# Activation of a supercooled liquid confined in a nanopore.

Felix Mercier  Laboratoire de Photonique d’Angers EA 4464, Université
d’Angers, Physics Department, 2 Bd Lavoisier, 49045 Angers, France  Gaetan
Delhaye  Laboratoire de Photonique d’Angers EA 4464, Université d’Angers,
Physics Department, 2 Bd Lavoisier, 49045 Angers, France  Victor Teboul
<EMAIL_ADDRESS>Laboratoire de Photonique d’Angers EA 4464,
Université d’Angers, Physics Department, 2 Bd Lavoisier, 49045 Angers, France

###### Abstract

It is well stablished that confinement of supercooled liquids in nano-pores
induces various effects as a strong modification of the dynamics and a
layering of the local structure. In this work we raise the issue as how these
confinement effects are modified when the liquid is out of equilibrium. To
answer that question, we use molecular dynamics simulations to investigate the
effect of confinement on a supercooled liquid activated by the periodic
folding of a molecular motor. We find that the motor’s opening angle controls
the activation of the medium and use that result to study the effect of
different activations on the confined supercooled liquid. We observe an
increase of the activation effect on the dynamics when the medium is confined.
We find that the confinement slowing down dependence with the pore radius
depends on the activation of the liquid. We argue that these findings result
from a modification of dynamic correlation lengths with activation. In this
picture the activation permits to control the liquid correlation length and
dynamics inside the pore. Studying the local structure we observe a
modification of the layering organization induced by the activation. We also
find that the mobility inside the pore depends on the layers, being larger
where the local density is small.

dynamic heterogeneity,glass-transition

###### pacs:

64.70.pj, 61.20.Lc, 66.30.hh

##  I  Introduction

Liquids and solids constituted or comprising molecules like molecular motors
that are able to move by themselves are called activeactive1 ; active5 ;
active-confi . An example is cells of living biological organisms that are
constituted of active soft materials. Active matter opens a window to the
statistical physics of non equilibrium systems and to biological physics
making it a fascinating new domain of research that develops rapidlyactive1 ;
active2 ; active3 ; active4 ; active5 ; active6 ; active7 ; active8 ; active9
; active10 ; active11 ; active12 ; active13 ; active14 ; active15 ; active16 ;
active17 ; active18 ; active-confi . The different statistical physics
associated with non equilibrium systems, induces a number of new phenomena, as
for example non-equilibrium phase transitionsactive4 .

When liquids are cooled fast enough to remain liquid below their melting
temperature $T_{m}$, they are subject to a large increase of their viscosity.
The viscosity below $T_{m}$ then increases at least exponentially when the
temperature drops, leading eventually to a glass at the glass-transition
temperature $T_{g}$. That dramatic increase of the viscosity and the
associated decrease of most dynamical properties like the diffusion
coefficient, strangely appears without significant structural modifications of
the liquid. The nature of that transition from the liquid to the glass, known
as the long standing glass-transition problem, is still puzzling
scientistsanderson ; gt0 ; gt1 ; gt2 ; dh0 ; c3 ; c4 ; md16 . As phase
transitions are usually associated with the divergence of correlation lengths
and the apparition of cooperative mechanisms, increasing correlation lengths
have been searched extensively in supercooled liquids. Confinement of the
liquid inside nano-pores has been seen as a way to cutoff the correlation
lengths and obtain indirect informations on them by studying the associated
dynamical modifications in the liquid. For that fondamental reason as well as
multiple practical applications, confinement has been the subject of a number
of studies and is still a subject of active researchconff1 ; conff2 ; conff3 ;
conff4 ; conff5 ; conff6 ; conf-1 ; conf0 ; conf1 ; conf2 ; conf3 ; conf4 ;
conf5 ; conf6 ; confine ; conf7 ; active-confi . Indeed, in supercooled
liquids, confinement induces various effects. As suggested by the theoretical
perspective described above, confinement in most cases induces a slowing down
of the dynamics reminiscent of the slowing down induced by a temperature
decrease. It also induces a structural organization called layering due to the
fixation of the local structure in front of the immobile walls of the pore. In
this work we raise the question to how the confinement effects are modified
for active matter. In order to answer that question, we use a simple
butterfly-like flat molecular motormotoro1 ; motoro2 ; motoro3 ; motoro4 ;
motoro5 ; motoro6 ; motoro7 ; motoro8 ; motoro9 ; motoro10 ; motoro11  (Figure
1), to activate with its periodic folding the supercooled liquid surrounding
it, confined in nano-pores of various radii (Figures 2 and 3). Experimentally
our motor can be seen as a simplification of the azobenzene molecule or
derivatives, which undergoes a photo-isomerization processaz1 ; az2 ; az3 ;
az4 ; az5 ; az6 ; az7 ; az8 ; az9 ; cage ; md16 ; az10  when subject to a
light stimulus. We study the effect of different activations induced by
different opening angles of our motor (Figure 1). We observe an increase of
the activation effect on the dynamics when the medium is confined. Studying
the local structure we observe a modification of the layering organization
induced by the activation. We also find that the motility is dependent on the
layers, being larger where the local density is small.

Figure 1: (color online) Picture of the folded and unfolded motor molecule.
Through this work we use the parameter $d_{z}$ to tune and quantify the
activation of the medium by the motor’s folding. We validate that choice at
posteriori in the excitation density section. Figure 2: (color online)
Snapshots of simulations for pores of radii (a) $R=5$Å, (b) $R=8$Å, (c)
$R=10$Å, and (d) $R=12.5$Å. The colors are arbitrary. $T=500K$. The
simulations use $8$ pores of radii varying from $5$ to $12.5$Å separated by
steps of $1$ Å ($1.5$ Å for the last step).  Figure 3: (color online) Snapshot
of a simulation observed from the side to show the motor. The pore radius is
$R=10$Å. $T=500K$.

##  II Calculation

In addition to theoretical and experimental methods, simulations in particular
molecular dynamics and Monte Carlo simulationsmd1 ; md2 ; md2b ; md4  together
with model systemsms1 ; ms2 ; ms3 ; ms4 ; ms5  are now widely used to unravel
unsolved problems in condensed matter and complex systems physicskeys ; md3 ;
md4b ; md6 ; md7 ; md8 ; md9 ; ee2 ; md11 ; md12 ; md13 ; md14 ; md15 ; ee1 ;
finite1 ; u1 . The reader will find details on our simulation procedure in
previous paperspccp ; prefold ; ariane , however for convenience we will
resume it here. Our simulations use one motor molecule (see Figure 1 for its
description) imbedded inside a medium constituted of $1760$ linear molecules,
in a parallelepiped box $31.1$ Å wide and $62.2$ Å high. After aging the
simulation box during $10ns$ at the temperature of studies, we create the pore
by suddenly freezing molecules located at a distance larger than $R$ from the
center axis of our simulation box. This procedure is intended to minimize the
structure modification of the liquid by the confinement. After their sudden
freezing, the glassy molecules of the pores are no more allowed to move,
leading to pure elastic interactions with the confined liquid. Consequently
the liquid temperature is not modified by the interactions with the pore
walls. After that procedure, as the dynamics changes with the confinement, the
liquid is again aged during $10ns$ before the beginning of any study.

We integrate the equations of motion using the Gear algorithm with a
quaternion decompositionmd1  and a time step $\Delta t=10^{-15}s$. When the
motor is active our simulations are out of equilibrium, because the motor’s
folding periodically release energy into the system. However our system is in
a steady state and does not age as explained below. For that purpose we use
the Berendsen thermostatberendsen  to evacuate from the system the energy
created by the motor’s folding. Notice that the thermostat applies to the
medium’s molecules but not to the motor. Thus the motor’s cooling results only
from the molecular interactions with the medium’s molecules. We use the NVT
canonic thermodynamic ensemble as approximated by Berendsen thermostat (see
ref.finite2  for an evaluation of the effect of the thermostat on our
calculations), and periodic boundary conditions. The molecules of the medium
(host)ariane  are constituted of two rigidly bonded atoms ($i=1,2$) at the
fixed interatomic distance  $l_{h}$$=1.73$Å. These atoms interact with atoms
of other molecules with the following Lennard-Jones potentials:

$V_{ij}=4\epsilon_{ij}((\sigma_{ij}/r)^{12}-(\sigma_{ij}/r)^{6})$ (1)

with the parametersariane : $\epsilon_{11}=\epsilon_{12}=0.5KJ/mol$,
$\epsilon_{22}=0.4KJ/mol$, $\sigma_{11}=\sigma_{12}=3.45$Å,
$\sigma_{22}=3.28$Å. The mass of the motor is $M=540g/mole$ (constituted of
$18$ atoms, each one of mass $30g/mole$) and the mass of the host molecule is
$m=80g/mole$ ($2$ atoms with a mass of $40g/mole$ each). We model the motor
with $18$ atoms in a rectangular shape constituted of two rows of $9$ rigidly
bonded atoms. The width of the swimmer is $L_{s}=4.4$Å and its length
$l_{s}=15.4$Å. The length of the host molecule is $l_{h}=5.09$Å and its width
$L_{h}=3.37$Å. The motor’s atoms interact with the medium’s atoms using mixing
rules and a Lennard-Jones interatomic potential on each atom of the motor,
defined by the parameters: $\epsilon_{33}=1.88KJ/mol$, $\sigma_{33}=3.405$Å.
We use the following mixing rules mix1 ; mix2 :

$\epsilon_{ij}=(\epsilon_{ii}.\epsilon_{jj})^{0.5};\sigma_{ij}=(\sigma_{ii}.\sigma_{jj})^{0.5}$
(2)

for the interactions between the motor and the host atoms. Our medium is a
fragile liquidfragile1 ; fragile2  that falls out of equilibrium in bulk
simulations below $T=340K$, i.e. $T=340$ K is the smallest temperature for
which we can equilibrate the bulk liquid when the motor is not active.
Consequently the bulk medium above that temperature behaves as a viscous
supercooled liquid in our simulations while below it behaves as a solid (as
$t_{simulation}<\tau_{\alpha}$). Notice however that when active, the motor’s
motions induce a fluidization of the medium around itmd16 ; flu1 ; flu2 ; flu3
; flu4 ; carry ; flu5 ; rate , an effect that has been experimentally
demonstratedflu1 ; flu2 ; flu3 ; flu4 ; flu5  with azobenzene photo-
isomerizing molecules embedded in soft matter. We evaluate the glass
transition temperature $T_{g}$ in the bulk to be slightly smaller
$T_{g}\approx 250K$. Notice that as they are modeled with Lennard-Jones atoms,
the host and motor potentials are quite versatile. Due to that property, a
shift in the parameters $\epsilon$ will shift all the temperatures by the same
amount, including the glass-transition temperature and the melting temperature
of the material. Each folding is modeled as continuous, using a constant
quaternion variation, with a folding time $\tau_{f}=0.3ps$. The total cycle
period is constant and fixed at $\tau_{p}=400ps$. The folded and unfolded part
of the cycle have the same duration equal to $\tau_{p}/2=200ps$ including the
$0.3ps$ folding or unfolding. Our system, while out of equilibrium, is in a
steady state and is not aging. That behavior is obtained because the energy
released by the motor into the medium is small enough and the time lapse
between two stimuli large enough for the system to relax before a new stimuli
appears. In other words we are in the linear response regimepccp . Through
this work we use the mean square displacement to obtain informations on the
molecules mobility and diffusion inside the pore. The mean square displacement
is defined asmd1 :

$\displaystyle{<r^{2}(t)>={1\over N.N_{t_{0}}}\sum_{i,t_{0}}\mid{{\bf
r}_{i}(t+t_{0})-{\bf r}_{i}(t_{0})}}\mid^{2}$ (3)

From the time evolution of the mean square displacement we then calculate the
diffusion coefficient $D$ for diffusive displacements using the Stokes-
Einstein equation:

$\displaystyle{\lim_{t\to\infty}<r^{2}(t)>=6Dt}$ (4)

Finally we will use the Non Gaussian parameter $\alpha_{2}(t)$ to measure the
extent of cooperative motions inside the medium.

$\displaystyle{\alpha_{2}(t)=\frac{3<r^{4}(t)>}{5<r^{2}(t)>^{2}}-1}$ (5)

Unless otherwise specified the results presented in this work correspond to a
temperature $T=500K$ at which our model liquid is supercooled. Notice however
that our system is a model system with Lennard-Jones interactions only.
Therefore, due to the properties of the Lennard-Jones potential, one can shift
the temperatures by any factor $\alpha$ ($T^{\prime}=\alpha T$) creating a new
material. That new material will have all the $\epsilon$ parameters of the LJ
potentials (including for the motor) modified in the same way
$\epsilon^{\prime}=\alpha\epsilon$. The results corresponding to that new
material will have to be translated from the results of our work by a shift of
time $t^{\prime}=t/\alpha^{0.5}$. Similarly, the size of the atoms can be
scaled by a factor $\gamma$ ($\sigma^{\prime}=\gamma\sigma$) provided that all
the distances of the system are scaled by the same factor. Our results will in
that case have to be translated by a shift of distance $\gamma$
($r^{\prime}=\gamma r$) and a shift of time $\gamma$ ($t^{\prime}=\gamma t$).
In that way our simulations are valid for a large number of materials although
approximately, and experimentalists can adjust our data to their system of
concern.

##  III Results and discussion

###  III.1 Diffusion inside the pore

Figure 4: (color online) Mean square displacement of the medium molecules
located at a distance $r<10$ Å from the motor at time $t=0$, for different
pore radii $R$. The activation parameter $d_{z}$ is different for each Figure.
$Z$ is the direction of the pore axis and $z$ the direction perpendicular to
the motor’s plane in the motor’s molecular frame. In Figure (a) $d_{z}=0$
meaning that there is no activation and the motor acts as a passive probe.

Figures 4a 4b 4c and 4d show the mean square displacement of the confined
medium for different pore radii ranging from the Bulk to a pore radius $R=4$Å,
each Figure corresponding to a different activation quantified by the motion
of the arms $d_{z}$ (Figure 1). For a purely passive motor molecule (Figure
4a), the pore effect is important. As the pore size decreases, the diffusion
inside the medium decreases (i.e. the viscosity increases) leading for pores
of radii smaller than 11Å to a solid non diffusive medium. This confinement
effect is due to the decrease of the probability of molecules near the wall to
escape the cage created by the molecules surrounding them, as part of their
cage is created by the pore wall and thus never opens. Notice that, this well
established confinement effect conf-1 ; conf0 ; conf1 ; conf2 ; conf3 ; conf4
; conf5 ; conf6 ; confine ; conf7 , displays a number of similarities with the
effect of a decrease in temperature in supercooled liquids. As expected, when
the motor activates the medium in Figures 4b to 4d, the diffusion inside the
pores increases. However, surprisingly it increases more in the small pores
(see for example the rapid increase for $R=5$Å), than in the larger ones,
leading to a decrease of the difference in diffusions. Eventually, for the
larger activation investigated in Figure 4d, the diffusion inside the
different pores is rather similar. Thus we observe in Figure 4 that the
activation progressively suppresses the effect of confinement.

Figure 5: (color online) Diffusion coefficient of the medium molecules for
different pore radii $R$ and activation parameters $d_{z}$.

The evolution of the diffusion coefficient in Figure 5 resumes that behavior.
The upper circles on the Figure show the effect of the larger activation by
the motor on the medium’s diffusion for various pore radii. The diffusion
coefficients appear roughly constant for that large activation, even
increasing slightly when the pore radius decreases. As discussed before, the
activation of the medium inside the pore washes out the slowing down induced
by the confinement. For an activation slightly weaker the second rank of
circles from the top show a constant diffusion for the different pores. When
the activation decreases the pore slowing down effect increases but is still
much smaller in the Figure than for the non-activated medium ($d_{z}=0$).

###  III.2 Activation effect

Figure 5 shows that the diffusion coefficients follow exponentially decreasing
laws with the inverse of the pore radius $1/R$.

$\displaystyle{D=D_{0}exp(-{\zeta}/R)}$ (6)

Where $\zeta$ is a correlation length that depends on the activation parameter
$d_{z}$. As a result, one can explain the decrease of the pore effect with the
activation, by the decrease of that correlation length of the medium when it
is activated. For our largest activation the correlation length $\zeta$ tends
to zero and the pore effect disappears.

Figure 6: (color online) Mean square displacement of the medium molecules for
various activation parameters $d_{z}$; (a) in the bulk ($R=\infty$); (b) for a
pore of radius $R=5$Å. The activation dependence appears much larger in the
pore.

Figures 6(a) and (b) compare the effect of the activation in the bulk and in a
pore of radius $R=5$Å. Figure 6 shows a much larger activation effect on the
medium’s displacements inside the small pore than in the bulk. For a time
lapse of $10ns$ the mean square displacement in the pore varies indeed from
$10^{-1}$Å2 to $4.10^{1}$Å2, that is by a factor $400$ depending on the
activation, while in the bulk it varies by a mere factor $3$. This result
agrees well with the findings of previous section that the diffusion for large
activation doesn’t depend on the pore size. As the pore radius decreases, the
slowing down by the pore wall increases, but the activation effect also
increases compensating that slowing down. Notice that as the pore radius
decreases, the relative number of motor to free molecules increases due to the
geometry. While results we show take only into account molecules a distance
$r<10$Å apart from the motor, reducing that geometrical effect, it doesn’t
eliminate it. Geometry appears consequently as a possible cause for the
increase of the activation impact on the diffusion for small pores. In order
to conclude on that possibility we calculated the mean square displacements at
a smaller distance $r<5$Å from the motor. The results (not shown) were very
similar to the results of Figure 6, showing that the increase of the
activation effect on diffusion for small pores, is not arising from the
geometry (i.e. from the mean distance to the motor).

It is interesting to point out however that the increase of the effect of
activation when the pore size decreases doesn’t increase the diffusion to
values larger than bulk values but tends instead to reach the bulk values from
below. That point comes in support of the decrease of the correlation lengths
explanation.

###  III.3 Cooperative motions

Figure 7: (color online) Non Gaussian parameter
$\alpha_{2}(t)=\frac{3<r^{4}(t)>}{5<r^{2}(t)>^{2}}-1$ around the motor
($r<10$Å) for various activation parameters $d_{z}$ and for a pore of radius
$R=5$Å, at $T=500K$.

Spontaneous (thermal)dh0  and activation-induced cooperative motionsmd16 ; c3
; c4  are present in supercooled liquids and have been well documented.
Because cooperativity is directly related to the physical correlation length
of the liquid, its increase or decrease can give us information on that
correlation length. Therefore, to shed some light on the correlation length
explanation, we now study the extent of cooperative motions in our system. For
that purpose we display in Figure 7 the Non Gaussian parameter $\alpha_{2}(t)$
evolution with the activation. Cooperative motions in supercooled liquids
induce a tail in the Van Hove self correlation function, leading to its
departure from the Gaussian shape. For that reason the departure from the
Gaussian shape of the Van Hove, measured by the Non Gaussian coefficient, has
been used extensively as a measure of the cooperative motions.

In Figure 7 on the right $\alpha_{2}(t)$ shows peaks at times corresponding to
multiple of the motor’s folding period, that are the result of the excitations
induced by the motor’s folding on the medium. The activation-induced
cooperativity as measured by $\alpha_{2}(t)$ decreases when we increase the
activation parameter $d_{z}$ from $3.3$Å to $4.9$Å. That result comes again in
support of the decrease of the medium’s correlation length with the
activation. Notice however that as expected when the activation parameter is
null the induced cooperativity disappears as there is no more activation. This
result suggests an increase of the cooperativity for small activations and an
optimum.

###  III.4 Excitation density

To better understand previous results, we now turn our attention to the
density of excitations inside the pores. Following Keys et al. keys  we define
excitations as elementary diffusion processes. In our work, excitations are
molecules that move in a time lapse $\Delta t=10ps$ a distance larger than
$\Delta r=1$ Å from their previous position. Figure 8 shows the excitation
density for various pore radii and excitation processes. We find in Figure 8
that the density of excitation increases with the activation whatever the pore
radius. That result validates at posteriori our choice of the activation
parameter as the displacement of the motor’s arm $d_{z}$.

Interestingly the maximum excitation density is not always located at the pore
center. This effect could be due to layering inside the pores. Layering
induces oscillations of the density inside the pores and low densities promote
excitations as cages are then loose, letting molecules escape easier. We will
test that interpretation in Figure 9.

Figure 8 shows that when the pore size decreases the excitation density
decreases much less rapidly for large activations. Therefore the excitation
density is larger than expected for small pores when the activation is on, in
agreement with the activation effect described in a previous section. That
result suggests that the larger than expected diffusion observed for small
pores (the activation effect) is due to a larger density of excitations that
is a larger number of mobile molecules as opposed to larger motions of some
molecules.

Figure 8: (color online) Excitation density $\rho_{excitation}$ as a function
of the distance $r$ from the pore central axis, for different pore radii $R$
and activation parameters $d_{z}$. Excitations are here defined as medium’s
molecules motion larger than $\Delta r=1$ Å occuring during a time lapse
$\Delta t=10ps$.

###  III.5 Structural organization

We will now investigate the local structure inside the pores. In Figure 9 we
display the local density inside the pores as a function of the distance $r$
from the pore center. That is, $r$ is the radius of cylindrical coordinates,
$z$ being the pore axis. The Figure shows a clear layering effect, that we
highlight with a line when the activation is off. The local density is in most
cases as expected maximum at the center of the pore ($r=0$) and minimum around
the wall. However this is not always the case. For instance for a pore of
$R=5$Å radius, the maximum density is not located in the center of the pore,
and around the wall for large activations the density is not null. For a pore
of $R=8$Å radius, there is a maximum density at the pore center but it
corresponds to a thin peak, and we observe a wider maximum apart from the
center.

As expected, the layering explains the excitation density maxima shifted from
the center of the pore, as the local density is actually smaller at the
location of excitation density maxima. Moreover an unexpected effect appears,
the layering changes with the activation. Not only it decreases slightly but
the oscillations locations are also modified. Work is in progress to better
understand that interesting effect.

Figure 9: (color online) Density $\rho$ of medium molecules inside the pore as
a function of the distance $r$ from the pore central axis. The Figures show
that result for different pore radii $R$ and activation parameters $d_{z}$.
The values are normalized by the average density inside the pore $\rho_{0}$.
The dashed line displays the layering when the motor is not active. The
figures show that the layering is strongly modified with the activation.

##  IV Conclusion

In supercooled liquids, nano-confinement modifies the dynamical properties of
the liquid usually inducing an important slowing down of the dynamics.
Confinement also induces a structural organization called layering. In this
work we investigated how these confinement effects are modified when the
liquid is out of equilibrium. Results show that the dynamical slowing down
induced by the confinement is strongly modified by the activation, leading to
a diffusion that may not depend anymore on the pore diameter. In agreement
with that finding our results show that the activation effect is much larger
upon confinement than in the bulk, increasing rapidly when the pore size
decreases. We interpret these effects as arising from an activation induced
decrease of the dynamical correlation lengths of the liquid. In agreement with
that picture we observe a decrease of the cooperative motions upon activation.
While the structure of the liquid is unchanged, we observe a decrease of the
layering inside the pore but also a qualitative modification of the layering
with the activation. Eventually we observe a relation between the layering and
the elementary diffusion processes called excitations, a result strongly
suggesting that the layering triggers an increase of the mobility in the less
dense parts of the pore.

Conflict of interest

There are no conflict of interest to declare.

Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

  * (1) O. Dauchot, H. Lowen,  J. Chem. Phys. 151, 114901 (2019).
  * (2) C. Bechinger, R. Di Leonardo, H. Lowen, C. Reichhardt, G. Volpe, G. Volpe,  Rev. Mod. Phys. 88, 045006 (2016).
  * (3) P. Liu, H. Zhu, Y. Zeng, G. Du, L. Ning, D. Wang, K. Chen, Y. Lu, N. Zheng, F. Ye, M. Yang,  P.N.A.S. 117, 11901 (2020).
  * (4) A.P. Solon, J. Stenhammar, R. Wittkowski, M. Kardar, Y. Kafri, M.E. Cates, J. Tailleur,  Phys. Rev. Lett. 114, 198301 (2015).
  * (5) A.P. Solon, J. Stenhammar, M.E. Cates, Y. Kafri, J. Tailleur,  Phys. Rev. E 97, 020602(R) (2018).
  * (6) T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen, O. Shochet,  Phys. Rev. Lett. 75, 1226 (1995).
  * (7) J. Elgeti, R.G. Winkler, G. Gompper,  Rep. Prog. Phys. 78, 056601 (2015).
  * (8) E.W. Burkholder, J.F. Brady,  J. Chem. Phys. 150, 184901 (2019).
  * (9) G. Junot, G. Briand, R. Ledesma-Alonso, O. Dauchot,  Phys. Rev. Lett. 119, 028002 (2017).
  * (10) J.U. Klamser, S. Kapfer, W. Krauth,  J. Chem. Phys. 150, 144113 (2019).
  * (11) N. Nikola, A.P. Solon, Y. Kafri, M. Kardar, J. Tailleur, R. Voituriez,  Phys. Rev. Lett. 117, 098001 (2016).
  * (12) H. Wang, T. Qian, X. Xu,  Soft Matter 17, 3634 (2021).
  * (13) S. Dal Cengio, D. Levis, I. Paganobarraga,  J. Stat. Mech. 043201 (2021).
  * (14) H.G. Wood, J.A. Hanna,  Soft Matter 17, 3137 (2021).
  * (15) C. Reichhardt, C.J.O. Reichhardt,  Phys. Rev. E 103, 022602 (2021).
  * (16) J. Denk, E. Frey,  P.N.A.S. 117, 31623 (2020).
  * (17) V. Teboul, G. Rajonson,  J. Chem. Phys. 150, 214505 (2019).
  * (18) P. Herrera, M. Sandoval,  Phys. Rev. E 103, 012601 (2021).
  * (19) D. Dattler, G. Fuks, J. Heiser, E. Moulin, A. Perrot, X. Yao, N. Giuseppone,  Chem. Rev. 120, 310 (2020).
  * (20) P.W. Anderson,  Science 267, 1610 (1995)
  * (21) K. Binder, W. Kob, Glassy materials and disordered solids  World Scientific (Singapore) 2011
  * (22) P.G. Debenedetti, Metastable liquids  Princeton University Press (Princeton) 1996
  * (23) P.G. Wolynes, V. Lubchenko, Structural glasses and supercooled liquids  Wiley (Hoboken) 2012
  * (24) L. Berthier, G. Biroli, J.P. Bouchaud, L. Cipelletti, W. Van Saarloos, Dynamical heterogeneities in glasses, colloids and granular media, Oxford Science Publications, Oxford 2011.
  * (25) V. Teboul, J. B. Accary,  Phys. Rev. E 89, 012303 (2014).
  * (26) V. Teboul, J. B. Accary, M. Chrysos,  Phys. Rev. E 87, 032309 (2013).
  * (27) V. Teboul, M. Saiddine, J.M. Nunzi,  Phys. Rev. Lett. 103, 265701 (2009).
  * (28) A. Jani, M. Busch, J.B. Mietner, et al., J. Chem. Phys. 154, 094505 (2021).
  * (29) R. Mhanna, P. Catrou, S. Dutta, et al., J. Phys. Chem. B 124, 3152 (2020).
  * (30) B. Malfait, A. Pouessel, A. Jani, D. Morineau, J. Phys. Chem. Lett. 11, 5763 (2020).
  * (31) S. Fei, W.L. Hsu, J.J. Delaunay, et al., J. Chem. Phys. 154, 144503 (2021).
  * (32) P. Bampoulis, J. Chem. Phys. 154, 134705 (2021).
  * (33) P. Sepulveda-Medina, M. Tyagi, C. Wang, et al., J. Chem. Phys. 154, 154903 (2021).
  * (34) L. Stefanutti, L.E. Bove, G. Lelong, M.A. Ricci, A.K. Soper, F. Bruni, Phys. Chem. Chem. Phys. 21, 4931 (2019).
  * (35) J. Swenson, Phys. Chem. Chem. Phys. 20, 30095 (2018).
  * (36) A.A. Milischuk, B.M. Ladanyi, J. Chem. Phys. 135, 174709 (2011).
  * (37) N. Kuon, A.A. Milischuk, B. M. Ladanyi, E. Flenner, J. Chem. Phys. 146, 214501 (2017).
  * (38) J. Swenson, H. Jansson, R. Bergman, Phys. Rev. Lett. 96, 247802 (2006).
  * (39) S.M. Saparov, J.R. Pfeifer, L. Al-Momani, G. Portella, B.L. deGroot, U. Koert, P. Pohl, Phys. Rev. Lett. 96, 148101 (2006).
  * (40) V. Teboul, Int. J. Nanotechnol. 5, 851 (2008).
  * (41) D.J. Mann, M.D. Halls, Phys. Rev. Lett. 90, 195503 (2003).
  * (42) M.C. Bellissent-Funel, J. Phys.: Condens. Matter 13, 9165-9177 (2001).
  * (43) R. Bergman, J. Swenson, Nature 403, 283-286 (2000).
  * (44) I. Aprahamian,  ACS Central Science 6, 347 (2020).
  * (45) J. Wang,  Nanomachines: Fundamentals and Applications, Wiley, Weinheim 2013.
  * (46) A.P. Davis, Nature 401, 120-121 (1999).
  * (47) J.P. Sauvage, _Molecular machines and motors_ , Springer, Berlin, 2001.
  * (48) J.R. Howse, R.A.L. Jones, A.J. Ryan, T. Gough, R. Vafabakhsh, R. Golestanian,  Phys. Rev. Lett. 99, 048102 (2007).
  * (49) H. Karani, G.E. Pradillo, P.M. Vlahovska,  Phys. Rev. Lett. 123, 208002 (2019).
  * (50) F. Novotny, M. Pumera,  Scien. Rep. 9, 13222 (2019).
  * (51) X. Arque, A. Romero-Rivera, F. Feixas, T. Patino, S. Osuna, S. Sanchez,  Nature Comm. 10, 2826 (2019).
  * (52) A.M. Brooks, M. Tasinkevych, S. Sabrina, D. Velegol, A. Sen, K.J.M. Bishop,  Nature Comm. 10, 495 (2019).
  * (53) P. Pietzonka, E. Fodor, C. Lohrmann, M.E. Cates, U. Seifert,  Phys. Rev. X 9, 041032 (2019).
  * (54) C. Calero, J. Garcia-Torres, A. Ortiz-Ambriz, F. Sagues, I. Pagonabarraga, P. Tierno,  Nanoscale 11, 18723 (2019).
  * (55) A. Natansohn, P. Rochon,  Chem. Rev. 102, 4139-4175 (2002).
  * (56) J.A. Delaire, K. Nakatani,  Chem. Rev. 100, 1817 (2000).
  * (57) G.S. Kumar, D.C. Neckers,  Chem. Rev. 89, 1915 (1989).
  * (58) K.G. Yager, C.J. Barrett,  Curr. Opin. Solid State Mater. Sci. 5, 487 (2001).
  * (59) T.G. Pedersen, P.M. Johansen,  Phys. Rev. Lett. 79, 2470-2473 (1997).
  * (60) T.G. Pedersen, P.M. Johansen, N.C.R. Holme, P.S. Ramanujam, S. Hvilsted,  Phys. Rev. Lett. 80, 89-92 (1998).
  * (61) J. Kumar, L. Li, X.L. Jiang, D.Y. Kim, T.S. Lee, S. Tripathy,  Appl. Phys. Lett. 72, 2096-2098 (1998).
  * (62) C.J. Barrett, P.L. Rochon, A.L. Natansohn,  J. Chem. Phys. 109, 1505-1516 (1998).
  * (63) C.J. Barrett, A.L. Natansohn, P.L. Rochon,  J. Phys. Chem. 100, 8836-8842 (1996).
  * (64) V. Teboul, M. Saiddine, J.M. Nunzi, J.B. Accary,  J. Chem. Phys. 134, 114517 (2011).
  * (65) P. Lefin, C. Fiorini, J.M. Nunzi,  Pure Appl. Opt. 7, 71-82 (1998).
  * (66) M.P. Allen, D.J. Tildesley, _Computer Simulation of Liquids_ , Oxford University Press, New York 1990.
  * (67) M. Griebel, S. Knapek, G. Zumbusch, _Numerical Simulation in Molecular Dynamics_ , Springer-Verlag, Berlin 2007.
  * (68) D. Frenkel, B. Smit, _Understanding Molecular Simulation_ , Academic Press, San Diego 1996.
  * (69) S. Chaussedent, V. Teboul, A. Monteil,  Curr. Opin. Solid State Mater. Science 7, 111-116 (2003).
  * (70) F. Ritort, P. Sollich, Adv. Phys. 52, 219 (2003).
  * (71) G. H. Fredrickson, H. C. Andersen, Phys. Rev. Lett. 53, 1244 (1984).
  * (72) J. Jackle, S. Eisinger, Z. Phys. B 84, 115 (1991).
  * (73) W. Kob and H. C. Andersen, Phys. Rev. E 48, 4364 (1993).
  * (74) V. Teboul,  J. Chem. Phys. 141, 194501 (2014).
  * (75) C.F.E. Schroer, A. Heuer, Phys. Rev. Lett. 110, 067801 (2013).
  * (76) J.B. Accary, V. Teboul,  J. Chem. Phys. 136, 0194502 (2012).
  * (77) A. Furukawa, K. Kim, S. Saito, H. Tanaka, Phys. Rev. Lett. 102, 016001 (2009).
  * (78) T. Iwashita, T. Egami, Phys. Rev. Lett. 108, 196001 (2012).
  * (79) T. Gleim, W. Kob, K. Binder, Phys. Rev. Lett. 81, 4404 (1998).
  * (80) E.J. Saltzman, K.S. Schweitzer, J. Chem. Phys. 125, 044509 (2006).
  * (81) Y. Le Duff, V. Teboul,  Phys. Lett. A 157, 44 (1991).
  * (82) V. Teboul, Y. Le Duff, T. Bancewicz,  J. Chem. Phys. 103, 1384 (1995).
  * (83) B. Uralcan, I.A. Aksay, P.G. Debenedetti, D. Limmer,  J. Phys. Chem. Lett. 7, 2733 (2016).
  * (84) J. Zhang, F. Muller-Plathe, F. Leroy,  Langmuir 31, 7544 (2015).
  * (85) E. Flenner, G. Szamel,  J. Phys. Chem. B 119, 9188 (2015).
  * (86) E. Flenner, G. Szamel,  Nature Comm. 6, 7392 (2015).
  * (87) L.C. Fai, V. Teboul, A. Monteil, S. Maabou, I. Nsangou,  Cond. Matt. Phys. 8, 639 (2005).
  * (88) Y.S. Elmatad, A.S. Keys,  Phys. Rev. E 85, 061502 (2012).
  * (89) V. Teboul,  Eur. Phys. J. B 51, 111 (2006).
  * (90) A. S. Keys, L. O. Hedges, J.P. Garrahan, S.C. Glotzer, D. Chandler,  Phys. Rev. X 1, 021013 (2011).
  * (91) A.P. Kerasidou, Y. Mauboussin, V. Teboul, Chem. Phys. 450, 91 (2015).
  * (92) S. Ciobotarescu, S. Bechelli, G. Rajonson, S. Migirditch, B. Hester, N. Hurduc, V. Teboul,  Phys. Rev. E 96, 062614 (2017).
  * (93) S. Ciobotarescu, N. Hurduc, V. Teboul,  Phys. Chem. Chem. Phys. 18, 14654 (2016).
  * (94) H.J.C. Berendsen, J.P.M. Postma, W. Van Gunsteren, A. DiNola, J.R. Haak,  J. Chem. Phys. 81, 3684-3690 (1984).
  * (95) S. Taamalli, J. Hinds, S. Migirditch, V. Teboul,  Phys. Rev. E 94, 052604 (2016).
  * (96) R.J. Good, C.J. Hope,  J. Chem. Phys. 53, 540 (1970).
  * (97) J. Delhommelle, P. Millie,  Mol. Phys. 99, 619 (2001).
  * (98) C.A. Angell,  Science 267, 1924-1935 (1995).
  * (99) N.A. Mauro, M. Blodgett, M.L. Johnson, A.J. Vogt, K.F. Kelton,  Nature Comm. 5, 4616 (2014).
  * (100) M. Saiddine, V. Teboul, J.M. Nunzi,  J. Chem. Phys. 133, 044902 (2010).
  * (101) V. Teboul, R. Barille, P. Tajalli, S. Ahmadi-Kandjani, H. Tajalli, S. Zielinska, E. Ortyl,  Soft Matt. 11, 6444 (2015).
  * (102) J.B. Accary, V. Teboul, J. Chem. Phys. 139, 034501 (2013).
  * (103) P. Karageorgiev, D. Neher, B. Schulz, B. Stiller, U. Pietsch, M. Giersig, L. Brehmer, Nature Mater. 4, 699-703 (2005).
  * (104) G.J. Fang, J.E. Maclennan, Y. Yi, M.A. Glaser, M. Farrow, E. Korblova, D.M. Walba, T.E. Furtak, N.A. Clark, Nature Comm. 4, 1521 (2013).
  * (105) N. Hurduc, B.C. Donose, A. Macovei, C. Paius, C. Ibanescu, D. Scutaru, M. Hamel, N. Branza-Nichita, L. Rocha, Soft Mat. 10, 4640-4647 (2014).
  * (106) J. Vapaavuori, A. Laventure, C.G. Bazuin, O. Lebel, C. Pellerin, J. Amer. Chem. Soc. 137, 13510 (2015).