Medium-density amorphous ice unveils shear rate as a new dimension in water's phase diagram

Abstract

Recent experiments revealed a new amorphous ice phase, medium-density amorphous ice (MDA), formed by ball-milling ice I_h at 77 K [Rosu-Finsen et al., Science 379, 474-478 (2023)]. MDA has density between that of low-density amorphous (LDA) and high-density amorphous (HDA) ices, adding to the complexity of water's phase diagram, known for its glass polyamorphism and two-state thermodynamics. The nature of MDA and its relation to other amorphous ices and liquid water remain unsolved. Here, we use molecular simulations under controlled pressure and shear rate at 77 K to produce and investigate MDA. We find that MDA formed at constant shear rate is a steady-state nonequilibrium shear-driven amorphous ice (SDA), that can be produced by shearing ice I_h, LDA, or HDA. Our results suggest that MDA could be obtained by ball-milling water glasses without crystallization interference. Increasing the shear rate at ambient pressure produces SDAs with densities ranging from LDA to HDA, revealing shear rate as a new thermodynamic variable in the nonequilibrium phase diagram of water. Indeed, shearing provides access to amorphous states inaccessible by controlling pressure and temperature alone. SDAs produced with shearing rates as high as 10⁶ s^-1 sample the same region of the potential energy landscape than hyperquenched glasses with identical density, pressure, and temperature. Intriguingly, SDAs obtained by shearing at ~10⁸ s^-1 have density, enthalpy, and structure indistinguishable from those of water "instantaneously" quenched from room temperature to 77 K over 10 ps, making them good approximants for the "true glass" of ambient liquid water.

Keywords: amorphization; glasses; polyamorphism; shear; water.

Fig. 1.

MD simulation snapshots of polycrystalline ice samples. We start with (A) five ice grains, each about 15 nm in size or (B) with approximately 500 ice grains, with an average size of 3 nm. These grains are represented by different colors, indicating their orientations, while the gray spheres denote disordered regions. To identify the ice grains, we use the grain segmentation algorithm implemented in Ovito (22, 23). The total amount of amorphous and ice are identified by CHILL+ (24). Subsequently, we reach the configuration equivalent to a strain of $\gamma =1$, at which the density has become stationary. We define $\gamma =L_{\mathit{yz}}/L_z$, where $L_{\mathit{yz}}$ is the change in length along the shearing direction and $L_z$ is the initial length of the box. The shear deformation is imposed at a constant shear rate of $\dot{\gamma }= {10}^8 {\mathrm{s}}^{-1}$. See SI Appendix, sections I and II for details.

Fig. 2.

Density of and enthalpy of the SDA ices prepared at 77 K at constant shear rate and pressure. Panels (A) and (B) present the density as a function of pressure for amorphous phases of ML-BOP and TIP4P/Ice at 77 K. The densities of SDAs produced by shearing at a constant shear rate of $\dot{\gamma }= {10}^8 {\mathrm{s}}^{-1}$ ice are shown with blue squares, by shearing LDA with green triangles, and by shearing HDA with red circles. The green dotted line is the density of the liquid obtained along the melting line T_m without applied shear (27, 28). The dashed magenta line in (A) indicates the density of the liquid at the TMD under no shear. The cyan diamonds in (A), (B), and (D) indicate the density and absolute enthalpy of the glass produced by instantaneous quenching of liquid water from 280 K to 77 K at 0.1 MPa. The gray region in (A) encompass all SDAs made by shearing at $\dot{\gamma }$ from 10⁵ s⁻¹ to 10¹² s⁻¹ (SI Appendix, Figs. S10 and S11): the red crosses and red diamonds show the data at the slowest and fastest shear rates at the lower and upper boundaries, respectively. The red circles indicate the densities at $\dot{\gamma }= {10}^8 {\mathrm{s}}^{-1}$. The purple dashed line in (A) represents the density of the hyperquenched glass (HQG) after isobarically cooling the liquid at 10 K ns⁻¹ (29). The black lines correspond to the compression-induced LDA-to-HDA transformation and decompression of HDA, all at 77 K in the absence of shear. The empty teal circles in panel (B) show the density upon instantaneous decompression of high-pressure SDAs to which they are connected by dashed orange arrows. (C) Density of SDAs of ML-BOP as a function of shear rate at various pressures. (D) Enthalpy of the SDAs obtained by shearing ML-BOP. The gray area represents all the range of enthalpies of SDAs sheared at rates 10⁵ s⁻¹ to 10¹² s⁻¹, the SDAs produced by shearing at 10⁸ s⁻¹ (red circles) have the same enthalpy as HDA (thin black line). The enthalpy of LDA is shown with a thick black line, the enthalpy of the liquid at the melting line with a green dashed line, and the enthalpy of the hyperquenched glasses produced by isobaric cooling of the liquid at 10 K ns⁻¹ by a dashed purple line. We find that the enthalpy of HQG is quite similar to that of the most stable LDA and HDA states, indicating that hyperquenching can produce more stable glasses compared to compression/expansion at 77 K. The simulation details are presented in SI Appendix, sections II, III, and IV.

Fig. 3.

Sketch of the T-p-$\dot{\gamma }$ amorphous phase diagram of ML-BOP and TIP4P/Ice which have a LLT line at $\dot{\gamma }=0$ (blue to red colored line). p and T determine the phase behavior of the one component system in equilibrium. The application of a constant shear rate adds a new dimension to the phase diagram, producing distinct SDA nonequilibrium stationary states that may not be accessed through changes in temperature and pressure. The dashed black lines represent the glass transition temperatures T_g(p), which have opposite slope for LDA and MDA in the models (26, 32) and experiments (7), and above which fast ice formation hinders the study of the LLT in water. The shaded magenta region represents the p, $\dot{\gamma }$ area at 77 K, well below T_g(p), investigated in this study. The sketch is based on data of the LLT of ML-BOP, which ends at a critical point located at 170 ± 10 MPa and 181 ± 3 K (26, 33). The equilibrium between LDL and HDL has been extrapolated to occur at p_LL = 410 MPa at 77 K for ML-BOP (26). Similarly, the LLT of TIP4P/Ice has a critical point at 173.9 ± 0.6 MPa and 188.6 ± 1 MPa (34). Fig. 2 presents the density of the shear-driven phases at 77 K as a function of p and $\dot{\gamma }$. The question mark stresses the unknown impact of shear on the metastable LLT.

Fig. 4.

Sketch of the density regions of the amorphous phases produced at 77 K under varying pressure and shear rates. The black dashed line refers to the hypothetical LDL and HDL at 77 K if these liquids could be equilibrated and ice crystallization could be avoided; the vertical dashed line indicates the first-order transition between these two phases. The blue area indicates the density region accessible via both shearing and cooling methods. We conjecture that the blue region may be fully reachable with slower shear rates and cooling rates. Conversely, the magenta region is exclusively accessible with high shear rates, as cooling the liquid is constrained by its maximum density (indicated by the yellow line). The density of HQG and SDA phases approach the density of the stable liquid, LDL, or HDL, when the cooling or shear rates are very slow (Fig. 2C), may be approaching the inverse of the relaxation time at 77 K. The thin black lines illustrate ρ(p) of the SDA at three different shear rates. The liquid–liquid coexistence temperature T_LL for water at 77 K in the absence of shear is expected to occur at p_LL ≈ 250 MPa (7), although it has also been proposed to occur at pressures below p_max (42). For ML-BOP at 77 K, p_LL is extrapolated to be at 410 MPa (26). Also in the absence of shear, the maximum (extremum) in the TMD to the newest equation of state of supercooled water (42) is ρ_max = 1.08 g cm⁻³ at p_max = 151 MPa and 177 K, while for ML-BOP ρ _max= 1.185 g cm⁻³ at p_max = 430 MPa and 128 K (SI Appendix, Fig. S12). At pressures above p_max the liquid does not display a density anomaly, and consequently the most stable state can be only reached by slower cooling rates. The white dotted line indicates the possible relaxation for MDA obtained in ball-milling experiments at high pressures to the experimentally recovered MDA with density 1.07 ± 0.02 g cm⁻³. Fig. 2C indicates that to reach such density at ambient pressure the shear rate should be ~10¹⁰ s⁻¹, well above the rates reached by ball-milling.

Fig. 5.

Structure factor S(q) at T = 77 K, p = 300 MPa, and ρ = 1.109 g cm⁻³ glass made by isobaric cooling a simulation cell with 64,000 ML-BOP molecules at 25 K ns⁻¹ (blue squares) and MDA made by shearing a cell of HDA with the same number of molecules at constant shear rate $\dot{\gamma }= {10}^6 {\mathrm{s}}^{-1}$ (red circles). The lines represent a fit of a parabola to the data, from which we obtain S(0) = 0.076 for MDA and S(0) = 0.069 for the HQG. The black dotted line indicates 4 π/L, where L = 7 nm is the box length. The Inset shows that S(q) of the sheared and hyperquenched amorphous states agree over a larger range of q values.