# Distinct elastic properties and their origins in glasses and gels

Yinqiao Wang  Research Center for Advanced Science and Technology, The
University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan  Michio
Tateno  Research Center for Advanced Science and Technology, The University of
Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan  Hajime Tanaka
<EMAIL_ADDRESS>Research Center for Advanced Science and Technology,
The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan
Department of Fundamental Engineering, Institute of Industrial Science, The
University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan

###### Abstract

Glasses and gels, widely encountered amorphous solids with diverse industrial
and everyday applications, share intriguing similarities such as rigidity
without crystalline order and dynamic slowing down during aging. However, the
underlying differences between these two fascinating materials have remained
elusive. Here we uncover distinct elastic properties concerning observation
and aging times in glasses and gels, while delving into the underlying
mechanisms. In glasses, we observe a gradual decrease in the shear modulus,
while the bulk modulus remains constant throughout the observation time. In
contrast, gels exhibit a decrease in both the shear and bulk moduli over the
observation time. Additionally, during aging, glasses exhibit a steady trend
of stiffening, while gels demonstrate initial stiffening followed by
softening. By unravelling the intricate relationship between structure,
dynamics, and elasticity, we attribute these differences to mechanisms that
minimize free energy: structural ordering in glasses and interface reduction
in gels. Our work not only uncovers the distinct behaviors of glasses and gels
but also sheds light on the origin and evolution of elasticity in non-
equilibrium disordered solids, offering significant implications for the
application and design of amorphous materials.

##  I Introduction

Glasses, including metallic, polymer, colloidal, and granular glasses, are
formed through rapid quenching or densification processes Berthier and Biroli
(2011); Wang (2012). These materials are characterized by a homogeneous
density field at a scale beyond individual particles. Their disordered yet
homogeneous nature imparts unique mechanical properties distinct from
crystals, leading to diverse and widespread applications. In contrast, low-
density solids characterized by space-spanning particle networks emerge from
the dynamical arrest of phase separation Zaccarelli (2007); Royall  _et al._
(2021), which are often called physical gels. These materials exhibit
extraordinary soft elasticity, which is vital in various applications within
biological systems, food products, pharmaceuticals, and cosmetics.

Here we consider glasses and gels of spherical particles interacting with
isotropic potentials. Glasses with such interactions sterically self-organize
their structure while gaining vibrational entropy at the expense of
configurational entropy Tanaka  _et al._ (2019), as widely recognized for hard
spheres. Colloidal gels, composed of particles with short-range attractions
and interparticle bonds that can form and break reversibly, are widely studied
as model systems to investigate the formation, aging, and yielding processes
of physical gels Zaccarelli (2007); Royall  _et al._ (2021); Bonn  _et al._
(2017). The mechanisms responsible for gelation have been a subject of intense
debate. One proposed mechanism suggests the dense phase evolving glass
transition Lu  _et al._ (2008); Zia  _et al._ (2014); Testard  _et al._ (2011,
2014), suggesting a connection between glasses and gels as members of the same
class of amorphous solids. Conversely, another mechanism proposes the
percolation of local rigid structure Whitaker  _et al._ (2019); Tsurusawa  _et
al._ (2019); Hsiao Lilian  _et al._ (2012); Patrick Royall  _et al._ (2008);
Zhang  _et al._ (2019); Tsurusawa and Tanaka (2023). In this mechanism, gels
are formed through a process of hierarchical ordering driven by local
‘potential energy’ Tsurusawa and Tanaka (2023). This indicates a fundamental
difference between gels and glasses, as the latter’s formation is primarily
driven by ‘entropy’ Tanaka  _et al._ (2019).

Apart from the ongoing debates about the formation of glasses and gels, there
is still uncertainty about how their material properties compare and differ
once they are formed. One fundamental material property of interest is the
elastic modulus, particularly the shear modulus, denoted as $G$, which is
linked to nonergodicity and thus crucial in distinguishing between liquids and
solids. In crystals, the origin of $G$ is well-established, arising from
broken translational symmetry Alexander (1998). However, in amorphous solids,
such as glasses, gels, or jammed materials, the structural changes during the
transition from a liquid-like to a solid-like state are minimal and do not
involve translational ordering.

For athermal amorphous solids or inherent states of supercooled liquids, the
shear modulus $G$ has been attributed to the mechanical balance of forces
acting within the system DeGiuli (2018); Lemaître (2018); Nampoothiri  _et
al._ (2020); Wang  _et al._ (2020). In amorphous solids at finite temperature,
the emergence of $G$ has been explained using a mean-field theory approach
Yoshino and Mézard (2010); Yoshino and Zamponi (2014); Szamel and Flenner
(2011); Parisi and Zamponi (2010) or through transient mechanical force
balance Yanagishima  _et al._ (2017); Tong  _et al._ (2020). In a practical
context, the shear modulus $G$ has also been linked to the mean-squared
displacements, rescaled either by density Ding  _et al._ (2016) or by the
infinite-frequency modulus Saw and Harrowell (2016), which quantify the
available configurational space.

Glasses and gels, being far from thermodynamic equilibrium states, undergo an
aging process where their properties slowly change over time. The dynamics of
amorphous solids generally exhibit slowing down during aging, indicating a
decrease of available configurational space, which is expected to increase the
shear modulus $G$. Such an age-stiffening phenomenon is observed in glasses
Greinert  _et al._ (2006); Wang (2012). However, for physical gels, many
rheology experiments C. K. Poon  _et al._ (1999); Teece  _et al._ (2014);
Bartlett  _et al._ (2012); Kamp and Kilfoil (2009); Clarke (2021); Fenton  _et
al._ (2023) demonstrate a different behavior: while $G$ initially increases
gradually during aging, it eventually decreases. This phenomenon is related to
the delayed collapse of gels under the influence of gravity C. K. Poon  _et
al._ (1999); Teece  _et al._ (2014); Bartlett  _et al._ (2012); Kamp and
Kilfoil (2009); Clarke (2021); Fenton  _et al._ (2023). Moreover, researchers
have extensively investigated the connection between the elastic properties
and the static structure of colloidal gels Zhang  _et al._ (2019); Bouzid  _et
al._ (2018); Bantawa  _et al._ (2023). However, these studies often neglect
the influence of thermal fluctuations, leaving the combined impact of both
static structure and dynamics on the elasticity of amorphous materials,
including glasses and gels, at finite temperatures unclear even at a
fundamental level.

To address these critical issues, we conducted dynamic simulations on glass
formers and colloidal gels with isotropic interactions. First, we observed
distinct behaviors in the observation-time-dependent moduli of glasses and
gels. Specifically, as the observation time increases, the shear modulus $G$
of glasses exhibited a decrease while the bulk modulus $K$ remained constant;
however, both $G$ and $K$ of gels showed decreasing trends. Then, selecting
the plateau modulus as a meaningful measure of elasticity, we monitored its
time evolution during the aging process of both systems. In glasses, we
observed an increase in $G$ with a slight decrease in $K$ during aging. In
contrast, for gels, we observed a gradual increase in both $G$ and $K$,
followed by a subsequent decrease. By comparing these results to the behaviors
of the corresponding elastic moduli $G^{\rm IS}$ and $K^{\rm IS}$ of the
inherent states, which were obtained by minimizing the corresponding age-
dependent structure at finite temperature, we found that the shear modulus at
finite temperature, $G$, is primarily influenced by the shear modulus at
inherent states, $G^{\rm IS}$, and the vibrational mean-squared displacement.
This connection also holds true for the bulk modulus of gels but not for the
bulk modulus of glasses, as it is mainly controlled by the volumetric
constraint. Furthermore, we revealed two distinct aging mechanisms in glasses
and gels. In glasses, the decrease in bulk free energy leads to structural
ordering, resulting in a slowing down of dynamics and an increase in $G$,
while $G^{\rm IS}$ remained relatively constant. On the other hand, in gels,
the decrease in interfacial free energy promotes the reduction of the
interface area of a network, leading to an increase in particle-scale
connectivity ($Z$) and a decrease in network-scale connectivity ($N_{\rm
loop}$). The former slows down dynamics and increases the elasticity of the
inherent state, while the latter significantly reduces it, ultimately
resulting in a peak in elastic moduli ($G$ and $K$) at finite temperatures for
gels. These findings shed light on the complex and contrasting mechanical
behaviors of glasses and gels during their aging processes, and highlight the
importance of considering both static and dynamic factors in understanding the
elasticity of amorphous materials at finite temperatures.

##  II Results and discussion

Figure 1: aging dynamics and observation-time-dependent elasticity in glasses
and gels. Illustration of the typical structure of simple glass formers (a)
and colloidal gels (d), where particles are colour-coded based on their local
volume fraction $\phi_{l}$, representing the ratio of particle volume to the
Voronoi volume. (b) Cage-relative mean-squared displacement (MSD)
$\langle\Delta r^{2}\rangle_{\rm rel}$ versus observation time $\Delta t$ for
different ages $t_{\rm w}$ of glasses. (e) MSD of all particles $\langle\Delta
r^{2}\rangle$ and only isostatic particles $\langle\Delta r^{2}\rangle_{\rm
iso}$ versus $\Delta t$ for different $t_{\rm w}$ of gels. (c) Observation-
time-dependent shear modulus $G$ and bulk modulus $K$ of glasses. The short
horizontal lines indicate the affine moduli $G_{\rm A}$ and $K_{\rm A}$. (f)
Observation-time-dependent shear modulus $G$ and bulk modulus $K$ of gels.
Inset: the same data as the main panel with the vertical axis zoomed out to
show the affine moduli $G_{\rm A}$ and $K_{\rm A}$, indicated by the short
horizontal lines. Here $\tau_{\rm B}$ denotes the Brownian time.

We conducted numerical simulations on two distinct systems. The first one
involved molecular dynamics simulations of a 2D binary mixture of particles
with harmonic potentials, which is a well-studied glass-forming system Tong
and Tanaka (2018, 2019); Tong  _et al._ (2020). It is worth mentioning that
this system exhibits consistent behaviors both in 2D and 3D Tong and Tanaka
(2018, 2019). We selected the 2D system to allow for extended observation and
aging times. The second one involved Langevin dynamics simulations of a 3D
polydisperse mixture of particles with Morse potentials Griffiths  _et al._
(2017); Zia  _et al._ (2014). This system represents a colloidal gel system
that closely emulates the conditions observed in experimental colloid-polymer
systems Tateno  _et al._ (2022); Tsurusawa  _et al._ (2019); Tsurusawa and
Tanaka (2023). In both systems, we initially equilibrated the particles at
high-temperature liquid states. Subsequently, we instantaneously quenched the
systems from temperatures well above the glass transition temperature $T_{\rm
g}$ or the demixing critical temperature $T_{\rm c}$ to temperatures below
these thresholds. The time elapsed from the quench is referred to as the
waiting time or aging time, denoted as $t_{\rm w}$. Further simulation details
can be found in the Methods section. Figure 1a and d display typical structure
of glasses and gels, respectively. Glasses exhibit a homogeneous density field
beyond a scale above a few particle diameters (Fig. 1a). Conversely, gels
exhibit a highly inhomogeneous density field and possess a large amount of
interface area between the colloid-rich and colloid-poor phases (Fig. 1d).

Figure 1b and e, along with Fig. S1, illustrate the age-dependent dynamics of
both glasses and gels, revealing a noticeable slowing down in both systems.
Glasses exhibit a clear two-step relaxation behavior, and their mean-squared
displacement (MSD) displays a plateau region, which represents the vibrational
MSD, akin to the Debye-Waller factor Widmer-Cooper and Harrowell (2006a, b).
Here, we use the cage-relative MSD denoted as $\langle\Delta r^{2}\rangle_{\rm
rel}$ to eliminate long-wavelength Mermin-Wagner fluctuations characteristic
of 2D systems Flenner and Szamel (2015); Shiba  _et al._ (2016); Vivek  _et
al._ (2017); Illing  _et al._ (2017). On the other hand, the MSD of gels shows
a crossover from subdiffusion to normal diffusion, but it retains a plateau
region when considering only locally rigid (isostatic) particles with a
contact number $Z\geq 6$. This observation indicates that the contribution of
highly mobile particles around the interface Zia  _et al._ (2014) masks the
plateau in MSD. Here, it is worth mentioning that isostatic particles are
associated with the emergence of elasticity in the gel network Tsurusawa  _et
al._ (2019); Hsiao Lilian  _et al._ (2012).

###  II.1 Observation-time-dependent elasticity in glasses and gels

For determining the elasticity of materials, two methods are often utilized
Mizuno and Ikeda (2022); Saw and Harrowell (2016); Kriuchevskyi  _et al._
(2017); Zia  _et al._ (2014); Bouzid  _et al._ (2018); Lemaître and Maloney
(2006). The first method involves directly applying external strain and
measuring its stress response. For example, one can obtain viscoelastic moduli
from oscillatory shear experiments Zia  _et al._ (2014); Bouzid  _et al._
(2018); Bantawa  _et al._ (2023). The second method is the fluctuation
formulation based on linear response theory Saw and Harrowell (2016);
Kriuchevskyi  _et al._ (2017); Mizuno and Ikeda (2022); Lemaître and Maloney
(2006). This approach considers the total modulus, which is the difference
between the affine and nonaffine moduli. The affine component represents the
energy increase when particles conform to homogeneous or affine deformation,
while the nonaffine component arises from particle displacements that deviate
from affine deformation. In thermal systems, the nonaffine modulus can be
described using stress fluctuations Tong  _et al._ (2020); Saw and Harrowell
(2016); Kriuchevskyi  _et al._ (2017); Mizuno and Ikeda (2022). On the other
hand, for athermal systems, it can be directly expressed using the Hessian
matrix Mizuno and Ikeda (2022); Lemaître and Maloney (2006). Detailed
formulations of these methods are provided in the Methods section.

To determine the elastic moduli $G$ and $K$ of glasses, we utilize the stress
fluctuation formalism. On the other hand, for gels, we employ oscillatory
shear and compression to obtain the frequency-dependent shear and bulk moduli,
respectively (see Fig. S3). As depicted in Fig. 1c and f, the elasticity
depends on the observation time and exhibits distinct trends between glasses
and gels.

In glasses, the observation time $\Delta t$ represents the time scale used to
analyse stress fluctuations. As $\Delta t$ increases, $G$ decreases from the
infinite-frequency, or affine value $G_{\rm A}$, reaches a plateau, and
eventually vanishes as particles become diffusive. This plateau is known as
the plateau modulus, corresponding to the plateau region observed in the MSD,
as shown in Fig. 1b. In contrast, the bulk modulus $K$ of glasses remains
constant after decreasing from $K_{\rm A}$ within a very short time at a fast
$\beta$ timescale.

In gels, the observation time $\Delta t$ refers to the period of oscillatory
deformation. Both $G$ and $K$ decrease from their respective affine values,
reach plateau regions, and eventually vanish at large $\Delta t$.

The distinct observation-time dependence of $G$ and $K$ in glasses is
attributed to the different responses of materials to volumetric and shear
deformation. The bulk modulus $K$ arises from pressure or density changes upon
volumetric deformation, but for a uniform material like glasses, these changes
cannot be relaxed. As a result, $K$ remains relatively constant even as the
observation time increases. On the other hand, the shear modulus $G$ arises
from stress anisotropy upon shear deformation. When the system undergoes
particle rearrangements and loses the memory of the initial anisotropic state,
it returns to an isotropic state, and $G$ eventually vanishes. Thus, the bulk
modulus is mainly determined by volumetric constraints, such as density and
mean contact number, while the shear modulus is controlled by configurational
constraints, representing the configurational space the system can explore.

In contrast, for gels, due to the existence of a large void space, even under
compression, the gel network has space to dilate, allowing the increased
pressure to be released through particle rearrangement, similar to the shear
case. Therefore, the configurational constraints affect $G$ and $K$ of gels in
the same manner, leading to the observed similar observation-time dependence
for both moduli. More detailed discussions on the influence of configurational
constraints on $G$ and $K$ in gels will be provided later.

###  II.2 Age-dependent elasticity in glasses and gels

Figure 2: Age-dependent elasticity in glasses and gels. The plateau moduli
$G$, $K$, $G/K$ in finite temperature (red circles) and inherent moduli
$G^{\rm IS}$, $K^{\rm IS}$, $G^{\rm IS}/K^{\rm IS}$ (blue squares) of glasses
are plotted against waiting time $t_{\rm w}$ in (a), (b), and (c). The same
elastic moduli of gels are displayed in (d), (e), and (f). The time $\Delta t$
used to measure these plateau moduli are marked by the vertical dashed lines
in Fig. 1.

To assess age-dependent elasticity, we employ the plateau modulus as a
representative measurement. This modulus represents the timescale during which
particle vibration is permitted while diffusion is prohibited. The plateau
modulus characterizes how materials respond to small-amplitude external
deformation, in which particles undergo nonaffine displacements while
maintaining their neighbors. While the infinite-frequency modulus (also known
as the affine modulus) can be easily determined by the mean number of contacts
or bonds, and the zero-frequency modulus diminishes as particles become
diffusive, the plateau modulus is a manifestation of the intrinsic disordered
structure inherent in amorphous solids and holds profound physical
significance. Here it is worth noting that the terminology for the plateau
modulus may vary across different communities. In the context of metallic
glasses, it is often referred to as the high- or infinite-frequency modulus
Wang (2012); Saw and Harrowell (2016), whereas for colloidal gels, it is
commonly denoted as the low- or zero-frequency modulus Rocklin  _et al._
(2021). This variation arises due to the relative position of measuring ranges
of instruments and the characteristic vibrational frequency of materials.

For glasses, we choose the modulus at $\Delta t=400$ as the plateau timescale,
which is indicated by dashed lines in Fig. 1b and c. Similar trends are
observed for other timescales near the plateau region, as shown in Fig. S2. As
for gels, we select the period of oscillatory deformation as $\Delta
t/\tau_{\rm B}=343$, marked by dashed lines in Fig. 1e and f.

To investigate the age-dependent static structure at finite temperatures, we
derive the inherent states by eliminating thermal fluctuations using
conjugate-gradient minimization. Then, we determine the elastic moduli of
these inherent states using the Hessian matrix (see Methods), denoted as
inherent elasticity or inherent moduli $G^{\rm IS}$ and $K^{\rm IS}$.

Figure 2 presents the age-dependent thermal elasticity $M$ and inherent
elasticity $M^{\rm IS}$ for both glasses and gels, revealing distinct trends.
In glasses, the shear modulus $G$ shows a monotonically increasing behavior
over time, while it exhibits a peak in gels. Interestingly, in glasses, the
bulk modulus $K$ experiences a slight decrease over time, in contrast to the
trend observed for $G$. However, in gels, both $K$ and $G$ demonstrate similar
trends. One shared characteristic between glasses and gels is the increasing
trend of the modulus ratio $G/K$, which indicates a decrease in Poisson’s
ratio. Regarding the inherent moduli $G^{\rm IS}$ and $K^{\rm IS}$, they
remain relatively stable for different ages in glasses, unlike the changes
observed in the thermal state. This finding suggests that the variation in the
thermal elasticity of glasses during aging is primarily influenced by thermal
fluctuations. On the other hand, in gels, the inherent elasticity already
exhibits a peak, indicating that the static structural evolution alone can
lead to a peak in thermal elasticity. This implies that the thermal
fluctuations have a more significant impact on the elasticity of glasses
compared to gels. The contribution of the static structure and thermal
fluctuations to the thermal elasticity will be further discussed in the
following sections.

###  II.3 Role of structure and dynamics on elasticity

Figure 3: Role of inherent elasticity and mean-squared displacement on thermal
elasticity. (a) Inverse cage-relative MSD $\langle\Delta
r^{2}\rangle^{-1}_{\rm rel}$ of glasses for different observation time $\Delta
t$ plotted against waiting time $t_{\rm w}$. (b) $G/G^{\rm IS}$ and $K/K^{\rm
IS}$ calculated using different statistic times $\Delta t$ for stress
fluctuations in glasses. (c) $G/G^{\rm IS}$ versus $\langle\Delta
r^{2}\rangle^{-1}_{\rm rel}$ for different $\Delta t$ in glasses. (d) Inverse
MSD of isostatic particles ($Z\geq 6$) $\langle\Delta r^{2}\rangle^{-1}_{\rm
iso}$ of gels for different $\Delta t$ plotted against $t_{\rm w}$. (e)
$G/G^{\rm IS}$ and $K/K^{\rm IS}$ of gels measured from different periods
$\Delta t$ of oscillatory deformation. (f) $G/G^{\rm IS}$ and $K/K^{\rm IS}$
of gels versus $\langle\Delta r^{2}\rangle^{-1}_{\rm iso}$ for different
$\Delta t$.

As mentioned in the introduction, the shear modulus $G$ of the glass-liquid
system follows a single proportional function of $\langle\Delta
r^{2}\rangle^{-1}$ when rescaled by the infinite-frequency value
$G_{\infty}=G_{\rm A}$ Saw and Harrowell (2016). However, upon plotting
$G/G_{\rm A}$ against $\langle\Delta r^{2}\rangle^{-1}$ in Fig. S4a and b, we
observe that while this approach works well for glasses, it fails for gels. In
gels, $\langle\Delta r^{2}\rangle^{-1}$ increases monotonically, similar to
glasses, as shown in Fig. 3a and d, which naturally leads to a failure in
predicting the peak behavior of $G/G_{\rm A}$.

Notably, since the inherent elasticity $G^{\rm IS}$ of gels also exhibits a
peak, we are inspired to rescale $G$ by $G^{\rm IS}$ instead of $G_{\rm A}$.
As shown in Fig. 3b and e, we plot the thermal elasticity rescaled by the
inherent elasticity against the waiting time $t_{\rm w}$, which demonstrates a
monotonic increase.

The bulk modulus of glasses appears insensitive to the configurational
constraints, as discussed in the previous section. On the other hand, the
shear modulus of glasses as well as the shear and bulk moduli of gels, exhibit
similar trends with respect to $\langle\Delta r^{2}\rangle^{-1}$ when rescaled
by the corresponding inherent moduli. To illustrate this, we plot $M/M^{\rm
IS}$ as a function of $\langle\Delta r^{2}\rangle^{-1}$ in Fig. 3c and f,
resulting in excellent collapses of all the curves. This rescaled elasticity
shows a monotonic increase with aging, resembling the age-stiffening
phenomenon observed in gels formed by van der Waals forces Bonacci  _et al._
(2020). However, it is important to note that, unlike that case, the
stiffening in our case is not due to an increase in contact constraints, but
rather it results from an increase of constraints in configurational space.

As mentioned above, for 2D glasses, we employ the cage-relative MSD denoted as
$\langle\Delta r^{2}\rangle_{\rm rel}$ to eliminate long-wavelength Mermin-
Wagner fluctuations Flenner and Szamel (2015); Shiba  _et al._ (2016); Vivek
_et al._ (2017); Illing  _et al._ (2017). As shown in Fig. S4c, using the
original MSD does not lead to the collapse of $G/G^{\rm IS}$. Similarly, in
the case of 3D gels, employing the original MSD also fails to achieve a
collapse of $M/M^{\rm IS}$, as illustrated in Fig. S4d. This discrepancy can
be attributed to the fast motion of low-coordination interface particles,
which contributes little to the network’s elasticity. Considering this point,
we utilize the MSD of isostatic particles with a contact number $Z\geq 6$,
denoted as $\langle\Delta r^{2}\rangle_{\rm iso}$, as shown in Fig. 3f.

Furthermore, we observe that even the cage-relative MSD of isostatic particles
for gels fails to result in a collapse of $M/M^{\rm IS}$, as shown in Fig.
S4e. This suggests that the vibrations in gels involve not only cage-relative
motion but also collective motion of network strands at longer length-scales.
Both of these types of motions contribute to the thermal elasticity of gels.
This behavior is distinct from glasses, where collective motion only occurs at
larger time scales, such as $\alpha$-relaxation time Tong and Tanaka (2018).
The presence of collective motion in gels further adds to the complexity of
their mechanical properties.

Based on the collapse of $G/G^{\rm IS}$ versus $\langle\Delta
r^{2}\rangle^{-1}$ for different observation times $\Delta t$, we can conclude
that the shear modulus $G$ is universally determined by the inherent modulus
$G^{\rm IS}$ and the vibrational MSD $\langle\Delta r^{2}\rangle^{-1}$.
Therefore, the vibrational MSD effectively serves as a significant measure of
configurational constraints in the material. In contrast to glasses, the bulk
modulus $K$ of gels shares the same configurational constraint with the shear
modulus $G$, as demonstrated by the collapse of $K/K^{\rm IS}$ and $G/G^{\rm
IS}$ in Fig. 3f, as well as the collapse of $G/K$ and $G^{\rm IS}/K^{\rm IS}$
in Fig. 2f. This suggests that the configurational constraints affecting the
shear and bulk moduli in gels are strongly related and similarly influence
their mechanical properties.

In the following sections, we will delve into the underlying mechanisms that
drive the evolution of structure and dynamics during aging. We aim to
elucidate how these factors control the evolution of the shear modulus $G$ and
the bulk modulus $K$ in glasses and gels.

###  II.4 Structural ordering in glasses and interface reduction in gels

Figure 4: Age-dependent potential energy and structure in glasses and gels.
(a) Age-dependent potential energy per particle, $E$, and per contact, $2E/Z$,
in glasses. (b) Orientational order parameter $\Theta$ and anisotropy
parameter $\beta$ of Voronoi cells in glasses, where $\Theta=0$ indicates
perfect order and $\beta=1$ indicates isotropic cells. (c) Mean contact number
$Z$ and pressure $p$ in glasses. (d) Age-dependent potential energy per
particle, $E$, and per contact, $2E/Z$, in gels. (e) Age-dependent mean
contact number $Z$ in gels. (f) Age-dependent loop number $N_{\rm loop}$ in
gels.

Figure 5: Illustration of the origin of elasticity and distinct aging
mechanisms in glasses and gels. The cartoon insets demonstrate two strategies
for achieving a good harvest that corresponds to the decrease in potential
energy in our problem. One strategy, illustrated for glasses, involves
increasing the size of fruits (thus reducing the potential energy per contact)
while keeping the number of fruits constant (the number of contacts). The
other strategy, shown for gels, involves increasing the number of fruits
(thereby increasing the number of contacts), even if the size of individual
fruits (the potential energy per contact) decreases slightly on average.
Additionally, we provide a schematic representation of the impacts of static
structure and dynamics on elasticity for glasses (left) and gels (right).

The fundamental driving force behind the aging process in non-equilibrium
systems is their tendency to decrease free energy and approach a thermodynamic
equilibrium state. Due to the different structure in glasses and gels, with
uniform and non-uniform density, respectively, as shown in Fig. 1a and d, the
thermodynamic driving force is dominated by the bulk free energy for glasses
and the interfacial free energy for gels. As measuring the absolute entropy
poses challenges, we focus on the potential energy $E$, which holds
significance for both soft-sphere glasses and attractive gels.

The potential energy $E$ can be expressed as the product of the mean contact
number $Z$ and the potential energy per contact $2E/Z$. Here, we denote $E$ as
the energy and $2E/Z$ as the contact energy. By utilising this decomposition,
we identify two distinct ways to decrease the total energy in glasses and
gels. As depicted in Fig. 4a, in glasses, the decrease in contact energy
contributes to over 90% of the total energy decrease. However, in gels, the
contact energy even increases by around 10%, as shown in Fig. 4d. This implies
that the increase in the contact number contributes approximately 110% to the
overall decrease in the total energy.

Based on these observations, we can draw the following conclusions. For
glasses, with little change in the number of contacts, the structural
evolution aims to decrease the contact energy or optimize contacts. In
contrast, for gels, the system gains more contacts to decrease the energy,
even though, on average, the contact energy increases slightly. Next, we will
explore how these two ways of energy decrease drive the evolution of
structure, dynamics, and elasticity in both glasses and gels in a coupled
manner.

In glasses, the decrease in contact energy signifies an increase in packing
capability, which can be quantified by an agnostic structural order parameter
$\Theta$ Tong and Tanaka (2018, 2019); Tanaka  _et al._ (2019). This parameter
measures the deviation of the local packing structure from the perfect packing
structure and has been found to strongly correlate with the slowing dynamics
in glass-forming liquids Tong and Tanaka (2018, 2019); Tanaka  _et al._
(2019). As illustrated in Fig. 4b, $\Theta$ decreases logarithmically during
the aging, indicating structural ordering, and leads to the slowing down of
dynamics and an increase in the shear modulus $G$. Moreover, the decrease in
$\Theta$ may also contribute to the increase in vibrational entropy in simple
glass formers, at least in hard-sphere systems Tong and Tanaka (2018, 2019);
Tanaka  _et al._ (2019). This highlights the relationship between structural
ordering, dynamics, and mechanical properties in glasses.

Recent studies have also explored the relationship between the slowing down of
dynamics and a non-agnostic structural parameter Zhang  _et al._ (2023), known
as the cage anisotropy $\beta$. The cage anisotropy $\beta$ measures the ratio
between the smallest and largest eigenvalues of the inertial moment of the
Voronoi cell. As depicted in Fig. 4b, the Voronoi cells become more isotropic
over time. The decrease in the Voronoi cell’s anisotropy also contributes to
the slowing down of dynamics, as particles tend to move maximally along the
long axis of the Voronoi cell.

In our simulations conducted under constant volume, an increase in structural
ordering or packing capability leads to a slight decrease in volumetric
constraints, such as pressure $p$ and mean contact number $Z$, as shown in
Fig. 4c. Consequently, this leads to a slight decrease in $K$, as shown in
Fig. 2b. This finding is not contradictory to the increase in $K$ observed in
experiments Greinert  _et al._ (2006); Wang (2012). In experiments, structural
ordering leads to densification under isobaric (constant-pressure) conditions,
giving rise to the increase of $K$, which cannot occur in constant-volume
simulations.

To disentangle the effects of static structure and thermal fluctuations on
elasticity, we also analyse the structure and elasticity of glasses in
inherent states. Interestingly, we observe that $K^{\rm IS}$ and $G^{\rm IS}$
remain almost constant, which is consistent with the minimal variation of $Z$
in inherent states, as shown in Fig. S5b. This finding aligns with the
existing understanding that both shear and bulk moduli in athermal jammed
systems are determined by the contact number $Z$ O’Hern  _et al._ (2003). As
presented in Fig. S5a, the variations of $\Theta$ and $\beta$ in inherent
states are comparable to those in thermal states. These results indicate that
the structural order parameters $\Theta$ or $\beta$ do not directly control
the shear modulus, but instead determine the dynamics or the configurational
constraints in the thermal state. These configurational constraints, in turn,
influence the shear modulus $G$.

Our results provide an explanation for the observed significant changes in the
shear modulus $G$ with minimal alterations in the bulk modulus $K$ for
different processing histories of glasses Cheng and Ma (2009). The reason lies
in the fact that the structural order, particularly the orientational order,
can undergo substantial variations with minimal impact on volumetric
constraints, such as density or mean contact number.

In gels, the decrease in interfacial energy drives the increase in the contact
number $Z$, as shown in Fig. 4e. As discussed earlier with respect to the
vibrational MSD and the rescaled moduli $M/M^{\rm IS}$ in gels, the
vibrational MSD encompasses both cage-relative local vibrations and vibrations
on a strand-scale. It is evident that particles with more contacts exhibit
lower mobility Zia  _et al._ (2014). Additionally, as the thickness of strands
increases, the vibration of strands is expected to decrease. Thus, the
increase in $Z$ leads to a monotonically increasing configurational
constraint. The peak modulus, as demonstrated, originates from static
structure, and another impact of the increase in the contact number is the
stiffening of the inherent structure. However, as the system decreases in
interface area, thin strands are likely to rupture and be absorbed into the
main part of the gel. Consequently, this process decreases the connectivity on
a network scale, quantified by the loop number $N_{\rm loop}$ of the coarse-
grained network in Fig. 4f, subsequently softening the inherent structure.
Both $Z$ and $N_{\rm loop}$ show a linear increase and decrease, respectively,
against the logarithmic timescale. This suggests that the relative increase in
$Z$ will become weaker over time, while the relative decrease in $N_{\rm
loop}$ will become stronger, ultimately leading to the softening of gels. The
increase in the modulus ratio $G^{\rm IS}/K^{\rm IS}$ or $G/K$, as shown in
Fig. 2f, indicates that the reduction in loop number has a stronger impact on
volumetric deformation than on shear deformation.

##  III Outlook

In summary, our study illustrates the intricate interplay among structure,
dynamics, and elastic properties in non-equilibrium disordered systems by
investigating two typical amorphous solids: simple glass formers and colloidal
gels. Despite sharing many similarities as non-equilibrium amorphous solids,
we reveal fundamental differences in their elastic properties between glasses
and gels and elucidate the underlying mechanisms driving their respective
behaviors. For clarity, we present a logic diagram in Fig. 5.

In the aging process of glasses, with uniform density fields, the dominant
driving force is the bulk free energy. In contrast, gels exhibiting non-
uniform density fields are governed by interfacial free energy. Consequently,
the structural evolutions involved in decreasing free energy differ
fundamentally between these two systems.

In glasses, the decrease in bulk free energy is accomplished through
structural ordering, which can be quantified by the orientational order
parameter $\Theta^{-1}$. As a result, an increase in $\Theta^{-1}$ leads to
the slowing down of dynamics, resulting in an elevation of the configurational
constraint $\langle\Delta r^{2}\rangle^{-1}$ and, ultimately, an increase in
the shear modulus $G$. Furthermore, the structural ordering enhances the
packing capability, resulting in a slight reduction in the volumetric
constraint $Z$, which, in turn, causes a decrease in the bulk modulus $K$.

Upon investigating the inherent states, obtained by removing the thermal
fluctuations of the corresponding age-dependent structures, we observe that
$K^{\rm IS}$ remains constant due to the minimal variation in $Z$. Similarly,
$G^{\rm IS}$ remains constant, indicating that the influence of the order
parameter $\Theta^{-1}$ on the shear modulus $G$ is indirect, exerting its
effect through the slowing down of dynamics, or configurational constraints.
These findings shed light on the underlying mechanisms linking structural
order, dynamics, and elastic properties in glasses, contributing to a deeper
understanding of their complex behavior during aging.

In gels, the reduction of free energy is achieved by the reduction of
interfacial area rather than structural ordering in the colloid-rich phase.
Consequently, there is an increase in particle-scale connectivity (contact
number $Z$) and a decrease in network-scale connectivity (loop number $N_{\rm
loop}$). The increase in $Z$ elevates the configurational constraint
$\langle\Delta r^{2}\rangle^{-1}$ and increases the elasticity of inherent
states. On the other hand, the decrease in $N_{\rm loop}$ tends to reduce the
elasticity of inherent states. The competition between $Z$ and $N_{\rm loop}$
gives rise to a peak in the age-dependent $G^{\rm IS}$ and $K^{\rm IS}$. The
elasticity at finite temperature, which is a product of inherent elasticity
$M^{\rm IS}$ (structure) and configurational constraint $\langle\Delta
r^{2}\rangle^{-1}$ (dynamics), also exhibits a peak during aging, although the
decreasing trend at later stages is less pronounced compared to the elasticity
in the inherent states. These findings highlight the intricate interplay
between interface effects, structural connectivity, and dynamics in gels,
leading to a complex evolution of their elastic properties during the aging
process.

Notably, in glasses with uniform density, the bulk modulus is primarily
affected by the bulk constraint. However, in gels with highly non-uniform
density, the bulk modulus is also influenced by the configurational
constraint. Consequently, an intriguing result emerges: at long observation
times, when particle diffusion takes place, glasses behave like liquids with a
finite bulk modulus and zero shear modulus, whereas gels resemble gases with
zero bulk modulus and zero shear modulus.

Our work not only provides valuable insights into fundamental non-equilibrium
physics but also has significant implications for materials science. It offers
guidance on designing and manufacturing amorphous solids with desired elastic
properties, by understanding the interplay between structural order,
configurational constraints, and material response. These findings open up new
possibilities for tailoring the mechanical behavior of amorphous materials,
paving the way for innovative applications in various engineering and
industrial fields.

###### Acknowledgements.

H.T. acknowledges the support by the Grant-in-Aid for Specially Promoted
Research (JSPS KAKENHI Grant No. JP20H05619) from the Japan Society for the
Promotion of Science (JSPS). M.T. acknowledges the support from JSPS KAKENHI
(Grant No. JP20K14424). Y.W. acknowledges the support from Shanghai Jiao Tong
University via the scholarship for outstanding PhD graduates.

## Methods

Simulation methods for simple glass formers. We conducted molecular dynamics
simulations on a 2D binary mixture of particles interacting with harmonic
potentials in a square box under the $NVT$ ensemble with a Nose-Hoover
thermostat. The pairwise potential $u(r_{ij})$ between particles $i$ and $j$
is described by a harmonic function:

$u(r_{ij})=\frac{\epsilon}{2}\left(1-\frac{r_{ij}}{d_{ij}}\right)^{2},\quad
r_{ij}<d_{ij},$ (1)

where $r_{ij}$ is the distance between particles $i$ and $j$, and $d_{ij}$ is
the sum of their radii.

The system comprises $N=4096$ particles, with an equal number of large and
small particles and a diameter ratio of 1.4. The diameter $d_{\rm s}$ of the
small particles is set as the length unit, and both the small and large
particles possess the same mass $m$. The energy, time, and temperature are
scaled in units of $\epsilon$, $\sqrt{md_{\rm s}^{2}/\epsilon}$, and
$10^{-3}\epsilon/k_{\rm B}$, respectively, where $k_{\rm B}$ represents the
Boltzmann constant. The packing fraction of the system is fixed at
$\phi=0.91$.

Prior studies have thoroughly investigated the glassy dynamics of these glass
formers Tong and Tanaka (2018, 2019); Tong  _et al._ (2020). The onset
temperature $T_{\rm on}$ is approximately 2.1, the mode-coupling-theory (MCT)
temperature $T_{\rm mct}$ is around 1.21, and the ideal glass transition
temperature $T_{0}$ is roughly 0.63. In this study, we first equilibrated the
system at $T=3.0$ and then rapidly quenched it to $T=1.0$ to observe the
system’s relaxation.

A total of 504 independent trajectories were performed for our investigation.

Simulation methods for colloidal gels. We conducted Langevin dynamics
simulations on a 3D polydisperse mixture of particles featuring short-range
attractive potentials. The pairwise potential $u(r_{ij})$ between particles
$i$ and $j$ is described by a Morse pair potential, given by:

$u(r_{ij})=\epsilon\exp{[\rho_{M}(d_{ij}-r_{ij})]}(\exp{[\rho_{M}(d_{ij}-r_{ij})]}-2),$
(2)

where the potential depth $\epsilon$ and the interaction range parameter
$\rho_{M}$ can be independently controlled. Here, $r_{ij}$ represents the
distance between particles $i$ and $j$, and $d_{ij}$ is the sum of their
radii.

The system consists of $N=10000$ particles with a polydispersity of
$\Delta/d=0.04$. The average particle diameter $d$ is taken as the length
unit, and all particles have the same mass $m$. The energy, time, and
temperature are scaled in units of $\epsilon$, $\sqrt{md^{2}/\epsilon}$, and
$\epsilon/k_{\rm B}$, respectively, where $k_{\rm B}$ denotes the Boltzmann
constant. The time step is set as $0.002$, and the damping factor is
$\gamma=0.1$. The Brownian time $\tau_{\rm B}=(d/2)^{2}/(6D)$ is used to
rescale the waiting time $t_{\rm w}$, where $D=\gamma k_{\rm B}T$ represents
the self-diffusion constant for an isolated particle.

To align with typical PMMA colloid-polystyrene polymer experiments, we set the
parameters as follows: packing fraction $\phi=0.12$, temperature $T=0.143$,
and interaction range $\rho_{M}=48.63$. The range parameter $\rho_{M}$ is
determined based on the second virial coefficients of the Asakura-Osawa (AO)
potential Tateno and Tanaka (2019) with an attraction range of
$\delta/d=0.13$, and the Morse potential is truncated and shifted at $r_{\rm
cut}=1.13d_{ij}$.

We first equilibrate the system at a temperature of $T=1.0$, significantly
higher than the demixing critical temperature of $T_{c}\sim 0.3$ Griffiths
_et al._ (2017). Subsequently, we instantaneously lower the temperature to the
target value of $T=0.143$, thereby facilitating the aggregation of particles
to form space-spanning networks.

A total of 23 independent trajectories were performed for our study.

Measuring elastic moduli in thermal systems via stress-fluctuations. The
elastic modulus can be decomposed to the affine and nonaffine components,
expressed as

$C^{\alpha\beta\kappa\chi}=C_{\rm A}^{\alpha\beta\kappa\chi}-C_{\rm
NA}^{\alpha\beta\kappa\chi}.$ (3)

The affine modulus is expressed as Mizuno and Ikeda (2022):

$\displaystyle C_{\rm A}^{\alpha\beta\kappa\chi}=$ $\displaystyle C_{\rm
B}^{\alpha\beta\kappa\chi}+C_{\rm K}^{\alpha\beta\kappa\chi}+C_{\rm
C}^{\alpha\beta\kappa\chi},$ (4) $\displaystyle C_{\rm
B}^{\alpha\beta\kappa\chi}=$
$\displaystyle\frac{1}{V}\sum_{i<j}(r_{ij}k_{ij}+f_{ij})r_{ij}n_{ij}^{\alpha}n_{ij}^{\beta}n_{ij}^{\kappa}n_{ij}^{\chi},$
(5) $\displaystyle C_{\rm K}^{\alpha\beta\kappa\chi}=$ $\displaystyle 2\rho
T(\delta_{\alpha\kappa}\delta_{\beta\chi}+\delta_{\alpha\chi}\delta_{\beta\kappa}),$
(6) $\displaystyle C_{\rm C}^{\alpha\beta\kappa\chi}=$
$\displaystyle\frac{1}{2}(2\langle\sigma_{\alpha\beta}\rangle\delta_{\kappa\chi}-\langle\sigma_{\alpha\kappa}\rangle\delta_{\beta\chi}-\langle\sigma_{\alpha\chi}\rangle\delta_{\beta\kappa}$
(7)
$\displaystyle-\langle\sigma_{\beta\kappa}\rangle\delta_{\alpha\chi}-\langle\sigma_{\beta\chi}\rangle\delta_{\alpha\kappa}),$
(8) $\displaystyle\sigma_{\alpha\beta}=$ $\displaystyle\rho k_{\rm
B}T\delta_{\alpha\beta}+\frac{1}{V}\sum_{i<j}f_{ij}^{\alpha}r_{ij}^{\beta},$
(9)

where $k_{ij}=\partial^{2}u_{ij}/\partial r_{ij}^{2}$ is the spring constant,
$f_{ij}=-\partial u_{ij}/\partial r_{ij}$ is the interparticle force,
$r_{ij}^{\alpha}=r_{i}^{\alpha}-r_{j}^{\alpha}$ is the position vector,
$n_{ij}^{\alpha}=r_{ij}^{\alpha}/r_{ij}$, $\langle\rangle$ denotes the
ensemble average, $V$ is the area or volume of the system, $\rho$ is the
number density, $\sigma_{\alpha\beta}$ is the stress tensor. Here $C_{\rm
B}^{\alpha\beta\kappa\chi}$ is the so-called Born term, $C_{\rm
K}^{\alpha\beta\kappa\chi}$ is the kinetic term, and $C_{\rm
C}^{\alpha\beta\kappa\chi}$ is the pre-stress term. The kinetic and pre-stress
terms are often negligible for low-temperature amorphous solids.

The nonaffine modulus can be expressed as the fluctuation of stress, given by

$C_{\rm
NA}^{\alpha\beta\kappa\chi}=\frac{V}{T}\left[\langle\sigma_{\alpha\beta}\sigma_{\kappa\chi}\rangle-\langle\sigma_{\alpha\beta}\rangle\langle\sigma_{\kappa\chi}\rangle\right].$
(10)

Measuring elastic moduli in athermal systems via the Hessian matrix. In
athermal systems, the affine modulus is calculated using the same formula as
in the thermal scenario (Eq. (4)-(8)), excluding the kinetic term. The
nonaffine modulus is represented using the Hessian matrix $\mathbf{H}$
Lemaître and Maloney (2006); Mizuno and Ikeda (2022).

The Hessian matrix $\mathbf{H}$ is defined as

$\mathbf{H}=\frac{\partial^{2}U}{\partial\mathbf{r}_{i}\partial\mathbf{r}_{j}},$
(11)

where $U$ represents the total interaction potential:

$U(\\{r_{ij}\\})=\sum_{i>j}u_{ij}(r_{ij}).$ (12)

The off-diagonal terms of $\mathbf{H}$ are given by

$\displaystyle H_{ij}^{\alpha\beta}$
$\displaystyle=\frac{\partial^{2}u(r_{ij})}{\partial r_{i}^{\alpha}\partial
r_{j}^{\beta}}$ (13)
$\displaystyle=-(k_{ij}+\frac{f_{ij}}{r_{ij}})n_{ij}^{\alpha}n_{ij}^{\beta}+\frac{f_{ij}}{r_{ij}}\delta_{\alpha\beta},$
(14)

where $k_{ij}=\partial^{2}u_{ij}/\partial r_{ij}^{2}$ is the spring constant,
$f_{ij}=-\partial u_{ij}/\partial r_{ij}$ is the interparticle force,
$r_{ij}^{\alpha}=r_{i}^{\alpha}-r_{j}^{\alpha}$ is the position vector, and
$r_{ij}$ is the distance between particles $i$ and $j$,
$n_{ij}^{\alpha}=r_{ij}^{\alpha}/r_{ij}$. The diagonal terms of $\mathbf{H}$
are $H_{ii}^{\alpha\beta}=-\sum_{j\neq i}H_{ij}^{\alpha\beta}$.

The total force exerted on individual particles post the affine deformation,
known as the affine force field $\mathbf{\Xi}_{\kappa\chi}$, is given by

$\mathbf{\Xi}_{\kappa\chi}=\Xi_{i,\kappa\chi}^{\alpha}=-\sum_{j}(r_{ij}k_{ij}+f_{ij})n_{ij}^{\alpha}n_{ij}^{\kappa}n_{ij}^{\chi}.$
(15)

The nonaffine modulus tensor can subsequently be expressed as

$C_{\rm
NA}^{\alpha\beta\kappa\chi}=\frac{1}{V}\mathbf{\Xi}_{\alpha\beta}\mathbf{H}^{-1}\mathbf{\Xi}_{\kappa\chi}.$
(16)

Measuring elastic moduli via oscillatory shear. The elastic moduli of a
material can be ascertained by employing oscillatory deformation in
simulations, a method analogous to the experimental approach of determining a
material’s viscoelastic response with a rheometer.

The induced strain is represented by a sinusoidal function:

$\epsilon(t)=\epsilon_{0}\sin{(\omega t)}$ (17)

where $\epsilon_{0}$ specifies the amplitude, and $\omega$ indicates the
frequency of the oscillatory deformation.

The stress responses to this strain can be expressed as follows:

$\begin{split}\sigma(t)&=\sigma_{0}\sin{(\omega t+\delta)}+\sigma_{\rm c}\\\
&=\sigma_{0}\cos{\delta}\sin{(\omega t)}+\sigma_{0}\sin{\delta}\cos{(\omega
t)}+\sigma_{\rm c}.\end{split}$ (18)

The storage modulus, represented by the in-phase term
$M^{\prime}=(\sigma_{0}\cos{\delta})/\epsilon_{0}$, and the loss modulus,
denoted by the out-of-phase term
$M^{\prime\prime}=(\sigma_{0}\sin{\delta})/\epsilon_{0}$, can be derived
accordingly. The variable $\sigma_{\rm c}$ denotes the initial stress within
the material, typically zero for shear stress but finite in the case of
pressure.

The complex shear modulus $G^{\prime}+iG^{\prime\prime}$ can be obtained by
implementing a shear strain $\epsilon_{xy}$ and measuring the correlated shear
stress response $\epsilon_{xy}$. Similarly, by applying a volumetric strain
$\epsilon_{\rm b}$ and evaluating the corresponding pressure response $p$, the
complex bulk modulus $K^{\prime}+iK^{\prime\prime}$ can be calculated.

Initially, we utilize an amplitude sweep to identify the linear-response
region. Upon selecting an amplitude within this region, we are able to
ascertain the frequency-dependent storage and loss moduli, as illustrated in
Fig. S3.

In this context, our primary interest is the storage modulus. Therefore, in
the main text, $G$ and $K$ refer exclusively to the storage moduli.

Anisotropy parameter of Voronoi cells in glasses. In the case of polydisperse
or bidisperse mixtures, the radical Voronoi tessellation is employed to
acquire the Voronoi cells. The anisotropy of the Voronoi cells is assessed by
calculating the ratio of the smallest to largest eigenvalues of the Minkowski
tensors Schröder-Turk  _et al._ (2010). In this context, the Minkowski tensor
$\boldsymbol{\rm W}^{20}_{0}=\int_{K}\boldsymbol{r}^{2}{\rm d}V$ is utilized,
representing the inertial tensor of the Voronoi cell. Subsequently, the
anisotropy parameter is expressed as $\beta=I_{\min}/I_{\max}$, where $I$
represents the eigenvalue of $\boldsymbol{\rm W}^{20}_{0}$. This anisotropy
parameter $\beta$ quantifies the degree of asymmetry of the Voronoi cells.

Orientational order parameter in glasses. The generalized orientational order
parameter $\Theta$ Tong and Tanaka (2018, 2019) measures the extent to which a
local packing deviates from the ideal arrangement where neighboring particles
can be maximally efficiently packed around the central particle. For a
particle $o$ and its adjacent neighbors $i$ and $j$, the angle between $oi$
and $oj$ is denoted as $\theta_{ij}^{(1)}$. Here the neighbors are given by
the radical Voronoi tessellation. When particle $o$, $i$, and $j$ are in
contact, the angle between $oi$ and $oj$ is denoted as $\theta_{ij}^{(2)}$.
The orientational order parameter $\Theta_{o}$ for particle $o$ is defined as
follows:

$\Theta_{o}=\frac{1}{N_{o}}\sum_{\langle
ij\rangle}|\theta_{ij}^{(1)}-\theta_{ij}^{(2)}|,$ (19)

where $N_{o}$ represents the number of neighbor pairs, which is equal to the
number of neighbors of particle $o$, and the summation includes all adjacent
neighbor pairs. Then the orientational order parameter of the whole system is
given by $\Theta=\sum_{o=1}^{N}\Theta_{o}/N$.

Loop number for a coarse-grained network in gels. To capture the topological
feature of the network on the strand scale, we binarize the coarse-grained
density field of the largest cluster. The coarse-grained density field is
given by $\rho_{\rm
cg}(\mathbf{r})=\sum_{i}\exp{(-|\mathbf{r}-\mathbf{R}_{i}|^{2}/2\Delta_{\rm
cg}^{2})}$, where $\Delta_{\rm cg}=d/2$ is the standard deviation of the
Gaussian filter, and $\mathbf{R}_{i}$ is the position of particle $i$
belonging to the largest cluster. The threshold for the binarisation is set as
$1/2$.

This allows us to calculate the Euler number $\chi$ of the coarse-grained
network and determine the number of loops (tunnels), using $N_{\rm
loop}=N_{\rm obj}+N_{\rm hole}-\chi$. Here, $N_{\rm obj}=1$ is the number of
objects, and the number of holes, $N_{\rm hole}$, can be neglected.

Relaxation dynamics in glasses and gels. In 2D glass formers, we analyse the
particle dynamics using cage-relative motion to eliminate the long-wavelength
Mermin-Wagner fluctuations Flenner and Szamel (2015); Shiba  _et al._ (2016);
Vivek  _et al._ (2017); Illing  _et al._ (2017). This motion is defined as:

$\underline{\boldsymbol{r}}_{j}(t)=\boldsymbol{r}_{j}(t)-\frac{1}{n_{j}}\sum_{l}\boldsymbol{r}_{l}(t),$
(20)

where $\boldsymbol{r}_{j}(t)$ represents the position vector of particle $j$,
$l$ iterates over its Voronoi neighbors, and $n_{j}$ denotes the number of
radical Voronoi neighbors of particle $j$. The self-intermediate scattering
function is defined as:

$F_{\rm
s}(k,t)=\left\langle\frac{1}{N}\sum_{j}\exp\left(i\boldsymbol{k}\cdot\left[\underline{\boldsymbol{r}}_{j}(t)-\underline{\boldsymbol{r}}_{j}(0)\right]\right)\right\rangle,$
(21)

where $k=|\boldsymbol{k}|$ is selected as the first peak of the static
structure factor. The mean squared displacement is defined as:

$\langle\Delta
r^{2}(t)\rangle=\left\langle\frac{1}{N}\sum_{j}\left[\underline{\boldsymbol{r}}_{j}(t)-\underline{\boldsymbol{r}}_{j}(0)\right]^{2}\right\rangle.$
(22)

In 3D gels, the original particle positions are used to calculate the $F_{\rm
s}(k,t)$ and $\langle\Delta r^{2}\rangle$.

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Figure S1: Self-intermediate scattering function of glasses and gels. (a)
Cage-relative self-intermediate scattering function $F_{\rm s}(k,\Delta t)$
for glasses at different ages ($t_{\rm
w}=10^{2},10^{3},10^{4},10^{5},10^{6}$), depicted from left to right. (b)
$F_{\rm s}(k,\Delta t)$ of all particles (dashed lines) and only isostatic
particles (solid lines) for gels at different ages ($t_{\rm w}/\tau_{\rm
B}=10^{2},10^{3},10^{4},10^{5},10^{6}$), shown from left to right. The
vertical dashed line in (a) indicates the time scale used to calculate the
stress fluctuations in glasses, while the vertical dashed line in (b)
indicates the period of oscillatory deformation used in measuring the moduli
of gels.

Figure S2: Age-dependent moduli for different observation times in glasses.
(a) Schematic representation of the observation-time or frequency-dependent
shear modulus. Age-dependent (b) shear modulus and (c) bulk modulus obtained
using different statistical times $\Delta t$ for calculating stress
fluctuations.

Figure S3: Amplitude and frequency sweep moduli in gels. (a) Shear stress
$\sigma_{xy}$ and shear strain $\epsilon_{xy}$ versus time $t/\Delta t$ during
oscillatory shear, where $\Delta t$ is the period time. The black points
represent individual $\sigma_{xy}$, and the red curve represents the fitting
results using $\sigma_{xy}=\sigma_{0}\sin{(\omega t+\delta)}$. (b) Amplitude
$\gamma$ sweep of storage shear modulus $G^{\prime}$ and loss shear modulus
$G^{\prime\prime}$. (c) Frequency $\omega=2\pi/(\Delta t/\tau_{\rm B})$ sweep
of $G^{\prime}$ and $G^{\prime\prime}$. (d) Pressure $p$ and volumetric strain
$\epsilon_{b}$ versus $t/\Delta t$ during oscillatory compression. The red
curve represents the fitting results using $p=p_{0}\sin{(\omega
t+\delta)}+p_{\rm c}$, where $p_{\rm c}$ is the initial pressure of the
system. (e) Amplitude $\gamma$ sweep of storage bulk modulus $K^{\prime}$ and
loss bulk modulus $K^{\prime\prime}$. (f) Frequency $\omega=2\pi/(\Delta
t/\tau_{\rm B})$ sweep of $K^{\prime}$ and $K^{\prime\prime}$. The vertical
dashed lines mark the amplitude and frequency used to measure the moduli in
Fig. 2. In this work, $G$ and $K$ of gels refer specifically to the storage
moduli $G^{\prime}$ and $K^{\prime}$.

Figure S4: Rescaled moduli versus mean-squared displacement (MSD) in glasses
and gels. (a) Shear modulus $G$ rescaled by the affine modulus $G_{\rm A}$
(infinite-frequency modulus) versus the inverse of cage-relative MSD
$\langle\Delta r^{2}\rangle^{-1}_{\rm rel}$ in glasses. (b) Shear modulus $G$
rescaled by $G_{\rm A}$ versus the inverse of isostatic MSD $\langle\Delta
r^{2}\rangle^{-1}_{\rm iso}$ in gels. (c) Rescaled shear modulus $G/G^{\rm
IS}$ versus original MSD $\langle\Delta r^{2}\rangle^{-1}$ in glasses. (d)
Rescaled shear modulus $G/G^{\rm IS}$ and bulk modulus $K/K^{\rm IS}$ versus
original MSD of all particles $\langle\Delta r^{2}\rangle^{-1}$ in gels. (e)
Rescaled shear modulus $G/G^{\rm IS}$ and bulk modulus $K/K^{\rm IS}$ versus
cage-relative MSD of isostatic particles $\langle\Delta r^{2}\rangle^{-1}_{\rm
iso,\ rel}$.

Figure S5: Age-dependent structure of glasses in inherent states. (a)
Orientational order parameter $\Theta$ and cage anisotropy parameter $\beta$.
(b) Mean contact number $Z$ and pressure $p$. Note that the scales of all
vertical axes are the same as those shown for the corresponding structural
parameters in Fig. 4 in the main text.