# The role of charge content in the two-step deswelling of
Poly(N-isopropylacrylamide)-based microgels

Rajam Elancheliyan  Laboratoire Charles Coulomb, UMR 5221, CNRS–Université de
Montpellier, F-34095 Montpellier, France  Giovanni Del Monte  National
Research Council–Institute for Complex Systems (CNR-ISC), Sapienza University
of Rome, 00185 Rome, Italy  Edouard Chauveau  Laboratoire Charles Coulomb, UMR
5221, CNRS–Université de Montpellier, F-34095 Montpellier, France  Simona
Sennato  National Research Council–Institute for Complex Systems (CNR-ISC),
Sapienza University of Rome, 00185 Rome, Italy  Emanuela Zaccarelli  National
Research Council–Institute for Complex Systems (CNR-ISC), Sapienza University
of Rome, 00185 Rome, Italy<EMAIL_ADDRESS>Domenico Truzzolillo
Laboratoire Charles Coulomb, UMR 5221, CNRS–Université de Montpellier, F-34095
Montpellier, France<EMAIL_ADDRESS>
###### Abstract

Poly(N-isopropylacrylamide)-based microgels are soft colloids undergoing a
Volume Phase Transition (VPT) close to ambient temperature. Although widely
employed for fundamental research and application purposes, the modifications
of the microgels internal structure occurring at the VPT are not yet
completely understood, especially concerning the role of electrostatics. Here
we study in detail, both experimentally and numerically, the effect of the
addition of acrylic acid (AAc) co-monomer on the microgel deswelling process.
By combining viscosimetry, light scattering and electrophoresis, we show that
the progressive addition of AAc increases the microgel mass and suppresses the
occurrence of the VPT, progressively shifting the microgel collapse to higher
temperatures. Most importantly, it also highly enhances the two-step
deswelling of these submicron-sized networks, so that the inner core collapses
at temperatures always lower than those marking the transition of the outer
corona. These results indicate that a net increase of the charge density
mismatch between the bulk and the surface of the microgels takes place.
Numerical simulations fully confirm this scenario and clarify the impact of
the charge distribution on the two-step deswelling, with mobile counterions
efficiently screening the charges within the inner core, while leaving more
monomers ionized on the surface. Our work unambiguously shows how
electrostatic interactions influence the behavior of thermosensitive microgels
in aqueous environment close to the VPT.

###### keywords:

Poly(N-isopropylacrylamide-co-acrylic acid), Light scattering

Department of Physics, Sapienza University of Rome, 00185 Rome, Italy
Department of Physics, Sapienza University of Rome, 00185 Rome, Italy
Department of Physics, Sapienza University of Rome, 00185 Rome, Italy +39 (0)6
49913524 +33 (0)467 143589 PNIPAm, AAc, $f_{AAc}$, SLS, DLS

##  1 Introduction

Microgels are soft colloidal particles consisting of crosslinked polymer
networks whose size ranges from few tens of nanometers up to few microns. They
are widely studied for a plethora of purposes spanning many research fields,
such as biomedicine 1, 2, sensors3 and 3D bioprinting4, and they are currently
used in many industrial applications as biocompatible additives5, delivery
vehicles 6 and color-changing systems 7. Stimuli-responsive microgels have
gathered much attention due to the fact that they display a phase change or an
abrupt single-particle volume change in response to external stimuli, such as
a temperature, pH and/or ionic strength variation 8, 9, 10, and many
applications have been based on their tunability2, 11, 12, 13.

Among polymeric microgels, those made of poly(N-isopropylacrylamide) (PNIPAm)
are particularly interesting, since they undergo a Volume Phase Transition
(VPT) at ambient temperature: below a characteristic temperature, $T_{c}$
$\approx 32^{\circ}C$, these colloidal networks exist in a swollen, fully
hydrated state, while above $T_{c}$ they collapse due to their increased
hydrophobicity. PNIPAm microgels can be synthesized using standard emulsion
polymerization in aqueous media14 and their size, shape and structure have
been investigated using various methods including x-ray, light and neutron
scattering 15, 16, 17, 18, and electron or atomic force microscopy 18. The
combination of many different experimental techniques to investigate the VPT
of PNIPAm microgels made it possible to grasp a lot of information on this
phenomenon, which is one of the most studied in soft condensed matter
nowadays.

The transition temperature as well as the physical nature of the volume phase
transition can be modified by additives 19, 20, like salts 21 or surfactants
22, through changes in the solvent composition either by chemical insertion
23, e.g. by embedding charged groups within the networks24, 25, 26, or by
physical interpenetration of different polymer species 27. Despite the intense
research effort, one basic aspect of the volume phase transition of microgels
in aqueous media, namely the influence of electrostatic interactions, still
remains relatively unexplored and poorly understood.

Very recently, some of us 28 have shown that crosslinked networks of PNIPAm
homopolymers undergo a two-step volume phase transition, where the residual
charges carried by the ionic initiators, mainly located in the outer corona,
play a pivotal role. During an ascending temperature ramp, PNIPAm microgels
deswell with their initiator-free core collapsing first, namely at
temperatures lower than the initiator-rich corona. The fingerprint of such
two-step deswelling has been identified in the onset of a minimum of the ratio
$R_{g}/R_{H}$ between the gyration and the hydrodynamic radius of the
microgels as a function of temperature 28. Simulations have further shown that
this feature does not take place in the absence of charges28.

Previous works have also demonstrated that the properties of the VPT, in
particular the critical temperature $T_{c}$, can be altered by incorporating a
charged co-monomer such as Acrylic acid (AAc) 29 into the PNIPAm network. In
the presence of acrylic acid and for pH$>$pKa(AAc), the swelling/deswelling
behaviour of microgels depends on the subtle balance between hydrophobic
attraction and electrostatic repulsion due to the carboxylate groups borne by
the charged co-monomers. Introducing electric charge by co-polymerization
hence offers the possibility to embed these charges within the microgel and
not simply at one chain end, as for the case of the ionic groups carried by
the initiator.

AAc units can be incorporated in microgels following three different synthesis
strategies: i) Poly(N-isopropylacrylamide-co-acrylic acid) microgels can be
synthesized using emulsion polymerization technique 25, 30, 31, giving rise _a
priori_ to random co-polymerization; ii) random distributions of AAc units can
also be obtained by incorporating AAc monomers into PNIPAm networks in the
form of cross-linker rather than simple co-monomers32; iii) polymer
interpenetration—IPN (Interpenetrated Polymer Network) can also be employed to
produce pH responsive PNIPAm/PAAc microgels that have been widely
investigated33, 27. Among these methods, random co-polymerization (i) can be
considered as the natural extension of the more standard radical
polymerization of homopolymers towards the formation of chemically
heterogeneous polymeric colloids or polymers. Indeed, this route does not
assign any particular structural feature to the charged units, either in terms
of topology or of connectivity, with respect to the neutral monomers. However,
despite this strategy has been widely used to tune the size and the
responsiveness of microgels, its impact on the microgel structure across the
VPT remains currently unknown.

The aim of the present work is to investigate the role played by the AAc
fraction on the two-step deswelling of Poly(N-isopropylacrylamide-co-acrylic
acid) microgels. By means of viscosimetry, light scattering and
electrophoresis, we show that the presence of AAc monomers in the polymer
structure increases the microgel mass and suppresses the VPT, progressively
shifting the microgel collapse to higher temperatures. In addition, it also
largely enhances the two-step deswelling phenomenon. In order to isolate the
role played by the charge content on the volume phase transition of the
PNIPAm-co-AAc microgels, we have studied microgel suspensions in the absence
of added salts, acids and bases, with all carboxylic groups borne by the
acrylic acid monomers in their deprotonated state. In this regime a unique
continuous deswelling for ascending temperature ramps is observed in both the
hydrodynamic size and the gyration radius of the particles in agreement with
other syntheses 25, 26. To have a closer look into the mechanisms driving such
a phenomenon we combine the experiments with state-of-the-art computer
simulations using a coarse-grained bead-spring model, based on a realistic
representation of the polymer network at a monomer-resolved level. Simulations
further shed light on the role played by the total charge and its spatial
distribution around the VPT, corroborating the enhancement of the two-step
deswelling observed in experiments.

The remainder of this paper is organized as follows. In section 2 we give
details of the synthesis, the experimental techniques and the numerical
methods used to investigate single microgel structure and their VPT. In
section 3 we report both experimental and numerical results. Finally, in
Section 4 we make some concluding remarks and summarize the key results of the
work.

##  2 Materials and Methods

###  2.1 Synthesis

Poly(N-isopropylacrylamide-co-acrylic acid) microgels were synthesized using
emulsion polymerization34. 150 mL of ultra-pure water was introduced in a
250mL three-necked flask, and degassing was carried out using vacuum/argon
cycles. Vacuum was achieved by a vane pump and the argon/vacuum sequence was
repeated 6 times. Finally, argon was bubbled for 15 min. After having
completed degassing, 3 to 4 mL of the (degassed) water was withdrawn via a
syringe to dissolve the initiator (potassium peroxodisulfate, KPS – purchased
from Sigma Aldrich and used without further purification) that was added in a
later stage. Once the bubbling was stopped and the mechanical agitation was
set up (via Teflon rotating anchor), the two side necks were plugged. Sodium
Dodecyl Sulfate (SDS – purchased from Sigma Aldrich and used without further
purification) was added in the flask right before the solution was heated up
to the desired temperature ($T_{s}=70\pm 1$ ∘C). SDS/NIPAm molar ratio has
been fixed to 0.0085. The largest relative variation of the SDS mass added
during the different syntheses amounts to 5%. A relative increase of the
swollen microgel size 35 that does not exceed 4%, was expected due to such
differences and is barely detectable.

The target quantities of i) N-isopropylacrylamide (NIPAm) (from Sigma Aldrich,
used without further purification), ii) N,N-methylene-bis-acrylamide (BIS)
(from Sigma Aldrich, used without further purification) and iii) acrylic acid
(AAc) (from Sigma Aldrich and used without further purification) were
introduced in turn as reported in table (Table 1 in SM). The AAc mass content
has been chosen in order to vary its monomer fraction, namely the ratio
between the number of AAc monomers and the total number of monomers
participating to the radical polymerization, from 0% to 5% (Coded as AC0, AC1,
AC3 and AC5 in Table 1). During the heating ramp, the initiator (KPS) was
dissolved in 3-4 mL of deionized degassed water and injected slowly once the
temperature of the batch reached $T_{s}$. The mixture was left under stirring
at $T_{s}$ for 6 hours. The polymerization terminated spontaneously. All
samples have been purified via three consecutive centrifugation/supernatant
removal cycles 36, 28 and 19.5 mg (2 mM) of Sodium Azide have been added to
prevent bacterial growth. Finally we point out that the pH of all samples
prepared for viscosimetry, light scattering and electrophoresis experiments
was never less than 6. At these pH values, deprotonation of the carboxylic
groups carried by AAC monomers occurs, resulting in a net electrostatic
repulsion between the negatively charged carboxylate residues
(pKa${}^{(AAc)}\simeq$4.25)29. On the contrary, the nominal pH of the samples
during the synthesis and well before the complete polymerization, that can be
calculated on the basis of the amount of the added AAc monomers, ranges from
3.52 (5% AAc) to 3.87 (1% AAc), at which the acidic monomers are mainly
protonated (neutral).

###  2.2 Viscosimetry

We performed viscosimetry experiments to measure the colloidal volume fraction
of the microgels and to discern the effect of acrylic acid addition on the
mass of a single microgel, as detailed in section 3.

The colloidal volume fraction is defined as $\varphi$ = $n_{p}v_{p}$, where
$n_{p}$ is the number density of colloids and $v_{p}=\pi d^{3}/6$ the volume
of a single particle of diameter $d$ at infinite dilution. Experimentally,
only the concentration $c$ (mg/ml) of a (purified) suspension can be directly
measured, by weighting a small aliquot of the sample before and after removing
the solvent by evaporation. Since the microgels are highly swollen, their mass
density is essentially the same as that of the solvent; consequently, $c$ and
$\varphi$ are proportional, $\varphi=kc$. We determined the constant $k$
matching the $c$ dependence of the zero-shear viscosity of the purified
suspensions $\eta$ with the values expected from the Einstein equation37,

$\frac{\eta}{\eta_{0}}=1+2.5kc,$ (1)

where $\eta_{0}$ is the viscosity of the solvent. By fitting $\eta/\eta_{0}$
to a straight line, we obtained $k$.

Such a procedure has been repeated for all samples. The measurements were
performed at $T=20$ ∘C using an Anton Paar Lovis 2000 ME micro-viscosimeter in
the range 0.02 $mg/ml$ $<c<$ 1.6 $mg/ml$, where $\eta$ increases linearly with
$c$. To double-check that the mutual interactions between microgels did not
affect viscosity in this concentrantion range we calculated a posteriori the
ratio $d_{c}/\lambda_{D}$ between the center-to-center distance $d_{c}$
between microgels and the Debye length $\lambda_{D}$ of the suspensions taking
into consideration only ions coming from the dissociated sodium azide. We
obtained always $d_{c}/\lambda_{D}>100$, confirming that the suspended
particles interact only rarely due to collisions and their mutual interaction
do not participate to the increased viscosity.

###  2.3 Light scattering

Light scattering measurements were performed to study the structure of PNIPAm-
AAc microgels as a function of temperature. A laser source ($\lambda$ = 532
nm) was used for all the scattering measurements that were performed via an
Amtec-goniometer allowing to collect the light at scattering angles in the
range $16^{\circ}\leq\theta\leq 150^{\circ}$ corresponding to scattering wave-
vectors $4.4$ $\mu m^{-1}$ $\leq q\leq 30.3$ $\mu m^{-1}$. All scattering
experiments have been performed in dilute samples: an aliquot of each mother
batch was diluted 50 times in deionized water, which resulted in a final
microgel volume fraction $\varphi\leq 0.5\%$. The pH of the samples at 20 ∘C
ranged from 6.2 (5$\%$AAc) to 6.9 (0% AAc). In such a small slightly acidic pH
interval the size of PNIPAm-co-AAc micro- and macrogels is not remarkably
affected by pH variation 38, 39. Before each data acquisition, the samples
were thermostatted for 10 mins via a recirculating bath with 0.1 ∘C accuracy.

The hydrodynamic radius, $R_{H}$, and the size distribution of the microgels
were characterized by means of dynamic light scattering. The scattered light
intensity was collected at a fixed scattering angle ($\theta=70^{\circ}$)
correspondent to a scattering vector $q_{DLS}=18$ $\mu m^{-1}$ and analyzed
using a digital autocorrelator. The time decay of the autocorrelation function
$F_{s}(\vec{q},t)^{2}$ was then fitted by a second-order cumulant expansion to
extract the diffusion coefficient $D$ as shown below 40:

$F_{s}(\vec{q}_{DLS},t)^{2}\propto
exp\left(-2q^{2}_{DLS}Dt\right)\left[1+\frac{\mu_{2}t^{2}}{2!}+o(t^{3})\right]^{2}$
(2)

where $\mu_{2}$ is related to the second moment of the distribution of the
diffusion coefficients of the suspended particles. The average diffusion
coefficient is then used to estimate the hydrodynamic radius using the Stokes-
Einstein relation: $D=K_{B}T/6\pi\eta_{o}R_{H}$, where $K_{B}$ is the
Boltzmann constant, $T$ is the bath temperature and $\eta_{0}$ is the zero-
shear viscosity of the solvent. The corresponding size dispersion and
polydispersity index are respectively
$\sigma_{R_{H}}=\sqrt{\mu_{2}}R_{H}/(Dq^{2}_{DLS})$ and
$\gamma=\mu_{2}/D^{2}q_{DLS}^{4}$. The polydispersity index for all the
samples investigated here never exceeded 0.23 (See SI).

The gyration radius, $R_{g}$, was measured by collecting the intensity of the
light $I(q)$ scattered by the microgel samples at different scattering angles.
The scattered light was subsequently fitted to the Guinier equation41 to
extract $R_{g}$:

$I(q)=I(0)\exp\left[-\frac{(qR_{g})^{2}}{3}\right],$ (3)

where $I(0)$ is a constant depending on the number of particles in the
scattering volume and on the scattering factor of a single particle. The
Guinier regime for all samples was attained in the range $0.5\leq qR_{g}\leq
2.5$, coherently with previously reported microgel syntheses 42, 32. The
uncertainty on $R_{g}$ is given by the fit error, the latter being less than
$2.5\%$ of the best-fit value. For the sample with the largest acrylic acid
content (AC5 hereafter) most of the data points lied in range $1\leq
qR_{g}\leq 2.5$, hence we employed an extended Guinier approximation as
reported in 43.

$I(q)=I(0)\exp\left[-\frac{(qR_{g})^{2}}{3}-Cq^{4}\right].$ (4)

where the constant $C$ quantifies the $O(q^{4})$ term in the expansion of the
form factor of the microgels around $q=0$. In all cases this term resulted
negligible, since $R_{g}$ values extracted via equation 4 deviated less than
3% from the values extracted by employing equation 3.

###  2.4 Electrophoresis

The electrophoretic mobility of the microgels has been measured by means of a
NanoZetaSizer apparatus (MALVERN), that is integrated with a laser Doppler
electrophoresis technique. The average electrophoretic mobility is determined
using the Phase Analysis Light Scattering (PALS) method 44, which is useful
especially at high ionic strengths, where mobilities are low. In these cases
the setup in the PALS configuration can measure mobilities two orders of
magnitudes lower than traditional light scattering methods based on the
shifted frequency spectrum (spectral analysis). We measured the
electrophoretic mobility at 20 ∘C and 60 ∘C after a proper thermalization to
monitor the effect of acrylic acid addition onto the mobility of swollen and
collapsed microgels.

###  2.5 Numerical simulations

Numerical simulations have been performed to unveil the microscopic mechanisms
underlying the structural swelling-deswelling transition in PNIPAm-co-AAc
microgels. We rely on a coarse-grained bead-spring model45 of the polymer
network composing the particles, wherein single polymer chains are modelled as
sequences of beads representing single uncorrelated polymer segments. Beads
interact among each other through a repulsive steric interaction modelled with
the Weeks-Chandler-Anderson (WCA) potential:

$V_{\text{WCA}}(r)=\begin{cases}4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right]+\epsilon&\quad\text{if}\quad
r\leq 2^{1/6}\sigma\\\ 0&\quad\text{if}\quad r>2^{1/6}\sigma\end{cases}$ (5)

where $\epsilon$ and $\sigma$ are the energy and length units, respectively.
Connected beads also interact via the finitely extensible nonlinear elastic
potential (FENE):

$V_{\text{FENE}}(r)=-\epsilon
k_{F}{R_{0}}^{2}\log\left[1-{\left(\frac{r}{R_{0}\sigma}\right)}^{2}\right],\quad
r<R_{0}\sigma$ (6)

where $R_{0}=1.5\sigma$ is the maximum bond distance and $k_{F}=15$ is a
stiffness parameter influencing the rigidity of the bond. The values of
$R_{0}$ and $k_{F}$ have been chosen in order to preserve the topology of the
network during the simulation. These bonds cannot break during the simulation,
mimicking strong covalent bonding. Beads that are linked via FENE potential to
two neighbours represent polymer segments containing NIPAm or AAc monomers
only, while those mimicking crosslinkers have fourfold valence.

Our simulations focus on the study of single microgel particles of $N\sim
42000$ beads, obtained through a recently developed protocol 46 able to
generate fully-bonded, disordered polymer networks. In order to accurately
describe the internal structure of microgels, we adopt an appropriate tuning
of the radial distribution of the crosslinkers, as established in previous
work 47 through a quantitative comparison of the microgel form factors across
the VPT. In this way, we ensure that the simulated networks have a realistic
core-corona structure similar to the experimental one. In this work, we
analyse for each simulated conditions a single microgel topology having a
crosslinker fraction similar to the experimental one, i.e. $c=$5.0%.

Within the model we also include electrostatic interactions to appropriately
describe the presence of ions in the polymer backbone stemming both from
dissociated AAc monomers and from SO4^- molecular groups, that result from the
thermal splitting of the persulfate ions S2O8^2- (stemming from the solvation
of KPS) and initiate free-radical polymerisation. To model the presence of
these ionic species, we consider a fraction $f$ of the beads as having a
negative charge and insert an equivalent number of counterions with positive
charge to preserve the overall electro-neutrality. Following our previous
works48, 49, counterions sterically interact among each other and with
microgel beads through the WCA potential, and their diameter is set to
$\sigma_{c}=0.1\sigma$, while electrostatic interactions are given by the
Coulomb potential:

$V_{\rm coul}(r_{ij})=\frac{q_{i}q_{j}\sigma}{e^{*2}r_{ij}}\epsilon$ (7)

where $q_{i}$ and $q_{j}$ are the charges of the beads, and
$e^{*}=\sqrt{4\pi\varepsilon_{0}\varepsilon_{r}\sigma\epsilon}$ is its reduced
unit, embedding the vacuum and relative dielectric constants,
$\varepsilon_{0}$ and $\varepsilon_{r}$. $q_{i}$ and $q_{j}$ are set to the
values $-e^{*}$ or $+e^{*}$ whether it refers to charged beads or counterions,
respectively.

To match with the experimental systems, we study different charge fractions
$f$ and distributions. In particular, we consider that charged groups coming
from the initiator are distributed within the corona of the microgels 50, 28,
so that we assign a fraction of charged beads $f_{s}=3.2\%$, representing
initiator molecules, to be located at a distance from the microgel center of
mass greater than its radius of gyration. On the other hand, in principle, we
do not know where the AAc monomers are located. Therefore, we first consider
that they can be found anywhere in the network and we distribute a varying
fraction of them, $f_{r}=0\%,1\%,3\%,4\%$, mimicking the experimental values,
randomly throughout the microgel. However, since part of the AAc monomers
could be also preferentially located in the microgel periphery to minimize
their mutual electrostatic repulsion during the polymerization process, we
additionally investigate the case with $f=f_{s}=7.2\%$, where all charges
(both coming from initiator and from AAc monomers) are located on the surface
as the most extreme condition. This case will be directly compared with that
having $f_{s}=3.2\%$ and $f_{r}=4\%$, which should be in closer agreement with
the experimental synthesis conditions discussed in the Methods, in order to
assess the effect of charges distribution on the swelling properties.

In all simulations crosslinkers are always set to be neutral and the solvent
is implicitly taken into account through an effective potential which mimics
the change in the polymer-solvent affinity by raising temperature:

$V_{\alpha}(r)=\begin{cases}-\epsilon\alpha&\text{if }r\leq 2^{1/6}\sigma\\\
\frac{1}{2}\alpha\epsilon\left\\{\cos\left[\gamma{\left(\frac{r}{\sigma}\right)}^{2}+\beta\right]-1\right\\}&\text{if
}2^{1/6}\sigma<r\leq R_{0}\sigma\\\ 0&\text{if }r>R_{0}\sigma\end{cases}$ (8)

where $\alpha$ is the solvophobicity parameter representing the effective
temperature, which is varied from $\alpha=0$ (good solvent conditions) to
$\alpha=1.40$ (bad solvent conditions) to reproduce the swelling curve from
the swollen state to the most collapsed ones 46. The constants
$\gamma=\pi\left(\frac{9}{4}-2^{1/3}\right)^{-1}$ and
$\beta=2\pi-\frac{9}{4}\gamma$ determine the functional shape of the potential
51, which gives an effective short-range attraction. Only neutral monomers
interact with the additional $V_{\alpha}$ potential, to consider the
solvophilic character that charged monomers keep at all temperatures 49.

The equations of motion are integrated through a Nosé-Hoover thermostat in the
constant NVT ensemble for the equilibration, and through a Velocity-Verlet
algorithm in the constant-energy ensemble for the production runs. The
integration time-step is $\Delta t=0.002\tau$, where
$\tau=\sqrt{m\sigma^{2}/\epsilon}$ is the reduced time unit. All simulations
are performed with the LAMMPS package 52 at fixed temperature
$K_{B}T/\epsilon=1.0$. The particle-particle-particle-mesh method 53 has been
employed to compute long-range Coulomb interactions. The equilibration of each
system is carried out for $2000\tau$, followed by a production run of
$6000\tau$ from which we extract the equilibrium averages of the main
observables.

#### Main calculated observables

To assess the swelling properties of the microgels we compute their gyration
radius:

$R_{\text{g}}=\sqrt{\sum_{i=1}^{N}\frac{(\vec{r}_{\text{i}}-\vec{r}_{\text{cm}})^{2}}{N}}$
(9)

where $\vec{r}_{\text{i}}$ is the position of the i-th among the $N$ beads and
$\vec{r}_{\text{cm}}$ the position of the centre of mass of the whole
microgel. $R_{\text{g}}$ gives the overall extension of the mass distribution
of the particle and we previously showed that it mainly describes the
extension of the core region of the microgel 28. To assess the extension of
the full particle, we also calculate the hydrodynamic radius $R_{\text{H}}$ of
the microgel, as 54

$\zeta=\frac{K_{B}T}{\bar{D}}=6\pi\eta R_{H}\equiv 6\pi\eta C_{\Omega}$ (10)

where $\zeta$ is the friction felt by the particle, $\bar{D}$ its diffusion
coefficient, $\eta$ the solvent viscosity and $C_{\Omega}$ an electrostatic
capacitance, that is mathematically equivalent to the hydrodynamic radius, as
a good approximation, due to the similarity among the Navier-Stokes equation
for the momentum flux density and the Poisson’s equation for electrostatics
54. To use this relation in simulations, we approximate the shape of the
microgel as an effective ellipsoid 55, to take into account the instantaneous
asphericity acquired from the microgel due to the presence of long external
chains, and then calculate the hydrodynamic friction for rigid ellipsoids 54,
28:

$R_{H}=C_{\Omega}=2\left[\int_{0}^{\infty}\frac{1}{\sqrt{(a^{2}+\theta)(b^{2}+\theta)(c^{2}+\theta)}}d\theta\right]^{-1}$
(11)

where $a$, $b$, $c$ are the principal semiaxes.

Since we became aware of a different method used in the literature to estimate
the hydrodynamic radius of macromolecules, also recently applied to nanogels
56, that is based on the ZENO algorithm 57, we also use this approach to
evaluate $R_{H}$ for all investigated microgels. It is important to note that
we find an overall consistency between the two methods, so that both appear to
be valid approaches to estimate $R_{H}$ from implicit solvent simulations.

To study the internal structure of microgel we also evaluate the radial
density profiles, defined as the average densities of a given species s at a
fixed distance from the centre of mass:

$\rho_{\text{s}}(r)=\left\langle\sum_{i=1}^{N_{\text{s}}}\delta(|\vec{r}_{i}-\vec{r}_{\text{cm}}|-r)\right\rangle.$
(12)

In particular, we monitor the density profiles of all the monomers
$\rho_{\text{m}}(r)$, of charged beads only $\rho_{\text{i}}(r)$, of the
counterions $\rho_{\text{c}}(r)$ and of the net charge
$\rho_{\text{Q}}(r)\equiv\rho_{\text{c}}(r)-\rho_{\text{i}}(r)$.

##  3 Results and Discussion

###  3.1 Experimental results

We first discuss the zero-shear viscosity of different microgel suspensions in
the dilute regime since it allows to characterize the role played by AAc
monomers during the synthesis in the monomer-to-microgel conversion. As
predicted by the Einstein’s equation (eq. 1), $\eta$ increases linearly with
particle concentration. All data collapse on the same master curve when the
viscosity is appropriately normalized. We thus define a normalized viscosity
$\eta^{*}$ as:

$\eta^{*}=\frac{\eta/\eta_{o}-1}{2.5k}.$ (13)

Figure 1(a) reports $\eta^{*}$ as a function of total polymer concentration
for all investigated samples at 20 ∘C, while Fig. 1(b) shows that the
proportionality constant $k$ extracted from $\eta(c)$ increases when the
fraction of AAc content is augmented in the microgel, similarly to what
observed for IPN microgels58, suggesting that for a given number of monomers
introduced during the synthesis a higher volume fraction is obtained when the
fraction of acrylic acid units increases. At this stage we cannot discern
whether this is due to a higher number of microgels or to a larger microgel
mass. To overcome such difficulty we can couple the so obtained $k$ values to
the hydrodynamic size measured via DLS at the same temperature as in
viscosimetry tests (20 ∘C, Table 1) and measure the relative variation of
their molecular weight with respect to a reference sample. In more detail,
since $k=\varphi/c$, $\varphi=N_{m}V_{m}/V$ and $c=M/V$, where $N_{m}$ is the
total number of suspended microgels, $V_{m}=(4/3\pi R_{H}^{3})$ the volume of
one microgel, $V$ the volume of the suspension and $M$ the total mass of
monomers introduced during the synthesis, we straightforwardly get

$\frac{V_{m}}{k}=\frac{M}{N_{m}}=M_{w}^{E}$ (14)

where the ratio $\frac{M}{N_{m}}$ is nothing but an effective molecular weight
of a microgel, whose value cannot be considered strictly correct, since it
leverages on equation 1, where the factor 2.5 holds for hard spheres, while
here we have penetrable and deformable colloids. Nonetheless we can compare
results obtained for $M_{w}^{E}$ with respect one reference sample, AC0 here,
and define the molecular weight ratio
$\rho_{M_{w}}=\frac{M_{w}^{E}(f_{AAc})}{M_{w}^{E}(f_{AAc}=0)}$, that is not _a
priori_ dependent on the specific first order correction to the viscosity of
the solvent due to the presence of penetrable microgels for a given microgel
structure. Moreover, despite a softening due to less compact structures of the
microgels and a reduction of the hydrodynamic friction coefficient for
increasing $f_{AAc}$ is still possible, this would give rise to a decrease of
$k$, that is not emerging from our analysis.

The molecular weight ratio $\rho_{M_{w}}$ is reported in figure 1(c) as a
function of the acrylic acid fraction. We find that $\rho_{M_{w}}$ increases
as a function of the AAc content in the microgels, implying that the mass of a
single particle increases for increasing $f_{AAc}$. This occurs despite the
total number of monomers is kept constant in all the syntheses (see SI). Such
an increase in the microgel mass can be explained only if i) the number of
microgels and nucleation sites decreases or ii) the number of unreacted
monomers progressively decreases with increasing $f_{AAc}$. We rule out the
second hypothesis since we do not detect any significant variation of the
residual mass obtained by drying an aliquot of supernatant resulting after one
centrifugation cycle: the unreacted mass was always about 1% of the total
NIPAm dissolved in water during the synthesis. Therefore we do not expect an
impact of monomer conversion on the final mass of a single microgel. This
suggests that the observed behavior is due to a decrease in the number of
microgels. These findings are consistent with the fact that, at the synthesis
temperature (70 ∘C), the propagation rate of NIPAm monomers is much higher
than that of acrylate units 59, 60, the latter being more hydrophilic and
partially charged at all temperatures in water. We thus speculate that this
difference makes AAc monomers less reactive units during the synthesis,
decreasing in turn the number of nucleation sites for the chain growth and
hence increasing the final molecular weight of the crosslinked networks. The
larger molecular weight and hydrophilicity of highly charged microgels is
further corroborated by the comparison between the hydrodynamic radii measured
before and after purification (Figure 1-(d)). The difference between the two
sizes progressively increases for increasing $f_{AAC}$, showing that
purification induces a larger swelling in microgels containing more charged
groups and characterized by a larger molecular weight. We ascribe such an
enhanced swelling mainly to a net reduction of the electrostatic screening and
AAc protonation due to the removal of leftover SDS, unreacted AAc and NIPAm
monomers and part of their respective free counterions.

Figure 1: (a) Normalized relative viscosity $\eta^{*}$ (see Eq. 13) of all investigated microgels as a function of their mass concentration; (b) Proportionality constant $k$ as a function of AAc fraction; (c) Relative average molecular weight of a single microgel as a function of AAc fraction; (d) Hydrodynamic radius before and after the purification process at $T=20^{\circ}$. Figure 2: Normalized scattered intensity $I(q)$ from static light scattering (panels, a,b,c,d) and autocorrelation functions $g_{2}-1$ from dynamic light scattering (panels e,f,g,h) for all studied microgels (AC0, AC1, AC3, AC5) at different temperatures. Table 1: Summary of the results from static and dynamic light scattering for all the microgel samples: Hydrodynamic radii at 20 ∘C, critical temperatures $(T_{c}^{H},T_{c}^{g})$ and sharpness parameters $(s^{H},s^{g})$ obtained by fitting $R_{H}(T)$ and $R_{g}(T)$ via equation 15. sample code | $f_{AAc}$ |  $R^{20}_{H}$ (nm) |  $T_{c}^{H}$ (∘C) |  $T_{c}^{g}$ (∘C) | sH (${}^{\circ}C^{-1}$) | sg (${}^{\circ}C^{-1}$)
---|---|---|---|---|---|---
AC0 | 0% | 200.5 | $33.70\pm 0.03$ | $33.06\pm 0.07$ | $0.71\pm 0.03$ | $1.39\pm 0.30$
AC1 | 1% | 233.6 | $38.62\pm 0.06$ | $36.94\pm 0.04$ | $0.32\pm 0.01$ | $0.50\pm 0.02$
AC3 | 3% | 284.6 | $46.81\pm 0.14$ | $43.52\pm 0.05$ | $0.24\pm 0.01$ | $0.49\pm 0.02$
AC5 | 5% | 390.8 | - | - | - | -
Table 2: Summary of results from mobility measurements: mobility $\mu$ and normalized mobility $\mu\eta_{0}/\epsilon$ measured at 20 ∘C and 60 ∘C for all the investigated samples. The reported errors are the standard deviations obtained over multiple runs. sample code |  $\mu^{20}$ ($10^{-8}m/Vs$) |  $\mu^{60}$ ($10^{-8}m/Vs$) |  $\mu^{20}\eta_{0}/\epsilon$ ($10^{-8}m$ $Pa/Vs$) |  $\mu^{60}\eta_{0}/\epsilon$ ($10^{-8}m$ $Pa/Vs$)
---|---|---|---|---
AC0 | -0.87 $\pm$ 0.04 | -5.90 $\pm$ 0.29 | -0.010 $\pm$ 0.001 | -0.041 $\pm$ 0.002
AC1 | -0.58 $\pm$ 0.03 | -6.05 $\pm$ 0.30 | -0.007 $\pm$ 0.001 | -0.042 $\pm$ 0.002
AC3 | -1.16 $\pm$ 0.06 | -5.85 $\pm$ 0.29 | -0.014 $\pm$ 0.001 | -0.041 $\pm$ 0.002
AC5 | -2.40 $\pm$ 0.12 | -5.13 $\pm$ 0.26 | -0.030 $\pm$ 0.002 | -0.036 $\pm$ 0.002

Figure 2 shows the results obtained from both dynamic and static light
scattering experiments. The top panels (a,b,c,d) report $I(q)/I_{0}$ as a
function of $q^{2}$ for selected temperatures and all investigated AAc
fractions. In all cases we observe an exponential decay in a quite large
window of scattering vectors allowing to fit the intensity via equation 3 and
extract $R_{g}$. The presence of the microgel VPT, signalled by a large
variation of the slope of $log(I/I_{0})$, is evident for all the microgels but
one, namely the AC5 sample, where $I(q)/I_{0}$ does not appreciably change
with temperature. We also note that $I(q)/I_{0}$ decays more rapidly for
increasing wave vectors $q$ as the AAc content of microgels grows, pointing to
a net increase of the gyration radius of the microgels embedding more acrylic
acid units. This occurs essentially at low temperatures where all microgels
are swollen.

The autocorrelation functions ($g_{2}(t)-1$) obtained via dynamic light
scattering for samples with different AAc content are reported in figure 2
(bottom panels, e,f,g,h). The lag time $t$ is normalized by $K_{B}T/\eta$ to
filter out the trivial speeding up of the dynamics due to the temperature
dependence of the solvent viscosity and to the thermal agitation. Hence, the
obtained shift at different temperatures is only due to the change in $R_{H}$
of the microgels. Once again, in agreement with the SLS data discussed above,
for $f_{AAc}=5\%$ the intensity autocorrelation functions do not show any
remarkable variation with temperature, indicating that $R_{H}$ of the
microgels does not appreciably change in the accessible range of temperatures.

The resulting $R_{g}(T)$ and $R_{H}(T)$ for all investigated microgels are
shown in figure 3. Both quantities increase below $T_{c}$ for increasing AAc
content, showing that acrylic acid addition enhances the overall
hydrophilicity of the microgels and it is consistent with an increase of their
mass as well. We may in principle also hypothesize that, since the synthesis
of the microgels occurs at $70$ ∘C, i.e. a temperature at which the PNIPAm
monomers become partially hydrophobic, and a fraction, even very small, of AAc
monomers always stay deprotonated also at low pH, more AAc units could tend to
occupy the periphery of the microgel (see also discussion below). However we
anticipate here that this does not result in a detectable change of the
swollen microgel structure as witnessed by the nearly constant ratio of the
two measured sizes ($R_{g}/R_{H}\simeq$ 0.7) at 25 ${}^{\circ}C$ for all
acrylic acid fraction. The thermal evolution of such a ratio will be discussed
afterwards. A strongly uneven distribution of charged acidic groups therefore
has to be excluded, suggesting that the already mentioned increase of $R_{H}$,
once the synthesis leftovers are removed by centrifugation/supernatant removal
cycles (figure 1-d), must be attributed to the larger mass and the augmented
electrostatic self-energy of the entire AAc-rich microgels, and not only to
the swelling of the peripheral corona.

Figure 3: Hydrodynamic radii RH and gyration radii Rg of PNIPAm-co-AAc
microgels as a function of temperature for varying concentration of acrylic
acid (fAAc). (a) AC0, (b) AC1, (c) AC3 and (d) AC5

As the temperature is increased, microgels undergo a volume phase transition
due to the presence of a finite Low Critical Solution Temperature (LCST) for
PNIPAm. However, the electrostatic repulsion among the AAc-rich chains,
together with their hydrophilic character, tends to keep the microgel swollen,
thus shifting the occurrence of the VPT to higher $T$-values. The critical
temperatures ($T_{c}$) at which the microgels undergo the transition are
estimated by fitting the both $R_{g}$ and $R_{H}$ to an auxiliary function 28:

$R_{H,g}=[R_{0}-\Delta R_{H,g}\tanh(s(T-T_{c}^{H,g}))]+A(T-T_{c}^{H,g})$ (15)

where, $R_{0}$ is the radius of the microgel at the VPT, $\Delta R_{H,g}$ is
the amplitude of the VPT and the parameter $s$ quantifies its sharpness. The
obtained values of $T_{c}$ for the different samples are reported in Table 1
for both $R_{g}$ and $R_{H}$, while the best-fitting curves are shown in SI.
As foreshadowed by Fig. 2, $T_{c}$ shifts to higher temperature as the AAc
content increases, until for $f_{AAc}=5\%$ neither $R_{g}$ nor $R_{H}$ show
any sign of collapse: it appears that the gain in translational entropy of
water that would result in the collapse of these microgels is not sufficient
anymore to overcome electrostatic repulsion between AAc-rich chains. Such a
scenario is further supported by the electrophoretic mobility data $20$ ∘C and
$60$ ∘C, reported in Table 2, which confirm the presence of AAc monomers and
charged initiators within the microgels and are sensitive to microgel
collapse. We observe a net increase (in modulus) of the absolute mobility
$\mu$ for the AC3 and AC5 samples when compared to AC0, while we ascribe the
weak decrease of the mobility in AC1 sample to the enhanced screening due to
the presence of confined AAc counterions. Whether the electrophoretic mobility
of swollen PNIPAm-co-AAc microgels shows a non-monotonic behavior in function
of the acrylic acid content due to the subtle balance between electric and
frictional forces of different origin61, 62 is currently unknown, although a
non-trivial behavior of the mobility has already been reported as a function
of salt concentration37. This represents an interesting aspect _per se_ and
will not be investigated further here. In all cases we observe a large
increase of mobility at high temperature. To discern whether this is due to
their collapse or simply to the relative variation of the viscosity $\eta_{0}$
and/or the relative permittivity $\epsilon$ of the solvent63, we also report
the normalized mobility $\mu\eta_{0}/\epsilon$ for the two temperatures. We
unambiguously observe that the normalized mobility for AC0, AC1 and AC3
increases by factor whose value is comprised between 3 and 4, going from 20 ∘C
to 60 ∘C, while for the AC5 sample it stays almost unaltered, corroborating
the absence of significant deswelling for this sample. Another important
feature due to the addition of AAc is the decrease of the ”sharpness”
parameter $s$ for increasing acrylic acid fraction $f_{AAc}$ (Table 1). This
can be attributed to the increase of repulsive units into the temperature-
sensitive NIPAm structure, pointing to a relatively even distribution of
acidic ”defects” within the microgels, which reflect a net decrease of the
cooperativity of the volume phase transition: the latter is characterized by a
larger spectrum of critical temperatures for increasing AAc content. It is
also worth noting that, despite we observe a clear broadening of the volume
phase transition, we do not observe a successive sequence of transitions as
those described in previous works25, 30 for larger $f_{AAc}$, which reported
first a collapse of PNIPAm-rich domains, followed by the collapse of the AAc-
rich domains at higher temperatures. Such a scenario has been previously
rationalized by proton migration from the solution to the microgels after the
first collapse 25, inducing a large counterion condensation on the carboxylic
groups and a second microgel deswelling. Our results therefore suggest a
homogeneous distribution of acrylic acid segments and continuous transitions,
as discussed in 26 for PNIPAm-co-Methacrylic acid microgels. Transition
broadening and VPT suppression have been observed for increasing fraction of
deprotonated acrylic acid groups also in very similar syntheses of PNIPAm-co-
AAc microgels 39, 64. In addition, since during the synthesis (before
purification) the nominal pH of the samples is less than pKa(AAc) (see section
2), their high protonation degree could increase their solubility into the
NIPAm droplets formed at 70 ∘C and homogenize their distribution within the
emerging network. This said, we cannot exclude that two distinct collapses of
PNIPAm-rich and AAc-rich domains might occur at higher pH ($>$6.9), namely for
larger fractions of deprotonated AAc monomers. This hypothesis will be a
subject of future investigation.

Figure 4: Ratio of Rg/RH of PNIPAm-co-AAc microgels as a function of
temperature for all the investigated samples as indicated in the panels.

As clearly visible in Fig. 3, $R_{g}$ is systematically lower than $R_{H}$ for
all samples and the decrease of both quantities with temperature is more and
more decoupled upon increasing the fraction $f_{AAc}$. The decoupling in the
VPT for $R_{H}$ and $R_{g}$ can be clearly seen by looking at the difference
between the critical temperatures ($\Delta T_{c}=T^{H}_{C}-T^{g}_{C}$)
obtained for the two quantities. For AC0 microgels, $\Delta T_{c}$ is $\leq
0.7^{\circ}C$, and it increases up to $\Delta T_{c}\approx 3.3$ ∘C for AC3
(see Table 1). Therefore, for increasing $f_{AAc}$, the transitions of $R_{H}$
and $R_{g}$ become more and more distant in temperature. Such an enhanced
decoupling results in the presence of a minimum in the ratio $R_{g}/R_{H}$ as
a function of temperature, as shown in Figure 4. At low temperatures, all
samples exhibit a ratio $0.6\leq Rg/R_{H}\leq 0.8$ until $T_{C}$ is reached.
Since the transition in $R_{g}$ anticipates that in $R_{H}$, a minimum appears
when $R_{g}$ collapses, with a minimum depth depending on $f_{AAC}$,
coherently with the $\Delta T_{c}$ trend. The added charges thus enhance the
inhomogeneous deswelling and the two-step character of the volume phase
transition of thermosensitive microgels, a feature that up to now has been
identified only in pure PNIPAm microgel suspensions 28. Up to our knowledge
this is the first experimental evidence of the influence of charged AAc co-
monomers on the structural change across the continuous VPT of PNIPAm-based
microgels.

###  3.2 Simulations

To provide a better understanding of the role played by the presence of co-
polymerized ionizable groups in the VPT of microgels, we performed molecular
dynamics simulations as described in section 2.5.

Figure 5: Top: Swelling curves of (a) $R_{g}$, (b) $R_{H}$ computed with the
convex hull method and (c) $R_{H}$ computed with the ZENO algorithm. Bottom:
Swelling ratios among $R_{g}$ and $R_{H}$, as computed with the (d) convex
hull method and (e) the ZENO algorithm.

Fig. 5 shows the results for (top row) the radius of gyration (a) and for the
hydrodynamic radius, calculated through the convex hull method (b) and the
ZENO algorithm (c), respectively, In agreement with experiments, for all
charge contents and distributions, the gyration radius of the microgels is
smaller than the hydrodynamic radius at all the temperatures, and the increase
of charge results in a net increment of the microgel size, with the
hydrodynamic radius being more affected than the gyration one. Most
importantly, the hydrodynamic volume phase transition temperature experiences
a larger shift as the fraction of ionized beads is increased with respect to
the same transition temperature extracted from the gyration radius data: the
two transitions decouple because of the charge addition. We note that the
results obtained for $R_{H}$ from the two methods are in full qualitative
agreement, giving us confidence in both of them for a reliable estimate of the
hydrodynamic radius of macromolecules in solvent-free molecular dynamics
simulations.

Such an agreement is also observed in the behavior of the ratio $R_{g}/R_{H}$
ratio, plotted as a function of the solvophobicity parameter $\alpha$ in Fig.
5(d,e), respectively for the convex hull and the Zeno methods. The observed
decoupling in the transitions of $R_{g}$ and $R_{H}$ is thus reflected in the
appearance and in the successive deepening of a minimum in their ratio. We
remark that this feature is strictly absent for neutral microgels for both
methods, stressing once more the importance of an appropriate charge modeling
to reproduce the experimental behavior even in pure PNIPAm homopolymer
microgels. At this point, it is worth noticing that in experiments it is hard
to obtain strictly neutral PNIPAm-based microgels in aqueous environments,
since non-ionic initiators are i) poorly soluble in water, making the
synthesis problematic and hardly reproducible, and ii) often show successive
unwanted deprotonation, as in the case of 2-2’-azobisisobutirronitrile (AIBN)
that produces anionic microgels as already discussed in other microgel
synthesis procedures65. In this respect, simulations appear to be crucial
since they put forward the evidence that the two-step deswelling is strictly
an electrostatic phenomenon. In addition, the minimum is further affected by
the spatial charge distribution. Despite the presence of only 3.2% of charged
beads within the outermost corona of the microgels is enough to trigger the
aforementioned decoupling and the subsequent minimum in $R_{g}/R_{H}$, the
increase of the amount of ionic beads enhances such a decoupling mechanism and
deepens the minimum. This effect is already present when additional charged
groups are randomly distributed within the microgel, but becomes much more
evident when charges are all distributed on the surface. A tentative
explanation of this effect, when adding charges randomly throughout the
network, would be that those groups located in the core are always well
screened by counterions, thus barely affecting the $R_{g}$ transition, while
the fraction of them that still goes to the surface results in an increase of
the charged layer of the corona, as Figure 7 shows, in turn increasing its
swelling and the transition temperature of $R_{H}$. The more charged the
surface, the more inhomogeneous is the structural swelling transition. This is
confirmed by the results obtained with 7.2% of ionic beads all distributed
within the corona: in this case, the decoupling between $R_{g}$ and $R_{H}$ is
enhanced, allowing us to conclude that charge accumulation in the corona is
the main driving mechanism for the observation of the two-step deswelling, as
also shown by the large increase of the minimum depth for this particular case
(Figure 5-d,e). It is important to note that this feature is also accompanied
by a shift of the minimum position to larger solvophobic parameter values,
hence effective temperature, in good agreement with experiments (Figure 3) for
increasing amount of acrylic acid monomers. In simulations we also observe a
mild decrease of the ratio $R_{g}/R_{H}$ at low effective temperatures
(swollen microgels) between the two configurations just mentioned and more
generally with increasing $f$. This is not observed in experiments and could
be due to the fact that our charged groups are added to the same microgel
topology, assembled in silico in the absence of charges, which may result in a
slightly different structure of the simulated networks in the swollen state.
However, this aspect goes beyond the scope of the present work and will not be
investigated further. Our simulations thus confirm a general scenario where
electrostatic charges in thermosensitive microgels give rise to a two-step
deswelling process with the core collapsing first than the outer corona and
the ratio $R_{g}/R_{H}$ starting from values comprised in the interval 0.6-0.7
at low temperatures, passing through a minimum and finally tending to the hard
sphere limit ($R_{g}^{HS}/R_{H}^{HS}=0.77$) when the collapsed state is
approached.

Figure 6: (a) Monomers, (b) ions, (c) counterions and (d) net charge radial
density profiles for the microgels as a function of the distance from the
microgel center of mass $r$ for $\alpha=0$. Inset in (d): snapshots
corresponding to the microgels with surface charge $f_{s}=3.2\%$, with surface
charge $f_{s}=3.2\%$ and random charge $f_{r}=4\%$, and with surface charge
$f=7.2\%$, from left to right.

In order to have a closer look to the effect of the VPT on the charge
distribution (and viceversa), we report in Figs. 6, 7 and 8 the density
profiles of all monomers (a), charged ions (b), counterions (c) and net charge
(d) at three representative temperatures, $\alpha=0,0.80$ and $1.20$
corresponding to the fully swollen, nearly critical and collapsed state of the
microgels, respectively.

For swollen microgels ($\alpha=0$) the monomer density (panels (a)) is only
weakly affected by ion embedding. We observe that the tail of the distribution
become slightly fatter for increasing $f$ or net charge mismatch between the
corona and the core, while its $(r\rightarrow 0)$ value of the density profile
decreases. Such a weak, albeit detectable effect, can be attributed to the
augmented microgel size. This feature also holds for the other two values of
$\alpha$ shown in Figs. 7,8(a), being more evident for near-critical microgels
($\alpha=0.8$) where the effect of the added charge is magnified, along with
inhomogeneities in the structure.

Figure 7: (a) Monomers, (b) ions, (c) counterions and (d) net charge radial
density profiles for the microgels as a function of the distance from the
microgel center of mass $r$ for $\alpha=0.8$. Inset in (d): snapshots
corresponding to the microgels with surface charge $f_{s}=3.2\%$, with surface
charge $f_{s}=3.2\%$ and random charge $f_{r}=4\%$, and with surface charge
$f=7.2\%$, from left to right.

Turning to the ion density distributions (panels (b)), we find that for all
temperatures they are always peaked at the microgel periphery, where most of
the charges are located, while a plateau at short distances is found to
increase with the amount of randomly distributed beads.The peak height, as
expected, increases when microgels become more hydrophobic, since the average
inter-charge distance decreases, thus producing a net increment of the charge
density of the microgels (panels (d)). Such an augmented charge density has
been already widely discussed in experiments 66, 37, 18. The cases of
microgels with only surface charges display a different behavior, where a net
positive charge develops within the core of the particles due to the
accumulation of counterions (panels (c)). Such a trend is not present when
random charges are added, so that the net charge inside the core tends to
zero, as visible at not too small distances from the centre of mass (where
statistical noise becomes rather large for $r\rightarrow 0$). In addition the
net charge distribution (panels (d)) displays the largest (negative) peaks for
the microgels with surface charges only. This is due to the nearly zero
electric field in the microgel interior when all the charged beads are
confined in its periphery. Therefore counterions are still able to penetrate
and diffuse within the microgel, tending to repel further counterions that
would neutralize the fixed charge distribution carried by the chains. When
charges are randomly added throughout the network, this repulsive electric
field decreases, and the surface peak of negative beads increases, allowing
counterions to better screen them even in the corona, resulting in a decrease
of the peaks in the net charge. The fact that charged beads are spatially
constrained by network bonds, while counterions are free to move, generates an
asymmetry resulting in a net peak of the counterion density, that is always
more smeared with respect to that of charged monomers. This is also evidenced
by the net charge overshoots close to the main peaks, especially in collapsed
microgels (Figure 8).

Snapshots of equilibrium configurations of the microgels with $f=3.2\%$
charges on the surface, with added $f_{r}=4\%$ charges randomly distributed,
and with $f=7.2\%$ surface charges are also shown in panels (d). They allow us
to visualize the fact that the increase of the random charges induce a larger
structural inhomogeneity, a feature that is further enhanced by moving all
fixed charges (monomers) within the corona. Hence, the microgel progressively
acquires a more open structure and a larger amount of dangling chains that
contribute to the net increase of the hydrodynamic size of the particle. All
the discussed features are maintained as $\alpha$ increases and it is
particularly evident that, even well above the VPT temperature, the more
charged the microgel and the more asymmetric its charge distribution, the
higher is the degree of anisotropy of the collapsed state, which maintains
residual swollen chains.

This explains why a further increase of $f$, especially if charges are only
distributed within the corona region, will eventually prevent the full
microgel collapse or even the occurrence of the VPT in the usual range of
investigated temperatures in experiments, as found for the AC5 microgels in
Fig. 3 and in previous works 24, 25, 64.

Figure 8: (a) Monomers, (b) ions, (c) counterions and (d) net charge radial
density profiles for the microgels as a function of the distance from the
microgel center of mass $r$ for $\alpha=1.2$. Inset in (d): snapshots
corresponding to the microgels with surface charge $f_{s}=3.2\%$, with surface
charge $f_{s}=3.2\%$ and random charge $f_{r}=4\%$, and with surface charge
$f=7.2\%$, from left to right.

##  4 Conclusions

By means of experiments and numerical simulations we have shown that the
addition of acrylic acid monomers in PNIPAm microgels via random co-
polymerization results in a large accentuation of the two-step deswelling
previously reported for pure PNIPAm microgels only 28. Experimentally, the AAc
addition is found to i) increase microgel mass, ii) shift, smooth out and
eventually suppress the occurrence of the VPT and iii) favour the thermal
decoupling between the collapse of the microgel core and that of its outer
corona. Simulation results fully confirm this scenario, further highlighting
the crucial role of the charge distribution within the network. Indeed, they
demonstrate that the two-step deswelling is enhanced by the charge
preferential location in the outer corona, with a non-trivial net charge
distribution arising from strongly asymmetric counterions spatial
distributions. To sum up, the (random or peripheral) addition of charged beads
promotes the formation of microgels with more open structures, with the outer
region being more affected than the inner part of the networks. This results
in an increase of the hydrodynamic size of the microgels that is more
pronounced as compared to the increase of the corresponding gyration radius.
Such an effect, present in swollen microgels, reflects in their thermal
behavior: their external, looser and more charged corona regions collapse
progressively at higher temperatures than the corresponding cores, for
increasing fraction of charged monomers.

Although our work sheds light on the effect of AAc co-polymerization on the
two-step deswelling of PNIPAm microgels, many questions still remain open. In
particular, a direct proof of charge location in experiments would be very
valuable. To this purpose, super-resolution microscopy experiments 36 could be
helpful, when combined with selective labeling of the initiators and/or of
charged co-monomers. Additionally, one could rely on neutron scattering
measurements, where differentiation of the monomer components can be achieved
via contrast variation 67, 50.

Another important question concerns whether larger charge contents or pH
values would trigger different kinds of transitions (e.g. discontinuous or
multi-step), as observed in previous works 25. To this aim, it would be
interesting to assess how this will affect the two-step deswelling mechanism
discussed in the present work. Hence, for such very charged microgels, a
simultaneous measurement of $Rg$ and $R_{H}$ would again be very valuable.

Finally, the outcomes of this work are relevant for future investigations of
the collective behavior and dynamics of PNIPAm-based microgels, especially at
high temperatures and for near-critical conditions, where
numerical/theoretical works are still lacking. In fact, though the effective
potential for swollen microgels can be described as a soft repulsion 68, at
high temperatures both hydrophobic and electrostatic interactions should
become important 69, 70 . It will be therefore necessary to appropriately
assess their relative contribution to the collective and rheological behavior
of dense microgel suspensions. Last but not least, polyelectrolyte and/or
nanoparticle adsorption onto soft colloids, such as microgels, is very often
determined by electrostatic forces that are tightly related to the swelling
state of the adsorbing substrate 37, 18. Understanding how randomly
distributed charges and the enhanced two-step deswelling influence the
adsorption of oppositely charged objects onto soft ionic microgels represents
another important open question having potential important implications in
applications such as water remediation protocols and sensors. We acknowledge
financial support from the Agence Nationale de la Recherche (Grant
ANR-20-CE06-0030-01; THELECTRA) and from the European Research Council (ERC
Consolidator Grant 681597; MIMIC). DT thanks Luca Cipelletti for illuminating
discussions.

We report: i) the mass of each compound introduced in all the free radical
polymerization syntheses; ii) the best fits (Equation 15) obtained for both
$R_{H}(T)$ and $R_{g}(T)$ of all the microgels showing a VPT (AC0,AC1,AC3);
iii) the polydispersity index $\gamma$ obtained via cumulant analysis as
detailed in section 2 for all the samples and temperatures.

## References

  * Lima et al. 2016 Lima, L. H.; Morales, Y.; Cabral, T. Ocular Biocompatibility of Poly-N-Isopropylacrylamide (pNIPAM). _J. Ophthalmol._ 2016, _2016_ , 1–6
  * Garcia-Pinel et al. 2020 Garcia-Pinel, B.; Ortega-Rodríguez, A.; Porras-Alcalá, C.; Cabeza, L.; Contreras-Cáceres, R.; Ortiz, R.; Díaz, A.; Moscoso, A.; Sarabia, F.; Prados, J.; López-Romero, J. M.; Melguizo, C. Magnetically active pNIPAM nanosystems as temperature-sensitive biocompatible structures for controlled drug delivery. _Artif Cells Nanomed Biotechnol_ 2020, _48_ , 1022–1035
  * Islam et al. 2014 Islam, M.; Ahiabu, A.; Li, X.; Serpe, M. Poly (N-isopropylacrylamide) Microgel-Based Optical Devices for Sensing and Biosensing. _Sensors_ 2014, _14_ , 8984–8995
  * Highley et al. 2019 Highley, C. B.; Song, K. H.; Daly, A. C.; Burdick, J. A. Jammed Microgel Inks for 3D Printing Applications. _Adv. Sci._ 2019, _6_ , 1801076
  * Compaan et al. 2020 Compaan, A. M.; Song, K.; Chai, W.; Huang, Y. Cross-Linkable Microgel Composite Matrix Bath for Embedded Bioprinting of Perfusable Tissue Constructs and Sculpting of Solid Objects. _ACS Appl. Mater. Interfaces_ 2020, _12_ , 7855–7868
  * Murthy et al. 2003 Murthy, N.; Xu, M.; Schuck, S.; Kunisawa, J.; Shastri, N.; Fréchet, J. M. J. A macromolecular delivery vehicle for protein-based vaccines: Acid-degradable protein-loaded microgels. _Proc. Natl. Acad. Sci. U.S.A._ 2003, _100_ , 4995–5000
  * Debord et al. 2002 Debord, J.; Eustis, S.; Byul Debord, S.; Lofye, M.; Lyon, L. Color-Tunable Colloidal Crystals from Soft Hydrogel Nanoparticles. _Adv. Mater._ 2002, _14_ , 658–662
  * Keidel et al. 2018 Keidel, R.; Ghavami, A.; Lugo, D. M.; Lotze, G.; Virtanen, O.; Beumers, P.; Pedersen, J. S.; Bardow, A.; Winkler, R. G.; Richtering, W. Time-resolved structural evolution during the collapse of responsive hydrogels: The microgel-to-particle transition. _Sci. Adv._ 2018, _4_ , eaao7086
  * Debord and Lyon 2003 Debord, J. D.; Lyon, L. A. Synthesis and Characterization of pH-Responsive Copolymer Microgels with Tunable Volume Phase Transition Temperatures. _Langmuir_ 2003, _19_ , 7662–7664
  * Wypysek et al. 2020 Wypysek, S. K.; Scotti, A.; Alziyadi, M. O.; Potemkin, I. I.; Denton, A. R.; Richtering, W. Tailoring the Cavity of Hollow Polyelectrolyte Microgels. _Macromol. Rapid Commun._ 2020, _41_ , 1900422
  * Murthy et al. 2003 Murthy, N.; Xu, M.; Schuck, S.; Kunisawa, J.; Shastri, N.; Fréchet, J. M. J. A macromolecular delivery vehicle for protein-based vaccines: Acid-degradable protein-loaded microgels. _Proc. Natl. Acad. Sci. U.S.A._ 2003, _100_ , 4995–5000
  * Nagayama et al. 1995 Nagayama, H.; Maeda, Y.; Shimasaki, C.; Kitano, H. Catalytic properties of enzymes modified with temperature-responsive polymer chains. _Macromol. Chem. Phys._ 1995, _196_ , 611–620
  * Bergbreiter et al. 1998 Bergbreiter, D. E.; Case, B. L.; Liu, Y.-S.; Caraway, J. W. Poly( N -isopropylacrylamide) Soluble Polymer Supports in Catalysis and Synthesis. _Macromolecules_ 1998, _31_ , 6053–6062
  * Pelton and Chibante 1986 Pelton, R.; Chibante, P. Preparation of aqueous latices with N-isopropylacrylamide. _Colloids and Surfaces_ 1986, _20_ , 247–256
  * Seelenmeyer et al. 2000 Seelenmeyer, S.; Deike, I.; Dibgenouts, N.; Rosenfeldt, S.; Norhausen, C.; Ballauff, M.; Narayanan, T. Analysis of the volume phase trasition in thermosensitive core-shell particles by synchrotron small-angle X-ray scattering. _J. Appl. Crystallogr._ 2000, _33_ , 574–576
  * Seelenmeyer et al. 2001 Seelenmeyer, S.; Deike, I.; Rosenfeldt, S.; Norhausen, C.; Dingenouts, N.; Ballauff, M.; Narayanan, T.; Lindner, P. Small-angle x-ray and neutron scattering studies of the volume phase transition in thermosensitive core–shell colloids. _J. Chem. Phys._ 2001, _114_ , 10471–10478
  * Stieger et al. 2004 Stieger, M.; Richtering, W.; Pedersen, J. S.; Lindner, P. Small-angle neutron scattering study of structural changes in temperature sensitive microgel colloids. _J. Chem. Phys._ 2004, _120_ , 6197–6206
  * Sennato et al. 2021 Sennato, S.; Chauveau, E.; Casciardi, S.; Bordi, F.; Truzzolillo, D. The Double-Faced Electrostatic Behavior of PNIPAm Microgels. _Polymers_ 2021, _13_ , 1153
  * Ishidao et al. 1993 Ishidao, T.; Akagi, M.; Sugimoto, H.; Iwai, Y.; Arai, Y. Swelling behaviors of poly(N-isopropylacrylamide) gel in poly(ethylene glycol)-water mixtures. _Macromolecules_ 1993, _26_ , 7361–7362
  * Kim et al. 1995 Kim, Y.-H.; Kwon, I. C.; Bae, Y. H.; Kim, S. W. Saccharide Effect on the Lower Critical Solution Temperature of Thermosensitive Polymers. _Macromolecules_ 1995, _28_ , 939–944
  * Park and Hoffman 1993 Park, T. G.; Hoffman, A. S. Sodium chloride-induced phase transition in nonionic poly(N-isopropylacrylamide) gel. _Macromolecules_ 1993, _26_ , 5045–5048
  * Zhang et al. 1992 Zhang, Y.-Q.; Tanaka, T.; Shibayama, M. Super-absorbency and phase transition of gels in physiological salt solutions. _Nature_ 1992, _360_ , 142–144
  * Inomata et al. 1995 Inomata, H.; Wada, N.; Yagi, Y.; Goto, S.; Saito, S. Swelling behaviours of N-alkylacrylamide gels in water: effects of copolymerization and crosslinking density. _Polymer_ 1995, _36_ , 875–877
  * Shibayama et al. 1996 Shibayama, M.; Fujikawa, Y.; Nomura, S. Dynamic Light Scattering Study of Poly(N -isopropylacrylamide- co -acrylic acid) Gels. _Macromolecules_ 1996, _29_ , 6535–6540
  * Kratz et al. 2000 Kratz, K.; Hellweg, T.; Eimer, W. Influence of charge density on the swelling of colloidal poly(N-isopropylacrylamide-co-acrylic acid) microgels. _Colloids Surf. A Physicochem. Eng. Asp._ 2000, _170_ , 137–149
  * Karanastasis et al. 2020 Karanastasis, A. A.; Kenath, G. S.; Andersen, D.; Fokas, D.; Ryu, C. Y.; Ullal, C. K. One-pot surfactant-free modulation of size and functional group distribution in thermoresponsive microgels. _J. Colloid Interface Sci._ 2020, _568_ , 264–272
  * Nigro et al. 2021 Nigro, V.; Angelini, R.; Bertoldo, M.; Buratti, E.; Franco, S.; Ruzicka, B. Chemical-Physical Behaviour of Microgels Made of Interpenetrating Polymer Networks of PNIPAM and Poly(acrylic Acid). _Polymers_ 2021, _13_ , 1353
  * Del Monte et al. 2021 Del Monte, G.; Truzzolillo, D.; Camerin, F.; Ninarello, A.; Chauveau, E.; Tavagnacco, L.; Gnan, N.; Rovigatti, L.; Sennato, S.; Zaccarelli, E. Two-step deswelling in the Volume Phase Transition of thermoresponsive microgels. _Proc Natl Acad Sci USA_ 2021, _118_ , e2109560118
  * Snowden et al. 1996 Snowden, M. J.; Chowdhry, B. Z.; Vincent, B.; Morris, G. E. Colloidal copolymer microgels of N-isopropylacrylamide and acrylic acid: pH, ionic strength and temperature effects. _Faraday Trans._ 1996, _92_ , 5013
  * Huo et al. 2006 Huo, D.; Li, Y.; Qian, Q.; Kobayashi, T. Temperature–pH sensitivity of bovine serum albumin protein-microgels based on cross-linked poly(N-isopropylacrylamide-co-acrylic acid). _Colloids Surf. B_ 2006, _50_ , 36–42
  * Burmistrova et al. 2011 Burmistrova, A.; Richter, M.; Eisele, M.; Üzüm, C.; von Klitzing, R. The Effect of Co-Monomer Content on the Swelling/Shrinking and Mechanical Behaviour of Individually Adsorbed PNIPAM Microgel Particles. _Polymers_ 2011, _3_ , 1575–1590
  * Clara-Rahola et al. 2012 Clara-Rahola, J.; Fernandez-Nieves, A.; Sierra-Martin, B.; South, A. B.; Lyon, L. A.; Kohlbrecher, J.; Fernandez Barbero, A. Structural properties of thermoresponsive poly( N -isopropylacrylamide)-poly(ethyleneglycol) microgels. _J. Chem. Phys._ 2012, _136_ , 214903
  * Mattsson et al. 2009 Mattsson, J.; Wyss, H. M.; Fernandez-Nieves, A.; Miyazaki, K.; Hu, Z.; Reichman, D. R.; Weitz, D. A. Soft colloids make strong glasses. _Nature_ 2009, _462_ , 83–86
  * Senff and Richtering 1999 Senff, H.; Richtering, W. Temperature sensitive microgel suspensions: Colloidal phase behavior and rheology of soft spheres. _J. Chem. Phys._ 1999, _111_ , 1705–1711
  * Misra et al. 2020 Misra, C.; Behera, S. K.; Bandyopadhyay, R. Influence of particle size on the thermoresponsive and rheological properties of aqueous poly(N-isopropylacrylamide) colloidal suspensions. _Bull Mater Sci_ 2020, _43_ , 182
  * Conley et al. 2019 Conley, G. M.; Zhang, C.; Aebischer, P.; Harden, J. L.; Scheffold, F. Relationship between rheology and structure of interpenetrating, deforming and compressing microgels. _Nat Commun_ 2019, _10_ , 2436
  * Truzzolillo et al. 2018 Truzzolillo, D.; Sennato, S.; Sarti, S.; Casciardi, S.; Bazzoni, C.; Bordi, F. Overcharging and reentrant condensation of thermoresponsive ionic microgels. _Soft Matter_ 2018, _14_ , 4110–4125
  * Pei et al. 2004 Pei, Y.; Chen, J.; Yang, L.; Shi, L.; Tao, Q.; Hui, B.; Li, J. The effect of pH on the LCST of poly(N-isopropylacrylamide) and poly(N-isopropylacrylamide-co-acrylic acid). _J. Biomater. Sci. Polym. Ed._ 2004, _15_ , 585–594
  * Al-Manasir et al. 2009 Al-Manasir, N.; Zhu, K.; Kjøniksen, A.-L.; Knudsen, K. D.; Karlsson, G.; Nyström, B. Effects of Temperature and pH on the Contraction and Aggregation of Microgels in Aqueous Suspensions. _J. Phys. Chem. B_ 2009, _113_ , 11115–11123
  * Frisken 2001 Frisken, B. J. Revisiting the method of cumulants for the analysis of dynamic light-scattering data. _Appl. Opt._ 2001, _40_ , 4087
  * Guinier 1955 Guinier, A. Small angle scattering of X-rays. 1955, 276
  * Gasser et al. 2014 Gasser, U.; Hyatt, J. S.; Lietor-Santos, J.-J.; Herman, E. S.; Lyon, L. A.; Fernandez-Nieves, A. Form factor of pNIPAM microgels in overpacked states. _J. Chem. Phys._ 2014, _141_ , 034901
  * Mohammed et al. 2019 Mohammed, A. S. A.; Carino, A.; Testino, A.; Andalibi, M. R.; Cervellino, A. A dilute gold nanoparticle suspension as small-angle X-ray scattering standard for an absolute scale using an extended Guinier approximation. _J Appl Crystallogr_ 2019, _52_ , 344–350
  * Tscharnuter 2001 Tscharnuter, W. W. Mobility measurements by phase analysis. _Appl. Opt._ 2001, _40_ , 3995–4003
  * Grest and Kremer 1986 Grest, G. S.; Kremer, K. Molecular dynamics simulation for polymers in the presence of a heat bath. _Phys. Rev. A_ 1986, _33_ , 3628
  * Gnan et al. 2017 Gnan, N.; Rovigatti, L.; Bergman, M.; Zaccarelli, E. In Silico Synthesis of Microgel Particles. _Macromolecules_ 2017, _50_ , 8777–8786
  * Ninarello et al. 2019 Ninarello, A.; Crassous, J. J.; Paloli, D.; Camerin, F.; Gnan, N.; Rovigatti, L.; Schurtenberger, P.; Zaccarelli, E. Modeling Microgels with a Controlled Structure across the Volume Phase Transition. _Macromolecules_ 2019, _52_ , 7584–7592
  * Del Monte et al. 2019 Del Monte, G.; Ninarello, A.; Camerin, F.; Rovigatti, L.; Gnan, N.; Zaccarelli, E. Numerical insights on ionic microgels: Structure and swelling behaviour. _Soft Matter_ 2019, _15_ , 8113–8128
  * Del Monte et al. 2020 Del Monte, G.; Camerin, F.; Ninarello, A.; Gnan, N.; Rovigatti, L.; Zaccarelli, E. Charge affinity and solvent effects in numerical simulations of ionic microgels. _J. Condens. Matter Phys._ 2020, _33_ , 084001
  * Scotti et al. 2018 Scotti, A.; Brugnoni, M.; Rudov, A. A.; Houston, J. E.; Potemkin, I. I.; Richtering, W. Hollow microgels squeezed in overcrowded environments. _J. Chem. Phys._ 2018, _148_ , 174903
  * Soddemann et al. 2001 Soddemann, T.; Dünweg, B.; Kremer, K. A generic computer model for amphiphilic systems. _Eur. Phys. J. E_ 2001, _6_ , 409–419
  * Plimpton 1995 Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular Dynamics. _J. Comput. Phys._ 1995, _117_ , 1–19
  * Deserno and Holm 1998 Deserno, M.; Holm, C. How to mesh up Ewald sums. I. A theoretical and numerical comparison of various particle mesh routines. _J. Chem. Phys._ 1998, _109_ , 7678–7693
  * Hubbard and Douglas 1993 Hubbard, J. B.; Douglas, J. F. Hydrodynamic friction of arbitrarily shaped Brownian particles. _Phys. Rev. E_ 1993, _47_ , R2983
  * Rovigatti et al. 2019 Rovigatti, L.; Gnan, N.; Ninarello, A.; Zaccarelli, E. Connecting elasticity and effective interactions of neutral microgels: The validity of the Hertzian model. _Macromolecules_ 2019, _52_ , 4895–4906
  * Chremos et al. 2022 Chremos, A.; Horkay, F.; Douglas, J. F. Influence of network defects on the conformational structure of nanogel particles: From “closed compact” to “open fractal” nanogel particles. _J. Chem. Phys._ 2022, _156_ , 094903
  * Douglas et al. 1994 Douglas, J. F.; Zhou, H.-X.; Hubbard, J. B. Hydrodynamic friction and the capacitance of arbitrarily shaped objects. _Phys. Rev. E_ 1994, _49_ , 5319
  * Franco et al. 2021 Franco, S.; Buratti, E.; Ruzicka, B.; Nigro, V.; Zoratto, N.; Matricardi, P.; Zaccarelli, E.; Angelini, R. Volume fraction determination of microgel composed of interpenetrating polymer networks of PNIPAM and polyacrylic acid. _J. Phys.: Condens. Matter_ 2021, _33_ , 174004
  * Kröger et al. 2017 Kröger, L. C.; Kopp, W. A.; Leonhard, K. Prediction of Chain Propagation Rate Constants of Polymerization Reactions in Aqueous NIPAM/BIS and VCL/BIS Systems. _J. Phys. Chem. B_ 2017, _121_ , 2887–2895
  * Beuermann and Buback 2002 Beuermann, S.; Buback, M. Rate coefcients of free-radical polymerization deduced from pulsed laser experiments. _Prog. Polym. Sci._ 2002, 64
  * O’Brien and White 1978 O’Brien, R. W.; White, L. R. Electrophoretic mobility of a spherical colloidal particle. _J. Chem. Soc., Faraday Trans. 2_ 1978, _74_ , 1607
  * Zhou and Schmid 2015 Zhou, J.; Schmid, F. Computer simulations of single particles in external electric fields. _Soft Matter_ 2015, _11_ , 6728–6739
  * Hunter 2013 Hunter, R. J. _Zeta potential in colloid science: principles and applications_ ; Academic press, 2013; Vol. 2
  * Farooqi et al. 2017 Farooqi, Z. H.; Khan, H. U.; Shah, S. M.; Siddiq, M. Stability of poly(N-isopropylacrylamide-co-acrylic acid) polymer microgels under various conditions of temperature, pH and salt concentration. _Arab. J. Chem._ 2017, _10_ , 329–335
  * Bonham et al. 2017 Bonham, J. A.; Waggett, F.; Faers, M. A.; van Duijneveldt, J. S. The role of initiator on the dispersibility of polystyrene microgels in non-aqueous solvents. _Colloid Polym Sci_ 2017, _295_ , 479–486
  * Pelton et al. 1989 Pelton, R. H.; Pelton, H. M.; Morphesis, A.; Rowell, R. L. Particle sizes and electrophoretic mobilities of poly(N-isopropylacrylamide) latex. _Langmuir_ 1989, _5_ , 816–818
  * Mohanty et al. 2017 Mohanty, P. S.; Nöjd, S.; van Gruijthuijsen, K.; Crassous, J. J.; Obiols-Rabasa, M.; Schweins, R.; Stradner, A.; Schurtenberger, P. Interpenetration of polymeric microgels at ultrahigh densities. _Sci Rep_ 2017, _7_ , 1487
  * Bergman et al. 2018 Bergman, M. J.; Gnan, N.; Obiols-Rabasa, M.; Meijer, J.-M.; Rovigatti, L.; Zaccarelli, E.; Schurtenberger, P. A new look at effective interactions between microgel particles. _Nat Commun_ 2018, _9_ , 5039
  * Howe et al. 2009 Howe, A. M.; Desrousseaux, S.; Lunel, L. S.; Tavacoli, J.; Yow, H. N.; Routh, A. F. Anomalous viscosity jump during the volume phase transition of poly(N-isopropylacrylamide) particles. _Adv. Colloid Interface Sci._ 2009, _147-148_ , 124–131
  * Chaudhary et al. 2021 Chaudhary, G.; Ghosh, A.; Kang, J. G.; Braun, P. V.; Ewoldt, R. H.; Schweizer, K. S. Linear and nonlinear viscoelasticity of concentrated thermoresponsive microgel suspensions. _J. Colloid Interface Sci._ 2021, _601_ , 886–898


we may restrict their sign.

For example, if $n=2$, then we can build a block-diagonal matrix of signature
$(2,1)$ from elementary blocks as follows:

  * •

$(0,1)\oplus(1,0)\oplus(1,0)$

  * •

$(0,1)\oplus(2,0)$

  * •

$(1,1)\oplus(1,0)$

  * •

$(2,1)$

This results in the following normal forms for matrices in
$\mathfrak{so}(2,1)$:

$\begin{pmatrix}0&0&0\\\ 0&0&0\\\
0&0&0\end{pmatrix}\qquad\begin{pmatrix}0&1&0\\\ 1&0&\pm 1\\\ 0&\pm
1&0\end{pmatrix}\qquad\begin{pmatrix}0&0&0\\\ 0&0&\varphi\\\
0&-\varphi&0\end{pmatrix}\qquad\begin{pmatrix}0&\beta&0\\\ \beta&0&0\\\
0&0&0\end{pmatrix}.$ (D.1)

The two irreducible blocks cannot be related under the identity component
$SO(n,1)_{0}$, and neither can the sign of $\varphi$ be changed. However,
there is a proper orthochronous Lorentz transformation which changes the sign
of $\beta$ in the last block, so that we can actually take $\beta>0$. Because
the Killing form for $\mathfrak{so}(2,1)$ has signature $(2,1)$, the coadjoint
orbits of $\operatorname{SO}(2,1)_{0}$ coincide with the orbits in a three-
dimensional lorentzian vector $(V,\eta)$ space under the proper orthochronous
Lorentz transformations: namely,

  * •

the origin, corresponding to the zero matrix;

  * •

the future and past deleted lightcones $\eta(v,v)=0$, corresponding to the
matrices $\begin{pmatrix}0&1&0\\\ 1&0&\pm 1\\\ 0&\pm 1&0\end{pmatrix}$;

  * •

the upper and lower sheets of the two-sheeted hyperboloids
$\eta(v,v)=-\varphi^{2}$, corresponding to the matrices
$\begin{pmatrix}0&0&0\\\ 0&0&\varphi\\\ 0&-\varphi&0\end{pmatrix}$ for
$\pm\varphi>0$; and

  * •

the one-sheeted hyperboloid $\eta(v,v)=\beta^{2}$, corresponding to the
matrices $\begin{pmatrix}0&\beta&0\\\ \beta&0&0\\\ 0&0&0\end{pmatrix}$, for
$\beta>0$.

This of course is completely elementary and is the lorentzian analogue of the
decomposition of three-dimensional euclidean space under the group of
rotations into the origin and the spheres of radius $r$ for $r>0$.

To illustrate further, let us now consider $n=3$ and discuss the coadjoint
orbits of $\operatorname{SO}(3,1)_{0}$. Using the elementary blocks in Table
8, we can build a block diagonal matrix of signature $(3,1)$ as follows:

  * •

$(0,1)\oplus(1,0)\oplus(1,0)\oplus(1,0)$

  * •

$(2,1)\oplus(1,0)$

  * •

$(1,1)\oplus(2,0)$

  * •

$(1,1)\oplus(1,0)\oplus(1,0)$

  * •

$(0,1)\oplus(1,0)\oplus(2,0)$

This results in the following normal forms for matrices in
$\mathfrak{so}(3,1)$:

$\begin{pmatrix}0&1&0&0\\\ 1&0&\pm 1&0\\\ 0&\mp 1&0&0\\\
0&0&0&0\end{pmatrix}\qquad\begin{pmatrix}0&\beta&0&0\\\ \beta&0&0&0\\\
0&0&0&\varphi\\\
0&0&-\varphi&0\end{pmatrix}\qquad\begin{pmatrix}0&\beta&0&0\\\ \beta&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\end{pmatrix}\qquad\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\
0&0&0&\varphi\\\ 0&0&-\varphi&0\end{pmatrix}$ (D.2)

in addition to the zero matrix. Now conjugation with
$\operatorname{SO}(3,1)_{0}$ relates the versions of the first matrix with the
different signs, so we need only consider one of them. Similarly, in the last
two normal forms, we need only keep $\beta>0$ and $\varphi>0$. In the second
normal form, conjugation changes the signs of $\beta$ and $\varphi$
simultaneously, hence we can take $\beta>0$ and $\varphi$ nonzero but
otherwise unconstrained.

## References

  * [1] S. A. Hartnoll, A. Lucas, and S. Sachdev, “Holographic quantum matter,” arXiv:1612.07324 [hep-th].
  * [2] N. Iqbal, H. Liu, and M. Mezei, “Lectures on holographic non-Fermi liquids and quantum phase transitions,” in Theoretical Advanced Study Institute in Elementary Particle Physics: String theory and its Applications: From meV to the Planck Scale.  10, 2011.  arXiv:1110.3814 [hep-th].
  * [3] I. Arav, S. Chapman, and Y. Oz, “Lifshitz Scale Anomalies,” JHEP 02 (2015) 078, arXiv:1410.5831 [hep-th].
  * [4] M. Taylor, “Lifshitz holography,” Class. Quant. Grav. 33 (2016), no. 3, 033001, arXiv:1512.03554 [hep-th].
  * [5] M. Mohammadi and A. Sheykhi, “Lifshitz scaling effects on the holographic p-wave superconductors coupled to nonlinear electrodynamics,” Eur. Phys. J. C 80 (2020), no. 10, 928, arXiv:2010.07105 [hep-th].
  * [6] I. Arav, Y. Oz, and A. Raviv-Moshe, “Holomorphic Structure and Quantum Critical Points in Supersymmetric Lifshitz Field Theories,” JHEP 11 (2019) 064, arXiv:1908.03220 [hep-th].
  * [7] C. Charmousis, B. Gouteraux, B. S. Kim, E. Kiritsis, and R. Meyer, “Effective Holographic Theories for low-temperature condensed matter systems,” JHEP 11 (2010) 151, arXiv:1005.4690 [hep-th].
  * [8] E. J. Brynjolfsson, U. H. Danielsson, L. Thorlacius, and T. Zingg, “Black Hole Thermodynamics and Heavy Fermion Metals,” JHEP 08 (2010) 027, arXiv:1003.5361 [hep-th].
  * [9] Y. Cao, D. Chowdhury, D. Rodan-Legrain, O. Rubies-Bigorda, K. Watanabe, T. Taniguchi, T. Senthil, and P. Jarillo-Herrero, “Strange metal in magic-angle graphene with near planckian dissipation,” Phys. Rev. Lett. 124 (Feb, 2020) 076801.
  * [10] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-Herrero, “Correlated insulator behaviour at half-filling in magic-angle graphene superlattices,” Nature 556 (Apr., 2018) 80–84, arXiv:1802.00553 [cond-mat.mes-hall].
  * [11] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, “Unconventional superconductivity in magic-angle graphene superlattices,” Nature 556 (2018), no. 7699, 43–50.
  * [12] S. Sachdev, “Condensed Matter and AdS/CFT,” Lect. Notes Phys. 828 (2011) 273–311, arXiv:1002.2947 [hep-th].
  * [13] G. A. Inkof, J. M. C. Küppers, J. M. Link, B. Goutéraux, and J. Schmalian, “Quantum critical scaling and holographic bound for transport coefficients near Lifshitz points,” JHEP 11 (2020) 088, arXiv:1907.05744 [cond-mat.str-el].
  * [14] M. R. Mohammadi Mozaffar and A. Mollabashi, “Holographic quantum critical points in Lifshitz space-time,” JHEP 04 (2013) 081, arXiv:1212.6635 [hep-th].
  * [15] J. Erdmenger, I. Matthaiakakis, R. Meyer, and D. Rodríguez Fernández, “Strongly coupled electron fluids in the Poiseuille regime,” Phys. Rev. B 98 (2018), no. 19, 195143, arXiv:1806.10635 [cond-mat.mes-hall].
  * [16] H.-S. Jeong, Y. Ahn, D. Ahn, C. Niu, W.-J. Li, and K.-Y. Kim, “Thermal diffusivity and butterfly velocity in anisotropic Q-Lattice models,” JHEP 01 (2018) 140, arXiv:1708.08822 [hep-th].
  * [17] O. Trovarelli, C. Geibel, S. Mederle, C. Langhammer, F. M. Grosche, P. Gegenwart, M. Lang, G. Sparn, and F. Steglich, “${\mathrm{ybrh}}_{2}{\mathrm{si}}_{2}$: Pronounced non-fermi-liquid effects above a low-lying magnetic phase transition,” Phys. Rev. Lett. 85 (Jul, 2000) 626–629.
  * [18] J. A. N. Bruin, H. Sakai, R. S. Perry, and A. P. Mackenzie, “Similarity of scattering rates in metals showing t-linear resistivity,” Science 339 (2013), no. 6121, 804–807, https://science.sciencemag.org/content/339/6121/804.full.pdf.
  * [19] M. Baggio, J. de Boer, and K. Holsheimer, “Anomalous Breaking of Anisotropic Scaling Symmetry in the Quantum Lifshitz Model,” JHEP 07 (2012) 099, arXiv:1112.6416 [hep-th].
  * [20] S. A. Hartnoll, “Lectures on holographic methods for condensed matter physics,” Class. Quant. Grav. 26 (2009) 224002, arXiv:0903.3246 [hep-th].
  * [21] S. Sachdev, “What can gauge-gravity duality teach us about condensed matter physics?,” Ann. Rev. Condensed Matter Phys. 3 (2012) 9–33, arXiv:1108.1197 [cond-mat.str-el].
  * [22] J. Zaanen, Y. Liu, Y.-W. Sun, and K. Schalm, Holographic Duality in Condensed Matter Physics.  Cambridge University Press, 2015.
  * [23] H. Gong, P. Liu, G. Fu, X.-M. Kuang, and J.-P. Wu, “Informational properties of holographic Lifshitz field theory,” Chin. Phys. C 45 (2021), no. 6, 065101, arXiv:2009.00450 [hep-th].
  * [24] M. R. Mohammadi Mozaffar and A. Mollabashi, “Entanglement in Lifshitz-type Quantum Field Theories,” JHEP 07 (2017) 120, arXiv:1705.00483 [hep-th].
  * [25] T. He, J. M. Magan, and S. Vandoren, “Entanglement Entropy in Lifshitz Theories,” SciPost Phys. 3 (2017), no. 5, 034, arXiv:1705.01147 [hep-th].
  * [26] G. T. Horowitz, “Introduction to Holographic Superconductors,” Lect. Notes Phys. 828 (2011) 313–347, arXiv:1002.1722 [hep-th].
  * [27] M. Ammon and J. Erdmenger, Gauge/Gravity Duality: Foundations and Applications.  Cambridge University Press, 2015.
  * [28] S. Kachru, X. Liu, and M. Mulligan, “Gravity Duals of Lifshitz-like Fixed Points,” Phys.Rev. D78 (2008) 106005, arXiv:0808.1725 [hep-th].
  * [29] J. Hartong, E. Kiritsis, and N. A. Obers, “Schrödinger Invariance from Lifshitz Isometries in Holography and Field Theory,” Phys. Rev. D 92 (2015) 066003, arXiv:1409.1522 [hep-th].
  * [30] J. Figueroa-O’Farrill and S. Prohazka, “Spatially isotropic homogeneous spacetimes,” JHEP 01 (2019) 229, arXiv:1809.01224 [hep-th].
  * [31] J. M. Figueroa-O’Farrill, “Conformal Lie algebras via deformation theory,” J. Math. Phys. 60 (2019), no. 2, 021702, arXiv:1809.03603 [hep-th].
  * [32] L. Bianchi, “Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti,” Memorie di Matematica e di Fisica della Societa Italiana delle Scienze, Serie Terza, Tomo XI (1898) 267–352.
  * [33] J.-M. Souriau, Structure of dynamical systems, vol. 149 of Progress in Mathematics.  Birkhäuser Boston, Inc., Boston, MA, 1997.  A symplectic view of physics, Translated from the French by C. H. Cushman-de Vries, Translation edited and with a preface by R. H. Cushman and G. M. Tuynman.
  * [34] N. Jokela, J. Järvelä, and A. V. Ramallo, “Non-relativistic anyons from holography,” Nucl. Phys. B 916 (2017) 727–768, arXiv:1605.09156 [hep-th].
  * [35] J. Figueroa-O’Farrill, R. Grassie, and S. Prohazka, “Geometry and BMS Lie algebras of spatially isotropic homogeneous spacetimes,” JHEP 08 (2019) 119, arXiv:1905.00034 [hep-th].
  * [36] H. Weyl, The classical groups.  Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1997.  Their invariants and representations, Fifteenth printing, Princeton Paperbacks.
  * [37] A. Beckett and J. Figueroa-O’Farrill, “Symplectic actions and central extensions,” arXiv:2203.nnnn [math.DG].
  * [38] W. T. van Est, “Group cohomology and Lie algebra cohomology in Lie groups. I, II,” Nederl. Akad. Wetensch. Proc. Ser. A. 56 = Indagationes Math. 15 (1953) 484–492, 493–504.
  * [39] W. T. van Est, “On the algebraic cohomology concepts in Lie groups. I, II,” Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955) 225–233, 286–294.
  * [40] E. Meinrenken and M. A. Salazar, “Van Est differentiation and integration,” Math. Ann. 376 (2020), no. 3-4, 1395–1428.
  * [41] J. Gomis, “Lifshitz and Schrödinger Algebras and Dynamical Realizations.” 2011 Melbourne Seminar.
  * [42] B. Oblak, BMS Particles in Three Dimensions.  PhD thesis, Brussels U., 2016.  arXiv:1610.08526 [hep-th].
  * [43] P. Arathoon and J. Montaldi, “Hermitian flag manifolds and orbits of the Euclidean group,” arXiv:1804.09463 [math.RT].
  * [44] A. Galajinsky, “Dynamical realizations of the Lifshitz group,” Phys. Rev. D 105 (2022), no. 10, 106023, arXiv:2201.10187 [hep-th].
  * [45] R. M. Nandkishore and M. Hermele, “Fractons,” Ann. Rev. Condensed Matter Phys. 10 (2019) 295–313, arXiv:1803.11196 [cond-mat.str-el].
  * [46] M. Pretko, X. Chen, and Y. You, “Fracton Phases of Matter,” Int. J. Mod. Phys. A 35 (2020), no. 06, 2030003, arXiv:2001.01722 [cond-mat.str-el].
  * [47] K. T. Grosvenor, C. Hoyos, F. Peña Benitez, and P. Surówka, “Space-Dependent Symmetries and Fractons,” Front. in Phys. 9 (2022) 792621, arXiv:2112.00531 [hep-th].
  * [48] A. Nicolis, R. Rattazzi, and E. Trincherini, “The Galileon as a local modification of gravity,” Phys. Rev. D 79 (2009) 064036, arXiv:0811.2197 [hep-th].
  * [49] T. Griffin, K. T. Grosvenor, P. Horava, and Z. Yan, “Multicritical Symmetry Breaking and Naturalness of Slow Nambu-Goldstone Bosons,” Phys. Rev. D 88 (2013) 101701, arXiv:1308.5967 [hep-th].
  * [50] K. Hinterbichler and A. Joyce, “Goldstones with Extended Shift Symmetries,” Int. J. Mod. Phys. D 23 (2014), no. 13, 1443001, arXiv:1404.4047 [hep-th].
  * [51] T. Griffin, K. T. Grosvenor, P. Horava, and Z. Yan, “Scalar Field Theories with Polynomial Shift Symmetries,” Commun. Math. Phys. 340 (2015), no. 3, 985–1048, arXiv:1412.1046 [hep-th].
  * [52] L. Bidussi, J. Hartong, E. Have, J. Musaeus, and S. Prohazka, “Fractons, dipole symmetries and curved spacetime,” arXiv:2111.03668 [hep-th].
  * [53] A. Jain and K. Jensen, “Fractons in curved space,” SciPost Phys. 12 (2022) 142, arXiv:2111.03973 [hep-th].
  * [54] A. Gromov, “Towards classification of Fracton phases: the multipole algebra,” Phys. Rev. X 9 (2019), no. 3, 031035, arXiv:1812.05104 [cond-mat.str-el].
  * [55] Y. Hirono, M. You, S. Angus, and G. Y. Cho, “A symmetry principle for gauge theories with fractons,” arXiv:2207.00854 [cond-mat.str-el].
  * [56] J. Figueroa-O’Farrill, R. Grassie, and S. Prohazka.  In preparation.
  * [57] J. M. Figueroa-O’Farrill and J. Simón, “Supersymmetric Kaluza-Klein reductions of AdS backgrounds,” Adv. Theor. Math. Phys. 8 (2004), no. 2, 217–317, arXiv:hep-th/0401206.