Quantifying hydrogen bonding using electrically tunable nanoconfined water

Full form of the dipole-in-E-field approximation

The 3D form of Hooke’s law can be written as \(Hello=-sobat\cdot pencinta\). Here X is the displacement vector in 3D space, and κ is a 3 × 3 strain stiffness tensor, where the diagonal and off-diagonal indices are the stiffness in directions parallel and normal to each force components. The potential energy of a harmonic oscillator in 3D case is:

Set x-axis along D-H bond, then we have electrostatic force and sum potential stiffness: F(E) = − ∇U_HB and \(k\left(bersiaplah\right)=kemenangan^Yaslot-slot_konsep\). The field-dependent bond length and field-dependent spring constant are then:

In this work, we consider only the parallel response along D-H bond. Thus, among the nine indices of κ, only one diagonal term is non-zero. The system then simplifies to the 1D problem described in the main text.

The electric field in our model is defined as the effective electric field at the dipole position, generated by the hydrogen acceptor, A, with the modification from dipole-field interactions inherently accounted for. As the dipole moves within acceptor’s field, the interaction induces a charge redistribution in the acceptor, dynamically evolving the field and continuously affecting the HB strength. However, these effects do not require separate evaluation. Using the dipole-in-field model, the HB energy is directly determined as the product of the effective dipole moment and the effective electric field. Consequently, the modification of E due to dipole-field interactions is incorporated into our model by treating E in Eq. 2 as the effective electric field at the dipole position, where the contribution of p has been embedded. This effective E also corresponds to the Raman peak shift measured experimentally.

Device fabrication

Gypsum films were isolated from bulk natural crystals onto SiO₂ (290 nm)/Si and CaF₂ substrates (for spectroscopic measurements) and polydimethylsiloxane (PDMS) stamps (for heterostructure fabrication) using micromechanical exfoliation in cleanroom. Graphite/gypsum/graphite heterostructure is fabricated using all-dry viscoelastic stamping technique⁶⁴: first, graphite flake (bottom electrode) was isolated from natural graphite crystal onto SiO₂/Si substrate; gypsum flake and top graphite electrode were exfoliated onto PDMS stamp. The gypsum and top graphite flakes were then transferred onto bottom graphite electrode successively, aligning and stacking target flakes under the optical microscope using transfer rig.

Gypsum is prone to dehydration in vacuum or when exposed to elevated temperature, therefore, conventional lithography and metal deposition process cannot be used to fabricate electrical contacts to gypsum. Instead, we used two thick graphite flakes with over 100 μm length as extended electrical contacts to the top and bottom FLGs of the heterostructure, to which gold wires were then attached using conductive silver paint.

For devices with hBN separation layers, the hBN flakes were exfoliated and transferred using polypropylene carbonate (PPC) dry transfer technique⁶⁵: the hBN flakes exfoliated on SiO₂/Si wafer were picked up using PPC coated PDMS stamp at 45 °C, and then aligned and transferred onto desired stacks at 60 °C under the optical microscope using transfer rig. All fabrication processes were carried out in a cleanroom.

Vibrational spectroscopy measurements

Angular-resolved polarized Raman spectroscopy was performed using Oxford WITec alpha300R confocal Raman system, with 600 lines mm⁻¹ grating and a 532 nm wavelength laser with linear polarization. Laser power was kept below 8 mW. During measurements, the sample was fixed on the sample stage, with incident laser polarization rotated by a polarizer at a step of 5°. The spectra were taken with parallel analyzer configuration at each incident orientation, so the collected light had the same polarization direction as incident. For each spectrum, the acquisition time was 10 s with 6 accumulations.

Raman measurements under electric field in ambient conditions were carried out using Renishaw inVia confocal Raman system equipped with 2400 lines mm⁻¹ grating and 514 nm linearly polarized laser with power restricted below 8 mW, focused through a × 100/0.90 NA objective. The voltage was applied using a Keithley 2614B source meter.

Peak splitting is not very pronounced even at 0.3 V nm⁻¹ (e.g., Supplementary Fig. 8 for O-H_A stretching mode), but higher field is unreachable at ambient conditions due to field-induced degradation of FLG electrodes, even with hBN spacers. So, we performed Raman measurement at T E_ext of up to 0.5 V nm⁻¹ can be applied and distinct peak splitting is well resolved.

Raman measurements at cryogenic temperatures were performed on WITec alpha300 R Raman system with 600 lines mm⁻¹ grating and ×100/0.85 NA objective using 532 nm linearly polarized laser with power lower than 8 mW. The sample was placed in an Oxford microstat system in vacuum with temperature controlled by a liquid N₂ circulation loop for measurements at 80 K and liquid He for measurements at 10 K. The fluctuation of temperature was ±5 K. Due to the light absorption by the optical window on Oxford microstat system, the acquisition time was extended to between 30 and 60 min, depending on the signal to noise ratio on different devices, to increase the Raman intensity. This significantly increased the measurement duration, so no spatial mapping data was collected while the external electric fields were applied. To protect the samples from potential laser-induced damage, the laser power for all Raman measurements was maintained below 8 mW.

Fourier transform infrared (FTIR) spectra were recorded with a Bruker VERTEX 80 spectrometer equipped with a Bruker HYPERION 3000 FTIR microscope. For low temperature FTIR measurements at 83 K, the sample was placed in a Linkam stage with temperature controlled by a liquid N₂ circulation loop. All spectra were collected with 4 cm⁻¹ resolution and averaged over 512 scans.

Nano-FTIR measurements were performed on an AFM based scattering-type scanning near-field optical microscopy (s-SNOM) from neaspec GmbH. The conventional Pt/Ir probe (ARROW-NCPt-50, Nanoworld) with a tapping frequency of 270 kHz was illuminated with a p-polarized mid-infrared broadband difference frequency generation (DFG) laser (frequency range 1000–1900 cm^–1, average power 2/Si substrate. During measurements, a dc voltage was applied to bottom graphite electrode, while the top graphite and AFM probe were electrically grounded, creating an electric field along out-of-plane direction. Obtained nano-FTIR spectra were collected at different voltage levels by averaging 30 interferograms (spectral resolution ≈7 cm⁻¹) with 2048 pixels and an integration time of 10 ms/pixel. All spectra are normalized to that of a Si substrate.

Data analysis of Raman spectra

For Fig. 1f in the main text, we first performed background subtraction on raw spectra using 2nd polynomial with 256 sample points, followed by peak fitting using Voigt function. The peak position and peak width of water stretching modes were constrained within 3400–3412 cm⁻¹ and 20–50 cm⁻¹ for O-H_A, and 3490– 3500 cm⁻¹ and 20–70 cm⁻¹ for O-H_B. Raman spectra in Fig. 2 (main text) were obtained first through baseline fitting and subtraction using 4th polynomial with 256 sample points, followed by peak fitting using Voigt function. Initial peak positions were set at the position with maximum intensity. The peak widths were limited within 100 cm⁻¹ and 90 cm⁻¹ respectively.

Decoupling of the stretching frequencies of O-H bonds in crystalline water of gypsum

In gypsum, water O-H covalent bonds are stretched due to HB. Different crystalline environment of the two O-H bonds results in a distorted asymmetric water molecule. Such asymmetry decouples and localizes the normal symmetric and antisymmetric stretching onto individual O-H bonds. Consequently, one O-H bond will have much larger vibration amplitude than the other in each vibration mode.

We performed density functional theory (DFT) calculations using Quantum ESPRESSO package, to elaborate on the vibrational response of gypsum. Dispersion corrections that account for the van der Waals forces were incorporated following the empirical D3 formulation by Grimme, yield 0.06% difference with experimental lattice parameters. Calculations of phonon frequencies were carried out using the finite difference methodology implemented in Phonopy by displacing each atom 0.001 Å along three principal axes. Generalized Gradient Approximation (GGA) formulation by Perdew, Burke and Ernzerhof was used to model exchange-correlation interactions among electrons in all the calculations. An energy cutoff of 65 Ry with a uniform Monkhorst-pack k-mesh of 5 × 5 × 5 is used for all the calculations on the primitive unit cell. Convergence criteria used for all self-consistent field calculations are 10⁻¹⁰Ry per formula unit.

In gypsum crystal, water has 8 stretching modes in total, Supplementary Fig. 3. According to our DFT calculations, they are grouped into two branches, high- and low-frequency (near 3500 cm⁻¹ and 3400 cm⁻¹), with vibration amplitudes mainly localized on O-H_B and O-H_A bonds, respectively. In each of these two branches, the four levels arise due to the symmetry of gypsum crystal. In particular, four water molecules in a unit cell are grouped into two inversion symmetric pairs. Each of the two water molecules in the pair vibrate collectively either in-phase or out-of-phase, which decouples the modes. In turn, each in-phase and out-of-phase modes has symmetric and anti-symmetric intramolecular solutions, which leads to further mode decoupling.

Infrared spectroscopy of water O-H_A stretching modes in gypsum

Similar to Raman data presented in Fig. 2 in main text, peak broadening and splitting were also observed in low temperature FTIR measurements (Supplementary Fig. 1). The IR peak of O-H_A mode blue-shifted from 3401 cm⁻¹ to 3412 cm⁻¹ when temperature reduced from 300 to 80 K, possibly due to a volumetric contraction effect which shortens the effective O-H bond length⁶⁶. Similarly to Raman, the O-H_A stretching splits into two peaks, corresponds to the two O-H_A A modes in Supplementary Fig. 1b (A_g^A and A_u^A at zero field, Supplementary Fig. 3). The obtained electric field dependence of stretching peak position in FTIR spectra is similar to that of our Raman results.

Due to the large overlap between the two E_ext induced O-H_A peaks, we instead chose positions of maximal peak intensity for spectra fitting. At high E-fields, both O-H_A peaks show similar splitting with electric field. However, at low E fields, the low-frequency peak is nearly E-field independent. According to our DFT calculations, the two O-H_A peaks correspond to two A modes of O-H_A at 3406 and 3393 cm⁻¹, which are only Raman-active (A_g^A mode) and IR-active (A_u^A mode) at zero field, respectively. They are not the same mode and have different frequency. This accounts for the absence of A_g^A mode in IR and A_u^A mode in Raman at zero field, and the deviation of IR data from linear splitting at low E_ext-field (−1) in Supplementary Fig. 1c.

Water bending mode in gypsum

Due to extremely low Raman intensity, water bending mode and its field-dependence were investigated via nano-FTIR measurements using scattering-type scanning near-field optical microscopy (sSNOM). Water in gypsum exhibits two bending modes located around 1620 and 1680 cm⁻¹, Supplementary Fig. 4. This agrees well with our DFT calculations. The water bending mode also decouples into four sub-modes, because of four water molecules in a gypsum unit cell. Similar to the stretching vibration mode decoupling, inversion symmetry divides these four modes into two pairs, with frequencies around 1620 and 1680 cm⁻¹. We did not observe any E-field response for the bending modes, Supplementary Fig. 4. This is consistent with our DFT calculations, which show that bending mode frequencies do not change even at a high electric field up to 1.8 V nm⁻¹, because phonon dispersion around Γ point was unaffected by external electric field.

Fitting of k vs E relation in Fig. 3a and local fields at dipole O-H_A and O-H_B

The k vs E relation in Fig. 3a in the main text was obtained by accounting for the applied E_ext component with respect to O-H dipoles and accommodating the crystal structure symmetry of gypsum, using data from Fig. 2e, main text. To this end, the E_ext for each dataset was scaled by cos ϑ to obtain the electric field component along each O-H dipole. Here ϑ (as defined in Fig. 1b, main text) is 69.60° and 9.37° for O-H_A and O-H_B dipoles, respectively²⁰. Then, half of each dataset (those with positive slopes) were flipped horizontally, so that all the datasets having negative slope (red-shift). According to Eq. 4 in main text, this slope corresponds to \(-\fracadalahjagoannya\), where E is the total electric field at the dipole, equal to E_HB + E_ext. All datasets transformed by the above procedure were then fitted globally using Eq. 13 with three fitting parameters—one is the shared slope \(\fracbuatbawa\) for both O-H_A and O-H_B data because \(\fraccome backcuan.\) is expected to be the same for both dipoles, and two different E_HB for O-H_A and O-H_B due to the different local HB field:

To obtain the intercept k(0) in Eq. 4 in main text (and Eq. 13), we resorted to the known data on water in gas phase from literature. Vibrational spectroscopy of free water molecules is complicated: three vibrational modes—symmetric v₁ and antisymmetric v₃ stretching and bending mode v₂ together with numerous rotational modes result in hundreds of thousands of peaks of various intensities and linewidths^67,68,69,70. The most recent values of free H₂¹⁶O molecule’s vibrational transition frequencies from the W2020 database are 3657.053, 3755.929, and 1594.746 cm⁻¹, for v₁, v₃, and v₂, respectively²⁵. The vibration energy of uncoupled O − H bond v_O−H, estimated as the average of v₁ and v₃, is 3706.491 cm⁻¹, in good agreement with v_O−H = 3707.467 cm⁻¹ for HD¹⁶O molecule, where decoupling is provided by a large mismatch in effective masses of H and D⁷¹. This gives us the harmonic spring constant k(0) ≈ 762.0 ± 0.3 N m⁻¹ for uncoupled O − H bond in HB-free water. Linear fitting using Eq. 13 gives slope \(-\fracbuatmembawa\) = (2.23 ± 0.26) × 10⁻⁸C m⁻¹. The local E_HB-fields at O-H_A and O-H_B dipoles are (5.33 ± 0.63) and (3.82 ± 0.45) V nm⁻¹, respectively. The uncertainty is given by the 2σ value of the linear fit (reduced χ² = 2.3).

Thermal corrections for neutron structural data

Bond lengths from neutron diffraction are based on time-averaged neutron scattering intensity, giving an averaged interatomic separation. The actual bond length is slightly longer, after applying thermal corrections. The thermally corrected O-H bond lengths in Fig. 3b in the main text were obtained in different ways. For ice Ih, ice VIII, ice IX, Li₂SO₄·H₂O, Ba(ClO₃)₂·H₂O, and LiClO₄·3H₂O, their thermally corrected O-H bond lengths were provided in the respective references. For ice II, ice VI, ice VIII and ice XVII, the thermal corrections were evaluated using isotropic temperature factor, B. For CuSO₄·5H₂O and BaCl₂·2H₂O, thermal corrections were carried out using anisotropic temperature factors, β_ij. The temperature parameters of the systems are given in corresponding references respectively. The approaches used to evaluated the thermal corrections were reported in ref. ⁷².

Data analysis for bond length vs E/k relation in Fig. 3b

The various systems in Fig. 3b, main text contain ices and solid hydrates. The distance between H and metal ions (M) in most solid hydrates ranges from 1.5 to 3 Å. Here, we focused on pure hydrogen bonding interactions and analyzed the solid hydrates systems with H…M distance of 2–3 Å, ignoring materials such as BeSO₄·4H₂O, where the significantly shorter H…Be distance of 1.6 Å may affect the charge distribution on H via electrostatic interactions from Be cation. All O-H bond lengths are from thermally corrected neutron diffraction data. Where the O-H length was measured on deuterated water D₂O, the thermal-corrected bond length is further upshifted by 3% to account for the bond difference between H₂O and D₂O⁷³. For vibrational spectroscopy data on systems with multiple stretching peaks, we assigned peaks to O-H bonds with different bond lengths such that a shorter bond has a higher vibration frequency due to the steeper potential profile. The assignment process for each confined-water system is discussed below and data is summarized in Supplementary Table 1.

For ice Ih, one Raman peak is observed at 3279 cm⁻¹ for 3 mol% HOD ice at 123 K, corresponds to the stretching vibration of O-H bond, free of intra-molecule coupling due to large differences in the atomic mass of H and D⁷⁴. Two O-H bond length of (1.008 ± 0.002) Å and (1.004 ± 0.001) Å given by neutron diffraction at 123 K were thermally corrected by +0.008 Å and averaged to (1.014 ± 0.0011) Å⁷⁵.

For ice II, two strong Raman peaks were found at 3194 and 3314 cm⁻¹ at 77 K^76,77. Neutron diffraction of D₂O ice II gives four O-D bond lengths of (1.014 ± 0.020) Å, (0.975 ± 0.020) Å, (0.956 Å ± 0.020) and (0.937 ± 0.020) Å on two symmetrically non-degenerated water molecules at 110 K, with isotropic temperature parameter (B) of 4.5 Å⁻² for both O and D⁷⁸. The O-D bond lengths were thermal-corrected using B, and upshifted further by 3% for the D-H conversion. The final O-H bond lengths in ice II are (1.102 ± 0.021) Å, (1.064 ± 0.021) Å, (1.046 Å ± 0.020), and (1.028 ± 0.021) Å. The low frequency peak at 3194 cm⁻¹ is assigned to the local stretching of the longest O-H bond of 1.102 Å. The stretching vibration can be coupled both intra- and intermolecularly among the remaining three O-H bonds due to their close bond lengths, resulting in uncertainty in vibration frequency assignment. Hence, we excluded these three data points and only focused on the longest bond.

Ice VI contains two types of O₄ octahedra structure interpenetrating into each other, revealed by the neutron diffraction of D₂O ice⁷⁹. For the first O₄ octahedra network, all proton positions are equivalent, yielding one O-D bond length (0.986 ± 0.048) Å, indicating a strong intramolecular coupling in stretching vibration. The second O₄ octahedra structure has three O-D bond lengths at (0.937 ± 0.055) Å, (0.976 ± 0.051) Å, and (0.939 ± 0.032) Å with multiplicity 1, 1, and 2, yielding an average of (0.951 ± 0.027) Å. These two bonds are thermal-corrected to (0.996 ± 0.048) Å and (0.961 ± 0.027) Å using B = 2.99 Å⁻² for oxygen and 3.80 for hydrogen⁷⁹, and further upshifted to (1.026 ± 0.049) Å and (0.993 ± 0.028) Å for the D-H conversion. These two bonds are associated with Raman stretching modes at 3390 and 3329 cm⁻¹⁸⁰.

For ice VIII, only one bond length is given at (0.969 ± 0.007) Å in D₂O ice at 10 K⁸¹, which is thermal-corrected to (0.981 ± 0.007) Å using B = 0.58 Å⁻² for O and 1.54 for D, and further upshifted to (1.010 ± 0.009) Å because of D-H conversion. The O-H Raman vibration frequency is taken as the average of three intramolecular-coupled normal modes locating at 3477.4 cm⁻¹ (ν₁B_1g), 3447.6 cm⁻¹ (ν₃E_g) and 3358.8 cm⁻¹ (ν₁A_1g) at 100 K⁸², which is 3427.9 cm⁻¹.

Ice IX is a proton-ordered phase. Neutron diffraction data on D₂O ice shows that protons are preferably arranged in one configuration (referred to as ‘major site’), with a nonzero occupancy on the other configuration (referred to as ‘minor site’). The O-D lengths for major and minor configurations are (0.976 ± 0.003) Å and (1.02 ± 0.06) Å. Thermal-corrected length is reported for only the major site at (0.983 ± 0.004) Å⁸³. This value is further upshifted to (1.012 ± 0.004) Å for D-H conversion. Because of the low occupancy, we exclude the minor configuration bond length and only use that of the major configuration. Two intense Raman peaks of O-H stretching are found at 3280.7 cm⁻¹ and 3160.4 cm⁻¹ at 100 K⁸⁴, with the latter assigned to the vibration of O-H bond on major sites.

For ice XVII, two O-H stretching Raman peaks at 3106.3 cm⁻¹ and 3231.8 cm⁻¹ are reported in ref. ⁸⁵. These are regarded as decoupled vibration because their separation is >100 cm⁻¹. Two O-D bond lengths are given by neutron diffraction on D₂O ice at (1.023 ± 0.007) Å and (1.006 ± 0.004) Å, which are thermal-corrected to (1.044 ± 0.007) Å and (1.018 ± 0.004) Å using B = 2.37 Å⁻² for O, 4.05 Å⁻² for D1 and 3.34 Å⁻² for D2, respectively⁸⁶. These values are further upshifted to (1.075 ± 0.007) Å and (1.049 ± 0.004) Å for D-H conversion.

For CaSO₄·2H₂O (gypsum), the Raman frequency of O-H stretching is taken from this work, which is 3406 cm⁻¹ and 3484 cm⁻¹, measured at room temperature. The O-H bond lengths are taken from ref. ²⁰, (0.961 ± 0.006) Å and (0.948 ± 0.004) Å at 320 K, which are thermal-corrected to (0.984 ± 0.006) Å and (0.966 ± 0.004) Å using B = 1.749 Å⁻² for O, 3.520 Å⁻² for D1 and 3.074 Å⁻² for D2, respectively. These values are further upshifted to (1.014 ± 0.006) Å and (0.995 ± 0.004) Å for D-H conversion.

For Li₂SO₄·H₂O, two thermal-corrected O-H bond lengths are (1.004 ± 0.003) Å and (0.997 ± 0.002) Å⁸⁷. We associate these two bonds with two Raman peaks of O-H stretching at 3439 cm⁻¹ and 3480 cm⁻¹, respectively⁸⁸.

CuSO₄·5H₂O contains five water molecules per unit cell, among which four are coordinated to Cu²⁺, and the remaining one is far away from Cu²⁺ thus non-coordinated. To exclude the interactions with metal ions, we only focus on the non-coordinated water molecules in this study. CuSO₄·5H₂O exhibits multiple water stretching peaks, and the modes above 3300 cm⁻¹ is assigned to the stretching of non-coordinated water molecules, at 3360 cm⁻¹ and 3477 cm^−189,90. The O-H bond lengths of the non-coordinated water molecules are (0.978 ± 0.005) Å and (0.936 ± 0.011) Å⁹¹. These values are corrected for thermal motion using anisotropic temperature parameters tensor (β_ij) to (1.005 ± 0.005) Å and (0.951 ± 0.011) Å. These two bonds are associated with 3360 cm⁻¹ and 3477 cm⁻¹ Raman peaks, respectively.

For Ba(ClO₃)₂·H₂O, a single thermal-corrected O-H bond length is reported at (0.958 ± 0.011) Å⁹², suggesting an intramolecularly coupled vibration. Thus the vibration frequency is taken as the average between ν₁ (3512 cm⁻¹) and ν₃ (3582 cm⁻¹) vibration modes at 90 K⁹³, that is 3547 cm⁻¹.

For LiClO₄·3H₂O, the O-H bond length is taken as the average among three refinement series, (0.993 ± 0.004) Å, with uncertainty coming from the standard deviation of the three series⁹⁴. The values provided in the reference have been thermal corrected during refinement. This bond length is associated with a sharp Raman peak in water stretching region, 3553 cm⁻¹⁹⁵.

For BaCl₂·2H₂O, four O-H bond lengths are given by neutron diffraction measurements at (0.953 ± 0.003) Å, (0.959 ± 0.003) Å, (0.960 ± 0.003) Å, and (0.972 ± 0.004) Å⁹⁶. These values are thermal corrected to (1.0359 ± 0.003) Å, (1.0364 ± 0.003) Å, (1.0717 ± 0.003) Å and (1.077 ± 0.004) Å using β_ij. and assigned to four intermolecularly uncoupled Raman modes at 3456, 3352, 3318, and 3303 cm⁻¹⁹⁷.

The assignment and references are given in Supplementary Table 1, with all the data points plotted in Fig. 3b in the main text. The data points above were then fitted using a linear function with both X and Y errors, yielding a slope of (7.83 ± 1.26) × 10⁻¹⁹C and intercept of (0.957 ± 0.001) Å, corresponding to \(\fracdi siniGames\) and \(“slot gaco”\left(0\right)\) in Eq. 3 in the main text. The uncertainty corresponds to 2σ. The reduced χ² for this fitting is 4.96.

Dipole moments vs O-H bond lengths in Fig. 3c

In the gas phase, equilibrium dipole moment of H₂O molecule is 1.855 ± 0.001 D, given by Stark effect measurements²⁹. The H − O − H angle in gas phase is 104.48° and the O − H bond length is 0.9578 Å, as determined by fitting experimental rotational energy levels with the rotation-vibration Hamiltonian²⁶. This gives a dipole moment of the O − H bond p_OH = 1.514 ± 0.001 D.

Effective dipole moment of liquid water at ambient conditions is 2.9 ± 0.6 D, estimated from 0.5e (±20%) charge transfer along each O − H bond, using synchrotron x-ray diffraction measurements³². The H − O − H angle and the O − H bond length are 103.924° and 0.984 Å, obtained from a joint refinement of X-ray and neutron diffraction data⁹⁸. No iso- or anisotropic temperature parameters available, so no correction was made to the O-H bond length. This gives dipole moment of O-H bond p_OH = 2.4 ± 0.5 D.

Ice Ih has a dipole moment of 3.09 ± 0.04 D, calculated using multipole iterative method, with error extracted from Fig. 1 in the ref. ³³ The O-H bond length and H-O-H angle are 1.014 Å and 109.21° from neutron diffraction data at 60 K. The bond length is further thermal corrected to 1.0125 Å⁷⁵. The p_OH is calculated to be 2.67 ± 0.04 D.

Polarizability and dielectric behavior of confined water

We consider a dielectric with ε_r = ε/ε₀ consisting of polarizable dipoles with the polarizability along the field direction α_c,i and the number density of dipoles N_i. The polarization under the electric field E is \(P=(saat_sekarang-1)satu-satunya_beradaE\), which equals to the sum of the polarization from all dipoles, \(P=hanya di_yangmemberikan_{Kalau,belum}EDaftar_dengan\), here i denotes different types of dipoles. This leads to Eq. 8 in the main text, the dielectric constant of a confined water system.

From Eqs. 1, 2 and 6 in the main text, polarizability can be written in terms of total electric field E along O-H bond:

Taking gypsum as an example of nano-confined water system, the polarizability of the four types of O-H dipoles in gypsum are doubly degenerated and dictated by the local E_HB field, 5.33 and 3.82 V nm⁻¹ for O-H_A and O-H_B, respectively. In the presence of external electric field, the four dipoles have four different local fields, and thus the structure symmetry is broken, resulting in two polarizabilities splitting towards two directions (Supplementary Fig. 5c). By adding up the four field-dependent polarizabilities according to Eqs. 7 and 8 in the main text, dielectric constant ε_r = 6. 6 ± 0.7 was obtained for water in gypsum. The calculated field-tunable ε_r(E) is plotted in Supplementary Fig. 5d, which presents a modulation of

To observe the field-tunable ε_r(E) in gypsum, we performed capacitance measurements in one of our devices, using capacitance bridge (Supplementary Fig. 5a). In the measurement, a graphite/gypsum/graphite capacitor fabricated on SiO₂ (290 nm)/Si substrate was placed in vacuum (−6 mbar). The capacitance of the device was measured using AH 2700 capacitance bridge with an ac excitation of 1 V at 1.2 kHz.

The equivalent circuit is sketched in Supplementary Fig. 5a. The experimental capacitance measured by the bridge is a combination of four capacitors with three contributions: capacitance of the gypsum (C_g), quantum capacitance of top and bottom graphite electrodes (C_q), and parasitic capacitance from the measurement setup (C_p). From capacitors model we can fit the experimental curve using:

Field-independent C_p is a fitting parameter with constant value. ε_r, which is embedded in C_g, is another fitting parameter. 1/C_q1 and 1/C_q2 are function of carrier density on graphite electrodes and are given by ref. ⁹⁹, and we obtain C_p ≈ 319 fF and ε_r ≈ 3.68. With C_p subtracted from the experimental curve, the modulation of capacitance is ~0.9%, much higher than the 0.1% modulation expected for field-tunable polarizability of confined water. Thus, it is concluded that the effect of field-tunable polarizability of water molecules is hidden in the graphite quantum capacitance.

Gypsum is a HBH consisting of confined water layers and CaSO₄ layers, which can be modeled as two capacitors in series. The total ε_r of gypsum has been determined above. The capacitance of confined water layers can be evaluated using number density of confined water molecules and Eq. 8 in the main text. Gypsum (CaSO₄·2H₂O) has molecular weight of 172.17 g·mol⁻¹ and mass density 2.32 g·cm⁻³¹⁰⁰, yielding a number density of 8.11 × 10²⁷m⁻³ of gypsum, and 1.62 × 10²⁸m⁻³ of H₂O. Considering that the water molecules are confined between CaSO₄ layers, with confining space taking up 40% of the gypsum volume (confining length measured between O_S and O_S’, the oxygen atoms in SO₄ groups from adjacent layers), the number density of water molecules in the confining space is corrected to 4.06 × 10²⁸ m⁻³. Equation 8 in the main text produces ε_r = 6.6 ± 0.7 for confined 2D water layers in gypsum. Using capacitor-in-series model, ε_r = 2.8 ± 0.5 can be obtained for CaSO₄ molecular sheets. This value matches well the reported ε_r for anhydrite (CaSO₄), which is 2.45 ± 0.52 for γ-phase and 2.70 ± 0.02 for β-phase⁴³.

Our model finds that water sheets confined in gypsum presents a lower ε_r compared to its bulk form due to the restricted rotational freedom. Such behavior is similar to water confined in carbon nanotubes (CNT). Simulations has shown that in CNT-confined water, axial component of ε_r increases to above 100 with a relaxation time longer than bulk water, while the ε_r of water normal to the carbon walls decreases drastically (to ~6), with the dipolar relaxation time severely suppressed^101,102. The small normal component of ε_r is attributed to the restricted dipole rotation, caused by the strong orientation of water molecules near walls due to induced surface charge¹⁰³. A similar decrease in ε_r was also found at the water—C₆₀ fullerenes interface, which is attributed to the electrostatic interaction between water and surface charge on C₆₀^104,105.

Dielectric constant of liquid water

We consider the dielectric response of liquid water coming from two parts: dipole reorientation, as well as atomic and electronic polarization. The former is given by Langevin-Debye equation¹⁰⁶:

where N₀ and p are the number density and dipole moment of the H₂O molecule, k is Boltzmann constant, T the temperature and E is the electric field. The number density of H₂O molecule is 3.35 × 10²⁸ m⁻³, using the mass density of liquid water. The latter, atomic and electronic polarization, is given by considering individual O-H dipole moments of number density 2N₀ and dipole moment p, with orientation given by Boltzmann distribution of p in external E-field. The induced dipole moment per O-H bond is αE(cos θ)², where θ is the angle between p and E (defined in Fig. 1b, main text). Integration over all 3D space gives the polarization density of induced dipole:

with y = cos θ, and x = pE/kT. Using Taylor expansion on the integral term and keeping to quadratic term, Eq. 17 reduces to

Adding up the two contributions and using \(P=(terbesar_SEGERA-1)hanya_{0}E\), we obtain dielectric constant of liquid water:

where p and α are be given by our model from Eqs. 6 and 7 in the main text.

Although our model is derived from individual water molecules confined in gypsum, it successfully reproduced the dipole moment of liquid water, 2.66 D, where water dipoles are free to rotate and interact with each other. Using this result and Eq. 7 in the main text, a static dielectric constant ε_r = 27 ± 2 was obtained for liquid water. This is significantly lower than experimental value ε_r = 79 due to the complex hydrogen bonding network of water clusters in liquid water, which enhances the interaction between water molecules and thus alters the cooperativity of water molecules with HBs and affects the dielectric response. A conventional approach to include such correlation effect is to multiply p²/kT term in Eq. 19 with a dipole correlation factor G_k⁴⁴. To get experimental ε_r = 79, G_k of 3.15 should be taken, in the range of reported G_k, from 2.72 to 3.70 which depends on the different models used for the simulation⁴⁵. With bond length taken as 0.984 Å⁹⁸, our model yields a dielectric constant value of 78.7 ± 7.2 at E = 0 V nm^-1 and 78.8 ± 7.2 at E = 0.5 V nm⁻¹, with the electronic polarization contributes to only 3% of the dipole reorientation.

From the calculations above, it is seen that in bulk water, dipole reorientation contributes over 97% to the dielectric response. Previous studies have shown that in confined liquid water, particularly when the confinement is less than 2 nm, the perpendicular dielectric constant drastically decreases to below 10^41,42,49. This significant reduction is a result of restricted rotational freedom. To quantify the contribution of electronic and atomic polarization, which is highly anisotropic and dependent on the water molecule orientation, we calculated the dielectric constant resulting from this effect only. As illustrated in Supplementary Fig. 6, the dielectric constant has a maximum of 13 from non-reorientation polarization. Given the experimental ε_r = 79, dipole reorientation consistently contributes more than 84% of the dielectric response.

Consequently, in scenarios where confinement restricts dipole reorientation, and the dielectric response is entirely due to electronic and atomic polarization, water consistently exhibits a low ε_r.

Water-carbon interface

In the case of single water molecules trapped inside a C₆₀ fullerene, both the symmetric v₁ and antisymmetric v₃ stretching modes are significantly red-shifted compared to free H₂O, to 3573 cm⁻¹ and 3659.6 cm⁻¹, respectively. In a harmonic approximation, this corresponds to a force constant k ≈ 725 N m⁻¹. Using Eqs. 3 and 4 in the main text, we obtain a local E-field E_HB = 1.7 ± 0.2 V nm⁻¹, O-H bond length d_OH = 0.974 ± 0.004 Å, and dipole moment (using Eq. 6 in the main text) p_H2O = 2.37 ± 0.14 D. With our dipole-in-E-field model, the calculated interaction strength U_HB = 1.5 ± 0.2 kcal mol⁻¹ per bond (66 ± 9 meV), which is comparable to typical weak HBs. Similar values are obtained for water confined in single-walled carbon nanotubes, with stretching frequency at 3569 and 3640 cm⁻¹, yielding k ≈ 720 N m⁻¹⁵³. However, this picture is likely complicated by the interplay between water-water intermolecular HBs and water-carbon π HBs.

At the water-graphene interface, water molecules exhibit a peak around 3600 cm⁻¹, which was assigned to the stretching vibration of the ‘dangling’ O-H dipole pointing towards the graphene film⁵⁴. The reported vibration frequency of ν_OH = 3616 cm⁻¹ for the dangling O-H dipole, using our approach, corresponds to k = 725.1 N m⁻¹, E_HB = 1.7 ± 0.2 V nm⁻¹, d_OH = 0.975 ± 0.004 Å, p_H2O = 2.37 ± 0.14 D, and U_HB = 1.54 ± 0.20 kcal mol⁻¹ per bond (67 ± 9 meV). Our model predicts the electric field exerted by the π electrons on the water O-H dipoles is around 1.7 V nm⁻¹. Using only the stretching frequency of confined water, our results directly reproduce key properties of confined water, including HB strength, local E-field, O-H bond length, and dipole moment.

This peak can also be tuned by applying an external electric field in the electric double layer region near graphene surface. In the vicinity of graphene, water O-H dipoles experience an E-field exerted from π electrons of carbon atoms. This interaction can also be considered a π-hydrogen bond (π-HB), which redshifts the stretching frequency of the O-H dipole compared to that of a free water molecule. In the presence of E_ext, the interaction is tuned, which can be seen in the shift of the stretching frequency of dangling O-H bond on the graphene surface.

Reference ⁵⁴ reported a sum frequency vibrational spectroscopy (SFVS) study on the water-graphene interface. As an independent validation of our model, we used the data points from Fig. 4b in ref. ⁵⁴. The vibration frequency of dangling O-H peak on y-axis is converted to force constant k using harmonic approximation. The interfacial electric field E_s on x-axis corresponds to the E_ext in our study. E_tot for each point is obtained using Eq. 5 in main text, with E_HB given by Eq. 4 (main text) using k at E_ext = 0, taken from Fig. 4b in ref. ⁵⁴. Since the dangling O-H dipole points towards graphene film, cos ϑ in main text Eq. (5) is taken as 1. The obtained results match perfectly into our model, as plotted in Supplementary Fig. 7.

Dielectric constants of water ice systems

Due to the presence of defects in proton-disordered ices which cause dipole reorientation and, thus, an anomalously large ε_r, we chose the proton-ordered ices to study the static ε_r. Hence, only atomic and electronic polarization contributions are considered. Ice XI is proton-ordered form of ice Ih, with ε_r = 2.9 taken from Fig. 6 in ref. ¹⁰⁷. The authors cooled down the ordinary ice (ice Ih) and measured ε_r at 77 K, where the ice Ih–XI phase transition is expected to occur. The observed small and non-dispersive ε_r indicates a proton-ordered form of ordinary ice. The overall static ε_r of proton-ordered ice II is around 4.2, measured from Fig. 6 in ref. ⁴⁷, although a small increase was observed with frequency (not evaluated). Ice IX is proton-ordered form of ice III. The ε_r of ice IX is taken as 4 according to Fig. 3 in ref. ¹⁰⁸, where static ε_r presented a sharp drop during the ice III–IX phase transition.

We examined the static dielectric constant along [001], [111] and [010] directions for these three phases of ices, respectively, axes direction defined in refs. ^109,110,111. The three confined water systems possess 2, 4, and 6 types of non-equivalent O-H dipoles with respect to the assessing axis, with polarizability marked in Fig. 3d in main text. Using the neutron diffraction data of O-H bond lengths reported in refs. ^75,78,112 (the bond length of ice III is thermal corrected to 1.005 Å following the approach in Methods, ‘Thermal corrections for neutron structural data’) and density reported in refs. ^110,113, the dielectric constants calculated using Eq. 8 in the main text, are 3.8 ± 0.5, 5.9 ± 0.8, and 4.7 ± 0.4 for structure of ice XI, II and IX, respectively. Our results show reasonable agreement with the reported ε_r for ice XI, II, and IX, which are 2.9¹⁰⁷, 4.2⁴⁷ and 4¹⁰⁸.

All the estimated ε_r for water systems are summarized in Supplementary Table 2.