoare, L. A. Lyon, J. J. Crassous, D. Suzuki, R. A. Gumerov, S. Schneider, I. I. Potemkin, and W. Richtering, Langmuir 35, 6231 (2019).
  * Scotti  _et al._ (2022) A. Scotti, M. F. Schulte, C. G. Lopez, J. J. Crassous, S. Bochenek, and W. Richtering, Chem. Rev. 122, 11675 (2022).
  * Pelton and Hoare (2011) R. Pelton and T. Hoare, in  _Microgel Suspensions: Fundamentals and Applications_ , edited by A. Fernández-Nieves, H. Wyss, J. Mattsson, and D. A. Weitz (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2011) pp. 3–32.
  * Sierra-Martin  _et al._ (2011) B. Sierra-Martin, J. J. Lietor-Santos, A. Fernandez-Barbero, T. T. Nguyen, and A. Fernandez-Nieves, “Swelling thermodynamics of microgel particles,” in  _Microgel Suspensions: Fundamentals and Applications_ (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2011) pp. 71–116.
  * Flory (1953) P. J. Flory,  _Principles of Polymer Chemistry_ (Cornell University Press, Ithaca, 1953).
  * Wyss  _et al._ (2011) H. M. Wyss, J. Mattsson, T. Franke, A. Fernández-Nieves, and D. A. Weitz, in  _Microgel Suspensions: Fundamentals and Applications_ , edited by A. Fernández-Nieves, H. Wyss, J. Mattsson, and D. A. Weitz (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2011) pp. 311–326.
  * Fernández-Nieves  _et al._ (2003) A. Fernández-Nieves, A. Fernández-Barbero, B. Vincent, and F. J. de las Nieves, J. Chem. Phys. 119, 10383 (2003).
  * Likos (2011) C. N. Likos, in  _Microgel Suspensions: Fundamentals and Applications_ , edited by A. Fernández-Nieves, H. Wyss, J. Mattsson, and D. A. Weitz (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2011) pp. 165–193.
  * Romeo  _et al._ (2012) G. Romeo, L. Imperiali, J.-W. Kim, A. Fernández-Nieves, and D. A. Weitz, J. Chem. Phys. 136, 124905 (2012).
  * Pashkovski (2011) E. Pashkovski, in  _Microgel Suspensions: Fundamentals and Applications_ , edited by A. Fernández-Nieves, H. Wyss, J. Mattsson, and D. A. Weitz (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2011) pp. 423–450.
  * Malmsten (2011) M. Malmsten, in  _Microgel Suspensions: Fundamentals and Applications_ , edited by A. Fernández-Nieves, H. Wyss, J. Mattsson, and D. A. Weitz (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2011) pp. 375–405.
  * Ben  _et al._ (2011) Y. Ben, I. Robb, P. Tonmukayakul, and Q. Wang, in  _Microgel Suspensions: Fundamentals and Applications_ , edited by A. Fernández-Nieves, H. Wyss, J. Mattsson, and D. A. Weitz (Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2011) pp. 407–422.
  * Tavakoli and Tang (2017) J. Tavakoli and Y. Tang, Polymers 9, 364 (2017).
  * Hamidi  _et al._ (2008) M. Hamidi, A. Azadi, and P. Rafiei, Advanced Drug Delivery Reviews 60, 1638 (2008).
  * Ohmine and Tanaka (1982) I. Ohmine and T. Tanaka, J. Chem. Phys. 77, 5725 (1982).
  * Ricka and Tanaka (1984) J. Ricka and T. Tanaka, Macromol. 17, 2916 (1984).
  * Siegel and Firestone (1988) R. A. Siegel and B. A. Firestone, Macromol. 21, 3254 (1988).
  * Barrat  _et al._ (1992) J.-L. Barrat, J.-F. Joanny, and P. Pincus, J. Phys. II France 2, 1531 (1992).
  * Denton and Tang (2016) A. R. Denton and Q. Tang, J. Chem. Phys. 145, 164901 (2016).
  * Denton and Alziyadi (2019) A. R. Denton and M. O. Alziyadi, J. Chem. Phys. 151, 074903 (2019).
  * Fernández-Nieves  _et al._ (2000) A. Fernández-Nieves, A. Fernández-Barbero, B. Vincent, and F. J. de las Nieves, Macromol. 33, 2114 (2000).
  * Mohanty and Richtering (2008) P. S. Mohanty and W. Richtering, J. Phys. Chem. B 112, 14692 (2008).
  * Holmqvist  _et al._ (2012) P. Holmqvist, P. S. Mohanty, G. Nägele, P. Schurtenberger, and M. Heinen, Phys. Rev. Lett. 109, 048302 (2012), 117, 179901(E) (2016).
  * Braibanti  _et al._ (2016) M. Braibanti, C. Haro-Pérez, M. Quesada-Pérez, L. F. Rojas-Ochoa, and V. Trappe, Phys. Rev. E 94, 032601 (2016).
  * Stieger  _et al._ (2004) M. Stieger, W. Richtering, J. S. Pedersen, and P. Lindner, J. Chem. Phys. 120, 6197 (2004).
  * Liétor-Santos  _et al._ (2010) J. J. Liétor-Santos, U. Gasser, R. Vavrin, Z. B. Hu, and A. Fernández-Nieves, J. Chem. Phys. 133, 034901 (2010).
  * Nöjd  _et al._ (2018) S. Nöjd, P. Holmqvist, N. Boon, M. Obiols-Rabasa, P. S. Mohanty, R. Schweins, and P. Schurtenberger, Soft Matter 14, 4150 (2018).
  * Zhou  _et al._ (2023) B. Zhou, U. Gasser, and A. Fernandez-Nieves, Nat. Commun. 14, 3827 (2023).
  * Pelaez-Fernandez  _et al._ (2015) M. Pelaez-Fernandez, A. Souslov, L. A. Lyon, P. M. Goldbart, and A. Fernández-Nieves, Phys. Rev. Lett. 114, 098303 (2015).
  * Scotti  _et al._ (2021) A. Scotti, M. Pelaez-Fernandez, U. Gasser, and A. Fernandez-Nieves, Phys. Rev. E 103, 012609 (2021).
  * Mohanty  _et al._ (2016) P. S. Mohanty, S. Nöjd, M. J. Bergman, G. Nägele, S. Arrese-Igor, A. Alegria, R. Roa, P. Schurtenberger, and J. K. G. Dhont, Soft Matter 12, 9705 (2016).
  * Borrega  _et al._ (1999) R. Borrega, M. Cloitre, I. Betremieux, B. Ernst, and L. Leibler, Euro. Phys. Lett. 47, 729 (1999).
  * Cloitre  _et al._ (2003) M. Cloitre, R. Borrega, F. Monti, and L. Leibler, C. R. Physique 4, 221 (2003).
  * Levin (2002) Y. Levin, Phys. Rev. E 65, 036143 (2002).
  * Riest  _et al._ (2012) J. Riest, P. Mohanty, P. Schurtenberger, and C. N. Likos, Z. Phys. Chem. 226, 711 (2012).
  * Colla  _et al._ (2014) T. Colla, C. N. Likos, and Y. Levin, J. Chem. Phys. 141, 234902 (2014).
  * Moncho-Jordá (2013) A. Moncho-Jordá, J. Chem. Phys. 139, 064906 (2013).
  * Denton (2003) A. R. Denton, Phys. Rev. E 67, 011804 (2003), 68, 049904(E) (2003).
  * Gottwald  _et al._ (2005) D. Gottwald, C. N. Likos, G. Kahl, and H. Löwen, J. Chem. Phys. 122, 074903 (2005).
  * Weyer and Denton (2018) T. J. Weyer and A. R. Denton, Soft Matter 14, 4530 (2018).
  * Aguirre-Manzo  _et al._ (2019) L. A. Aguirre-Manzo, M. Ledesma-Motolinía, L. F. Rojas-Ochoa, V. Trappe, J. Callejas-Fernández, C. Haro-Pérez, and P. González-Mozuelos, Phys. Rev. E 100, 032602 (2019).
  * Brito  _et al._ (2019) M. E. Brito, A. R. Denton, and G. Nägele, J. Chem. Phys. 151, 224901 (2019).
  * Hedrick  _et al._ (2015) M. M. Hedrick, J. K. Chung, and A. R. Denton, J. Chem. Phys. 142, 034904 (2015).
  * Kobayashi and Winkler (2014) H. Kobayashi and R. G. Winkler, Polymers 6, 1602 (2014).
  * Ghavami and Winkler (2017) A. Ghavami and R. G. Winkler, ACS Macro Lett. 6, 721 (2017).
  * Mohanty  _et al._ (2014) P. S. Mohanty, D. Paloli, J. J. Crassous, E. Zaccarelli, and P. Schurtenberger, J. Chem. Phys. 140, 094901 (2014).
  * Gnan  _et al._ (2017) N. Gnan, L. Rovigatti, M. Bergman, and E. Zaccarelli, Macromol. 50, 8777 (2017).
  * Schneider and Linse (2002) S. Schneider and P. Linse, Eur. Phys. J. E 8, 457 (2002).
  * Quesada-Pérez and Martín-Molina (2013) M. Quesada-Pérez and A. Martín-Molina, Soft Matter 9, 7086 (2013).
  * Hofzumahaus  _et al._ (2018) C. Hofzumahaus, P. Hebbeker, and S. Schneider, Soft Matter 14, 4087 (2018).
  * Urich and Denton (2016) M. Urich and A. R. Denton, Soft Matter 12, 9086 (2016).
  * Mohanty  _et al._ (2012) P. S. Mohanty, A. Yethiraj, and P. Schurtenberger, Soft Matter 8, 10819 (2012).
  * Boon and Schurtenberger (2017) N. Boon and P. Schurtenberger, Phys. Chem. Chem. Phys. 19, 23740 (2017).
  * Scotti  _et al._ (2016) A. Scotti, U. Gasser, E. S. Herman, M. Pelaez-Fernandez, J. Han, A. Menzel, L. A. Lyon, and A. Fernández-Nieves, Proc. Nat. Acad. Sci. 113, 5576 (2016).
  * Scotti  _et al._ (2017) A. Scotti, U. Gasser, E. S. Herman, J. Han, A. Menzel, L. A. Lyon, and A. Fernández-Nieves, Phys. Rev. E 96, 032609 (2017).
  * Gasser  _et al._ (2019) U. Gasser, A. Scotti, and A. Fernández-Nieves, Phys. Rev. E 99, 042602 (2019).
  * Zhou  _et al._ (2012) J. Zhou, J. Wei, T. Ngai, L. Wang, D. Zhu, and J. Shen, Macromolecules 45, 6158 (2012).
  * Deserno and Holm (2001) M. Deserno and C. Holm, in  _Electrostatic Effects in Soft Matter and Biophysics_ , NATO Advanced Studies Institute, Series II: Mathematics Physics and Chemistry, Vol. 46, edited by C. Holm, P. Kékicheff, and R. Podgornik (Kluwer, Dordrecht, 2001) pp. 27–50.
  * Denton (2014) A. R. Denton, in  _Electrostatics of Soft and Disordered Matter_ , edited by D. S. Dean, J. Dobnikar, A. Naji, and R. Podgornik (Pan Stanford, Singapore, 2014) pp. 201–215.
  * Wypysek  _et al._ (2019) S. K. Wypysek, A. Scotti, M. O. Alziyadi, I. I. Potemkin, A. R. Denton, and W. Richtering, Macromol. Rapid Comm. 41, 1900422 (2019).
  * Alziyadi and Denton (2021) M. O. Alziyadi and A. R. Denton, J. Chem. Phys. 155, 214904 (2021).
  * Ashcroft and Mermin (1976) N. W. Ashcroft and N. D. Mermin,  _Solid State Physics_ (Holt-Saunders, Fort Worth, 1976).
  * Kittel (2004) C. Kittel,  _Introduction to Solid State Physics_ (Wiley, Hoboken, 2004).
  * Marcus (1955) R. A. Marcus, J. Chem. Phys. 23, 1057 (1955).
  * Claudio  _et al._ (2009) G. C. Claudio, K. Kremer, and C. Holm, J. Chem. Phys. 131, 094903 (2009).
  * Wennerström  _et al._ (1982) H. Wennerström, B. Jönsson, and P. Linse, J. Chem. Phys. 76, 4665 (1982).
  * Flory and Rehner (1943a) P. J. Flory and J. Rehner, J. Chem. Phys. 11, 512 (1943a).
  * Flory and Rehner (1943b) P. J. Flory and J. Rehner, J. Chem. Phys. 11, 521 (1943b).
  * Trizac and Hansen (1997) E. Trizac and J.-P. Hansen, Phys. Rev. E 56, 3137 (1997).
  * Widom  _et al._ (2009) A. Widom, J. Swain, and J. Silverberg, Phys. Rev. E 80, 016301 (2009).
  * Löwen  _et al._ (1992) H. Löwen, P. A. Madden, and J.-P. Hansen, Phys. Rev. Lett. 68, 1081 (1992).
  * (74) http://lammps.sandia.gov.
  * Plimpton (1995) S. Plimpton, J. Comp. Phys. 117, 1 (1995).
  * Denton (2010) A. R. Denton, J. Phys.: Condens. Matter 22, 364108 (2010).