^{3}scaling of the vibrational density of states in quasi-2D nanoconfined solids

The vibrational properties of crystalline bulk materials are well described by Debye theory, which successfully predicts the quadratic ^{2} low-frequency scaling of the vibrational density of states. However, the analogous framework for nanoconfined materials with fewer degrees of freedom has been far less well explored. Using inelastic neutron scattering, we characterize the vibrational density of states of amorphous ice confined inside graphene oxide membranes and we observe a crossover from the Debye ^{2} scaling to an anomalous ^{3} behaviour upon reducing the confinement size ^{3} law results from the geometric constraints on the momentum phase space induced by confinement along one spatial direction. Finally, we predict that the Debye scaling reappears at a characteristic frequency _{×} =

A description of the vibrational properties of amorphous ice confined in graphene oxide membranes, as an exemplary nanoconfined material, is presented. Inelastic neutron scattering experiments and molecular dynamics simulations show anomalous deviations from standard bulk behavior.

These authors contributed equally: Yuanxi Yu, Chenxing Yang.

These authors jointly supervised this work: Matteo Baggioli, Liang Hong.

Describing the vibrational and thermodynamic properties of matter is one of the long-standing tasks of solid-state physics^{1}. In the case of crystalline bulk solids, the problem has been solved a long-time ago, in 1912, by Peter Debye with his celebrated model^{2}. Debye’s theory correctly predicts the quadratic ~ ^{2} low-frequency scaling of the vibrational density of states (VDOS) of bulk solids and the corresponding ~ ^{3} low-temperature scaling of their specific heat, in perfect agreement with many experimental observations^{1,3}.

The Debye model relies only on two fundamental assumptions: (I) the low-energy vibrational dynamics is governed by a set of propagating Goldstone modes^{4}, i.e. acoustic phonons, with linear dispersion relation _{D}, the Debye wavevector. The first assumption is notably violated in liquids in which the late-time (low energy) dynamics is dominated by diffusion^{5}, which induces a constant contribution to the density of states observed both in molecular dynamics simulations and in experiments. Additionally, in liquids, normal modes coexist at low energy with a large number of unstable instantaneous normal modes (INMs) which appear as negative eigenvalues of the Hessian or dynamical matrix and they reflect the presence of numerous saddle points in the highly anharmonic potential landscape^{6–8}. Consequently, even when the diffusive contribution is neglected, the density of states in liquids grows linearly with the frequency instead of quadratically as predicted by Debye theory^{9}. At the same time, the specific heat does not grow with temperature but rather decreases^{10}. Both these peculiar behaviours, which remarkably deviate from Debye’s theory, can be explained in terms of additional low energy degrees of freedom, the unstable INMs^{11,12}. In a nutshell, paraphrasing Stratt^{7}, the deviations from Debye’s theory in liquids are simply due to the fact that “liquids are not held together by springs," and strong anharmonicities play a dominant role. The first assumption of Debye’s theory is also notably violated in glasses, which present additional quasi-localized modes and exhibit well-known anomalies in the vibrational and thermodynamic properties with respect to their ordered counterparts^{13}.

Despite the known examples of amorphous systems (liquids and glasses), whether the Debye model, and indeed even the continuum theory of elasticity, work in ordered solids under strong spatial confinement is largely unknown. As we will show, geometric confinement could indeed lead to a violation of the assumption (II) presented above and a deviation from the Debye scaling even in ordered solids, without the need of having additional low energy modes as in liquids or glasses.

Nanometer confinement is ubiquitous in the frontiers of biotechnology, electronic engineering, and material sciences to achieve unprecedented advantage, e.g. applications of graphene or graphene-based materials for electron conduction^{14–16}, seawater desalination^{17,18} and biosensoring^{19,20}. Atomic vibrations in confined environments play a crucial role in a plethora of phenomena such as facilitating electron conduction in graphene-based electronic devices^{21,22}, enhancing proton delivery through the ion channel across the cell membrane^{23,24} and conducting energy via the power enzymatic motion of protein molecules^{25,26}. The effects of strong confinement have been investigated in nanoconfined liquids^{27–31} where liquid-to-solid transitions have been found^{32–34}. More importantly, the role of confinement (specially in nanopores geometries with confinement along two spatial directions)in the VDOS of glasses have been discussed in several works with particular attention to the fate of the boson peak anomaly and to the effects on the glassy relaxational dynamics. In the case of hard confinement, when the dynamics of the confining matrix is slower than that of the confined material, an ubiquitous reduction of the low-frequency part of the VDOS below the boson peak frequency has been observed experimentally in polymers^{35}, metallic glasses^{36}, glass-forming liquid salol^{37}, molecular glass former dibutyl phthalate/ferrocene^{38} and liquid crystals E7^{39}. In the alternative scenario of soft confinement, the behavior is opposite and the weight is shifted towards lower frequencies upon confinement, as shown experimentally in propylene glycol^{40}, discotic liquid crystals^{41}. In the context of amorphous solids, the difference between soft and hard confinement and the importance of the boundary conditions have been established in^{42}. As we will see, our findings are in qualitative agreement with those of ref. ^{42}. We refer the reader to ref. ^{43} for a comprehensive review of the influence of spatial confinement on the dynamics of glass-forming systems. Finally, a crossover between 2D and 3D Debye law has been observed experimentally in gold nanostructures^{44} and more recently in MD simulations data^{45}. Although the importance of reduced dimensionality to nanoscience has long been appreciated, most of the previous vibrational studies have focused on 0D or 1D confinement in nanopores and a qualitative deviation from Debye law at low frequency has never been observed. To the best of our knowledge, the effects of nano-confinement on the VDOS of solids confined in slab geometries have not been considered so far and they are indeed the topic of this manuscript.

In this work, we performed both inelastic neutron scattering and molecular dynamics simulations on quasi-2D nanoconfined crystalline and amorphous ice where the length scale of confinement can be well controlled between 7 Å and 20 Å. We found that the low-energy vibrational density of states of the confined solids exhibits a robust ~^{3} power law, faster than the expected ~^{2} Debye’s law (see Fig. ^{3} scaling and the standard Debye law appears at a characteristic wavevector _{×} = 2^{35–39,41–43,46}, claiming a low-energy suppression of the VDOS due to confinement, have been performed in nano-confined amorphous systems, this scaling and its theoretical foundations have been never mentioned nor discussed in the past.

The (low frequency) VDOS ^{2} scaling as predicted by Debye theory. Liquids, once the diffusive contribution is removed, shows a peculiar linear scaling due to the presence of unstable instantaneous normal modes. As demonstrated in this work, slab-confined solids, whether amorphous or ordered, exhibits a ^{3} scaling which is visible both in experiments and simulations and it can be explained analytically by considering simple geometric constraints on the wavevector phase space of low-energy phonon modes.

Using inelastic neutron scattering, we have measured the VDOS of water at 120 K (solid state) softly confined^{47} between graphene oxide membranes (GOM) with different levels of hydration. The detailed information on the sample preparation, the experimental methods and the data analysis are provided in the Methods.

As a reference, in Fig. ^{48}. At low frequency, approximately below 4 meV, we observe a leveling-off of the reduced density of states, ^{2}, and a well-defined Debye level, in agreement with the standard Debye model expectation, ^{2}. This is further supported by the VDOS of bulk crystalline and disordered ice samples derived from MD simulations (see next section). Hence, the Debye model is a valid description of the low-energy vibrational dynamics in bulk solids, as expected. In sharp contrast, in the sample with lowest hydration level (0.1 g water per gram GOM), where the distance between the neighboring GOM layers is only 7Å (Fig. ^{3} (Fig. ^{49}) and thus the corresponding data are not presented. In order to make sure that this anomalous scaling does not stem from the GOM structure itself, we measured the VDOS of the dry GOM without any water in between (shown in Fig. ^{3} scaling in hydrated GOM must result from the nano-confined water. Further X-ray diffraction measurements reveal that the water confined in GOM at this hydration level is amorphously packed as no characteristic Bragg peak of the crystalline ice is present (see Fig.

^{2} for high density amorphous (HDA) ice, low density amorphous ice (LDA) obtained from ref. ^{48} and bulk crystalline ice measured in this work at 120K. The horizontal dashed lines indicate the Debye level for each sample. The curves have been manually shifted vertically for better presentation and the vertical axes has arbitrary units. ^{2} for the hydrated GOM sample at ^{3} scaling due to confinement. The inset shows the original VDOS data in log–log scale. The

^{2} vs. frequency for bulk ice (green), GOM sample at ^{3} low frequency scaling. The blue data correspond to those reported in the panel (b) of Fig. ^{3} scaling and the standard quadratic Debye's law. The DSC data showing the mass proportion of bulk ice for the corresponding hydration levels (blue).

To further explore this ^{3} scaling found in the experimental data, we measured a series of samples with different levels of hydration. The results of the VDOS measurements on the various samples are shown in Fig. ^{50,51}. This picture is supported by the small-angle X-ray scattering results (see Supplementary Fig. ^{52,53}. This observation is further supported by the DSC measurement, where no first-order transition is observed at ^{3} scaling (assumed to be the same as the one discovered at ^{2}. Therefore, by fitting to a single power law, we obtain a smooth interpolation of the power from 3 to 2 by increasing ^{3}. Such a cubic power law has also been recently reported in ref. ^{54} for a 2D model glass system. Despite the tempting similarities, our setup is different with respect to that of ref. ^{54} in various aspects. First, our is a quasi-2D system, rather than a small 2D system, with excitations also in the vertical ^{3} power law is not an exclusive property of amorphous systems but it appears also in crystalline ones under strong confinement. All in all, the nature of the scaling discussed in this manuscript appears to be profoundly different with respect to the one reported in ref. ^{54}.

Before moving on, let us comment on another interesting outcome of the experimental analysis. The bulk sample at 120 K displays a sharp resonance peak in the reduced VDOS at ≈7 meV (Fig. ^{55–57}. By decreasing the level of hydration, and therefore increasing the strength of confinement, we observe a shift of this peak towards lower frequencies and a broadening of its linewidth. This observation suggests that the effects of confinement decrease the lifetime of this mode and renormalize the energy of the optical modes similarly to what disorder or anharmonicity would do. Indeed, one could argue that confinement is itself a source of anharmonicity. Interestingly, the observed red-shift of the spectrum induced by the soft confinement is consistent with the results of ref. ^{42}.

In order to confirm the universality of this power-law ~ ^{3} for nano-confined solids and to prove that, as expected, it is not a peculiar property of the confined amorphous state, we perform all-atom molecular dynamics (MD) simulations on bulk ice, slab ice, supercooled water and slab supercooled water at 120 K, where the thickness of the slab along the ^{2}. The Debye level is reasonably lower in the crystalline sample as compared to the bulk amorphous case because of a much larger value of the speed of sound. In the slab (confined) samples (Methods and ^{3} scaling shows up, consistently with the neutron scattering experimental findings. In summary, the analysis of the MD simulations data confirms the outcomes of the experiments and it also provides additional evidence for the universality of the phenomenon which does attain to any confined solid regardless of the crystalline or amorphous structure.

^{3} low-frequency scaling. The

We provide a concise analytic derivation of the ^{3} scaling observed for nano-confined solids both in the experimental data and MD simulations. Let us consider a three-dimensional system of linear size

In our system, however, the smooth hard-wall BCs do not apply because the confining boundaries are graphene oxide sheets, in which the basal plane is highly screwed and possesses different and spatially random-distributed oxide groups (hydroxyl, expoxy, and carboxy groups) (see the scanning electron microscope (SEM) image in Fig. ^{35,42}.

Going back to the main discussion, the number of states with wavenumber between _{k} = (2^{3}/^{3} is the ^{2} scaling in Eq. (^{1,3}. Here, we have considered a single sound mode and we have neglected the existence of different polarizations as in realistic solids. Given the additive property of the density of states

Let us now turn to a different situation in which one of the three bulk dimensions is confined but the atoms are still free to vibrate along it. This corresponds exactly to the setup considered in the experimental and simulation parts discussed above. We follow closely the framework presented in^{58}, and we shall work without hard-wall BCs consistent with the discussion above. In particular, let us consider a geometry which is rotational invariant in the (_{z} ≪ _{x}, _{y}. We use spherical coordinates, measuring the polar angle _{×} ≡ 2_{×}. Also, the effects of confinement are relevant only for wavevectors below the crossover scale _{×}. At higher wavevectors, the phase space is not affected since no minimum value nor angular dependence appears (see panel b of Fig.

_{z} ≪ _{x}, _{y}. The system is assumed to be rotational invariant in the 2-dimensional (_{×} defined in the main text. _{×} as a function of the inverse confinement length 2_{TA} ≈ 1300 m s^{−1}.

As a result of this geometric constraint on _{×}, the integration over the polar angles, which gave the 4_{×}. In particular, in that limit, the lower limit of integration tends to zero. Hence, we obtain that, below a certain threasold _{×}, the number of states with wavenumber in the range ∈ [

Equation (^{2} law which arises because of the geometric constraints on the momentum phase space induced by spatial confinement in real space. This analysis is in agreement with the experimental and simulations outcomes of the previous sections and it is able to explain with a simple argument the ^{3} universal scaling in solid systems under slab-confinement. Importantly, the same results would not be obtained for geometries with confinement along two spatial directions (cylinder) or three spatial directions (sphere) where, at least at low frequency, Debye’s law is expected to work, as also confirmed in refs. ^{35,37–39,41,43}.

Interestingly, our theoretical framework is also able to predict the crossover scale above which the Debye scaling re-appears. The latter is indeed given by _{×} = 2_{×} ≡ 2^{2}. In the bulk sample (yellow markers), a clear Debye level-off is visible at low frequencies. By decreasing the size of the confined region ^{3} scaling as reported in the previous sections both in experiments and simulations. Importantly, using the data from simulations we are able to track the frequency at which the scaling of the density of state changes from the cubic scaling to the more standard Debye one (indicated with a × in panel c of Fig. _{TA} ≈ 1300 m s^{−1} within a 14% error(see the inset of Fig. _{LA} ≈ 3900 m s^{−1}, an even better agreement with the value derived from Eq. (

In summary, the results from simulations confirm our theoretical framework and also prove that, despite the simplicity of the model, even its quantitative predictions are accurate.

In this work, we have reported the experimental observation of a low-frequency anomalous scaling in the vibrational density of states of nano-confined solids which violates the well-known Debye’s law for bulk solid systems. In particular, using inelastic neutron scattering experiments on amorphous ice at 120 K nano-confined inside graphene oxide membranes, we have observed a low-frequency ^{3} scaling law which substitutes the quadratic behaviour expected from Debye theory at low frequencies. This interesting experimental finding has been further confirmed by all-atom molecular dynamics simulations on confined ice in both crystalline and amorphous phases. Moreover, using a simple geometric analytical argument, a generalized law for the vibrational density of states of systems confined along one spatial direction has been derived. The appearance of this scaling is a consequence of the restricted wavevector phase space due to the geometric constraints imposed by confinement, while the nature of the low-energy vibrational modes does not change. Our picture is compatible with the idea that strong confinement produces a depletion of the low-energy part of the VDOS spectrum observed in several amorphous systems confined in nanopores^{38,39,41,42,46}, where nevertheless Debye’s law is still obeyed at low frequencies.

Furthermore, our theory predicts that the Debye quadratic scaling re-appears above a characteristic frequency given by _{×} = 2^{3} scaling but it also provides a good quantitative estimate of its frequency window. Finally, we stress that the nature of this scaling is not linked to the appearance of additional low-energy quasi-localized modes typical of amorphous systems as in^{54} but it results from the geometric effects of confinement on the phase space of acoustic phonons.

Our analysis provides a universal answer to the fundamental question of the vibrational properties of quasi-2D nanoconfined solids and it paves a new path towards the understanding and study of the mechanical properties of condensed matter systems under confinement^{59–61}. A direct consequence of this ^{3} scaling in VDOS is to shift the acoustic modes towards the higher energies. Thus, the phonon-assisted transportation of energy, electron or proton in various electronic devices and biological systems of nano-meter confinement will be inevitably carried out more by the high-energy short-wavelength phonon modes. More specific functional changes of materials due to this new mechanism are left to be discovered. They may be significant to several fields including nano-mechanical systems, transport phenomena at the nano-scale and nano-scale manipulation of biological systems.

The GOM sample was synthesized using the modified Hummers’ method^{62}. The GOM sample was first dehydrated by heating it from room temperature to 313 K and then annealed at this temperature for 12 h in a vacuum to the dry condition. The oxidation rate of the GOM sample is 28%, which is determined by XPS. The dehydrated sample was sealed in a desiccator and exposed to the water vapor to allow water molecules to adsorb to the surface and the interlamination of the GOM sheets. The hydration levels were controlled by adjusting the expose time of the sample and the final values of hydration levels were determined by measuring the weight before and after the water adsorption.

Differential scanning calorimetry (DSC) was used to measure the ratio between bulk ice and confined amorphous ice in GOM sample at low temperature. The DSC results of GOM at different hydration levels were performed by the DSC1 (METTLER TOLEDO). The samples were first annealed at 293

The powder X-ray diffraction data for GOM at different hydration levels were collected using a Rigaku Mini Flex600 X-ray diffractometer, with a Cu K^{∘}/min from 10^{∘} to 60^{∘}. The PXRD data were analyzed by MDI Jade software.

SAXS characterizations were employed to monitor the interlayer distance evolution in GOM with temperature decreasing and ice freezing. The SAXS measurements were carried out at the BL16B1 beamline of the Shanghai Synchrotron Radiation Facility (SSRF). The wavelength of the X-ray was 1.24 Å. The SAXS patterns were collected by using a Pilatus 2M detector with a resolution of 1475 pixels × 1679 pixels and a pixel size of 172

SEM images were taken by a MIRA 3 FE-SEM with a 5 kV accelerating voltage.

The dynamic neutron scattering is described in terms of the intermediate scattering function, _{inc}(_{coh}(_{j,inc} and _{j,coh} are the incoherent and coherent scattering lengths of a given atom _{j} is the coordination vector of that atom, the bracket <⋯> denotes an ensemble and orientation average and _{inc}(_{coh}(_{2}O are dominated by incoherent intermediate scattering function, _{inc}(_{coh}, becomes the static structure factor,

As the incoherent scattering cross-section of hydrogen is at least 1 order of magnitude larger than incoherent and coherent scattering cross-sections of other elements, the neutron signals are dominated by incoherent scattering function, and primarily reflect the self-motion of the water molecules. The experimental vibrational density of states (DOS) ^{63}:_{B} is the Boltzmann constant, and _{2}O with ^{49}) and the energy ranges up to 24.8 meV with the energy gain mode used in the ^{−1} to 4.5 Å^{−1}. The incident energy is 14.9 meV with the wavelength of 2.345 Å. The samples were contained inside aluminum foils in a solid form and sealed in aluminum sample cans in a helium atmosphere. The empty can signal was subtracted at each temperature. The detector efficiency in the data was normalized using a vanadium standard. All steps were performed with standard routines within the LAMP software package^{64} and the scripts are available upon request. The experiment for pure dry GOM was conducted by using a high-intensity Fermi-chopper spectrometer 4SEASONS at J-PARC in Japan^{65}. The measurement was done with multi-incident energies^{66} and the data with incident energy 27.1 meV was chosen to cover the energy range up to 17.2 meV and the ^{−1} to 7 Å^{−1}. The energy resolution at the elastic line is Δ^{67} and Mslice software packages.

The MD simulations were performed using LAMMPS^{68} to simulate ice and supercooled water at 120 K. The supercooled water is simply obtained by simulating bulk liquid water at room temperature and then cooling it down to 120 K. A temperature of 120 K is low enough to freeze out the translational degrees of freedom; therefore, one can consider the disorderly packed water at such temperature as amorphous ice. The equilibration of the MD systems was performed in constant temperature and constant pressure ensemble, using the Nosé-Hoover thermostat and Parrinello–Rahman barostat to control the temperature and pressure, and then switched to NVT ensemble to calculate dynamical properties. The timestep is set as 2 fs. The inter-molecule potential of H_{2}O used is TIP4P/2005^{69}, which shows good accuracy for ice and supercooled water^{70}. To reduce the finite-size effect, the simulation are performed using 360,000 and 216,000 molecules for the ordered and disordered structures respectively, with the configuration edge sizes of 180 and 200 Å. Both crystal and amorphous structures were equilibrated at 120 K at 1 atm. The slab structures were then cut from the bulk system to a thickness of 30 Å. We freeze the bottom and top layer (~3 Å) of the slab systems to force the structure to remain 2D confined during the whole simulation. In order to find the optimal volume of the ice slab, we performed a simulation for ice slab in the NPT ensemble. The position of the top and bottom layers are rescaled to new positions when the simulation box changes. A snapshot of the slab geometry can be found in Fig.

The VDOS is calculated by the Fourier transform of the oxygen velocity autocorrelation function:

The authors thank Reiner Zorn, Lijin Wang, and Jie Zhang for useful discussions. The authors thank Dr. Xiaran Miao from BL16B1 beamline of Shanghai Synchrotron Radiation Facility (SSRF) for help with synchrotron X-ray measurements. We also appreciate the assistance from the Instrumental Analysis Center of Shanghai Jiao Tong University for SEM, PXRD, DSC measurements. M.B. acknowledges the support of the Shanghai Municipal Science and Technology Major Project (Grant No.2019SHZDZX01). C.Y. acknowledges the support of the NSF China (11904224). This work was supported by NSF China (11504231, 31630002, and 22063007), the Innovation Program of Shanghai Municipal Education Commission and the FJIRSM&IUE Joint Research Fund (No. RHZX-2019-002). The neutron experiment at the Materials and Life Science Experimental Facility of the J-PARC was performed under a user program (Proposal No. 2020I0001). The beam time supported by ANSTO through the proposal number P7273.

Y.Y. performed the experimental measurements; M.B. and L.H. conceived the idea of this work, C.Y. implemented the MD simulations, L.Z. made the experimental sample, A.E.P., M.B., and A.Z. developed the theoretical model; R.K., M.N., and D.Y. helped with the inelastic neutron scattering experiments in the Japan and Australia, respectively; Y.Y. and C.Y. contributed equally to this work. M.B. and L.H. jointly supervised this work. All the authors contributed to the writing of the manuscript.

The datasets generated and analysed during the current study are available upon reasonable request by contacting the corresponding authors.

The code that supports the findings of this study is available upon reasonable request by contacting the corresponding authors.

The authors declare no competing interests.

^{−3}finite-size effects in the viscoelasticity of amorphous systems

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