Simulations clarify when supercooled water freezes into glassy structures

Although liquid water is a ubiquitous substance and its properties are crucial for all living species, the precise understanding of these properties is still a matter of active scientific research. One rather mysterious aspect concerns the conditions when undercooled water freezes not into ice crystals but into glass-like structures. Based on a rather novel type of computer simulation approach, in PNAS, Limmer and Chandler (1) propose a nonequilibrium phase diagram that attempts to clarify the conditions (temperature, pressure, cooling protocol) under which one should observe transitions from undercooled water to different forms of amorphous ice.

Many aspects of the phase behavior of condensed matter systems in thermal equilibrium, i.e., liquids and crystalline solids, are well understood at least on a qualitative level. However, this is not so for the glassy, amorphous state of condensed matter and for the glass transition that leads to this state from the undercooled liquid. The theoretical understanding of the glass transition was termed one of the “grand challenge problems of physics” decades ago; however, when one consults recent textbooks on the subject, the conclusion is evident that the matter is still highly controversial (2, 3). There is a lack of understanding of whether or not there is a deep structural difference between an amorphous solid and the corresponding supercooled liquid. It is debated whether one can find some (hidden) “order parameter” distinguishing these states, just as different equilibrium phases of a material can be distinguished by order parameters (3). This has also hampered the understanding of supercooled water and amorphous ice [or ices, respectively: there is evidence that several distinct nonequilibrium, low temperature frozen states of water exist (4)]. These states are of great physical significance in many contexts (e.g., nucleation processes in the atmosphere, frozen biomolecules to which water molecules are attached, and freezing of interfacial water confined in porous rocks). Also, the intriguing idea has been advocated that many of the anomalies in the thermal properties of water are linked to a (second-order) phase transition occurring in deeply supercooled water (5). Of course, supercooled water is a metastable state, and one must address questions such as (i) how large is its lifetime before it decays via crystallization and (ii) if we cool water down fast enough, when do we get amorphous ice, and what is its structure?

This last issue is addressed by Limmer and Chandler by computer simulations of the so-called mW model (6) of water, using the recently developed theory of glassification based on the “s-ensemble”: this is used as guiding principle to bias the evolution of the system toward the formation of amorphous ice structures that are supposed to resemble the amorphous ice structures found in nature [and would require cooling procedures that are many orders of magnitude slower than would be feasible in conventional molecular dynamics (MD) simulations].

Next, they give the reader a flavor of what is the “s-ensemble”: this is an interpretation of the glass transition as a kind of first-order transition phenomenon in the trajectory space of the system [space-time thermodynamics (7)]. Denoting this ensemble of (equivalent) trajectories in space symbolically as {x(t)}, t being the time, one can consider the number of (essential) configuration changes in each particular trajectory. This number K{x(t)} is defined in practice, e.g., by enduring displacements (i.e., lasting over a properly chosen time interval). The key step of the theory then is to define a nonequilibrium partition function

$$Z_K\left(s,t\right)={\displaystyle \sum _{\left\{x\left(t\right)\right\}}P\left\{x\left(t\right)\right\}\exp \left[-sK\left\{x\left(t\right)\right\}\right]},$$ [1]

where P{x(t)} is a probability density functional for the trajectories, and s is an (intensive) variable (thermodynamically) conjugate to the variable K (that is extensive in space-time). The analogy to real equilibrium problems (e.g., an Ising magnet where the partition function contains a factor exp(−hM), h being the magnetic field and M being the (extensive) magnetization), is pretty obvious, whereas the physical meaning of s is less straightforward. However, this description has received ample support from (admittedly rather abstract) constrained kinetic models on lattices, and the first applications of this formalism to standard models of glass formers also look promising.

Dealing with water, the choice of an appropriate model potential is a major stumbling block: quantum effects of both the electrons (responsible for the formation of the network of bonds due to bridging hydrogen atoms) and the quantum effect of the protons cannot be fully captured rigorously by an effective potential to be used in purely classical MD simulations. There are many competing proposals for water potentials, each having some problems. Limmer and Chandler choose to use the mW potential (6), which describes observable thermal properties of water reasonably well; whatever choice one makes, one simulates a model, “computer water,” and there is never a guarantee that some important properties of real supercooled water or amorphous ices would not differ from the model predictions. However, other limitations of MD (limited length of the generated trajectories; limited numbers of trajectories samples; and small size of the system, in the present case only n = 216 molecules in a cubic box with periodic boundary conditions are used), presumably are also important.

The key idea of the work by Limmer and Chandler is that by suitable variation of the parameter s, one can sample the nonequilibrium probability distribution of the density and a second parameter, measuring mobility, and establish from this information a kind of nonequilibrium phase diagram of their model (see figure 1 in ref. 1). In the space of reduced pressure (p) and reduced temperature (T), three nonequilibrium phases are revealed: supercooled liquid, high-density amorphous ice (HDA), and low-density amorphous ice (LDA). The locations of the glass transitions are not precisely defined but depend on a nonequilibrium length scale that depends on the rate with which the liquid is quenched to lower temperatures. It is suggested that by varying the cooling rate in the range of 10⁴–10⁸ K/s, an appreciable shift of the glass transition curve in the p–T plane occurs, but its general shape stays the same (experimental cooling rates need to be in the quoted range to avoid crystallization). The most interesting aspect of the work by Limmer and Chandler clearly is the prediction of the first-order HDA-LDA transition line in the p–T plane; reasonable predictions for the structural properties of these amorphous ice states are made, and the kinetics of transitions between these amorphous states is elucidated.

The most interesting aspect of the work by Limmer and Chandler clearly is the prediction of the first order HDA-LDA transition line in the p–T plane.

Does this work clarify the situation of the hypothetical second critical point in the “no man’s land” of undercooled water (5)? The answer is no, for two reasons: (i) the mW model is lacking some of the signatures of incumbent liquid–liquid phase separation suggested to occur for other models; and (ii) there is a problem of principle, where only for small finite systems can one have arbitrary long-lived liquid states far below the melting temperature, because finite size can suppress nucleation [see the well-studied analogous problem of supersaturated vapor that is perfectly stable in simulation boxes of fixed size (8)]. However, a critical point means that there is a correlation length ξ that diverges, and associated critical slowing down occurs [relaxation time τ_ξ ∝ ξ^z, z being some dynamic exponent (3)]. A study by finite size scaling then requires variation of the box linear dimensions L over a range. Since there is need to equilibrate the system over times τ that grow as L^z (3), and for large L, will inevitably exceed the lifetime of the metastable deeply undercooled liquid. Simulating systems that contain only a few hundred molecules hence would be hardly adequate to settle such an issue.

To provide a better insight into this issue of relevant time scales, Fig. 1A provides a sketch of the structural relaxation time τ_R of a very good glass former as a function of temperature: at the melting temperature, it is [in water (9)] of the order of 10 ps, and in the supercooled regime, it increases by many orders of magnitude [up to about 10⁴ ps in water (9) and up to 10¹⁴ ps in glass formers such as orthoterphenyl or SiO₂ (2, 3)]. The lifetime τ_ms of the supercooled metastable fluid is controlled by crystal nucleation and hence diverges (10) at T_m, but decreases monotonically with progressing supercooling (when measured in units of τ_R; Fig. 1B). On a physical time scale (such as picoseconds), due to the rapid increase of τ_R, a minimum τ_min of τ_ms at some temperature T_min must occur (Fig. 1C). When we cool the system, decreasing T(t) with time according to a linear protocol, T(t) = T_m(1 − t/τ_cool), the condition τ_cool << τ_min will ensure that crystallization is avoided and amorphous structures are reached. Unfortunately, the reliable theoretical prediction of τ_ms(T) is also a very hard, and in general unsolved, problem. If in the metastable fluid a correlation length ξ of liquid-liquid phase separation is growing, this can only be observed as long as τ_ξ ∼ τ_R(ξ/a)^z (where a is a molecular length, z = 3) does not exceed τ_cool. This problem must prevent the observation of actual critical singularities, even if liquid–liquid phase separation would occur.

Fig. 1.

(A) Schematic variation of the structural relaxation time τ_R of a very good glass former with temperature. The glass transition temperature is empirically defined (1, 2) by τ_R ∼ 100 s. At the melting temperature T_m, τ_R is many orders of magnitude smaller. (B) Variation of the scaled lifetime τ_ms of the metastable state, τ_ms/τ_R, with temperature. (C) Schematic plot of τ_ms vs. temperature: the cooling time τ_cool should not exceed τ_min; when τ_cool ∼ τ_R, the system falls out of metastable equilibrium and forms a (nonequilibrium) glassy structure.