Unravelling Water Anomalies Using a Percolation Approach

Abstract

Computational studies suggest that the maximum density of water at 277 K and the divergence from experimentally measured compressibility at 228 K at 1 bar arise from the competition between low and high-density molecular populations. However, the relationships among these anomalies, including the minimum density at 203 K, are still debated. Using molecular dynamics with the TIP4P/2005 model, we investigate the pressure–temperature behavior of metastable water along isothermal and isobaric lines to show that in both cases the density and compressibility anomalies result from the emergence, coexistence and extinction of low and high-density percolating clusters. We connect these behaviors with distinct stages of metastable water phase separation, which occur in regions of steep increase in sample density. Our results are consistent with liquid–liquid critical point scenarios while establishing a phenomenological percolation-based framework that systematically organizes water anomalies across diverse temperatures and pressures. This framework may explain percolation transitions observed in oxide and metallic glasses, providing a perspective to describe amorphous transformations in pure substances.

1. Introduction

Water’s anomalies shape Earth, and sustain life in multiple ways, making their understanding crucial across science and technology. Water liquid cooling triggers a density maximum at 277 K at atmospheric pressure attributed to the expansion of the second coordination shell via non-bonded interactions. Just below the homogeneous nucleation temperature T _H (deep supercooling) and supported by computational calculations, the prevailing hypothesis attributes the isothermal compressibility κ _T divergence at 228 K to a maximum in the compressibility originated from strong density fluctuations between low-density liquid (LDL) and high-density liquid (HDL). , A phenomenon that extends to low temperatures T and positive pressures p, outlining the Widom line, often operationally identified through maxima of response functions such as the isothermal compressibility, which ends at a metastable liquid–liquid critical point (LLCP) from which it emanates. − The hypothetical LLCP marks one end of a coexistence line, − confirmed despite challenges accessing the “no-man’s land” due to rapid crystallization in the deep supercooling. , Quenching the liquid to even lower T drives the system out of equilibrium and leads to a dynamic arrest at the glass transition temperature T _g, resulting in either low-density amorphous (LDA) or high-density amorphous (HDA) ice, depending on the quenching route. , Upon pressure, LDA ice transitions to HDA, and HDA to very-high density amorphous (VHDA) ice. , The conjectured LDA–HDA boundary emanating from the hypothetical LLCP, is supported by experiments on amorphous ices and ultraviscous liquid in the 115–150 K range. ,

Low-density (LD) populations exhibit nearly ideal tetrahedral arrangements of H₂O molecules, whereas high-density (HD) populations are characterized by distorted configurations. , This distinction provides a crucial insight into understanding water’s anomalies, as some of these arise from local structural changes between the first and second coordination shells when thermodynamic conditions are varied. ,, It has facilitated the drawing of parallels between crystalline ice, glassy and liquid water, − although the relationship between anomalies, phase separation and their criticality remains an active subject of investigation.

The coarse-grained model introduced by Franzese and Stanley and further developed by Franzese and collaborators has shed light on the critical behavior underlying the low-density to high-density liquid transition in the supercooled regime. A recent study , employing this model, which encodes the essential cooperative nature of the hydrogen bond network, captures the thermodynamic hallmarks of the liquid–liquid phase transition and yields strong numerical evidence, via histogram reweighting and finite-size extrapolation of the critical parameters, that the liquid–liquid critical point is consistent with the 3D Ising universality classa result governed by the dimensionality and symmetry of the order parameter rather than by the specific details of the molecular interactions.

Studies on metallic and oxide glasses have revealed the presence of long-range percolating clusters within their structures. , Notably, simulations of SiO₂ glasses show that under compression, a large cluster with increased polyhedricity emerges and percolates, indicating the onset of a dominant “phase”. For water, early phenomenological percolation approaches were originally proposed by Stanley and co-workers in the context of supercooled water and subsequently tested experimentally and numerically. − More recent studies suggest that supercooled and supercritical water undergo percolation-type transitions. − A geometric perspective on this transition is provided by Bianco and Franzese, , who mapped a supercooled water monolayer onto a Hamiltonially correlated site-bond percolation problem. The percolation threshold tracks the liquid–liquid phase coexistence line, coincides with the liquid–liquid critical point, and follows the Widom line at subcritical pressures, offering a unified description across all three regimes. At the critical point, the percolation critical behavior is consistent with the 2D Ising universality class, while along the Widom line a percolation transition persists in the absence of a thermodynamic divergence. However, the absence of systematic studies in atomistic molecular dynamics simulations makes its ubiquity challenging to evaluate. Moreover, the relationship between percolation and phase separation in binary mixtures is well established and correlated percolation mappings have been developed for coarse grained models of water, ,,, but a systematic atomistic study across broad thermodynamic conditions remains to be performed. In this work, percolation is employed as a geometrical and phenomenological framework to analyze phase changes driven by connectivity, rather than as a Hamiltonian-based correlated percolation theory. Using molecular dynamics with the TIP4P/2005 model, we explore a broad range of temperatures and pressures to demonstrate that percolation transitions correlate with key water anomalies and can delineate the low-density/high-density phase separation of liquid and glassy water under metastable conditions. Our results are consistent with scenarios involving a liquid–liquid critical point in water, while showing that a unified percolation description captures the main metastable phase transformations over a wide thermodynamic domain.

2. Methods

Molecular dynamics simulations of water samples were performed on samples containing 8192; 2825; 896, and 336 H₂O molecules in a wide range of temperatures T and pressures p using the TIP4P/2005 force field in the NPT ensemble. The leapfrog integrator (2 fs time step), stochastic velocity-rescaling thermostat ensuring proper canonical sampling of kinetic energy within the NPT ensemble, and Parrinello–Rahman barostat were implemented. Bond and angle constraints were enforced with a sixth-order LINCS algorithm, and the van der Waals and real-space Coulombic interaction cutoffs were 12 Å (9 Å for the smallest sample). A particle-mesh Ewald scheme was used to treat long-range electrostatic interactions, with a Fourier grid spacing of 1.6 Å (1 Å for the smallest sample). At different pressures, the samples were first equilibrated in the liquid phase at 300 K and then cooled isobarically to 80 K in steps of 1 K each 1 ns, except samples containing 896 molecules at 1 bar that were cooled up to 10 K. In addition, samples were isothermally compressed up to 15 kbar in steps of 0.1 kbar each 1 ns at 300 K, 240 K, 220 K, 172 K, 148 K, 124 K, 100 and 80 K. For 8192 molecules, part of the cooling and compressed data was taken from refs and The cooling and compression rates were carefully selected based on previous TIP4P/2005 studies, , ensuring the convergence of the results while keeping the computational CPU time duration affordable. In addition, isobaric cooling/heating and compression/decompression cycles around the Widom line were performed for samples containing 896 H₂O molecules. For the cooling/heating cycles, supercooled samples at 1 bar and 1.5 kbar were heated in 1 K increments every 1 ns, starting from 180 K. For compression/decompression cycles, samples compressed to 5 kbar were decompressed in 0.1 kbar steps every 1 ns, with those at 180 K reaching negative pressures.

Molecules are classified by the number of neighbors within a cutoff distance, consistent with other indicators. ,, Molecules with four hydrogen-bonded O neighbors (Z = 4) in the first coordination shell form near-perfect tetrahedral arrangements characteristic of low-density (LD) populations, whereas distorted tetrahedra containing interstitial O atoms displaced from the second coordination shell are associated with high-density (HD) populations. ,,,− Here, HD populations are defined as molecules having between five and seven neighbors (Z = 5 – 7), while very-high density populations (VHD) are linked to higher coordination (Z ≥ 8). A 3.5 Å cutoff distance is employed, supported by the flat minimum between the first and second peaks of the pair-distribution function g(r) at 3.0 and 3.5 Å for low T and p (Supporting Information, Figures S1a,b). In these conditions, a large proportion of tetrahedrically arranged O atoms is expected, maintaining a reasonable separation distance (∼0.7 Å) between the first and second coordination shell. This cutoff distance has been widely justified for the identification of LD and HD populations, regardless of the formulation of the specific structural indicator, ,− and is consistent with the well-known symmetric bimodal distribution of these populations along the Widom line, where the isothermal compressibility peaks. ,,,, In addition, most of the proposed indicators have been used both to distinguish between LD and HD populations, and as order parameters. However, their effectiveness as order parameters is limited by their lack of scale invariance, since they only account for neighboring molecules and fail to capture geometric or symmetry information at larger length scales. As a result, they are inadequate for describing the critical behavior expected at a phase transition threshold. Instead, we propose using the percolation order parameter P _∞, which requires a connectivity analysis to map clusters of LD and HD populations formed by connected neighboring molecules of the same species, while monitoring their growth, percolation, and extinction. Details of the numerical procedure used for cluster analysis are documented in the Supporting Information (Cluster analysis). Following the heuristic percolation ideas of Stanley and co-workers, − both approaches address correlated connectivity beyond random percolation, but differ in that correlations are introduced probabilistically via hydrogen-bond fractions f _j in the Stanley framework, whereas here they emerge directly from the molecular dynamics, leading to a less restrictive definition of LD environments and a distinct characterization of HD populations.

The glass transition temperatures T _g were estimated by fitting lines to the high- and low-temperature branches of the enthalpy–temperature curves along the cooling ramps for different pressures p as done elsewhere (Supporting Information, Figures S2d,e). Isothermal compressibility κ _T and other response functions, were calculated using Gromacs tools. For the MD calculations, the GROMACS v2018.466 package was used. Up to ten samples were simulated for each cooling process and five samples for the compression cases. The averages of all indicated quantities were calculated over the final 0.9 ns of each trajectory, using the last configuration every 0.1 ns. Standard deviations were calculated to estimate error bars.

3. Results

3.1. Percolation in Water upon Cooling

Figure a illustrates different stages of water phase separation upon isobaric cooling at 1 bar. At room conditions, the liquid is a connected HD network that expands along the box, with a minor proportion of finite-sized LD clusters inside. The percolation order parameter P _∞ is defined as the probability that a molecule belongs to a percolating cluster, i.e., the fraction of LD or HD water molecules that are part of the spanning cluster (see Figure b and Supporting Information for details). By construction, P _∞ = 1 for a fully percolating LD or HD network and P _∞ = 0 when only finite LD or HD clusters are present. Upon cooling, LD clusters grow fractally from multiple nucleation points, as suggested for crystal nucleation in deep supercooling. The connectivity correlation length ξ, defined as the characteristic distance over which H₂O molecules remain statistically connected, quantifies geometrical percolation here (see Supporting Information, Cluster Analysis). For LD clusters, ξ­(LD) increases with decreasing temperature T, while cluster sizes s (number of water molecules) scale as $R_{\mathrm{g}}^{D_f}$ , where R _g is the gyration radius and D _f is the fractal dimension (top-left image in Figure a). The fractal behavior spans all scales at T = 277 K, where ξ­(LD) and the sample density ρ reaches a maximum. Such maximum of ξ­(LD) indicates percolation of an LD cluster (see Figure S3 in the Supporting Information). A linear regression yields D _f equal to 2.2 ± 0.1, which differs from ordinary percolation.

1.

Snapshots in panel (a) depict the transformation from high-density liquid (HDL) to low-density amorphous (LDA) ice at 1 bar as temperature decreases (right to left). Red and yellow rods represent connected low-density (LD) populations in the largest and smaller LD clusters, respectively. Similarly, blue and cyan rods indicate the largest and smaller high-density (HD) clusters, respectively. The lower images display the percolating LD and HD clusters (percolation order parameter P _∞ ≈ 1), while upper images show remaining clusters. In all cases, the percolating clusters display isotropic three-dimensional morphologies, with no evidence of planar or layered structures. Panel (b) shows population fractions ϕ (dashed lines) and P _∞ (circles) against the temperature. The blue and red areas represent the dominant HD and LD phase, respectively, and the white area denotes percolation coexistence between both states. Panel (c) displays the correlation lengths ξ of emerging LD (red) and collapsing HD (blue) clusters, the sample density ρ (black curve) and isothermal compressibility κ _T (green curve). In (c) κ _T has been rescaled for display; its actual values range from 0.1 to 0.65 GPa^–1. The glass transition temperature T _g is indicated by the arrow. The samples contain 8192 H₂O molecules.

2.

Top images of panels (a,b) show log–log plots of the average radius of gyration R _g versus cluster size s (number of H₂O molecules), excluding the largest cluster when it percolates. Bottom images of (a,b) depict the cluster size distributions (number of clusters of size s). In (a), results correspond to samples cooled at 1 bar: left images for LD clusters and the right image for HD clusters, while the different colors identify the corresponding temperatures. In (b), results correspond to samples compressed at 124 K: left images for HD clusters and the right image for LD clusters, while the different colors identify different pressures. Solid lines show linear regressions. For cooling data, fits were performed at 277 K (percolation onset) for the HD clusters and at 220 K (percolation extinction). For compression data, fits were applied at 4.8 kbar (HD percolation onset) and 6.8 kbar (LD extinction). The linear fits of n(s) include values larger than 1, since a lower value of n(s) means the absence of s sizes in some of the samples. The samples contain 8192 H₂O molecules.

Upon supercooling, P _∞(LD) increases continuously from 0 to 1, which we link to the source of sample expansion of the metastable liquid, marking the onset of phase separation between HDL and LDL as well as their percolation coexistence (white region in Figure b). At lower temperatures, small LD clusters merge with the percolating LD network, reducing the overall number of LD clusters (Supporting Information, Figure S4). At deep supercooling, P _∞(LD) = P _∞(HD) (≈1) when population fractions ϕ­(LD) = ϕ­(HD). These events are spread over a narrow range around ∼228–233 K, where κ _T and other response functions exhibit a peak (Figure c and Figure S2 in the Supporting Information), with small offsets attributable to finite-size effects.

Cooling further, the percolating LD cluster coarsens at the expense of HD populations, losing any trace of fractality. Consequently, the HDL phase fragments, inducing an increase of the number of HD clusters (Figure S4 in Supporting Information). This occurs as P _∞(HD) drops to 0, while ξ­(HD) peaks near the temperature of the minimum density ρ­(T)^min, marking the HDL percolation limit and the end of percolation coexistence. At lower T, the low-density states fully dominates the structure, while population fractions ϕ­(LD) and ϕ­(HD) tend to constant values. In addition, at temperatures where the LD clusters begin to percolate and the HD clusters cease to percolate, both cluster size distributions n _s scale as s ^–τ, a hallmark of percolation transition criticality (bottom images in Figure a). Linear regressions yield an exponent τ equal to −2.1 (±0.1), close to the ordinary percolation value (2.18). Note that, although the exponents (D _f ≈ 2.2, τ ≈ 2.1) lie between 2D (D _f = 1.896, τ = 2.055) and 3D (D _f = 2.523, τ = 2.189) ordinary percolation values, this does not indicate quasi-2D clustering. The planar geometry of individual water molecules promotes tetrahedral hydrogen-bonding arrangements, allowing water to form fully three-dimensional networks that extend from the molecular to the macroscopic scale. We instead speculate that the deviation from standard 3D percolation exponents may be related to hydrogen-bond directionality in the hydrogen-bond network, which may induce correlations in the formation of Z-clusters as the average hydrogen-bond coordination per molecule approaches 4 upon cooling (see the dotted curve in the top panel of Figure S1c in the Supporting Information).

From the correlation length results for LD and HD clusters, three distinct situations emerge during isobaric cooling at 1 bar (see Figure c). Here, the maxima of the percolation correlation lengths ξ­(LD) and ξ­(HD) reflect a geometrical percolation criticality, consistent with a Kertész-like line, i.e., a critical transition in connectivity without an associated thermodynamic singularity. First, at 277 K, the correlation length ξ­(LD) and the sample density ρ both reach a maximum. Second, around 230 K, the population fractions ϕ of LD and HD clusters become equal, coinciding with the maximum in compressibility κ _T , a temperature associated with the experimentally observed divergence of compressibility (228 K). Finally, slightly above the sample’s minimum density, ξ­(HD) reaches its maximum at 215 K, signaling the extinction of the HD percolating cluster.

Similar trends are observed when water is cooled at positive pressures p below the critical pressure P _c of the hypothetical LLCP (P _c = 1.6 ± 0.3 kbar and T _c = 183 ± 11 K for TIP4P/2005 ,, ). As p tends to P _c , the temperature at which ξ­(LD) reaches its maximum follows the trend of ρ­(T)^max, with a slight deviation as pressure approaches P _c (see Figure ). The deviation between these temperatures may reflects the onset of progressively slower, near-vitreous dynamics as the deeply supercooled regime is approached near P _c , which can broaden and shift correlation length maxima relative to density extrema. In addition, enhanced critical slowing down, finite size and glass-like relaxation effects may also contribute to these deviations near P _c .

6.

Percolation results for cooled and compressed water are compiled in the pressure–temperature diagram. For T > T _c , the figure shows the pressure dependence of the maxima of the percolation correlation lengths ξ^max(LD) (red symbols) and ξ^max(HD) (blue symbols), together with the locus of maximum sample density ρ­(T)^max (thick black line). Open and filled symbols correspond to cooling and compression data, respectively, which tend to approach each other at higher temperatures, away from the vicinity of the critical point. In this temperature range, the P _∞(LD/HD) crossovers coincide with the peaks of the isothermal compressibility κ _T (green curve), which operationally delineate the Widom line. Although ξ^max(LD) and ρ­(T)^max do not coincide exactly, they exhibit similar trends for T > T _c , consistent with a Kertész-like geometrical percolation scenario not associated with a thermodynamic singularity. Near and below T _c , the two thermodynamic protocols probe markedly different behavior. For isothermal compression, ξ^max(LD) (filled symbols) progressively deviate toward pressures p > P _c , away from the hypothetical liquid–liquid transition line P _LL(T), and aligns with the peaks of κ _T , signaling a qualitative change from Kertész-like geometrical criticality to a regime dominated by vitrification of the system: as it is pressurized, our simulated system traverses LD, HD, and VHDA states and moves increasingly farther from metastable equilibrium. By contrast, upon isobaric cooling, the P _∞(LD) = P _∞(HD) crossover points (open square green symbols) remain closer to metastable equilibrium and delineate P _LL(T) itself, consistent with the locus of maxima of the zero-wavevector structure factor S(0) reported by Gartner III et al. and with the trend directly observed upon cooling in Figure . Dashed lines depict isochores for liquid water at room conditions (1 g/cm³), and for HDA (1.17 g/cm³) and VHDA (1.26 g/cm³) ices obtained through compression, while T _g obtained from the cooling data is shown with dotted line.

At the same time, the temperatures of the P _∞(LD/HD) and ϕ­(LD/HD) crossovers coincide with those of the κ _T peaks (operationally defining the Widom line), and converge to T _c (Figure ). Here, the Widom line is used in an operational sense based on response-function extrema; a rigorous definition based on mixed-field scaling and the associated mixing coefficient lies beyond the scope of the present structural–percolation analysis. In contrast, when the sample is cooled at pressure above P _c , the sample density no longer exhibits a maximum or a minimum, and the HD states dominate during cooling, while P _∞ and other quantities remain constant, although κ _T decreases at T _g, indicating an HDL to HDA ice transition (Figure c).

3.

For isobaric cooling, in panel (a), the top plot highlights the alignment between the crossovers of P _∞ and the population fractions ϕ of LD (red dashed line) and HD clusters (blue dashed line), as water cools to pressures p below the critical pressure P _c . The blue area represents the dominant HD phase, the red area indicates the dominant LD phase, and the white area denotes the percolation coexistence between the two states. In addition, the bottom plot reveals the alignments of P _∞(LD/HD) and ϕ­(LD/HD) crossovers with the isothermal compressibility peak κ _T (green curve, arbitrary units). It also plots the sample density ρ and correlation lengths ξ of the emerging LD (red symbols) and extincting HD (blue symbols) clusters. The panel (b) and (c) plot the same quantities as (a), but for p ≈ P _c and p > P _c , respectively. Small arrows indicate the resulting glass transition temperature T _g. Samples consist of 8192 H₂O molecules.

3.2. Pressure-Driven Percolation in Water

Isothermal compression from 1 bar triggers percolation transitions from LDL to HDL at deeply supercooled conditions for T > T _g, and from LDA to HDA ice for T < T _g. At low pressure p, LDL liquid and LDA ice dominate, while finite-sized HD clusters grow fractal-like as p increases (left-top image in Figure b). As pressure increases, P _∞(HD) rises from 0 to 1 (Figure a) alongside a sharp increase in sample density, while the coordination number Z increases continuously (Figures b and ). Simultaneously, the intensity of the response function κ _T also amplifies, indicating the onset of LD/HD percolation coexistence and phase separation (green curve in Figure ).

4.

Panel (a) plots κ _T and P _∞ for LD, HD, and VHD states, for compressed liquid water and ice, arranged from top to bottom for decreasing T. The red area denotes the dominant percolating LD “phase”, the blue area for HD, and the white and violet areas indicate the percolation coexistence of LD/HD and HD/VHD states, respectively. Panel (b) shows the sample densities ρ­(p) at which different characteristic loci are identified at the temperatures of panel (a), including the P _∞(LD/HD) crossover (cyan symbols), the peak of κ _T (green symbols), and the maxima of the percolation correlation length ξ associated with the collapsing LD (red symbols) and emerging HD (blue symbols) clusters. These loci are shown together for comparison and do not necessarily coincide, reflecting the distinct geometrical and thermodynamic nature of the corresponding quantities. The samples consist of 8192 H₂O molecules.

5.

Panels plot (a) the pressure p, (b) its density derivative dp/dρ, (c) the oxygen–oxygen coordination number Z _OO, and (d) the population fractions ϕ for LDA and HDA ices, as a function of density ρ. Panel (b) exhibits a series of dips in the density range 1.24–1.32 g/cm³, which has been used to identify the HDA-VHDA ice conversion. In panels (c) and (d), HD is defined by those molecules having more than four neighbors (Z _OO > 4, dotted curves), and are compared to the case when eight or more O atom neighbors are excluded (dashed curves). The vertical lines serve as guides for the eyes, identifying the reported densities for high-density amorphous (HDA, blue), very-high-density amorphous (VHDA, cyan) ices, and ice VI (orange). The samples consist of 8192 H₂O molecules.

The κ _T peak aligns more closely with the P _∞(LD/HD) crossover at 220 and 240 K during cooling. However, for T ≲ T _c , it instead coincides with the ξ­(LD) peak marking the percolation limit of the LD state, as previously observed. This alignment between the maxima of κ _T and ξ­(LD) provides an operational criterion for identifying a spinodal-like limit of stability. Indeed, upon compression and for T < T _c , the κ _T peaks are commonly used to delineate a branch-dependent spinodal-like boundary emerging from the percolation analysis and associated with the hypothetical LLCP, marking the limit of stability of the LD network under compression in a nonequilibrium, glassy regime. In addition, at pressures where HD clusters begin to percolate and LD clusters cease to percolate, the cluster size distributions for both HD and LD clusters follow power-law scaling, indicative of a percolation transition (bottom images in Figure b).

In the percolation coexistence region (white region in Figure a), and in the pressure range between the P _∞(LD/HD) crossover and the ξ­(LD) maximum, the HD percolating cluster becomes dominant and coarsens at the expense of LD clusters, which fragment as p increases. In fact, the HDL and HDA ice states experience homogeneous growth, in a process akin to the final coarsening stage of pressure- or temperature-induced spinodal decomposition in binary mixtures, , where percolation has proven useful in identifying one of the stages of the phase separation process. ,

Beyond the spinodal-like line, HD states dominate the structure at all scales, but a conversion from HDA to VHDA ice is expected at higher pressures and below T _g. For compressed samples, pressure–density derivatives are used to detect dips related to the presence of VHD populations (Figure b). These dips appear in the density range 1.24–1.30 g/cm³, and have been associated with a multistep conversion, consistent with the reported density of VHDA ice (1.26 g/cm³ ). In fact, when molecules with eight or more neighbors are labeled as VHD populations, a VHD cluster emerges and percolates in that density range, coexisting with a percolating HD cluster with lower coordination (see Figure a).

3.3. Pressure–Temperature Diagram

For T > T _c , comparison of the P _∞(LD/HD) crossover of the cooling and compression data in the p–T diagram reveals their alignment with the Widom line derived operationally from the κ _T peaks (Figure ). In this regime, the percolation transition is geometrically critical, with diverging connectivity length scales, but it does not correspond to a thermodynamic singularity. It can therefore be interpreted as a Kertész-like line. In addition, isobaric cooling/heating cycles around the Widom line at 1 bar and 1.5 kbar reveal that the heating-induced LDL-to-HDL transformation follows the same pathway as the cooling-driven HDL-to-LDL transformation, with only minor differences near the crossover region (Figure a). This reversibility around the Widom line aligns with prior experimental findings.

7.

Panel (a) shows isobaric cooling/heating cycles results at 1 bar and 1.5 kbar, while (b) displays isothermal compression/decompression cycles at 210, 200, 190, and 180 K. Upward triangles denote cooling (a) and compression (b) branches, whereas downward triangles denote heating and decompression branches. From top to bottom, the panels display the density ρ, the correlation lengths ξ (LD or HD) for cooling and compression, and ξ′ (LD or HD) for heating and decompression. The primed notation is used consistently, including for the percolation order parameter P _∞. In these simulations, samples contain 896 H₂O molecules, with the curves averaged over five samples and nine trajectory configurations for each.

We also performed compression/decompression cycles around the Widom line to show that reversibility is preserved above 200 K, but not for samples compressed/decompressed at 200 K or below (Figure b). For T < T _c , the sample density exhibits hysteresis, as previously reported, which correlates with the hysteresis observed in the percolation and thermodynamic quantities, similarly to what has been found in compressed/decompressed SiO₂ glasses. The HD states remain dominant along the decompression branch until the decompression branch of the correlation length ξ′(HD) peaks at p ≲ P _c , marking the percolation limit of the HDL liquid or HDA ice (i.e., a spinodal-like boundary of percolation origin), while LD states become dominant at all scales upon further decompression (Figure b).

When water samples are compressed below T _c , the κ _T peak (green curve in Figure ) progressively deviates from the hypothetical liquid–liquid transition line defining P _LL(T), delineated in Figure by the P _∞(LD) = P _∞(HD) crossover points below the LLCP. This deviation is characteristic of a compression branch in a thermodynamic cycle and identifies the LD spinodal-like boundary associated with the loss of stability of the LD network under compression in a non-equilibrium, glassy regime. Indeed, during compression, the pressure at which ξ­(LD) peaks shifts to values higher than P _LL(T), whereas upon decompression the ξ′(HD) peak occurs at pressures lower than P _LL(T). Moreover, the compression and decompression branches exhibit an approximate symmetry: the maximum of ξ­(LD) coincides with the density inflection upon compression, whereas the maximum of ξ′(HD) aligns with the density inflection upon decompression (Figure b), consistent with branch-dependent spinodal-like limits of network stability under the applied thermodynamic protocol. Note that for T < T _c the κ _T peak aligns with the ξ­(LD) maximum rather than with the P _∞(LD/HD) crossover observed for T > T _c (Figure ).

Below T _c , the compression/decompression hysteresis loops mask the underlying liquid–liquid boundary, which in the temperature range explored here extends into the amorphous domain and manifests as a low-to-high density first-order transformation under non-equilibrium conditions. − The coexistence line is often estimated in the literature by the midpoint of hysteresis loops. In the present work, the hysteresis analysis is used qualitatively to identify symmetry between compression and decompression branches rather than to provide a precise determination of the coexistence line. Here, we use the population fraction crossover ϕ­(LD/HD) during isobaric cooling as an operational indicator of the coexistence line (see Figure ), which is consistent with the convergence of ϕ­(LD) and ϕ­(HD) when cooling below T _c for p ≈ P _c (top panel of Figure b). In Figure , the vertical dashed line marks an approximate critical pressure P _c (1.75 kbar), emphasizing the crossover of the order parameter P _∞(LD/HD) and the population fraction ϕ­(LD/HD) near P _c for T < 200 K.

8.

Figures illustrate the LD and HD population fractions ϕ and the percolation order parameter P _∞ as functions of pressure, derived from cooling results at various temperatures. The vertical dashed line marks an approximate critical pressure P _c (1.75 kbar), emphasizing the LD-HD crossovers in ϕ and P _∞ near P _c for temperatures below 200 K. The solid lines represent data from samples containing 896 H₂O molecules, while the symbols denote samples with 8192 H₂O molecules.

In addition, a corresponding mapping for the LD/HD percolation loci above T _c is also provided by the alignment of the P _∞(LD/HD) crossovers with the κ _T -based Widom line and the isochores near ρ ≈ 1 g/cm³ (Figures , and ), indicating that low-Z percolation underlies the density maximum and the onset of LD–HD phase separation. The glass transition temperature T _g(p) shown in Figure , extracted from enthalpy–temperature curves, is determined from finite-rate cooling and therefore reflects kinetic arrest rather than a thermodynamic or percolation critical threshold. We do not observe the emergence of a distinct Z-cluster percolation transition at T _g. The LD–HD transformation extends across T _g, indicating that Z-cluster percolation captures connectivity changes associated with density evolution, but does not define the glass transition itself. Furthermore, T _g(p) falls in the same p–T region where the density isochores for HDA and VHDA (ρ ≃ 1.17 and 1.26 g/cm³) intersect the loci of HD and VHD percolation, respectively (Figure ). , This mirrors observations in compressed SiO₂ glasses, where the onset of a dominant high-coordination percolating cluster marks the emergence of a new amorphous state.

Finally, we note that the percolation analysis can also be performed by varying p in systems obtained via isobaric cooling from 300 K. The out-of-equilibrium nature of the low-temperature states becomes evident when comparing these results with the isothermal compression data, where a clear discrepancy between the two protocols is observed below T _c . Nevertheless, even with the limited data shown in Figure it is possible to construct the locus of the P _∞(LD/HD) crossovers as a function of pressure, shown directly in Figure as the P _∞(LD) = P _∞(HD) crossover points below the LLCP. Notably, despite the limited resolution, it compares favorably with the locus of S(0) maxima (zero-wavevector structure factor) reported numerically by Gartner III et al. in this region.

3.4. Critical Population Fractions for Percolation

The results reveal that the critical fractions ϕ _c (where LD clusters begin to percolate or HD clusters cease percolation) scale inversely with coordination number Z (Figure a,c), as observed in compressed SiO₂ glasses and regular networks. Figure c, also includes results obtained for the percolation of VHD clusters when the ice sample is compressed. Notably, ϕ _c (LD), ϕ _c (HD) and ϕ _c (VHD) approximate the critical percolation thresholds for regular bond structures with coordination numbers Z = 4 (ϕ _c = 0.39) and Z = 6 (ϕ _c = 0.25), and Z = 8 (ϕ _c = 0.19), respectively (vertical dashed lines in Figure a,c). However, while cooling and compression yield similar HD percolation thresholds, the LD threshold broadens under compression. For the latter, the effect was magnified for T ≲ T _c (Figure a,c).

9.

Panels plots the order parameter P _∞ against LD and HD population fractions ϕ­(LD) and ϕ­(HD) of water samples isobarically cooled (a) and subjected to a cooling–heating cycle (b), at different pressures. In 9b, triangles pointing downward and upward correspond to cooling and heating the sample, respectively. Panels (c,d) report results corresponding to water isothermally compressed (c) and subjected to a compression–decompression cycle (d). Panel (c) includes results for VHD clusters. In panel (d), triangles pointing upward and downward correspond to compression and decompression, respectively. Vertical lines indicate bond percolation thresholds reported for 3D regular lattices with coordination number Z = 4 (red), 6 (blue), and 8 (violet). Sample in (a,c) contain 8192 H₂O molecules, and (b,d) 896 molecules.

In Figure , panels (b) and (d) plot the corresponding dependence of P _∞ on population fractions ϕ­(LD) and ϕ­(HD) upon cooling/heating and compression/decompression thermodynamic cycles, respectively. Figure d shows that, when plotted as a function of the population fraction ϕ, the order parameter P _∞ displays a strong collapse between compression and decompression branches, with only a weak residual hysteresis persisting at the lowest temperatures, in contrast to the pronounced hysteresis observed when plotted as a function of pressure (Figure b). This suggests that the percolation transitions are associated with a specific critical population fraction ϕ _c that seems to mainly depend on the coordination number Z of the LD and HD clusters, regardless of the thermodynamic process that water sample undergoes.

3.5. Analogies between Amorphous and Crystalline Ices

From a configurational energy landscape perspective, each liquid or glassy state can be associated with a broad energy megabasin containing a deep minimum corresponding to a crystalline polymorph. , Around this minimum, long-range order is lost, yielding disordered states with a broad distribution of local densities and coordination numbers resembling those of the polymorph associated with that energy minimum. Changes in thermodynamic conditions can drive the system to explore different energy megabasins corresponding to more open or more compact network packings. Within this framework, as previously applied to SiO₂ glasses, the observed consistency between the sample density and coordination number Z of amorphous ices and those of crystalline ice polymorphs provides a basis for exploring their structural analogies. However, it is important to note that in bonded glasses (such as SiO₂), connectivity can be defined without ad hoc criteria owing to the directional nature of covalent and/or ionic bonding. In contrast, for amourphous ice, a non-bonded approach is used, i.e. coordination may be estimated by introducing a cutoff distance between oxygen atoms. Indeed, based on the arguments presented here and the rationale for the chosen cutoff value, it is possible to establish a meaningful analogy between amorphous and crystalline ices. Aside from the sample density, this comparison can incorporate local packing by quantifying the coordination number Z.

Crystalline ice polymorphs exhibit densities ranging from 0.9 to 1.4 g/cm³. The first coordination shell distance is fixed by the characteristic O–O hydrogen bond length (∼2.8 Å), while the second-shell distance decreases with increasing density. Moreover, second-shell oxygen atoms are not equidistant from the central molecule; rather, subsets approach closer, reflecting the distinct interpenetration and packing of structural units in different ice polymorphs. In this light, LDL and LDA have densities close to that of ice I _h (ρ = 0.92 g/cm³) and are composed predominantly of tetrahedral environments compatible with this ice I _h polymorph in agreement with experimental findings.

Neutron diffraction experiments have shown that HDA ice retains a fully hydrogen-bonded tetrahedral network but exhibits a markedly collapsed second shell, with substantial interstitial occupancy between the first and second O–O peaks and a corresponding increase of the effective first-shell coordination above four, in contrast to LDA and ice I _h . X-ray scattering and O–O pair-distribution functions further indicate that HDA’s intermediate-range order and interstitial population place it structurally between LDA/ice Ih and high-pressure polymorphs such as ice II (ρ = 1.17 g/cm³), ice IV (ρ = 1.27 g/cm³), ice IX (ρ = 1.16 g/cm³), and ice XII (ρ = 1.30 g/cm³), without matching any single polymorph uniquely. In this context, local environments with Z = 5–7 neighbors within the cutoff considered here (3.5 Å) can be interpreted as moderately collapsed tetrahedral motifs in which one to three O atoms from the nominal second shell penetrate into the first-shell distance range, consistent with diffraction-based evidence for enhanced interstitial population in HDA ice and with its intermediate density (about 1.15–1.20 g/cm³) relative to LDA and dense crystalline ices such as ice II and ice IV. ,,

For VHDA ice, neutron diffraction studies have shown significantly higher interstitial occupancy than in HDA ice, effectively increasing the number of oxygens occupying positions in the 3.0–3.5 Å region and reinforcing the view that VHDA is a densified, annealed continuation of HDA rather than a topologically distinct network. , Disordered-crystal modeling reproduces VHDA structure factors and O–O pair-distribution functions by introducing positional disorder into combinations of dense crystalline frameworks, notably ice IV, ice VI (ρ = 1.31 g/cm³), and ice XII, whose split second shells also contain subsets of oxygens at distances consistent with the coordination number Z ≥ 8 used here to define VHDA populations within a cutoff (3.5 Å). ,− This picture supports the view of VHDA ice as a very-high-density amorphous state with local coordination comparable to that of the densest crystalline ices.

4. Discussion

For T > T _c , regardless of whether the sample is cooled or compressed, the percolation of the LD and HD clusters, and their competition, leads to strong density fluctuations in the coexistence region. This unified percolation description provides a phenomenological framework to interpret both cooling- and compression-induced transformations within a common connectivity-based picture. The coexistence region corresponds to where the sample density increases sharply with rising temperature or pressure. In this regime, compressibility κ _T exhibits a maximum when the order parameter P _∞ and the fractions of LD and HD populations become similar. However, for T < T _c , when the water sample is compressed, the peak of κ _T narrows and shifts toward the pressure associated with the percolation threshold, where the correlation length ξ­(LD) reaches a maximum. This marks a qualitative change from a Kertész-like geometrical criticality for T > T _c to a spinodal-like behavior associated with the loss of stability of the LD network under compression for T < T _c .

The observed divergence of the isothermal compressibility κ _T , which reflects enhanced density fluctuations, could be qualitatively discussed within the framework of topological constraint theory, or rigidity percolation. , This theory describes a floppy–to–rigid transition of a network as the number of mechanical constraints increases, leading to the emergence of a rigid spanning cluster and primarily affecting elastic moduli, enabling the material to resist deformation. Note that in supercooled water, competing states LD and HD have been linked to a liquid of strong and fragile units , as well as rigid/nonrigid units. Unlike ordinary percolation, which is governed by connectivity alone, rigidity percolation is defined by mechanical constraints and the resulting elastic response. As the system approaches the rigidity percolation threshold, it progressively loses its ability to resist density variations induced by local rearrangements. This leads to pronounced density fluctuations and structural transformations that signal the onset of a new regime of mechanical stability. In percolation rigidity, the elastic moduli such as the bulk modulus decrease continuously and follow a power-law scaling ${(p-p_c)}^f$ at the critical percolation threshold, where p is the probability (or fraction) that a given bond (or connection) in the network is present, p _c is the critical percolation probability, and f is a critical exponent dependent on the details of the network. − Although the divergence of κ _T suggests a link to the critical behavior of the bulk modulus expected from percolation rigidity, their well-known inverse relation between these two physical quantities does not necessarily hold for water. This relation is valid only in homogeneous media, whereas water’s structural heterogeneity, with coexisting Z-clusters of different coordination, breaks this assumption. Thus, the analogy between bulk modulus variations and density fluctuations remains to be tested. As a future perspective, decomposing the bulk modulus into contributions from distinct Z-clusters and examining whether a percolating cluster follows rigidity-type behavior could provide valuable insight.

Percolation concepts, central to our analysis of water’s amorphous structures under cooling, have also illuminated atomic organization in other amorphous materials prepared differently. In situ X-ray diffraction and tomography, combined with computational simulations, reveal the emergence of fractal percolation clusters in metallic glasses during cooling. These liquid-phase atomic networks progressively rigidify as the system approaches the glass transition. In addition, compression-induced percolation transitions in SiO₂ glasses have been identified, with structural changes linked to the emergence of an infinite cluster of higher-coordination polyhedra. Several experimentally observed anomalies coincide with the emergence of percolating networks of SiO₂ polyhedra characterized by specific local coordination. This correspondence enabled direct comparison with crystalline SiO₂ polymorphs, based on their contrasting densities and local connectivity. From this perspective, percolation theory could bridge covalently/ionically bonded glasses (e.g., SiO₂), metallic glasses, and molecular glasses (e.g., amorphous ices). Future work could unify these insights by investigating universal scaling laws or crossover behaviors in percolation during cooling, compression, and other pathways, thereby revealing shared rigidity transitions across diverse amorphous materials.

The cutoff distance used to identify LD, HD, and VHD environments can be rationalized in terms of the effective range of non-bonded interactions that stabilize local structures as molecules migrate from the second to the first coordination shell under changing thermodynamic conditions. In particular, van der Waals forces have been shown to tune the occupancy of interstitial positions between these shells. Furthermore, in several ice polymorphs second-shell O atoms approach the first shell without forming additional hydrogen bonds (see Section ). Therefore, our choice of an O–O distance cutoff of about 3.5 Å may reflect the length scale over which non-bonded interactions modulate local density and coordination.

In previous studies of supercritical water, percolation across the liquid–gas Widom line was analyzed by defining connectivity via an O–O cutoff of 3.5 Å, so that any pair of molecules with r _OO < 3.5 Å was considered connected. , In our terminology, this defines a purely geometrical or “physically bonded” cluster: a connected network of water molecules linked only by an O–O distance criterion, independently of whether the pairs satisfy a hydrogen-bond definition. , By contrast, in the present work we adopt a more restrictive hydrogen-bond criterion than previously considered , and find that a percolating hydrogen-bond network of O atoms persists over the range of low and moderate T and p studied, with most nearest-neighbor O–O distances below 3 Å (Figure S1a,b). Rare separations above 3 Å and below 3.25 Å of the hydrogen bonded water molecules are associated with transiently distorted tetrahedra having 3 or 5 hydrogen-bonded neighbors and do not indicate a change in bonding topology. Although the cutoff-based coordination number Z (with a 3.5 Å O–O cutoff) can exceed 4 for HD and VHD populations, the average number of hydrogen-bonded neighbors per molecule remains equal to four, consistent with the ice rule (Figures S1c,d). Accordingly, a robust tetrahedral hydrogen-bond network is identified within the Z-clusters, with Z serving as a descriptor of local packing and non-bonded neighbors. This distinction between geometric (physically bonded) and hydrogen-bond connectivity is central to our LLCP-focused analysis: our results are consistent with the hypothesis that percolation and phase separation in metastable water are governed not only by the persistence of the hydrogen-bond network, but also by the emergence and percolation of dense Z-clusters driven by non-bonded interactions.

Interestingly, Simões et al. showed that perturbing the van der Waals equation for a simple fluid with a third-order term induces a LLCP. This term represents an effective many-body contribution that accounts for correlations beyond the pairwise interaction level. While pairwise interactions describe simple van der Waals fluids, higher-order terms can phenomenologically capture cooperative effects relevant to network-forming liquids exhibiting an LLCP. In water, such cooperativity arises from hydrogen bonding, which introduces strong directional and many-body correlations. In this sense, the third-order term introduced by Simões et al. (key to generating the LLCP) is analogous to the explicit three-body angular interaction enforcing tetrahedral order in the mW model, and to the implicit many-body hydrogen-bond correlations in the TIP4P model considered here, induced by orientation-dependent electrostatics despite its pairwise-additive potential.

The link between percolation and thermal transitions has been of long-standing interest. Fisher conjectured that if this connection holds, critical points and their universality class should coincide. An early study showed that near the hypothetical LLCP, the thermal critical exponents align with those of the Ising model, as observed near the liquid–gas critical point LGCP. Furthermore, such finding has been theoretically supported by Hamiltonian models based on the hydrogen-bond network description developed by Franzese and collaborators, , in which the LLCP is associated with a correlated percolation process emerging from the underlying interactions rather than from purely geometrical connectivity. Our preliminary finite size scaling analysis of percolating clusters, based on established scaling relations (see Supporting Information, Critical exponents and finite size scaling), shows a distinct critical behavior. For cooling data, we find γ/ν = 1.6 ± 0.2, a result that differs significantly from both Ising model (γ/ν = 1.97) and ordinary percolation (γ/ν = 2.05) (see Figure and Table ).

10.

Figures show the scaling of the largest cluster size s ^big, the maxima of the correlation length ξ^max and the cluster average size S ^max, with respect to the box length L. Panel (a) displays results for isobaric cooling across varying pressures, while panel (b) shows isothermal compression at different temperatures. Using finite-size scaling analysis, we relate the linear regression slopes to their respective scaling exponents (Table , additional details in the Supporting Information). Error bars are displayed when the absolute percentage value exceeds 10%.

1. Finite Size Scaling and Critical Exponents .

scaling relation	exponent	percolation	ising	cooling		compressing
(ϕ → ϕ c )		(3D)	(3D)	LD	HD	LD	HD
n(s)–s –τ	τ	2.18	-	2.2	2.0	1.9	2.3
$$s-R_{g,s}^{D_f}$$	D f	2.53	-	2.2	2.2	2.1	2.2
ξmax–L a	a	1	1	1.0	1.0	1.1(1.3)	1.2(1.0)
s big–L D f	D f	2.53	-	2.2	2.1	-	-
S max–L γ/ν	γ/ν	2.05	1.97	1.7	1.4	-	-

^aThe table compares the exponents derived from scaling relations at the threshold for the LD and HD percolation transitions during cooling and compression, with ordinary percolation. The exponent τ for the size distribution n(s) and the fractal dimension D _f were obtained as described in Figure in the main text. Additionally, the table presents results for D _f derived from scaling relations based on finite size scaling assumptions, valid when the peak of the correlation length ξ^max scales linearly with the box size L. The critical exponent ratio γ/ν was calculated under these conditions and are also compared with Ising results. The values reported in parentheses correspond to resulting exponent ratio when excluding the smallest size in the linear regression of the compressing data (see Figure ).

In the Fisher’s scenario, correlated percolation becomes relevant. Unlike random site occupancy in ordinary percolation, correlated occupancy emerges due to interactions between entities, analogous to those observed in Ising spins or molecular systems. A theoretical framework demonstrates that it is possible to define percolating clusters that account for correlations and thus provide an accurate description of the emerging phase in thermal transitions. This behavior was validated in magnetic systems, Lennard–Jones fluids, and lattice–gas models, where correlated percolation develops along a Kertész line rather than along the Widom line, in close analogy with the present water results for T > T _c . Along this line, percolation exhibits ordinary percolation criticality, except at the thermal critical point where the universality class crosses over to that of the Ising model. In contrast, within the correlated–percolation framework introduced by Bianco and Franzese, percolation signatures are found to track the Widom line and become critical only at the LLCP, reflecting the dominant role of interaction–induced correlations. , The deviation of the effective critical behavior observed here from both ordinary and correlated percolation universality classes should not be interpreted as a statement on the underlying equilibrium physics. Rather, it reflects that the cluster definition employed in this work does not account for the thermodynamic correlations relevant to either universality class. This same geometrical definition does not exclude the ordinary-percolation universality class far from the LLCP; its failure to recover this behavior in the present analysis most likely reflects limitations intrinsic to simulations of metastable supercooled water, including finite-size effects, limited statistical sampling, and the rapid growth of relaxation times, consistent with the nonmonotonic behavior of the response function maxima documented in Figure S2 and previously discussed in this context by Gartner III et al. We note that these limitations would equally hinder an accurate determination of the critical exponents even under a Hill-type energetic connectivity criterion, as they stem from the simulation protocol itself rather than from the specific choice of cluster definition.

In the correlated (Fortuin–Kasteleyn) percolation picture for an Ising-like system, two Kertész lines one for each sign of the order parameterflank the Widom line (h = 0) above T _c and converge at the critical point, where the Widom line itself becomes the first-order coexistence line below T _c . Our percolation analysis reproduces this same topology: the two branches ξ^max(LD) and ξ^max(HD) track the κ _T -based Widom line for T > T _c , approach each other as T _c is approached, and merge at (P _c , T _c ) into the locus that, below T _c , delineates the liquid–liquid (amorphous) coexistence boundary shown in Figure . This structural correspondence with the Kertész-line topology expected for a genuine second-order critical point does not, by itself, demonstrate the existence of a thermodynamic LLCP, but our results, although they do not demonstrate it, support the LLCP scenario.

Finally, the cluster definition employed in this work is purely geometrical and based on intermolecular distances. The Z-clusters are heuristic geometrical constructs based on local coordination and distance criteria, and should not be viewed as thermodynamic correlation volumes. This choice contributes to deviations from ideal correlated–percolation behavior. While this choice does not alter the identification of percolation thresholds or the overall connectivity scenario, it may contribute to deviations from ideal correlated–percolation behavior. More refined criteria, such as Hill-type energetic connectivity criteria that explicitly compare relative kinetic energies with intermolecular potential energies, could in principle sharpen the identification of correlated bonds. Such refinements are not expected to qualitatively modify the phase behavior reported here, but they may slightly affect the precise location of percolation thresholds and the associated universality class.

5. Conclusion

In this work we have employed a geometrical and phenomenological percolation model as a unifying framework to describe metastable transformations in water across a broad pressure–temperature domain. Within this framework, the emergence, coexistence, and extinction of low-density (LD) and high-density (HD) percolating clusters can be interpreted as distinct stages of phase separation under varying thermodynamic conditions. Above T _c , the percolation transitions correlate with the Widom line and the main density and compressibility anomalies of water, in a regime where the system reaches metastable equilibrium, in close analogy with Ising-like correlated percolation scenarios. Below T _c , the same percolation framework captures the out-of-equilibrium phenomenology of amorphous water: the LD correlation-length maxima coincide with a spinodal-like boundary under compression, hysteresis loops are consistent with the first-order character of the LDA–HDA transition, and the HD–VHD percolation transition is consistent with the densification toward very-high-density amorphous ice. Furthermore, the locus of P _∞ (LD/HD) crossovers obtained from systems cooled isobarically from 300 K and then compressed compares favorably with the locus of S(0) maxima reported numerically by Gartner et al. near P _c , although a more exhaustive finite-size analysis remains to be performed.

These findings are consistent with scenarios involving a liquid–liquid critical point (LLCP) and provides a phenomenological description linking percolation and metastable phase separation. However, the percolating LD or HD cluster near the LLCP could be improved by a Hill-type criterion to reproduce the Ising universality class. Regardless of the outcome, since our percolation findings are intrinsically linked to water’s phase separation into distinct states, this approach could also admit an LLCP-free scenario, ,, and offers a phenomenological perspective for understanding percolation behaviors observed in SiO₂ and metallic glasses. , Thus, a comparison of the universality classes of percolation could shed light on similarities between anomalies in water and other pure substances, with the potential to unveil useful insights for describing amorphous–amorphous transformations, where transitions occur between distinct energy megabasins rather than localized energy minima. ,

Figures and corresponding data sets (agr format), as well as sample trajectories at selected pressures and temperatures are available at Zenodo (10.5281/zenodo.21880316).

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.6c03046.

• Includes details on the theory and methods used here for the cluster analysis and for the estimation of percolation critical exponents, and Figures S1–S5 (PDF)

The authors declare no competing financial interest.