Introduction

Water’s surface tension underpins a wide range of physical, chemical, and biological phenomena1,2,3,4,5. It enables insects like water striders to walk on water6,7, shapes liquid droplets8,9, stabilises liquid bridges between solids10, and drives capillary flow in confined spaces such as plant vessels11,12,13. By tuning surface tension, it is possible to control bubble formation in gas-evolving reactions14,15 and manipulate fluid motion via the Bénard–Marangoni effect, where temperature gradients create surface tension gradients that drive flow16.

The physical origin of surface tension is the stress anisotropy created near the liquid surface2,17,18. At the interface, cohesive forces on molecules are unbalanced due to the reduced number of neighbouring molecules in the vapour phase compared to the liquid. This imbalance creates a tensile force that pulls molecules inward to minimise interfacial area. The magnitude of surface tension γ depends on the strength of intermolecular interactions. At ambient conditions, water has a relatively high surface tension of γ ≈ 72 mN/m19,20, significantly higher than most organic liquids, such as benzene (γ ≈ 28 mN/m)21. The high surface tension of water arises from the significant energy cost associated with disrupting the hydrogen-bonding network when a surface is formed. Because hydrogen bonds are substantially stronger than typical van der Waals or dispersion interactions, the loss of even a small number of hydrogen bonds at the air–water interface contributes disproportionately to the surface energy density. This explains why water exhibits a much higher surface tension compared with most organic liquids.

While most simple liquids exhibit a nearly linear increase in surface tension with decreasing temperature3, water deviates markedly from this trend. Experiments and simulations show that the rate of increase in γ gradually slows below ~275 K (see, e.g., ref. 22). More notably, in the deeply supercooled regime (T ≲ 250K), simulations have revealed the emergence of a second inflection point, characterised by an accelerated rise in γ upon further cooling23,24,25. Experimental evidence for this anomaly remains scarce due to the difficulty of probing water at such low temperatures. However, recent high-precision measurements by Vinš et al.26, using a modified capillary rise technique down to 241.8 K, report subtle but systematic deviations from the smooth temperature dependence observed at higher temperatures, suggesting the onset of interfacial anomalies in this regime.

This reentrant, nonlinear behaviour highlights that water exhibits fundamentally distinct interfacial properties from those of simple liquids at low temperatures. Previous studies have proposed diverse explanations, such as stronger and longer-lived hydrogen bonds25, preferential adsorption of high-density liquid (HDL)24, and a minimum in surface excess entropy linked to compact interfacial structuring23. While these interpretations offer important insights into interfacial thermodynamics and kinetics, they do not directly link molecular-scale structural transformations to the local stress anisotropy that defines surface tension, nor do they clarify how these features evolve spatially across the interfacial region. Moreover, previous interpretations have overlooked an intermediate temperature regime where the increase of γ with decreasing T is slower than in the high-T linear regime.

In contrast, our approach is grounded in the two-state model27,28,29, which describes water as a mixture of locally favoured tetrahedral (S-) and disordered (ρ-) state molecules. We show that the anomalous temperature dependence of surface tension arises from a structural competition between surface-induced OH/dipolar ordering—more easily adopted by the ρ-state—and the formation of tetrahedral S-state—stabilised in the subsurface region by negative pressure due to its larger specific volume. This competition governs interfacial stress anisotropy, the microscopic origin of surface tension.

In this work, using atomistic molecular dynamics simulations, we show that water’s surface tension is governed by the interplay of ρ- and S-state anisotropies. At intermediate temperatures, the ρ-state contribution saturates while interfacial S-states remain largely orientationally disordered and thus weakly anisotropic, moderating the growth of γ. Upon deeper supercooling, however, dipole–dipole interactions drive orientational ordering of S-states near the interface, amplifying anisotropy and producing the reentrant rise in γ. This surface-induced S-ordering also promotes the formation of Ice-0-like motifs with five- and seven-membered rings, thereby facilitating ice nucleation at air–water interfaces30. Our findings thus provide a unified microscopic framework for understanding the nonlinear temperature dependence of water’s surface tension, rooted in two-state structural competition. More broadly, this approach offers general insight into interfacial anomalies and phase transitions in complex liquids exhibiting local structural ordering31,32, with important implications for crystal nucleation and related interfacial phenomena.

Results

Nonlinear T-dependence of water’s surface tension

Our system consists of TIP4P/2005 water molecules33 forming a water film within a three-dimensional periodic box (Fig. 1a; see “Methods” section for additional details). We examine the T-dependence of γ. The TIP4P/2005 water model33 accurately produce γ across a wide temperature range, from T = 220 K to T = 550 K Fig. 1a, aligning well with prior simulation results17,25. Although minor deviations from experimental data remain, the TIP4P/2005 model reproduces the overall trend and magnitude of experimental γ(T)19,20,22, including the inflection behaviour, with satisfactory accuracy. Notably, a clear deviation from the linear trend in γ appears around 275 K, where its temperature dependence becomes nonlinear. Moreover, we reproduce the characteristic more rapid growth in γ for T ≲ 250 K (Fig. 1a), as reported in previous experimental26 and simulation studies23,24,25. This anomalous, nonlinear behaviour of γ contrasts with that of simple liquids such as Lennard-Jones model fluids34.

Fig. 1: Temperature T-dependence of surface tension γ and two-state features of interfacial water.
figure 1

a The surface tension γ(T) from our MD simulations (black) compared with experiment (purple)20,22 and previous simulations25 (green). The calculated γ (black) is decomposed into S-state (red) and ρ-state (blue) contributions in the corresponding simulations at various temperatures. Unlike simple liquids, the surface tension of water exhibits a distinctive temperature dependence: it first shows a slowdown in its rate of increase (light orange zone), followed by a reentrant rise in the deeply supercooled regime (light green zone). Error bars denote standard errors obtained from 200 ~ 400 ns simulations. Error bars are smaller than the symbols. Inset: simulation setup of a water film with the z-axis normal to the interface. b Fraction s of tetrahedral S-state water as a function of T, identified via θavg35,36. This matches the empirical relation s = 1 − gOO(rH), where gOO is the oxygen-oxygen radial distribution function at rH = 3.5 Å37,38. The interfacial region of the water film (brown line; z = 14–22 Å) shows reduced s compared to the bulk, with a sharp rise near T ≈ 250 K. Insets show a typical S-state water tetrahedral structure. c Bimodal distribution P(ζ, θavg) for parameters ζ37 and θavg35,36 for interfacial water (z = 14–22 Å) at T = 300 K.

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Distinct structural states of interfacial water

Hereafter, we analyse the surface tension anomaly based on the two-state model. As T decreases, many properties of water deviate from the normal behaviour of simple liquids39. These deviations, known as water’s anomalies, are widely attributed to the formation of locally favoured tetrahedral structures (LFTS) stabilised by hydrogen bonding27,37,40,41,42,43,44,45,46,47. The competition between two structural forms, LFTS (S-state; see the inset of Fig. 1b) and a disordered structure (ρ-state), has been incorporated into the two-state model27,28,29, providing physical explanations for the origin of water’s anomalies.

We classify water molecules into S-state and ρ-state using the translational order parameter ζ and orientational order parameter θavg. Here, ζ is defined as the difference between the minimum distance to a non-hydrogen-bonded neighbour and the maximum distance to a hydrogen-bonded neighbour37, while θavg represents the average angle formed between an oxygen atom and its neighbouring oxygens35,36. The parameter θavg captures orientational order within the first coordination shell35,36, while ζ reflects translational order in the second coordination shell37. Water is considered hydrogen-bonded if the donor (D)-acceptor (A) distance is less than 3.5 Å and the H-D-A angle is less than 30∘ (Luzar-Chandler parameters)48.

To demonstrate the presence of two distinct structural states in interfacial water, we analyse the bivariate distribution P(ζ, θavg)38 for water molecules located at z = 14 −22 Å [Fig. 1c], where z denotes the distance from the centre of the water film. Consistent with findings in bulk TIP4P/2005 water38, we observe two distinct peaks separated by a pronounced valley [Fig. 1c for T = 300 K; also see Fig. S1 for T = 230 − 400 K], highlighting the two-state character of interfacial water. The peak near ζ ≈ 0.5 Å and θavg ≈ 109∘ corresponds to the S-state, while the other peak signifies the ρ-state. We remark that the probability P(ζ, θavg) is averaged over all water molecules and time frames. Our earlier work43,49 further analysed the rapid exchange dynamics between the ρ- and S-states, confirming that these two predominant motifs coexist in dynamic equilibrium. Considering the fundamental relation between the free-energy landscape and probability distributions,

$$F(\zeta,{\theta }_{{{{\rm{avg}}}}})=-{k}_{{{{\rm{B}}}}}TlnP(\zeta,{\theta }_{{{{\rm{avg}}}}}),$$
(1)

where kB is the Boltzmann constant, the clear bimodality of P(ζ, θavg) observed in Fig. 1 of ref. 38 and Fig. 1c supports the two-state description of both bulk and interfacial water as coarse-grained yet physically consistent representations. This indicates a strong correlation between translational and orientational ordering, supporting a two-order-parameter description31. The free-energy landscape \(F=-{k}_{{{{\rm{B}}}}}TlnP(\zeta,{\theta }_{{{{\rm{avg}}}}})\) (Fig. S1a) provides a thermodynamic basis for our two-state framework. Importantly, the bimodality of the distribution P(ζ, θavg) and the associated two-state features persist when an instantaneous liquid interface is employed (Willard–Chandler interface; Fig. S2)50. We further emphasise that these two-state features are not artefacts of sampling water molecules exclusively at the air–water interface. Even in the bulk liquid, where there is no net orientational ordering, we consistently observe the coexistence of two distinct structural motifs (Fig. S2), in agreement with our work38. Together, these results demonstrate that the S-state and ρ-state are intrinsic structural features of liquid water, present in both bulk and interfacial environments.

We note that our two-state model serves as a conceptual framework for partitioning fluctuating water structures into two distinct motifs, a framework long used to explain water’s thermodynamic anomalies27,28,29. Applied to surface tension, this approach allows us to disentangle the partial contributions of these motifs. Importantly, the partitioning is statistical: water molecules are not permanently assigned to a single state. Rather, the respective contributions are obtained by long-time averaging, ensuring thermodynamic consistency and robustness. Although recent simulation studies have questioned the two-state description at room temperature51,52,53,54, our work focuses on the supercooled regime, where the distinction between S- and ρ-states becomes clear and the dominant role of the S-state becomes critical.

Roles of ρ- and S-states in the surface tension anomaly

To probe the structural origin of the nonlinear γ(T), we decompose the surface tension into contributions from the S- and ρ-states (Fig. 1a). The S-state contributes substantially less than the ρ-state, even as its fraction increases at lower T, confirming its tension-reducing role. For example, at T = 250 K, the S-state accounts for only 13% of the total γ (Fig. 1a), despite representing approximately 37% of the bulk population (Fig. 1b). This disparity highlights that the symmetric hydrogen-bonded structure of the S-state, being mechanically rigid, produces lower interfacial stress as long as its dipoles remain randomly oriented.

The slowdown of γ between 275–250 K primarily reflects the saturation of the ρ-state contribution (Fig. 1a), as the surface layer becomes fully populated with aligned ρ-states whose anisotropy no longer increases upon cooling (see the inset of panel c in Figs. S3–S7). In this regime, the interfacial s varies only weakly (Fig. 1b), so the S-state contribution also remains nearly constant. Together, these effects lead to the plateau-like behaviour of γ(T).

Moreover, we reproduce the more rapid growth in γ for T ≲ 250 K (Fig. 1a), as reported in previous experimental26 and simulation studies23,24,25. This behaviour closely matches the inflection point at which the contribution from the S-state increases sharply (red line in Fig. 1a), suggesting that the stress anisotropy associated with the S-state is significantly enhanced in the deeply supercooled regime. We also note that the interfacial region consistently shows a lower S-state fraction (s) compared to the bulk (brown line in Fig. 1b), in agreement with experimental observations of reduced tetrahedral coordination at water interfaces55. Importantly, a sharp increase in interfacial s is observed around T ≈ 250 K (Fig. 1a), which coincides with the onset of the faster rise in γ(T). This reflects the fact that S-states, which generate little stress when randomly oriented, begin to develop orientational order in this regime (as discussed below), thereby enhancing anisotropy and contributing to the reentrant increase of γ.

Spatial distribution of tetrahedral water

To understand the spatial distribution of the S-state near the interface, we decompose the water density profile ρw(z) into contributions from the S- and ρ-states (Fig. 2a). This decomposition allows us to determine the fraction of the S-state, s(z), as a function of the distance z from the centre of the water film (Fig. 2b). Notably, we observe that s peaks at z ≈ 16 Å and decreases significantly at the interface. Furthermore, the variation of γ(z) with respect to z reveals that the stress anisotropy, which is the origin of surface tension2,3, extends down to 10 Å below the interface (Fig. 2c). In the bulk, stress anisotropy is absent due to the isotropic environment. The interfacial thickness estimated from the γ(z) profile (~10 Å) is slightly larger than the 6–8 Å reported by MB-pol simulations and second-order spectroscopic measurements56, and also larger than the 3.5 Å corresponding to two surface layers identified by DFT-based MD simulations57. This discrepancy may arise because our approach quantifies stress anisotropy, a mechanical property that reflects collective, non-local correlations, rather than structural anisotropy probed by optical methods. Stress-based measures may therefore exhibit a longer decay length compared to structurally defined interfacial thickness. Notably, we find that the S-state contributes significantly less to surface tension compared to the ρ-state at the interface. This is not solely because the S-state is less prevalent; we also confirm that the surface tension contribution per water molecule, γw, is consistently smaller for the S-state (Fig. 2d). These results confirm that the stable, symmetric tetrahedral structure of the S-state contributes to a balanced stress environment and thus reduces surface tension, provided their orientations remain random. We perform a similar analysis as in Fig. 2 (at T = 300 K) across a broader temperature range (T = 210 − 400 K), revealing qualitatively similar roles of the S-state (Figs. S3–S7). Water molecules in the S-state consistently exhibit weaker spatial stress anisotropy (smaller γw) and contribute less to the surface tension compared to those in the ρ-state.

Fig. 2: Characterisation of water structure and surface tension at 300 K.
figure 2

a Water number density profile ρw(z) vs. the distance z from the centre of the water film, with black, orange, and blue lines representing total water, ρ-state, and S-state water, respectively. b Fraction s of S-state water vs. z, with the dashed line indicating the bulk s value. c Surface tension profile γ(z) for total water, ρ-state, and S-state. Inset: profile of \(\langle \cos (\theta )\rangle\), where θ is the angle between the water dipole and the z-direction. d Surface tension per molecule γw(z) vs. z. Inset: normal pressure Pz and lateral pressure Pxy.

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However, we observe that below T ≲ 250 K, the S-state exhibits a sharp increase in γw(z) (Figs. S3–S7). This indicates that the S-state no longer mitigates the rise in surface tension. The trend is further supported by the evolution of the maximum peak γ* with respect to T for both states. Specifically, γ* grows markedly with decreasing T, particularly below T ≲ 250 K (Fig. S8f), consistent with the onset of accelerated growth in γ under deeply supercooled conditions. We attribute this enhancement of S-state stress anisotropy to the emergence of orientational ordering at low temperatures. Details will be discussed in the following sections.

Negative pressure enhancing tetrahedral ordering

We note that the local enhancement in tetrahedral order s in the interfacial region can be expressed as δs = speak − sbulk ≈ 1.3%, where speak and sbulk are the peak and bulk values of the profile Δs(z) (Fig. 3a). This enhancement can be partially attributed to the subsurface negative pressure P, located slightly beneath the surface [see the inset of Fig. 2d for P]. This mechanical negative pressure promotes formation of the S-state relative to the bulk, where P ≈ 0 and sbulk ≈ 24%. However, δs remains nearly constant across temperatures, even as the magnitude of the negative pressure increases at lower T (Fig. 3b). We attribute this constancy of δs to enhanced OH- and dipole-orientational orderings at lower T (see Fig. 3c; details of these orderings will be discussed later), which compete with the negative-pressure induced S-ordering. This scenario also explains the slight inward shift in the peak position of s (Fig. 2b) compared to the position of the negative pressure (the inset of Fig. 2d). Thus, the inward shift of the peak position—while its magnitude remains nearly constant—indicates that the balance between negative pressure and OH ordering is not determined solely by their relative amplitudes but also by their depth-dependent contributions across the interface.

Fig. 3: Mechanical negative pressure and dipolar ordering of water near air interfaces.
figure 3

a The variation in the fraction s(z) of the S-state, represented by Δs(z) = s − sbulk, is plotted as a function of distance z for temperatures from T = 210 K to 400 K. The local maximum enhancement of the S-state, δs = speak − sbulk, remains relatively stable across temperatures, with an increase of about 1% to 2%. b The temperature dependence of the minimum negative pressure, Pneg, is shown for both in-plane (Pxy) and perpendicular (Pz) components of pressure. c The maximum peak \({\mu }_{z}^{*}\) (normalised by the dipole of a single water molecule, 2.305D) of the dipole profile μz(z) as a function of T for the S-state and ρ-state. Above 250 K, the S-state contributes significantly less to the net dipole moment than the ρ-state. Below this temperature, however, its contribution increases sharply due to the alignment of S-state molecules driven by dipole-dipole interactions. d Representative configurations of central molecules only (thus appearing dilute) of interfacial S-states at T = 210 K. Dashed squares mark full S-states (central molecule with four nearest neighbours). Black arrows indicate dipoles. e Same visualisation as (d), but at 300 K. At T = 300 K, the S-states are randomly oriented, consistent with the slower growth of γ upon cooling, whereas at 210 K the emerging alignment enhances stress anisotropy and contributes to the second inflection point.

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Together, Figs. 2 and 3 reveal a spatial pattern of competing molecular orders at the air-water interface. Near the surface (z ≈ 20–22 Å), water exhibits strong OH-orientational and dipolar alignment, which is enhanced by broken symmetry and reduced local density. This ordering frustrates the formation of tetrahedral hydrogen-bond ordering, thereby suppressing the S-state in this region. Slightly deeper into the subsurface layer (z ≈ 16–20 Å), the local negative pressure stabilises S-state structures, although residual dipolar alignment still imposes orientational constraints. In the bulk-like region, both the OH-orientational ordering and stress anisotropy vanish, and S-states dominate with random orientations.

In the intermediate temperature range (275–250 K), the behaviour of the S-state reflects a balance between opposing effects. The extent of the S-enriched subsurface layer contracts (Fig. 3a), reducing the fraction of molecules that contribute to isotropy. At the same time, the fraction of S-states within the remaining layer rises moderately, mirroring the bulk T-dependence of s (Figs. 1b and S3b–S7b). This competition yields only a weak T-dependence of interfacial s. In contrast, once T falls below 250 K, the sharp increase of bulk s drives a corresponding rise in interfacial s (Fig. 1b).

Competition between tetrahedral and proton ordering

Two distinct ordering tendencies compete at the air–water interface. Broken surface symmetry promotes OH/dipolar alignment, favouring distorted ρ-states. In contrast, the larger specific volume of tetrahedral S-states is stabilised by subsurface negative pressure. Understanding how these competing orders interact is central to explaining water’s surface-tension anomaly.

To examine this competition, we analyse OH-bond orientations using the joint distribution \(P(\cos {\theta }_{1},\cos {\theta }_{2})\) for molecules in successive interfacial layers. Figure 2a shows the formation of ρ-state dominates near the interface, where the intrinsic S-ordering is weakened. Previous spectroscopy experiments have demonstrated that water near air interface exhibits OH-orientational ordering and forms dipole layers58,59. A broken symmetry at the surface induces OH-orientational order to increase hydrogen bonding with neighbouring molecules and align dipoles to minimise free energy. For example, sum-frequency generation (SFG) spectroscopy experiments indicate that the top layer exhibits an average dipole orientation pointing towards the air, while dipoles slightly below the surface are on average oriented towards the bulk60,61. This average behaviour arises from a broad distribution of configurations, with the most common motif featuring only a single O–H bond directed outward at the outermost surface. SFG experiments have also shown that the average orientation of interfacial water at the air–water interface remains essentially unchanged within the accessible temperature range of 283 K to 303 K62. Indeed, we observe the existence of two surface layers where the dipolar orientation is opposite (the inset of Fig. 2c), consistent with earlier findings59,60,62,63. Notably, we find that the S-state exhibits weaker dipolar ordering compared to the ρ-state (the inset of Fig. 2c) and contributes significantly less to the creation of net dipole (Fig. S8). This raises an intriguing question: how are S-ordering and OH-orientational and dipolar orderings coupled at the air-water interface?

To evaluate orientational ordering of OH bonds, we calculate the bivariate distribution \(P(\cos ({\theta }_{1}),\cos ({\theta }_{2}))\)64 for consecutive 1 Å-thick layers from the topmost surface to the bulk water (Fig. S9; T = 300 K), where θ1 and θ2 are the angles formed by the two OH bonds of a water molecule with the z-axis. For the topmost layers [z = 20 − 22 Å where \(\left\langle \cos (\theta )\right\rangle < 0\); Fig. S9a, b] we observe that most water molecules exhibit a structure with dangling OH bonds oriented upward towards the air, which is consistent with previous studies60,61. Notably, in this region, we also observe a small portion of in-plane water structures [\(\cos ({\theta }_{1})\approx \cos ({\theta }_{2})\approx 0\)]. This can be understood in the context of dielectric effects: water has a higher dielectric constant than air, resulting in polarisation that prefers in-plane dipole orientations, which are energetically more favourable than dipoles oriented upward or downward65. However, unlike in dipolar systems near dielectric interfaces65, such in-plane motifs are not predominant in water. While these configurations may be energetically stabilised by dielectric effects, their abundance is limited by entropic penalties and by the preference for orientations that permit stronger hydrogen bonding with the bulk liquid. As we delve deeper below the surface [z = 17 − 20 Å where \(\langle \cos (\theta )\rangle > 0\); Fig. S9c–f], we observe that most water molecules exhibit dangling OH bonds oriented downward towards the bulk, resulting in positive dipoles, which also aligns with previous studies60,61,66. We conduct similar analysis for lower temperatures T = 250 K (Fig. S10) and T = 230 K (Fig. S11) and observe a more pronounced OH-orientational order.

The two structural motifs show distinct orientational behaviour near the interface. ρ-states, favoured by surface symmetry breaking, are strongly aligned with one OH pointing outward and the dipole oriented on average towards air. By contrast, tetrahedral S-states remain orientationally disordered, thereby reducing their contribution to stress anisotropy. Thus, at moderate temperatures, interfacial stress is dominated by aligned ρ-states, while S-states act to soften it.

This can be confirmed by comparing \(P(\cos ({\theta }_{1}),\cos ({\theta }_{2}))\) between the S- and ρ-states in regions where \(\left\langle \cos (\theta )\right\rangle\) displays a negative peak (Fig.4) and a positive peak (Fig. S12). We find that S-state consistently exhibits less pronounced OH-orientational order, characterised by a more dispersed \(P(\cos ({\theta }_{1}),\cos ({\theta }_{2}))\) (for example, compare Fig. 4c, e with Fig. 4d, f for S- and ρ-states in the same region at the same temperature). In the S-state, the values of \(\cos ({\theta }_{1})\) and \(\cos ({\theta }_{2})\) are broadly distributed across the full configuration space, rather than concentrated in distinct regions. This indicates weaker dipolar ordering compared to the ρ-state and accounts for the smaller positive peak of the S-state in \(\left\langle \cos (\theta )\right\rangle\) [the inset of Fig. 2c]. It should be noted that an isotropic distribution of molecular orientations does not appear as a uniform circular profile in the \(\cos ({\theta }_{1})\)–\(\cos ({\theta }_{2})\) representation (for an alternative representation, the tilt–twist Euler angle formalism can be used; see, e.g., ref. 67.) In our analysis, the focus is on the relative spread of the distributions—specifically, the degree of orientational disorder—between the S-state and the ρ-state, rather than on the absolute shape of the profiles. Upon deeper supercooling, however, S-state distributions narrow markedly, signalling the onset of orientational alignment in the subsurface region.

Fig. 4: Characterisation of proton orientational ordering for S-state and ρ-state in the negative dipole region.
figure 4

Distribution \(P(\cos ({\theta }_{1}),\cos ({\theta }_{2}))\) for S-state (a) and ρ-state (b) water at z = 21–22 Å and T = 210 K. Distribution \(P(\cos ({\theta }_{1}),\cos ({\theta }_{2}))\) for S-state (c) and ρ-state (d) water at z = 21–22 Å and T = 250 K. Distribution \(P(\cos ({\theta }_{1}),\cos ({\theta }_{2}))\) for S-state (e) and ρ-state (f) water at z = 21–22 Å and T = 300 K. g Representative configurations of water molecules labelled #1 through #5, corresponding to regions in \(P(\cos ({\theta }_{1}),\cos ({\theta }_{2}))\) plot in (c).

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Indeed, analysis of orientational order parameters (Figs. 3c, 4 and S8a–e, S12) reveals that S-state molecules become progressively aligned along the surface normal upon cooling. Visualisations of central S-state molecules at the interface illustrate this crossover clearly: at T = 300 K, S-state molecules are randomly oriented (Fig. 3e), whereas at T = 210 K they display strong alignment (Fig. 3d). This transition from disordered to ordered S-states amplifies stress anisotropy, producing the pronounced rise in γ at low T (Fig. 1a), concomitant with the increase in interfacial s (Fig. 1b). Consequently, the balance of stress contributions shifts: γ softens at intermediate temperatures when S-states are disordered, but increases rapidly again once orientational order sets in. This structural crossover explains the anomalous progression from a gradual to a reentrant increase in surface tension. We also calculate the distribution of \(\cos ({\theta }_{{{{\rm{Hb}}}}})\), where θHb is the angle between a hydrogen-bonded O–H bond and the surface normal. For water in the outermost layer, the distribution peaks at \(\cos ({\theta }_{{{{\rm{Hb}}}}}) < 0\) (Fig. 5a), indicating that these OH groups preferentially point towards the bulk. This observation is consistent with experimental evidence for hydrogen bonding at the outermost surface55. Additional characterisations of hydrogen bonds further demonstrate a 2D-like bonding character at interface (Fig. S13).

Fig. 5: Hydrogen bonding and formation of nucleation precursors near water interfaces at 300 K.
figure 5

a The distribution of \(\cos ({\theta }_{{{{\rm{Hb}}}}})\), where θHb is the angle between the OH bonds involved in hydrogen bonding and the z-axis across different water layers. b Number density profile of Ice-0-like water molecules (black line: total; blue line: S-state; orange line: ρ-state) identified by the ring analysis30. The surface is enriched with Ice-0-like water molecules. c Fraction profile of Ice-0-like water molecules within the total, S-state, and ρ-state water at T = 300 K, showing that the S-state favours the formation of Ice-0-like structures. d Fraction profile f of water involved in m-membered rings (m = 4 − 7). e Fraction profiles fs and fρ of S-state and ρ-state water involved in m-membered rings (m = 4 − 7), where f = fs + fρ. f Relative proportions fs/f and fρ/f of S-state and ρ-state among water molecules in m-membered rings (m = 4 − 7). The dashed orange lines indicate the average fraction s and 1 − s of S-state and ρ-state, respectively.

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Collectively, these results reveal a competition between distinct ordering tendencies at the air-water interface. Surface-induced symmetry breaking promotes OH-orientational (proton) ordering, which is more readily accommodated by ρ-state water and disrupts the formation of tetrahedral hydrogen-bond networks. This frustrates the development of S-state structures, whose highly symmetrical geometry favours the formation of approximately four well-oriented hydrogen bonds. The anisotropic interfacial environment is inherently incompatible with such symmetry, leading to a suppression of S-state formation near the surface. In contrast, the subsurface region, characterised by negative pressure, stabilises the S-state—consistent with its larger specific volume compared to the ρ-state27,28,29.

Previous work66 has shown that non-ideal hydrogen-bond geometries are required to account for the depth-dependent variation of orientational polarisation at the air–water interface at room temperature and identified triangular three-body hydrogen-bond defects as key structural elements. We propose that the two states identified in our work provide a natural structural basis for these observations. The S-state, representing the highly ordered, low-density component, acts as a structural precursor that largely preserves ideal tetrahedral hydrogen bonding. In contrast, the ρ-state (the high-density component) corresponds to disordered local environments characterised by non-ideal hydrogen-bond geometries and associated defects, which contribute significantly to the net orientational polarisation at the interface. In this sense, our two-state framework extends previous structural interpretations by systematically exploring their temperature dependence down to the deeply supercooled regime and by providing a structural foundation for understanding the anomalous behaviour of water’s surface tension.

Previous studies have also characterised the air–water interface through analyses of molecular orientation60,61,66. Our results are fully consistent with these findings while offering a complementary, topological perspective. Whereas orientational metrics quantify how individual molecules align relative to the interface normal, the S-state and ρ-state framework captures the connectivity and local hydrogen-bonding environment. For example, the broken hydrogen bonds associated with perpendicular or “dangling" water molecules are characteristic of the disordered ρ-state. Thus, our two-state description deepens earlier orientational analyses60,61,66 by providing a thermodynamic and structural link between orientational ordering and local tetrahedral ordering at the air–water interface.

Microscopic mechanism of the water’s surface tension anomaly

Here, we summarise the microscopic origin of water’s anomalous surface tension by emphasising the interplay between two structural motifs with distinct symmetry. Distorted ρ-states, favoured by surface symmetry breaking, possess strong OH/dipolar alignment that enhances stress anisotropy, while tetrahedral S-states, stabilised subsurface by negative pressure, are mechanically rigid but remain orientationally disordered at moderate temperatures. This contrast in symmetry underlies their opposing roles: ρ-states increase γ, whereas disordered S-states soften it.

As T decreases into the 275–250 K regime, the ρ-state contribution saturates once the surface layer is fully aligned, while the S-state contribution changes little owing to a balance between narrowing of the S-rich zone and a modest rise in local S fraction. This compensation leads to the observed slowdown of γ(T). At still lower temperatures ( ≲ 250 K), dipole–dipole correlations promote orientational ordering of S-states, breaking isotropy and amplifying stress anisotropy, which drives the reentrant rise.

Thus, the two inflection points of γ(T) emerge from the coupled influence of motifs with contrasting symmetry: surface-induced ρ-state alignment and subsurface tetrahedral S-states that transition from isotropic to ordered upon deep supercooling.

Impact of tetrahedral ordering on nucleation

We remark that at low temperatures, our previous study has demonstrated that the surface region of water is enriched with nucleation precursors in the form of Ice-0-like structural motifs with 5-, 6-, and 7-membered rings30. Crucially, we find that the S-state is more favourable for the formation of these nucleation precursors (Fig. 5b–f for T = 300 K; also see Figs. S14–S17 for other T). Our data show that the relative occurrence of the S-state is enhanced in 5-, 6-, and 7-membered rings compared to its global average population s, indicating a thermodynamic preference of the S-state for these ring motifs and Ice-0-like environments. By contrast, the ρ-state is relatively depleted in these ring types, despite its larger absolute fraction overall.

Interestingly, we find that the dipole vectors of tetrahedral (S-state) water molecules exhibit short-range orientational correlations, with an average alignment of \(\left\langle {{{{\boldsymbol{d}}}}}_{{{{\rm{c}}}}}\cdot {{{{\boldsymbol{d}}}}}_{{{{\rm{n}}}}}\right\rangle \sim 0.3\), between a central dipole unit vector dc and those of its four nearest neighbours, dn. This value is comparable to that observed in bulk ice (Fig. S18), where long-range dipolar order is frustrated by proton disorder—the statistical randomness of hydrogen orientations in the hydrogen-bond network68,69,70. The persistence of such short-range dipolar correlations in the S-state suggests that these tetrahedral units inherently carry a well-defined dipole moment, forming a locally polar structural motif within both supercooled liquid water and ice.

There is a growing body of evidence suggesting the possible existence of a ferroelectric phase transition in supercooled water71,72,73,74. Early theoretical work proposed that supercooled water may support a ferroelectric state, driven by collective dipolar ordering within the hydrogen-bond network71. More recently, molecular dynamics studies based on the classical TIP4P/Ice model have investigated the ferroelectric origins of bulk supercooled water, proposing that the ferroelectric transition and the liquid–liquid phase transition (LLPT) can be viewed as two facets of the same underlying phenomenon governed by coupling between density and polarisation degrees of freedom74. Consistent with this picture, simulations employing deep neural-network potentials have shown that the low-density liquid exhibits a strong propensity toward spontaneous polarisation, providing further support for the connection between tetrahedral ordering and dipolar correlations73. Our results offer a complementary, structurally resolved perspective on this emerging picture. We find that a robust short-range dipolar correlation consistently exists for the tetrahedral S-state (the low-density component) in both liquid water and ice (Fig. S18). The magnitude of this correlation is comparable to that observed in crystalline ice, indicating that S-state water molecules form locally polar structural motifs even in the absence of long-range ferroelectric order. While this local ordering does not imply macroscopic polarisation, it raises the intriguing possibility that, under conditions of deep supercooling or confinement—where orientational frustration is reduced—S-state clusters may serve as structural precursors for ferroelectric ordering. Importantly, these dipolar correlations between S-structures can propagate in space at low temperatures, where the S-state becomes the dominant species, thereby facilitating the emergence of longer-ranged ferroelectric ordering. This scenario is consistent with recent reports of emerging ferroelectric correlations in supercooled water72,73,74, and points to a potential link between local tetrahedral ordering and emergent collective polarisation.

Thus, our results provide a plausible structure-based explanation for the reported ferroelectric ordering in supercooled water72,73,74, without interfering with the existence of a liquid–liquid critical point (LLCP). Within this framework, ferroelectric correlations and liquid–liquid criticality can naturally coexist, with the tetrahedral S-state acting as the microscopic structural motif that links tetrahedral ordering, dipolar correlations, and macroscopic polarisation tendencies. Moreover, ferroelectric ordering can be viewed not merely as a byproduct of the liquid–liquid transition, but as an active driving mechanism that promotes it through the cooperative growth of dipolar correlations, pointing to a unified physical picture in which polarisation and density ordering are intrinsically coupled and mutually reinforcing.

Discussion

Relation to existing interpretations

Having established the microscopic mechanism of the anomaly, we now compare our results with previous interpretations. Explanations for the anomalous surface tension of supercooled water have invoked excess entropy23, preferential adsorption of high-density liquid (HDL)24, or hydrogen-bond kinetics25. While these capture certain thermodynamic or kinetic aspects, they do not directly link molecular-scale structural motifs to the stress anisotropy that defines surface tension, nor do they account for the intermediate regime of slowed γ(T) growth. In contrast, our framework identifies stress anisotropy as the unifying physical basis of the anomaly, arising from the interplay of surface-induced ρ-state alignment and subsurface S-state ordering. The hierarchical sequence of structural changes—from tetrahedral formation to dipolar alignment—thus provides a physically grounded and unified explanation of both inflection points in γ(T).

Our mechanism for surface tension anomalies is also distinct from criticality-based explanations involving a liquid-liquid critical point (LLCP)75,76. Unlike LLCP scenarios, which attribute anomalies in γ(T) to diverging correlation lengths and mesoscopic thermodynamic fluctuations near the Widom line, our framework roots the anomalous behaviour in spatially resolved stress anisotropy arising from the local competition between surface-induced dipolar alignment and subsurface tetrahedral hydrogen-bond ordering. In this picture, interfacial hydrogen-bonding motifs—namely, OH/dipolar ordering and S-state formation—directly govern stress imbalance and thus surface tension. This mechanism is inherently real-space and structural, operating without the need for diverging response functions or bulk criticality. Although the temperature at which the surface tension anomaly emerges (~230 K) coincides with the Widom line estimated for TIP4P/2005 water38,77, our results demonstrate that such anomalies can arise solely from interfacial structural heterogeneity and spatially localised stress anisotropy.

Importantly, our two-state description—more precisely, a two-order-parameter model78 in which the S-state order parameter is non-conserved—is conceptually equivalent to the interconversion model based on two species79, and remains fully compatible with LLCP-based thermodynamics27,28,29,37,38,80. These frameworks may describe different facets of the same underlying physics: while the LLCP scenario emphasises bulk criticality and diverging correlation lengths, our model highlights local structural competition and stress anisotropy near interfaces. The two mechanisms may coexist or even reinforce each other in different regimes. Our work thus provides a complementary microscopic foundation that links molecular ordering to macroscopic anomalies, offering a bridge between real-space interfacial structure and thermodynamic response in supercooled water. This connection between microscopic interfacial ordering and the bulk critical framework is intriguing, and may help establish a unified understanding of water’s anomalies across scales.

We conclude that the anomaly is an intrinsic property of the air–water interface, arising from the interplay between the anisotropies of ρ- and S-state motifs. This mechanism provides a microscopic structural foundation for the anomaly across the full temperature range, independent of critical fluctuations75,76 or HDL adsorption24. By directly linking molecular ordering to macroscopic interfacial mechanics, our model offers fresh conceptual insight into water’s interfacial anomalies.

Model limitations

Finally, we note that the TIP4P/2005 model used in this study does not include nuclear quantum effects or polarizability, which become increasingly relevant in the deeply supercooled regime. These effects can influence hydrogen-bond geometry, dipole alignment, and orientational fluctuations at the interface. Nevertheless, TIP4P/2005 reproduces key experimental properties of interfacial water, including its orientational structure and surface tension, supporting the robustness of our findings. We also note that simulations with polarisable or quantum water models are typically restricted to nanosecond timescales, which limits reliable sampling of interfacial tension in the deeply supercooled regime. For example, recent simulations using machine-learning potentials (MLPs) for water with quantum-mechanical accuracy have reported that water molecules in the outermost layer in contact with air tend to be positively charged, while those in a sub-interfacial layer are negatively charged, reflecting interfacial charge transfer effects81. These simulations are particularly valuable in that they provide microscopic structural information at the quantum-mechanical level while being significantly faster than conventional density functional theory calculations.

Although our results are obtained using a classical water model, we observe qualitatively consistent interfacial behaviour: the net dipole moment near the interface points toward the vapour phase (inset of Fig. 2c), and the outermost layer is enriched in dangling OH bonds (Fig. S9 at 300 K). The presence of these dangling OH bonds is structurally consistent with the positive charging tendency reported in ref. 81, as the exposed hydrogen atoms carry partial positive charge. Together with the close quantitative agreement between our calculated surface tension γ and experimental values (Fig. 1a), these results support the suitability of the TIP4P/2005 model33 for investigating the structural and thermodynamic properties of interfacial water. We note that the MLP study81 focused on interfacial structure at temperatures of 300 K and above and did not report surface-tension calculations. In contrast, the present work addresses the temperature dependence of surface tension over a broader range, including the deeply supercooled regime down to 210 K. While we expect the qualitative picture of structural competition and its link to stress anisotropy to be robust, future simulations with polarisable or quantum models will be valuable for refining quantitative predictions, particularly concerning orientational ordering at low temperatures.

Physical picture

We have shown that the anomalous, nonlinear temperature dependence of water’s surface tension originates from the interplay between two structural motifs at the air–water interface: surface-induced ρ-states with strong OH/dipolar alignment and tetrahedral S-states stabilised by subsurface negative pressure. The two-step deviation from linear γ(T) reflects distinct regimes governed by their anisotropies. Between 275–250 K, the ρ-state contribution saturates once the surface layer is fully aligned, while the S-state contribution changes little owing to compensating effects of a narrowed subsurface zone and a modest local increase in S fraction. This balance yields only weak evolution of net anisotropy, slowing the growth of γ(T). At lower T ( ≲ 250 K), however, dipole–dipole correlations drive orientational ordering of S-states, which amplifies anisotropy and produces the reentrant rise. This sequence explains both inflection points and shows how directional bonding in tetrahedral liquids can generate anomalies absent in simple liquids.

Beyond resolving a long-standing puzzle of water’s interfacial behaviour, these findings demonstrate how structural competition, amplified by surface symmetry breaking, governs hydrogen-bond networks, interfacial stress, and crystallisation. Linking tetrahedral ordering to Ice-0–like nucleation precursors, we further show that S-state water molecules contribute to surface-facilitated nucleation. These insights open new avenues for controlling nucleation and interfacial phenomena in atmospheric, cryogenic, and biological contexts.

More broadly, our work establishes a general framework for connecting local structural motifs to interfacial stress and thermodynamic behaviour in network-forming liquids. In nonpolar tetrahedral systems such as silica, silicon, and carbon82,83,84, anomalies arise from the competition between open tetrahedral and more compact disordered configurations, which strongly influences interfacial properties and phase behaviour. Water is distinct in that its tetrahedral units carry permanent molecular dipoles, allowing structural competition to couple directly to dipolar correlations and stress anisotropy. This additional orientational degree of freedom suggests the possibility of emergent ferroelectric-like ordering in the supercooled regime. Exploring both the similarities and differences between polar and nonpolar tetrahedral liquids will be an important future direction, helping to disentangle universal structural mechanisms from dipole-specific effects.

While our simulations reveal a clear structural mechanism for water’s anomalous surface tension, direct experimental validation remains challenging. Resolving spatially localised stress anisotropy and separating interfacial S- and ρ-state populations are currently beyond direct experimental reach. Nevertheless, our predictions may be indirectly tested via surface-specific spectroscopies: enhanced tetrahedral ordering and dipole alignment below 250 K should produce characteristic changes in OH-stretch vibrational signatures and orientational anisotropy detectable by sum-frequency generation (SFG) or second-harmonic generation (SHG) spectroscopy (see, e.g., refs. 58,62). Future work could involve analysing the depth-dependent second-order susceptibility χ(2) at low temperatures, with separate contributions from S- and ρ-state water. Such state-resolved spectra would provide a valuable basis for experimental validation of the proposed structural mechanism.

Methods

Water modelling

We perform atomistic simulations of TIP4P/2005 water33, implemented in the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS; version 29Aug2024)85. Our system contains Nw = 2112 water molecules arranged in a slab geometry: a 4 nm-thick water film confined in a three-dimensional periodic box of dimensions Lx = Ly = 4 nm and Lz = 12 nm (Fig. 1a). This setup creates two water-air interfaces along the z-axis, separated by an 8 nm vacuum region. The vacuum spacing is sufficiently large to eliminate spurious interactions between the two surfaces, allowing each interface to be treated as effectively independent. Surface symmetry breaking arises naturally from the abrupt density gradient at the interface and the anisotropic boundary conditions imposed along the z-direction. We use the SHAKE algorithm86 to constrain bond lengths and angles, ensuring the rigidity of water. The system is evolved using the velocity-Verlet algorithm with a time step 1 fs. Electrostatic interactions and dispersion interactions are calculated via the particle–particle particle–mesh algorithm87 with a relative accuracy of 10−5. The temperature T is maintained using the Nosé–Hoover thermostat with damping parameter 102 fs88 in the canonical (NVT) ensemble. We conduct simulations for 1 ns to equilibrate the system, followed by the production run of 9ns. For temperatures of T ≤ 250 K, we reduce Lx = Ly by a factor of \(\sqrt{2}\) and decrease the number of water molecules to Nw = 1056, while maintaining the film thickness and box size Lz. The system is equilibrated for 20 ns, followed by an extended production run of 200 ns (220 K ≤ T ≤ 250 K) and 400 ns (T = 210 K). Configurations are recorded every 103fs for sampling the equilibrium properties. The surface tension is calculated using the mechanical definition17,89,

$$\gamma=(1/2){L}_{z}[\left\langle {\sigma }_{zz}\right\rangle -(1/2)\left(\left\langle {\sigma }_{xx}\right\rangle+\left\langle {\sigma }_{yy}\right\rangle \right)].$$
(2)

Here, \({\sigma }_{\alpha \alpha }=(1/V){\sum }_{i}{\sigma }_{\alpha \alpha }^{i}\), where \({\sigma }_{\alpha \alpha }^{i}\) are the diagonal components of the per-atom stress tensor of atom i and V = LxLyLz. The microscopic stress tensor was calculated for each atom using the compute stress/atom command in LAMMPS (version 29Aug2024)85 (https://docs.lammps.org/compute_stress_atom.html), which implements the Irving–Kirkwood (IK) formulation. The details of how LAMMPS computes stress under periodic boundary conditions are described in refs. 90,91. We acknowledge that the local stress profile is not uniquely defined and depends on the specific method used to localise intermolecular forces92,93. However, our primary conclusions are based on the qualitative features and temperature dependence of the stress profile rather than on the precise quantitative value at any single spatial position. As a result, modest quantitative variations in γ(z) arising from different definitions of the local stress do not affect the overall physical interpretation or the conclusions of this work.

Ring analysis

Ring analysis94 was performed by first constructing a hydrogen-bond network connecting each oxygen atom to its nearest neighbours. For each water molecule i, six triplets are defined, each consisting of i and a pair of its hydrogen-bonded neighbours. A ring is then identified as the shortest closed path that includes all molecules within a triplet. The local topology around a molecule is characterised by the size distribution of these rings94. Ice-0 is distinguished by two characteristic local environments around the central water: (5,5,5,5,6,6) and (5,5,5,7,6,6)30,94. A molecule is classified as “Ice-0–like” if at least five of the six surrounding rings match these signatures.