# Active gels: from chemical microreactor to polymeric actuator

Baptiste Blanc  Department of Physics, Brandeis University, 415 South St.,
Waltham, MA 02454, United States  Zhenkun Zhang  Key Laboratory of Functional
Polymer Materials of Ministry of Education, Institute of Polymer Chemistry,
College of Chemistry, Nankai University, 300071 Tianjin, China  Eric Liu
Department of Chemical and Biological Engineering, Tufts University, Medford,
Massachusetts 02155, United States  Ning Zhou  Department of Chemistry,
Brandeis University, 415 South St., Waltham, MA 02454, United States  Hyunmin
Yi  Department of Chemical and Biological Engineering, Tufts University,
Medford, Massachusetts 02155, United States  Seth Fraden  Department of
Physics, Brandeis University, 415 South St., Waltham, MA 02454, United States

###### Abstract

We report on a new synthesis protocol, experimental characterization and
theoretical modeling of Belousov-Zhabotinsky (BZ) hydrogels. BZ gels combine
the ability to convert chemical energy into actuation with a sensing
capability. Our new two-step synthesis technique allows independent
optimization of the chemical and the mechanical properties of BZ gels. We
demonstrate that the strain of these BZ gels can be made arbitrarily large by
decreasing their bulk modulus, an important result for the design of shape-
changing hydrogels. In addition to strain, we argue that the stress developed
by BZ gels is a better metric for assessing the utility of a given gel for
chemomechanical energy conversion. We highlight the role of the surrounding
media for the occurrence of hydrogel oscillation, which is determined by the
Damköhler number, the ratio of chemical reaction to diffusion rates. These
experimental findings are supported by a Flory-Huggins theory for
polyelectrolyte gels coupled to a Vanag-Epstein model of BZ chemistry.

###### pacs:

.

##  I Introduction

The existence of living organisms is evidence that functional motile
autonomous materials can be built robustly using macromolecules. However,
assembling polymers that possess these properties remains an open challenge.
Current endeavors in this field can be classified in two main categories. The
first one consists in constructing life-like materials from building blocks
that are taken from the living realm, for instance proteins Litschel ; He ;
Qian , molecular motors DogicFraden ; Waltherreview ; Dogicreview ; Lee , and
cells Ricotti ; Parker , or from synthetic molecules and colloidal particles
Wang2020  because these elements have the coveted capability of autonomously
converting chemical energy into mechanical work. The second one couples
passive engineered polymeric materials such as colloids, hydrogels and
elastomers to external power sources Wehner ; Christianson ; Shin . Although
both of these approaches have shown promising applications, the design of
synthetic functional chemomechanical materials is in its infancy and could
lead to the creation of life-like machines with augmented performance compared
to what evolution has created Katsonis .

In this work, we present a synthetic mechanically active hydrogel driven by a
chemical oscillator in which the mechanical and chemical properties can be
separately optimized. Inspired by Yoshida’s work Yoshida1996(2) , we create
acrylamide hydrogels doped with the catalyst of the Belousov Zhabotinsky (BZ)
reaction Zhabotinsky1964 . When placed in an aqueous solution containing the
ingredients of the BZ reaction, these gels cyclically convert chemical energy
into mechanical work. The simplest description of the BZ gel mechanism
consists in separating the BZ chemical reaction from its actuation. In the BZ
reaction, an autocatalytic activator drives the catalyst to its oxidized state
and generates an inhibitor that returns the catalyst to its reduced state
Noyes1972 . The cyclic change of the oxidation state of the catalyst induces
the oscillatory swelling of the gel.

Over the past 25 years, material scientists have strived to maximize the
oscillatory strain amplitude of BZ gels. Controlling strain is crucial for the
design of shape changing functional elements, such as bimorphs Na2015 ;
Levin2020 ; Li2021 . The composition of the BZ gels Aizenberg2018 ; Kramb2015
; Masuda2015 , the architecture of the catalyst Aizenberg2018 ; Zhang2013 ,
the charge of the monomer Maeda2007 ; Aizenberg2018 , the porous structure of
the hydrogel Aizenberg2018 ; Takeoka2003 ; Suzuki2012 ; Hu2017 , the shape of
the gels Yuan2013 ; Chen2011  and the chemical environment of the gel
Buskohl2016  have been reported to influence the swelling of the BZ gel. Here,
we provide experimental and theoretical evidence supporting that the strain of
a BZ gel can be made arbitrarly large by decreasing its bulk modulus,
elucidating a system-independant mechanism for increasing the strain of active
gels.

Figure 1: (a) Schematic of the gel synthesis process. (b) Bright-field image
of pre-gel drops on a PDMS mold before UV exposure. The drops are an aqueous
solution of 15$\%$ acrylamide (AAm), 1.5$\%$ acrylic acid (AAc), 1$\%$
Bisacrylamide (Bis). The outer solution is hexadecane. (c) Relation between
the volume of the micropatterned wells of the PDMS mold and the volume of the
crosslinked gel. The line represents the theoretical prediction assuming that
the volume of the gel is proportional to the volume of the micropatterned
wells of the PDMS. (d) Bright-field image of 200$\mu$m diameter gels of
composition 15$\%$$\mathrm{AAm}$, 1.5$\%$$\mathrm{AAc}$, 1$\%$$\mathrm{Bis}$
coupled with the ruthenium catalyst. (e) Chemical structure of the ruthenium
catalyst synthesized in this study:
Bis(2,$2^{{}^{\prime}}$-bipyridine)-$4^{{}^{\prime}}$-methyl-4-amido
ethylamine-ruthenium bis(hexafluorophosphate), noted
$\mathrm{Ru(bpy)_{2}(bpyEtAm)(PF_{6})_{2}}$.

However, to optimize the properties of an active gel for engineering
applications, we argue that it is necessary to characterize both the strain
and the stress fields. To understand why, conceptualize the active gel as a
spring in which an active force ($F$) drives a displacement ($x$) that is
proportional to the passive restoring force of the gel ($k$), e.g. $F=kx$.
When a BZ gel experiences a large strain, e.g. $x=F/k$, upon oxidation of the
ruthenium catalyst, one can not distinguish if this is caused by a gel having
a low bulk modulus (small $k$) or by a large active stress (large $F$) caused
by the oxidation of the catalyst. Measuring the stress and the strain allows
one to disentangle the role of the chemical driving forces from the elastic
restorative forces of the gel. Therefore the stress, a quantity so far
understudied in the field of BZ gels, is of central importance because it
directly couples to molecular properties of the polymer Hara2014 . Here we
simultaneously measure the gels’ strain and stress and we demonstrate that
varying the catalyst content in the gel allows tuning of the stress.

We also study several out-of-equilibrium properties of oscillating BZ gels and
demonstrate experimentally and theoretically that the radius of a spherical
gel affects the period of the BZ reaction and even controls whether or not the
gel even oscillates. We measured the timescale of BZ gel swelling as a
function of radius and demonstrate that the factor limiting the strain
amplitude is the diffusive nature of gel swelling, which has a long timescale
compared to the shorter period of the BZ oscillation. We finally discuss a
simple way to overcome this impediment.

##  II Results and discussion

Spherical BZ hydrogels are fabricated through a two-step process (Figure
1)Masuda2015 ; Smith2012 ; Kramb2014 . First, we create a hydrogel by
simultaneously polymerizing mixtures of inert and functional monomers in
predetermined stoichiometric ratios and crosslinking the polymers together.
Second, we attach a pendant BZ catalyst to the functional groups. This two-
step process is designed to allow the production of BZ gels with independent
control over their mechanical properties, their chemically oscillating
behavior and their geometry. The stiffness of the gel can be systematically
controlled by varying the cross-link density before introducing the catalyst,
while the catalyst concentration is set by the concentration of functional
monomers. Control over the size of the spherical gel is accomplished by
combining an emulsion polymerization approach and a molding process Liu2016 .
Using this protocol, we created gels from 100$\mu$m to 600$\mu$m in diameter
with a bulk modulus as soft as 10 kPa. Notable desirable features of the two-
step process are the ability to homogeneously incorporate the ruthenium
trisbipyridine catalyst in the gel and to create gels with a higher catalyst
concentration than the solubility limit. Here we report homogeneously
incorporating synthetized $\mathrm{Ru(bpy)_{2}(bpyEtAm)(PF_{6})_{2}}$ in the
gel with catalyst concentrations up to 12mM, which is significantly higher
that the solubility limit in water (Supplementary SM 2, SM 3).

###  II.1 Optimizing the strain upon oxidation of BZ gels

The strain of a BZ gel is the traditional metric to assess the chemomechanical
energy conversion because the mechanical strain energy density depends
quadratically on this quantity and because shape changing elements such as
bimorphs rely on bending deformations that are controlled by the strain Na2015
. Therefore, we first investigate the BZ hydrogels’ composition that maximizes
strain upon oxidation of the ruthenium catalyst. Gels are immersed in
oxidative solutions made of sodium bromate and sulfuric acid at
concentrations, [NaBrO3]=0.08M and [H2SO4]=0.4M, for which the ruthenium
catalyst is strongly oxidized. The strain induced by the oxidation of the BZ
gel is measured by recording its radius change when its catalyst concentration
and its crosslinking density are varied independently. We choose the maximal
radius during swelling as a metric of responsiveness of the hydrogel upon
oxidation and assume this quantity to be the equilibrium size of the hydrogel
when oxidized (Supplementary SM 4, SM 5). With this metric, the strain of a
gel upon oxidation is shown to increase with catalyst concentration (Figure
2-a), while this strain decreases with increasing crosslinker density in the
solution (Figure 2-b).

We interpret these experimental results in light of the Flory-Huggins
equilibrium theory for polyelectrolyte gels Doi2013 . In this model we
conceptualize the gel as an entropic spring that balances the solvation forces
generated by the polymer’s interaction with water and the osmotic stress from
the counterions. We imagine that the solvation and counterion pressures vary
with the oxidation state of the catalyst of the BZ chemical reaction, but the
properties of the entropic spring do not.

Specifically, we consider that the equilibrium size of a gel results from a
balance between its entropic elasticity
$\it{G_{\mathrm{0}}(\frac{\it{\Phi}}{\it{\Phi_{\mathrm{0}}}})^{\frac{1}{3}}}$
and the sum of the osmotic pressure of the polymer
$\it{\Pi_{\mathrm{pol}}=\frac{\mathrm{k_{B}}T}{\nu_{v}}\mathrm{(\frac{1}{2}}-\chi\mathrm{)}\Phi^{2}}$
and of that of the counterions
$\it{\Pi_{\mathrm{ions}}=\mathrm{k_{B}}T\frac{c_{\mathrm{b}}^{\mathrm{2}}}{\mathrm{4}c_{\mathrm{s}}}}$,
with $\it{G_{\mathrm{0}}}$ the shear modulus of the gel in the reference
state, $\it{\Phi_{\mathrm{0}}}$ the volume fraction of the gel in the
reference state, $\it{\Phi}$ the volume fraction of the gel, $\mathrm{k_{B}}$
the Boltzmann constant, $\it{T}$ the temperature, $\mathrm{\nu_{\mathrm{v}}}$
the volume of a solvent molecule or polymer, assumed to be the same here, and
thus independent of the oxidative state of the ruthenium catalyst, $\it{\chi}$
the adimensional parameter characterizing the interaction between the solvent
and the polymer, $\it{c_{\mathrm{b}}}$ the number concentration of charges of
the polymer and $\it{c_{\mathrm{s}}}$ the salt number concentration.

We derive two expressions for the elastic stress $\it{\Pi_{\mathrm{elastic}}}$
of a spherical gel of radius $\it{R}$ as a function of its radial change
$\Deltaup\it{R}$ during oxidation of the ruthenium catalyst. The first one
relates $\it{\Pi_{\mathrm{elastic}}}$ to the bulk modulus of the gel
$\it{\kappa}$ (Eqn. 1). The second one is deduced from the Flory-Huggins
theory and states that $\it{\Pi_{\mathrm{elastic}}}$ is the sum of the change
of counterions osmotic stress $\it{\mathrm{\Delta}\Pi_{\mathrm{ions}}}$ and of
the change of polymer osmotic stress $\it{\mathrm{\Delta}\Pi_{\mathrm{pol}}}$
caused by the oxidation of the ruthenium catalyst (Eqn. 2,3,4) (Supplementary
SM 9):

$\it{\Pi_{\mathrm{elastic}}}=\mathrm{3\it{\kappa}\frac{\mathrm{\Delta}\it{R}}{\it{R}}}$
(1)

$\it{\Pi_{\mathrm{active}}}=\it{\mathrm{\Delta}\Pi_{\mathrm{ions}}}+\it{\mathrm{\Delta}\Pi_{\mathrm{pol}}}$
(2)

$\it{\Pi_{\mathrm{active}}}=\mathrm{k_{B}}T\frac{\mathrm{5}c_{\mathrm{Ru^{2+}}}^{\mathrm{2}}}{\mathrm{4}c_{\mathrm{s}}}+\mathrm{k_{B}}\it{T}\frac{\it{\Phi_{\mathrm{red}}^{2}}}{\nu_{v}}(\chi_{\mathrm{red}}-\chi_{\mathrm{ox}})$
(3)

$\it{\Pi_{\mathrm{elastic}}}=\it{\Pi_{\mathrm{active}}}$ (4)

with $\it{\Phi_{\mathrm{red}}}$ its volume fraction when the catalyst is in
its reduced state, $\it{c_{\mathrm{Ru^{2+}}}}$ the concentration of ruthenium
catalyst in the gel, $\it{\chi_{\mathrm{red}}}$ and $\it{\chi_{\mathrm{ox}}}$,
the $\it{\chi}$ parameters of the polymer containing the reduced and oxidized
form of the ruthenium catalyst, respectively.

We experimentally verified the relationship of Eqn. 1,
$\it{\Pi_{\mathrm{elastic}}}=\mathrm{3\it{\kappa}\frac{\mathrm{\Delta}\it{R}}{\it{R}}}$,
by varying $\it{\Pi_{\mathrm{elastic}}}$ using large molecular weight Dextran
molecules of known concentration and tabulated osmotic pressure and measuring
$\frac{\mathrm{\Delta}\it{R}}{\it{R}}$. We observed a linear dependence
between $\it{\Pi_{\mathrm{elastic}}}$ and
$\frac{\mathrm{\Delta}\it{R}}{\it{R}}$ and deduced $\it{\kappa}$ from the
slope (see Materials and Methods)Dolega2017 ; Bonnet-Gonnet1994 . For these
experimental parameters, the gel behaves as an elastic spring.

To a first approximation, Eqn. 4 is consistent with the experimental
observation that the swelling of the gel upon its oxidation is inversely
proportional to its bulk modulus $\it{\kappa}$ (Figure 2). Optimizing the
strain can thus be simply achieved by making softer gels. This is an important
result for the design of autonomous shape changing composite hydrogels, where
a bending deformation can be induced by the differential swelling of two
materials Na2015 . A well-known example of such a material is a bilayer, made
from a swellable gel anchored to a thin rigid object. In this case, the
deformation of the material, here a change of curvature, is solely controlled
by its strain Maeda2007 ; Smith2013 .

###  II.2 Optimizing the stress upon oxidation of BZ gels

By definition, the mechanical strain energy density is the product of the
stress and strain. Therefore measuring the stress generated during the
oxidation of different BZ gels is an important quantity to characterize and
compare their chemomechanical energy performance. Moreover, note that the
right hand side of Eqn. 3 does not depend on the bulk modulus $\it{\kappa}$,
which is a passive elastic property of the gel. Thus measurements of stress
$(\it{\Pi_{\mathrm{elastic}}})$ do not depend on $\it{\kappa}$, while
measurements of strain alone, $(\frac{\mathrm{\Delta}\it{R}}{\it{R}})$, do. In
other words, when a BZ gel experiences a large strain
$(\frac{\mathrm{\Delta}\it{R}}{\it{R}})$ upon oxidation of the ruthenium
catalyst, one can not distinguish if this is caused by a gel having a low bulk
modulus or by a large active stress caused by the oxidation of the catalyst.
Measuring $\it{\Pi_{\mathrm{elastic}}}$ allows one to disentangle the role of
the chemical driving forces embodied by the terms involving solvation
$(\chi_{\mathrm{red}}-\chi_{\mathrm{ox}})$ and charged ions
$(\mathrm{Ru^{2+}})$ from the elastic restorative forces characterized by
$\it{\kappa}$.

Figure 2: (a) Relative maximal radius change
$\mathrm{\frac{\Delta\it{R}}{\it{R}}(\%)}$ as a function of catalyst
concentration in mM for the concentration of the crosslinker in
(w/v)(0.05$\mathrm{\%}$) in the pregel solution. The dash line represents the
equation $\mathrm{y\propto x^{2}}$. (b) Relative maximal radius change
$\mathrm{\frac{\Delta\it{R}}{\it{R}}(\%)}$ as a function of the concentration
of the crosslinker in (w/v)($\mathrm{\%}$) in the pregel solution for a
catalyst concentration of 9mM . The bulk modulus is proportional to the cross-
linker concentration (Supplementary SM 8). The dash line represents the
equation $\mathrm{y\propto\frac{1}{x}}$.

We thus first examine the influence of the elasticity of the gel on the stress
generated by the oxidation of the catalyst when the catalyst content is kept
constant at 9mM. We observe that $\it{\Pi_{\mathrm{elastic}}}$ does not depend
strongly on the elasticity of the gel and is limited to 30kPa (Figure 3-a).
The softness of the gel increases both the strain and the chemomechanical
energy conversion, but does not increase much the stress generated by the
oxidation of the catalyst.

We next study the influence of the gel’s catalyst content on
$\it{\Pi_{\mathrm{elastic}}}$. The catalyst concentration is determined by
measuring the absorbance of the gel (see Materials and Methods). We
demonstrate that $\it{\Pi_{\mathrm{elastic}}}$ increases with the catalyst
concentration for a concentration of up to 12mM (Figure 3-b). The stress,
strain and the chemomechanical energy conversion generated by the oxidation of
the BZ gel can therefore be engineered by tuning the catalyst concentration.

We now seek to quantify the role of the two chemical driving forces
$\it{\mathrm{\Delta}\Pi_{\mathrm{pol}}}$ and $\Delta\it{\Pi}_{\mathrm{ions}}$.
In all the analysis that follows, we compute
$\Delta\it{\Pi}_{\mathrm{ions}}=\mathrm{k_{B}}\it{T}\frac{\mathrm{5}\it{c_{\mathrm{Ru^{2+}}}^{2}}}{\mathrm{4}\it{c_{s}}}$
from the measurements of $\it{c_{\mathrm{Ru}^{2+}}}$ and $\it{c_{s}}$. We
assume the veracity of Eqn. 4,
$\it{\Pi_{\mathrm{elastic}}}=\it{\mathrm{\Delta}\Pi_{\mathrm{ions}}}+\it{\mathrm{\Delta}\Pi_{\mathrm{pol}}}$,
and having measured $\it{\Pi_{\mathrm{elastic}}}$, deduce the polymer osmotic
stress,
$\mathrm{\Delta}\it{\Pi_{\mathrm{pol}}}=\mathrm{k_{B}}\it{T}\frac{\it{\Phi_{\mathrm{red}}^{2}}}{\it{\nu_{c}}}(\it{\chi_{\mathrm{red}}}-\it{\chi_{\mathrm{ox}}})$,
which is plotted in Figure 3-a for gels containing 9mM of catalyst. We observe
a slight monotonic variation with the crosslinking concentration. For a gel
with a bulk modulus of 10kPa, $\it{\mathrm{\Delta}\Pi_{\mathrm{pol}}}$
increases with the catalyst concentration, as shown in Figure 3-b. More
importantly, for the polymer under study, we determine that
$\it{\mathrm{\Delta}\Pi_{\mathrm{pol}}}$ is much larger than
$\it{\mathrm{\Delta}\Pi}_{\mathrm{ions}}$, thus contributions to the measured
elastic stress from the counterions is negligible compared to the change of
solvation energy. This result yields insights into the mechanism responsible
for the change of the $\it{\chi}$ parameter of the gels. If one assumed that
charges in the polymer gels interact via a repulsive Debye-Huckel electric
potential, one can calculate the contribution of their electrostatic
repulsions to the $\it{\chi}$ parameter of the hydrogel. This theoretical
argument predicts that an electrostatic osmotic stress
$\it{\Pi}_{\mathrm{elec}}$ is generated by the electrostatic interactions of
the ruthenium catalyst when the ruthenium catalyst loses an electron and is
equal to $\it{\mathrm{\Delta}\Pi}_{\mathrm{ions}}$ in magnitude (Supplementary
SM 9). However, $\it{\mathrm{\Delta}\Pi}_{\mathrm{ions}}$ is much lower than
the experimentally measured $\it{\mathrm{\Delta}\Pi_{\mathrm{pol}}}$. This
proves that more complicated interactions than electrostatic repulsion, e.g.
solvation forces, control the change of $\it{\chi}$ parameter of the BZ gel.

There are also two additional measurements made on these gels which suggest
interactions besides charge play a role in the gel chemomechanics. First, the
bulk modulus of soft gels ($\it{\kappa}\sim\mathrm{10}$kPa) increases linearly
with the catalyst concentration (Supplementary SM 8). This observation
demonstrates that the ruthenium monomer causes the polymer behavior to deviate
from that of an ideal Gaussian polymer chain, as the elasticity of a
crosslinked ideal Gaussian polymer chain is simply proportional to the
concentration of crosslinker. The catalyst is thought to be connected to the
polymer backbone as a pendant chain and therefore is not expected to affect
the gel elasticity. Second, the equilibrium radius of the hydrogel decreases
as the ruthenium catalyst content increases (Supplementary SM 10). This
observation emphasizes the unexpected character of the ruthenium
trisbipyridine molecule. A higher concentration of ruthenium catalyst in the
gel is expected to both generate a higher concentration of counterions and a
higher electrostatic repulsive interaction between ruthenium molecules. These
effects should induce swelling of the gel, in stark contradiction with our
experimental observations. Previous studies similarly highlight the importance
of forces other than simple electrostatic interactions in the equilibrium
properties of the gels. Indeed, BZ gels have been shown to shrink upon
oxidation when multivalent ruthenium catalyst is used as a crosslinker, or
when specific monomers compose the gels Aizenberg2018 ; Zhang2013 ; Yuan2013 ;
Kramb2015 . Taken together, these observations spotlight the crucial role of
the gel’s molecular characteristics in its swelling behavior and the daunting
challenge of explaining its thermodynamic equilibrium properties with entropy
and electrostatic interactions as the only two physical effects.

Nonetheless, in the framework of the Flory-Huggins model, the effect of these
additional molecular interactions on the macroscopic properties of BZ gels can
be accounted for by introducing the $\chi$ parameter. Therefore, we
experimentally determine a quantitative estimate of the change of the
$\it{\chi}$ parameter of the gel during its oxidation,
$\it{\delta\chi=\chi_{\mathrm{red}}-\chi_{\mathrm{ox}}}$, as a function of the
catalyst and polymer concentration (Figure 3-c). To do so, we divide the
osmotic stress generated by the polymer by the square of the density of the
gel $\it{\rho}$. Estimations of gel’s density as a function of the crosslinker
concentration in solution and as a function of its catalyst concentration are
respectively obtained by recording the free fall of the hydrogel in a water
column and by measuring its relative change of equilibrium radius
(Supplementary SM 10). For gels that have a catalyst concentration of
$\sim\mathrm{9mM}$, we measure that $\it{\delta\chi}$ is nearly constant with
the polymer concentration, ie the gel density (Inset-Figure 3-c). For gels
that have a density of $\sim$20$\mathrm{kg.m^{-3}}$, which corresponds to a
concentration of polymer of $\mathrm{0.4M}$, we find that $\it{\delta\chi}$
increases linearly with the catalyst concentration (Figure 3-c). Assuming that
$\it{\delta\chi}=\it{\delta\chi}_{Ru}\times p$ with p the fraction of
ruthenium monomer to acrylamide monomer, we find that the order of magnitude
of the energy change per molecules of ruthenium catalyst due to these
molecular attractive interactions $\it{\delta\chi}_{Ru}$ is around
$4\mathrm{k_{B}}\it{T}$.

As a conclusion to this part, we demonstrated that the strain generated during
the oxidation of BZ gels can be tuned by increasing its catalyst concentration
and decreasing its bulk modulus, an important result for the design of shape
changing hydrogels. Additionally, we demonstrated that the stress (eq. 3)
generated by BZ gels during the oxidation of the ruthenium catalyst is an
appropriate metric for the assessment of actuation performance because the
stress is directly related to the active chemical processes and does not
depend on the passive bulk modulus of the gel, in contrast to the strain.
Furthermore, the active stress upon oxidation of the gel is shown to be mainly
controlled by solvation energy change, of entropic and/or enthalpic origins,
captured by a change of the gel’s $\mathrm{\chi}$ parameter.

Figure 3: The stresses defined in this figure are related as
$\it{\Pi_{\mathrm{elastic}}}=\Delta\it{\Pi_{\mathrm{ions}}}+\Delta\it{\Pi_{\mathrm{pol}}}$.
(a) Elastic stress $\it{\Pi_{\mathrm{elastic}}}$ (red), osmotic stress of the
polymer $\Delta\it{\Pi_{\mathrm{pol}}}$ (blue), expected osmotic stress of the
counterions $\Delta\it{\Pi_{\mathrm{ions}}}$ (black) generated by the
oxidation of the ruthenium catalyst as a function of crosslinker concentration
in solution for gels with a catalyst concentration of 9mM. (b) Elastic stress
$\it{\Pi_{\mathrm{elastic}}}$ (red), osmotic stress of the polymer
$\Delta\it{\Pi_{\mathrm{pol}}}$ (blue), expected osmotic stress of the
counterions $\Delta\it{\Pi_{\mathrm{ions}}}$ (black) generated by the
oxidation of the ruthenium catalyst as a function of catalyst concentration
for gels with a quantity of crosslinker in solution of $0.05\%$. (c)
$\mathrm{\delta\chi=\chi_{\mathrm{red}}-\chi_{\mathrm{ox}}}$ as a function of
the catalyst concentration in the BZ hydrogel.
Inset:$\mathrm{\delta\chi=\chi_{\mathrm{red}}-\chi_{\mathrm{ox}}}$ as a
function of the polymer concentration in the BZ hydrogel.

###  II.3 Influence of hydrogel size on BZ oscillatory chemistry

We next focus on optimizing the autonomous cyclic swelling of the BZ gel. For
a BZ hydrogel to fully swell during each BZ chemical cycle, one needs to
ensure that the swelling timescale of the gel is shorter than the time the gel
stays in the oxidized and reduced state, subject to the additional constraint
that the BZ reaction keeps oscillating. The swelling dynamics of a hydrogel
follows a diffusion-like process consisting of the polymer network diffusing
in a viscous solvent, with a characteristic time proportional to the square of
the diameter of the gel Doi2009 . Therefore, decreasing the radius of the BZ
gel reduces its timescale of swelling. However, supporting chemical
oscillation within gel beads requires that the rate of reaction producing
chemicals inside the gel is faster than the rate of loss of those chemicals
due to diffusion, a term that can depend on the gel’s geometrical properties.
In other words, nondimensional numbers made from the ratio of these two rates,
e.g. Damköhler numbers,
$Da=\frac{\mathrm{diffusion~{}rate}}{\mathrm{reaction~{}rate}}$ predicts that
gels oscillate when $Da<1$ Yoshikawa1998 .

We first investigate how the dynamics of the Belousov Zhabotinsky reaction is
affected by the size of the BZ gel for various BZ solution composition, in
particular the concentration of the oxidizer, sodium bromate. BZ beads of 3
different radii $\mathrm{100\mu m}$, $\mathrm{220\mu m}$ and $\mathrm{285\mu
m}$ composed of $15\%$$\mathrm{AAm}$, $1.5\%$$\mathrm{AAc}$, $1\%$
$\mathrm{Bisacrylamide}$, functionalized with the ruthenium catalyst
($\it{c_{\mathrm{Ru^{2+}}}}\sim$10mM) were immersed in a BZ solution
containing [$\mathrm{H_{2}SO_{4}}$]=0.6M sulfuric acid, [$\mathrm{MA}$]=0.2M
malonic acid, [$\mathrm{NaBr}$]=0.1M sodium bromide, and sodium bromate of
concentration varying from 0.16M to 0.42M. The change of color of the catalyst
was recorded as a function of time (Figure 4-a). We observed that when the
bromate concentration is below a critical threshold of
$\it{c}\mathrm{=0.24M}$, the gel does not chemically oscillate but rather
remains in the reduced state (Figure 4-c). For gel beads of radius larger than
$\mathrm{200\mu m}$, the period of oscillation decreases as the bromate
concentration increases. Moreover, we find that a gel of $\mathrm{100\mu m}$
radius never oscillates, even when the bromate concentration is increased
(Figure 4-c). Therefore, we postulate that there exists a critical radius
below which the gel does not oscillate.

To test our hypothesis, we construct a theoretical reaction-diffusion model in
which the gel is coupled to its environment. We combine the Vanag-Epstein
model for the BZ reaction kinetics with an additional loss term due to the
efflux out of the gel of the mobile species [$\mathrm{Br_{2}}$],
[$\mathrm{Br^{-}}$] and [$\mathrm{HBrO_{2}}$] produced in the gel during the
BZ reaction Vanag2009 . This loss term,
$\mathrm{-\alpha\frac{\it{c}_{i}}{\it{R}}}$, is proportional to the
concentration of the species $\mathrm{\it{c}_{i}}$, and inversely proportional
to the radius of the bead $\it{R}$ (see Materials and Methods)Yoshikawa1998 .
In agreement with our experimental observations, this theoretical model
predicts that the period of oscillation decreases when the bromate
concentration increases (Figure 4-d). The rate of change of the oscillation
period increases when the size of the gel bead is smaller. Additionally, when
the gel radius is below a critical threshold, the gel stops chemically
oscillating and remains in the reduced state. The condition to observe
oxidation of the beads simplifies to
$\mathrm{k_{4}=42\times}[\mathrm{H_{2}SO_{4}}]\times[\mathrm{NaBrO_{3}}]=25.2\times[\mathrm{NaBrO_{3}}]>\frac{\alpha}{R}$
(equation 2) (Inset of Figure 4-d). This mathematical expression compares the
production rate of the activator $\mathrm{HBrO_{2}}$ during the oxidation of
the ruthenium catalyst with its loss rate due to its diffusion out of the gel
bead. This model predicts that the critical size arises due to the loss of
activator $\mathrm{HBrO_{2}}$ by diffusion and is consistent with experiment
(Figure 4).

Figure 4: (a) Intensity profile of a 220$\mathrm{\mu m}$ radius BZ gel bead as
a function of time in a BZ solution with a concentration of sodium bromate of
0.42M. (b) The catalyst concentration as a function of time obtained from a
Vanag Epstein model with a diffusive flux coefficient
$\it{\frac{\alpha}{R}}\mathrm{=5s^{-1}}$ and with a concentration of sodium
bromate of 0.42M. The model shows steady oscillation. (c) Experiment. Period
of the BZ gel chemical oscillation as a function of the bromate concentration
for different size of beads. Gels of 100$\mathrm{\mu m}$, 220$\mathrm{\mu m}$,
285$\mathrm{\mu m}$ radius are represented respectively in red, green and
blue. Disks and crosses represent chemically oscillating and non oscillating
beads respectively. (d) Theory. Period of the BZ gel oscillation as a function
of bromate concentration for various amplitudes of the diffusive flux
$\it{\frac{\alpha}{R}}$ obtained from the Vanag-Epstein model
($\it{\frac{\alpha}{R}}\mathrm{=7.15s^{-1}}$,
$\it{\frac{\alpha}{R}}\mathrm{=5s^{-1}}$,
$\it{\frac{\alpha}{R}}\mathrm{=2s^{-1}}$,
$\it{\frac{\alpha}{R}}\mathrm{=0s^{-1}}$ are respectively represented in red,
green, blue and black). Inset: Critical bromate concentration
$[\mathrm{NaBrO_{3}}]_{c}$ as a function of the diffusive flux
$\it{\frac{\alpha}{R_{\mathrm{c}}}}$ (Dashed line, numerical results; full
line, theoretical prediction (Equation 2)).

More precisely, the coupling via diffusion to the environment can be written
as $\it{\alpha=\frac{D}{L}}$ with $\it{D}$ the diffusion coefficient of the
chemicals, and $\it{L}$ the scale over which the chemical gradient is built.
$\it{L}$ can be controlled either by diffusion only or by diffusion and
reaction, in case the chemical leaving the gel reacts in the media exterior to
the gel. In the diffusion controlled case, $\it{L}=\it{R}$, whereas in the
reaction-diffusion limited case, $\it{L}=\sqrt{\frac{\it{D}}{\it{k}}}$, with
$\it{k}$ the rate constant of consumption of the activator (SM 1-reaction A.2
of the Vanag-Epstein model with $\it{k_{1}}\mathrm{=2\times 10^{6}M.s^{-1}}$).
The reaction-diffusion lengthscale is estimated to be smaller than the
diffusion lengthscale for these experimental conditions, and we thus conclude
that the outward flux of the activator is controlled by its reaction-diffusion
property.

###  II.4 The oscillatory strain of BZ gels is controlled by the timescale of
gel swelling

Next, we measure the mechanical oscillations of such oscillating gels. Cyclic
actuation of rigid gels as studied in Figure 4 is not detectable. Under the
same chemical conditions, a softer gel ($\mathrm{\kappa\sim 10kPa}$) of
$\mathrm{230\mu m}$ radius, just above the critical size for sustained
chemical oscillation, displays a swelling amplitude of approximately
$\mathrm{1\%}$ (Supplementary SM 11), one order of magnitude smaller than the
relative radius change in oxidative solution.

To interpret this result, we measure the timescale of gel swelling by
recording the radius of the gel as a function of time when an osmotic stress
is applied. The experimental set up is identical to the one used to quantify
the bulk modulus of the gel (Material and Methods). We measure that the
swelling timescale is proportional to the square of the diameter of the gel
(Supplementary SM 12). For the $\mathrm{400\mu m}$ diameter gel studied, the
swelling timescale is around 100s. The oxidation spike is at most 10s. The
expected volume change during the oxidation spike is then
$\mathrm{\frac{\Delta\it{R}_{\mathrm{ox-BZ}}}{\it{R}_{\mathrm{red-
BZ}}}=\frac{\Delta\it{R}_{\mathrm{ox}}}{\it{R}_{\mathrm{red}}}(1-\exp(-\frac{\it{T}_{\mathrm{ox}}}{\it{T}_{\mathrm{{swelling}}}}))\sim
2\%}$, in good agreement with the experimental value. The modest swelling
amplitude in the oscillating BZ condition is caused by the slow timescale of
gel swelling compared to the period of the BZ oscillation.

To further support this conclusion, we perform additional experiments with BZ
gels that have longer periods of oscillation. The following composition of the
BZ solution is used to lengthen transiently the period of the BZ gel chemical
oscillations:([$\mathrm{H_{2}SO_{4}}$]=0.4M, [$\mathrm{NaBrO_{3}}$]=0.08M,
[MA]=0.06M) Maeda2007 ; Smith2013 ; Yoshida1996(2) ; Aizenberg2018 ; Zhang2013
; Kramb2015 ; Chen2011  (Supplementary SM 13). We observe a significant cyclic
volume change in the gels with a long period of oxidation and reduction. The
magnitude of both the swelling and the shrinking amplitude of oscillating gels
is now comparable to the strain obtained during their sole oxidation. An
increase of the catalyst concentration and a decrease of the gel elasticity
generates a larger strain, in agreement with the results obtained for
experiments where the BZ gel is only oxidized. These results definitely prove
that the discrepancy between the swelling timescale of the gel and the period
of the BZ oscillation is responsible for the modest actuation of the gel in
the steady state regime. A straightforward solution to this limitation is to
create porous BZ gels, where the chemical lengthscale is controlled by the
size of the bead, and the swelling timescale is controlled by the size of the
gel microstructure Takeoka2003 ; Suzuki2012 ; Hu2017 .

##  III Conclusion

We demonstrated a versatile two-step synthesis process of Belousov Zhabotinsky
hydrogels that allows independent control of the BZ catalyst concentration,
which is responsible for the active properties of the gel and the cross-linker
concentration, which is primarily responsible for the passive gel elasticity.
We observed that the swelling of BZ gels increases when increasing the
catalyst concentration and when decreasing the cross-linker concentration, an
important result for the design of autonomous shape-changing hydrogels.

Additionally, we introduced a new metric for assessing the chemomechanical
energy conversion of active gels by quantifying their strain and their stress
upon their oxidation. We argued that these two measurements are necessary to
characterize the chemomechanical properties of a BZ oscillating gel. By
definition, the mechanical strain energy density is the product of the stress
and the strain. Moreover, we showed experimentally that the stress driving BZ
gel swelling does not depend on the passive bulk modulus of the material.
Thus, measuring the stress responsible for swelling the BZ gel allows
separation of the contributions from the chemical driving forces and the
restorative forces of the bulk modulus. These experimental results are
supported by a Flory-Huggins model for polyelectrolytes, where we considered
the source of the chemomechanics as arising from the hydration parameter
$\chi$ that varied with oxidation state of the BZ catalyst. We also deduced
that interactions at the molecular level, which are more complex than simple
electrostatic interactions, are responsible for the variation in the Flory-
Huggins $\chi$ parameter that occurs when the concentration of ruthenium
incorporated in the gel is varied and when the ruthenium is oxidized.

Furthermore, we experimentally and theoretically studied how the radius of a
spherical gel affects the period of the BZ reaction and controls even whether
or not the gel oscillates. We measured the timescale of BZ gel swelling and
demonstrated that the factor limiting the strain amplitude is the long
swelling timescale of the gel compared to the shorter period of the BZ
oscillation.

The methodology developed here of measuring both strain and stress in gels can
be used to firstly, isolate the molecular composition of the BZ hydrogels that
optimizes their chemomechanical energy conversion and secondly, design the
shape and/or the microstructure of the gels that enhance the gel swelling
timescale with respect to the period of the BZ reaction.

##  IV Materials and methods

#### Materials

Acrylamide (AAm) (99.9$\%$), 1-ethyl-3-(3-(dimethylamino)propyl)carbodiimide
HCl (EDC), N-hydroxysuccinimide (NHS), 2-(4-morpholino)ethanesulfonic acid
(MES), hexadecane, 2-propanol, sodium phosphate monobasic anhydrous (99$\%$),
sodium phosphate dibasic anhydrous ($\geq 99\%$), Tween 20,
poly(dimethylsiloxane) (PDMS) elastomer kits (Sylgard 184) and sodium
hydroxide (NaOH) were purchased from Thermo Fisher Scientific (Waltham, MA).
Acrylic acid (AA) anhydrous (180-200ppm MEHQ inhibitor,$99\%$),
2-hydroxy-2-methylpropiophenone (Darocur 1173, photoinitiator), saline sodium
citrate (SSC) buffer (20× concentrate, molecular biology grade), N-Boc
ethylene diamine, hydroxybenzotriazole (HOBt), N,N-dimethylformamide (DMF),
Tetrahydrofuran (THF), N,N’-Diisopropylcarbodiimide (DIC) were purchased from
Sigma-Aldrich (St. Louis, MO), N,N’-Methylenebis(acrylamide) (Bis) were
purchased from Sigma Aldrich. 4-Methyl-4’-carboxy-2,2’-bipyridine was
purchased from Carbosynth. All chemicals were of analytical grade and used
without further purification.

#### Micromolding-based fabrication of monodisperse polyacrylamide gels

The p(AAm-co-AA) microspheres in this study were fabricated according to
methods in recent reports Liu2016  . Briefly, the compositions of the aqueous
prepolymer solutions were 15$\%$ (w/v) AAm, $0.5\%$ to $10\%$ (w/v) AAc,
$0.05\%$ to $1\%$ (w/v) MBAM, and the composition of the wetting fluid was
99$\%$ (v/v) hexadecane and $1\%$ (v/v) Darocur 1173 photoinitiator. As shown
in Figure 1-a, the prepolymer solution was placed into a micropatterned PDMS
mold, which was formed with Sylgard 184 containing 10$\%$ (w/w) curing agent
following overnight incubation at 65 °C on a silicon master template. Bubbles
in the microwells were removed by rubbing the surface of the mold with a
disposable pipet tip. Excess prepolymer solution was removed via pipetting,
and the wetting fluid was then placed on top of the mold in order to lead to
surface tension-induced droplet formation. For beads with a diameter greater
than $\mathrm{300\mu m}$, applying manually pressure around the hexagonal mold
helps droplet formation. The mold was then carefully placed on an aluminum
mirror (Thorlabs, Newton, NJ) and exposed to low-intensity 365 nm UV light
with an 8 W hand-held UV lamp (Spectronics Corp., Westbury, NY) for 15 min to
initiate radical chain polymerization. The microspheres were collected via
pipetting, transferred to a microcentrifuge tube, and rinsed to remove any
wetting fluid and unreacted chemicals as follows: mixing the microspheres in
2-propanol by pipetting, allowing them to settle to the bottom, and removing
the supernatant. After rinsing at least three times with 2-propanol, the
rinsing procedure was repeated at least three times with deionized water
containing 0.05$\%$ (v/v) Tween 20 and at least twice with 20 mM MES buffer
(adjusted to pH 6 with 1 M NaOH) containing 0.05$\%$ (v/v) Tween 20 or
deionized water.

#### Catalyst synthesis

The ruthenium catalyst is synthesised in our lab with the following procedure.
The starting material is commercially available
4-Methyl-4’-carboxy-2,2’-bipyridine from Carbosynth. The
4-Methyl-4’-carboxy-2,2’-bipyridine is further treated with an N-Boc ethylene
diamine molecule to obtain the ligand
N-(2-aminoethyl)-[2,2’-Bipyridine]-4-carboxamide. Then the catalyst is
obtained by mixing this ligand in a 50 mL two-neck round-bottom flask
connected to a condenser charged with dichloro(p-cymene)-ruthenium(II) dimer
(100 mg, 0.163 mmol). We added 2 mL of distilled water and 18 mL of ethanol.
Nitrogen was bubbled through the solution for 40 min. The mixture was
distilled in darkness for 3 days. After the solvent was removed, the crude
mixture was purified on a Sephadex-LH 20 column.

#### Functionalizing the Hydrogel bead with the ruthenium catalyst via
carbodiimide chemistry

The second step of the fabrication process is to functionalize the hydrogel
created above with a ruthenium catalyst complex. To do so, we use the
reactivity of the carboxylic function with an amine group, to create an amide
bond via EDC/NHS coupling. The carboxylate groups in the microspheres are
first converted into reactive NHS ester groups. 400 mM EDC and 100 mM NHS are
added to microspheres in 20 mM MES buffer (pH 6) containing 0.05$\%$ (v/v)
Tween 20, mixed via pipetting, and placed on a rotator for 15 min at room
temperature. Unreacted EDC and NHS are removed by rinsing the microspheres
with 20 mM MES buffer (pH 6) containing 0.05$\%$ (v/v) Tween 20 at least three
times and then 100mM sodium phosphate buffer adjusted to pH 8 by adding 1M of
NaOH solution, at least twice. For ruthenium catalyst conjugation, p(AAm-co-
AAc) microspheres activated with EDC/NHS (roughly 50 particles) are reacted
with 10$\mu$L of a solution of 35mM ruthenium catalyst dissolved in DMF in
100$\mu$L of bead solution overnight on a rotator at room temperature in the
sodium phosphate buffer adjusted to pH 8. Unreacted ruthenium BZ catalyst are
removed by rinsing the microspheres 3 times with deionized water+0.05$\%$
(v/v) and at least 3 times with 5$\times$SSC buffer (pH 7) containing 0.05$\%$
(v/v) Tween 20. The BZ catalyst show good reactivity with the carboxylic
function. We provide a proof of the covalent bonding between the carboxylic
function and the amine group of the catalyst by doing a serie of test
experiments either not using EDC, or NHS, or not using both of them. When
washed with the same conditions, we observe that the BZ beads created under
these test conditions do not show any color, proving that this washing
procedure removes any unreacted catalyst. The synthetized BZ gel bead
(acrylamide, acrylic acid) do not show any non specific adsorption. Finally,
we can check the homogeneity of the catalyst incorporation inside the gel
bead, with a confocal microscope (SM 2).

#### Experimental set up for characterizing the oscillating hydrogels

Each experiment is conducted in a container made of a glass slide on the
bottom and on the top, with a PDMS lateral wall of a few millimeters in height
and a centimeter in diameter. The solution is kept under pressure with a clamp
to avoid evaporation and the creation of carbon dioxide bubbles. The change of
color of the catalyst is recorded as a function of time. Ruthenium-conjugated
microspheres were visualized with an epifluorescence microscope (Olympus BX51
equipped with a DP70 microscope digital camera, Center Valley, PA) or a Leica
S4 E Stereo Zoom Microscope with a MARLIN F-131C color camera. The images were
then analyzed with the MATLAB software to obtain the temporal evolution of the
intensity and the radius change of a gel.

#### Details of the model

The Vanag-Epstein (VE) model simplifies the BZ reaction to a model with only
four variables, each representing a chemical concentration. The four
concentration variables are $c_{1}$=[HBrO2], $c_{2}$=[Br-],$c_{3}$=[oxidized
catalyst]=[Ru3+] and $c_{4}$=[Br2]. HBrO2, Br- and Br2 are chemicals which can
diffuse out of the gel, whereas the Ru3+ is bound to the gel.
The mathematical expression of the loss term can be justified as follows
Yoshikawa1998 . Consider the number of moles of a chemical product inside the
bead $N_{i,in}$. This product is diffusing out of the gel with a flux $j$
proportional to its concentration, $j=\alpha^{{}^{\prime}}c$. The variation of
the number of moles of product inside the bead $\mathrm{d}N_{i,in}$ due to
diffusion can be written:

$\displaystyle\frac{\mathrm{d}N_{i,in}}{\mathrm{d}t}$ $\displaystyle=-j\pi
R^{2}$ $\displaystyle\frac{\mathrm{d}(\frac{4}{3}\pi
R^{3}c_{i})}{\mathrm{d}t}$ $\displaystyle=-\alpha^{{}^{\prime}}\pi c_{i}R^{2}$

The size of a rigid bead is constant, so the temporal variation of the
concentration of this product takes the following form:

$\frac{\mathrm{d}c_{i}}{\mathrm{d}t}=-\frac{3\alpha^{{}^{\prime}}}{4R}c_{i}$

The diffusion acts then as an additional effective first order chemical
reaction on the temporal evolution of the chemical concentration. We define
$\alpha=\frac{3\alpha^{{}^{\prime}}}{4}$.

$\displaystyle\frac{\mathrm{d}c_{1}}{\mathrm{d}t}$
$\displaystyle=-k_{1}c_{1}c_{2}+k_{2}c_{2}-2k_{3}c_{1}^{2}+k_{4}c_{1}\frac{(c_{3,0}-c_{3})}{c_{3,0}-c_{3}+c_{min}}$
$\displaystyle\qquad{}\qquad{}\qquad{}\qquad{}\qquad{}\qquad{}-\frac{\alpha}{R}c_{1}$
$\displaystyle\frac{\mathrm{d}c_{2}}{\mathrm{d}t}$
$\displaystyle=-3k_{1}c_{1}c_{2}-2k_{2}c_{2}-k_{3}c_{1}^{2}+k_{7}c_{4}+k_{9}c_{3}-\frac{\alpha}{R}c_{2}$
$\displaystyle\frac{\mathrm{d}c_{3}}{\mathrm{d}t}$
$\displaystyle=2k_{4}c_{1}\frac{(c_{3,0}-c_{3})}{c_{3,0}-c_{3}+c_{min}}-k_{9}c_{3}-k_{10}c_{3}$
$\displaystyle\frac{\mathrm{d}c_{4}}{\mathrm{d}t}$
$\displaystyle=2k_{1}c_{1}c_{2}+k_{2}c_{2}+k_{3}c_{1}^{2}-k_{7}c_{4}-\frac{\alpha}{R}c_{4}$

The rate constant of reaction $k_{i}$ and the constant concentration $c_{min}$
are given in the supplementary material (SM 1).
We solve this coupled non linear differential equation using the MATLAB solver
ode45.

#### Measuring the catalyst concentration

The catalyst concentration is measured by squeezing the gel into a flat slab
between two glass plate separated by a gap of known thickness, created by a
double sticky tape. In parallel, a microfluidic chamber of the same thickness
with a scale of known concentration of catalyst in solution is created. We
measure the transmitted light I through the sample, for the same light
condition for the BZ bead and the solution. The liquid sample of known
concentration enables us to convert the transmitted light intensity into real
concentration value, via a fitting function in agreement with a Beer Lambert
Law (SM 3).

#### Measuring the Bulk modulus by applying an osmotic stress with high
molecular weight polymer Dextran

To achieve this measurement, we design a microfluidic chamber made of double
sticky tape of thickness 500$\mu$m and drill a hole of 1mm diameter and
500$\mu$m height on a Plexiglas sheet. We flow successively solutions of
sulfuric acid at a concentration of 0.4M with different Dextran concentration.
We use a high molecular weight Dextran (MW=500 000g.mol-1, MW=2 000
000g.mol-1), to avoid the molecule diffusing inside the hydrogel. We measure
the radius of the gel at mechanical equilibrium for each dextran
concentration.
The osmotic stress $\it{\Pi}_{\mathrm{dex}}$ is given by:

$\log_{10}[\it{\Pi}_{\mathrm{dex}}\mathrm{(\it{w}\mathrm{)]=1.75+1.03}\it{w}^{\mathrm{0.383}}}$
(5)

with $\it{w}$ the weight pourcent of Dextran in the solution Parsegian1986 ;
Monnier2016 .

#### Timescale of gel swelling

To achieve this measurement, we design a microfluidic chamber made of double
sticky tape of thickness 500$\mu$m and drill a hole of 1mm diameter and
500$\mu$m height on a Plexiglas sheet. We flow successively solutions of
sulfuric acid at a concentration of 0.4M with a concentration of 5$\%$ Dextran
concentration. We use a high molecular weight Dextran (MW=500 000g.mol-1, MW=2
000 000g.mol-1), to avoid the molecule diffusing inside the hydrogel. We
record the evolution of the diameter of the gel $\it{D}$ as a function of
time. We then fit these data with an exponential function having a
characteristic time $\tau$. We finally verify that the characteristic time of
swelling $\tau$ is proportional to the square of the gel diameter $\it{D}$ (SM
12).

#### Measuring the density

To measure the density of micrometric hydrogels, we determine their steady
state velocity during their free fall in a water column
$\it{V_{\mathrm{\infty}}}$. The water column is a plastic cuvette used for
absorbance measurement graduated every 5mm.

$\it{V_{\mathrm{\infty}}}$ is directly related to the difference of density
between the surrounding medium and the object.

$\mathrm{\Delta}\rho=\rho_{g}-\rho_{w}=\frac{9}{2}\frac{\eta
V_{\infty}}{\mathrm{g}R^{2}}\hskip 14.22636pt\text{with
$\mathrm{\Delta\rho=\rho_{\mathrm{g}}-\rho_{\mathrm{w}}}$}$ (6)

with $\Delta\mathrm{\rho}$, $\mathrm{\eta}$, $\it{R}$, $V_{\infty}$ being,
$\mathrm{g}$ respectively the difference of density between the hydrogel
$\rho_{\mathrm{g}}$ and the water $\rho_{\mathrm{w}}$, the dynamic viscosity
of water, the radius of the bead, the velocity of the beads in the steady
regime and the gravitational acceleration on Earth. Additional information are
presented in the supplementary materials (SM 10).

##  V Data availability

All data generated for this study are included in the main and supplementary
figures. Other data that support the findings of this study are available on
reasonable request from the corresponding author.

##  VI Acknowledgment

We thank A.Agvhami for help with confocal imaging, B.Xu for sharing chemical
tools, I.Dellatolas for critical reading of the manuscript and D.Blanc for
help with the figures. We acknowledge useful discussions with M.M.Norton,
G.Duclos, B.Rogers, R.Hayward, H.Kim, T.Emrick and I.Epstein.

##  VII Authors contribution

B.B., Z.Z., S.F devised the experiments. B.B., E.L., H.Y. fabricated the gels,
N.Z., B.B synthesized the ruthenium catalyst, B.B developed the models, B.B,
S.F. wrote the manuscript. The experiments were done at Brandeis University
and mainly supported by NSF DMREF-1534890. B.B acknowledges support of the
Brandeis Provost research award. E.L., H.Y. gratefully acknowledge financial
support by U.S. National Science Foundation (grant NSF CBET-1703547). The
authors declare no competing interests.

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