# Hot Brownian Motion of thermoresponsive microgels in optical tweezers shows
discontinuous volume phase transition and bistability

Miguel Angel Fernandez-Rodriguez1,2∗ Sergio Orozco-Barrera1 Wei Sun1,3
Francisco Gámez1,4 Carlos Caro5 María L. García-Martín5,6 Raúl Alberto Rica1,∗
1Universidad de Granada, Nanoparticles Trapping Laboratory, Department of
Applied Physics and Research Unit “Modeling Nature” (MNat), Faculty of
Sciences, Campus de Fuentenueva s/n, 18071 Granada, Spain
2Laboratory of Surface and Interface Physics, Department of Applied Physics,
Faculty of Sciences, Universidad de Granada, Campus de Fuentenueva s/n, 18071
Granada, Spain
3Department of Physics, Yanshan University, Qinhuangdao, 066004, China
4Department of Physical Chemistry, Faculty of Chemical Sciences, Complutense
University of Madrid, 28040 Madrid, Spain
5Instituto de Investigación Biomédica de Málaga y Plataforma en Nanomedicina
(IBIMA Plataforma BIONAND), C/ Severo Ochoa, 35, 29590 Málaga, Spain
6Biomedical Research Networking Center in Bioengineering, Biomaterials &
Nanomedicine (CIBER-BBN), 28029 Madrid, Spain
∗To whom correspondence should be addressed: E-mail<EMAIL_ADDRESS>(MAFR<EMAIL_ADDRESS>(RAR).

###### Abstract

Microgels are soft microparticles that often exhibit thermoresponsiveness and
feature a transformation at a critical temperature, referred to as the volume
phase transition temperature. The question of whether this transformation
occurs as a smooth or as a discontinuous one is still a matter of debate. This
question can be addressed by studying individual microgels trapped in optical
tweezers. For this aim, composite particles were obtained by decorating pNIPAM
microgels with iron oxide nanocubes. These composites become self-heating when
illuminated by the infrared trapping laser, featuring Hot Brownian Motion
within the trap. Above a certain laser power, a single decorated microgel
features a volume phase transition that is discontinuous, while the usual
continuous sigmoidal-like dependence is recovered after averaging over
different microgels. The collective sigmoidal behavior enables the application
of a power-to-temperature calibration and provides the effective drag
coefficient of the self-heating microgels, thus establishing these composite
particles as potential micro-thermometers and micro-heaters. Moreover, the
self-heating microgels also exhibit an unexpected and intriguing bistability
behavior above the critical temperature, probably due to partial collapses of
the microgel. These results set the stage for further studies and the
development of applications based on the Hot Brownian Motion of soft
particles.

Introduction. Microgels are soft systems comprised of crosslinked hydrogels
that often exhibit thermoresponsiveness and collapse above a volume phase
transition temperature ($VPTT$) Camerin  _et al._ (2019); Fernandez-Rodriguez
_et al._ (2021). The proximity of the $VPTT$ to physiological temperatures
(e.g. around 32 $\degree$C for pNIPAM) together with their capability to carry
cargo make them very well suited to develop interesting applications,
including their use as drug-delivery carriers Wu  _et al._ (2022); Agrawal and
Agrawal (2018), their potential applicability in the exploration of synthetic
cell research Wang  _et al._ (2020) and, more generally, to manufacture
thermoresponsive surfaces and materials Li  _et al._ (2019); Jia  _et al._
(2019).

There are still open questions regarding the thermoresponsiveness of
microgels, e.g., how the mechanical properties change across the $VPTT$ Tagit
_et al._ (2008), or why there are multiple values of $VPTT$ reported for
pNIPAM in the literature Halperin  _et al._ (2015); Tayebi  _et al._ (2022). A
more fundamental question remains unsolved, namely, whether the transition,
i.e., collapse/swelling upon heating/cooling above/below the $VPTT$, is a
sudden discontinuous or a smooth continuous process. One of the more usual
ways to characterize thermoresponsiveness is to measure the variation of the
average hydrodynamic size of microgel suspensions with temperature by dynamic
light scattering (DLS). This variation is typically reported to follow a
sigmoidal smooth curve Camerin  _et al._ (2019). Only recently, some works
dealt with the characterization of the $VPTT$ transition of thermoresponsive
microgels by optical tweezers, i.e., measuring properties in a one-microgel-
at-a-time basis while externally heated Kannan  _et al._ (2011); Gupta  _et
al._ (2016). The results were disparate and in apparent contradiction. While
the first work reported a discontinuous transition Kannan  _et al._ (2011),
the latter accounted for a smooth sigmoidal one Gupta  _et al._ (2016), but
being unspecified if the sigmoidal behavior were obtained by averaging results
over several microgels. Moreover, the same authors found a rather
discontinuous transition for pH-responsive microgels upon pH changes Dhanapal
_et al._ (2016), so the available results are inconclusive.

Alternative à la carte synthetic routes widen the applications spectra of
microgels by, for instance, adding co-monomers during the synthesis able to
induce pH-responsiveness Bhattacharya  _et al._ (2007) or embedding Lu  _et
al._ (2022); Hormeño  _et al._ (2014) or decorating Hain  _et al._ (2008);
Juarez and Liz-Marzán (2015); Karg  _et al._ (2007) the microgels with
inorganic nanoparticles that transfer their properties to the microgel to some
extent Liu  _et al._ (2015). Hence, rather than being externally heated, self-
heating microgels owning an absorbing core Lu  _et al._ (2022); Hormeño  _et
al._ (2014) or decorated with absorbing nanoparticles Hormeño  _et al._
(2011); Xiao  _et al._ (2022) might be assembled. These approaches reported
continuous transitions too. From fundamental grounds, by adding absorbing
nanoparticles in optical tweezers, we enter the interesting domain of Hot
Brownian Motion (HBM) Schmidt  _et al._ (2021); Holubec  _et al._ (2020);
Rings  _et al._ (2012, 2010), an intrinsically out-of-equilibrium process in
which a particle is hotter than the surrounding medium, and gradients of
temperature and viscosity develop. In this situation, the characterization of
the temperature and other properties of the self-heating trapped particle
becomes very difficult Geiß and Kroy (2020); Srivastava and Chakraborty
(2018), and exotic properties in both the particle and the surrounding liquid
can be observed Lu  _et al._ (2020).

In this work, we decorated thermoresponsive pNIPAM microgels with iron oxide
nanocubes. The dynamics and thermal behavior of the optically trapped
composite were evaluated with the aim of both responding to the fundamental
question raised before on the nature of the thermoresponsive transition and
providing new insights and useful routes for characterizing the HBM of
absorbing particles under optical trapping conditions. We found unexpected
phenomena with potential broad applications as the accurate and localized
heating and micro-thermometry of biologically relevant systems, and, in
parallel, very intriguing bistability that could serve as a unique playground
for statistical physics Ricci  _et al._ (2017) and soft active systems Bley
_et al._ (2022).

Results and Discussion. The pNIPAM microgels used in this work were
synthesized by precipitation polymerization with a 4.86 %m crosslinking
density, resulting in a more crosslinked core and a less crosslinked shell
Camerin  _et al._ (2019); Fernandez-Rodriguez  _et al._ (2021). We
characterized the hydrodynamic diameter $d_{H}$ of the microgel suspensions by
DLS as a function of the temperature $T$ (see Figure 1a). The results followed
the usual smooth transition that can be fitted by the sigmoidal function Lu
_et al._ (2022); Hormeño  _et al._ (2014):

$d_{H}(T)=d_{H,collapsed}+\frac{d_{H,swollen}-d_{H,collapsed}}{1+\exp\left(\frac{T-VPTT}{\tau}\right)}$
(1)

where the fitting parameters are the hydrodynamic diameters in the swollen
($d_{H,swollen}=911\pm 14$ nm) and collapsed ($d_{H,collapsed}=556\pm 13$ nm)
states, the decay coefficient $\tau=3.0\pm 0.5$ K, and the $VPTT=303.9\pm 0.6$
K. In order to characterize the microgel morphology in detail, we deposited a
monolayer at the water/air interface at a surface pressure of $\Pi$=5
mN$\cdot$m-1, and transferred it onto a flat silicon substrate to measure
their height by Atomic Force Microscopy (AFM, see Figure S1a and the
Experimental Section for further details). This measurement allowed us to
characterize the individual topology of the microgels, and the core-shell
morphology was reflected in the Gaussian profile exhibited in the AFM images
(see Figure S1b). This property is ascribed to the microgels expansion at the
interface caused by the interfacial tension and only counterbalanced by the
internal elasticity, which is proportional to the crosslinking densityCamerin
_et al._ (2019). In general, the size of the microgels at an interface is
different from their size in bulk.

In order to make microgels moderately absorbing at the wavelength of our
optical tweezers setup (wavelength of the trapping laser $\lambda=1064$ nm),
we decorated them with iron oxide nanocubes of 12$\pm$2 nm of space diagonal
length and 31$\pm$2 nm of $d_{H}$ with a polydispersity index of 0.30 (see the
Experimental Section and Figure S1c-d for further details about the size
distribution and optical properties of the nanocubes). The electrostatically
driven decoration was possible thanks to the positive charge of the microgels
(see the red curve in Figure 1d) and the negative charge of the iron oxide
nanocubes ($\zeta$-potential=–18$\pm$2 mV). The nanocubes got stuck on the
polymer matrix, leading to composite particles (see Figure 1b) that are both
thermoresponsive and self heat upon external illumination, as we demonstrate
below. We tested three different relative fractions of nanocubes per microgel.
For this purpose, we produced three samples labeled $\mu gel_{High}$, $\mu
gel_{Med}$, and $\mu gel_{Low}$, in which we used a given amount of iron oxide
nanocubes, half, and a quarter, respectively (see the Experimental Section for
further details). The charge of the composite was fully inverted with respect
to the bare microgels in all cases (see Figure 1c). This fact might point out
specific interactions that not only induce electrostatic screening but a
charge inversion Moncho-Jordá and Adroher-Benítez (2014). Despite this fact,
the microgels preserved in all cases their thermoresponsiveness upon heating
the whole dispersion, as it is shown in Figure 1c. In this case, we can see
that the charges per unit area, and hence the electrophoretic mobility
$\mu_{e}$, increase in absolute value upon temperature-induced collapse. As
compared to the bare microgels, the hydrodynamic diameter of the ones
decorated with the highest amount of nanocubes ($\mu gel_{High}$) were reduced
to $d_{H,swollen}=770\pm 240$ nm and $d_{H,collapsed}=366\pm 76$ nm, where the
deviations account for the full width at half maximum (FWHM) of the
corresponding measured distributions of $d_{H}$ (see Figure S1e). The widening
in the distribution of $d_{H}$ might result from differences in the decoration
with iron oxide nanocubes. Since the values of $\mu_{e}$ were similar in all
decorated microgels, we considered that their $d_{H}$ were also similar.

Figure 1: (a) Hydrodynamic diameter $d_{H}$ of the microgels measured by DLS
against temperature. (b) TEM image of a $\mu gel_{High}$ decorated with the
iron oxide nanocubes. (c) Electrophoretic mobility of the microgels against
$T$ for the different prepared samples. The solid lines stand for sigmoidal
fittings.

The hydrodynamic diameter of _individual_ particles can be characterized in
our optical tweezers device, thus avoiding averaging effects Tolić-Nørrelykke
_et al._ (2006). As we show below, this averaging becomes crucial in the
interpretation of the collapse of microgels. Therefore, we studied single
microgels in the optical tweezers device after dilution of the microgel
dispersions in MilliQ water to a concentration down to a few particles per
cm3, in order to ensure that a single microgel was trapped at a time (see the
scheme in Figure 2a). The evaluation of the diffusion coefficient
$D=k_{B}T/\gamma$ of trapped particles is a well-established technique Tolić-
Nørrelykke  _et al._ (2006); Jones  _et al._ (2015), where $k_{B}$ is the
Boltzmann constant, $T$ is the temperature of the environment and
$\gamma=3\pi\eta\cdot d_{H}$ is the friction coefficient of a sphere of
hydrodynamic diameter $d_{H}$ in a liquid with viscosity $\eta$. A piezo stage
that oscillates sinusoidally can be used to apply a known drag force on the
trapped particle, providing direct measurements for the corner frequency of
the particle in the trap $f_{c}$ and the diffusion coefficient (see the
Experimental Section and Figure S2 for further details). We assumed that the
light absorption by the bare microgels was negligible in the laser power $P$
range considered here, i.e., 8 to 351 mW at 1064 nm. This is reasonable since
it is accepted that a 1064 nm laser heats water at a rate of 0.008
K$\cdot$mW-1 when trapping a 500 nm silica nanoparticle Peterman  _et al._
(2003), corresponding to an increase of 3 K at our maximum $P$, which ensures
that we do not reach the $VPPT$ from room temperature ($T=299\pm 2$K).
Following this methodology, we evaluated the hydrodynamic diameter of 10
individual bare microgels at room temperature, obtaining an average value of
$d_{H,single}=849\pm 70$ nm. This result is in fair agreement with the average
obtained for bare microgels from DLS measurements in the swollen state.

Decorated microgels trapped in optical tweezers presented clear signatures of
heating due to the absorption of laser light, consisting in the observation of
HBM and eventually collapsing upon increasing laser power. In order to
demonstrate these effects, we resort to the evaluation of the diffusion
coefficient of the decorated microgels, which can provide information about
both effects. The measurement of the diffusion coefficient was performed
following the same procedure described before for bare microgels. In Figure
2b, we plotted $D$ as a function of $P$ for three individual self-heating
microgel composites of the $\mu gel_{High}$ sample. The diffusion coefficient
increases strongly with laser power, reaching a 5-fold increase at the maximum
power we tested. There is a concomitant increasing trend of $f_{c}$ with $P$
which shows non-linear dependence, as expected for a sample that heats up upon
laser power increase. It is also interesting to note that above the
transition, the values of $f_{c}$ seem to coincide for the three microgels,
pointing to an equivalent final state for all of them. A second striking
feature in the dependence of the diffusion coefficient is a clear
discontinuous transition at values $P\simeq 150$mW, slightly different for
each microgel (see Figure 2b). Regardless of the $P$ at which the transition
was observed, the character is always discontinuous even when dozens of
composites of each sample were evaluated. Interestingly, Figure S3a-b shows
that the behavior of $D$ was reversible, as expected for the temperature-
induced collapse of pNIPAM.

Figure 2: (a) Scheme of the optically trapped microgel decorated with iron
oxide nanocubes below and above the $VPTT$. The inset shows a trapped microgel
visualized from the optical camera. (b) Diffusivity $D$ of individual
microgels and corner frequency $f_{c}$ as a function of the laser power $P$.
The plots of $D$ are displaced vertically for each sample for the sake of
clarity as detailed in the corresponding legends. Arrows depict the
discontinuous transition for each microgel.

To rule out the possibility that this discontinuous transition is an artifact
of a particular way of measuring, complementary measurements of the
electrophoretic mobility of the trapped composites $\mu_{e,trap}$ were
performed. For this aim, we placed two parallel electrodes made of gold wires
within the trap cell and subsequently applied a AC signal of $V_{pp}$=4 V at
20 kHz (see Figure S3c). Despite both measurements of $D$ and $\mu_{e,trap}$
being independent, discontinuous transitions at the same $P$ are derived for
both parameters (see Figure S3d). To sum up, intriguingly, the one-by-one
characterization of the temperature dependence of the diffusion coefficient
and the electrophoretic mobility of optically trapped self-heating microgels
leads to a discontinuous transition, which we proved to be reversible during a
heating/cooling cycle. Previous results also showed discontinuous transitions
for optically trapped non self-heating microgels Kannan  _et al._ (2011),
where the full cell was heated externally. For our self-heating microgels, we
cannot distinguish if the dispersion in critical values of $P$ are due to
slight heterogeneities in the crosslinking profiles or in the number of
nanocubes embedded within the microgel.

So far, we proved that there is a critical laser power above which a decorated
microgel experiences a transformation. However, we still have not discussed
whether this transition occurs at the $VPTT$, since no direct access to the
local temperature $T$ of the self-heating microgel has been figured out yet.
Moreover, a sigmoidal fitting does not render a good description for the
transition of _individual_ self-heating microgels. The situation becomes
different when the results are averaged over several self-heating microgels.
In Figure 3a, the variation of $D$ and $f_{c}$ against $P$ is plotted as
obtained by averaging the power dependence over twelve individual microgels
for each sample. Bare microgels feature a slight increase of $D$ with $P$ that
might be related to the ability of the 1064 nm laser to heat the polymer and
water, as discussed before. This was also evidenced by a departure from
linearity in $f_{c}$ at higher $P$. In the case of self-heating microgels,
averaging $D$ over several particles leads to a smooth sigmoidal-like
transition, as it is typically observed in DLS measurements. It is hence
tempting to argue that averaging over slightly different discontinuous
transitions leads to a sigmoidal behaviour when averaging is performed either
directly, like in DLS, or a posteriori, as it is the case of our one-by-one
experiments. Moreover, we see that the value of $f_{c}$ for the three samples
of decorated microgels coincide at low powers with that of the bare ones,
suggesting that the contribution to the polarizability of the iron oxide
nanocubes within the polymer matrix is negligible and that all the observed
effects could be attributed to self-heating.

Figure 3: (a) Averaged $D$ and $f_{c}$ of 12 microgels for each sample. The dashed and solid lines are linear and Gaussian-like fittings, respectively, that allow to calibrate the $T$ corresponding to each $P$. The red line in the $f_{c}$ plot is the linear fitting y(x)=a$\cdot$x for the first points of the red data, to stress the deviation from linearity for higher $P$. (b) $d_{H}$ of the microgels as a function of $T$ by using the data fitted in (a) and Equation 1 and 2. The DLS size data from Figure 1b was added for comparison, and the red dashed line and shaded region correspond to the $VPTT$=303.9$\pm$0.6 K. Table 1: Fitting parameters of the linear $T(P)=a\cdot P+b$, and Gaussian-like $T(P)=a\cdot\exp\left(-2\left(\frac{d_{H}(T(P))\cdot r}{2w}\right)^{2}\right)P+b$ models, and drag-modification coefficient $\phi$, for the curves in Figure 2c. $\chi_{\nu}^{2}$ reported for each fitting. Sample | Fit |  $a$ (K$\cdot$mW-1) |  $b$ (K) | $\phi$ | $\chi_{\nu}^{2}$
---|---|---|---|---|---
$\mu gel_{High}$ | lin. | $0.124\pm 0.022$ | $288.4\pm 2.6$ | $0.45\pm 0.05$ | 0.13
$\mu gel_{Med}$ | lin. | $0.118\pm 0.012$ | $290.9\pm 1.3$ | $0.57\pm 0.03$ | 0.03
$\mu gel_{Low}$ | lin. | $0.074\pm 0.008$ | $292.6\pm 1.8$ | $0.52\pm 0.05$ | 0.03
$\mu gel_{High}$ | Gauss. | $0.127\pm 0.021$ | $293.8\pm 1.8$ | $0.45\pm 0.05$ | 0.11
$\mu gel_{Med}$ | Gauss. | $0.128\pm 0.011$ | $295.5\pm 0.8$ | $0.59\pm 0.03$ | 0.02
$\mu gel_{Low}$ | Gauss. | $0.085\pm 0.010$ | $297.5\pm 1.2$ | $0.58\pm 0.05$ | 0.03

Let us now develop a simple model that captures the main features of the
observed behavior after averaging over experiments. The model should include
both HBM due to the fact that the decorated microgels are hotter that the
surrounding liquid, and the eventual collapse above the $VPTT$. We shall
notice that a model for the HBM of soft particles is lacking, and beyond the
scope of the present work. We will resort to a minimal model that allows us to
estimate the transition temperature, in order to compare it with the reported
values for $VPTT$. For this aim, we consider that the diffusion coefficient of
a decorated microgel has a complex dependence on the temperature since both
the size of the decorated microgel and the viscosity of water depend on it.
The available models developed for hard spheres consider that the diffusion
coefficient of a hot Brownian particle can be described by modified
temperature $T_{\rm HBM}$ and viscosity $\eta_{\rm HBM}$, i.e., $D_{\rm
HBM}=k_{B}T_{\rm HBM}/3\pi\eta_{\rm HBM}d_{H}$, where $T_{\rm HBM}$ relates
the temperature at the hydrodynamic surface of the particle $T_{s}$ with that
in the bulk far from the hot particle, $T_{0}$ Rings  _et al._ (2010, 2011).
However, an alternative description is needed for our experiments, since the
meaning of $T_{s}$ is lost in the case of microgels, and we need to account
for its internal temperature, which will be non-uniform. Since no such model
is available, we propose here a phenomenological model to capture both the HBM
of a soft particle (SHBM) and the temperature-induced collapse:

$D_{\rm SHBM}(T,\phi)=\frac{k_{B}T}{\phi\cdot 3\pi\eta(T)d_{H}(T)},$ (2)

where the temperature is assumed to be the average internal temperature of the
microgel and the factor $\phi$ accounts for the modified drag due to the
gradient of temperature around the microgel and the concomitant viscosity one
Rings  _et al._ (2010); Oppenheimer  _et al._ (2016); Geiß and Kroy (2020). We
propose to combine the sigmoidal function in Equation 1 with Equation 2 to fit
our experimental data. Since the temperature dependence of the viscosity
$\eta$ of water is well known Reid  _et al._ (1987) (see Figure S4a), we can
use the measured values $d_{H,swollen}$ and $d_{H,collapsed}$ with DLS for the
decorated microgels to obtain a non-linear expression of $D$ that increases
with $T$ as shown in Figure S4b.

The missing ingredient that allows us to fit the data in Figure 3a is the
relation between the laser power and the effective temperature. The
temperature increase is due to the absorption of laser light by the iron oxide
nanocubes decorating the microgel. Two different calibrations are presented
thereupon. Our first approximation assumes a linear increase of $T$ with $P$
($T(P)=a\cdot P+b$), as it has been considered in several previous works
Hormeño  _et al._ (2014); Rodríguez-Sevilla  _et al._ (2018); Rodríguez-
Rodríguez  _et al._ (2020). Using this relation together with Eqs. 1 and 2, we
can fit the data in Figure 3a, obtaining the values of $a$, $b$, and $\phi$
presented in Table 1 and the dashed lines in Figure 3a. The first test of
consistency is that $b$ should be the room temperature of the cell as $P=0$.
Slightly lower values are in general obtained, featuring a very mild increase
with the number of iron oxide nanocubes, and a similar increasing trend was
obtained for the heating parameter $a$, compatible with reported values
measured for NIR heating of multiple optically trapped iron oxide nanocubes of
similar size at $P$=100 mW Rodríguez-Rodríguez  _et al._ (2020). Our fitted
$a$ contains the contribution from water heating due to the laser as discussed
before, in the absence of self-heating nanoparticles, reported to be around
0.008 K$\cdot$mW-1 for 500 nm silica nanoparticles Peterman  _et al._ (2003).
Moreover, for more absorbing gold nanoparticles of 50 nm the heating
coefficient was reported to be around 0.266 K$\cdot$mW-1 for a single
nanoparticle Seol  _et al._ (2006). Therefore, the iron oxide nanocubes seem
to be a nice compromise to decorate the microgels as they become moderately
self-heating, enabling us to observe the transition within the optical
tweezers laser power range, and avoiding extreme phenomena such as cavitation.

The linear dependence of $T$ with $P$ might be a good approximation for self-
heating hard spheres, and below 320 K, where anomalous diffusion effects are
reported Lu  _et al._ (2020). However, it has been proven that the efficiency
of heat absorption changes upon the collapse of a microgel shell over an
inorganic absorbing core Lu  _et al._ (2022), and probably the effect is more
dramatic in our case, where the absorbing nanoparticles decorate the shell of
the microgel. In fact, upon collapse, the nanoparticles are brought closer to
the center of the Gaussian profile of the laser beam Rings  _et al._ (2011),
where the light intensity is higher, and therefore the absorption should also
increase. Moreover, it is reasonable to consider that the magnetite cubes will
sit where the polymer density is higher. We can estimate this via a ratio
between the collapsed and swollen states for the decorated microgels
$r=d_{H,collapsed}/d_{H,swollen}\simeq 0.5$. Then, $T$ would better be
described using a Gaussian dependence of the form
$T(P)=a\cdot\exp\left(-2\left(\frac{d_{H}(T(P))\cdot
r}{2w}\right)^{2}\right)P+b$, where $d_{H}$ still changes with $T$ according
to Equation 1 and $w=$280 nm is the beam radius of the trapping laser
Rodríguez-Rodríguez  _et al._ (2020). Since $d_{H}$ depends on the unknown
value of $T$, the sigmoidal dependence of $T$ with $P$ was recursively solved
using the results for the linear dependence to obtain the initial seed for
$d_{H}(T(P))$. The sigmoidal fitting parameters are presented in Table 1 and
as solid lines in Figure 3a. It is apparent that the consistency test is
better than when the linear dependence is assumed, with an average value of
$\overline{b}$=296$\pm$2 K, in agreement with our room temperature.
Nevertheless, the fittings did not significantly differ in practical terms as
shown in Figure 3a and S5a. For example, at $VPTT$=303.9 K, the difference of
$P$ between the linear and Gaussian fittings is $|\Delta P|\simeq$4.5 mW,
$\simeq$1.5 mW, and $\simeq$13.9 mW, for the $\mu gel_{High}$, $\mu
gel_{Med}$, and $\mu gel_{Low}$, respectively. Consequently, we converted the
$D(P)$ data from Figure 3 to $D(T)$ with both calibration approaches (see
Figure S5b)). By using the calibrated $D(T)$ we can retrieve a value of
$d_{H}(T)$ by using Equation 2, results that are presented in Figure 3b and
S5c. These plots need to be taken with a grain of salt, as they involve
different simple and incomplete models relating $P$ with $T$.

The first interesting feature we observe in Figure S5b is that the curves
$D(T)$ show rather discontinuous transitions around the $VPTT$, also for the
ensemble of microgels, i.e., after appropriate averaging. The maximum and
minimum values of $d_{H}$ in Figure 3b show a decreasing trend with the
content of iron oxide nanocubes, which showcases the ability of this technique
to characterize the size of thermoresponsive soft particles when there is not
enough sample to acquire DLS measurements. Nevertheless, we cannot capture the
upper part of the sigmoidal curve as there is heating as soon as the particle
is optically trapped. Furthermore, the Gaussian fit seems to provide better
results, with better discrimination between samples and a value for $d_{H}$
above the $VPTT$ that remains rather constant, as shown in Figure 3b and S5c.
The results indicate that the decoration with iron oxide nanocubes does not
seem to significantly alter the value of the $VPPT$, as also shown in Figure
1c from DLS measurements. It also gives value to our phenomenological model,
as the transition is observed at the $VPPT$ and thus it seems reasonable to
assume that the $T$ in Equation 2 is related to the real internal temperature
of the microgel, and not only to the temperature at the hydrodynamic surface
of the hot particle. Finally, we estimated the stiffness of the trap as a
function of $P$ and $T$ as $\kappa=2\pi f_{c}\gamma_{\rm
SHBM}=6\pi^{2}f_{c}\phi\eta(T)d_{H}(T)$ for both calibrations (see Figure S6).
There are mild changes in trend above the $VPPT$ that are clearer for the
Gaussian-like calibration, and as discussed for $f_{c}$, $\kappa\neq 0$ for
$P=0$ mW which points out to the complex SBHM behavior.

The agreement of our observations with trapped microgels and the $VPTT$
reinforces the assumption that the free parameter $\phi$ accounts for HBM
effects only related to drag. The average value of $\phi$ is 0.51$\pm$0.06 for
the linear fitting (0.54$\pm$0.08 for the Gaussian fitting), which means that
the effective $\gamma_{\rm SHBM}\simeq$ $\gamma$/2, i.e., a $\simeq$ 50 $\%$
smaller with respect to the drag of a sphere. This result benchmarks the
effects of HBM on a soft particle and could be compared with the changes in
drag coefficient due to the deformation of a soft spherical particle to a
pancake shape, estimated to decrease $\gamma$ by $\simeq$ 60 $\%$ when it
completely flattens Srivastava  _et al._ (2012). This would not be supported
by the optical images of the trapped microgel, whose size was observed to be
compatible with the one from DLS, and would likely be in conflict with the
collapse of the microgel that we observe. Summing up, it has been demonstrated
that the self-heating microgels prepared here anew showed a discontinuous
transition when isolated in an optical trap, while a sigmoidal-like continuous
transition was obtained averaging over a dozen of experiments. In the latter
situation, a fitting of $T$ at each laser power density can be established,
enabling (i) their use as both local micro-heaters and micro-thermometers with
potential applicability in biological studies, and (ii) providing a new way to
measure the effective $\gamma_{\rm SHBM}$ that we expect to be helpful to
delve into the HBM of both hard and soft particles from direct experimental
data.

Figure 4: X raw traces for a $\mu gel_{High}$ in blue, and running average
over 400 points in orange, below and above the $VPTT$. (b) Switching times for
the bistability, rise, and fall in the X trace as a function of $f_{c}$ and
$t=(2\pi f_{c})^{-1}$. (c) PSD of the X, Y, and Z traces for a $\mu gel_{Med}$
above the $VPTT$. The Gaussian fitting was used to estimate $T$ in (a-c). (d)
Comparison between the features visible in the traces in volts (top) and by
video tracking (bottom) of the X and Y centroid position, and the radius of
gyration of the microgel brightness.

Finally, we turn our attention back to the individual behaviour of decorated
microgels to uncover an unexpected feature of their dynamics while trapped in
optical tweezers. Taking a look at the individual traces we observed that,
above the $VPTT=303.9\pm 0.6$ K, a kind of bistability appeared in the time
traces (see Figure 4a), similar to the behavior reported for bistable trapping
potentials Rondin  _et al._ (2017); Zijlstra  _et al._ (2020); Schnoering and
Genet (2015). It is worth having in mind that the difference between the
linear and Gaussian fittings at both 115 and 151 mW is $|\Delta T|<$ 0.8 K.
Thus, we could make a reasonable claim that we were below and above the $VPTT$
for those laser powers, respectively. The bistability was typically observed
in both X and Y signals of the photodiodes, which correspond with the trapping
plane, being clearly correlated (see Figure S7-8 for several time traces at
different powers and their corresponding PSDs, respectively). Less clear was
the presence of bistability in the Z signal, which corresponds with the
direction of the laser beam. It is not clear if this is a fundamental property
of the phenomenon or if this is an artifact of the detection along this axis,
which is generally accepted to be less reliable Tolić-Nørrelykke  _et al._
(2006). We extracted the characteristic switching times as a function of
$f_{c}$ (and thus the characteristic relaxation time in the trap $t=(2\pi
f_{c})^{-1}$), finding no clear dependence with it (see Figure 4b). This is
true even if the typical residence time clearly increases with $P$, and hence
with $f_{c}$ (see Figure S7).

Interestingly, this behavior was accompanied by an increase in the plateau of
the PSDs at low frequencies (see Figure 4c). This feature was previously
reported in microgels with absorbing inorganic nanoparticles in the shell
Hormeño  _et al._ (2014). We believe this might be a signature of self-heating
in optical trapping, and it would be interesting to understand if the
bistability appears also in their self-heating microgels. For comparison
purposes, in Figure S9a we present the PSD for both a non-absorbing silica
microparticle (1 $\mu$m SiO2, Sigma Aldrich) and a self-heating hard
microparticle (0.883 $\mu$m ProMag 1 with 26.5 $\,wt\%$ magnetite content,
Bangs Laboratories). The latter displayed a rise in the PSD at low
frequencies. Nevertheless, in the case of a hard self-heating microparticle we
did not observe bistability.

In order to ensure that this effect is not an artifact from the detection with
photodiodes, we recorded a movie with a CMOS camera of a microgel under
bistability conditions and compared the traces from the photodiodes and the
position and radius of gyration of the brightness $R_{g}$ after tracking the
self-heating microgel position from the movie (see Figure 4d and Movie S1).
The same trends were observed in both the position and $R_{g}$. The jumps in
the traces in volts correspond to very subtle movements of the microgel that
are not visible in Movie S1 by the naked eye since they correspond to
variations of the centroid position below $\simeq$50 nm. Furthermore, the PSD
for the different steps of bistability in Figure 4d were plotted in Figure
S9b) and we found it to be similar to those PSDs in Figure 4c. This fact
discarded that the increase of the PSDs at low frequencies was only due to the
bistability.

We do not have enough evidence to elucidate the mechanism behind this
bistability. We identified as possible hypotheses: i) displacements of the
microgel within the trap, ii) spatial reconfigurations of the adsorbed iron
oxide nanocubes within the soft microgel, iii) an alternation of the
orientation between two birefringence axes of the collapsed microgel, and/or
iv) partial collapses of the microgel, a phenomenon that it is currently under
study Nothdurft  _et al._ (2021); Del Monte  _et al._ (2021); Kyrey  _et al._
(2019). The first hypothesis would be supported by characteristic switching
times of the order of the characteristic relaxation time in the trap $(2\pi
f_{c})^{-1}\simeq$ [0.01-0.05] ms, but we found them to be $\simeq$ 12 ms,
independently of $P$ and hence of $f_{c}$ (see Figure 4b). For the second
hypothesis, we believe that the reconfiguration of the iron oxide nanocubes
would lead to continuous changes rather than bistability, since it is
difficult to imagine the nanocubes jumping between two configurations within
the microgel matrix. For the birefringence hypothesis, we estimated that the
scattering force provided by the iron oxide nanocubes is twice the one
necessary to deform the microgel by $\Delta V/V$ by 10% (see Figure S10),
which would support the idea of a microgel deformed to the point of exhibiting
birefringence. Nevertheless, in this case, we would also expect a dependence
of the switching time with $f_{c}$, as it would be related to the torque
exerted by the linearly polarized laser, and hence should also decrease with
$P$ Rings  _et al._ (2012); Li  _et al._ (2016), contrary to what we observe.
The latter hypothesis on partial collapses might be tested if the changes in
size were big enough to show differences in $f_{c}$, but when we checked the
$f_{c}$ of the different segments of the bistable traces (see Figure S9b) they
were not significantly different. Therefore, further studies are needed to
elucidate the origin of this bistability. Regardless of the origin, this
behavior constitutes an interesting model for statistical physics and poses a
challenge for scientists in the search for answers in related fields Bley  _et
al._ (2022).

Conclusions. By combining thermoresponsive pNIPAM microgels with iron oxide
nanocubes, we obtained a self-heating soft microparticle system that was
interrogated by optical tweezers. Both the diffusion coefficient $D$ and the
corner frequency $f_{c}$ of the microgels were obtained under an assumption-
free experimental approach. We found that single trapped self-heating
microgels showed a discontinuous volume transition upon heating, while the
usual sigmoidal-like transition was recovered when averaging over dozens of
measurements. These results provide new insights into the discussion of the
nature of the thermoresponsive transition of pNIPAM microgels. In particular,
we provide experimental evidence that the sigmoidal-like thermal behaviour of
self-heating microgels might come from collective effects due to subtle
heterogeneity features in the size or chemical compositions of the
microparticles. This evidence also explains the conclusions aroused when bulk
techniques, such as DLS, are employed to monitor the $VPTT$. The collective
sigmoidal-like behavior enabled us to model the dependence of $T$ on $P$, by
either assuming a linear or a Gaussian-like dependence. This procedure turned
the self-heating microgels into a dual system that simultaneously acts as both
local micro-heaters and micro-thermometers. This feature is highly desired in
nanomedicine and material sciences. At the same time, they provide a new way
of characterizing the effective drag $\gamma_{\rm SHBM}$, a parameter that
might be helpful for studying the Hot Brownian Motion of soft particles.
Furthermore, we found an intriguing bistability in the time trace obtained
above the $VPTT$ for single microgels that might be ascribed to partial
collapses of the self-heating microgels, among other working hypotheses.

Finally, we hope that the potential applications and fascinating bistability
features of soft self-heating microgels demonstrated in this work shall spark
the interest of a broad spectrum of the scientific community, from
experimentalists focused on applied science to fundamental statistical
physicists.

##  I MATERIALS AND METHODS

###  I.1 Synthesis and characterization of pNIPAM microgels

The microgels were synthesized by precipitation polymerization as described in
our previous works Fernandez-Rodriguez  _et al._ (2021). First, we prepared
MilliQ water at pH 4 by adding HCl with a glass pipette (Sigma Aldrich, ACS
reagent 37 %). Then, we added it (55 mL) to a round bottom glass flask and
further added n-isopropylacrylamide (1.108 g, NIPAM, Sigma Aldrich, 99 $\%$)
monomer and bis-acrylamide (0.077 g, BIS, Sigma Aldrich, 99.5 $\%$)
crosslinker, obtaining a crosslinking density of 4.86 $\%$m. Then, we gently
shaked the flask until all the powder was dissolved in the water, and we
placed a portion of this solution (5 mL) in a glass vial that we sealed with a
rubber septum. Next, we added a teflon-coated magnetic stirrer and sealed the
flask with another rubber septum, heating in an oil bath (80 ∘C) under
magnetic stirring and N2 bubbling for 1 h to remove the oxygen in the flask.
In parallel, we had the vial (5 mL) with the monomer and crosslinker under N2
bubbling and we prepared another glass vial (6 mL) with pH 4 MilliQ water and
added 2,2’-azobis(2-methylpropionamidine)dihydrochloride (0.088 g, V50, Acros,
98$\%$) initiator, and sealed the glass vial with another rubber septum,
bubbling N2 for 1 h. Next, we added the V50 initiator solution (2 mL) with a
syringe and in a few minutes the solution turned white. In order to increase
the size of the microgels, 15 min later we added more monomer+crosslinker
solution (1.25 mL) from the separated glass vial and the V50 initiator (1 mL)
every 10 min, for a total of 4 aliquots. Next, we kept the flask for 5 h at 80
∘C under N2 atmosphere. The dispersion of microgels was then filtered with
Whatmann paper (16$\,\mu$m pore size) and a glass funnel. In order to clean
the sample from unreacted species we performed 5 cycles of centrifugation
(39800 rcf) for 1 h each, removing the supernatant and redispersing the
microgels in MilliQ water.

After this, we diluted it to 0.24 $\,wt\%$ with MilliQ water. In order to
characterize the microgels, first we prepared a 4:1 microgel
dispersion:isopropanol (Sigma Aldrich, HPLC grade) and added drops with a
glass microsyringe to a water/air interface in a Langmuir trough (KSV NIMA,
Biolin Scientific), where we previously immersed a silicon substrate (2
$\times$ 1 cm2, $\left\langle 100\right\rangle$ orientation, p-type, Boron
doped, 1-10 $\Omega\cdot$cm, University Wafer) cut by laser (Laser E-20 SHG
II, Rofin) and supported in a dipper. We added the dispersion at the interface
until the surface pressure reached a value of $\Pi$=5 mN$\cdot$m-1, and a
Langmuir-Blodgett film was then deposited the by raising the dipper at 0.5 mm
per min, keeping $\Pi$ constant with the movable barriers. After this, the
monolayer was characterized by AFM (Dimension 3000) in tapping mode
(Tap300Al-G cantilevers, 300 kHz, 40 N$\cdot$m-1, BudgetSensors). We also
characterized their electrophoretic mobility $\mu_{e}$ by Laser Doppler Micro-
electrophoresis (ZetaSizer NanoZ, Malvern), and their $d_{H}$ by Dynamic Light
Scattering with a capillary cell that allows to measure big nanoparticles
(DLS, Zetasizer Ultra, Malvern).

###  I.2 Synthesis of iron oxide nanocubes.

Iron oleate precursor was synthesized as previously described in reference
Pernia Leal  _et al._ (2017). Briefly, a mixture of iron chloride (10.8 g, 40
mm) and sodium oleate (36.5 g, 120 mm) were dissolved in ethanol (80 mL),
distilled water (60 mL) and hexane (140 mL). The resulting solution was heated
(60 ∘C) and let for 4 h in a reflux of hexane and in inert atmosphere. At that
time, the reaction was cooled down to room temperature and two phases can be
distinguished: a lower aqueous phase and an upper organic phase with the iron
oleate. The organic phase was washed 3 times with distilled water and the
hexane was evaporated in the rotavapor. Once the precursor is synthesized, a
thermal decomposition approach was selected for the synthesis of iron oxide
nanoparticles. Following this method, iron oleate (1 g) and oleic acid (0.3 g)
were mixed with octadecene (5.15 g) in a round bottom flask (50 mL) under
vigorous stirring. The reaction was performed under inert atmosphere. The
temperature was increased at a constant heating rate (5 ∘C per min) from 30 ∘C
to 320 ∘C. The reaction was kept 1 h at this temperature and then cooled down
to room temperature. 3 washing steps with ethanol/acetone (1/1) were
conducted. Finally, the nanocubes were resuspended in toluene.

The chemical synthesis of PEGylated ligands and the subsequent ligand exchange
procedure were reported previously in references Pernia Leal  _et al._ (2017);
Pozo-Torres  _et al._ (2020), respectively. In brief, a reaction solution was
prepared in a round bottom flask with polyethylene glycol (3 g, PEG-3000),
gallic acid (170 mg) and 4-dimethylaminopyridine (24 mg) in tetrahydrofuran
(100 mL) and dichloromethane (10 mL). A solution of DCC (1 g) in
tetrahydrofuran (10 mL) was added dropwise to the reaction solution under
inert N2 atmosphere. The mixture was stirred overnight at room temperature.
The reaction mixture was filtered through a filter paper and the solvents were
rotaevaporated. Solvents were supplied by Acros organics and the remaining
reagents by Sigma Aldrich and used without further purification. The final
solution contains a Fe concentration of 0.6 mg$\cdot$mL-1 as determined with
standard protocols by Inductively Coupled Plasma Hight Resolution Mass
spectroscopy (ICPMD, NexION ICP-HRMS from Perkin-Elmer). 1H nucler magnetic
resonance (NMR Bruker Ascend 400 MHz spectometer) spectroscopy confirmed the
desired product gallol-PEG-OH. 1H NMR (400 MHz, CDCl3) $\delta$ (ppm): 7.22
(s,2H), 4.43-4.40 (m,2H), 3.85-3.45 (m,CH2-PEG,-OH).

For the ligand exchange, in a glass vial a solution with nanocubes (1 mL) at
concentration of 10 g$\cdot$L-1 of Fe, we added the PEGylated ligand (1.0 mL)
in a concentration of 0.1 M in CHCl3. Triethylamine (50 $\mu$L) was added
subsequently. The mixture was sonicated for 1 h and kept 4 h at 50 ∘C. At this
point, it was diluted with a mixture of toluene (5 mL), MilliQ water (5 mL)
and acetone (10 mL). The nanoparticles were then transferred into the aqueous
phase. After that, the aqueous phase was collected in a round-bottom flask and
the residual organic solvents were rotaevaporated. The gallol-derived
nanocubes were purified in centrifuge filters with a molecular weight cutoff
of 100 kDa at 450 rcf. In each centrifugation step, the functionalized
nanoparticles were resuspended in MilliQ water. The purification step was
repeated several times until the filtered solution was clear. After the
purification, the gallol-derived nanoparticles were resuspended in MilliQ
water. Finally, to promote colloidal stability and monodispersivity, the
suspension was centrifuged (150 rcf) for 5 min before being placed onto a
permanent magnet (0.6 T) for 5 min previous to supernatant collection.

###  I.3 Physicochemical characterization of iron oxide nanocubes.

The UV-Vis-NIR spectrum was recorded in a UV-VIS-NIR Spectrophotometer (Jenway
6705) with a quartz tray having a light path of 1 cm. Nanocubes dispersion
were used at a concentration of $\sim$0.04 mg$\cdot$mL-1. The size
distribution and $\zeta$-potential measurements of the gallol-derived
nanocubes were performed on a Zetasizer Nano ZS90 (Malvern, USA). The
nanoparticles were dispersed in MilliQ water at a concentration of 100
mg$\cdot$L-1 of Fe. The measurements were done on a cell type ZEN0118-low
volume disposable sizing cuvette, setting 2.42 as refractive index with 173∘
Backscatter (NIBS default) as angle of detection. The measurement duration was
set as automatic and three as the number of measurements. TEM images were
obtained on a FEI Tecnai G2 Twin microscope operated at an accelerating
voltage of 100 kV. TEM samples were prepared by dropping a preconcentrated
nanocubes suspension at $\sim$1 g Fe per liter on a carbon-coated copper grid
and letting the solvent evaporate. The nanocube size were calculated by
averaging a hundred of nanoparticles.

###  I.4 Decoration of microgels with iron oxide nanocubes

The decoration of the positively charged microgels with negatively charged
iron oxide nanocubes was performed by adding the microgel dispersion (10
$\mu$L) to an Eppendorf with MilliQ water (1 mL), and three different volumes
of iron oxide nanocubes dispersion: 40 $\mu$L, 20 $\mu$L, and 8 $\mu$L, to
produce the $\mu gel_{High}$, $\mu gel_{Med}$, and $\mu gel_{Low}$ samples,
respectively. The three Eppendorfs were placed in a shaker overnight. The
dispersion was centrifuged in an Eppendorf (8000 rpm) for 15 min and the
supernatant was removed and replaced by MilliQ water. The decorated microgels
were characterized by TEM (Thermo Fisher Scientific TALOS F200X), obtaining
also their $\mu_{e}$ and $d_{H}$ as described in the previous section.

###  I.5 Optical tweezers trapping

We used the NanoTracker-II optical tweezers (JPK-Bruker) equipped with a 1064
nm laser. The fluidic chambers were built by depositing two fine seams of
vacuum seal grease (Korasilon paste, Kurt Obermeier GmbH) on a #1.5 glass
slide and covered with a #1.5 cover slip. Next, the sample was deposited in
between the two glasses by capillarity and the cell was sealed with vacuum
seal grease. Laser powers were selected in the range 8-358 mW. The trap focus
was allocated $\simeq$ 10 $\mu$m above the bottom glass of the cell to avoid
spurious capture of microgels from the cone of light below the trap and to
make hydordynamic interaction with the wall negligible Jones  _et al._ (2015).
The photons that are forward scattered by a trapped microgel carry information
about its motion within the trap, and are detected by photodiodes to track the
X, Y and Z time traces. The setup allows for an easy detection of multi-
trapping events that are visualized both in the live video as a sudden
brightening of the newly trapped microgel and as a step-wise increase in the
time traces, coupled with a significant increase in the corner frequency
$f_{c}$ obtained from the power spectral density (PSD) of the traces. If any
of these features was detected, we discarded the experiment and looked for an
alternative particle. For each trapped microgel, we calibrated the trap by
performing a Stokes sinusoidal drag calibration Tolić-Nørrelykke  _et al._
(2006); Jones  _et al._ (2015). In this calibration method, the piezo-stage
performed sinusoidal oscillation in the X-direction with an amplitude of
$X_{stage}=$340 nm at $f_{stage}=$20 Hz, and the traces for the X, Y and Z
coordinates in volts were recorded from the four quadrant photodiodes for X-Y,
and a photodiode for the Z direction, at a rate of 50 kHz for 5 s. The
standard procedure to analyse these data consists in the calculation of the
power spectral density (PSD) of the time trace, which is composed by a
Lorentzian function plus a peak corresponding to the forced oscillation of the
trapped particle:

$PSD(f)=\frac{D}{2\pi^{2}(f^{2}+f_{c}^{2})}+\frac{X_{stage}^{2}}{4(1+f_{c}^{2}/f_{stage}^{2})}\delta(f-f_{stage}),$
(3)

where the first addend in the rhs comes from the Brownian motion of the
particle in the trap, while the second one is due to the forced oscillation,
with $\delta(x)$ the Dirac delta function of argument $x$Tolić-Nørrelykke  _et
al._ (2006). Fitting the experimentally obtained PSD to Equation 3, we obtain
estimates for the corner frequency $f_{c}$, the calibration factor
$S_{V\rightarrow m}\left[\frac{m}{V}\right]$, and the diffusion coefficient
without any assumption on the temperature, viscosity of the media or the
$d_{H}$ of the microgel (see Supplemental Material). This process was
performed for each microgel and $P$ in triplicate to calculate averages and
standard deviations. For the bare microgels, we could assume $\eta$ to be the
viscosity of water at room temperature and obtain the value of the
hydrodynamic diameter $d_{H}$.

Videos at a frame rate of 92 fps were also recorded to enable direct
comparisons with the traces in volts stored during the tracking of the
microgel with the Trackpy software Crocker and Grier (1996). As a result of
the tracking, the centroid position in the XY plane and the radius of gyration
of the brightness of the trapped microgel were obtained.

Finally, the electrophoretic mobility $\mu_{e,trap}$ measurements within the
optical trap were performed by placing two gold nanowires electrodes of
diameter 50 $\mu$m separated $d\simeq$0.5 mm Pesce  _et al._ (2013); Ma  _et
al._ (2017). Then, a voltage of $V_{pp}$=4 V was applied at a frequency of
$f_{EF}=$20 kHz with a function generator (RSDG 2082X), while keeping the
optical trap fixed with a microgel in between the two electrodes. By fitting
the area of the peak that appears in the PSD at 20 kHz due to electrophoresis
$W_{exp,EF}$, we calculated the value of $\mu_{e,trap}$ for each microgel at
each $P$ value using Equation 5 Pesce  _et al._ (2013); Ma  _et al._ (2017),
which in analogy to the ”Stokes calibration” does not depend on $T$, $\eta$,
or the size of the trapped microgel. The expression relating optical tweezers
parameters with electrophoretic mobility is

$|\mu_{e,trap}|=\frac{4\pi
f_{c}}{E}\sqrt{W_{exp,EF}\left(1+\frac{f_{c}^{2}}{f_{EF}^{2}}\right)},$ (4)

where the electric field $E$ is considered to be constant and given by
$E=V_{pp}/2d$ (see Supplementary material for further details). Triplicate
experiments were computed to obtained averages and standard deviations (see
Figure S4b-c).

Acknowledgements

This work was supported by the projects EQC2018-004693-P,
PID2021-127427NB-I00, PID2020-116615RA-I00, PID2020-118448RB-C21, and
PID2019-105195RA-I00, and the grant IJC2018-035946-I funded by MCIN/AEI/
10.13039/501100011033/ FEDER, UE, and by the project P18-FR-3583 and EMERGIA
grant with reference EMC21_00008 funded by Consejería de Universidad,
Investigación e Innovación de la Junta de Andalucía.
†M.A.F.R and S.O.B. contributed equally to this work.

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Table of Contents

Figure 5: Self-heating microgels trapped in optical tweezers show
discontinuous transitions and bistability when heated above the volume phase
transition temperature. This new system can be used as local micro-heaters and
micro-thermometers.

Supplementary Information

Figure S1: (a) AFM image of a microgel monolayer deposited at $\Pi$=5
mN$\cdot$m-1. (b) Gaussian profile of the height of a microgel corresponding
to the cyan section in the AFM image inset. (c) TEM image of the magnetite
nanocubes. Inset with the distribution of diagonal lengths. (d) UV-vis-NIR
extinction spectrum of the magnetic nanocubes in water. (e) Distribution of
$d_{H}$ measured by DLS of microgels decorated with magnetite nanocubes $\mu
gel_{High}$ in water.

## Stokes active calibration

Calibration of the QPD signal to operative length units is required. The
commercial optical tweezers provide with an automatic passive calibration
procedure that adjust the PSD of a trapped particle and compare it with
theoretical expressions. However, this method assumes constant local
temperature $T_{0}$, viscosity and particle size, assumed to be spherical. As
our study is focused on self-heating soft microparticles, none of those
assumptions are valid, and an alternative calibration method is required. In
this work, an active calibration method was selected. Within this approach,
the voltage-to-length constant is derived from viscous drag (Stokes)
measurements without any premise about the properties of the medium or the
trapped particle Jones  _et al._ (2015). To that aim, the uncalibrated
particle’s temporal traces were recorded while applying a sinusoidal movement
of amplitude $X_{stage}$ and frequency $f_{stage}$ described by:

$x(t)=X_{stage}\sin{(2\pi f_{stage}t)}.$ (S1)

The so-obtained PSDexp,V is formed by the superposition of the a peak
corresponding to the sinusoidal trace over the Lorentzian curve at $f_{stage}$
(see Figure S2):

$PSD_{exp}^{volts}(f)\left[\frac{V^{2}}{Hz}\right]=\frac{A}{(f^{2}+f_{c}^{2})}+W_{exp}t_{msr}\delta_{Kronecker}(f-f_{stage}).$
(S2)

Experimentally, the measurement time $t_{msr}$ is finite, and therefore the
Dirac delta in Equation 3 of the main text here becomes a Kronecker delta.
Therefore, from a fitting to the experimental data we obtain the parameters
$A$, $f_{c}$, and $W_{exp}$ without any assumption (see Figure S2a). Comparing
with Equation 3 in the main text, we obtain the calibration factor:

$S_{V\rightarrow
m}\left[\frac{m}{V}\right]=\sqrt{\frac{X_{stage}^{2}t_{msr}}{4W_{exp}(1+f_{c}^{2}/f_{stage}^{2})}},$
(S3)

and from it we can calibrate the PSD in length units:

$PSD_{exp}^{length}=S_{V\rightarrow m}^{2}\cdot PSD_{exp}^{volts},$ (S4)

and a measurement of the diffusion coefficient:

$D=\frac{k_{B}T}{\gamma}=2\pi^{2}A\cdot S_{V\rightarrow m}^{2}.$ (S5)

Figure S2: (a) Typical PSD of a trapped bare microgel ($P$=79 mW). (b) PSD of
the same microgel in (a) but subjected to a sinusoidal drag of the stage of
340 nm amplitude and a frequency of 20 Hz, keeping the trap fixed. (c) X trace
of the sinusoidal drag corresponding to the PSD in (b).

## Electrophoretic mobility of trapped microparticles

In order to correlate the diffusion coefficient $D$ of the optically trapped
microparticles with their electrophoretic mobility $\mu_{e,trap}$, we use the
method described in the Experimental Section derived from Pesce  _et al._
(2013); Ma  _et al._ (2017). The method is similar to the ”Stokes active
calibration” described before. For each trapped microparticle and laser power
$P$ we measured by triplicate the microparticle temporal trace $x(t)$ while
applying a sinusoidal electric field of amplitude $A_{EF}$ generated by a
peak-to-peak voltage $V_{pp}=$4 V at $f_{EF}=$20 kHz. If the electric field
$E$ is considered to be constant across the distance between electrodes $d$:

$E=\frac{V_{pp}}{2d}.$ (S6)

Then, the expression for PSDexp,EF can be approximated by:

$PSD_{exp,EF}(f)=\frac{A_{exp}^{length}}{f^{2}+f_{c}^{2}}+\frac{A_{EF}^{length}}{4(1+f_{c}^{2}/f_{EF}^{2})}\delta(f-f_{EF}),$
(S7)

which is quite similar to Equation S2, just changing the origin of the
sinusoidal movement and taking the values in length units as obtained by the
calibration procedure described above.

In equilibrium, the force due to the electric field $F_{EF}$, given by:

$F_{EF}=Q_{eff}E,$ (S8)

is counterbalanced by the optical restoring force $F_{trap}$ determined by the
stiffness $\kappa$:

$F_{trap}=\kappa A_{EF}^{length}.$ (S9)

In Equation S8 $Q_{eff}$ is the effective charge of the trapped microparticle,
related to the electophoretic mobility by:

$Q_{eff}=|\mu_{e,trap}|\gamma.$ (S10)

It is straightforward to show that:

$A_{EF}^{length}=\frac{Q_{eff}E}{\kappa}.$ (S11)

While the particle is moving, $F_{EF}$ must be counterbalanced by the drag
force $F_{drag}$:

$F_{drag}=|\mu_{e,trap}|\gamma E.$ (S12)

and an operative form relating $A_{EF}^{length}$ with $\mu_{e,trap}$ Ma  _et
al._ (2017) can be obtained:

$A_{EF}^{length}=\frac{|\mu_{e,trap}|\gamma
E}{\kappa}=\frac{|\mu_{e,trap}|E}{2\pi f_{c}}.$ (S13)

Similarly to the active calibration, we obtain the area under the experimental
peak $W_{exp,EF}$ and compared to the theoretical one $W_{th,EF}$ given by:

$W_{th,EF}=\int_{0}^{f_{Nyquist}}PSD_{EF}(f)df=\frac{(A_{EF}^{length})^{2}}{4(1+f_{c}^{2}/f_{stage}^{2})}=\frac{\mu_{e,trap}^{2}E^{2}}{16\pi^{2}f_{c}^{2}(1+f_{c}^{2}/f_{stage}^{2})}.$
(S14)

Finally, we can obtain the absolute value of $\mu_{e,trap}$ as reported in
Equation 5 in the main text, reproduced here:

$|\mu_{e,trap}|=\frac{4\pi
f_{c}}{E}\sqrt{W_{exp,EF}\left(1+\frac{f_{c}^{2}}{f_{EF}^{2}}\right)}.$ (S15)

It must be emphasized that no assumption for $T$, $\eta$, nor the size or
shape of the trapped microparticle have been assumed during the derivation. As
discussed in the manuscript, these values, although in the correct units and
order of magnitude, are difficult to compare to the ones obtained by Laser
Doppler Micro-electrophoresis in Figure 1d. In the latter, a 10 times higher
electric field is applied, and the velocity is obtained by Doppler shift from
the laser pointed at the sample. In any case, we use them here as an
independent measurement to find the discontinuous transition of each trapped
self-heating microgel.

Figure S3: (a) $D$ of one microgel as a function of $P$ upon heating and then
cooling, to show the reversibility of the process. Arrows depict the
discontinuous transitions. (b) $f_{c}$ corresponding to the curves in (a). (c)
Scheme of the measurement of $|\mu_{e,trap}|$. (d) Figure 2b (diffusion
coefficients $D$) plotted next to the independent measurements of
$|\mu_{e,trap}|$ for each microgel. Arrows pointing to the discontinuous
transitions. Figure S4: (a) Viscosity $\eta$ of water as a function of $T$
Reid  _et al._ (1987), that can be approximated by the equation
$\eta(T)=2.414\cdot 10^{-5}\cdot 10^{247.8\rm{K}/(T-140\rm{K})}\,\rm Pa\cdot
s$ (b) Diffusion coefficient $D$ expected for a single $\mu gel_{High}$ as a
function of $T$ as obtained from Equation 1 and 2. Figure S5: (a) $T(P)$ for
the linear (dashed lines and data points with errors) and Gaussian-like (solid
lines) fittings in Figure 2c and Table 1. (b) $D$ from Figure 3a as a function
of $T$, calibrated with the linear and Gaussian-like fitting. (c) $d_{H}$ of
the microgels as a function of $T$ from (b). Bare microgels characterized by
DLS were added for comparison. The $VPTT=303.9$ K is represented by red dashed
lines. Light red shaded regions stand for the $\pm 0.6$ K interval. Figure S6:
Stiffness of the trap $\kappa=2\pi f_{c}\gamma_{\rm
SHBM}=6\pi^{2}f_{c}\phi\eta(T)d_{H}(T)$ as a function of (a) $P$ and (b) $T$
for the linear and Gaussian-like temperature calibrations. The $VPTT=303.9$ K
is represented by solid lines in (a) and red dashed lines in (b). Light red
shaded regions stand for the $\pm 0.6$ K interval. Figure S7: (a) Traces for
the same microgel as in Figure 4a at different $P$ and $T$ values. The
Gaussian fitting was used to estimate $T$. Figure S8: PSDs corresponding to
the measurements in Figure S7. Figure S9: (a) PSDs of the X trace of a non-
absorbing 1 $\mu$m hard silica particle ($P$=358 mW) and a self-heating hard
Promag microparticle ($P$=28 mW), and X trace showing no bistability for the
latter (smooth curve in orange). Promag microparticles are polymer-based
magnetite-loaded micro-spheres of 1 $\mu$m. (b) PSDs of the Y trace in volts
corresponding to the different domains in the bistability shown in Figure
3c-d.

## Scattering force vs bulk elasticity

We can estimate the scattering force provided by the iron oxide nanocubes
within the decorated microgel Pesce  _et al._ (2020) as:

$F_{scattering}=N\frac{n_{m}\sigma_{ext}}{c}I_{0}e^{-2\left(\frac{\rho}{w}\right)^{2}}$
(S16)

where $N$ is the number of iron oxide nanocubes per microgel. This was
estimated to be $N\simeq 1600$ in Figure 1b (tracked by imageJ) for a $\mu
gel_{High}$. The refractive index of the medium is the one of water
$n_{m}=1.33$, $c$ is the speed of light, $\sigma_{ext}\simeq 2.5$ nm2 for
quite similar iron oxide nanocubes Rodríguez-Rodríguez  _et al._ (2020) at
1064 nm, and $\rho$ is the distance of the nanocubes to the center of the
trap. $I_{0}=P/(\pi w^{2})$, where $P$ is the laser power and $w=280$ nm is
the beam radius of our laser. Finally, we used $\rho=d_{H,collapsed}\cdot
r/2=366/4$ nm, where $r\simeq 0.5$ accounted for the position of the iron
oxide nanocubes within the microgel as discussed in the main text.

Further, we estimate the force needed to deform the microgel by a relative
volume $\Delta V/V$ using the bulk compressibility modulus $K$, reported to be
$\simeq 1$ kPa above the $VPTT$ for very similar microgels in size and
composition, measured by AFM Burmistrova  _et al._ (2011). Similar results
were reported from osmotic pressure measurements Sierra-Martin  _et al._
(2011). The expression of $K$ is the following:

$K=-V\frac{dP}{dV}\simeq\frac{\Delta P}{\frac{\Delta V}{V}}$ (S17)

In Equation S18, the pressure $\Delta P$ can be seen as a force $F$ on the
decorated surface of the particle $\Delta P=F_{deformation}/(4\pi\rho^{2})$,
resulting in:

$F_{deformation}=4\pi\left(\frac{d_{H,collapsed}\cdot
r}{2}\right)^{2}K\frac{\Delta V}{V}$ (S18)

We plotted Equation S16 and S18 in Figure S10 as a function of $P$ and $\Delta
V/V$ for the $F_{scattering}$ and $F_{deformation}$, respectively, showing
that $F_{scattering}>F_{deformation}$. This result indicates that a
deformation of the microgel along the propagation direction of the beam is
feasible.

Figure S10: Comparison between the scattering force as a function of laser
power $P$ and the force required to deform the microgel as a function of the
deformation $\Delta V/V$.

Movie S1. Movie acquired at 92 fps for the trapped $\mu gel_{Med}$ at $P$=358
mW used to track the position and radius of gyration of the brightness of the
microgel in Figure 3d.