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# Emergence of solid-like Debye scaling in the vibrational density of states
of liquids under nanoconfinement

Yuanxi Yu1,2 Sha Jin1,3,4 Xue Fan5,6,7 Mona Sarter8 Dehong Yu9 Matteo
Baggioli1,3,4 Liang Hong1,2,5,10 1 School of Physics and Astronomy, Shanghai
Jiao Tong University, Shanghai 200240, China  2 Institute of Natural Sciences,
Shanghai Jiao Tong University, Shanghai 200240, China  3 Wilczek Quantum
Center, Shanghai Jiao Tong University, Shanghai 200240, China  4 Shanghai
Research Center for Quantum Sciences, Shanghai 201315, China  5 Shanghai
National Center for Applied Mathematics(SJTU Center), MOE-LSC, Shanghai Jiao
Tong University, Shanghai 200240, China  6 Materials Genome Institute,
Shanghai University, Shanghai 200444, China  7 School of Materials Science and
Engineering, Georgia Institute of Technology, Atlanta, GA 30332, United States
of America  8 Rutherford Appleton Laboratory, ISIS Neutron and Muon Facility,
Science and Technology Facilities Council, Didcot OX11 0QX, United Kingdom  9
Australian Nuclear Science and Technology Organisation, Lucas Heights, NSW
2232, Australia  10 Zhangjiang Institute for Advanced Study, Shanghai Jiao
Tong University, Shanghai 201203, China

###### Abstract

At frequencies higher than the inverse of the structural relaxation time
$\tau$, the dynamics of liquids display several solid-like properties,
including propagating collective shear waves and emergent elasticity. However,
in classical bulk liquids, where $\tau$ is typically of the order of 1 ps or
less, this solid-like behavior remains elusive in the low-frequency region of
the vibrational density of states (VDOS). Here, we provide compelling evidence
for the emergent solid-like nature of liquids at short distances through
inelastic neutron scattering measurements of the low-frequency VDOS in liquid
water and glycerol confined within graphene oxide membranes. In particular,
upon increasing the strength of confinement, we observe a transition from a
liquid-like VDOS (linear in the frequency $\omega$) to a solid-like behavior
(Debye law, $\sim\omega^{2}$) in the range of $1$-$4$ meV. Molecular dynamics
simulations confirm these findings and reveal additional solid-like features,
including propagating collective shear waves and a reduction in the self-
diffusion constant. Finally, we show that the onset of solid-like dynamics is
pushed towards low frequency along with the slowing-down of the relaxation
processes upon confinement, and that the scale at which solidity emerges is
qualitatively compatible with k-gap theory and the concept of gapped momentum
states. Our results provide a convincing experimental evidence of the
continuity between liquids and solids, as originally advocated by Frenkel and
Maxwell, and a deeper understanding of the dynamics of liquids across a wide
range of length scales.

##  I Introduction

From a structural point of view, liquids are profoundly different from solids.
They do not display long-range order and they cannot be defined using the
theoretical concept of spontaneous symmetry breaking and its direct
consequences, such as the existence of long-wavelength phonon modes and the
elasticity that emerges from thatChaikin and Lubensky (2000). This poses
fundamental questions in the description of the low-energy collective dynamics
of liquids which are predominantly relaxational, rather than vibrational, and
usually described with the theory of hydrodynamicsBoon and Yip (1991).

It comes then as no surprise that the dynamical properties of liquids and
solids at large distances and over long timescales – the hydrodynamic regime –
are substantially different. To make this distinction more precise, it is
instructive to introduce the concept of structural relaxation time $\tau$,
which determines the speed of atomic re-arrangements in a liquid and the
extent of the hydrodynamic regime. For bulk water at room temperature $\tau$
is of the order of $1$ ps Cunsolo  _et al._ (1999), close to the value for the
Maxwell relaxation time Malomuzh and Shakun (2019); O’Sullivan  _et al._
(2019). In the rest of the manuscript, we will use the term low-frequency to
indicate the range of energies below the inverse of the structural relaxation
time for the bulk liquid, in which we do not expect any solid-like features.
This corresponds to an energy of approximately $\hslash\omega<4$meV for water
at room temperature.

A direct manifestation of the contrast between solids and liquids can be
observed in the low-frequency behavior of their vibrational density of states
(VDOS), $g(\omega)$. 3D bulk solids display a characteristic quadratic scaling
at low frequency $g(\omega)\propto\omega^{2}$, which is known as the Debye
lawKittel (2021). The law indicates that the low-frequency dynamics of solids
are governed by collective vibrational and propagating modes known as phonons,
which can be identified with the Goldstone modes for the spontaneously broken
translational symmetryLeutwyler (1997). On the contrary, the low-frequency
VDOS of liquids exhibits two distinctive characteristics. First, the zero
frequency value of $g(\omega)$ does not vanish and directly relates to the
liquid relaxational dynamics, i.e., a non-zero self-diffusion constantHansen
and McDonald (1990). Second, the low-frequency scaling of the VDOS for 3D bulk
liquids is linear in frequency, $g(\omega)\propto\omega$, as derived from
theoretical arguments Keyes (1994); Zaccone and Baggioli (2021); Zürcher and
Keyes (1997); Wu and Loring (1993); Wan and Stratt (1994); Schirmacher  _et
al._ (2022); Keyes  _et al._ (2002); Keyes (2005); Xu  _et al._ (2023), by
instantaneous normal mode (INM) simulations Keyes (1994, 1997), and confirmed
by neutron scattering experiments Phillips  _et al._ (1989); Dawidowski  _et
al._ (2000); Stamper  _et al._ (2022a); Jin  _et al._ (2023); Achibat  _et
al._ (1995). For simplicity, in the rest of the manuscript, we will refer to
the quadratic Debye behavior as solid-like and to the linear scaling in
frequency as liquid-like (see insets in Figure 1). Since the dynamics of
liquids for frequencies $\omega<1/\tau$ do not involve any solid-like motion,
their VDOS cannot display any Debye-like behavior at low frequency, as
experimentally confirmed in Phillips  _et al._ (1989); Dawidowski  _et al._
(2000); Stamper  _et al._ (2022a); Jin  _et al._ (2023); Achibat  _et al._
(1995).

After acknowledging that the low-frequency dynamics of bulk liquids are
drastically different from those in solids, a question arises as whether this
sharp contrast persists at high frequencies or short scales. It has been early
realized by FrenkelFrenkel (1984) (see also Zwanzig Zwanzig (1967), or the
more recent reviews Trachenko and Brazhkin (2015); Baggioli  _et al._ (2020))
that liquid dynamics involve solid-like oscillatory motion around a position
of local equilibrium interrupted by diffusive jumps toward different potential
minima. These jumps constitute the fundamental origin of fluidity (liquid
flow) and they happen at an average rate $1/\tau$ (see Figure 1), where $\tau$
is the structural Maxwell relaxation time or the average time for a particle
to jump out of the cage formed by its neighbors. As a consequence, the
dynamics of liquids are expected to be solid-like for frequencies faster than
this relaxation rate, $\omega\gg 1/\tau$, simply because relaxation has no
time to take place. In other words, one does expect the appearance of solid-
like vibrational modes in the high-frequency dynamics of liquids. This is
indeed the case, as reported by many works Copley and Rowe (1974); Hosokawa
_et al._ (2009); Giordano and Monaco (2010); Hosokawa  _et al._ (2015);
Simeoni  _et al._ (2010); Scopigno  _et al._ (2005).

Figure 1: At short distances and short times, liquids exhibit solid-like
properties. First, below a critical length-scale, propagating shear waves are
expected in liquids, instead of the large wavelength shear diffusion. Second,
for times below a structural relaxation scale $\tau$, the dynamics is
oscillatory and confined at the bottom of the potential landscape basin.
Hence, the vibrational density of states of liquids is expected to be liquid-
like at large scales and to become solid-like (Debye law) at short scales.

Another way to model a continuous transition between a propagating oscillatory
frequency behavior and a low-frequency liquid-like collective diffusive
dynamics is by using the so-called telegrapher equation, or k-gap theory
Baggioli  _et al._ (2020):

$\omega^{2}+i\omega\tau_{g}^{-1}=v^{2}k^{2},$ (1)

where $v$ is the asymptotic speed of propagation of the collective waves at
$\omega\gg 1/\tau$, and $k,\omega$ are respectively the wave-vector and the
frequency of the collective mode. In the expression above, $\tau_{g}$ is a
timescale governing relaxation processes and expected to decrease with
temperature increasing. By using Maxwell interpolation, one can make the
assumption of identifying $\tau_{g}$ with the Maxwell relaxation time
$\tau_{M}\equiv\eta/G_{\infty}$, where $\eta$ and $G_{\infty}$ are
respectively the shear viscosity and the instantaneous shear modulus Trachenko
and Brazhkin (2015); Baggioli  _et al._ (2020). Despite this assumption
looking reasonable, a formal derivation of this equivalence does not exist
(see discussion in Baggioli  _et al._ (2022)). Only one instance, liquid
GaKhusnutdinoff  _et al._ (2020a), has been reported in which this equivalence
is quantitatively confirmed. Because of these reasons, at this stage we will
consider $\tau_{g}$ as an independent parameter which can be extracted from
the dispersion relation of gapped shear waves in liquids (see for example Yang
_et al._ (2017)).

Back to Eq.(1), in the low frequency limit, the first term can be neglected
and the dynamics is purely diffusive (liquid-like), with macroscopic diffusion
constant $\mathcal{D}=v^{2}\tau_{g}$ (not to be confused with the single-
particle self-diffusion constant $D$). In the opposite limit, the second term
can be neglected and the dynamics is purely vibrational and solid-like. A more
careful analysis reveals still a crucial difference between the large
frequency behavior arising from Eq.(1) and the dynamics of phonons in solids.
Indeed, expanding the solution of Eq.(1) at large frequency, we obtain a
dispersion of the type $\omega=vk-i/2\tau_{g}$, whose imaginary part is wave-
vector independent and therefore qualitatively different from that of phonons
in solids Chaikin and Lubensky (2000) (where at finite temperature and low
wave-vector is quadratic in $k$, i.e., Akhiezer damping). This distinction is
compatible with the observed broadening of high-frequency solid-like modes in
liquids, which arises physically from the distribution of the liquid local
structures Giordano and Monaco (2010).

Eq.(1) appears in the description of several physical systemsBaggioli  _et
al._ (2020) (e.g., Cattaneo heat conduction equation) and plays a fundamental
role in the understanding of the dynamics of collective shear waves in
liquids, where it has been relabeled as the k-gap equationTrachenko and
Brazhkin (2015). Besides modelling the high-frequency solid-like collective
motion in liquids, Eq.(1) presents another striking prediction. In particular,
it suggests the existence of shear waves with finite real frequency in liquids
above a certain wave-vector cutoff, known as k-gap, and given by
$k_{g}=1/(2v\tau_{g})$. This prediction has been verified in simulations of
classical liquidsYang  _et al._ (2017); Khusnutdinoff  _et al._ (2020b); Fomin
(2020); Guarini  _et al._ (2023); Yakovlev  _et al._ (2020); Kryuchkov  _et
al._ (2020) and plasmasKhrapak  _et al._ (2019); Goree  _et al._ (2012);
Murillo (2000); Ohta and Hamaguchi (2000), but experimentally confirmed only
in a two-dimensional Yukawa dusty plasma Nosenko  _et al._ (2006). In the
literature, it is often claimed that the presence of a k-gap is tantamount to
saying that below a critical length-scale $L_{g}=2\pi/k_{g}$, and
independently of the value of the frequency, the collective dynamics of
liquids is akin to that of solids as it presents propagating shear waves. A
more conservative approach requires that the real part of the frequency is at
least larger than its imaginary part, or in other words, that the wave is able
to propagate over a few wave-lengths before decaying. This second criterion
applied to the k-gap equation qualitatively agrees with Frenkel original
argument $\omega>1/\tau_{g}$ (see Appendix D). This liquid-like to overdamped
to propagating dynamics is in perfect agreement with the previous experimental
analysis for water presented in Sampoli  _et al._ (1997). There, it has been
shown that the dynamics of room temperature water is liquid-like
(hydrodynamic) for $q<2$nm-1 (to be compared with our value for $k_{g}$ given
by $\approx 2.2$nm-1), overdamped in the range $2$nm${}^{-1}<q<4$nm-1, and
solid-like above $q=4$nm-1.

In parallel to this discussion, in recent years there has been intense
progress in revealing and understanding the emergent short-scale elasticity of
liquids Kume  _et al._ (2021); Zaccone and Noirez (2021); Zaccone and
Trachenko (2020), which has ultimately led to the question of where the
hydrodynamic limit really isHansen (2021); Baggioli (2021). The solid-like
vibrational dynamics at short-scale are obviously connected to the emergence
of macroscopic rigidity at short distances. A large part of this research
program has been focused on disclosing the solid-like nature of liquids by
considering confined systems, and therefore directly probing their short-scale
dynamics. Confined liquid systems include liquids in nanoporous materials,
droplets, liquid films, and liquids affected by interfacial interactions on
the microscopic scale. Under these circumstances, the interactions between
atoms or molecules in confined liquids undergo significant changes due to
geometric constraints, resulting in distinct manifestations in the VDOS. The
effects of confinement on liquid structure, transport and dynamics have been
explored in several works, specially for the case of waterTripathy  _et al._
(2019); Calero and Franzese (2020); Li and Riedo (2008); Khan  _et al._
(2010); Corti  _et al._ (2021); Kuo  _et al._ (2023).

In confined liquids, the mechanical relaxation times have been shown to
increase dramatically as compared to bulk behaviorKhan  _et al._ (2010). A
strong enhancement of the thermal conductivity has also been observed under
confinement Frank and Drikakis (2017) and attributed, following Frenkel’s idea
Frenkel (1984), to the presence of additional transverse oscillations
occurring because of the larger liquid relaxation time. More closely to the
topic of this manuscript, the low-energy behavior of the VDOS of confined
water has been investigated in Chen  _et al._ (2019) and more recently in Kuo
_et al._ (2023). In Chen  _et al._ (2019) a distinctive difference between
fragile confined water, where $g(\omega)\propto\omega^{1.5}$, and strong
water, where $g(\omega)\propto\omega^{2}$, was observed and explained using
mode-coupling theory and the idea of a crossover between heteoregeneity
dominated dynamics to phonon-like excitations. In Kuo  _et al._ (2023), a
reduction of the fraction of unstable instantaneous normal modes and of the
diffusion constant as a function of the confinement width, indicating the
solid-like emergent dynamics, was reported. More importantly, the low-
frequency power-law of the instantaneous normal mode VDOS has been studied as
a function of the width and the direction. Interestingly, Kuo et al.Kuo  _et
al._ (2023) found that the power-law in the direction of confinement,which is
perpendicular to the confinement surface, grows gradually by reducing the
width and reaches a value around $\approx 1.7$, close to Debye law.

In summary, despite the effects of confinement on the structure and dynamics
of liquids being largely explored in the past, a convincing experimental proof
of the emergence of a solid-like collective dynamics under confinement is
still missing. Furthermore, it is not yet clear how to connect the emergence
of solid-like dynamics in bulk liquids at high-frequency to the same
phenomenon in confined liquids at low frequency.

In this work, we employed a combination of inelastic neutron scattering and
molecular dynamics simulations to analyze the low-frequency (below $4$ meV)
VDOS in liquid water and glycerol nano-confined within graphene oxide
membranes (GOM). GOM, prepared by the modified Hummer’s method Marcano  _et
al._ (2010), have $28\%$ oxidation and hydrophilic interfaces, enabling water
and glycerol molecules easy access to the interlayers. Additionally, the
sandwich-like structure of GOM, as illustrated in the inset of Figure 2c,
provides quasi-two-dimensional spatial confinement for the liquid. These
characteristics have made GOM a widely used spatial confinement material for
studying confined systems Romanelli  _et al._ (2016); Zhang  _et al._ (2022).
Inelastic neutron scattering has been widely used to investigate the
vibrational density of states (VDOS) in liquids and solids Frick  _et al._
(2005); Stamper  _et al._ (2022b); Buchsteiner  _et al._ (2006); Yu  _et al._
(2022); Trouw and Price (1999).

Utilizing such experimental techniques, we provide compelling evidence of the
evolution of the scaling for the low-frequency VDOS in these two systems from
a linear to quadratic Debye-like behavior by decreasing the confinement size.
The validation of this phenomenon in two different systems suggests that
emergent solid-like vibrational modes induced by confinement may be a
universal feature of confined liquids, consistent with the observation of
elastic response previously reported in the literature Gao  _et al._ (2007);
Ortiz-Young  _et al._ (2013a). Various microscopic quantities of the liquids,
including current correlation functions, transverse and longitudinal wave
dispersions, and self-diffusion constants are analyzed in order to
characterize this finding in more detail. Exploiting qualitative arguments
based on the k-gap theory Baggioli  _et al._ (2020), we attribute the
emergence of solid-like VDOS to the slowing down of relaxation processes under
nanoconfinement, which corroborates the original perspective of liquid
dynamics by Frenkel and Maxwell.

Figure 2: (a) The experimental non-normalized VDOS of confined water at
different hydration level (gram water/gram GOM). All datasets are fitted in
the same energy interval (from $0.8$ meV to $4.5$ meV) and rescaled such that
the first data points overlap for better comparison.The error bars are plotted
in the same colors as the samples. (b) Analogous experimental non-normalized
VDOS of confined glycerol with different ratio (gram glycerol/gram GOM).The
error bars are plotted in the same colors as the samples. (c) The XRD data of
GOM sample with different hydration levels. (d) The small angle X-ray
scattering for water confined in GOM signaling the absence of Bragg peaks at
temperature above the freezing point.

##  II Experimental results

Using inelastic neutron scattering, we measured the vibrational density of
states (VDOS) of water confined within GOM at $280$ K and glycerol confined
within GOM at $300$ K, for various confinement sizes. More details regarding
sample preparation and experimental methods are provided in Appendix A. See
also Yu  _et al._ (2022); Zhang  _et al._ (2022) for a similar setup.

As a reference, we present the data for bulk water at $280$ K and glycerol at
$300$ K in Figure 2a  and Figure 2b  respectively. The corresponding dynamical
structure factors could be found in Figure 3. Both systems exhibit a linear
scaling of the VDOS at low frequencies, consistent with previous experimental
results Phillips  _et al._ (1989); Stamper  _et al._ (2022a); Jin  _et al._
(2023). We note that the measured energy range of the experimental VDOS for
the confined glycerol begins at larger energy($\sim 1.8$ meV) than water due
to the different observable energy range of instruments. Notice that the
results for water are consistent with previous estimates for the structural
relation time, or equivalently Maxwell relaxation time, which at room
temperature is around $1$ ps Cunsolo  _et al._ (1999); Malomuzh and Shakun
(2019); O’Sullivan  _et al._ (2019). This implies that the dynamics of water
at room temperature is expected to be fully liquid-like (linear in frequency)
below $\approx 4.13$ meV, consistent with our fits.

Figure 3: The average dynamical structure factor $S(q,\omega)$ from inelastic neutron scattering for water (a) and glycerol (b). Samples | GOM 0.2gly | GOM 0.4gly | GOM 0.8gly
---|---|---|---
$2\theta(^{\circ})$ | 8.9 | 7.4 | 5.3
Distance$({\textup{\AA}})$ | 10.0 | 12.0 | 16.6
Samples |  GOM 0.1H2O |  GOM 0.4H2O |  GOM 0.7H2O
$2\theta(^{\circ})$ | 11.4 | 9.2 | 6.9
Distance$({\textup{\AA}})$ | 7.8 | 9.6 | 12.8
Table 1: The interlayer distance for the experimental samples under different
hydration levels. The corresponding XRD data for confined water are shown in
Figure 2c.

These findings are further supported by molecular dynamics simulations (see
next section). In stark contrast, under extremely confined conditions, where
the water (or glycerol) to GOM weight ratio reaches $0.1$ ($0.2$)
respectively, the scaling of the low-frequency VDOS deviates significantly
from the linear liquid-like scaling. The VDOS displays a power law behavior
$\omega^{\alpha}$ where the exponent $\alpha$ lies between the quadratic Debye
law for solids and the linear in frequency law for liquids. The relationship
between the weight ratio (liquid to GOM) and the confinement size is examined
by X-ray diffraction in Figure 2c, and the detailed results are presented in
Table 1, where the inter-layer distance in GOM varies from 7 Å to 13 Å for
water when this ratio changes from 0.1 to 0.7 while it varies from 1 nm to 1.6
nm for glycerol when the ratio changes from 0.2 to 0.8. The extreme
confinement size provided by GOM for both liquids is approximately 7 Å (see
Figure 2c). At that scale, the power $\alpha$ reaches the value of $\approx
1.58$ and $\approx 1.89$ respectively for water and glycerol. By fitting with
a power-law function, we find that the scaling of the low-frequency VDOS of
confined water and glycerol gradually deviates from the linear scaling of the
bulk liquid as the confinement size is reduced. This suggests that under
unidirectional confinement, the influence on low-frequency vibrational modes
in liquids positively correlates with the degree of confinement, and the
scaling tends to approach the Debye value typical of solids.

In our previous work Yu  _et al._ (2022), we have demonstrated that the VDOS
of GOM itself does not contribute significantly to the VDOS of the confined
system in this frequency range. Moreover, we conducted inelastic neutron
scattering on water at the surface of reduced graphene oxide(rGO) with the
absence of any confinement effect. As shown in Figure 4, with only oxygen-
containing groups and surface effects Liu  _et al._ (2020), the low-frequency
scaling of VDOS for water exhibits minimal differences. The scaling of the
low-frequency VDOS for three different water contents on rGO samples remains
around 1 at temperature above freezing point. We note that even in the case of
the lowest water content (h=0.1), there is no significant change in the
scaling compared with pure water. Therefore, the low-frequency scaling just
reported does not arise from the interactions between water and GOM.
Additionally ,the small angle X-ray scattering (SAXS) for water confined in
GOM, shown in Figure 2d , is employed to provide proof that the liquid remains
in the liquid phase under confinement and no crystalline structure is formed.
The presence of Bragg peaks, which symbolize crystalline solid ice, only
occurs at lower temperatures, below $270$ K. At temperature above the freezing
point, proved by the absence of characteristic Bragg peaks, water confined
within GOM remains in the liquid state. The diffusive behavior of the water
confined in GOM has also been reported in our previous work Zhang  _et al._
(2022). Therefore, the low-frequency scaling does not originate from a
crystalline phase but rather from the confined liquid water itself.

Figure 4: The experimental non-normalized VDOS of water on rGO with different
hydration level. The error bars are plotted in the same colors as the samples.

To confirm the experimental results of this faster-than-linear scaling in
confined liquids, we conducted molecular dynamics (MD) simulations for water
and glycerol. A snapshot of the two setups is provided in the top panels of
Figure 5. In the MD simulations, we utilized additional confinement sizes to
validate the experimental results in depth. More details regarding the MD
simulations can be found in Appendix A.6. For the water and glycerol systems,
the confinement sizes ranged respectively from $10\,\textup{\AA}$ to
$100\,\textup{\AA}$ and $15\,\textup{\AA}$ to $100\,\textup{\AA}$ (see Figures
5a  and 5b). In both of the systems, the largest size is considered as a bulk
sample.

As shown in Figures 5c  and 5d, the results of confined water and glycerol
from the MD simulations are consistent with the experimental observations. In
order to be comparable with neutron scattering experiments, only the
trajectories of hydrogen atoms are calculated when calculating VDOS, which has
been shown to qualitatively coincide with the results from all atoms in Figure
9 in Appendix A.6. More precisely, the low-frequency VDOS exhibits a larger
scaling as the confinement size is reduced, strongly deviating from the linear
scaling of the bulk state. For the maximum confinement simulated, we observe
respectively a power law scaling $\omega^{1.36}$ for water and $\omega^{1.44}$
for glycerol, which are compatible but slightly smaller than the experimental
values. This indicates that the simulations can qualitatively reproduce the
experimental conditions but they underestimate the confinement strength.

Interestingly, both in experiments and simulations we find that for the same
level of confinement the power law for glycerol is larger than the one of
water. This implies that glycerol exhibits a solid-like dynamics on larger
scales with respect to water. Assuming that the critical scale for the
emergence of solidity is roughly given by $L_{c}\propto v\tau$, we can, at
least qualitatively, explain this observation. In particular, by following
Maxwell’s idea and interpreting $\tau$ with the Maxwell relaxation time, the
critical length becomes proportional to the viscosity of the liquid. That
said, glycerol is more viscous than water, $\approx 1400$ times larger at room
temperature and therefore the solid-like dynamics are expected to emerge on
larger scales, as observed in both experiments and simulations.

Figure 5: The snapshot of the confined system for water (a) and glycerol (b).
(c): The simulation low-frequency VDOS for confined water under different
confinement sizes $L$ (reported in the inset). (d): The simulation low-
frequency VDOS for confined glycerol under different confinement sizes $L$
(reported in the inset). In both panels, only the hydrogen atoms are
considered to calculate the VDOS. The dashed lines indicate the fitted values
for the low-frequency scaling at the lowest and highest degree of confinement.

##  III Explaining the low-frequency solidity of liquids under confinement

In the previous section, by means of experiments and simulations, we have
shown that by decreasing the confinement size from several nanometers to
$7\textup{\AA}$ the low-frequency behavior of the vibrational density of
states of water and glycerol drastically changes. In particular, we observed a
continuous transition in the vibrational density of states (VDOS) from a
linear scaling in frequency dependence in bulk liquids to a solid-like Debye
law in strongly nanoconfined liquids.

In order to explain this behavior, we rely on the interpretation of liquid
dynamics originally proposed by Maxwell and Frenkel Frenkel (1984) (see also
Zwanzig Zwanzig (1967)), and later re-interpreted by Trachenko and
collaborators Trachenko and Brazhkin (2015); Baggioli  _et al._ (2020). The
main idea, as previously introduced, is that the dynamics of liquids are
characterized by solid-like quasi-harmonic vibrations around local minima of
the potential landscape interrupted by thermally activated re-arrangements
corresponding to jumps over the barrier towards a different minimum (see inset
in Figure 1). This relaxational processes happen at an average rate
$\tau^{-1}$, where $\tau$ is a so far undetermined re-arrangement timescale.
Following this simple picture, one could immediately argue that for times
shorter than $\tau$ a liquid should not be too different from a solid, i.e.,
the Frenkel criterion.

Mathematically, this framework can be described using the telegrapher
equation, Eq.(1), which determines the dynamics of the collective (shear)
modes in liquids, up to the lowest order in the frequency $\omega$ and the
wave-vector $k$ and provides a simple interpolation between a solid-like
behavior at high frequencies and a liquid-like dynamics at low frequencies.

By solving Eq.(1), and focusing on the real part of the frequency, one finds
the simple expression:

$\mathrm{Re}(\omega)=v\sqrt{k^{2}-k_{g}^{2}}\qquad\text{where}\qquad
k_{g}\equiv\left(2v\tau_{g}\right)^{-1},$ (2)

where we have kept a distinction between the structural relaxation time $\tau$
($\approx 1$ps for bulk water) and the timescale $\tau_{g}$ appearing in the
k-gap equation above. In bulk liquids, $\tau_{g}$ decreases with temperature,
and therefore $k_{g}$ can be taken as a measure of “fluidity”.

Using MD simulations,we obtained the dispersion relation of the collective
shear waves in liquid water, as shown in Figure 6a. Unfortunately, the
statistical quality of the data points and the fitting curve in the low-
wavenumber region is poor. This limitation is attributed to the finite size of
our simulations ($100\textup{\AA}$). For the bulk sample at $280$ K, we find
that $k_{g}\approx 0.23\,\textup{\AA}^{-1}$. which is of the same order as
liquid gallium at $313$ K Khusnutdinoff  _et al._ (2020b) and liquid sodium at
$393$ K Giordano and Monaco (2010). Most importantly, the obtained value is in
quantitative agreement with a previous estimate for water at room temperature,
where the system has been proven to be fully hydrodynamic below a value of
$\approx 0.2\,\textup{\AA}^{-1}$ Sampoli  _et al._ (1997).

It should be noted that, as derived in Sampoli  _et al._ (1997), the dynamics
of bulk water at room temperature becomes underdamped as is the case in solids
only above $\approx 0.4\,\textup{\AA}^{-1}$. This implies that the requirement
of having the wave-vector larger than the k-gap $k_{g}$ is not enough to have
a well-defined underdamped solid-like dynamics in liquids. As explained in
Appendix D, this is consistent with k-gap theory. More precisely, above but
near the k-gap, $\mathrm{Re}(\omega)<\mathrm{Im}(\omega)$, meaning that the
dynamics are not liquid-like but still overdamped. This is consistent with the
range of $0.2\,\textup{\AA}^{-1}<k<0.4\,\textup{\AA}^{-1}$ found
experimentally in Sampoli  _et al._ (1997). More interestingly, at a
quantitative level, $0.4\,\textup{\AA}^{-1}$ is approximately twice of the
k-gap, and converted in frequency scale is very close to the Frenkel criterion
(see Appendix D for more details about this).

In order to prove the emergence of solidity under confinement, in Figure 6a,
we analyze the dispersion relation of the collective shear waves in the water
sample by reducing the confinement size. The detailed results of the fit can
be found in Table 2 in Appendix D. We observe that the momentum gap of the
collective modes decreases with decreasing confinement size. In other words,
under strong confinement the gap in the wave-vector closes and the dispersion
relation shifts towards the solid-like one $\mathrm{Re}(\omega)=vk$. Before
analyzing this behavior in detail, we seek further confirmation of this trend
by examining the experimental and simulation diffusion constants as a function
of the confinement size.

As shown in Figure 6b, the diffusion constant obtained from the mean square
displacement (see Figure 8) increases monotonically with the confinement size
$L$ and approaches the value for bulk water for larger $L$. A decrease of the
self-diffusion constant with confinements size, and a dramatic increase of the
viscous forces have been already reported in simulations and experiments Zhang
_et al._ (2022); Li and Riedo (2008); Kuo  _et al._ (2023); Ortiz-Young  _et
al._ (2013b). For comparison, in the same panel, we also show the behavior of
the diffusion constant for confined glycerol at $300$ K. The diffusion
constant in glycerol is approximately two orders of magnitude smaller than
that of water, as the dynamics therein are much slower and the effective
viscosity is much larger.

We then extract the parameters $v$ and $\tau_{g}$. Despite the precision of
the fitting is limited, specially for very low values of $L$, the analysis can
still provide a useful qualitative estimate. In Figure 6c, we show the
behavior of $k_{g}$ and $\tau_{g}$ as a function of the confinement size $L$.
As explicitly verified in Appendix D (see Table 2), the shear waves speed (and
the speed of longitudinal sound as well) are to a first approximation constant
with respect to $L$, and therefore their effect can be neglected. On the
contrary, the relaxation time $\tau_{g}$ and the momentum gap $k_{g}$ are very
sensitive to the confinement scale. The relaxation time $\tau_{g}$ grows
approximately linearly with $L$, and the momentum gap follows approximately
the inverse behavior $k_{g}\propto 1/\tau_{g}$. From a physical perspective,
these results indicate that,with the increasing the strength of the
confinement, collective shear waves in the liquid may start to propagate at
lower frequencies and the solid-like dynamics extends lower wave-vectors.
Moreover, the dynamics and the local-rearrangements are substantially slowed
down, as a further evidence that the liquidity of the system is hindered at
short distances.

We notice that for the bulk water the relaxation time $\tau_{g}$ is $\approx
2$ps. This is about a factor of two larger than the structural relaxation time
and the Maxwell relaxation time reported for bulk water at room temperature
Cunsolo  _et al._ (1999); Malomuzh and Shakun (2019); O’Sullivan  _et al._
(2019). While we cannot derive a conclusive relation, the qualitative
consistency suggests $\tau_{g}$ and $\tau$ could both characterize certain
relaxation processes in liquids. Though they do not quantitatively agree, at
least in water. As a final remark, the k-gap analysis and the behavior of the
diffusion constant confirm a close connection between the retardation of
relaxation dynamics and the tendency toward solid-like vibrational modes. The
same conclusion has been reached in several MD simulations by changing the
temperature of the system Khusnutdinoff  _et al._ (2020a); Yang  _et al._
(2017).

Figure 6: (a) Dispersion curves for transverse waves in confined water under
different confinement sizes. (b) Self-diffusion constant increasing with the
confinement size in confined water and confined glycerol. (c) The momentum gap
extracted from the dispersion curves in panel (a) and the corresponding
relaxation time $\tau_{g}$ as a function of the confinement size $L$. (d) The
linear correlation between $k_{g}$ and the diffusion constant.

In Figure 6d, we reveal a direct linear correlation between the size of the
momentum gap for collective shear waves and the single-particle self-diffusion
constant. The two quantities show an approximate linear relation of the type
$k_{g}\propto D$ confirming that they are both measurements of the fluidity of
a system. The larger the gap of shear waves, the larger the self-diffusion
constant and the ability of the system to flow as a fluid. To the best of our
knowledge, this direct correlation between the k-gap and the diffusion
constant has not been revealed before, and it certainly deserves further
investigation.

At this point, it is reasonable to ask how the transition from the linear in
frequency liquid-like behavior to the quadratic Debye law happens upon moving
the confinement size $L$ and how it is related to the re-appearance of
collective shear waves and the behavior of the momentum gap $k_{g}$. In order
to reach a qualitative argument, a simplified toy model based on the k-gap
equation Eq.(1) can be constructed (see Appendix D for more details). By
neglecting the effects of the other excitations and concentrating on the
gapped shear waves, and by assuming a spherical phase space for their wave-
vector unaffected by the confinement size, one can derive a simple expression
for the contribution of the shear waves to the VDOS given by:

$g_{\text{shear}}(\omega)\propto\omega\sqrt{\omega^{2}+\frac{1}{4\tau_{g}^{2}}}\,.$
(3)

Despite the simplicity of this derivation, one striking prediction emerges.
Precisely, one would expect a transition from the liquid-like linear VDOS to a
Debye law at frequencies $\omega\gg 1/\tau_{g}$, which is another
manifestation of the Frenkel criterion. In other words, we expect that the
deviations from the purely linear behavior will appear only at frequencies
larger than the inverse relaxation time $\tau_{g}^{-1}$. Since $\tau_{g}$
increases by reducing the confinement size $L$, as shown in Figure 6c, one
would expect the Debye scaling to reach lower and lower frequencies upon
confining the liquid.

In order to check this prediction, in Figure 7a, we revisit the experimental
data for confined water under this new perspective and we estimate the
frequency at which the linear behavior is lost as a function of the
confinement size $L$. We denote such an energy scale as the crossover
frequency $\omega_{\times}$. In bulk water at $280$ K we do not observe any
crossover and the low-frequency VDOS is perfectly linear below $4$ meV. This
is in perfect agreement with the idea that the solid-like dynamics can appear
only above $1/\tau$. For bulk water, $\tau=1$ ps and therefore the cutoff for
solid-like vibrations is around $4.15$ meV.

As proved in Figure 6, under confinement the relaxation time strongly
increases, moving the onset for solid-like vibrations in confined liquids to
energies below the bulk cutoff of $4$ meV. Indeed, for higher degrees of
confinement – shorter $L$ – we start observing a crossover between a linear
behavior, that appear within the blue region in Figure 7a, and a quadratic
Debye-like one that emerges above $\omega_{\times}$, in the white region in
Figure 7a. These two different scalings are emphasized in Fig.7a by dashed
black lines. The crossover frequency becomes smaller by decreasing the
confinement size $L$, ranging from $\approx 2.8$ meV at $L=12.8\,\textup{\AA}$
to $1.9$ meV at $L=7.8\,\textup{\AA}$.

This is compatible with the relaxation time becoming longer and longer under
the enhanced confinement and the onset of solid-like dynamics, $\sim 1/\tau$,
appearing at lower frequencies, below the experimental cutoff of $\approx 4$
meV (see Figure 7b  for a representation of this phenomenon).
This simple argument also suggests that the fractional power laws extracted in
the previous section from the experimental data using a single power fitting
function might be the result of a combination of a liquid-like linear term and
a Debye quadratic contribution. Qualitatively, the smaller the confinement
size $L$, the more the Debye contribution enters at low frequency and the
larger the power extracted from a single power fitting law. This also explains
why a perfect Debye scaling is not recovered for small values of $L$. This
would happen only in the unrealistic limit $\tau\rightarrow\infty$ in which
the crossover frequency goes to zero and the relaxational dynamics are
completely frozen. For the shortest confinement scale we can achieve,
$L\approx 7.8\,\textup{\AA}$, we see a crossover frequency of
$\omega_{\times}\approx 1.9$ meV, implying that below such a scale a residual
linear in frequency liquid-like contribution is still present.

Figure 7: (a) A re-visitation of the experimental data in Figure 2. The
colored region represents the liquid-like regime, where solid-like vibrational
modes cannot exist. The black dashed lines indicate the two different scalings
$g(\omega)\sim\omega$ (in the blue region) and $g(\omega)\sim\omega^{2}$,
above $\omega_{\times}$ in the white region. (b) A pictorial representation of
the underlying physics based on k-gap theory (see Appendix D). The lines from
black to red corresponds to smaller confinement size $L$. The colored disks
locates the position of the crossover between linear behavior and quadratic
behavior – the onset of solid-dynamics – upon confinement. The orange region
indicates the low-frequency regime, where the power-law scalings can be
experimentally detected, which for water at room temperature extends up to
$\approx 4$meV. For bulk water at room temperature, solid-like vibrations are
expected only above that scale, within the magenta region.

##  IV Conclusions

In this study, we investigated the vibrational density of states (VDOS) of
water and glycerol liquids nano-confined in multiple-layer graphene oxide
membranes (GOM) using inelastic neutron scattering and molecular dynamics
simulations. We focused on the low-frequency (below $4$meV) scaling of the
VDOS in confined liquids, its variation with the confinement length and its
relation with the relaxational dynamics and the dispersion of gapped shear
waves. Experimental observations revealed that the low-frequency VDOS of the
confined liquid evolves from a linear behavior in the bulk state to a faster
scaling, close to the Debye-like quadratic behavior, under strong confinement.

Through molecular dynamics simulations, we explored a broad range of
confinement sizes and confirmed the gradual solid-like evolution of both
liquids as the confinement size decreases. Analyzing the vibrational modes in
the different directions, we found that emergent solid-like vibrational modes
primarily arise along the confined direction, in agreement with previous
simulation results Kuo  _et al._ (2023), and that spatial confinement strongly
affects the dispersion relation of transverse waves in the liquid. As the
confinement size decreases, both the diffusive processes and the structural
rearrangements are slowed down, and the transverse dispersion curves
progressively resemble those of a solid, with no wave-vector gap. Upon
confinement, the longitudinal waves of the liquid are not significantly
affected. On the contrary, the momentum gap for collective shear waves,
typical of liquids, closes by decreasing $L$ and beautifully correlates with
the behavior of the self-diffusion constant $D$. In a nutshell, decreasing the
confinement size $L$ is akin to decreasing the temperature $T$ and going
towards the solid phase in a continuous fashion. In addition, we have found
that the relaxation time $\tau_{g}$ extracted from the k-gap equation is
strongly slowed down upon confinement.

Building upon Frenkel-Maxwell’s ideas Frenkel (1984) and the more recent k-gap
theory Baggioli  _et al._ (2020), we provided a simple physical picture for
the experimental and simulation observations. In particular, we have
corroborated that the appearance of solid-like dynamics at low frequency  in
water, below $4$meV, is highly correlated with slow down of the relaxation
time under confinement. This confirms the idea that at frequencies above the
inverse of the relaxation time $1/\tau$, solid-like vibrational modes appear.
Under nanoconfinement conditions, the corresponding energy scale becomes
smaller than $4$meV and strongly alters the low-frequency VDOS of liquids.
This mechanism is accompanied by the reduction of the gap for transverse shear
waves, which is again controlled, according to k-gap theory, by the inverse of
the relaxation scale. In other words, we have shown that the closing of the
k-gap under confinement is the microscopic origin behind the solid-like low-
frequency VDOS of liquids under confinement and it strogly correlates with the
decrease of the self-diffusion constant. This picture gives a direct
verification of the Frenkel criterion and of the idea of continuity between
the liquid and solid phases of matter.

In conclusion, our study elucidates the impact of spatial confinement on the
vibrational dynamics and on the onset of solid-like vibrational modes in
liquids. Our findings contribute to a better understanding of the behavior of
liquids under confinement, and on the short-scale dynamics of liquids in
general, providing important insights into the fundamental aspects of their
vibrational properties. Further investigations are necessary to explore
different confinement geometries, liquid systems, and environmental conditions
to deepen our knowledge of liquid dynamics and their implications for various
fields such as materials science, nanotechnology, and energy storage.

###### Acknowledgements.

We would like to thank H. Xu, A. Zaccone, K. Trachenko, L. Noirez and T. Keyes
for fruitful discussions on the topic of liquids. We thank Dr. Victoria Garcia
Sakai from ISIS Neutron and Muon Facility for assistance with the inelastic
neutron scattering. We acknowledge the Instrumental Analysis Center of
Shanghai Jiao Tong University for assistance with structural characterization
via SEM and XRD. We thank Dr. Xiaran Miao from BL16B1 beamline of Shanghai
Synchrotron Radiation Facility (SSRF) for help with synchrotron X-ray
measurements. This work was supported by the National Natural Science
Foundation of China (11974239), the Innovation Program of Shanghai Municipal
Education Commission(2019-01-07-00-02-E00076),and Shanghai Jiao Tong
University Scientific and Technological Innovation Funds (21X010200843). M.B.
acknowledges the support of the Shanghai Municipal Science and Technology
Major Project (Grant No.2019SHZDZX01) and the sponsorship from the Yangyang
Development Fund. We acknowledge the support of the Australian Centre for
Neutron Scattering, ANSTO and the Australian Government through the National
Collaborative Research Infrastructure Strategy, in supporting the neutron
research infrastructure used in this work via ACNS proposal P7273.

##  Appendix A Experimental and simulation method

###  A.1 Sample preparation

The GOM sample was synthesized using the modified Hummers’ methodMarcano  _et
al._ (2010).Initially, the sample was dehydrated by heating it from room
temperature to 313 K and subsequently annealed at this temperature for 12
hours under vacuum conditions to achieve a dry state. The oxidation rate of
the GOM sample was determined to be $28\%$ using X-ray Photoelectron
Spectroscopy (XPS). After dehydration, the GOM sample was sealed in a
desiccator and exposed to water vapor to facilitate the adsorption of water
molecules onto the surface and between the interlayers of the GOM sheets. The
exposure time of the sample was adjusted to control the hydration levels. The
final hydration levels were determined by measuring the weight of the sample
before and after water adsorption, providing a quantitative assessment of the
absorbed water content.The dehydrated GOM is soaked in glycerol for varying
amounts of time and dried at 40 ${}^{\circ}C$ for 24 hours after removal to
thoroughly remove the water. The specific gravity of glycerol is determined by
the soaking time of the samples.

###  A.2 Powder X-ray Diffraction (PXRD)

The powder X-ray diffraction (PXRD) data for GOM at various hydration levels
were obtained using a Rigaku Mini Flex600 X-ray diffractometer. The instrument
was equipped with a Cu K$\alpha$ source ($\lambda$ = 1.5406 Å) and operated at
40 kV and 15 mA. The data were collected over a scanning range of 10$\degree$
to 60$\degree$ at a scan rate of 10$\degree$/min. The analysis of the PXRD
data was performed using MDI Jade software.

###  A.3 Small Angle X-ray Scattering (SAXS)

SAXS measurements were utilized to track the changes in interlayer distance
within graphene-oxide membranes (GOM) during temperature reduction and ice
formation. The experiments were conducted at the BL16B1 beamline of the
Shanghai Synchrotron Radiation Facility (SSRF) using X-rays with a wavelength
of $1.24\textup{\AA}$. The SAXS patterns were captured using a Pilatus 2M
detector, featuring a resolution of 1475 pixels $\times$ 1679 pixels and a
pixel size of $172\,\mu m\times 172\,\mu m$. To ensure accurate data
collection, each SAXS frame had an acquisition time of 10 s. The sample-to-
detector distance for SAXS measurements was maintained at 258 mm.

###  A.4 Scanning Electron Spectroscopy (SEM)

The SEM images were acquired using a MIRA 3 FE-SEM operating at a 5 kV
accelerating voltage.

###  A.5 Inelastic Neutron Scattering (INS).

Due to the significantly larger incoherent scattering cross section of
hydrogen atomZheng  _et al._ (2022), the intensity measured in neutron signals
in the system are primarily influenced by the incoherent scattering function,
which predominantly reflects the self-motion of water molecules within the
sample(the contribution from the graphene oxide will be very smallYu  _et al._
(2022)). The experimental vibrational density of states (DOS), denoted as
$g(\omega)$, can be derived from the dynamic structure factor, $S(q,\omega)$,
using the approachTaraskin and Elliott (1997):

$\displaystyle
g(\omega)\,=\,\int\frac{\hslash\omega}{{q}^{2}}S(q,\omega)\left(1-e^{-\frac{\hslash\omega}{k_{B}T}}\right)dq,$
(4)

where $\hslash$ is the Planck constant, $q$ is the scattering wavevector,
$\omega$ is the frequency which related to the energy transfer, $k_{B}$ is the
Boltzmann constant, and T is the temperature. For water confined in GOM, three
samples were fabricated with different weight ratio h (gram water/gram GOM) :
0.1, 0.4 and 0.7. The inelastic neutron scattering experiment were conducted
by the time-of-flight cold neutron spectrometer - PELICAN at Australian
Nuclear Science and Technology Organisation (ANSTO). The incident neutron
energy is 14.9 meV with an energy resolution $\Delta E=0.5$ meV at the elastic
peakYu  _et al._ (2013). The $q$ range covered is from
$0.08\,\textup{\AA}^{-1}$ to $4.5\,\textup{\AA}^{-1}$. The samples were
contained inside aluminum foils which were sealed in aluminum sample cans
under helium atmosphere. The empty can signal was subtracted as background at
each temperature. The detector efficiency was normalized using a vanadium
standard. Data reduction and DOS extraction were performed by LAMP software
packageRichard  _et al._ (1996) and the scripts are available upon request.
The experiment for glycerol confined in GOM was conducted using the cold
neutron multi-chopper spectrometer LET at ISIS in UKBewley  _et al._ (2011).
The confined glycerol samples were also fabricated in three different weight
ratio h (gram glycerol/gram GOM): 0.2,0.4 and 0.8 The measurement was done
with incident energy $22.78$ meV which covers the energy transfer range up to
$13.5$ meV and the $q$ range from $0.34\,\textup{\AA}^{-1}$ to
$7.08\,\textup{\AA}^{-1}$. Data reduction was performed on Mantid software
packagesArnold  _et al._ (2014).

Figure 8: The mean-square displacement for confined water (a) and confined
glycerol (b) as a function of time and different degrees of confinement. The
dashed lines indicate the late-time linear fit which allows to determine the
value of the self-diffusion constant reported as inset. The obtained data for
the self-diffusion constant as a function of the confinement size $L$ are
reported in the main text in Figure 6.

###  A.6 Molecular dynamics (MD) simulations

The classical MD simulation was performed via the large-scale atomic/molecular
massively parallel simulator (LAMMPS)Plimpton (1995) to simulate water and
glycerol at $280$ K and $300$ K, respectively. Water molecules are simulated
with the widely used four-point TIP4P/2005Abascal and Vega (2005) model where
the energy parameter $\epsilon_{O-O}$ is $0.008$ eV and the size parameter
$\sigma_{O-O}$ is $3.16$ Å. The interactions potential of water molecules is
modeled by 12-6 Lennard-Jones (LJ) potential with a cutoff radius of $12$ Å
and the electrostatic forces are calculated with PPPM algorithm with an
accuracy of $10^{-4}$. The system of glycerol was modeled using Material
Studio and simulated using the AMBER force field parameters, which have been
proven to accurately describe the dynamics and structure of the glycerol
system Chelli  _et al._ (1999). All initial structures were relaxed at given
temperatures with the isothermal-isobaric (NPT) ensemble for 300 ps. The Nosé-
Hoover thermostat and Parrinello-Rahman barostat to control the temperature
and pressure, and then switched to NVT ensemble at 100 ps (equilibrium time)
to calculate dynamical properties. The Newton equation of motion was
integrated using the velocity-Verlet algorithm with a time step of $1$ fs per
step. We freeze the bottom and top layer ($\sim 3$ Å) of the slab systems to
force the structure to remain 2D confined during the whole simulation.

A snapshot of the slab geometry can be found in Figure 5 a,b in the main text.
The VDOS is calculated by the Fourier transform of the velocity auto-
correlation function:

$C_{v}(\omega)=\int_{-\infty}^{\infty}C_{v}(t)\exp(-i\omega t)dt\,.$ (5)

The velocity auto-correlation function (VACF) is defined as:

$C_{v}(t)=<\textbf{v}(0)\cdot\textbf{v}(t)>$ (6)

where $\textbf{v}(0)$ are the oxygen velocities. In order to compare with the
VDOS from neutron experiments, which mainly comes from the signal of hydrogen
atoms, we calculated the VACF with the corresponding VDOS of hydrogen atoms
and all atoms(see Figure 5 and Figure 9), respectively.
The Diffusion constant is calculated by the mean square displacements of the
water molecules:

$\displaystyle
D=\frac{1}{6}\frac{d}{dt}\left<\left|\mathbf{r}_{i}(t)-\mathbf{r}_{i}(0)\right|^{2}\right>,$
(7)

where $\mathbf{r}_{i}(t)$ is the position of the $i^{th}$ particle at time
$t$.

The current density is a vector quantity which can be decomposed into a
longitudinal part containing the component parallel to the
$\boldsymbol{q}$-vector and a transverse part containing the perpendicular
component, according to

$\displaystyle\boldsymbol{j}(\boldsymbol{q},t)=\boldsymbol{j}_{L}(\boldsymbol{q},t)+\boldsymbol{j}_{T}(\boldsymbol{q},t)$
(8)

where we have defined

$\displaystyle\boldsymbol{j}_{T}(\boldsymbol{q},t)=\sum_{i}^{N}(\boldsymbol{v}_{i}(t)\cdot\hat{\boldsymbol{q}})\hat{\boldsymbol{q}}e^{i\boldsymbol{q}\cdot\boldsymbol{r}_{i}(t)},$
(9)
$\displaystyle\boldsymbol{j}_{T}(\boldsymbol{q},t)=\sum_{i}^{N}[(\boldsymbol{v}_{i}(t)-(\boldsymbol{v}_{i}(t)\cdot\hat{\boldsymbol{q}})\hat{\boldsymbol{q}}]e^{i\boldsymbol{q}\cdot\boldsymbol{r}_{i}(t)},$
(10)

and use $\hat{\boldsymbol{q}}$ to denote the unit vector.
The current correlation functions can now be computed as:

$\displaystyle
C_{L}(\boldsymbol{q},t)=\frac{1}{N}\left\langle(\boldsymbol{j}_{L}(\boldsymbol{q},t)\cdot\boldsymbol{j}_{L}(\boldsymbol{-q},0)\right\rangle$
(11) $\displaystyle
C_{T}(\boldsymbol{q},t)=\frac{1}{N}\left\langle(\boldsymbol{j}_{T}(\boldsymbol{q},t)\cdot\boldsymbol{j}_{T}(\boldsymbol{-q},0)\right\rangle$
(12)

Then the velocity current spectra could be derived using:

$\displaystyle C_{L,T}(\boldsymbol{q},\omega)=\int dte^{i\omega
t}Re(C_{L,T}(\boldsymbol{q},t)).$ (13)

Since simple fluids are isotropic, we average $C_{L,T}(\boldsymbol{q},\omega)$
over all directions of the wavevector $\boldsymbol{q}$.

$\displaystyle
C_{L,T}(q,\omega)=\frac{1}{N_{q}}\sum_{|\boldsymbol{q}|=q}C_{L,T}(\boldsymbol{q},\omega)\,,$
(14)

where $N_{q}$ is the number of directions used for averaging.

Figure 9: The vibrational density of states calculated by all atoms for water
(a) and glycerol (b) with different confinement sizes. Figure 10: The
vibrational density of states (VDOS) of confined water along different
directions and for different confinement sizes: (a) 10 Å; (b) 30 Å; (c) 50 Å;
(d) bulk water as reference. Double logarithmic scale is used to highlight the
differences in the low energy scalings. The data are vertically shifted for
better visualization.

##  Appendix B Partial density of states and self-diffusion constant

In Figure 10, we present the simulation density of states of water under
different confinement conditions and decomposed into the three different
spatial directions x,y,z. At the smallest confinement size
($L=10\,\textup{\AA}$), the scaling of the vibrational density of states
(VDOS) in the z-direction, which represents the confined direction, is
significantly separated from the scaling of the other two components. It grows
faster compared to the horizontal components and its low-frequency power-law
is closer to the Debye value. In other words, and perhaps not surprisingly,
the the solid-like dynamics is more pronounced along the confined direction.
This is in agreement with the results recently appeared in Kuo  _et al._
(2023). The overall VDOS exhibits a scaling that falls between the faster one
along the $z$ direction and the on-plane one. This might also explains why the
full VDOS does not exhibit a quadratic Debye law even at the highest degree of
confinement. This is simply because the dynamics along the non-confined
directions are still mostly liquid-like. This phenomenon becomes less
pronounced as the confinement size increases. In the reference case of bulk
water, where the anisotropy is lost, the VDOS scaling for all three components
and the overall VDOS perfectly overlap.

Finally, in Figure 8, we report the behavior of the mean-square displacement
(MSD) as a function of time and for different degrees of confinement. By
fitting the late-time linear behavior we can directly obtain the self-
diffusion constant of the system for various confinement strengths. For both
confined water and glycerol, the diffusion constant decreases with confinement
as reported in the main text in Figure 6.

Figure 11: The dispersion relation of longitudinal sound waves in confined
water for different confinement sizes: (a) $30\textup{\AA}$ , (b)
$40\textup{\AA}$ , (c) $50\textup{\AA}$ , and (d) bulk water as a reference.

##  Appendix C Longitudinal and shear waves dispersion relations and fits

In Figure 6 in the main text, we have reported the simulation dispersion
relation for collective shear waves in confined water. We have fitted the
simulation data with the following formula obtained from k-gap theory:

$\mathrm{Re}(\omega)=\sqrt{v^{2}k^{2}-\frac{1}{4\tau_{g}^{2}}}\equiv
v\sqrt{k^{2}-k_{g}^{2}}$ (15)

and extracted the value of the phenomenological parameters $v,\tau_{g},k_{g}$
as a function of the confinement size $L$. The results of the fit are reported
in Table 2. The data for the k-gap $k_{g}$ and the relaxation time $\tau_{g}$
as a function of $L$ are shown in Figure 6 in the main text. Upon confinement,
the propagation speed of the collective shear waves varies only of $7\%$ and
it can therefore be considered in first approximation as constant.
Unlike shear waves, the dispersion relation of longitudinal waves in confined
water is not significantly affected by the variations in the confinement size.
The longitudinal speed of sound varies only of $\approx 3.5\%$ under
confinement and it can be approximated as well by a constant.

##  Appendix D The dynamics of shear waves in liquids and the k-gap model

Let us return to our initial telegrapher equation

$\omega^{2}+i\frac{\omega}{\tau_{g}}=v^{2}k^{2},$ (16)

which describes the dynamical crossover between a propagating wave at large
frequency/wave-vector to a overdamped diffusive motion at low energy. By
keeping the wave-vector real and assuming the frequency to be a complex
number, the solution of the above equation is given by

$\omega=-\frac{i}{2\tau_{g}}\pm\sqrt{v^{2}k^{2}-\frac{1}{4\tau_{g}^{2}}}\,.$
(17)

Clearly, the real part of the frequency becomes non-zero only above the so-
called k-gap wave-vector which is given by

$k_{g}=\frac{1}{2v\tau_{g}}\,.$ (18)

Below the k-gap, the mode with lowest energy is diffusive, with dispersion
relation $\omega=-iv^{2}\tau_{g}k^{2}$. Importantly, and often overlooked in
the related literature, the appearance of a non-zero real frequency is not
enough to have a propagating mode. Indeed, in order to have a well-defined
propagating excitation the real part of the frequency must be larger than its
imaginary part. This is nothing else than requiring the relaxation time of the
wave excitation to be larger than the period of oscillation, such that
oscillations might appear before being completely attenuated. In the opposite
scenario, where the imaginary part is too large, the wave decays too quickly
and no wave-like oscillation has time to take place. By identifying the
overdamped-to-underdamped crossover with the location at which imaginary and
real part are of the same order, one obtains a critical wave-vector/frequency
which are given respectively by

$k^{*}=\frac{1}{\sqrt{2}v\tau_{g}}\,,\qquad\omega^{*}=\frac{1}{2\tau_{g}}.$
(19)

Notice that this frequency cutoff is a factor $1/2$ smaller than the original
Frenkel criterion $\omega>\omega_{F}\equiv\tau_{g}^{-1}$. The latter
corresponds to a larger value of the wave-vector, given by

$k_{F}=\frac{\sqrt{5}}{2v\tau_{g}}.$ (20)

$L\,(\textup{\AA})$ 30 40 50 60 $k_{g}\,(\textup{\AA}^{-1})$ 0.06283 0.08886
0.1405 0.1405 $v$ (m/s) 1001 1011 1047 1035 $\tau_{g}\,(ps^{-1})$ 7.949 5.565
3.399 3.438 $L\,(\textup{\AA})$ 70 80 90 100 $k_{g}\,(\textup{\AA}^{-1})$
0.1539 0.1539 0.1777 0.2205 $v$ (m/s) 1070 1070 1071 1072
$\tau_{g}\,(ps^{-1})$ 3.036 3.036 2.626 2.114

Table 2: The results of the fit for the dispersion relation of the collective
shear waves shown in Figure 6a  using Eq.(15). Figure 12: The dispersion
relation of the collective waves arising from the telegrapher equation. The
different colors indicate the crossover between relaxational diffusive
dynamics (with $\mathrm{Re}(\omega)=0$), overdamped oscillations (with
$\mathrm{Re}(\omega)<\mathrm{Im}(\omega)$) and underdamped oscillations (with
$\mathrm{Re}(\omega)>\mathrm{Im}(\omega)$). For illustration, also the Frenkel
criterion $\omega>1/\tau_{g}$ is illustrated. Here, we have taken
$v=\tau_{g}=1$.

This hierarchy of scales is displayed in Figure 12 together with the real part
of the dispersion relation of the collective waves. Importantly, apart from
$\mathcal{O}(1)$ numerical factor (see for example the factor of $2$ between
$\tau_{g}$ for bulk water and the values reported in the literature for the
structural relaxation time Cunsolo  _et al._ (1999); Malomuzh and Shakun
(2019); O’Sullivan  _et al._ (2019)), which given the simplicity of the models
are likely irrelevant, the major outcome of this analysis is that in classical
liquids one can define a critical scale, $L_{c}\approx 2\pi v\tau$, below
which the dynamics is expected to be solid-like.
After discussing the dispersion relation of the collective shear waves
described via the telegrapher equation, let us briefly emphasize an
interesting outcome of this toy model (see Trachenko (2023) for a recent
preprint appeared during the completion of our manuscript and discussing a
similar idea). Let us assume that the wave-vector phase space of the shear
waves in a liquids is still given by a Debye sphere, as for phonons in solids.
Then, the density of states can be derived using the identity

$g(\omega)d\omega=V_{k}4\pi k^{2}dk,$ (21)

where $V_{k}$ is the volume in $k$-space. By neglecting the imaginary part of
the frequency, and identifying $\omega$ with $\mathrm{Re}(\omega)$, we can
immediately obtain the frequency dependent density of states which reads

$g(\omega)=\frac{2\pi
V_{k}\omega\sqrt{4\omega^{2}+\gamma^{2}}}{v^{3}}\qquad\text{where}\qquad\gamma\equiv
1/\tau_{g}\,.$ (22)

Figure 13: The vibrational density of states which can be naively derived from
a wave-like excitation whose dispersion follows from the telegrapher equation.
The VDOS displays a crossover between a liquid-like linear behavior at low
energy to a Debye law at large frequency. The crossover corresponds to the
scale at which the collective excitation passes from being overdamped to being
underdamped which approximately corresponds to the Frenkel frequency
$\omega_{F}\equiv 1/\tau_{g}$.

The behavior of this VDOS is shown in Figure 13. The VDOS displays a crossover
between a low frequency linear scaling $g(\omega)\propto\omega$ to a quadratic
Debye law at large frequency. Interestingly, the change in the behavior
coincides exactly with the crossover between the overdamped to underdamped
dynamics. Not surprisingly, when the wave is underdamped, the corresponding
VDOS is Debye-like as for solids. On the contrary, whenever the wave is
overdamped or even not-well defined (relaxational), the VDOS is liquid like.
At least in first approximation, this simple computation tells us that the
liquid-like linear behavior in liquids should disappear, and smoothly connect
with the quadratic Debye scaling, at a frequency scale which is approximately
of the order of the Frenkel frequency $\omega_{F}\equiv 1/\tau$. In other
words, for $\omega\gg 1/\tau$, even a liquid is expected to display a Debye-
like density of state $g(\omega)\propto\omega^{2}$. When the temperature is
high, then $1/\tau$ is very large and the Debye solid like scaling is likely
overwhelmed and dominated by other frequency modes. Nevertheless, when the
temperature is low, or like in our case when confinement makes $1/\tau$ small
enough, then one could possibly directly see the crossover between $\omega$
and $\omega^{2}$ in the low-frequency DOS of liquids. This is exactly what we
show in the main text in Figure 7 for the experimental data of confined water.
This is qualitatively in perfect agreement with k-gap theory and the original
Frenkel idea.

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